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Countably compact inverse semigroups and Nyikos problem
<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Countably compact inverse semigroups and Nyikos problem</title> <!--Generated on Mon Mar 17 19:13:17 2025 by LaTeXML (version 0.8.8) http://dlmf.nist.gov/LaTeXML/.--> <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <meta content="Nyikos problem, countably compact semigroup, compact inverse semigroup, continuity of inversion" lang="en" name="keywords"/> <base href="/html/2503.13666v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S1" title="In Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S2" title="In Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Compact-like semigroups with dense inverse subsemigroups and the automatic continuity of inversion</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3" title="In Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Chains in topological semilattices</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S4" title="In Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Nyikos semilattices</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S5" title="In Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Proof of the main result and final remarks</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line ltx_leqno"> <h1 class="ltx_title ltx_title_document">Countably compact inverse semigroups and Nyikos problem</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">S. Bardyla </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_address">S. Bardyla: Institute of Mathematics, University of Vienna, Austria. </span> <span class="ltx_contact ltx_role_email"><a href="mailto:sbardyla@gmail.com">sbardyla@gmail.com</a> </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract.</h6> <p class="ltx_p" id="id3.3">A regular separable first-countable countably compact space is called a <span class="ltx_text ltx_font_italic" id="id3.3.1">Nyikos</span> space. In this paper, we give a partial solution to an old problem of Nyikos by showing that each locally compact Nyikos inverse topological semigroup is compact. Also, we show that a topological semigroup <math alttext="S" class="ltx_Math" display="inline" id="id1.1.m1.1"><semantics id="id1.1.m1.1a"><mi id="id1.1.m1.1.1" xref="id1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="id1.1.m1.1b"><ci id="id1.1.m1.1.1.cmml" xref="id1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="id1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="id1.1.m1.1d">italic_S</annotation></semantics></math> that contains a dense inverse subsemigroup is a topological inverse semigroup, provided (i) <math alttext="S" class="ltx_Math" display="inline" id="id2.2.m2.1"><semantics id="id2.2.m2.1a"><mi id="id2.2.m2.1.1" xref="id2.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="id2.2.m2.1b"><ci id="id2.2.m2.1.1.cmml" xref="id2.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="id2.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="id2.2.m2.1d">italic_S</annotation></semantics></math> is compact, or (ii) <math alttext="S" class="ltx_Math" display="inline" id="id3.3.m3.1"><semantics id="id3.3.m3.1a"><mi id="id3.3.m3.1.1" xref="id3.3.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="id3.3.m3.1b"><ci id="id3.3.m3.1.1.cmml" xref="id3.3.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="id3.3.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="id3.3.m3.1d">italic_S</annotation></semantics></math> is countably compact and sequential. The latter result solves a problem of Banakh and Pastukhova and provides the automatic continuity of inversion in certain compact-like inverse semigroups.</p> </div> <div class="ltx_keywords"> <h6 class="ltx_title ltx_title_keywords">Key words and phrases: </h6>Nyikos problem, countably compact semigroup, compact inverse semigroup, continuity of inversion </div> <div class="ltx_classification"> <h6 class="ltx_title ltx_title_classification">2020 Mathematics Subject Classification: </h6>20M18, 22A15, 22A26, 54D30 </div> <div class="ltx_acknowledgements">The research of the author was funded in whole by the Austrian Science Fund FWF [10.55776/ESP399]. </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1. </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.4">All topological spaces in this paper are assumed to be <span class="ltx_text ltx_font_italic" id="S1.p1.4.1">Hausdorff</span>. A regular separable first-countable countably compact space is called a <span class="ltx_text ltx_font_italic" id="S1.p1.4.2">Nyikos</span> space. Recall that a space <math alttext="X" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><mi id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.p1.1.m1.1b"><ci id="S1.p1.1.m1.1.1.cmml" xref="S1.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">italic_X</annotation></semantics></math> is called <span class="ltx_text ltx_font_italic" id="S1.p1.4.3">separable</span> if <math alttext="X" class="ltx_Math" display="inline" id="S1.p1.2.m2.1"><semantics id="S1.p1.2.m2.1a"><mi id="S1.p1.2.m2.1.1" xref="S1.p1.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.p1.2.m2.1b"><ci id="S1.p1.2.m2.1.1.cmml" xref="S1.p1.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.p1.2.m2.1d">italic_X</annotation></semantics></math> has a countable dense subset; <span class="ltx_text ltx_font_italic" id="S1.p1.4.4">first-countable</span> if each point of <math alttext="X" class="ltx_Math" display="inline" id="S1.p1.3.m3.1"><semantics id="S1.p1.3.m3.1a"><mi id="S1.p1.3.m3.1.1" xref="S1.p1.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.p1.3.m3.1b"><ci id="S1.p1.3.m3.1.1.cmml" xref="S1.p1.3.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.p1.3.m3.1d">italic_X</annotation></semantics></math> has a countable open neighborhood base; <span class="ltx_text ltx_font_italic" id="S1.p1.4.5">countably compact</span> if each infinite subset of <math alttext="X" class="ltx_Math" display="inline" id="S1.p1.4.m4.1"><semantics id="S1.p1.4.m4.1a"><mi id="S1.p1.4.m4.1.1" xref="S1.p1.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.p1.4.m4.1b"><ci id="S1.p1.4.m4.1.1.cmml" xref="S1.p1.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.p1.4.m4.1d">italic_X</annotation></semantics></math> has an accumulation point. The following problem was posed by Nyikos in 1986 and listed among 20 central problems in Set-theoretic Topology by HruΕ‘ak and Moore <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib33" title="">33</a>]</cite> (for other occurrences of this problem see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib40" title="">40</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib41" title="">41</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib44" title="">44</a>]</cite>).</p> </div> <div class="ltx_theorem ltx_theorem_problem" id="S1.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem1.1.1.1">Problem 1.1</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem1.2.2"> </span>(Nyikos)<span class="ltx_text ltx_font_bold" id="S1.Thmtheorem1.3.3">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem1.p1"> <p class="ltx_p" id="S1.Thmtheorem1.p1.1"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem1.p1.1.1">Does ZFC imply the existence of a noncompact Nyikos space?</span></p> </div> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.8">Franklin and Rajagopalan <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib23" title="">23</a>]</cite> constructed a normal locally compact noncompact Nyikos space under the consistent assumption <math alttext="\omega_{1}=\mathfrak{t}" class="ltx_Math" display="inline" id="S1.p2.1.m1.1"><semantics id="S1.p2.1.m1.1a"><mrow id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml"><msub id="S1.p2.1.m1.1.1.2" xref="S1.p2.1.m1.1.1.2.cmml"><mi id="S1.p2.1.m1.1.1.2.2" xref="S1.p2.1.m1.1.1.2.2.cmml">Ο</mi><mn id="S1.p2.1.m1.1.1.2.3" xref="S1.p2.1.m1.1.1.2.3.cmml">1</mn></msub><mo id="S1.p2.1.m1.1.1.1" xref="S1.p2.1.m1.1.1.1.cmml">=</mo><mi id="S1.p2.1.m1.1.1.3" xref="S1.p2.1.m1.1.1.3.cmml">π±</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.1b"><apply id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1"><eq id="S1.p2.1.m1.1.1.1.cmml" xref="S1.p2.1.m1.1.1.1"></eq><apply id="S1.p2.1.m1.1.1.2.cmml" xref="S1.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S1.p2.1.m1.1.1.2.1.cmml" xref="S1.p2.1.m1.1.1.2">subscript</csymbol><ci id="S1.p2.1.m1.1.1.2.2.cmml" xref="S1.p2.1.m1.1.1.2.2">π</ci><cn id="S1.p2.1.m1.1.1.2.3.cmml" type="integer" xref="S1.p2.1.m1.1.1.2.3">1</cn></apply><ci id="S1.p2.1.m1.1.1.3.cmml" xref="S1.p2.1.m1.1.1.3">π±</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.1c">\omega_{1}=\mathfrak{t}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.1d">italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = fraktur_t</annotation></semantics></math> (for basic information about cardinal characteristics of the continuum we refer the reader to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib12" title="">12</a>]</cite>). Ostaszewski <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib43" title="">43</a>]</cite> constructed a perfectly normal locally compact hereditary separable noncompact Nyikos space assuming <math alttext="\diamondsuit" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><mi id="S1.p2.2.m2.1.1" mathvariant="normal" xref="S1.p2.2.m2.1.1.cmml">β’</mi><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><ci id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1">β’</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">\diamondsuit</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">β’</annotation></semantics></math> (for more about the diamond axiom see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib35" title="">35</a>, Chapter II.7]</cite>). Later van Douwen observed that Ostaszewskiβs arguments yield the existence of a locally compact noncompact Nyikos space under <math alttext="\mathfrak{b}=\mathfrak{c}" class="ltx_Math" display="inline" id="S1.p2.3.m3.1"><semantics id="S1.p2.3.m3.1a"><mrow id="S1.p2.3.m3.1.1" xref="S1.p2.3.m3.1.1.cmml"><mi id="S1.p2.3.m3.1.1.2" xref="S1.p2.3.m3.1.1.2.cmml">π</mi><mo id="S1.p2.3.m3.1.1.1" xref="S1.p2.3.m3.1.1.1.cmml">=</mo><mi id="S1.p2.3.m3.1.1.3" xref="S1.p2.3.m3.1.1.3.cmml">π </mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.3.m3.1b"><apply id="S1.p2.3.m3.1.1.cmml" xref="S1.p2.3.m3.1.1"><eq id="S1.p2.3.m3.1.1.1.cmml" xref="S1.p2.3.m3.1.1.1"></eq><ci id="S1.p2.3.m3.1.1.2.cmml" xref="S1.p2.3.m3.1.1.2">π</ci><ci id="S1.p2.3.m3.1.1.3.cmml" xref="S1.p2.3.m3.1.1.3">π </ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.3.m3.1c">\mathfrak{b}=\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.3.m3.1d">fraktur_b = fraktur_c</annotation></semantics></math>. Bardyla and Zdomskyy <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib10" title="">10</a>]</cite> constructed a consistent example of a Nyikos space which is <math alttext="\mathbb{R}" class="ltx_Math" display="inline" id="S1.p2.4.m4.1"><semantics id="S1.p2.4.m4.1a"><mi id="S1.p2.4.m4.1.1" xref="S1.p2.4.m4.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S1.p2.4.m4.1b"><ci id="S1.p2.4.m4.1.1.cmml" xref="S1.p2.4.m4.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.4.m4.1c">\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.4.m4.1d">blackboard_R</annotation></semantics></math>-rigid, i.e. admits only constant continuous real-valued functions. Note that <math alttext="\mathbb{R}" class="ltx_Math" display="inline" id="S1.p2.5.m5.1"><semantics id="S1.p2.5.m5.1a"><mi id="S1.p2.5.m5.1.1" xref="S1.p2.5.m5.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S1.p2.5.m5.1b"><ci id="S1.p2.5.m5.1.1.cmml" xref="S1.p2.5.m5.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.5.m5.1c">\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.5.m5.1d">blackboard_R</annotation></semantics></math>-rigid spaces, being not Tychonoff, are not locally compact. Moreover, Bardyla, Nyikos and Zdomskyy <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib9" title="">9</a>]</cite> showed that assuming <math alttext="\omega_{1}<\mathfrak{b}=\mathfrak{s}=\mathfrak{c}" class="ltx_Math" display="inline" id="S1.p2.6.m6.1"><semantics id="S1.p2.6.m6.1a"><mrow id="S1.p2.6.m6.1.1" xref="S1.p2.6.m6.1.1.cmml"><msub id="S1.p2.6.m6.1.1.2" xref="S1.p2.6.m6.1.1.2.cmml"><mi id="S1.p2.6.m6.1.1.2.2" xref="S1.p2.6.m6.1.1.2.2.cmml">Ο</mi><mn id="S1.p2.6.m6.1.1.2.3" xref="S1.p2.6.m6.1.1.2.3.cmml">1</mn></msub><mo id="S1.p2.6.m6.1.1.3" xref="S1.p2.6.m6.1.1.3.cmml"><</mo><mi id="S1.p2.6.m6.1.1.4" xref="S1.p2.6.m6.1.1.4.cmml">π</mi><mo id="S1.p2.6.m6.1.1.5" xref="S1.p2.6.m6.1.1.5.cmml">=</mo><mi id="S1.p2.6.m6.1.1.6" xref="S1.p2.6.m6.1.1.6.cmml">π°</mi><mo id="S1.p2.6.m6.1.1.7" xref="S1.p2.6.m6.1.1.7.cmml">=</mo><mi id="S1.p2.6.m6.1.1.8" xref="S1.p2.6.m6.1.1.8.cmml">π </mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.6.m6.1b"><apply id="S1.p2.6.m6.1.1.cmml" xref="S1.p2.6.m6.1.1"><and id="S1.p2.6.m6.1.1a.cmml" xref="S1.p2.6.m6.1.1"></and><apply id="S1.p2.6.m6.1.1b.cmml" xref="S1.p2.6.m6.1.1"><lt id="S1.p2.6.m6.1.1.3.cmml" xref="S1.p2.6.m6.1.1.3"></lt><apply id="S1.p2.6.m6.1.1.2.cmml" xref="S1.p2.6.m6.1.1.2"><csymbol cd="ambiguous" id="S1.p2.6.m6.1.1.2.1.cmml" xref="S1.p2.6.m6.1.1.2">subscript</csymbol><ci id="S1.p2.6.m6.1.1.2.2.cmml" xref="S1.p2.6.m6.1.1.2.2">π</ci><cn id="S1.p2.6.m6.1.1.2.3.cmml" type="integer" xref="S1.p2.6.m6.1.1.2.3">1</cn></apply><ci id="S1.p2.6.m6.1.1.4.cmml" xref="S1.p2.6.m6.1.1.4">π</ci></apply><apply id="S1.p2.6.m6.1.1c.cmml" xref="S1.p2.6.m6.1.1"><eq id="S1.p2.6.m6.1.1.5.cmml" xref="S1.p2.6.m6.1.1.5"></eq><share href="https://arxiv.org/html/2503.13666v1#S1.p2.6.m6.1.1.4.cmml" id="S1.p2.6.m6.1.1d.cmml" xref="S1.p2.6.m6.1.1"></share><ci id="S1.p2.6.m6.1.1.6.cmml" xref="S1.p2.6.m6.1.1.6">π°</ci></apply><apply id="S1.p2.6.m6.1.1e.cmml" xref="S1.p2.6.m6.1.1"><eq id="S1.p2.6.m6.1.1.7.cmml" xref="S1.p2.6.m6.1.1.7"></eq><share href="https://arxiv.org/html/2503.13666v1#S1.p2.6.m6.1.1.6.cmml" id="S1.p2.6.m6.1.1f.cmml" xref="S1.p2.6.m6.1.1"></share><ci id="S1.p2.6.m6.1.1.8.cmml" xref="S1.p2.6.m6.1.1.8">π </ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.6.m6.1c">\omega_{1}<\mathfrak{b}=\mathfrak{s}=\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.6.m6.1d">italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < fraktur_b = fraktur_s = fraktur_c</annotation></semantics></math>, every regular separable first-countable non-normal space of weight <math alttext="<\mathfrak{c}" class="ltx_Math" display="inline" id="S1.p2.7.m7.1"><semantics id="S1.p2.7.m7.1a"><mrow id="S1.p2.7.m7.1.1" xref="S1.p2.7.m7.1.1.cmml"><mi id="S1.p2.7.m7.1.1.2" xref="S1.p2.7.m7.1.1.2.cmml"></mi><mo id="S1.p2.7.m7.1.1.1" xref="S1.p2.7.m7.1.1.1.cmml"><</mo><mi id="S1.p2.7.m7.1.1.3" xref="S1.p2.7.m7.1.1.3.cmml">π </mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.7.m7.1b"><apply id="S1.p2.7.m7.1.1.cmml" xref="S1.p2.7.m7.1.1"><lt id="S1.p2.7.m7.1.1.1.cmml" xref="S1.p2.7.m7.1.1.1"></lt><csymbol cd="latexml" id="S1.p2.7.m7.1.1.2.cmml" xref="S1.p2.7.m7.1.1.2">absent</csymbol><ci id="S1.p2.7.m7.1.1.3.cmml" xref="S1.p2.7.m7.1.1.3">π </ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.7.m7.1c"><\mathfrak{c}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.7.m7.1d">< fraktur_c</annotation></semantics></math> embeds into an <math alttext="\mathbb{R}" class="ltx_Math" display="inline" id="S1.p2.8.m8.1"><semantics id="S1.p2.8.m8.1a"><mi id="S1.p2.8.m8.1.1" xref="S1.p2.8.m8.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S1.p2.8.m8.1b"><ci id="S1.p2.8.m8.1.1.cmml" xref="S1.p2.8.m8.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.8.m8.1c">\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.8.m8.1d">blackboard_R</annotation></semantics></math>-rigid Nyikos space. On the other hand, Weiss <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib55" title="">55</a>]</cite> proved that under Martinβs Axiom every perfectly normal Nyikos space is compact. Nyikos and Zdomskyy <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib42" title="">42</a>]</cite> showed that the Proper Forcing Axiom, or briefly PFA, implies that each normal Nyikos space is compact. More information about PFA can be found in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib38" title="">38</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib53" title="">53</a>]</cite>. For other fruitful applications of PFA in Topology see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib17" title="">17</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib18" title="">18</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib49" title="">49</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.3">A semigroup <math alttext="S" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><mi id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><ci id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">italic_S</annotation></semantics></math> endowed with a topology is called a <span class="ltx_text ltx_font_italic" id="S1.p3.3.1">topological semigroup</span> if the semigroup operation viewed as a map from <math alttext="S{\times}S" class="ltx_Math" display="inline" id="S1.p3.2.m2.1"><semantics id="S1.p3.2.m2.1a"><mrow id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml"><mi id="S1.p3.2.m2.1.1.2" xref="S1.p3.2.m2.1.1.2.cmml">S</mi><mo id="S1.p3.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S1.p3.2.m2.1.1.1.cmml">Γ</mo><mi id="S1.p3.2.m2.1.1.3" xref="S1.p3.2.m2.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.1b"><apply id="S1.p3.2.m2.1.1.cmml" xref="S1.p3.2.m2.1.1"><times id="S1.p3.2.m2.1.1.1.cmml" xref="S1.p3.2.m2.1.1.1"></times><ci id="S1.p3.2.m2.1.1.2.cmml" xref="S1.p3.2.m2.1.1.2">π</ci><ci id="S1.p3.2.m2.1.1.3.cmml" xref="S1.p3.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.1c">S{\times}S</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.1d">italic_S Γ italic_S</annotation></semantics></math> to <math alttext="S" class="ltx_Math" display="inline" id="S1.p3.3.m3.1"><semantics id="S1.p3.3.m3.1a"><mi id="S1.p3.3.m3.1.1" xref="S1.p3.3.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.p3.3.m3.1b"><ci id="S1.p3.3.m3.1.1.cmml" xref="S1.p3.3.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.3.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.p3.3.m3.1d">italic_S</annotation></semantics></math> is continuous. For the sake of brevity, we call a topological semigroup <span class="ltx_text ltx_font_italic" id="S1.p3.3.2">Nyikos</span> if its underlying space is Nyikos. Recall that each topological space possesses a compatible semigroup operation. Thus the existence of noncompact Nyikos topological semigroups is consistent with ZFC. In this paper, we consider the following natural problem:</p> </div> <div class="ltx_theorem ltx_theorem_problem" id="S1.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem2.1.1.1">Problem 1.2</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem2.p1"> <p class="ltx_p" id="S1.Thmtheorem2.p1.1"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem2.p1.1.1">Which Nyikos topological semigroups are compact?</span></p> </div> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.5">Note that any ZFC solution of Problem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S1.Thmtheorem2" title="Problem 1.2. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">1.2</span></a> can be viewed as a partial answer to Nyikos problem. Countably compact topological semigroups and groups are widely studied in Topological Algebra. In particular, much research has been done on convergent sequences in countably compact topological groups <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib13" title="">13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib15" title="">15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib16" title="">16</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib24" title="">24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib30" title="">30</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib32" title="">32</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib34" title="">34</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib51" title="">51</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib52" title="">52</a>]</cite>, and countably compact cancellative semigroups <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib25" title="">25</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib39" title="">39</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib47" title="">47</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib54" title="">54</a>]</cite>. Recall that a semigroup <math alttext="S" class="ltx_Math" display="inline" id="S1.p4.1.m1.1"><semantics id="S1.p4.1.m1.1a"><mi id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.p4.1.m1.1b"><ci id="S1.p4.1.m1.1.1.cmml" xref="S1.p4.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">italic_S</annotation></semantics></math> is called <span class="ltx_text ltx_font_italic" id="S1.p4.5.1">cancellative</span> if for every <math alttext="a,b,c\in S" class="ltx_Math" display="inline" id="S1.p4.2.m2.3"><semantics id="S1.p4.2.m2.3a"><mrow id="S1.p4.2.m2.3.4" xref="S1.p4.2.m2.3.4.cmml"><mrow id="S1.p4.2.m2.3.4.2.2" xref="S1.p4.2.m2.3.4.2.1.cmml"><mi id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml">a</mi><mo id="S1.p4.2.m2.3.4.2.2.1" xref="S1.p4.2.m2.3.4.2.1.cmml">,</mo><mi id="S1.p4.2.m2.2.2" xref="S1.p4.2.m2.2.2.cmml">b</mi><mo id="S1.p4.2.m2.3.4.2.2.2" xref="S1.p4.2.m2.3.4.2.1.cmml">,</mo><mi id="S1.p4.2.m2.3.3" xref="S1.p4.2.m2.3.3.cmml">c</mi></mrow><mo id="S1.p4.2.m2.3.4.1" xref="S1.p4.2.m2.3.4.1.cmml">β</mo><mi id="S1.p4.2.m2.3.4.3" xref="S1.p4.2.m2.3.4.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.2.m2.3b"><apply id="S1.p4.2.m2.3.4.cmml" xref="S1.p4.2.m2.3.4"><in id="S1.p4.2.m2.3.4.1.cmml" xref="S1.p4.2.m2.3.4.1"></in><list id="S1.p4.2.m2.3.4.2.1.cmml" xref="S1.p4.2.m2.3.4.2.2"><ci id="S1.p4.2.m2.1.1.cmml" xref="S1.p4.2.m2.1.1">π</ci><ci id="S1.p4.2.m2.2.2.cmml" xref="S1.p4.2.m2.2.2">π</ci><ci id="S1.p4.2.m2.3.3.cmml" xref="S1.p4.2.m2.3.3">π</ci></list><ci id="S1.p4.2.m2.3.4.3.cmml" xref="S1.p4.2.m2.3.4.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.2.m2.3c">a,b,c\in S</annotation><annotation encoding="application/x-llamapun" id="S1.p4.2.m2.3d">italic_a , italic_b , italic_c β italic_S</annotation></semantics></math> each of the equalities <math alttext="ab=ac" class="ltx_Math" display="inline" id="S1.p4.3.m3.1"><semantics id="S1.p4.3.m3.1a"><mrow id="S1.p4.3.m3.1.1" xref="S1.p4.3.m3.1.1.cmml"><mrow id="S1.p4.3.m3.1.1.2" xref="S1.p4.3.m3.1.1.2.cmml"><mi id="S1.p4.3.m3.1.1.2.2" xref="S1.p4.3.m3.1.1.2.2.cmml">a</mi><mo id="S1.p4.3.m3.1.1.2.1" xref="S1.p4.3.m3.1.1.2.1.cmml">β’</mo><mi id="S1.p4.3.m3.1.1.2.3" xref="S1.p4.3.m3.1.1.2.3.cmml">b</mi></mrow><mo id="S1.p4.3.m3.1.1.1" xref="S1.p4.3.m3.1.1.1.cmml">=</mo><mrow id="S1.p4.3.m3.1.1.3" xref="S1.p4.3.m3.1.1.3.cmml"><mi id="S1.p4.3.m3.1.1.3.2" xref="S1.p4.3.m3.1.1.3.2.cmml">a</mi><mo id="S1.p4.3.m3.1.1.3.1" xref="S1.p4.3.m3.1.1.3.1.cmml">β’</mo><mi id="S1.p4.3.m3.1.1.3.3" xref="S1.p4.3.m3.1.1.3.3.cmml">c</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.3.m3.1b"><apply id="S1.p4.3.m3.1.1.cmml" xref="S1.p4.3.m3.1.1"><eq id="S1.p4.3.m3.1.1.1.cmml" xref="S1.p4.3.m3.1.1.1"></eq><apply id="S1.p4.3.m3.1.1.2.cmml" xref="S1.p4.3.m3.1.1.2"><times id="S1.p4.3.m3.1.1.2.1.cmml" xref="S1.p4.3.m3.1.1.2.1"></times><ci id="S1.p4.3.m3.1.1.2.2.cmml" xref="S1.p4.3.m3.1.1.2.2">π</ci><ci id="S1.p4.3.m3.1.1.2.3.cmml" xref="S1.p4.3.m3.1.1.2.3">π</ci></apply><apply id="S1.p4.3.m3.1.1.3.cmml" xref="S1.p4.3.m3.1.1.3"><times id="S1.p4.3.m3.1.1.3.1.cmml" xref="S1.p4.3.m3.1.1.3.1"></times><ci id="S1.p4.3.m3.1.1.3.2.cmml" xref="S1.p4.3.m3.1.1.3.2">π</ci><ci id="S1.p4.3.m3.1.1.3.3.cmml" xref="S1.p4.3.m3.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.3.m3.1c">ab=ac</annotation><annotation encoding="application/x-llamapun" id="S1.p4.3.m3.1d">italic_a italic_b = italic_a italic_c</annotation></semantics></math> and <math alttext="ba=ca" class="ltx_Math" display="inline" id="S1.p4.4.m4.1"><semantics id="S1.p4.4.m4.1a"><mrow id="S1.p4.4.m4.1.1" xref="S1.p4.4.m4.1.1.cmml"><mrow id="S1.p4.4.m4.1.1.2" xref="S1.p4.4.m4.1.1.2.cmml"><mi id="S1.p4.4.m4.1.1.2.2" xref="S1.p4.4.m4.1.1.2.2.cmml">b</mi><mo id="S1.p4.4.m4.1.1.2.1" xref="S1.p4.4.m4.1.1.2.1.cmml">β’</mo><mi id="S1.p4.4.m4.1.1.2.3" xref="S1.p4.4.m4.1.1.2.3.cmml">a</mi></mrow><mo id="S1.p4.4.m4.1.1.1" xref="S1.p4.4.m4.1.1.1.cmml">=</mo><mrow id="S1.p4.4.m4.1.1.3" xref="S1.p4.4.m4.1.1.3.cmml"><mi id="S1.p4.4.m4.1.1.3.2" xref="S1.p4.4.m4.1.1.3.2.cmml">c</mi><mo id="S1.p4.4.m4.1.1.3.1" xref="S1.p4.4.m4.1.1.3.1.cmml">β’</mo><mi id="S1.p4.4.m4.1.1.3.3" xref="S1.p4.4.m4.1.1.3.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.4.m4.1b"><apply id="S1.p4.4.m4.1.1.cmml" xref="S1.p4.4.m4.1.1"><eq id="S1.p4.4.m4.1.1.1.cmml" xref="S1.p4.4.m4.1.1.1"></eq><apply id="S1.p4.4.m4.1.1.2.cmml" xref="S1.p4.4.m4.1.1.2"><times id="S1.p4.4.m4.1.1.2.1.cmml" xref="S1.p4.4.m4.1.1.2.1"></times><ci id="S1.p4.4.m4.1.1.2.2.cmml" xref="S1.p4.4.m4.1.1.2.2">π</ci><ci id="S1.p4.4.m4.1.1.2.3.cmml" xref="S1.p4.4.m4.1.1.2.3">π</ci></apply><apply id="S1.p4.4.m4.1.1.3.cmml" xref="S1.p4.4.m4.1.1.3"><times id="S1.p4.4.m4.1.1.3.1.cmml" xref="S1.p4.4.m4.1.1.3.1"></times><ci id="S1.p4.4.m4.1.1.3.2.cmml" xref="S1.p4.4.m4.1.1.3.2">π</ci><ci id="S1.p4.4.m4.1.1.3.3.cmml" xref="S1.p4.4.m4.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.4.m4.1c">ba=ca</annotation><annotation encoding="application/x-llamapun" id="S1.p4.4.m4.1d">italic_b italic_a = italic_c italic_a</annotation></semantics></math> implies <math alttext="b=c" class="ltx_Math" display="inline" id="S1.p4.5.m5.1"><semantics id="S1.p4.5.m5.1a"><mrow id="S1.p4.5.m5.1.1" xref="S1.p4.5.m5.1.1.cmml"><mi id="S1.p4.5.m5.1.1.2" xref="S1.p4.5.m5.1.1.2.cmml">b</mi><mo id="S1.p4.5.m5.1.1.1" xref="S1.p4.5.m5.1.1.1.cmml">=</mo><mi id="S1.p4.5.m5.1.1.3" xref="S1.p4.5.m5.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.5.m5.1b"><apply id="S1.p4.5.m5.1.1.cmml" xref="S1.p4.5.m5.1.1"><eq id="S1.p4.5.m5.1.1.1.cmml" xref="S1.p4.5.m5.1.1.1"></eq><ci id="S1.p4.5.m5.1.1.2.cmml" xref="S1.p4.5.m5.1.1.2">π</ci><ci id="S1.p4.5.m5.1.1.3.cmml" xref="S1.p4.5.m5.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.5.m5.1c">b=c</annotation><annotation encoding="application/x-llamapun" id="S1.p4.5.m5.1d">italic_b = italic_c</annotation></semantics></math>. It is well known that a metrizable countably compact space is compact. By the classical Birkhoff-Kakutani theorem, each first-countable topological group is metrizable. Then Corollary 5 from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib39" title="">39</a>]</cite> implies the following.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem3.1.1.1">Theorem 1.3</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem3.2.2"> </span>(Mukherjea, Tserpes)<span class="ltx_text ltx_font_bold" id="S1.Thmtheorem3.3.3">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem3.p1"> <p class="ltx_p" id="S1.Thmtheorem3.p1.1"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem3.p1.1.1">Each first-countable countably compact cancellative topological semigroup is compact.</span></p> </div> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.9">A commutative semigroup of idempotents is called a <span class="ltx_text ltx_font_italic" id="S1.p5.9.1">semilattice</span>. Groups and semilattices are included in a much larger class of inverse semigroups. Recall that a semigroup <math alttext="S" class="ltx_Math" display="inline" id="S1.p5.1.m1.1"><semantics id="S1.p5.1.m1.1a"><mi id="S1.p5.1.m1.1.1" xref="S1.p5.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.p5.1.m1.1b"><ci id="S1.p5.1.m1.1.1.cmml" xref="S1.p5.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.1d">italic_S</annotation></semantics></math> is called <span class="ltx_text ltx_font_italic" id="S1.p5.9.2">inverse</span> if for each <math alttext="x\in S" class="ltx_Math" display="inline" id="S1.p5.2.m2.1"><semantics id="S1.p5.2.m2.1a"><mrow id="S1.p5.2.m2.1.1" xref="S1.p5.2.m2.1.1.cmml"><mi id="S1.p5.2.m2.1.1.2" xref="S1.p5.2.m2.1.1.2.cmml">x</mi><mo id="S1.p5.2.m2.1.1.1" xref="S1.p5.2.m2.1.1.1.cmml">β</mo><mi id="S1.p5.2.m2.1.1.3" xref="S1.p5.2.m2.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.2.m2.1b"><apply id="S1.p5.2.m2.1.1.cmml" xref="S1.p5.2.m2.1.1"><in id="S1.p5.2.m2.1.1.1.cmml" xref="S1.p5.2.m2.1.1.1"></in><ci id="S1.p5.2.m2.1.1.2.cmml" xref="S1.p5.2.m2.1.1.2">π₯</ci><ci id="S1.p5.2.m2.1.1.3.cmml" xref="S1.p5.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.2.m2.1c">x\in S</annotation><annotation encoding="application/x-llamapun" id="S1.p5.2.m2.1d">italic_x β italic_S</annotation></semantics></math> there exists a unique <math alttext="y\in X" class="ltx_Math" display="inline" id="S1.p5.3.m3.1"><semantics id="S1.p5.3.m3.1a"><mrow id="S1.p5.3.m3.1.1" xref="S1.p5.3.m3.1.1.cmml"><mi id="S1.p5.3.m3.1.1.2" xref="S1.p5.3.m3.1.1.2.cmml">y</mi><mo id="S1.p5.3.m3.1.1.1" xref="S1.p5.3.m3.1.1.1.cmml">β</mo><mi id="S1.p5.3.m3.1.1.3" xref="S1.p5.3.m3.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.3.m3.1b"><apply id="S1.p5.3.m3.1.1.cmml" xref="S1.p5.3.m3.1.1"><in id="S1.p5.3.m3.1.1.1.cmml" xref="S1.p5.3.m3.1.1.1"></in><ci id="S1.p5.3.m3.1.1.2.cmml" xref="S1.p5.3.m3.1.1.2">π¦</ci><ci id="S1.p5.3.m3.1.1.3.cmml" xref="S1.p5.3.m3.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.3.m3.1c">y\in X</annotation><annotation encoding="application/x-llamapun" id="S1.p5.3.m3.1d">italic_y β italic_X</annotation></semantics></math> such that <math alttext="xyx=x" class="ltx_Math" display="inline" id="S1.p5.4.m4.1"><semantics id="S1.p5.4.m4.1a"><mrow id="S1.p5.4.m4.1.1" xref="S1.p5.4.m4.1.1.cmml"><mrow id="S1.p5.4.m4.1.1.2" xref="S1.p5.4.m4.1.1.2.cmml"><mi id="S1.p5.4.m4.1.1.2.2" xref="S1.p5.4.m4.1.1.2.2.cmml">x</mi><mo id="S1.p5.4.m4.1.1.2.1" xref="S1.p5.4.m4.1.1.2.1.cmml">β’</mo><mi id="S1.p5.4.m4.1.1.2.3" xref="S1.p5.4.m4.1.1.2.3.cmml">y</mi><mo id="S1.p5.4.m4.1.1.2.1a" xref="S1.p5.4.m4.1.1.2.1.cmml">β’</mo><mi id="S1.p5.4.m4.1.1.2.4" xref="S1.p5.4.m4.1.1.2.4.cmml">x</mi></mrow><mo id="S1.p5.4.m4.1.1.1" xref="S1.p5.4.m4.1.1.1.cmml">=</mo><mi id="S1.p5.4.m4.1.1.3" xref="S1.p5.4.m4.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.4.m4.1b"><apply id="S1.p5.4.m4.1.1.cmml" xref="S1.p5.4.m4.1.1"><eq id="S1.p5.4.m4.1.1.1.cmml" xref="S1.p5.4.m4.1.1.1"></eq><apply id="S1.p5.4.m4.1.1.2.cmml" xref="S1.p5.4.m4.1.1.2"><times id="S1.p5.4.m4.1.1.2.1.cmml" xref="S1.p5.4.m4.1.1.2.1"></times><ci id="S1.p5.4.m4.1.1.2.2.cmml" xref="S1.p5.4.m4.1.1.2.2">π₯</ci><ci id="S1.p5.4.m4.1.1.2.3.cmml" xref="S1.p5.4.m4.1.1.2.3">π¦</ci><ci id="S1.p5.4.m4.1.1.2.4.cmml" xref="S1.p5.4.m4.1.1.2.4">π₯</ci></apply><ci id="S1.p5.4.m4.1.1.3.cmml" xref="S1.p5.4.m4.1.1.3">π₯</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.4.m4.1c">xyx=x</annotation><annotation encoding="application/x-llamapun" id="S1.p5.4.m4.1d">italic_x italic_y italic_x = italic_x</annotation></semantics></math> and <math alttext="yxy=y" class="ltx_Math" display="inline" id="S1.p5.5.m5.1"><semantics id="S1.p5.5.m5.1a"><mrow id="S1.p5.5.m5.1.1" xref="S1.p5.5.m5.1.1.cmml"><mrow id="S1.p5.5.m5.1.1.2" xref="S1.p5.5.m5.1.1.2.cmml"><mi id="S1.p5.5.m5.1.1.2.2" xref="S1.p5.5.m5.1.1.2.2.cmml">y</mi><mo id="S1.p5.5.m5.1.1.2.1" xref="S1.p5.5.m5.1.1.2.1.cmml">β’</mo><mi id="S1.p5.5.m5.1.1.2.3" xref="S1.p5.5.m5.1.1.2.3.cmml">x</mi><mo id="S1.p5.5.m5.1.1.2.1a" xref="S1.p5.5.m5.1.1.2.1.cmml">β’</mo><mi id="S1.p5.5.m5.1.1.2.4" xref="S1.p5.5.m5.1.1.2.4.cmml">y</mi></mrow><mo id="S1.p5.5.m5.1.1.1" xref="S1.p5.5.m5.1.1.1.cmml">=</mo><mi id="S1.p5.5.m5.1.1.3" xref="S1.p5.5.m5.1.1.3.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.5.m5.1b"><apply id="S1.p5.5.m5.1.1.cmml" xref="S1.p5.5.m5.1.1"><eq id="S1.p5.5.m5.1.1.1.cmml" xref="S1.p5.5.m5.1.1.1"></eq><apply id="S1.p5.5.m5.1.1.2.cmml" xref="S1.p5.5.m5.1.1.2"><times id="S1.p5.5.m5.1.1.2.1.cmml" xref="S1.p5.5.m5.1.1.2.1"></times><ci id="S1.p5.5.m5.1.1.2.2.cmml" xref="S1.p5.5.m5.1.1.2.2">π¦</ci><ci id="S1.p5.5.m5.1.1.2.3.cmml" xref="S1.p5.5.m5.1.1.2.3">π₯</ci><ci id="S1.p5.5.m5.1.1.2.4.cmml" xref="S1.p5.5.m5.1.1.2.4">π¦</ci></apply><ci id="S1.p5.5.m5.1.1.3.cmml" xref="S1.p5.5.m5.1.1.3">π¦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.5.m5.1c">yxy=y</annotation><annotation encoding="application/x-llamapun" id="S1.p5.5.m5.1d">italic_y italic_x italic_y = italic_y</annotation></semantics></math>. The element <math alttext="y" class="ltx_Math" display="inline" id="S1.p5.6.m6.1"><semantics id="S1.p5.6.m6.1a"><mi id="S1.p5.6.m6.1.1" xref="S1.p5.6.m6.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S1.p5.6.m6.1b"><ci id="S1.p5.6.m6.1.1.cmml" xref="S1.p5.6.m6.1.1">π¦</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.6.m6.1c">y</annotation><annotation encoding="application/x-llamapun" id="S1.p5.6.m6.1d">italic_y</annotation></semantics></math> is called the <span class="ltx_text ltx_font_italic" id="S1.p5.9.3">inverse</span> of <math alttext="x" class="ltx_Math" display="inline" id="S1.p5.7.m7.1"><semantics id="S1.p5.7.m7.1a"><mi id="S1.p5.7.m7.1.1" xref="S1.p5.7.m7.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S1.p5.7.m7.1b"><ci id="S1.p5.7.m7.1.1.cmml" xref="S1.p5.7.m7.1.1">π₯</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.7.m7.1c">x</annotation><annotation encoding="application/x-llamapun" id="S1.p5.7.m7.1d">italic_x</annotation></semantics></math> and is denoted by <math alttext="x^{-1}" class="ltx_Math" display="inline" id="S1.p5.8.m8.1"><semantics id="S1.p5.8.m8.1a"><msup id="S1.p5.8.m8.1.1" xref="S1.p5.8.m8.1.1.cmml"><mi id="S1.p5.8.m8.1.1.2" xref="S1.p5.8.m8.1.1.2.cmml">x</mi><mrow id="S1.p5.8.m8.1.1.3" xref="S1.p5.8.m8.1.1.3.cmml"><mo id="S1.p5.8.m8.1.1.3a" xref="S1.p5.8.m8.1.1.3.cmml">β</mo><mn id="S1.p5.8.m8.1.1.3.2" xref="S1.p5.8.m8.1.1.3.2.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.p5.8.m8.1b"><apply id="S1.p5.8.m8.1.1.cmml" xref="S1.p5.8.m8.1.1"><csymbol cd="ambiguous" id="S1.p5.8.m8.1.1.1.cmml" xref="S1.p5.8.m8.1.1">superscript</csymbol><ci id="S1.p5.8.m8.1.1.2.cmml" xref="S1.p5.8.m8.1.1.2">π₯</ci><apply id="S1.p5.8.m8.1.1.3.cmml" xref="S1.p5.8.m8.1.1.3"><minus id="S1.p5.8.m8.1.1.3.1.cmml" xref="S1.p5.8.m8.1.1.3"></minus><cn id="S1.p5.8.m8.1.1.3.2.cmml" type="integer" xref="S1.p5.8.m8.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.8.m8.1c">x^{-1}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.8.m8.1d">italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. The map <math alttext="x\mapsto x^{-1}" class="ltx_Math" display="inline" id="S1.p5.9.m9.1"><semantics id="S1.p5.9.m9.1a"><mrow id="S1.p5.9.m9.1.1" xref="S1.p5.9.m9.1.1.cmml"><mi id="S1.p5.9.m9.1.1.2" xref="S1.p5.9.m9.1.1.2.cmml">x</mi><mo id="S1.p5.9.m9.1.1.1" stretchy="false" xref="S1.p5.9.m9.1.1.1.cmml">β¦</mo><msup id="S1.p5.9.m9.1.1.3" xref="S1.p5.9.m9.1.1.3.cmml"><mi id="S1.p5.9.m9.1.1.3.2" xref="S1.p5.9.m9.1.1.3.2.cmml">x</mi><mrow id="S1.p5.9.m9.1.1.3.3" xref="S1.p5.9.m9.1.1.3.3.cmml"><mo id="S1.p5.9.m9.1.1.3.3a" xref="S1.p5.9.m9.1.1.3.3.cmml">β</mo><mn id="S1.p5.9.m9.1.1.3.3.2" xref="S1.p5.9.m9.1.1.3.3.2.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.9.m9.1b"><apply id="S1.p5.9.m9.1.1.cmml" xref="S1.p5.9.m9.1.1"><csymbol cd="latexml" id="S1.p5.9.m9.1.1.1.cmml" xref="S1.p5.9.m9.1.1.1">maps-to</csymbol><ci id="S1.p5.9.m9.1.1.2.cmml" xref="S1.p5.9.m9.1.1.2">π₯</ci><apply id="S1.p5.9.m9.1.1.3.cmml" xref="S1.p5.9.m9.1.1.3"><csymbol cd="ambiguous" id="S1.p5.9.m9.1.1.3.1.cmml" xref="S1.p5.9.m9.1.1.3">superscript</csymbol><ci id="S1.p5.9.m9.1.1.3.2.cmml" xref="S1.p5.9.m9.1.1.3.2">π₯</ci><apply id="S1.p5.9.m9.1.1.3.3.cmml" xref="S1.p5.9.m9.1.1.3.3"><minus id="S1.p5.9.m9.1.1.3.3.1.cmml" xref="S1.p5.9.m9.1.1.3.3"></minus><cn id="S1.p5.9.m9.1.1.3.3.2.cmml" type="integer" xref="S1.p5.9.m9.1.1.3.3.2">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.9.m9.1c">x\mapsto x^{-1}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.9.m9.1d">italic_x β¦ italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> is called <span class="ltx_text ltx_font_italic" id="S1.p5.9.4">inversion</span>. An inverse topological semigroup with continuous inversion is called a <span class="ltx_text ltx_font_italic" id="S1.p5.9.5">topological inverse</span> semigroup. The algebraic theory of inverse semigroups is well developed, see the monographs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib36" title="">36</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib45" title="">45</a>]</cite>. Topological inverse semigroups were investigated in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib3" title="">3</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib8" title="">8</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib20" title="">20</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib26" title="">26</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib29" title="">29</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib37" title="">37</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.1">The following partial solution of Problems <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S1.Thmtheorem1" title="Problem 1.1 (Nyikos). β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">1.1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S1.Thmtheorem2" title="Problem 1.2. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">1.2</span></a> is the main result of this paper.</p> </div> <div class="ltx_theorem ltx_theorem_ltheorem" id="Thmltheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmltheorem1.1.1.1">Theorem A</span></span><span class="ltx_text ltx_font_bold" id="Thmltheorem1.2.2">.</span> </h6> <div class="ltx_para" id="Thmltheorem1.p1"> <p class="ltx_p" id="Thmltheorem1.p1.1"><span class="ltx_text ltx_font_italic" id="Thmltheorem1.p1.1.1">A locally compact Nyikos inverse topological semigroup is compact.</span></p> </div> </div> <div class="ltx_para" id="S1.p7"> <p class="ltx_p" id="S1.p7.1">By <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S1.p7.1.m1.1"><semantics id="S1.p7.1.m1.1a"><msub id="S1.p7.1.m1.1.1" xref="S1.p7.1.m1.1.1.cmml"><mi id="S1.p7.1.m1.1.1.2" xref="S1.p7.1.m1.1.1.2.cmml">Ο</mi><mn id="S1.p7.1.m1.1.1.3" xref="S1.p7.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.p7.1.m1.1b"><apply id="S1.p7.1.m1.1.1.cmml" xref="S1.p7.1.m1.1.1"><csymbol cd="ambiguous" id="S1.p7.1.m1.1.1.1.cmml" xref="S1.p7.1.m1.1.1">subscript</csymbol><ci id="S1.p7.1.m1.1.1.2.cmml" xref="S1.p7.1.m1.1.1.2">π</ci><cn id="S1.p7.1.m1.1.1.3.cmml" type="integer" xref="S1.p7.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.1.m1.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.p7.1.m1.1d">italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> we denote the first uncountable cardinal. The proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem1" title="Theorem A. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">A</span></a> is based on the following two results that are of independent interest. The first one characterizes compact Nyikos topological semilattices in terms of their chains.</p> </div> <div class="ltx_theorem ltx_theorem_ltheorem" id="Thmltheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmltheorem2.1.1.1">Theorem B</span></span><span class="ltx_text ltx_font_bold" id="Thmltheorem2.2.2">.</span> </h6> <div class="ltx_para" id="Thmltheorem2.p1"> <p class="ltx_p" id="Thmltheorem2.p1.1"><span class="ltx_text ltx_font_italic" id="Thmltheorem2.p1.1.1">For a Tychonoff Nyikos topological semilattice <math alttext="X" class="ltx_Math" display="inline" id="Thmltheorem2.p1.1.1.m1.1"><semantics id="Thmltheorem2.p1.1.1.m1.1a"><mi id="Thmltheorem2.p1.1.1.m1.1.1" xref="Thmltheorem2.p1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="Thmltheorem2.p1.1.1.m1.1b"><ci id="Thmltheorem2.p1.1.1.m1.1.1.cmml" xref="Thmltheorem2.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmltheorem2.p1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="Thmltheorem2.p1.1.1.m1.1d">italic_X</annotation></semantics></math> the following assertions are equivalent:</span></p> <ol class="ltx_enumerate" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S1.I1.i1.p1"> <p class="ltx_p" id="S1.I1.i1.p1.1"><math alttext="X" class="ltx_Math" display="inline" id="S1.I1.i1.p1.1.m1.1"><semantics id="S1.I1.i1.p1.1.m1.1a"><mi id="S1.I1.i1.p1.1.m1.1.1" xref="S1.I1.i1.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.1.m1.1b"><ci id="S1.I1.i1.p1.1.m1.1.1.cmml" xref="S1.I1.i1.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.1.m1.1d">italic_X</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i1.p1.1.1"> is compact;</span></p> </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S1.I1.i2.p1"> <p class="ltx_p" id="S1.I1.i2.p1.3"><span class="ltx_text ltx_font_italic" id="S1.I1.i2.p1.3.1">no chain in </span><math alttext="X" class="ltx_Math" display="inline" id="S1.I1.i2.p1.1.m1.1"><semantics id="S1.I1.i2.p1.1.m1.1a"><mi id="S1.I1.i2.p1.1.m1.1.1" xref="S1.I1.i2.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.1.m1.1b"><ci id="S1.I1.i2.p1.1.m1.1.1.cmml" xref="S1.I1.i2.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.1.m1.1d">italic_X</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i2.p1.3.2"> is isomorphic to </span><math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S1.I1.i2.p1.2.m2.2"><semantics id="S1.I1.i2.p1.2.m2.2a"><mrow id="S1.I1.i2.p1.2.m2.2.2.1" xref="S1.I1.i2.p1.2.m2.2.2.2.cmml"><mo id="S1.I1.i2.p1.2.m2.2.2.1.2" stretchy="false" xref="S1.I1.i2.p1.2.m2.2.2.2.cmml">(</mo><msub id="S1.I1.i2.p1.2.m2.2.2.1.1" xref="S1.I1.i2.p1.2.m2.2.2.1.1.cmml"><mi id="S1.I1.i2.p1.2.m2.2.2.1.1.2" xref="S1.I1.i2.p1.2.m2.2.2.1.1.2.cmml">Ο</mi><mn id="S1.I1.i2.p1.2.m2.2.2.1.1.3" xref="S1.I1.i2.p1.2.m2.2.2.1.1.3.cmml">1</mn></msub><mo id="S1.I1.i2.p1.2.m2.2.2.1.3" xref="S1.I1.i2.p1.2.m2.2.2.2.cmml">,</mo><mi id="S1.I1.i2.p1.2.m2.1.1" xref="S1.I1.i2.p1.2.m2.1.1.cmml">min</mi><mo id="S1.I1.i2.p1.2.m2.2.2.1.4" stretchy="false" xref="S1.I1.i2.p1.2.m2.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.2.m2.2b"><interval closure="open" id="S1.I1.i2.p1.2.m2.2.2.2.cmml" xref="S1.I1.i2.p1.2.m2.2.2.1"><apply id="S1.I1.i2.p1.2.m2.2.2.1.1.cmml" xref="S1.I1.i2.p1.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S1.I1.i2.p1.2.m2.2.2.1.1.1.cmml" xref="S1.I1.i2.p1.2.m2.2.2.1.1">subscript</csymbol><ci id="S1.I1.i2.p1.2.m2.2.2.1.1.2.cmml" xref="S1.I1.i2.p1.2.m2.2.2.1.1.2">π</ci><cn id="S1.I1.i2.p1.2.m2.2.2.1.1.3.cmml" type="integer" xref="S1.I1.i2.p1.2.m2.2.2.1.1.3">1</cn></apply><min id="S1.I1.i2.p1.2.m2.1.1.cmml" xref="S1.I1.i2.p1.2.m2.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.2.m2.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.2.m2.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i2.p1.3.3"> or </span><math alttext="(\omega_{1},\max)" class="ltx_Math" display="inline" id="S1.I1.i2.p1.3.m3.2"><semantics id="S1.I1.i2.p1.3.m3.2a"><mrow id="S1.I1.i2.p1.3.m3.2.2.1" xref="S1.I1.i2.p1.3.m3.2.2.2.cmml"><mo id="S1.I1.i2.p1.3.m3.2.2.1.2" stretchy="false" xref="S1.I1.i2.p1.3.m3.2.2.2.cmml">(</mo><msub id="S1.I1.i2.p1.3.m3.2.2.1.1" xref="S1.I1.i2.p1.3.m3.2.2.1.1.cmml"><mi id="S1.I1.i2.p1.3.m3.2.2.1.1.2" xref="S1.I1.i2.p1.3.m3.2.2.1.1.2.cmml">Ο</mi><mn id="S1.I1.i2.p1.3.m3.2.2.1.1.3" xref="S1.I1.i2.p1.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S1.I1.i2.p1.3.m3.2.2.1.3" xref="S1.I1.i2.p1.3.m3.2.2.2.cmml">,</mo><mi id="S1.I1.i2.p1.3.m3.1.1" xref="S1.I1.i2.p1.3.m3.1.1.cmml">max</mi><mo id="S1.I1.i2.p1.3.m3.2.2.1.4" stretchy="false" xref="S1.I1.i2.p1.3.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.3.m3.2b"><interval closure="open" id="S1.I1.i2.p1.3.m3.2.2.2.cmml" xref="S1.I1.i2.p1.3.m3.2.2.1"><apply id="S1.I1.i2.p1.3.m3.2.2.1.1.cmml" xref="S1.I1.i2.p1.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S1.I1.i2.p1.3.m3.2.2.1.1.1.cmml" xref="S1.I1.i2.p1.3.m3.2.2.1.1">subscript</csymbol><ci id="S1.I1.i2.p1.3.m3.2.2.1.1.2.cmml" xref="S1.I1.i2.p1.3.m3.2.2.1.1.2">π</ci><cn id="S1.I1.i2.p1.3.m3.2.2.1.1.3.cmml" type="integer" xref="S1.I1.i2.p1.3.m3.2.2.1.1.3">1</cn></apply><max id="S1.I1.i2.p1.3.m3.1.1.cmml" xref="S1.I1.i2.p1.3.m3.1.1"></max></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.3.m3.2c">(\omega_{1},\max)</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.3.m3.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_max )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i2.p1.3.4">;</span></p> </div> </li> <li class="ltx_item" id="S1.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(iii)</span> <div class="ltx_para" id="S1.I1.i3.p1"> <p class="ltx_p" id="S1.I1.i3.p1.3"><span class="ltx_text ltx_font_italic" id="S1.I1.i3.p1.3.1">no chain in </span><math alttext="X" class="ltx_Math" display="inline" id="S1.I1.i3.p1.1.m1.1"><semantics id="S1.I1.i3.p1.1.m1.1a"><mi id="S1.I1.i3.p1.1.m1.1.1" xref="S1.I1.i3.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i3.p1.1.m1.1b"><ci id="S1.I1.i3.p1.1.m1.1.1.cmml" xref="S1.I1.i3.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i3.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i3.p1.1.m1.1d">italic_X</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i3.p1.3.2"> is topologically isomorphic to </span><math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S1.I1.i3.p1.2.m2.2"><semantics id="S1.I1.i3.p1.2.m2.2a"><mrow id="S1.I1.i3.p1.2.m2.2.2.1" xref="S1.I1.i3.p1.2.m2.2.2.2.cmml"><mo id="S1.I1.i3.p1.2.m2.2.2.1.2" stretchy="false" xref="S1.I1.i3.p1.2.m2.2.2.2.cmml">(</mo><msub id="S1.I1.i3.p1.2.m2.2.2.1.1" xref="S1.I1.i3.p1.2.m2.2.2.1.1.cmml"><mi id="S1.I1.i3.p1.2.m2.2.2.1.1.2" xref="S1.I1.i3.p1.2.m2.2.2.1.1.2.cmml">Ο</mi><mn id="S1.I1.i3.p1.2.m2.2.2.1.1.3" xref="S1.I1.i3.p1.2.m2.2.2.1.1.3.cmml">1</mn></msub><mo id="S1.I1.i3.p1.2.m2.2.2.1.3" xref="S1.I1.i3.p1.2.m2.2.2.2.cmml">,</mo><mi id="S1.I1.i3.p1.2.m2.1.1" xref="S1.I1.i3.p1.2.m2.1.1.cmml">min</mi><mo id="S1.I1.i3.p1.2.m2.2.2.1.4" stretchy="false" xref="S1.I1.i3.p1.2.m2.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i3.p1.2.m2.2b"><interval closure="open" id="S1.I1.i3.p1.2.m2.2.2.2.cmml" xref="S1.I1.i3.p1.2.m2.2.2.1"><apply id="S1.I1.i3.p1.2.m2.2.2.1.1.cmml" xref="S1.I1.i3.p1.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S1.I1.i3.p1.2.m2.2.2.1.1.1.cmml" xref="S1.I1.i3.p1.2.m2.2.2.1.1">subscript</csymbol><ci id="S1.I1.i3.p1.2.m2.2.2.1.1.2.cmml" xref="S1.I1.i3.p1.2.m2.2.2.1.1.2">π</ci><cn id="S1.I1.i3.p1.2.m2.2.2.1.1.3.cmml" type="integer" xref="S1.I1.i3.p1.2.m2.2.2.1.1.3">1</cn></apply><min id="S1.I1.i3.p1.2.m2.1.1.cmml" xref="S1.I1.i3.p1.2.m2.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i3.p1.2.m2.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i3.p1.2.m2.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i3.p1.3.3"> or </span><math alttext="(\omega_{1},\max)" class="ltx_Math" display="inline" id="S1.I1.i3.p1.3.m3.2"><semantics id="S1.I1.i3.p1.3.m3.2a"><mrow id="S1.I1.i3.p1.3.m3.2.2.1" xref="S1.I1.i3.p1.3.m3.2.2.2.cmml"><mo id="S1.I1.i3.p1.3.m3.2.2.1.2" stretchy="false" xref="S1.I1.i3.p1.3.m3.2.2.2.cmml">(</mo><msub id="S1.I1.i3.p1.3.m3.2.2.1.1" xref="S1.I1.i3.p1.3.m3.2.2.1.1.cmml"><mi id="S1.I1.i3.p1.3.m3.2.2.1.1.2" xref="S1.I1.i3.p1.3.m3.2.2.1.1.2.cmml">Ο</mi><mn id="S1.I1.i3.p1.3.m3.2.2.1.1.3" xref="S1.I1.i3.p1.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S1.I1.i3.p1.3.m3.2.2.1.3" xref="S1.I1.i3.p1.3.m3.2.2.2.cmml">,</mo><mi id="S1.I1.i3.p1.3.m3.1.1" xref="S1.I1.i3.p1.3.m3.1.1.cmml">max</mi><mo id="S1.I1.i3.p1.3.m3.2.2.1.4" stretchy="false" xref="S1.I1.i3.p1.3.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i3.p1.3.m3.2b"><interval closure="open" id="S1.I1.i3.p1.3.m3.2.2.2.cmml" xref="S1.I1.i3.p1.3.m3.2.2.1"><apply id="S1.I1.i3.p1.3.m3.2.2.1.1.cmml" xref="S1.I1.i3.p1.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S1.I1.i3.p1.3.m3.2.2.1.1.1.cmml" xref="S1.I1.i3.p1.3.m3.2.2.1.1">subscript</csymbol><ci id="S1.I1.i3.p1.3.m3.2.2.1.1.2.cmml" xref="S1.I1.i3.p1.3.m3.2.2.1.1.2">π</ci><cn id="S1.I1.i3.p1.3.m3.2.2.1.1.3.cmml" type="integer" xref="S1.I1.i3.p1.3.m3.2.2.1.1.3">1</cn></apply><max id="S1.I1.i3.p1.3.m3.1.1.cmml" xref="S1.I1.i3.p1.3.m3.1.1"></max></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i3.p1.3.m3.2c">(\omega_{1},\max)</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i3.p1.3.m3.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_max )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i3.p1.3.4"> endowed with the order topology.</span></p> </div> </li> </ol> </div> </div> <div class="ltx_para" id="S1.p8"> <p class="ltx_p" id="S1.p8.1">The second result solves the problem of Banakh and Pastukhova <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib5" title="">5</a>, Problem 2.2]</cite> and complements <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib5" title="">5</a>, Proposition 2.1]</cite>, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib27" title="">27</a>, Theorem 7]</cite> and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib4" title="">4</a>, Corollary 1]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_ltheorem" id="Thmltheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmltheorem3.1.1.1">Theorem C</span></span><span class="ltx_text ltx_font_bold" id="Thmltheorem3.2.2">.</span> </h6> <div class="ltx_para" id="Thmltheorem3.p1"> <p class="ltx_p" id="Thmltheorem3.p1.2"><span class="ltx_text ltx_font_italic" id="Thmltheorem3.p1.2.2">Let <math alttext="S" class="ltx_Math" display="inline" id="Thmltheorem3.p1.1.1.m1.1"><semantics id="Thmltheorem3.p1.1.1.m1.1a"><mi id="Thmltheorem3.p1.1.1.m1.1.1" xref="Thmltheorem3.p1.1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="Thmltheorem3.p1.1.1.m1.1b"><ci id="Thmltheorem3.p1.1.1.m1.1.1.cmml" xref="Thmltheorem3.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmltheorem3.p1.1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="Thmltheorem3.p1.1.1.m1.1d">italic_S</annotation></semantics></math> be a topological semigroup that contains a dense inverse subsemigroup. Then <math alttext="S" class="ltx_Math" display="inline" id="Thmltheorem3.p1.2.2.m2.1"><semantics id="Thmltheorem3.p1.2.2.m2.1a"><mi id="Thmltheorem3.p1.2.2.m2.1.1" xref="Thmltheorem3.p1.2.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="Thmltheorem3.p1.2.2.m2.1b"><ci id="Thmltheorem3.p1.2.2.m2.1.1.cmml" xref="Thmltheorem3.p1.2.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmltheorem3.p1.2.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="Thmltheorem3.p1.2.2.m2.1d">italic_S</annotation></semantics></math> is a topological inverse semigroup provided one of the following conditions holds:</span></p> <ol class="ltx_enumerate" id="S1.I2"> <li class="ltx_item" id="S1.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S1.I2.i1.p1"> <p class="ltx_p" id="S1.I2.i1.p1.1"><math alttext="S" class="ltx_Math" display="inline" id="S1.I2.i1.p1.1.m1.1"><semantics id="S1.I2.i1.p1.1.m1.1a"><mi id="S1.I2.i1.p1.1.m1.1.1" xref="S1.I2.i1.p1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.I2.i1.p1.1.m1.1b"><ci id="S1.I2.i1.p1.1.m1.1.1.cmml" xref="S1.I2.i1.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I2.i1.p1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.I2.i1.p1.1.m1.1d">italic_S</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I2.i1.p1.1.1"> is compact;</span></p> </div> </li> <li class="ltx_item" id="S1.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S1.I2.i2.p1"> <p class="ltx_p" id="S1.I2.i2.p1.1"><math alttext="S" class="ltx_Math" display="inline" id="S1.I2.i2.p1.1.m1.1"><semantics id="S1.I2.i2.p1.1.m1.1a"><mi id="S1.I2.i2.p1.1.m1.1.1" xref="S1.I2.i2.p1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.I2.i2.p1.1.m1.1b"><ci id="S1.I2.i2.p1.1.m1.1.1.cmml" xref="S1.I2.i2.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I2.i2.p1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.I2.i2.p1.1.m1.1d">italic_S</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I2.i2.p1.1.1"> is countably compact and sequential.</span></p> </div> </li> </ol> </div> </div> <div class="ltx_para" id="S1.p9"> <p class="ltx_p" id="S1.p9.4">Recall that a space <math alttext="X" class="ltx_Math" display="inline" id="S1.p9.1.m1.1"><semantics id="S1.p9.1.m1.1a"><mi id="S1.p9.1.m1.1.1" xref="S1.p9.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.p9.1.m1.1b"><ci id="S1.p9.1.m1.1.1.cmml" xref="S1.p9.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.p9.1.m1.1d">italic_X</annotation></semantics></math> is called <span class="ltx_text ltx_font_italic" id="S1.p9.4.1">sequential</span> if for every non-closed subset <math alttext="A\subset X" class="ltx_Math" display="inline" id="S1.p9.2.m2.1"><semantics id="S1.p9.2.m2.1a"><mrow id="S1.p9.2.m2.1.1" xref="S1.p9.2.m2.1.1.cmml"><mi id="S1.p9.2.m2.1.1.2" xref="S1.p9.2.m2.1.1.2.cmml">A</mi><mo id="S1.p9.2.m2.1.1.1" xref="S1.p9.2.m2.1.1.1.cmml">β</mo><mi id="S1.p9.2.m2.1.1.3" xref="S1.p9.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.2.m2.1b"><apply id="S1.p9.2.m2.1.1.cmml" xref="S1.p9.2.m2.1.1"><subset id="S1.p9.2.m2.1.1.1.cmml" xref="S1.p9.2.m2.1.1.1"></subset><ci id="S1.p9.2.m2.1.1.2.cmml" xref="S1.p9.2.m2.1.1.2">π΄</ci><ci id="S1.p9.2.m2.1.1.3.cmml" xref="S1.p9.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.2.m2.1c">A\subset X</annotation><annotation encoding="application/x-llamapun" id="S1.p9.2.m2.1d">italic_A β italic_X</annotation></semantics></math> there exists a sequence in <math alttext="A" class="ltx_Math" display="inline" id="S1.p9.3.m3.1"><semantics id="S1.p9.3.m3.1a"><mi id="S1.p9.3.m3.1.1" xref="S1.p9.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.p9.3.m3.1b"><ci id="S1.p9.3.m3.1.1.cmml" xref="S1.p9.3.m3.1.1">π΄</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.p9.3.m3.1d">italic_A</annotation></semantics></math> that converges to some point of <math alttext="X\setminus A" class="ltx_Math" display="inline" id="S1.p9.4.m4.1"><semantics id="S1.p9.4.m4.1a"><mrow id="S1.p9.4.m4.1.1" xref="S1.p9.4.m4.1.1.cmml"><mi id="S1.p9.4.m4.1.1.2" xref="S1.p9.4.m4.1.1.2.cmml">X</mi><mo id="S1.p9.4.m4.1.1.1" xref="S1.p9.4.m4.1.1.1.cmml">β</mo><mi id="S1.p9.4.m4.1.1.3" xref="S1.p9.4.m4.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.4.m4.1b"><apply id="S1.p9.4.m4.1.1.cmml" xref="S1.p9.4.m4.1.1"><setdiff id="S1.p9.4.m4.1.1.1.cmml" xref="S1.p9.4.m4.1.1.1"></setdiff><ci id="S1.p9.4.m4.1.1.2.cmml" xref="S1.p9.4.m4.1.1.2">π</ci><ci id="S1.p9.4.m4.1.1.3.cmml" xref="S1.p9.4.m4.1.1.3">π΄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.4.m4.1c">X\setminus A</annotation><annotation encoding="application/x-llamapun" id="S1.p9.4.m4.1d">italic_X β italic_A</annotation></semantics></math>. Since each first-countable space is sequential, Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem3" title="Theorem C. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">C</span></a> implies the following.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S1.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem4.1.1.1">Corollary 1.4</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem4.p1"> <p class="ltx_p" id="S1.Thmtheorem4.p1.2"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem4.p1.2.2">Let <math alttext="S" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.1.1.m1.1"><semantics id="S1.Thmtheorem4.p1.1.1.m1.1a"><mi id="S1.Thmtheorem4.p1.1.1.m1.1.1" xref="S1.Thmtheorem4.p1.1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.1.1.m1.1b"><ci id="S1.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S1.Thmtheorem4.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.1.1.m1.1d">italic_S</annotation></semantics></math> be a Nyikos topological semigroup that contains a dense inverse subsemigroup. Then <math alttext="S" class="ltx_Math" display="inline" id="S1.Thmtheorem4.p1.2.2.m2.1"><semantics id="S1.Thmtheorem4.p1.2.2.m2.1a"><mi id="S1.Thmtheorem4.p1.2.2.m2.1.1" xref="S1.Thmtheorem4.p1.2.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem4.p1.2.2.m2.1b"><ci id="S1.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S1.Thmtheorem4.p1.2.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem4.p1.2.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem4.p1.2.2.m2.1d">italic_S</annotation></semantics></math> is a topological inverse semigroup.</span></p> </div> </div> <div class="ltx_para" id="S1.p10"> <p class="ltx_p" id="S1.p10.9">A space <math alttext="X" class="ltx_Math" display="inline" id="S1.p10.1.m1.1"><semantics id="S1.p10.1.m1.1a"><mi id="S1.p10.1.m1.1.1" xref="S1.p10.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.p10.1.m1.1b"><ci id="S1.p10.1.m1.1.1.cmml" xref="S1.p10.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.p10.1.m1.1d">italic_X</annotation></semantics></math> is called <span class="ltx_text ltx_font_italic" id="S1.p10.9.1">pseudocompact</span> if <math alttext="X" class="ltx_Math" display="inline" id="S1.p10.2.m2.1"><semantics id="S1.p10.2.m2.1a"><mi id="S1.p10.2.m2.1.1" xref="S1.p10.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.p10.2.m2.1b"><ci id="S1.p10.2.m2.1.1.cmml" xref="S1.p10.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.p10.2.m2.1d">italic_X</annotation></semantics></math> is Tychonoff and each continuous real-valued function on <math alttext="X" class="ltx_Math" display="inline" id="S1.p10.3.m3.1"><semantics id="S1.p10.3.m3.1a"><mi id="S1.p10.3.m3.1.1" xref="S1.p10.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.p10.3.m3.1b"><ci id="S1.p10.3.m3.1.1.cmml" xref="S1.p10.3.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.p10.3.m3.1d">italic_X</annotation></semantics></math> is bounded. A semigroup <math alttext="S" class="ltx_Math" display="inline" id="S1.p10.4.m4.1"><semantics id="S1.p10.4.m4.1a"><mi id="S1.p10.4.m4.1.1" xref="S1.p10.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.p10.4.m4.1b"><ci id="S1.p10.4.m4.1.1.cmml" xref="S1.p10.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.p10.4.m4.1d">italic_S</annotation></semantics></math> endowed with a topology is called <span class="ltx_text ltx_font_italic" id="S1.p10.9.2">topologically periodic</span> if for each <math alttext="x\in S" class="ltx_Math" display="inline" id="S1.p10.5.m5.1"><semantics id="S1.p10.5.m5.1a"><mrow id="S1.p10.5.m5.1.1" xref="S1.p10.5.m5.1.1.cmml"><mi id="S1.p10.5.m5.1.1.2" xref="S1.p10.5.m5.1.1.2.cmml">x</mi><mo id="S1.p10.5.m5.1.1.1" xref="S1.p10.5.m5.1.1.1.cmml">β</mo><mi id="S1.p10.5.m5.1.1.3" xref="S1.p10.5.m5.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p10.5.m5.1b"><apply id="S1.p10.5.m5.1.1.cmml" xref="S1.p10.5.m5.1.1"><in id="S1.p10.5.m5.1.1.1.cmml" xref="S1.p10.5.m5.1.1.1"></in><ci id="S1.p10.5.m5.1.1.2.cmml" xref="S1.p10.5.m5.1.1.2">π₯</ci><ci id="S1.p10.5.m5.1.1.3.cmml" xref="S1.p10.5.m5.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.5.m5.1c">x\in S</annotation><annotation encoding="application/x-llamapun" id="S1.p10.5.m5.1d">italic_x β italic_S</annotation></semantics></math> and open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S1.p10.6.m6.1"><semantics id="S1.p10.6.m6.1a"><mi id="S1.p10.6.m6.1.1" xref="S1.p10.6.m6.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S1.p10.6.m6.1b"><ci id="S1.p10.6.m6.1.1.cmml" xref="S1.p10.6.m6.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.6.m6.1c">U</annotation><annotation encoding="application/x-llamapun" id="S1.p10.6.m6.1d">italic_U</annotation></semantics></math> of <math alttext="x" class="ltx_Math" display="inline" id="S1.p10.7.m7.1"><semantics id="S1.p10.7.m7.1a"><mi id="S1.p10.7.m7.1.1" xref="S1.p10.7.m7.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S1.p10.7.m7.1b"><ci id="S1.p10.7.m7.1.1.cmml" xref="S1.p10.7.m7.1.1">π₯</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.7.m7.1c">x</annotation><annotation encoding="application/x-llamapun" id="S1.p10.7.m7.1d">italic_x</annotation></semantics></math> there exists <math alttext="n\geq 2" class="ltx_Math" display="inline" id="S1.p10.8.m8.1"><semantics id="S1.p10.8.m8.1a"><mrow id="S1.p10.8.m8.1.1" xref="S1.p10.8.m8.1.1.cmml"><mi id="S1.p10.8.m8.1.1.2" xref="S1.p10.8.m8.1.1.2.cmml">n</mi><mo id="S1.p10.8.m8.1.1.1" xref="S1.p10.8.m8.1.1.1.cmml">β₯</mo><mn id="S1.p10.8.m8.1.1.3" xref="S1.p10.8.m8.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p10.8.m8.1b"><apply id="S1.p10.8.m8.1.1.cmml" xref="S1.p10.8.m8.1.1"><geq id="S1.p10.8.m8.1.1.1.cmml" xref="S1.p10.8.m8.1.1.1"></geq><ci id="S1.p10.8.m8.1.1.2.cmml" xref="S1.p10.8.m8.1.1.2">π</ci><cn id="S1.p10.8.m8.1.1.3.cmml" type="integer" xref="S1.p10.8.m8.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.8.m8.1c">n\geq 2</annotation><annotation encoding="application/x-llamapun" id="S1.p10.8.m8.1d">italic_n β₯ 2</annotation></semantics></math> such that <math alttext="x^{n}\in U" class="ltx_Math" display="inline" id="S1.p10.9.m9.1"><semantics id="S1.p10.9.m9.1a"><mrow id="S1.p10.9.m9.1.1" xref="S1.p10.9.m9.1.1.cmml"><msup id="S1.p10.9.m9.1.1.2" xref="S1.p10.9.m9.1.1.2.cmml"><mi id="S1.p10.9.m9.1.1.2.2" xref="S1.p10.9.m9.1.1.2.2.cmml">x</mi><mi id="S1.p10.9.m9.1.1.2.3" xref="S1.p10.9.m9.1.1.2.3.cmml">n</mi></msup><mo id="S1.p10.9.m9.1.1.1" xref="S1.p10.9.m9.1.1.1.cmml">β</mo><mi id="S1.p10.9.m9.1.1.3" xref="S1.p10.9.m9.1.1.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p10.9.m9.1b"><apply id="S1.p10.9.m9.1.1.cmml" xref="S1.p10.9.m9.1.1"><in id="S1.p10.9.m9.1.1.1.cmml" xref="S1.p10.9.m9.1.1.1"></in><apply id="S1.p10.9.m9.1.1.2.cmml" xref="S1.p10.9.m9.1.1.2"><csymbol cd="ambiguous" id="S1.p10.9.m9.1.1.2.1.cmml" xref="S1.p10.9.m9.1.1.2">superscript</csymbol><ci id="S1.p10.9.m9.1.1.2.2.cmml" xref="S1.p10.9.m9.1.1.2.2">π₯</ci><ci id="S1.p10.9.m9.1.1.2.3.cmml" xref="S1.p10.9.m9.1.1.2.3">π</ci></apply><ci id="S1.p10.9.m9.1.1.3.cmml" xref="S1.p10.9.m9.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.9.m9.1c">x^{n}\in U</annotation><annotation encoding="application/x-llamapun" id="S1.p10.9.m9.1d">italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT β italic_U</annotation></semantics></math>. Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem3" title="Theorem C. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">C</span></a> is essential for the proof of the following result that complements <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib4" title="">4</a>, Theorem 2]</cite> and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib27" title="">27</a>, Theorem 1]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_ltheorem" id="Thmltheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmltheorem4.1.1.1">Theorem D</span></span><span class="ltx_text ltx_font_bold" id="Thmltheorem4.2.2">.</span> </h6> <div class="ltx_para" id="Thmltheorem4.p1"> <p class="ltx_p" id="Thmltheorem4.p1.2"><span class="ltx_text ltx_font_italic" id="Thmltheorem4.p1.2.2">Let <math alttext="S" class="ltx_Math" display="inline" id="Thmltheorem4.p1.1.1.m1.1"><semantics id="Thmltheorem4.p1.1.1.m1.1a"><mi id="Thmltheorem4.p1.1.1.m1.1.1" xref="Thmltheorem4.p1.1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="Thmltheorem4.p1.1.1.m1.1b"><ci id="Thmltheorem4.p1.1.1.m1.1.1.cmml" xref="Thmltheorem4.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmltheorem4.p1.1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="Thmltheorem4.p1.1.1.m1.1d">italic_S</annotation></semantics></math> be an inverse topological semigroup. Then inversion is continuous in <math alttext="S" class="ltx_Math" display="inline" id="Thmltheorem4.p1.2.2.m2.1"><semantics id="Thmltheorem4.p1.2.2.m2.1a"><mi id="Thmltheorem4.p1.2.2.m2.1.1" xref="Thmltheorem4.p1.2.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="Thmltheorem4.p1.2.2.m2.1b"><ci id="Thmltheorem4.p1.2.2.m2.1.1.cmml" xref="Thmltheorem4.p1.2.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmltheorem4.p1.2.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="Thmltheorem4.p1.2.2.m2.1d">italic_S</annotation></semantics></math> provided one of the following conditions hold:</span></p> <ol class="ltx_enumerate" id="S1.I3"> <li class="ltx_item" id="S1.I3.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S1.I3.i1.p1"> <p class="ltx_p" id="S1.I3.i1.p1.2"><math alttext="S" class="ltx_Math" display="inline" id="S1.I3.i1.p1.1.m1.1"><semantics id="S1.I3.i1.p1.1.m1.1a"><mi id="S1.I3.i1.p1.1.m1.1.1" xref="S1.I3.i1.p1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.I3.i1.p1.1.m1.1b"><ci id="S1.I3.i1.p1.1.m1.1.1.cmml" xref="S1.I3.i1.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I3.i1.p1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.I3.i1.p1.1.m1.1d">italic_S</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I3.i1.p1.2.1"> is Tychonoff and </span><math alttext="S{\times}S" class="ltx_Math" display="inline" id="S1.I3.i1.p1.2.m2.1"><semantics id="S1.I3.i1.p1.2.m2.1a"><mrow id="S1.I3.i1.p1.2.m2.1.1" xref="S1.I3.i1.p1.2.m2.1.1.cmml"><mi id="S1.I3.i1.p1.2.m2.1.1.2" xref="S1.I3.i1.p1.2.m2.1.1.2.cmml">S</mi><mo id="S1.I3.i1.p1.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S1.I3.i1.p1.2.m2.1.1.1.cmml">Γ</mo><mi id="S1.I3.i1.p1.2.m2.1.1.3" xref="S1.I3.i1.p1.2.m2.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.I3.i1.p1.2.m2.1b"><apply id="S1.I3.i1.p1.2.m2.1.1.cmml" xref="S1.I3.i1.p1.2.m2.1.1"><times id="S1.I3.i1.p1.2.m2.1.1.1.cmml" xref="S1.I3.i1.p1.2.m2.1.1.1"></times><ci id="S1.I3.i1.p1.2.m2.1.1.2.cmml" xref="S1.I3.i1.p1.2.m2.1.1.2">π</ci><ci id="S1.I3.i1.p1.2.m2.1.1.3.cmml" xref="S1.I3.i1.p1.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I3.i1.p1.2.m2.1c">S{\times}S</annotation><annotation encoding="application/x-llamapun" id="S1.I3.i1.p1.2.m2.1d">italic_S Γ italic_S</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I3.i1.p1.2.2"> is pseudocompact;</span></p> </div> </li> <li class="ltx_item" id="S1.I3.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S1.I3.i2.p1"> <p class="ltx_p" id="S1.I3.i2.p1.2"><math alttext="S" class="ltx_Math" display="inline" id="S1.I3.i2.p1.1.m1.1"><semantics id="S1.I3.i2.p1.1.m1.1a"><mi id="S1.I3.i2.p1.1.m1.1.1" xref="S1.I3.i2.p1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.I3.i2.p1.1.m1.1b"><ci id="S1.I3.i2.p1.1.m1.1.1.cmml" xref="S1.I3.i2.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I3.i2.p1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.I3.i2.p1.1.m1.1d">italic_S</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I3.i2.p1.2.1"> is regular, topologically periodic and </span><math alttext="S{\times}S" class="ltx_Math" display="inline" id="S1.I3.i2.p1.2.m2.1"><semantics id="S1.I3.i2.p1.2.m2.1a"><mrow id="S1.I3.i2.p1.2.m2.1.1" xref="S1.I3.i2.p1.2.m2.1.1.cmml"><mi id="S1.I3.i2.p1.2.m2.1.1.2" xref="S1.I3.i2.p1.2.m2.1.1.2.cmml">S</mi><mo id="S1.I3.i2.p1.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S1.I3.i2.p1.2.m2.1.1.1.cmml">Γ</mo><mi id="S1.I3.i2.p1.2.m2.1.1.3" xref="S1.I3.i2.p1.2.m2.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.I3.i2.p1.2.m2.1b"><apply id="S1.I3.i2.p1.2.m2.1.1.cmml" xref="S1.I3.i2.p1.2.m2.1.1"><times id="S1.I3.i2.p1.2.m2.1.1.1.cmml" xref="S1.I3.i2.p1.2.m2.1.1.1"></times><ci id="S1.I3.i2.p1.2.m2.1.1.2.cmml" xref="S1.I3.i2.p1.2.m2.1.1.2">π</ci><ci id="S1.I3.i2.p1.2.m2.1.1.3.cmml" xref="S1.I3.i2.p1.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I3.i2.p1.2.m2.1c">S{\times}S</annotation><annotation encoding="application/x-llamapun" id="S1.I3.i2.p1.2.m2.1d">italic_S Γ italic_S</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I3.i2.p1.2.2"> is countably compact.</span></p> </div> </li> </ol> </div> </div> <div class="ltx_para" id="S1.p11"> <p class="ltx_p" id="S1.p11.1">This paper is organized as follows. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S2" title="2. Compact-like semigroups with dense inverse subsemigroups and the automatic continuity of inversion β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">2</span></a> is devoted to compact-like semigroups with dense inverse subsemigroups and the automatic continuity of inversion. In particular, Theorems <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem3" title="Theorem C. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">C</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem4" title="Theorem D. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">D</span></a> are proven there. In Section <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3" title="3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3</span></a> we investigate chains in Nyikos topological semilattices and prove Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem2" title="Theorem B. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">B</span></a>. In Section <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S4" title="4. Nyikos semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">4</span></a> we show that each locally compact Nyikos topological semilattice is compact. This result is a milestone in the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem1" title="Theorem A. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">A</span></a> that is given in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S5" title="5. Proof of the main result and final remarks β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">5</span></a>. Also, there we show that Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem1" title="Theorem A. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">A</span></a> cannot be generalized over simple bands.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2. </span>Compact-like semigroups with dense inverse subsemigroups and the automatic continuity of inversion</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.6">Let <math alttext="A" class="ltx_Math" display="inline" id="S2.p1.1.m1.1"><semantics id="S2.p1.1.m1.1a"><mi id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.1b"><ci id="S2.p1.1.m1.1.1.cmml" xref="S2.p1.1.m1.1.1">π΄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.1d">italic_A</annotation></semantics></math> be a subset of a space <math alttext="X" class="ltx_Math" display="inline" id="S2.p1.2.m2.1"><semantics id="S2.p1.2.m2.1a"><mi id="S2.p1.2.m2.1.1" xref="S2.p1.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.p1.2.m2.1b"><ci id="S2.p1.2.m2.1.1.cmml" xref="S2.p1.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m2.1d">italic_X</annotation></semantics></math>. By <math alttext="\operatorname{cl}_{X}(A)" class="ltx_Math" display="inline" id="S2.p1.3.m3.2"><semantics id="S2.p1.3.m3.2a"><mrow id="S2.p1.3.m3.2.2.1" xref="S2.p1.3.m3.2.2.2.cmml"><msub id="S2.p1.3.m3.2.2.1.1" xref="S2.p1.3.m3.2.2.1.1.cmml"><mi id="S2.p1.3.m3.2.2.1.1.2" xref="S2.p1.3.m3.2.2.1.1.2.cmml">cl</mi><mi id="S2.p1.3.m3.2.2.1.1.3" xref="S2.p1.3.m3.2.2.1.1.3.cmml">X</mi></msub><mo id="S2.p1.3.m3.2.2.1a" xref="S2.p1.3.m3.2.2.2.cmml">β‘</mo><mrow id="S2.p1.3.m3.2.2.1.2" xref="S2.p1.3.m3.2.2.2.cmml"><mo id="S2.p1.3.m3.2.2.1.2.1" stretchy="false" xref="S2.p1.3.m3.2.2.2.cmml">(</mo><mi id="S2.p1.3.m3.1.1" xref="S2.p1.3.m3.1.1.cmml">A</mi><mo id="S2.p1.3.m3.2.2.1.2.2" stretchy="false" xref="S2.p1.3.m3.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.3.m3.2b"><apply id="S2.p1.3.m3.2.2.2.cmml" xref="S2.p1.3.m3.2.2.1"><apply id="S2.p1.3.m3.2.2.1.1.cmml" xref="S2.p1.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S2.p1.3.m3.2.2.1.1.1.cmml" xref="S2.p1.3.m3.2.2.1.1">subscript</csymbol><ci id="S2.p1.3.m3.2.2.1.1.2.cmml" xref="S2.p1.3.m3.2.2.1.1.2">cl</ci><ci id="S2.p1.3.m3.2.2.1.1.3.cmml" xref="S2.p1.3.m3.2.2.1.1.3">π</ci></apply><ci id="S2.p1.3.m3.1.1.cmml" xref="S2.p1.3.m3.1.1">π΄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.3.m3.2c">\operatorname{cl}_{X}(A)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.3.m3.2d">roman_cl start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_A )</annotation></semantics></math> or simply <math alttext="\overline{A}" class="ltx_Math" display="inline" id="S2.p1.4.m4.1"><semantics id="S2.p1.4.m4.1a"><mover accent="true" id="S2.p1.4.m4.1.1" xref="S2.p1.4.m4.1.1.cmml"><mi id="S2.p1.4.m4.1.1.2" xref="S2.p1.4.m4.1.1.2.cmml">A</mi><mo id="S2.p1.4.m4.1.1.1" xref="S2.p1.4.m4.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S2.p1.4.m4.1b"><apply id="S2.p1.4.m4.1.1.cmml" xref="S2.p1.4.m4.1.1"><ci id="S2.p1.4.m4.1.1.1.cmml" xref="S2.p1.4.m4.1.1.1">Β―</ci><ci id="S2.p1.4.m4.1.1.2.cmml" xref="S2.p1.4.m4.1.1.2">π΄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.4.m4.1c">\overline{A}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.4.m4.1d">overΒ― start_ARG italic_A end_ARG</annotation></semantics></math> we denote the closure of <math alttext="A" class="ltx_Math" display="inline" id="S2.p1.5.m5.1"><semantics id="S2.p1.5.m5.1a"><mi id="S2.p1.5.m5.1.1" xref="S2.p1.5.m5.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.p1.5.m5.1b"><ci id="S2.p1.5.m5.1.1.cmml" xref="S2.p1.5.m5.1.1">π΄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.5.m5.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.p1.5.m5.1d">italic_A</annotation></semantics></math> in <math alttext="X" class="ltx_Math" display="inline" id="S2.p1.6.m6.1"><semantics id="S2.p1.6.m6.1a"><mi id="S2.p1.6.m6.1.1" xref="S2.p1.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.p1.6.m6.1b"><ci id="S2.p1.6.m6.1.1.cmml" xref="S2.p1.6.m6.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.p1.6.m6.1d">italic_X</annotation></semantics></math>. Let</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="s(A)=\{y\in X\colon\hbox{ there exists a sequence }\{x_{n}:n\in\omega\}% \subseteq A\hbox{ such that }\lim_{n\in\omega}x_{n}=y\}." class="ltx_Math" display="block" id="S2.Ex1.m1.2"><semantics id="S2.Ex1.m1.2a"><mrow id="S2.Ex1.m1.2.2.1" xref="S2.Ex1.m1.2.2.1.1.cmml"><mrow id="S2.Ex1.m1.2.2.1.1" xref="S2.Ex1.m1.2.2.1.1.cmml"><mrow id="S2.Ex1.m1.2.2.1.1.4" xref="S2.Ex1.m1.2.2.1.1.4.cmml"><mi id="S2.Ex1.m1.2.2.1.1.4.2" xref="S2.Ex1.m1.2.2.1.1.4.2.cmml">s</mi><mo id="S2.Ex1.m1.2.2.1.1.4.1" xref="S2.Ex1.m1.2.2.1.1.4.1.cmml">β’</mo><mrow id="S2.Ex1.m1.2.2.1.1.4.3.2" xref="S2.Ex1.m1.2.2.1.1.4.cmml"><mo id="S2.Ex1.m1.2.2.1.1.4.3.2.1" stretchy="false" xref="S2.Ex1.m1.2.2.1.1.4.cmml">(</mo><mi id="S2.Ex1.m1.1.1" xref="S2.Ex1.m1.1.1.cmml">A</mi><mo id="S2.Ex1.m1.2.2.1.1.4.3.2.2" stretchy="false" xref="S2.Ex1.m1.2.2.1.1.4.cmml">)</mo></mrow></mrow><mo id="S2.Ex1.m1.2.2.1.1.3" xref="S2.Ex1.m1.2.2.1.1.3.cmml">=</mo><mrow id="S2.Ex1.m1.2.2.1.1.2.2" xref="S2.Ex1.m1.2.2.1.1.2.3.cmml"><mo id="S2.Ex1.m1.2.2.1.1.2.2.3" stretchy="false" xref="S2.Ex1.m1.2.2.1.1.2.3.1.cmml">{</mo><mrow id="S2.Ex1.m1.2.2.1.1.1.1.1" xref="S2.Ex1.m1.2.2.1.1.1.1.1.cmml"><mi id="S2.Ex1.m1.2.2.1.1.1.1.1.2" xref="S2.Ex1.m1.2.2.1.1.1.1.1.2.cmml">y</mi><mo id="S2.Ex1.m1.2.2.1.1.1.1.1.1" xref="S2.Ex1.m1.2.2.1.1.1.1.1.1.cmml">β</mo><mi id="S2.Ex1.m1.2.2.1.1.1.1.1.3" xref="S2.Ex1.m1.2.2.1.1.1.1.1.3.cmml">X</mi></mrow><mo id="S2.Ex1.m1.2.2.1.1.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.Ex1.m1.2.2.1.1.2.3.1.cmml">:</mo><mrow id="S2.Ex1.m1.2.2.1.1.2.2.2" xref="S2.Ex1.m1.2.2.1.1.2.2.2.cmml"><mrow id="S2.Ex1.m1.2.2.1.1.2.2.2.2" xref="S2.Ex1.m1.2.2.1.1.2.2.2.2.cmml"><mtext id="S2.Ex1.m1.2.2.1.1.2.2.2.2.4" xref="S2.Ex1.m1.2.2.1.1.2.2.2.2.4a.cmml"> there exists a sequence </mtext><mo id="S2.Ex1.m1.2.2.1.1.2.2.2.2.3" xref="S2.Ex1.m1.2.2.1.1.2.2.2.2.3.cmml">β’</mo><mrow id="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.2" xref="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.3.cmml"><mo id="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.2.3" stretchy="false" xref="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.3.1.cmml">{</mo><msub id="S2.Ex1.m1.2.2.1.1.2.2.2.1.1.1.1" xref="S2.Ex1.m1.2.2.1.1.2.2.2.1.1.1.1.cmml"><mi id="S2.Ex1.m1.2.2.1.1.2.2.2.1.1.1.1.2" xref="S2.Ex1.m1.2.2.1.1.2.2.2.1.1.1.1.2.cmml">x</mi><mi id="S2.Ex1.m1.2.2.1.1.2.2.2.1.1.1.1.3" xref="S2.Ex1.m1.2.2.1.1.2.2.2.1.1.1.1.3.cmml">n</mi></msub><mo id="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.3.1.cmml">:</mo><mrow id="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.2.2" xref="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.2.2.cmml"><mi id="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.2.2.2" xref="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.2.2.2.cmml">n</mi><mo id="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.2.2.1" xref="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.2.2.1.cmml">β</mo><mi id="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.2.2.3" xref="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.2.2.3.cmml">Ο</mi></mrow><mo id="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.2.5" stretchy="false" xref="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.3.1.cmml">}</mo></mrow></mrow><mo id="S2.Ex1.m1.2.2.1.1.2.2.2.4" xref="S2.Ex1.m1.2.2.1.1.2.2.2.4.cmml">β</mo><mrow id="S2.Ex1.m1.2.2.1.1.2.2.2.5" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.cmml"><mi id="S2.Ex1.m1.2.2.1.1.2.2.2.5.2" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.2.cmml">A</mi><mo id="S2.Ex1.m1.2.2.1.1.2.2.2.5.1" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.1.cmml">β’</mo><mtext id="S2.Ex1.m1.2.2.1.1.2.2.2.5.3" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.3a.cmml"> such that </mtext><mo id="S2.Ex1.m1.2.2.1.1.2.2.2.5.1a" lspace="0.167em" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.1.cmml">β’</mo><mrow id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.cmml"><munder id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.cmml"><mo id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.2" movablelimits="false" rspace="0.167em" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.2.cmml">lim</mo><mrow id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.3" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.3.cmml"><mi id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.3.2" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.3.2.cmml">n</mi><mo id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.3.1" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.3.1.cmml">β</mo><mi id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.3.3" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.3.3.cmml">Ο</mi></mrow></munder><msub id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.2" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.2.cmml"><mi id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.2.2" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.2.2.cmml">x</mi><mi id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.2.3" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.2.3.cmml">n</mi></msub></mrow></mrow><mo id="S2.Ex1.m1.2.2.1.1.2.2.2.6" xref="S2.Ex1.m1.2.2.1.1.2.2.2.6.cmml">=</mo><mi id="S2.Ex1.m1.2.2.1.1.2.2.2.7" xref="S2.Ex1.m1.2.2.1.1.2.2.2.7.cmml">y</mi></mrow><mo id="S2.Ex1.m1.2.2.1.1.2.2.5" stretchy="false" xref="S2.Ex1.m1.2.2.1.1.2.3.1.cmml">}</mo></mrow></mrow><mo id="S2.Ex1.m1.2.2.1.2" lspace="0em" xref="S2.Ex1.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex1.m1.2b"><apply id="S2.Ex1.m1.2.2.1.1.cmml" xref="S2.Ex1.m1.2.2.1"><eq id="S2.Ex1.m1.2.2.1.1.3.cmml" xref="S2.Ex1.m1.2.2.1.1.3"></eq><apply id="S2.Ex1.m1.2.2.1.1.4.cmml" xref="S2.Ex1.m1.2.2.1.1.4"><times id="S2.Ex1.m1.2.2.1.1.4.1.cmml" xref="S2.Ex1.m1.2.2.1.1.4.1"></times><ci id="S2.Ex1.m1.2.2.1.1.4.2.cmml" xref="S2.Ex1.m1.2.2.1.1.4.2">π </ci><ci id="S2.Ex1.m1.1.1.cmml" xref="S2.Ex1.m1.1.1">π΄</ci></apply><apply id="S2.Ex1.m1.2.2.1.1.2.3.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2"><csymbol cd="latexml" id="S2.Ex1.m1.2.2.1.1.2.3.1.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.3">conditional-set</csymbol><apply id="S2.Ex1.m1.2.2.1.1.1.1.1.cmml" xref="S2.Ex1.m1.2.2.1.1.1.1.1"><in id="S2.Ex1.m1.2.2.1.1.1.1.1.1.cmml" xref="S2.Ex1.m1.2.2.1.1.1.1.1.1"></in><ci id="S2.Ex1.m1.2.2.1.1.1.1.1.2.cmml" xref="S2.Ex1.m1.2.2.1.1.1.1.1.2">π¦</ci><ci id="S2.Ex1.m1.2.2.1.1.1.1.1.3.cmml" xref="S2.Ex1.m1.2.2.1.1.1.1.1.3">π</ci></apply><apply id="S2.Ex1.m1.2.2.1.1.2.2.2.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2"><and id="S2.Ex1.m1.2.2.1.1.2.2.2a.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2"></and><apply id="S2.Ex1.m1.2.2.1.1.2.2.2b.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2"><subset id="S2.Ex1.m1.2.2.1.1.2.2.2.4.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.4"></subset><apply id="S2.Ex1.m1.2.2.1.1.2.2.2.2.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.2"><times id="S2.Ex1.m1.2.2.1.1.2.2.2.2.3.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.2.3"></times><ci id="S2.Ex1.m1.2.2.1.1.2.2.2.2.4a.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.2.4"><mtext id="S2.Ex1.m1.2.2.1.1.2.2.2.2.4.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.2.4"> there exists a sequence </mtext></ci><apply id="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.3.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.2"><csymbol cd="latexml" id="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.3.1.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.2.2.2.3">conditional-set</csymbol><apply id="S2.Ex1.m1.2.2.1.1.2.2.2.1.1.1.1.cmml" 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xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.2">π΄</ci><ci id="S2.Ex1.m1.2.2.1.1.2.2.2.5.3a.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.3"><mtext id="S2.Ex1.m1.2.2.1.1.2.2.2.5.3.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.3"> such that </mtext></ci><apply id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4"><apply id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1"><csymbol cd="ambiguous" id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.1.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1">subscript</csymbol><limit id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.2.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.2"></limit><apply id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.3.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.3"><in id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.3.1.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.3.1"></in><ci id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.3.2.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.3.2">π</ci><ci id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.3.3.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.1.3.3">π</ci></apply></apply><apply id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.2.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.2"><csymbol cd="ambiguous" id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.2.1.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.2">subscript</csymbol><ci id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.2.2.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.2.2">π₯</ci><ci id="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.2.3.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.5.4.2.3">π</ci></apply></apply></apply></apply><apply id="S2.Ex1.m1.2.2.1.1.2.2.2c.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2"><eq id="S2.Ex1.m1.2.2.1.1.2.2.2.6.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.6"></eq><share href="https://arxiv.org/html/2503.13666v1#S2.Ex1.m1.2.2.1.1.2.2.2.5.cmml" id="S2.Ex1.m1.2.2.1.1.2.2.2d.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2"></share><ci id="S2.Ex1.m1.2.2.1.1.2.2.2.7.cmml" xref="S2.Ex1.m1.2.2.1.1.2.2.2.7">π¦</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m1.2c">s(A)=\{y\in X\colon\hbox{ there exists a sequence }\{x_{n}:n\in\omega\}% \subseteq A\hbox{ such that }\lim_{n\in\omega}x_{n}=y\}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m1.2d">italic_s ( italic_A ) = { italic_y β italic_X : there exists a sequence { italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n β italic_Ο } β italic_A such that roman_lim start_POSTSUBSCRIPT italic_n β italic_Ο end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_y } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p1.19">The <span class="ltx_text ltx_font_italic" id="S2.p1.19.1">sequential closure</span> <math alttext="\operatorname{cls}(A)" class="ltx_Math" display="inline" id="S2.p1.7.m1.2"><semantics id="S2.p1.7.m1.2a"><mrow id="S2.p1.7.m1.2.3.2" xref="S2.p1.7.m1.2.3.1.cmml"><mi id="S2.p1.7.m1.1.1" xref="S2.p1.7.m1.1.1.cmml">cls</mi><mo id="S2.p1.7.m1.2.3.2a" xref="S2.p1.7.m1.2.3.1.cmml">β‘</mo><mrow id="S2.p1.7.m1.2.3.2.1" xref="S2.p1.7.m1.2.3.1.cmml"><mo id="S2.p1.7.m1.2.3.2.1.1" stretchy="false" xref="S2.p1.7.m1.2.3.1.cmml">(</mo><mi id="S2.p1.7.m1.2.2" xref="S2.p1.7.m1.2.2.cmml">A</mi><mo id="S2.p1.7.m1.2.3.2.1.2" stretchy="false" xref="S2.p1.7.m1.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.7.m1.2b"><apply id="S2.p1.7.m1.2.3.1.cmml" xref="S2.p1.7.m1.2.3.2"><ci id="S2.p1.7.m1.1.1.cmml" xref="S2.p1.7.m1.1.1">cls</ci><ci id="S2.p1.7.m1.2.2.cmml" xref="S2.p1.7.m1.2.2">π΄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.7.m1.2c">\operatorname{cls}(A)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.7.m1.2d">roman_cls ( italic_A )</annotation></semantics></math> of <math alttext="A" class="ltx_Math" display="inline" id="S2.p1.8.m2.1"><semantics id="S2.p1.8.m2.1a"><mi id="S2.p1.8.m2.1.1" xref="S2.p1.8.m2.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.p1.8.m2.1b"><ci id="S2.p1.8.m2.1.1.cmml" xref="S2.p1.8.m2.1.1">π΄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.8.m2.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.p1.8.m2.1d">italic_A</annotation></semantics></math> is defined recursively as follows. Let <math alttext="\operatorname{cls}^{0}(A)=A" class="ltx_Math" display="inline" id="S2.p1.9.m3.2"><semantics id="S2.p1.9.m3.2a"><mrow id="S2.p1.9.m3.2.2" xref="S2.p1.9.m3.2.2.cmml"><mrow id="S2.p1.9.m3.2.2.1.1" xref="S2.p1.9.m3.2.2.1.2.cmml"><msup id="S2.p1.9.m3.2.2.1.1.1" xref="S2.p1.9.m3.2.2.1.1.1.cmml"><mi id="S2.p1.9.m3.2.2.1.1.1.2" xref="S2.p1.9.m3.2.2.1.1.1.2.cmml">cls</mi><mn id="S2.p1.9.m3.2.2.1.1.1.3" xref="S2.p1.9.m3.2.2.1.1.1.3.cmml">0</mn></msup><mo id="S2.p1.9.m3.2.2.1.1a" xref="S2.p1.9.m3.2.2.1.2.cmml">β‘</mo><mrow id="S2.p1.9.m3.2.2.1.1.2" xref="S2.p1.9.m3.2.2.1.2.cmml"><mo id="S2.p1.9.m3.2.2.1.1.2.1" stretchy="false" xref="S2.p1.9.m3.2.2.1.2.cmml">(</mo><mi id="S2.p1.9.m3.1.1" xref="S2.p1.9.m3.1.1.cmml">A</mi><mo id="S2.p1.9.m3.2.2.1.1.2.2" stretchy="false" xref="S2.p1.9.m3.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S2.p1.9.m3.2.2.2" xref="S2.p1.9.m3.2.2.2.cmml">=</mo><mi id="S2.p1.9.m3.2.2.3" xref="S2.p1.9.m3.2.2.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.9.m3.2b"><apply id="S2.p1.9.m3.2.2.cmml" xref="S2.p1.9.m3.2.2"><eq id="S2.p1.9.m3.2.2.2.cmml" xref="S2.p1.9.m3.2.2.2"></eq><apply id="S2.p1.9.m3.2.2.1.2.cmml" xref="S2.p1.9.m3.2.2.1.1"><apply id="S2.p1.9.m3.2.2.1.1.1.cmml" xref="S2.p1.9.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.p1.9.m3.2.2.1.1.1.1.cmml" xref="S2.p1.9.m3.2.2.1.1.1">superscript</csymbol><ci id="S2.p1.9.m3.2.2.1.1.1.2.cmml" xref="S2.p1.9.m3.2.2.1.1.1.2">cls</ci><cn id="S2.p1.9.m3.2.2.1.1.1.3.cmml" type="integer" xref="S2.p1.9.m3.2.2.1.1.1.3">0</cn></apply><ci id="S2.p1.9.m3.1.1.cmml" xref="S2.p1.9.m3.1.1">π΄</ci></apply><ci id="S2.p1.9.m3.2.2.3.cmml" xref="S2.p1.9.m3.2.2.3">π΄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.9.m3.2c">\operatorname{cls}^{0}(A)=A</annotation><annotation encoding="application/x-llamapun" id="S2.p1.9.m3.2d">roman_cls start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_A ) = italic_A</annotation></semantics></math> and assume that for each ordinal <math alttext="\xi<\alpha\leq\omega_{1}" class="ltx_Math" display="inline" id="S2.p1.10.m4.1"><semantics id="S2.p1.10.m4.1a"><mrow id="S2.p1.10.m4.1.1" xref="S2.p1.10.m4.1.1.cmml"><mi id="S2.p1.10.m4.1.1.2" xref="S2.p1.10.m4.1.1.2.cmml">ΞΎ</mi><mo id="S2.p1.10.m4.1.1.3" xref="S2.p1.10.m4.1.1.3.cmml"><</mo><mi id="S2.p1.10.m4.1.1.4" xref="S2.p1.10.m4.1.1.4.cmml">Ξ±</mi><mo id="S2.p1.10.m4.1.1.5" xref="S2.p1.10.m4.1.1.5.cmml">β€</mo><msub id="S2.p1.10.m4.1.1.6" xref="S2.p1.10.m4.1.1.6.cmml"><mi id="S2.p1.10.m4.1.1.6.2" xref="S2.p1.10.m4.1.1.6.2.cmml">Ο</mi><mn id="S2.p1.10.m4.1.1.6.3" xref="S2.p1.10.m4.1.1.6.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.10.m4.1b"><apply id="S2.p1.10.m4.1.1.cmml" xref="S2.p1.10.m4.1.1"><and id="S2.p1.10.m4.1.1a.cmml" xref="S2.p1.10.m4.1.1"></and><apply id="S2.p1.10.m4.1.1b.cmml" xref="S2.p1.10.m4.1.1"><lt id="S2.p1.10.m4.1.1.3.cmml" xref="S2.p1.10.m4.1.1.3"></lt><ci id="S2.p1.10.m4.1.1.2.cmml" xref="S2.p1.10.m4.1.1.2">π</ci><ci id="S2.p1.10.m4.1.1.4.cmml" xref="S2.p1.10.m4.1.1.4">πΌ</ci></apply><apply id="S2.p1.10.m4.1.1c.cmml" xref="S2.p1.10.m4.1.1"><leq id="S2.p1.10.m4.1.1.5.cmml" xref="S2.p1.10.m4.1.1.5"></leq><share href="https://arxiv.org/html/2503.13666v1#S2.p1.10.m4.1.1.4.cmml" id="S2.p1.10.m4.1.1d.cmml" xref="S2.p1.10.m4.1.1"></share><apply id="S2.p1.10.m4.1.1.6.cmml" xref="S2.p1.10.m4.1.1.6"><csymbol cd="ambiguous" id="S2.p1.10.m4.1.1.6.1.cmml" xref="S2.p1.10.m4.1.1.6">subscript</csymbol><ci id="S2.p1.10.m4.1.1.6.2.cmml" xref="S2.p1.10.m4.1.1.6.2">π</ci><cn id="S2.p1.10.m4.1.1.6.3.cmml" type="integer" xref="S2.p1.10.m4.1.1.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.10.m4.1c">\xi<\alpha\leq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.10.m4.1d">italic_ΞΎ < italic_Ξ± β€ italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> the set <math alttext="\operatorname{cls}^{\xi}(A)" class="ltx_Math" display="inline" id="S2.p1.11.m5.2"><semantics id="S2.p1.11.m5.2a"><mrow id="S2.p1.11.m5.2.2.1" xref="S2.p1.11.m5.2.2.2.cmml"><msup id="S2.p1.11.m5.2.2.1.1" xref="S2.p1.11.m5.2.2.1.1.cmml"><mi id="S2.p1.11.m5.2.2.1.1.2" xref="S2.p1.11.m5.2.2.1.1.2.cmml">cls</mi><mi id="S2.p1.11.m5.2.2.1.1.3" xref="S2.p1.11.m5.2.2.1.1.3.cmml">ΞΎ</mi></msup><mo id="S2.p1.11.m5.2.2.1a" xref="S2.p1.11.m5.2.2.2.cmml">β‘</mo><mrow id="S2.p1.11.m5.2.2.1.2" xref="S2.p1.11.m5.2.2.2.cmml"><mo id="S2.p1.11.m5.2.2.1.2.1" stretchy="false" xref="S2.p1.11.m5.2.2.2.cmml">(</mo><mi id="S2.p1.11.m5.1.1" xref="S2.p1.11.m5.1.1.cmml">A</mi><mo id="S2.p1.11.m5.2.2.1.2.2" stretchy="false" xref="S2.p1.11.m5.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.11.m5.2b"><apply id="S2.p1.11.m5.2.2.2.cmml" xref="S2.p1.11.m5.2.2.1"><apply id="S2.p1.11.m5.2.2.1.1.cmml" xref="S2.p1.11.m5.2.2.1.1"><csymbol cd="ambiguous" id="S2.p1.11.m5.2.2.1.1.1.cmml" xref="S2.p1.11.m5.2.2.1.1">superscript</csymbol><ci id="S2.p1.11.m5.2.2.1.1.2.cmml" xref="S2.p1.11.m5.2.2.1.1.2">cls</ci><ci id="S2.p1.11.m5.2.2.1.1.3.cmml" xref="S2.p1.11.m5.2.2.1.1.3">π</ci></apply><ci id="S2.p1.11.m5.1.1.cmml" xref="S2.p1.11.m5.1.1">π΄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.11.m5.2c">\operatorname{cls}^{\xi}(A)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.11.m5.2d">roman_cls start_POSTSUPERSCRIPT italic_ΞΎ end_POSTSUPERSCRIPT ( italic_A )</annotation></semantics></math> is defined. Then <math alttext="\operatorname{cls}^{\alpha}(A)=\bigcup_{\xi\in\alpha}\operatorname{cls}^{\xi}(A)" class="ltx_Math" display="inline" id="S2.p1.12.m6.4"><semantics id="S2.p1.12.m6.4a"><mrow id="S2.p1.12.m6.4.4" xref="S2.p1.12.m6.4.4.cmml"><mrow id="S2.p1.12.m6.3.3.1.1" xref="S2.p1.12.m6.3.3.1.2.cmml"><msup id="S2.p1.12.m6.3.3.1.1.1" xref="S2.p1.12.m6.3.3.1.1.1.cmml"><mi id="S2.p1.12.m6.3.3.1.1.1.2" xref="S2.p1.12.m6.3.3.1.1.1.2.cmml">cls</mi><mi id="S2.p1.12.m6.3.3.1.1.1.3" xref="S2.p1.12.m6.3.3.1.1.1.3.cmml">Ξ±</mi></msup><mo id="S2.p1.12.m6.3.3.1.1a" xref="S2.p1.12.m6.3.3.1.2.cmml">β‘</mo><mrow id="S2.p1.12.m6.3.3.1.1.2" xref="S2.p1.12.m6.3.3.1.2.cmml"><mo id="S2.p1.12.m6.3.3.1.1.2.1" stretchy="false" xref="S2.p1.12.m6.3.3.1.2.cmml">(</mo><mi id="S2.p1.12.m6.1.1" xref="S2.p1.12.m6.1.1.cmml">A</mi><mo id="S2.p1.12.m6.3.3.1.1.2.2" stretchy="false" xref="S2.p1.12.m6.3.3.1.2.cmml">)</mo></mrow></mrow><mo id="S2.p1.12.m6.4.4.3" rspace="0.111em" xref="S2.p1.12.m6.4.4.3.cmml">=</mo><mrow id="S2.p1.12.m6.4.4.2" xref="S2.p1.12.m6.4.4.2.cmml"><msub id="S2.p1.12.m6.4.4.2.2" xref="S2.p1.12.m6.4.4.2.2.cmml"><mo id="S2.p1.12.m6.4.4.2.2.2" xref="S2.p1.12.m6.4.4.2.2.2.cmml">β</mo><mrow id="S2.p1.12.m6.4.4.2.2.3" xref="S2.p1.12.m6.4.4.2.2.3.cmml"><mi id="S2.p1.12.m6.4.4.2.2.3.2" xref="S2.p1.12.m6.4.4.2.2.3.2.cmml">ΞΎ</mi><mo id="S2.p1.12.m6.4.4.2.2.3.1" xref="S2.p1.12.m6.4.4.2.2.3.1.cmml">β</mo><mi id="S2.p1.12.m6.4.4.2.2.3.3" xref="S2.p1.12.m6.4.4.2.2.3.3.cmml">Ξ±</mi></mrow></msub><mrow id="S2.p1.12.m6.4.4.2.1.1" xref="S2.p1.12.m6.4.4.2.1.2.cmml"><msup id="S2.p1.12.m6.4.4.2.1.1.1" xref="S2.p1.12.m6.4.4.2.1.1.1.cmml"><mi id="S2.p1.12.m6.4.4.2.1.1.1.2" xref="S2.p1.12.m6.4.4.2.1.1.1.2.cmml">cls</mi><mi id="S2.p1.12.m6.4.4.2.1.1.1.3" xref="S2.p1.12.m6.4.4.2.1.1.1.3.cmml">ΞΎ</mi></msup><mo id="S2.p1.12.m6.4.4.2.1.1a" xref="S2.p1.12.m6.4.4.2.1.2.cmml">β‘</mo><mrow id="S2.p1.12.m6.4.4.2.1.1.2" xref="S2.p1.12.m6.4.4.2.1.2.cmml"><mo id="S2.p1.12.m6.4.4.2.1.1.2.1" stretchy="false" xref="S2.p1.12.m6.4.4.2.1.2.cmml">(</mo><mi id="S2.p1.12.m6.2.2" xref="S2.p1.12.m6.2.2.cmml">A</mi><mo id="S2.p1.12.m6.4.4.2.1.1.2.2" stretchy="false" xref="S2.p1.12.m6.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.12.m6.4b"><apply id="S2.p1.12.m6.4.4.cmml" xref="S2.p1.12.m6.4.4"><eq id="S2.p1.12.m6.4.4.3.cmml" xref="S2.p1.12.m6.4.4.3"></eq><apply id="S2.p1.12.m6.3.3.1.2.cmml" xref="S2.p1.12.m6.3.3.1.1"><apply id="S2.p1.12.m6.3.3.1.1.1.cmml" xref="S2.p1.12.m6.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.p1.12.m6.3.3.1.1.1.1.cmml" xref="S2.p1.12.m6.3.3.1.1.1">superscript</csymbol><ci id="S2.p1.12.m6.3.3.1.1.1.2.cmml" xref="S2.p1.12.m6.3.3.1.1.1.2">cls</ci><ci id="S2.p1.12.m6.3.3.1.1.1.3.cmml" xref="S2.p1.12.m6.3.3.1.1.1.3">πΌ</ci></apply><ci id="S2.p1.12.m6.1.1.cmml" xref="S2.p1.12.m6.1.1">π΄</ci></apply><apply id="S2.p1.12.m6.4.4.2.cmml" xref="S2.p1.12.m6.4.4.2"><apply id="S2.p1.12.m6.4.4.2.2.cmml" xref="S2.p1.12.m6.4.4.2.2"><csymbol cd="ambiguous" id="S2.p1.12.m6.4.4.2.2.1.cmml" xref="S2.p1.12.m6.4.4.2.2">subscript</csymbol><union id="S2.p1.12.m6.4.4.2.2.2.cmml" xref="S2.p1.12.m6.4.4.2.2.2"></union><apply id="S2.p1.12.m6.4.4.2.2.3.cmml" xref="S2.p1.12.m6.4.4.2.2.3"><in id="S2.p1.12.m6.4.4.2.2.3.1.cmml" xref="S2.p1.12.m6.4.4.2.2.3.1"></in><ci id="S2.p1.12.m6.4.4.2.2.3.2.cmml" xref="S2.p1.12.m6.4.4.2.2.3.2">π</ci><ci id="S2.p1.12.m6.4.4.2.2.3.3.cmml" xref="S2.p1.12.m6.4.4.2.2.3.3">πΌ</ci></apply></apply><apply id="S2.p1.12.m6.4.4.2.1.2.cmml" xref="S2.p1.12.m6.4.4.2.1.1"><apply id="S2.p1.12.m6.4.4.2.1.1.1.cmml" xref="S2.p1.12.m6.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S2.p1.12.m6.4.4.2.1.1.1.1.cmml" xref="S2.p1.12.m6.4.4.2.1.1.1">superscript</csymbol><ci id="S2.p1.12.m6.4.4.2.1.1.1.2.cmml" xref="S2.p1.12.m6.4.4.2.1.1.1.2">cls</ci><ci id="S2.p1.12.m6.4.4.2.1.1.1.3.cmml" xref="S2.p1.12.m6.4.4.2.1.1.1.3">π</ci></apply><ci id="S2.p1.12.m6.2.2.cmml" xref="S2.p1.12.m6.2.2">π΄</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.12.m6.4c">\operatorname{cls}^{\alpha}(A)=\bigcup_{\xi\in\alpha}\operatorname{cls}^{\xi}(A)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.12.m6.4d">roman_cls start_POSTSUPERSCRIPT italic_Ξ± end_POSTSUPERSCRIPT ( italic_A ) = β start_POSTSUBSCRIPT italic_ΞΎ β italic_Ξ± end_POSTSUBSCRIPT roman_cls start_POSTSUPERSCRIPT italic_ΞΎ end_POSTSUPERSCRIPT ( italic_A )</annotation></semantics></math> if <math alttext="\alpha" class="ltx_Math" display="inline" id="S2.p1.13.m7.1"><semantics id="S2.p1.13.m7.1a"><mi id="S2.p1.13.m7.1.1" xref="S2.p1.13.m7.1.1.cmml">Ξ±</mi><annotation-xml encoding="MathML-Content" id="S2.p1.13.m7.1b"><ci id="S2.p1.13.m7.1.1.cmml" xref="S2.p1.13.m7.1.1">πΌ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.13.m7.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.p1.13.m7.1d">italic_Ξ±</annotation></semantics></math> is limit and <math alttext="\operatorname{cls}^{\alpha}(A)=s(\operatorname{cls}^{\gamma}(A))" class="ltx_Math" display="inline" id="S2.p1.14.m8.4"><semantics id="S2.p1.14.m8.4a"><mrow id="S2.p1.14.m8.4.4" xref="S2.p1.14.m8.4.4.cmml"><mrow id="S2.p1.14.m8.3.3.1.1" xref="S2.p1.14.m8.3.3.1.2.cmml"><msup id="S2.p1.14.m8.3.3.1.1.1" xref="S2.p1.14.m8.3.3.1.1.1.cmml"><mi id="S2.p1.14.m8.3.3.1.1.1.2" xref="S2.p1.14.m8.3.3.1.1.1.2.cmml">cls</mi><mi id="S2.p1.14.m8.3.3.1.1.1.3" xref="S2.p1.14.m8.3.3.1.1.1.3.cmml">Ξ±</mi></msup><mo id="S2.p1.14.m8.3.3.1.1a" xref="S2.p1.14.m8.3.3.1.2.cmml">β‘</mo><mrow id="S2.p1.14.m8.3.3.1.1.2" xref="S2.p1.14.m8.3.3.1.2.cmml"><mo id="S2.p1.14.m8.3.3.1.1.2.1" stretchy="false" xref="S2.p1.14.m8.3.3.1.2.cmml">(</mo><mi id="S2.p1.14.m8.1.1" xref="S2.p1.14.m8.1.1.cmml">A</mi><mo id="S2.p1.14.m8.3.3.1.1.2.2" stretchy="false" xref="S2.p1.14.m8.3.3.1.2.cmml">)</mo></mrow></mrow><mo id="S2.p1.14.m8.4.4.3" xref="S2.p1.14.m8.4.4.3.cmml">=</mo><mrow id="S2.p1.14.m8.4.4.2" xref="S2.p1.14.m8.4.4.2.cmml"><mi id="S2.p1.14.m8.4.4.2.3" xref="S2.p1.14.m8.4.4.2.3.cmml">s</mi><mo id="S2.p1.14.m8.4.4.2.2" xref="S2.p1.14.m8.4.4.2.2.cmml">β’</mo><mrow id="S2.p1.14.m8.4.4.2.1.1" xref="S2.p1.14.m8.4.4.2.cmml"><mo id="S2.p1.14.m8.4.4.2.1.1.2" stretchy="false" xref="S2.p1.14.m8.4.4.2.cmml">(</mo><mrow id="S2.p1.14.m8.4.4.2.1.1.1.1" xref="S2.p1.14.m8.4.4.2.1.1.1.2.cmml"><msup id="S2.p1.14.m8.4.4.2.1.1.1.1.1" xref="S2.p1.14.m8.4.4.2.1.1.1.1.1.cmml"><mi id="S2.p1.14.m8.4.4.2.1.1.1.1.1.2" xref="S2.p1.14.m8.4.4.2.1.1.1.1.1.2.cmml">cls</mi><mi id="S2.p1.14.m8.4.4.2.1.1.1.1.1.3" xref="S2.p1.14.m8.4.4.2.1.1.1.1.1.3.cmml">Ξ³</mi></msup><mo id="S2.p1.14.m8.4.4.2.1.1.1.1a" xref="S2.p1.14.m8.4.4.2.1.1.1.2.cmml">β‘</mo><mrow id="S2.p1.14.m8.4.4.2.1.1.1.1.2" xref="S2.p1.14.m8.4.4.2.1.1.1.2.cmml"><mo id="S2.p1.14.m8.4.4.2.1.1.1.1.2.1" stretchy="false" xref="S2.p1.14.m8.4.4.2.1.1.1.2.cmml">(</mo><mi id="S2.p1.14.m8.2.2" xref="S2.p1.14.m8.2.2.cmml">A</mi><mo id="S2.p1.14.m8.4.4.2.1.1.1.1.2.2" stretchy="false" xref="S2.p1.14.m8.4.4.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.p1.14.m8.4.4.2.1.1.3" stretchy="false" xref="S2.p1.14.m8.4.4.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.14.m8.4b"><apply id="S2.p1.14.m8.4.4.cmml" xref="S2.p1.14.m8.4.4"><eq id="S2.p1.14.m8.4.4.3.cmml" xref="S2.p1.14.m8.4.4.3"></eq><apply id="S2.p1.14.m8.3.3.1.2.cmml" xref="S2.p1.14.m8.3.3.1.1"><apply id="S2.p1.14.m8.3.3.1.1.1.cmml" xref="S2.p1.14.m8.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.p1.14.m8.3.3.1.1.1.1.cmml" xref="S2.p1.14.m8.3.3.1.1.1">superscript</csymbol><ci id="S2.p1.14.m8.3.3.1.1.1.2.cmml" xref="S2.p1.14.m8.3.3.1.1.1.2">cls</ci><ci id="S2.p1.14.m8.3.3.1.1.1.3.cmml" xref="S2.p1.14.m8.3.3.1.1.1.3">πΌ</ci></apply><ci id="S2.p1.14.m8.1.1.cmml" xref="S2.p1.14.m8.1.1">π΄</ci></apply><apply id="S2.p1.14.m8.4.4.2.cmml" xref="S2.p1.14.m8.4.4.2"><times id="S2.p1.14.m8.4.4.2.2.cmml" xref="S2.p1.14.m8.4.4.2.2"></times><ci id="S2.p1.14.m8.4.4.2.3.cmml" xref="S2.p1.14.m8.4.4.2.3">π </ci><apply id="S2.p1.14.m8.4.4.2.1.1.1.2.cmml" xref="S2.p1.14.m8.4.4.2.1.1.1.1"><apply id="S2.p1.14.m8.4.4.2.1.1.1.1.1.cmml" xref="S2.p1.14.m8.4.4.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.p1.14.m8.4.4.2.1.1.1.1.1.1.cmml" xref="S2.p1.14.m8.4.4.2.1.1.1.1.1">superscript</csymbol><ci id="S2.p1.14.m8.4.4.2.1.1.1.1.1.2.cmml" xref="S2.p1.14.m8.4.4.2.1.1.1.1.1.2">cls</ci><ci id="S2.p1.14.m8.4.4.2.1.1.1.1.1.3.cmml" xref="S2.p1.14.m8.4.4.2.1.1.1.1.1.3">πΎ</ci></apply><ci id="S2.p1.14.m8.2.2.cmml" xref="S2.p1.14.m8.2.2">π΄</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.14.m8.4c">\operatorname{cls}^{\alpha}(A)=s(\operatorname{cls}^{\gamma}(A))</annotation><annotation encoding="application/x-llamapun" id="S2.p1.14.m8.4d">roman_cls start_POSTSUPERSCRIPT italic_Ξ± end_POSTSUPERSCRIPT ( italic_A ) = italic_s ( roman_cls start_POSTSUPERSCRIPT italic_Ξ³ end_POSTSUPERSCRIPT ( italic_A ) )</annotation></semantics></math> if <math alttext="\alpha=\gamma+1" class="ltx_Math" display="inline" id="S2.p1.15.m9.1"><semantics id="S2.p1.15.m9.1a"><mrow id="S2.p1.15.m9.1.1" xref="S2.p1.15.m9.1.1.cmml"><mi id="S2.p1.15.m9.1.1.2" xref="S2.p1.15.m9.1.1.2.cmml">Ξ±</mi><mo id="S2.p1.15.m9.1.1.1" xref="S2.p1.15.m9.1.1.1.cmml">=</mo><mrow id="S2.p1.15.m9.1.1.3" xref="S2.p1.15.m9.1.1.3.cmml"><mi id="S2.p1.15.m9.1.1.3.2" xref="S2.p1.15.m9.1.1.3.2.cmml">Ξ³</mi><mo id="S2.p1.15.m9.1.1.3.1" xref="S2.p1.15.m9.1.1.3.1.cmml">+</mo><mn id="S2.p1.15.m9.1.1.3.3" xref="S2.p1.15.m9.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.15.m9.1b"><apply id="S2.p1.15.m9.1.1.cmml" xref="S2.p1.15.m9.1.1"><eq id="S2.p1.15.m9.1.1.1.cmml" xref="S2.p1.15.m9.1.1.1"></eq><ci id="S2.p1.15.m9.1.1.2.cmml" xref="S2.p1.15.m9.1.1.2">πΌ</ci><apply id="S2.p1.15.m9.1.1.3.cmml" xref="S2.p1.15.m9.1.1.3"><plus id="S2.p1.15.m9.1.1.3.1.cmml" xref="S2.p1.15.m9.1.1.3.1"></plus><ci id="S2.p1.15.m9.1.1.3.2.cmml" xref="S2.p1.15.m9.1.1.3.2">πΎ</ci><cn id="S2.p1.15.m9.1.1.3.3.cmml" type="integer" xref="S2.p1.15.m9.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.15.m9.1c">\alpha=\gamma+1</annotation><annotation encoding="application/x-llamapun" id="S2.p1.15.m9.1d">italic_Ξ± = italic_Ξ³ + 1</annotation></semantics></math>. Finally, put <math alttext="\operatorname{cls}(A)=\operatorname{cls}^{\omega_{1}}(A)" class="ltx_Math" display="inline" id="S2.p1.16.m10.4"><semantics id="S2.p1.16.m10.4a"><mrow id="S2.p1.16.m10.4.4" xref="S2.p1.16.m10.4.4.cmml"><mrow id="S2.p1.16.m10.4.4.3.2" xref="S2.p1.16.m10.4.4.3.1.cmml"><mi id="S2.p1.16.m10.1.1" xref="S2.p1.16.m10.1.1.cmml">cls</mi><mo id="S2.p1.16.m10.4.4.3.2a" xref="S2.p1.16.m10.4.4.3.1.cmml">β‘</mo><mrow id="S2.p1.16.m10.4.4.3.2.1" xref="S2.p1.16.m10.4.4.3.1.cmml"><mo id="S2.p1.16.m10.4.4.3.2.1.1" stretchy="false" xref="S2.p1.16.m10.4.4.3.1.cmml">(</mo><mi id="S2.p1.16.m10.2.2" xref="S2.p1.16.m10.2.2.cmml">A</mi><mo id="S2.p1.16.m10.4.4.3.2.1.2" stretchy="false" xref="S2.p1.16.m10.4.4.3.1.cmml">)</mo></mrow></mrow><mo id="S2.p1.16.m10.4.4.2" xref="S2.p1.16.m10.4.4.2.cmml">=</mo><mrow id="S2.p1.16.m10.4.4.1.1" xref="S2.p1.16.m10.4.4.1.2.cmml"><msup id="S2.p1.16.m10.4.4.1.1.1" xref="S2.p1.16.m10.4.4.1.1.1.cmml"><mi id="S2.p1.16.m10.4.4.1.1.1.2" xref="S2.p1.16.m10.4.4.1.1.1.2.cmml">cls</mi><msub id="S2.p1.16.m10.4.4.1.1.1.3" xref="S2.p1.16.m10.4.4.1.1.1.3.cmml"><mi id="S2.p1.16.m10.4.4.1.1.1.3.2" xref="S2.p1.16.m10.4.4.1.1.1.3.2.cmml">Ο</mi><mn id="S2.p1.16.m10.4.4.1.1.1.3.3" xref="S2.p1.16.m10.4.4.1.1.1.3.3.cmml">1</mn></msub></msup><mo id="S2.p1.16.m10.4.4.1.1a" xref="S2.p1.16.m10.4.4.1.2.cmml">β‘</mo><mrow id="S2.p1.16.m10.4.4.1.1.2" xref="S2.p1.16.m10.4.4.1.2.cmml"><mo id="S2.p1.16.m10.4.4.1.1.2.1" stretchy="false" xref="S2.p1.16.m10.4.4.1.2.cmml">(</mo><mi id="S2.p1.16.m10.3.3" xref="S2.p1.16.m10.3.3.cmml">A</mi><mo id="S2.p1.16.m10.4.4.1.1.2.2" stretchy="false" xref="S2.p1.16.m10.4.4.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.16.m10.4b"><apply id="S2.p1.16.m10.4.4.cmml" xref="S2.p1.16.m10.4.4"><eq id="S2.p1.16.m10.4.4.2.cmml" xref="S2.p1.16.m10.4.4.2"></eq><apply id="S2.p1.16.m10.4.4.3.1.cmml" xref="S2.p1.16.m10.4.4.3.2"><ci id="S2.p1.16.m10.1.1.cmml" xref="S2.p1.16.m10.1.1">cls</ci><ci id="S2.p1.16.m10.2.2.cmml" xref="S2.p1.16.m10.2.2">π΄</ci></apply><apply id="S2.p1.16.m10.4.4.1.2.cmml" xref="S2.p1.16.m10.4.4.1.1"><apply id="S2.p1.16.m10.4.4.1.1.1.cmml" xref="S2.p1.16.m10.4.4.1.1.1"><csymbol cd="ambiguous" id="S2.p1.16.m10.4.4.1.1.1.1.cmml" xref="S2.p1.16.m10.4.4.1.1.1">superscript</csymbol><ci id="S2.p1.16.m10.4.4.1.1.1.2.cmml" xref="S2.p1.16.m10.4.4.1.1.1.2">cls</ci><apply id="S2.p1.16.m10.4.4.1.1.1.3.cmml" xref="S2.p1.16.m10.4.4.1.1.1.3"><csymbol cd="ambiguous" id="S2.p1.16.m10.4.4.1.1.1.3.1.cmml" xref="S2.p1.16.m10.4.4.1.1.1.3">subscript</csymbol><ci id="S2.p1.16.m10.4.4.1.1.1.3.2.cmml" xref="S2.p1.16.m10.4.4.1.1.1.3.2">π</ci><cn id="S2.p1.16.m10.4.4.1.1.1.3.3.cmml" type="integer" xref="S2.p1.16.m10.4.4.1.1.1.3.3">1</cn></apply></apply><ci id="S2.p1.16.m10.3.3.cmml" xref="S2.p1.16.m10.3.3">π΄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.16.m10.4c">\operatorname{cls}(A)=\operatorname{cls}^{\omega_{1}}(A)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.16.m10.4d">roman_cls ( italic_A ) = roman_cls start_POSTSUPERSCRIPT italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( italic_A )</annotation></semantics></math>. Recall that a space <math alttext="X" class="ltx_Math" display="inline" id="S2.p1.17.m11.1"><semantics id="S2.p1.17.m11.1a"><mi id="S2.p1.17.m11.1.1" xref="S2.p1.17.m11.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.p1.17.m11.1b"><ci id="S2.p1.17.m11.1.1.cmml" xref="S2.p1.17.m11.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.17.m11.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.p1.17.m11.1d">italic_X</annotation></semantics></math> is sequential if and only if <math alttext="\overline{A}=\operatorname{cls}(A)" class="ltx_Math" display="inline" id="S2.p1.18.m12.2"><semantics id="S2.p1.18.m12.2a"><mrow id="S2.p1.18.m12.2.3" xref="S2.p1.18.m12.2.3.cmml"><mover accent="true" id="S2.p1.18.m12.2.3.2" xref="S2.p1.18.m12.2.3.2.cmml"><mi id="S2.p1.18.m12.2.3.2.2" xref="S2.p1.18.m12.2.3.2.2.cmml">A</mi><mo id="S2.p1.18.m12.2.3.2.1" xref="S2.p1.18.m12.2.3.2.1.cmml">Β―</mo></mover><mo id="S2.p1.18.m12.2.3.1" xref="S2.p1.18.m12.2.3.1.cmml">=</mo><mrow id="S2.p1.18.m12.2.3.3.2" xref="S2.p1.18.m12.2.3.3.1.cmml"><mi id="S2.p1.18.m12.1.1" xref="S2.p1.18.m12.1.1.cmml">cls</mi><mo id="S2.p1.18.m12.2.3.3.2a" xref="S2.p1.18.m12.2.3.3.1.cmml">β‘</mo><mrow id="S2.p1.18.m12.2.3.3.2.1" xref="S2.p1.18.m12.2.3.3.1.cmml"><mo id="S2.p1.18.m12.2.3.3.2.1.1" stretchy="false" xref="S2.p1.18.m12.2.3.3.1.cmml">(</mo><mi id="S2.p1.18.m12.2.2" xref="S2.p1.18.m12.2.2.cmml">A</mi><mo id="S2.p1.18.m12.2.3.3.2.1.2" stretchy="false" xref="S2.p1.18.m12.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.18.m12.2b"><apply id="S2.p1.18.m12.2.3.cmml" xref="S2.p1.18.m12.2.3"><eq id="S2.p1.18.m12.2.3.1.cmml" xref="S2.p1.18.m12.2.3.1"></eq><apply id="S2.p1.18.m12.2.3.2.cmml" xref="S2.p1.18.m12.2.3.2"><ci id="S2.p1.18.m12.2.3.2.1.cmml" xref="S2.p1.18.m12.2.3.2.1">Β―</ci><ci id="S2.p1.18.m12.2.3.2.2.cmml" xref="S2.p1.18.m12.2.3.2.2">π΄</ci></apply><apply id="S2.p1.18.m12.2.3.3.1.cmml" xref="S2.p1.18.m12.2.3.3.2"><ci id="S2.p1.18.m12.1.1.cmml" xref="S2.p1.18.m12.1.1">cls</ci><ci id="S2.p1.18.m12.2.2.cmml" xref="S2.p1.18.m12.2.2">π΄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.18.m12.2c">\overline{A}=\operatorname{cls}(A)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.18.m12.2d">overΒ― start_ARG italic_A end_ARG = roman_cls ( italic_A )</annotation></semantics></math> for every <math alttext="A\subseteq X" class="ltx_Math" display="inline" id="S2.p1.19.m13.1"><semantics id="S2.p1.19.m13.1a"><mrow id="S2.p1.19.m13.1.1" xref="S2.p1.19.m13.1.1.cmml"><mi id="S2.p1.19.m13.1.1.2" xref="S2.p1.19.m13.1.1.2.cmml">A</mi><mo id="S2.p1.19.m13.1.1.1" xref="S2.p1.19.m13.1.1.1.cmml">β</mo><mi id="S2.p1.19.m13.1.1.3" xref="S2.p1.19.m13.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.19.m13.1b"><apply id="S2.p1.19.m13.1.1.cmml" xref="S2.p1.19.m13.1.1"><subset id="S2.p1.19.m13.1.1.1.cmml" xref="S2.p1.19.m13.1.1.1"></subset><ci id="S2.p1.19.m13.1.1.2.cmml" xref="S2.p1.19.m13.1.1.2">π΄</ci><ci id="S2.p1.19.m13.1.1.3.cmml" xref="S2.p1.19.m13.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.19.m13.1c">A\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S2.p1.19.m13.1d">italic_A β italic_X</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S2.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem1.1.1.1">Lemma 2.1</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem1.p1"> <p class="ltx_p" id="S2.Thmtheorem1.p1.5"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem1.p1.5.5">Let <math alttext="A" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.1.1.m1.1"><semantics id="S2.Thmtheorem1.p1.1.1.m1.1a"><mi id="S2.Thmtheorem1.p1.1.1.m1.1.1" xref="S2.Thmtheorem1.p1.1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.1.1.m1.1b"><ci id="S2.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem1.p1.1.1.m1.1.1">π΄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.1.1.m1.1d">italic_A</annotation></semantics></math> be a subsemigroup of a topological semigroup <math alttext="S" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.2.2.m2.1"><semantics id="S2.Thmtheorem1.p1.2.2.m2.1a"><mi id="S2.Thmtheorem1.p1.2.2.m2.1.1" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.2.2.m2.1b"><ci id="S2.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.2.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.2.2.m2.1d">italic_S</annotation></semantics></math>. Then <math alttext="\operatorname{cls}^{\alpha}(A)" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.3.3.m3.2"><semantics id="S2.Thmtheorem1.p1.3.3.m3.2a"><mrow id="S2.Thmtheorem1.p1.3.3.m3.2.2.1" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.2.cmml"><msup id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.cmml"><mi id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.2" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.2.cmml">cls</mi><mi id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.3" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.3.cmml">Ξ±</mi></msup><mo id="S2.Thmtheorem1.p1.3.3.m3.2.2.1a" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.2.cmml">β‘</mo><mrow id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.2" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.2.cmml"><mo id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.2.1" stretchy="false" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.2.cmml">(</mo><mi id="S2.Thmtheorem1.p1.3.3.m3.1.1" xref="S2.Thmtheorem1.p1.3.3.m3.1.1.cmml">A</mi><mo id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.2.2" stretchy="false" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.3.3.m3.2b"><apply id="S2.Thmtheorem1.p1.3.3.m3.2.2.2.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1"><apply id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1">superscript</csymbol><ci id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.2.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.2">cls</ci><ci id="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.3.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.2.2.1.1.3">πΌ</ci></apply><ci id="S2.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.1.1">π΄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.3.3.m3.2c">\operatorname{cls}^{\alpha}(A)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.3.3.m3.2d">roman_cls start_POSTSUPERSCRIPT italic_Ξ± end_POSTSUPERSCRIPT ( italic_A )</annotation></semantics></math> is a subsemigroup of <math alttext="S" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.4.4.m4.1"><semantics id="S2.Thmtheorem1.p1.4.4.m4.1a"><mi id="S2.Thmtheorem1.p1.4.4.m4.1.1" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.4.4.m4.1b"><ci id="S2.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.4.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.4.4.m4.1d">italic_S</annotation></semantics></math> for each <math alttext="\alpha\leq\omega_{1}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.5.5.m5.1"><semantics id="S2.Thmtheorem1.p1.5.5.m5.1a"><mrow id="S2.Thmtheorem1.p1.5.5.m5.1.1" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.cmml"><mi id="S2.Thmtheorem1.p1.5.5.m5.1.1.2" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.2.cmml">Ξ±</mi><mo id="S2.Thmtheorem1.p1.5.5.m5.1.1.1" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.1.cmml">β€</mo><msub id="S2.Thmtheorem1.p1.5.5.m5.1.1.3" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.3.cmml"><mi id="S2.Thmtheorem1.p1.5.5.m5.1.1.3.2" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.3.2.cmml">Ο</mi><mn id="S2.Thmtheorem1.p1.5.5.m5.1.1.3.3" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.5.5.m5.1b"><apply id="S2.Thmtheorem1.p1.5.5.m5.1.1.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.1.1"><leq id="S2.Thmtheorem1.p1.5.5.m5.1.1.1.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.1"></leq><ci id="S2.Thmtheorem1.p1.5.5.m5.1.1.2.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.2">πΌ</ci><apply id="S2.Thmtheorem1.p1.5.5.m5.1.1.3.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.5.5.m5.1.1.3.1.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem1.p1.5.5.m5.1.1.3.2.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.3.2">π</ci><cn id="S2.Thmtheorem1.p1.5.5.m5.1.1.3.3.cmml" type="integer" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.5.5.m5.1c">\alpha\leq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.5.5.m5.1d">italic_Ξ± β€ italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S2.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.1.p1"> <p class="ltx_p" id="S2.1.p1.27">Note that <math alttext="\operatorname{cls}^{0}(A)=A" class="ltx_Math" display="inline" id="S2.1.p1.1.m1.2"><semantics id="S2.1.p1.1.m1.2a"><mrow id="S2.1.p1.1.m1.2.2" xref="S2.1.p1.1.m1.2.2.cmml"><mrow id="S2.1.p1.1.m1.2.2.1.1" xref="S2.1.p1.1.m1.2.2.1.2.cmml"><msup id="S2.1.p1.1.m1.2.2.1.1.1" xref="S2.1.p1.1.m1.2.2.1.1.1.cmml"><mi id="S2.1.p1.1.m1.2.2.1.1.1.2" xref="S2.1.p1.1.m1.2.2.1.1.1.2.cmml">cls</mi><mn id="S2.1.p1.1.m1.2.2.1.1.1.3" xref="S2.1.p1.1.m1.2.2.1.1.1.3.cmml">0</mn></msup><mo id="S2.1.p1.1.m1.2.2.1.1a" xref="S2.1.p1.1.m1.2.2.1.2.cmml">β‘</mo><mrow id="S2.1.p1.1.m1.2.2.1.1.2" xref="S2.1.p1.1.m1.2.2.1.2.cmml"><mo id="S2.1.p1.1.m1.2.2.1.1.2.1" stretchy="false" xref="S2.1.p1.1.m1.2.2.1.2.cmml">(</mo><mi id="S2.1.p1.1.m1.1.1" xref="S2.1.p1.1.m1.1.1.cmml">A</mi><mo id="S2.1.p1.1.m1.2.2.1.1.2.2" stretchy="false" xref="S2.1.p1.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S2.1.p1.1.m1.2.2.2" xref="S2.1.p1.1.m1.2.2.2.cmml">=</mo><mi id="S2.1.p1.1.m1.2.2.3" xref="S2.1.p1.1.m1.2.2.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.1.m1.2b"><apply id="S2.1.p1.1.m1.2.2.cmml" xref="S2.1.p1.1.m1.2.2"><eq id="S2.1.p1.1.m1.2.2.2.cmml" xref="S2.1.p1.1.m1.2.2.2"></eq><apply id="S2.1.p1.1.m1.2.2.1.2.cmml" xref="S2.1.p1.1.m1.2.2.1.1"><apply id="S2.1.p1.1.m1.2.2.1.1.1.cmml" xref="S2.1.p1.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.1.p1.1.m1.2.2.1.1.1.1.cmml" xref="S2.1.p1.1.m1.2.2.1.1.1">superscript</csymbol><ci id="S2.1.p1.1.m1.2.2.1.1.1.2.cmml" xref="S2.1.p1.1.m1.2.2.1.1.1.2">cls</ci><cn id="S2.1.p1.1.m1.2.2.1.1.1.3.cmml" type="integer" xref="S2.1.p1.1.m1.2.2.1.1.1.3">0</cn></apply><ci id="S2.1.p1.1.m1.1.1.cmml" xref="S2.1.p1.1.m1.1.1">π΄</ci></apply><ci id="S2.1.p1.1.m1.2.2.3.cmml" xref="S2.1.p1.1.m1.2.2.3">π΄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.1.m1.2c">\operatorname{cls}^{0}(A)=A</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.1.m1.2d">roman_cls start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_A ) = italic_A</annotation></semantics></math> is a subsemigroup of <math alttext="S" class="ltx_Math" display="inline" id="S2.1.p1.2.m2.1"><semantics id="S2.1.p1.2.m2.1a"><mi id="S2.1.p1.2.m2.1.1" xref="S2.1.p1.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.1.p1.2.m2.1b"><ci id="S2.1.p1.2.m2.1.1.cmml" xref="S2.1.p1.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.2.m2.1d">italic_S</annotation></semantics></math>. Assume that for each <math alttext="\xi<\delta\leq\omega_{1}" class="ltx_Math" display="inline" id="S2.1.p1.3.m3.1"><semantics id="S2.1.p1.3.m3.1a"><mrow id="S2.1.p1.3.m3.1.1" xref="S2.1.p1.3.m3.1.1.cmml"><mi id="S2.1.p1.3.m3.1.1.2" xref="S2.1.p1.3.m3.1.1.2.cmml">ΞΎ</mi><mo id="S2.1.p1.3.m3.1.1.3" xref="S2.1.p1.3.m3.1.1.3.cmml"><</mo><mi id="S2.1.p1.3.m3.1.1.4" xref="S2.1.p1.3.m3.1.1.4.cmml">Ξ΄</mi><mo id="S2.1.p1.3.m3.1.1.5" xref="S2.1.p1.3.m3.1.1.5.cmml">β€</mo><msub id="S2.1.p1.3.m3.1.1.6" xref="S2.1.p1.3.m3.1.1.6.cmml"><mi id="S2.1.p1.3.m3.1.1.6.2" xref="S2.1.p1.3.m3.1.1.6.2.cmml">Ο</mi><mn id="S2.1.p1.3.m3.1.1.6.3" xref="S2.1.p1.3.m3.1.1.6.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.3.m3.1b"><apply id="S2.1.p1.3.m3.1.1.cmml" xref="S2.1.p1.3.m3.1.1"><and id="S2.1.p1.3.m3.1.1a.cmml" xref="S2.1.p1.3.m3.1.1"></and><apply id="S2.1.p1.3.m3.1.1b.cmml" xref="S2.1.p1.3.m3.1.1"><lt id="S2.1.p1.3.m3.1.1.3.cmml" xref="S2.1.p1.3.m3.1.1.3"></lt><ci id="S2.1.p1.3.m3.1.1.2.cmml" xref="S2.1.p1.3.m3.1.1.2">π</ci><ci id="S2.1.p1.3.m3.1.1.4.cmml" xref="S2.1.p1.3.m3.1.1.4">πΏ</ci></apply><apply id="S2.1.p1.3.m3.1.1c.cmml" xref="S2.1.p1.3.m3.1.1"><leq id="S2.1.p1.3.m3.1.1.5.cmml" xref="S2.1.p1.3.m3.1.1.5"></leq><share href="https://arxiv.org/html/2503.13666v1#S2.1.p1.3.m3.1.1.4.cmml" id="S2.1.p1.3.m3.1.1d.cmml" xref="S2.1.p1.3.m3.1.1"></share><apply id="S2.1.p1.3.m3.1.1.6.cmml" xref="S2.1.p1.3.m3.1.1.6"><csymbol cd="ambiguous" id="S2.1.p1.3.m3.1.1.6.1.cmml" xref="S2.1.p1.3.m3.1.1.6">subscript</csymbol><ci id="S2.1.p1.3.m3.1.1.6.2.cmml" xref="S2.1.p1.3.m3.1.1.6.2">π</ci><cn id="S2.1.p1.3.m3.1.1.6.3.cmml" type="integer" xref="S2.1.p1.3.m3.1.1.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.3.m3.1c">\xi<\delta\leq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.3.m3.1d">italic_ΞΎ < italic_Ξ΄ β€ italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\operatorname{cls}^{\xi}(A)" class="ltx_Math" display="inline" id="S2.1.p1.4.m4.2"><semantics id="S2.1.p1.4.m4.2a"><mrow id="S2.1.p1.4.m4.2.2.1" xref="S2.1.p1.4.m4.2.2.2.cmml"><msup id="S2.1.p1.4.m4.2.2.1.1" xref="S2.1.p1.4.m4.2.2.1.1.cmml"><mi id="S2.1.p1.4.m4.2.2.1.1.2" xref="S2.1.p1.4.m4.2.2.1.1.2.cmml">cls</mi><mi id="S2.1.p1.4.m4.2.2.1.1.3" xref="S2.1.p1.4.m4.2.2.1.1.3.cmml">ΞΎ</mi></msup><mo id="S2.1.p1.4.m4.2.2.1a" xref="S2.1.p1.4.m4.2.2.2.cmml">β‘</mo><mrow id="S2.1.p1.4.m4.2.2.1.2" xref="S2.1.p1.4.m4.2.2.2.cmml"><mo id="S2.1.p1.4.m4.2.2.1.2.1" stretchy="false" xref="S2.1.p1.4.m4.2.2.2.cmml">(</mo><mi id="S2.1.p1.4.m4.1.1" xref="S2.1.p1.4.m4.1.1.cmml">A</mi><mo id="S2.1.p1.4.m4.2.2.1.2.2" stretchy="false" xref="S2.1.p1.4.m4.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.4.m4.2b"><apply id="S2.1.p1.4.m4.2.2.2.cmml" xref="S2.1.p1.4.m4.2.2.1"><apply id="S2.1.p1.4.m4.2.2.1.1.cmml" xref="S2.1.p1.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S2.1.p1.4.m4.2.2.1.1.1.cmml" xref="S2.1.p1.4.m4.2.2.1.1">superscript</csymbol><ci id="S2.1.p1.4.m4.2.2.1.1.2.cmml" xref="S2.1.p1.4.m4.2.2.1.1.2">cls</ci><ci id="S2.1.p1.4.m4.2.2.1.1.3.cmml" xref="S2.1.p1.4.m4.2.2.1.1.3">π</ci></apply><ci id="S2.1.p1.4.m4.1.1.cmml" xref="S2.1.p1.4.m4.1.1">π΄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.4.m4.2c">\operatorname{cls}^{\xi}(A)</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.4.m4.2d">roman_cls start_POSTSUPERSCRIPT italic_ΞΎ end_POSTSUPERSCRIPT ( italic_A )</annotation></semantics></math> is a semigroup. If <math alttext="\delta" class="ltx_Math" display="inline" id="S2.1.p1.5.m5.1"><semantics id="S2.1.p1.5.m5.1a"><mi id="S2.1.p1.5.m5.1.1" xref="S2.1.p1.5.m5.1.1.cmml">Ξ΄</mi><annotation-xml encoding="MathML-Content" id="S2.1.p1.5.m5.1b"><ci id="S2.1.p1.5.m5.1.1.cmml" xref="S2.1.p1.5.m5.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.5.m5.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.5.m5.1d">italic_Ξ΄</annotation></semantics></math> is limit, then <math alttext="\operatorname{cls}^{\delta}(A)=\bigcup_{\xi<\delta}\operatorname{cls}^{\xi}(A)" class="ltx_Math" display="inline" id="S2.1.p1.6.m6.4"><semantics id="S2.1.p1.6.m6.4a"><mrow id="S2.1.p1.6.m6.4.4" xref="S2.1.p1.6.m6.4.4.cmml"><mrow id="S2.1.p1.6.m6.3.3.1.1" xref="S2.1.p1.6.m6.3.3.1.2.cmml"><msup id="S2.1.p1.6.m6.3.3.1.1.1" xref="S2.1.p1.6.m6.3.3.1.1.1.cmml"><mi id="S2.1.p1.6.m6.3.3.1.1.1.2" xref="S2.1.p1.6.m6.3.3.1.1.1.2.cmml">cls</mi><mi id="S2.1.p1.6.m6.3.3.1.1.1.3" xref="S2.1.p1.6.m6.3.3.1.1.1.3.cmml">Ξ΄</mi></msup><mo id="S2.1.p1.6.m6.3.3.1.1a" xref="S2.1.p1.6.m6.3.3.1.2.cmml">β‘</mo><mrow id="S2.1.p1.6.m6.3.3.1.1.2" xref="S2.1.p1.6.m6.3.3.1.2.cmml"><mo id="S2.1.p1.6.m6.3.3.1.1.2.1" stretchy="false" xref="S2.1.p1.6.m6.3.3.1.2.cmml">(</mo><mi id="S2.1.p1.6.m6.1.1" xref="S2.1.p1.6.m6.1.1.cmml">A</mi><mo id="S2.1.p1.6.m6.3.3.1.1.2.2" stretchy="false" xref="S2.1.p1.6.m6.3.3.1.2.cmml">)</mo></mrow></mrow><mo id="S2.1.p1.6.m6.4.4.3" rspace="0.111em" xref="S2.1.p1.6.m6.4.4.3.cmml">=</mo><mrow id="S2.1.p1.6.m6.4.4.2" xref="S2.1.p1.6.m6.4.4.2.cmml"><msub id="S2.1.p1.6.m6.4.4.2.2" xref="S2.1.p1.6.m6.4.4.2.2.cmml"><mo id="S2.1.p1.6.m6.4.4.2.2.2" xref="S2.1.p1.6.m6.4.4.2.2.2.cmml">β</mo><mrow id="S2.1.p1.6.m6.4.4.2.2.3" xref="S2.1.p1.6.m6.4.4.2.2.3.cmml"><mi id="S2.1.p1.6.m6.4.4.2.2.3.2" xref="S2.1.p1.6.m6.4.4.2.2.3.2.cmml">ΞΎ</mi><mo id="S2.1.p1.6.m6.4.4.2.2.3.1" xref="S2.1.p1.6.m6.4.4.2.2.3.1.cmml"><</mo><mi id="S2.1.p1.6.m6.4.4.2.2.3.3" xref="S2.1.p1.6.m6.4.4.2.2.3.3.cmml">Ξ΄</mi></mrow></msub><mrow id="S2.1.p1.6.m6.4.4.2.1.1" xref="S2.1.p1.6.m6.4.4.2.1.2.cmml"><msup id="S2.1.p1.6.m6.4.4.2.1.1.1" xref="S2.1.p1.6.m6.4.4.2.1.1.1.cmml"><mi id="S2.1.p1.6.m6.4.4.2.1.1.1.2" xref="S2.1.p1.6.m6.4.4.2.1.1.1.2.cmml">cls</mi><mi id="S2.1.p1.6.m6.4.4.2.1.1.1.3" xref="S2.1.p1.6.m6.4.4.2.1.1.1.3.cmml">ΞΎ</mi></msup><mo id="S2.1.p1.6.m6.4.4.2.1.1a" xref="S2.1.p1.6.m6.4.4.2.1.2.cmml">β‘</mo><mrow id="S2.1.p1.6.m6.4.4.2.1.1.2" xref="S2.1.p1.6.m6.4.4.2.1.2.cmml"><mo id="S2.1.p1.6.m6.4.4.2.1.1.2.1" stretchy="false" xref="S2.1.p1.6.m6.4.4.2.1.2.cmml">(</mo><mi id="S2.1.p1.6.m6.2.2" xref="S2.1.p1.6.m6.2.2.cmml">A</mi><mo id="S2.1.p1.6.m6.4.4.2.1.1.2.2" stretchy="false" xref="S2.1.p1.6.m6.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.6.m6.4b"><apply id="S2.1.p1.6.m6.4.4.cmml" xref="S2.1.p1.6.m6.4.4"><eq id="S2.1.p1.6.m6.4.4.3.cmml" xref="S2.1.p1.6.m6.4.4.3"></eq><apply id="S2.1.p1.6.m6.3.3.1.2.cmml" xref="S2.1.p1.6.m6.3.3.1.1"><apply id="S2.1.p1.6.m6.3.3.1.1.1.cmml" xref="S2.1.p1.6.m6.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.1.p1.6.m6.3.3.1.1.1.1.cmml" xref="S2.1.p1.6.m6.3.3.1.1.1">superscript</csymbol><ci id="S2.1.p1.6.m6.3.3.1.1.1.2.cmml" xref="S2.1.p1.6.m6.3.3.1.1.1.2">cls</ci><ci id="S2.1.p1.6.m6.3.3.1.1.1.3.cmml" xref="S2.1.p1.6.m6.3.3.1.1.1.3">πΏ</ci></apply><ci id="S2.1.p1.6.m6.1.1.cmml" xref="S2.1.p1.6.m6.1.1">π΄</ci></apply><apply id="S2.1.p1.6.m6.4.4.2.cmml" xref="S2.1.p1.6.m6.4.4.2"><apply id="S2.1.p1.6.m6.4.4.2.2.cmml" xref="S2.1.p1.6.m6.4.4.2.2"><csymbol cd="ambiguous" id="S2.1.p1.6.m6.4.4.2.2.1.cmml" xref="S2.1.p1.6.m6.4.4.2.2">subscript</csymbol><union id="S2.1.p1.6.m6.4.4.2.2.2.cmml" xref="S2.1.p1.6.m6.4.4.2.2.2"></union><apply id="S2.1.p1.6.m6.4.4.2.2.3.cmml" xref="S2.1.p1.6.m6.4.4.2.2.3"><lt id="S2.1.p1.6.m6.4.4.2.2.3.1.cmml" xref="S2.1.p1.6.m6.4.4.2.2.3.1"></lt><ci id="S2.1.p1.6.m6.4.4.2.2.3.2.cmml" xref="S2.1.p1.6.m6.4.4.2.2.3.2">π</ci><ci id="S2.1.p1.6.m6.4.4.2.2.3.3.cmml" xref="S2.1.p1.6.m6.4.4.2.2.3.3">πΏ</ci></apply></apply><apply id="S2.1.p1.6.m6.4.4.2.1.2.cmml" xref="S2.1.p1.6.m6.4.4.2.1.1"><apply id="S2.1.p1.6.m6.4.4.2.1.1.1.cmml" xref="S2.1.p1.6.m6.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S2.1.p1.6.m6.4.4.2.1.1.1.1.cmml" xref="S2.1.p1.6.m6.4.4.2.1.1.1">superscript</csymbol><ci id="S2.1.p1.6.m6.4.4.2.1.1.1.2.cmml" xref="S2.1.p1.6.m6.4.4.2.1.1.1.2">cls</ci><ci id="S2.1.p1.6.m6.4.4.2.1.1.1.3.cmml" xref="S2.1.p1.6.m6.4.4.2.1.1.1.3">π</ci></apply><ci id="S2.1.p1.6.m6.2.2.cmml" xref="S2.1.p1.6.m6.2.2">π΄</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.6.m6.4c">\operatorname{cls}^{\delta}(A)=\bigcup_{\xi<\delta}\operatorname{cls}^{\xi}(A)</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.6.m6.4d">roman_cls start_POSTSUPERSCRIPT italic_Ξ΄ end_POSTSUPERSCRIPT ( italic_A ) = β start_POSTSUBSCRIPT italic_ΞΎ < italic_Ξ΄ end_POSTSUBSCRIPT roman_cls start_POSTSUPERSCRIPT italic_ΞΎ end_POSTSUPERSCRIPT ( italic_A )</annotation></semantics></math> is a subsemigroup of <math alttext="S" class="ltx_Math" display="inline" id="S2.1.p1.7.m7.1"><semantics id="S2.1.p1.7.m7.1a"><mi id="S2.1.p1.7.m7.1.1" xref="S2.1.p1.7.m7.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.1.p1.7.m7.1b"><ci id="S2.1.p1.7.m7.1.1.cmml" xref="S2.1.p1.7.m7.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.7.m7.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.7.m7.1d">italic_S</annotation></semantics></math>, being an ascending union of semigroups. Suppose that <math alttext="\delta=\gamma+1" class="ltx_Math" display="inline" id="S2.1.p1.8.m8.1"><semantics id="S2.1.p1.8.m8.1a"><mrow id="S2.1.p1.8.m8.1.1" xref="S2.1.p1.8.m8.1.1.cmml"><mi id="S2.1.p1.8.m8.1.1.2" xref="S2.1.p1.8.m8.1.1.2.cmml">Ξ΄</mi><mo id="S2.1.p1.8.m8.1.1.1" xref="S2.1.p1.8.m8.1.1.1.cmml">=</mo><mrow id="S2.1.p1.8.m8.1.1.3" xref="S2.1.p1.8.m8.1.1.3.cmml"><mi id="S2.1.p1.8.m8.1.1.3.2" xref="S2.1.p1.8.m8.1.1.3.2.cmml">Ξ³</mi><mo id="S2.1.p1.8.m8.1.1.3.1" xref="S2.1.p1.8.m8.1.1.3.1.cmml">+</mo><mn id="S2.1.p1.8.m8.1.1.3.3" xref="S2.1.p1.8.m8.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.8.m8.1b"><apply id="S2.1.p1.8.m8.1.1.cmml" xref="S2.1.p1.8.m8.1.1"><eq id="S2.1.p1.8.m8.1.1.1.cmml" xref="S2.1.p1.8.m8.1.1.1"></eq><ci id="S2.1.p1.8.m8.1.1.2.cmml" xref="S2.1.p1.8.m8.1.1.2">πΏ</ci><apply id="S2.1.p1.8.m8.1.1.3.cmml" xref="S2.1.p1.8.m8.1.1.3"><plus id="S2.1.p1.8.m8.1.1.3.1.cmml" xref="S2.1.p1.8.m8.1.1.3.1"></plus><ci id="S2.1.p1.8.m8.1.1.3.2.cmml" xref="S2.1.p1.8.m8.1.1.3.2">πΎ</ci><cn id="S2.1.p1.8.m8.1.1.3.3.cmml" type="integer" xref="S2.1.p1.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.8.m8.1c">\delta=\gamma+1</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.8.m8.1d">italic_Ξ΄ = italic_Ξ³ + 1</annotation></semantics></math> for some <math alttext="\gamma<\omega_{1}" class="ltx_Math" display="inline" id="S2.1.p1.9.m9.1"><semantics id="S2.1.p1.9.m9.1a"><mrow id="S2.1.p1.9.m9.1.1" xref="S2.1.p1.9.m9.1.1.cmml"><mi id="S2.1.p1.9.m9.1.1.2" xref="S2.1.p1.9.m9.1.1.2.cmml">Ξ³</mi><mo id="S2.1.p1.9.m9.1.1.1" xref="S2.1.p1.9.m9.1.1.1.cmml"><</mo><msub id="S2.1.p1.9.m9.1.1.3" xref="S2.1.p1.9.m9.1.1.3.cmml"><mi id="S2.1.p1.9.m9.1.1.3.2" xref="S2.1.p1.9.m9.1.1.3.2.cmml">Ο</mi><mn id="S2.1.p1.9.m9.1.1.3.3" xref="S2.1.p1.9.m9.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.9.m9.1b"><apply id="S2.1.p1.9.m9.1.1.cmml" xref="S2.1.p1.9.m9.1.1"><lt id="S2.1.p1.9.m9.1.1.1.cmml" xref="S2.1.p1.9.m9.1.1.1"></lt><ci id="S2.1.p1.9.m9.1.1.2.cmml" xref="S2.1.p1.9.m9.1.1.2">πΎ</ci><apply id="S2.1.p1.9.m9.1.1.3.cmml" xref="S2.1.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="S2.1.p1.9.m9.1.1.3.1.cmml" xref="S2.1.p1.9.m9.1.1.3">subscript</csymbol><ci id="S2.1.p1.9.m9.1.1.3.2.cmml" xref="S2.1.p1.9.m9.1.1.3.2">π</ci><cn id="S2.1.p1.9.m9.1.1.3.3.cmml" type="integer" xref="S2.1.p1.9.m9.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.9.m9.1c">\gamma<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.9.m9.1d">italic_Ξ³ < italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Then <math alttext="\operatorname{cls}^{\delta}(A)=s(\operatorname{cls}^{\gamma}(A))" class="ltx_Math" display="inline" id="S2.1.p1.10.m10.4"><semantics id="S2.1.p1.10.m10.4a"><mrow id="S2.1.p1.10.m10.4.4" xref="S2.1.p1.10.m10.4.4.cmml"><mrow id="S2.1.p1.10.m10.3.3.1.1" xref="S2.1.p1.10.m10.3.3.1.2.cmml"><msup id="S2.1.p1.10.m10.3.3.1.1.1" xref="S2.1.p1.10.m10.3.3.1.1.1.cmml"><mi id="S2.1.p1.10.m10.3.3.1.1.1.2" xref="S2.1.p1.10.m10.3.3.1.1.1.2.cmml">cls</mi><mi id="S2.1.p1.10.m10.3.3.1.1.1.3" xref="S2.1.p1.10.m10.3.3.1.1.1.3.cmml">Ξ΄</mi></msup><mo id="S2.1.p1.10.m10.3.3.1.1a" xref="S2.1.p1.10.m10.3.3.1.2.cmml">β‘</mo><mrow id="S2.1.p1.10.m10.3.3.1.1.2" xref="S2.1.p1.10.m10.3.3.1.2.cmml"><mo id="S2.1.p1.10.m10.3.3.1.1.2.1" stretchy="false" xref="S2.1.p1.10.m10.3.3.1.2.cmml">(</mo><mi id="S2.1.p1.10.m10.1.1" xref="S2.1.p1.10.m10.1.1.cmml">A</mi><mo id="S2.1.p1.10.m10.3.3.1.1.2.2" stretchy="false" xref="S2.1.p1.10.m10.3.3.1.2.cmml">)</mo></mrow></mrow><mo id="S2.1.p1.10.m10.4.4.3" xref="S2.1.p1.10.m10.4.4.3.cmml">=</mo><mrow id="S2.1.p1.10.m10.4.4.2" xref="S2.1.p1.10.m10.4.4.2.cmml"><mi id="S2.1.p1.10.m10.4.4.2.3" xref="S2.1.p1.10.m10.4.4.2.3.cmml">s</mi><mo id="S2.1.p1.10.m10.4.4.2.2" xref="S2.1.p1.10.m10.4.4.2.2.cmml">β’</mo><mrow id="S2.1.p1.10.m10.4.4.2.1.1" xref="S2.1.p1.10.m10.4.4.2.cmml"><mo id="S2.1.p1.10.m10.4.4.2.1.1.2" stretchy="false" xref="S2.1.p1.10.m10.4.4.2.cmml">(</mo><mrow id="S2.1.p1.10.m10.4.4.2.1.1.1.1" xref="S2.1.p1.10.m10.4.4.2.1.1.1.2.cmml"><msup id="S2.1.p1.10.m10.4.4.2.1.1.1.1.1" xref="S2.1.p1.10.m10.4.4.2.1.1.1.1.1.cmml"><mi id="S2.1.p1.10.m10.4.4.2.1.1.1.1.1.2" xref="S2.1.p1.10.m10.4.4.2.1.1.1.1.1.2.cmml">cls</mi><mi id="S2.1.p1.10.m10.4.4.2.1.1.1.1.1.3" xref="S2.1.p1.10.m10.4.4.2.1.1.1.1.1.3.cmml">Ξ³</mi></msup><mo id="S2.1.p1.10.m10.4.4.2.1.1.1.1a" xref="S2.1.p1.10.m10.4.4.2.1.1.1.2.cmml">β‘</mo><mrow id="S2.1.p1.10.m10.4.4.2.1.1.1.1.2" xref="S2.1.p1.10.m10.4.4.2.1.1.1.2.cmml"><mo id="S2.1.p1.10.m10.4.4.2.1.1.1.1.2.1" stretchy="false" xref="S2.1.p1.10.m10.4.4.2.1.1.1.2.cmml">(</mo><mi id="S2.1.p1.10.m10.2.2" xref="S2.1.p1.10.m10.2.2.cmml">A</mi><mo id="S2.1.p1.10.m10.4.4.2.1.1.1.1.2.2" stretchy="false" xref="S2.1.p1.10.m10.4.4.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.1.p1.10.m10.4.4.2.1.1.3" stretchy="false" xref="S2.1.p1.10.m10.4.4.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.10.m10.4b"><apply id="S2.1.p1.10.m10.4.4.cmml" xref="S2.1.p1.10.m10.4.4"><eq id="S2.1.p1.10.m10.4.4.3.cmml" xref="S2.1.p1.10.m10.4.4.3"></eq><apply id="S2.1.p1.10.m10.3.3.1.2.cmml" xref="S2.1.p1.10.m10.3.3.1.1"><apply id="S2.1.p1.10.m10.3.3.1.1.1.cmml" xref="S2.1.p1.10.m10.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.1.p1.10.m10.3.3.1.1.1.1.cmml" xref="S2.1.p1.10.m10.3.3.1.1.1">superscript</csymbol><ci id="S2.1.p1.10.m10.3.3.1.1.1.2.cmml" xref="S2.1.p1.10.m10.3.3.1.1.1.2">cls</ci><ci id="S2.1.p1.10.m10.3.3.1.1.1.3.cmml" xref="S2.1.p1.10.m10.3.3.1.1.1.3">πΏ</ci></apply><ci id="S2.1.p1.10.m10.1.1.cmml" xref="S2.1.p1.10.m10.1.1">π΄</ci></apply><apply id="S2.1.p1.10.m10.4.4.2.cmml" xref="S2.1.p1.10.m10.4.4.2"><times id="S2.1.p1.10.m10.4.4.2.2.cmml" xref="S2.1.p1.10.m10.4.4.2.2"></times><ci id="S2.1.p1.10.m10.4.4.2.3.cmml" xref="S2.1.p1.10.m10.4.4.2.3">π </ci><apply id="S2.1.p1.10.m10.4.4.2.1.1.1.2.cmml" xref="S2.1.p1.10.m10.4.4.2.1.1.1.1"><apply id="S2.1.p1.10.m10.4.4.2.1.1.1.1.1.cmml" xref="S2.1.p1.10.m10.4.4.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.1.p1.10.m10.4.4.2.1.1.1.1.1.1.cmml" xref="S2.1.p1.10.m10.4.4.2.1.1.1.1.1">superscript</csymbol><ci id="S2.1.p1.10.m10.4.4.2.1.1.1.1.1.2.cmml" xref="S2.1.p1.10.m10.4.4.2.1.1.1.1.1.2">cls</ci><ci id="S2.1.p1.10.m10.4.4.2.1.1.1.1.1.3.cmml" xref="S2.1.p1.10.m10.4.4.2.1.1.1.1.1.3">πΎ</ci></apply><ci id="S2.1.p1.10.m10.2.2.cmml" xref="S2.1.p1.10.m10.2.2">π΄</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.10.m10.4c">\operatorname{cls}^{\delta}(A)=s(\operatorname{cls}^{\gamma}(A))</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.10.m10.4d">roman_cls start_POSTSUPERSCRIPT italic_Ξ΄ end_POSTSUPERSCRIPT ( italic_A ) = italic_s ( roman_cls start_POSTSUPERSCRIPT italic_Ξ³ end_POSTSUPERSCRIPT ( italic_A ) )</annotation></semantics></math>. Pick any points <math alttext="x,y\in\operatorname{cls}^{\delta}(A)" class="ltx_Math" display="inline" id="S2.1.p1.11.m11.4"><semantics id="S2.1.p1.11.m11.4a"><mrow id="S2.1.p1.11.m11.4.4" xref="S2.1.p1.11.m11.4.4.cmml"><mrow id="S2.1.p1.11.m11.4.4.3.2" xref="S2.1.p1.11.m11.4.4.3.1.cmml"><mi id="S2.1.p1.11.m11.2.2" xref="S2.1.p1.11.m11.2.2.cmml">x</mi><mo id="S2.1.p1.11.m11.4.4.3.2.1" xref="S2.1.p1.11.m11.4.4.3.1.cmml">,</mo><mi id="S2.1.p1.11.m11.3.3" xref="S2.1.p1.11.m11.3.3.cmml">y</mi></mrow><mo id="S2.1.p1.11.m11.4.4.2" xref="S2.1.p1.11.m11.4.4.2.cmml">β</mo><mrow id="S2.1.p1.11.m11.4.4.1.1" xref="S2.1.p1.11.m11.4.4.1.2.cmml"><msup id="S2.1.p1.11.m11.4.4.1.1.1" xref="S2.1.p1.11.m11.4.4.1.1.1.cmml"><mi id="S2.1.p1.11.m11.4.4.1.1.1.2" xref="S2.1.p1.11.m11.4.4.1.1.1.2.cmml">cls</mi><mi id="S2.1.p1.11.m11.4.4.1.1.1.3" xref="S2.1.p1.11.m11.4.4.1.1.1.3.cmml">Ξ΄</mi></msup><mo id="S2.1.p1.11.m11.4.4.1.1a" xref="S2.1.p1.11.m11.4.4.1.2.cmml">β‘</mo><mrow id="S2.1.p1.11.m11.4.4.1.1.2" xref="S2.1.p1.11.m11.4.4.1.2.cmml"><mo id="S2.1.p1.11.m11.4.4.1.1.2.1" stretchy="false" xref="S2.1.p1.11.m11.4.4.1.2.cmml">(</mo><mi id="S2.1.p1.11.m11.1.1" xref="S2.1.p1.11.m11.1.1.cmml">A</mi><mo id="S2.1.p1.11.m11.4.4.1.1.2.2" stretchy="false" xref="S2.1.p1.11.m11.4.4.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.11.m11.4b"><apply id="S2.1.p1.11.m11.4.4.cmml" xref="S2.1.p1.11.m11.4.4"><in id="S2.1.p1.11.m11.4.4.2.cmml" xref="S2.1.p1.11.m11.4.4.2"></in><list id="S2.1.p1.11.m11.4.4.3.1.cmml" xref="S2.1.p1.11.m11.4.4.3.2"><ci id="S2.1.p1.11.m11.2.2.cmml" xref="S2.1.p1.11.m11.2.2">π₯</ci><ci id="S2.1.p1.11.m11.3.3.cmml" xref="S2.1.p1.11.m11.3.3">π¦</ci></list><apply id="S2.1.p1.11.m11.4.4.1.2.cmml" xref="S2.1.p1.11.m11.4.4.1.1"><apply id="S2.1.p1.11.m11.4.4.1.1.1.cmml" xref="S2.1.p1.11.m11.4.4.1.1.1"><csymbol cd="ambiguous" id="S2.1.p1.11.m11.4.4.1.1.1.1.cmml" xref="S2.1.p1.11.m11.4.4.1.1.1">superscript</csymbol><ci id="S2.1.p1.11.m11.4.4.1.1.1.2.cmml" xref="S2.1.p1.11.m11.4.4.1.1.1.2">cls</ci><ci id="S2.1.p1.11.m11.4.4.1.1.1.3.cmml" xref="S2.1.p1.11.m11.4.4.1.1.1.3">πΏ</ci></apply><ci id="S2.1.p1.11.m11.1.1.cmml" xref="S2.1.p1.11.m11.1.1">π΄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.11.m11.4c">x,y\in\operatorname{cls}^{\delta}(A)</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.11.m11.4d">italic_x , italic_y β roman_cls start_POSTSUPERSCRIPT italic_Ξ΄ end_POSTSUPERSCRIPT ( italic_A )</annotation></semantics></math> and sequences <math alttext="\{x_{n}:n\in\omega\}\subset\operatorname{cls}^{\gamma}(A)" class="ltx_Math" display="inline" id="S2.1.p1.12.m12.4"><semantics id="S2.1.p1.12.m12.4a"><mrow id="S2.1.p1.12.m12.4.4" xref="S2.1.p1.12.m12.4.4.cmml"><mrow id="S2.1.p1.12.m12.3.3.2.2" xref="S2.1.p1.12.m12.3.3.2.3.cmml"><mo id="S2.1.p1.12.m12.3.3.2.2.3" stretchy="false" xref="S2.1.p1.12.m12.3.3.2.3.1.cmml">{</mo><msub id="S2.1.p1.12.m12.2.2.1.1.1" xref="S2.1.p1.12.m12.2.2.1.1.1.cmml"><mi id="S2.1.p1.12.m12.2.2.1.1.1.2" xref="S2.1.p1.12.m12.2.2.1.1.1.2.cmml">x</mi><mi id="S2.1.p1.12.m12.2.2.1.1.1.3" xref="S2.1.p1.12.m12.2.2.1.1.1.3.cmml">n</mi></msub><mo id="S2.1.p1.12.m12.3.3.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.1.p1.12.m12.3.3.2.3.1.cmml">:</mo><mrow id="S2.1.p1.12.m12.3.3.2.2.2" xref="S2.1.p1.12.m12.3.3.2.2.2.cmml"><mi id="S2.1.p1.12.m12.3.3.2.2.2.2" xref="S2.1.p1.12.m12.3.3.2.2.2.2.cmml">n</mi><mo id="S2.1.p1.12.m12.3.3.2.2.2.1" xref="S2.1.p1.12.m12.3.3.2.2.2.1.cmml">β</mo><mi id="S2.1.p1.12.m12.3.3.2.2.2.3" xref="S2.1.p1.12.m12.3.3.2.2.2.3.cmml">Ο</mi></mrow><mo id="S2.1.p1.12.m12.3.3.2.2.5" stretchy="false" xref="S2.1.p1.12.m12.3.3.2.3.1.cmml">}</mo></mrow><mo id="S2.1.p1.12.m12.4.4.4" xref="S2.1.p1.12.m12.4.4.4.cmml">β</mo><mrow id="S2.1.p1.12.m12.4.4.3.1" xref="S2.1.p1.12.m12.4.4.3.2.cmml"><msup id="S2.1.p1.12.m12.4.4.3.1.1" xref="S2.1.p1.12.m12.4.4.3.1.1.cmml"><mi id="S2.1.p1.12.m12.4.4.3.1.1.2" xref="S2.1.p1.12.m12.4.4.3.1.1.2.cmml">cls</mi><mi id="S2.1.p1.12.m12.4.4.3.1.1.3" xref="S2.1.p1.12.m12.4.4.3.1.1.3.cmml">Ξ³</mi></msup><mo id="S2.1.p1.12.m12.4.4.3.1a" xref="S2.1.p1.12.m12.4.4.3.2.cmml">β‘</mo><mrow id="S2.1.p1.12.m12.4.4.3.1.2" xref="S2.1.p1.12.m12.4.4.3.2.cmml"><mo id="S2.1.p1.12.m12.4.4.3.1.2.1" stretchy="false" xref="S2.1.p1.12.m12.4.4.3.2.cmml">(</mo><mi id="S2.1.p1.12.m12.1.1" xref="S2.1.p1.12.m12.1.1.cmml">A</mi><mo id="S2.1.p1.12.m12.4.4.3.1.2.2" stretchy="false" xref="S2.1.p1.12.m12.4.4.3.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.12.m12.4b"><apply id="S2.1.p1.12.m12.4.4.cmml" xref="S2.1.p1.12.m12.4.4"><subset id="S2.1.p1.12.m12.4.4.4.cmml" xref="S2.1.p1.12.m12.4.4.4"></subset><apply id="S2.1.p1.12.m12.3.3.2.3.cmml" xref="S2.1.p1.12.m12.3.3.2.2"><csymbol cd="latexml" id="S2.1.p1.12.m12.3.3.2.3.1.cmml" xref="S2.1.p1.12.m12.3.3.2.2.3">conditional-set</csymbol><apply id="S2.1.p1.12.m12.2.2.1.1.1.cmml" xref="S2.1.p1.12.m12.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.1.p1.12.m12.2.2.1.1.1.1.cmml" xref="S2.1.p1.12.m12.2.2.1.1.1">subscript</csymbol><ci id="S2.1.p1.12.m12.2.2.1.1.1.2.cmml" xref="S2.1.p1.12.m12.2.2.1.1.1.2">π₯</ci><ci id="S2.1.p1.12.m12.2.2.1.1.1.3.cmml" xref="S2.1.p1.12.m12.2.2.1.1.1.3">π</ci></apply><apply id="S2.1.p1.12.m12.3.3.2.2.2.cmml" xref="S2.1.p1.12.m12.3.3.2.2.2"><in id="S2.1.p1.12.m12.3.3.2.2.2.1.cmml" xref="S2.1.p1.12.m12.3.3.2.2.2.1"></in><ci id="S2.1.p1.12.m12.3.3.2.2.2.2.cmml" xref="S2.1.p1.12.m12.3.3.2.2.2.2">π</ci><ci id="S2.1.p1.12.m12.3.3.2.2.2.3.cmml" xref="S2.1.p1.12.m12.3.3.2.2.2.3">π</ci></apply></apply><apply id="S2.1.p1.12.m12.4.4.3.2.cmml" xref="S2.1.p1.12.m12.4.4.3.1"><apply id="S2.1.p1.12.m12.4.4.3.1.1.cmml" xref="S2.1.p1.12.m12.4.4.3.1.1"><csymbol cd="ambiguous" id="S2.1.p1.12.m12.4.4.3.1.1.1.cmml" xref="S2.1.p1.12.m12.4.4.3.1.1">superscript</csymbol><ci id="S2.1.p1.12.m12.4.4.3.1.1.2.cmml" xref="S2.1.p1.12.m12.4.4.3.1.1.2">cls</ci><ci id="S2.1.p1.12.m12.4.4.3.1.1.3.cmml" xref="S2.1.p1.12.m12.4.4.3.1.1.3">πΎ</ci></apply><ci id="S2.1.p1.12.m12.1.1.cmml" xref="S2.1.p1.12.m12.1.1">π΄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.12.m12.4c">\{x_{n}:n\in\omega\}\subset\operatorname{cls}^{\gamma}(A)</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.12.m12.4d">{ italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n β italic_Ο } β roman_cls start_POSTSUPERSCRIPT italic_Ξ³ end_POSTSUPERSCRIPT ( italic_A )</annotation></semantics></math> and <math alttext="\{y_{n}:n\in\omega\}\subset\operatorname{cls}^{\gamma}(A)" class="ltx_Math" display="inline" id="S2.1.p1.13.m13.4"><semantics id="S2.1.p1.13.m13.4a"><mrow id="S2.1.p1.13.m13.4.4" xref="S2.1.p1.13.m13.4.4.cmml"><mrow id="S2.1.p1.13.m13.3.3.2.2" xref="S2.1.p1.13.m13.3.3.2.3.cmml"><mo id="S2.1.p1.13.m13.3.3.2.2.3" stretchy="false" xref="S2.1.p1.13.m13.3.3.2.3.1.cmml">{</mo><msub id="S2.1.p1.13.m13.2.2.1.1.1" xref="S2.1.p1.13.m13.2.2.1.1.1.cmml"><mi id="S2.1.p1.13.m13.2.2.1.1.1.2" xref="S2.1.p1.13.m13.2.2.1.1.1.2.cmml">y</mi><mi id="S2.1.p1.13.m13.2.2.1.1.1.3" xref="S2.1.p1.13.m13.2.2.1.1.1.3.cmml">n</mi></msub><mo id="S2.1.p1.13.m13.3.3.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.1.p1.13.m13.3.3.2.3.1.cmml">:</mo><mrow id="S2.1.p1.13.m13.3.3.2.2.2" xref="S2.1.p1.13.m13.3.3.2.2.2.cmml"><mi id="S2.1.p1.13.m13.3.3.2.2.2.2" xref="S2.1.p1.13.m13.3.3.2.2.2.2.cmml">n</mi><mo id="S2.1.p1.13.m13.3.3.2.2.2.1" xref="S2.1.p1.13.m13.3.3.2.2.2.1.cmml">β</mo><mi id="S2.1.p1.13.m13.3.3.2.2.2.3" xref="S2.1.p1.13.m13.3.3.2.2.2.3.cmml">Ο</mi></mrow><mo id="S2.1.p1.13.m13.3.3.2.2.5" stretchy="false" xref="S2.1.p1.13.m13.3.3.2.3.1.cmml">}</mo></mrow><mo id="S2.1.p1.13.m13.4.4.4" xref="S2.1.p1.13.m13.4.4.4.cmml">β</mo><mrow id="S2.1.p1.13.m13.4.4.3.1" xref="S2.1.p1.13.m13.4.4.3.2.cmml"><msup id="S2.1.p1.13.m13.4.4.3.1.1" xref="S2.1.p1.13.m13.4.4.3.1.1.cmml"><mi id="S2.1.p1.13.m13.4.4.3.1.1.2" xref="S2.1.p1.13.m13.4.4.3.1.1.2.cmml">cls</mi><mi id="S2.1.p1.13.m13.4.4.3.1.1.3" xref="S2.1.p1.13.m13.4.4.3.1.1.3.cmml">Ξ³</mi></msup><mo id="S2.1.p1.13.m13.4.4.3.1a" xref="S2.1.p1.13.m13.4.4.3.2.cmml">β‘</mo><mrow id="S2.1.p1.13.m13.4.4.3.1.2" xref="S2.1.p1.13.m13.4.4.3.2.cmml"><mo id="S2.1.p1.13.m13.4.4.3.1.2.1" stretchy="false" xref="S2.1.p1.13.m13.4.4.3.2.cmml">(</mo><mi id="S2.1.p1.13.m13.1.1" xref="S2.1.p1.13.m13.1.1.cmml">A</mi><mo id="S2.1.p1.13.m13.4.4.3.1.2.2" stretchy="false" xref="S2.1.p1.13.m13.4.4.3.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.13.m13.4b"><apply id="S2.1.p1.13.m13.4.4.cmml" xref="S2.1.p1.13.m13.4.4"><subset id="S2.1.p1.13.m13.4.4.4.cmml" xref="S2.1.p1.13.m13.4.4.4"></subset><apply id="S2.1.p1.13.m13.3.3.2.3.cmml" xref="S2.1.p1.13.m13.3.3.2.2"><csymbol cd="latexml" id="S2.1.p1.13.m13.3.3.2.3.1.cmml" xref="S2.1.p1.13.m13.3.3.2.2.3">conditional-set</csymbol><apply id="S2.1.p1.13.m13.2.2.1.1.1.cmml" xref="S2.1.p1.13.m13.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.1.p1.13.m13.2.2.1.1.1.1.cmml" xref="S2.1.p1.13.m13.2.2.1.1.1">subscript</csymbol><ci id="S2.1.p1.13.m13.2.2.1.1.1.2.cmml" xref="S2.1.p1.13.m13.2.2.1.1.1.2">π¦</ci><ci id="S2.1.p1.13.m13.2.2.1.1.1.3.cmml" xref="S2.1.p1.13.m13.2.2.1.1.1.3">π</ci></apply><apply id="S2.1.p1.13.m13.3.3.2.2.2.cmml" xref="S2.1.p1.13.m13.3.3.2.2.2"><in id="S2.1.p1.13.m13.3.3.2.2.2.1.cmml" xref="S2.1.p1.13.m13.3.3.2.2.2.1"></in><ci id="S2.1.p1.13.m13.3.3.2.2.2.2.cmml" xref="S2.1.p1.13.m13.3.3.2.2.2.2">π</ci><ci id="S2.1.p1.13.m13.3.3.2.2.2.3.cmml" xref="S2.1.p1.13.m13.3.3.2.2.2.3">π</ci></apply></apply><apply id="S2.1.p1.13.m13.4.4.3.2.cmml" xref="S2.1.p1.13.m13.4.4.3.1"><apply id="S2.1.p1.13.m13.4.4.3.1.1.cmml" xref="S2.1.p1.13.m13.4.4.3.1.1"><csymbol cd="ambiguous" id="S2.1.p1.13.m13.4.4.3.1.1.1.cmml" xref="S2.1.p1.13.m13.4.4.3.1.1">superscript</csymbol><ci id="S2.1.p1.13.m13.4.4.3.1.1.2.cmml" xref="S2.1.p1.13.m13.4.4.3.1.1.2">cls</ci><ci id="S2.1.p1.13.m13.4.4.3.1.1.3.cmml" xref="S2.1.p1.13.m13.4.4.3.1.1.3">πΎ</ci></apply><ci id="S2.1.p1.13.m13.1.1.cmml" xref="S2.1.p1.13.m13.1.1">π΄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.13.m13.4c">\{y_{n}:n\in\omega\}\subset\operatorname{cls}^{\gamma}(A)</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.13.m13.4d">{ italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n β italic_Ο } β roman_cls start_POSTSUPERSCRIPT italic_Ξ³ end_POSTSUPERSCRIPT ( italic_A )</annotation></semantics></math> such that <math alttext="x=\lim_{n\in\omega}x_{n}" class="ltx_Math" display="inline" id="S2.1.p1.14.m14.1"><semantics id="S2.1.p1.14.m14.1a"><mrow id="S2.1.p1.14.m14.1.1" xref="S2.1.p1.14.m14.1.1.cmml"><mi id="S2.1.p1.14.m14.1.1.2" xref="S2.1.p1.14.m14.1.1.2.cmml">x</mi><mo id="S2.1.p1.14.m14.1.1.1" rspace="0.1389em" xref="S2.1.p1.14.m14.1.1.1.cmml">=</mo><mrow id="S2.1.p1.14.m14.1.1.3" xref="S2.1.p1.14.m14.1.1.3.cmml"><msub id="S2.1.p1.14.m14.1.1.3.1" xref="S2.1.p1.14.m14.1.1.3.1.cmml"><mo id="S2.1.p1.14.m14.1.1.3.1.2" lspace="0.1389em" rspace="0.167em" xref="S2.1.p1.14.m14.1.1.3.1.2.cmml">lim</mo><mrow id="S2.1.p1.14.m14.1.1.3.1.3" xref="S2.1.p1.14.m14.1.1.3.1.3.cmml"><mi id="S2.1.p1.14.m14.1.1.3.1.3.2" xref="S2.1.p1.14.m14.1.1.3.1.3.2.cmml">n</mi><mo id="S2.1.p1.14.m14.1.1.3.1.3.1" xref="S2.1.p1.14.m14.1.1.3.1.3.1.cmml">β</mo><mi id="S2.1.p1.14.m14.1.1.3.1.3.3" xref="S2.1.p1.14.m14.1.1.3.1.3.3.cmml">Ο</mi></mrow></msub><msub id="S2.1.p1.14.m14.1.1.3.2" xref="S2.1.p1.14.m14.1.1.3.2.cmml"><mi id="S2.1.p1.14.m14.1.1.3.2.2" xref="S2.1.p1.14.m14.1.1.3.2.2.cmml">x</mi><mi id="S2.1.p1.14.m14.1.1.3.2.3" xref="S2.1.p1.14.m14.1.1.3.2.3.cmml">n</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.14.m14.1b"><apply id="S2.1.p1.14.m14.1.1.cmml" xref="S2.1.p1.14.m14.1.1"><eq id="S2.1.p1.14.m14.1.1.1.cmml" xref="S2.1.p1.14.m14.1.1.1"></eq><ci id="S2.1.p1.14.m14.1.1.2.cmml" xref="S2.1.p1.14.m14.1.1.2">π₯</ci><apply id="S2.1.p1.14.m14.1.1.3.cmml" xref="S2.1.p1.14.m14.1.1.3"><apply id="S2.1.p1.14.m14.1.1.3.1.cmml" xref="S2.1.p1.14.m14.1.1.3.1"><csymbol cd="ambiguous" id="S2.1.p1.14.m14.1.1.3.1.1.cmml" xref="S2.1.p1.14.m14.1.1.3.1">subscript</csymbol><limit id="S2.1.p1.14.m14.1.1.3.1.2.cmml" xref="S2.1.p1.14.m14.1.1.3.1.2"></limit><apply id="S2.1.p1.14.m14.1.1.3.1.3.cmml" xref="S2.1.p1.14.m14.1.1.3.1.3"><in id="S2.1.p1.14.m14.1.1.3.1.3.1.cmml" xref="S2.1.p1.14.m14.1.1.3.1.3.1"></in><ci id="S2.1.p1.14.m14.1.1.3.1.3.2.cmml" xref="S2.1.p1.14.m14.1.1.3.1.3.2">π</ci><ci id="S2.1.p1.14.m14.1.1.3.1.3.3.cmml" xref="S2.1.p1.14.m14.1.1.3.1.3.3">π</ci></apply></apply><apply id="S2.1.p1.14.m14.1.1.3.2.cmml" xref="S2.1.p1.14.m14.1.1.3.2"><csymbol cd="ambiguous" id="S2.1.p1.14.m14.1.1.3.2.1.cmml" xref="S2.1.p1.14.m14.1.1.3.2">subscript</csymbol><ci id="S2.1.p1.14.m14.1.1.3.2.2.cmml" xref="S2.1.p1.14.m14.1.1.3.2.2">π₯</ci><ci id="S2.1.p1.14.m14.1.1.3.2.3.cmml" xref="S2.1.p1.14.m14.1.1.3.2.3">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.14.m14.1c">x=\lim_{n\in\omega}x_{n}</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.14.m14.1d">italic_x = roman_lim start_POSTSUBSCRIPT italic_n β italic_Ο end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="y=\lim_{n\in\omega}y_{n}" class="ltx_Math" display="inline" id="S2.1.p1.15.m15.1"><semantics id="S2.1.p1.15.m15.1a"><mrow id="S2.1.p1.15.m15.1.1" xref="S2.1.p1.15.m15.1.1.cmml"><mi id="S2.1.p1.15.m15.1.1.2" xref="S2.1.p1.15.m15.1.1.2.cmml">y</mi><mo id="S2.1.p1.15.m15.1.1.1" rspace="0.1389em" xref="S2.1.p1.15.m15.1.1.1.cmml">=</mo><mrow id="S2.1.p1.15.m15.1.1.3" xref="S2.1.p1.15.m15.1.1.3.cmml"><msub id="S2.1.p1.15.m15.1.1.3.1" xref="S2.1.p1.15.m15.1.1.3.1.cmml"><mo id="S2.1.p1.15.m15.1.1.3.1.2" lspace="0.1389em" rspace="0.167em" xref="S2.1.p1.15.m15.1.1.3.1.2.cmml">lim</mo><mrow id="S2.1.p1.15.m15.1.1.3.1.3" xref="S2.1.p1.15.m15.1.1.3.1.3.cmml"><mi id="S2.1.p1.15.m15.1.1.3.1.3.2" xref="S2.1.p1.15.m15.1.1.3.1.3.2.cmml">n</mi><mo id="S2.1.p1.15.m15.1.1.3.1.3.1" xref="S2.1.p1.15.m15.1.1.3.1.3.1.cmml">β</mo><mi id="S2.1.p1.15.m15.1.1.3.1.3.3" xref="S2.1.p1.15.m15.1.1.3.1.3.3.cmml">Ο</mi></mrow></msub><msub id="S2.1.p1.15.m15.1.1.3.2" xref="S2.1.p1.15.m15.1.1.3.2.cmml"><mi id="S2.1.p1.15.m15.1.1.3.2.2" xref="S2.1.p1.15.m15.1.1.3.2.2.cmml">y</mi><mi id="S2.1.p1.15.m15.1.1.3.2.3" xref="S2.1.p1.15.m15.1.1.3.2.3.cmml">n</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.15.m15.1b"><apply id="S2.1.p1.15.m15.1.1.cmml" xref="S2.1.p1.15.m15.1.1"><eq id="S2.1.p1.15.m15.1.1.1.cmml" xref="S2.1.p1.15.m15.1.1.1"></eq><ci id="S2.1.p1.15.m15.1.1.2.cmml" xref="S2.1.p1.15.m15.1.1.2">π¦</ci><apply id="S2.1.p1.15.m15.1.1.3.cmml" xref="S2.1.p1.15.m15.1.1.3"><apply id="S2.1.p1.15.m15.1.1.3.1.cmml" xref="S2.1.p1.15.m15.1.1.3.1"><csymbol cd="ambiguous" id="S2.1.p1.15.m15.1.1.3.1.1.cmml" xref="S2.1.p1.15.m15.1.1.3.1">subscript</csymbol><limit id="S2.1.p1.15.m15.1.1.3.1.2.cmml" xref="S2.1.p1.15.m15.1.1.3.1.2"></limit><apply id="S2.1.p1.15.m15.1.1.3.1.3.cmml" xref="S2.1.p1.15.m15.1.1.3.1.3"><in id="S2.1.p1.15.m15.1.1.3.1.3.1.cmml" xref="S2.1.p1.15.m15.1.1.3.1.3.1"></in><ci id="S2.1.p1.15.m15.1.1.3.1.3.2.cmml" xref="S2.1.p1.15.m15.1.1.3.1.3.2">π</ci><ci id="S2.1.p1.15.m15.1.1.3.1.3.3.cmml" xref="S2.1.p1.15.m15.1.1.3.1.3.3">π</ci></apply></apply><apply id="S2.1.p1.15.m15.1.1.3.2.cmml" xref="S2.1.p1.15.m15.1.1.3.2"><csymbol cd="ambiguous" id="S2.1.p1.15.m15.1.1.3.2.1.cmml" xref="S2.1.p1.15.m15.1.1.3.2">subscript</csymbol><ci id="S2.1.p1.15.m15.1.1.3.2.2.cmml" xref="S2.1.p1.15.m15.1.1.3.2.2">π¦</ci><ci id="S2.1.p1.15.m15.1.1.3.2.3.cmml" xref="S2.1.p1.15.m15.1.1.3.2.3">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.15.m15.1c">y=\lim_{n\in\omega}y_{n}</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.15.m15.1d">italic_y = roman_lim start_POSTSUBSCRIPT italic_n β italic_Ο end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="\operatorname{cls}^{\gamma}(A)" class="ltx_Math" display="inline" id="S2.1.p1.16.m16.2"><semantics id="S2.1.p1.16.m16.2a"><mrow id="S2.1.p1.16.m16.2.2.1" xref="S2.1.p1.16.m16.2.2.2.cmml"><msup id="S2.1.p1.16.m16.2.2.1.1" xref="S2.1.p1.16.m16.2.2.1.1.cmml"><mi id="S2.1.p1.16.m16.2.2.1.1.2" xref="S2.1.p1.16.m16.2.2.1.1.2.cmml">cls</mi><mi id="S2.1.p1.16.m16.2.2.1.1.3" xref="S2.1.p1.16.m16.2.2.1.1.3.cmml">Ξ³</mi></msup><mo id="S2.1.p1.16.m16.2.2.1a" xref="S2.1.p1.16.m16.2.2.2.cmml">β‘</mo><mrow id="S2.1.p1.16.m16.2.2.1.2" xref="S2.1.p1.16.m16.2.2.2.cmml"><mo id="S2.1.p1.16.m16.2.2.1.2.1" stretchy="false" xref="S2.1.p1.16.m16.2.2.2.cmml">(</mo><mi id="S2.1.p1.16.m16.1.1" xref="S2.1.p1.16.m16.1.1.cmml">A</mi><mo id="S2.1.p1.16.m16.2.2.1.2.2" stretchy="false" xref="S2.1.p1.16.m16.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.16.m16.2b"><apply id="S2.1.p1.16.m16.2.2.2.cmml" xref="S2.1.p1.16.m16.2.2.1"><apply id="S2.1.p1.16.m16.2.2.1.1.cmml" xref="S2.1.p1.16.m16.2.2.1.1"><csymbol cd="ambiguous" id="S2.1.p1.16.m16.2.2.1.1.1.cmml" xref="S2.1.p1.16.m16.2.2.1.1">superscript</csymbol><ci id="S2.1.p1.16.m16.2.2.1.1.2.cmml" xref="S2.1.p1.16.m16.2.2.1.1.2">cls</ci><ci id="S2.1.p1.16.m16.2.2.1.1.3.cmml" xref="S2.1.p1.16.m16.2.2.1.1.3">πΎ</ci></apply><ci id="S2.1.p1.16.m16.1.1.cmml" xref="S2.1.p1.16.m16.1.1">π΄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.16.m16.2c">\operatorname{cls}^{\gamma}(A)</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.16.m16.2d">roman_cls start_POSTSUPERSCRIPT italic_Ξ³ end_POSTSUPERSCRIPT ( italic_A )</annotation></semantics></math> is a semigroup, we get that the sequence <math alttext="\{x_{n}y_{n}:n\in\omega\}" class="ltx_Math" display="inline" id="S2.1.p1.17.m17.2"><semantics id="S2.1.p1.17.m17.2a"><mrow id="S2.1.p1.17.m17.2.2.2" xref="S2.1.p1.17.m17.2.2.3.cmml"><mo id="S2.1.p1.17.m17.2.2.2.3" stretchy="false" xref="S2.1.p1.17.m17.2.2.3.1.cmml">{</mo><mrow id="S2.1.p1.17.m17.1.1.1.1" xref="S2.1.p1.17.m17.1.1.1.1.cmml"><msub id="S2.1.p1.17.m17.1.1.1.1.2" xref="S2.1.p1.17.m17.1.1.1.1.2.cmml"><mi id="S2.1.p1.17.m17.1.1.1.1.2.2" xref="S2.1.p1.17.m17.1.1.1.1.2.2.cmml">x</mi><mi id="S2.1.p1.17.m17.1.1.1.1.2.3" xref="S2.1.p1.17.m17.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S2.1.p1.17.m17.1.1.1.1.1" xref="S2.1.p1.17.m17.1.1.1.1.1.cmml">β’</mo><msub id="S2.1.p1.17.m17.1.1.1.1.3" xref="S2.1.p1.17.m17.1.1.1.1.3.cmml"><mi id="S2.1.p1.17.m17.1.1.1.1.3.2" xref="S2.1.p1.17.m17.1.1.1.1.3.2.cmml">y</mi><mi id="S2.1.p1.17.m17.1.1.1.1.3.3" xref="S2.1.p1.17.m17.1.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S2.1.p1.17.m17.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.1.p1.17.m17.2.2.3.1.cmml">:</mo><mrow id="S2.1.p1.17.m17.2.2.2.2" xref="S2.1.p1.17.m17.2.2.2.2.cmml"><mi id="S2.1.p1.17.m17.2.2.2.2.2" xref="S2.1.p1.17.m17.2.2.2.2.2.cmml">n</mi><mo id="S2.1.p1.17.m17.2.2.2.2.1" xref="S2.1.p1.17.m17.2.2.2.2.1.cmml">β</mo><mi id="S2.1.p1.17.m17.2.2.2.2.3" xref="S2.1.p1.17.m17.2.2.2.2.3.cmml">Ο</mi></mrow><mo id="S2.1.p1.17.m17.2.2.2.5" stretchy="false" xref="S2.1.p1.17.m17.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.17.m17.2b"><apply id="S2.1.p1.17.m17.2.2.3.cmml" xref="S2.1.p1.17.m17.2.2.2"><csymbol cd="latexml" id="S2.1.p1.17.m17.2.2.3.1.cmml" xref="S2.1.p1.17.m17.2.2.2.3">conditional-set</csymbol><apply id="S2.1.p1.17.m17.1.1.1.1.cmml" xref="S2.1.p1.17.m17.1.1.1.1"><times id="S2.1.p1.17.m17.1.1.1.1.1.cmml" xref="S2.1.p1.17.m17.1.1.1.1.1"></times><apply id="S2.1.p1.17.m17.1.1.1.1.2.cmml" xref="S2.1.p1.17.m17.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.1.p1.17.m17.1.1.1.1.2.1.cmml" xref="S2.1.p1.17.m17.1.1.1.1.2">subscript</csymbol><ci id="S2.1.p1.17.m17.1.1.1.1.2.2.cmml" xref="S2.1.p1.17.m17.1.1.1.1.2.2">π₯</ci><ci id="S2.1.p1.17.m17.1.1.1.1.2.3.cmml" xref="S2.1.p1.17.m17.1.1.1.1.2.3">π</ci></apply><apply id="S2.1.p1.17.m17.1.1.1.1.3.cmml" xref="S2.1.p1.17.m17.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.1.p1.17.m17.1.1.1.1.3.1.cmml" xref="S2.1.p1.17.m17.1.1.1.1.3">subscript</csymbol><ci id="S2.1.p1.17.m17.1.1.1.1.3.2.cmml" xref="S2.1.p1.17.m17.1.1.1.1.3.2">π¦</ci><ci id="S2.1.p1.17.m17.1.1.1.1.3.3.cmml" xref="S2.1.p1.17.m17.1.1.1.1.3.3">π</ci></apply></apply><apply id="S2.1.p1.17.m17.2.2.2.2.cmml" xref="S2.1.p1.17.m17.2.2.2.2"><in id="S2.1.p1.17.m17.2.2.2.2.1.cmml" xref="S2.1.p1.17.m17.2.2.2.2.1"></in><ci id="S2.1.p1.17.m17.2.2.2.2.2.cmml" xref="S2.1.p1.17.m17.2.2.2.2.2">π</ci><ci id="S2.1.p1.17.m17.2.2.2.2.3.cmml" xref="S2.1.p1.17.m17.2.2.2.2.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.17.m17.2c">\{x_{n}y_{n}:n\in\omega\}</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.17.m17.2d">{ italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n β italic_Ο }</annotation></semantics></math> is contained in <math alttext="\operatorname{cls}^{\gamma}(A)" class="ltx_Math" display="inline" id="S2.1.p1.18.m18.2"><semantics id="S2.1.p1.18.m18.2a"><mrow id="S2.1.p1.18.m18.2.2.1" xref="S2.1.p1.18.m18.2.2.2.cmml"><msup id="S2.1.p1.18.m18.2.2.1.1" xref="S2.1.p1.18.m18.2.2.1.1.cmml"><mi id="S2.1.p1.18.m18.2.2.1.1.2" xref="S2.1.p1.18.m18.2.2.1.1.2.cmml">cls</mi><mi id="S2.1.p1.18.m18.2.2.1.1.3" xref="S2.1.p1.18.m18.2.2.1.1.3.cmml">Ξ³</mi></msup><mo id="S2.1.p1.18.m18.2.2.1a" xref="S2.1.p1.18.m18.2.2.2.cmml">β‘</mo><mrow id="S2.1.p1.18.m18.2.2.1.2" xref="S2.1.p1.18.m18.2.2.2.cmml"><mo id="S2.1.p1.18.m18.2.2.1.2.1" stretchy="false" xref="S2.1.p1.18.m18.2.2.2.cmml">(</mo><mi id="S2.1.p1.18.m18.1.1" xref="S2.1.p1.18.m18.1.1.cmml">A</mi><mo id="S2.1.p1.18.m18.2.2.1.2.2" stretchy="false" xref="S2.1.p1.18.m18.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.18.m18.2b"><apply id="S2.1.p1.18.m18.2.2.2.cmml" xref="S2.1.p1.18.m18.2.2.1"><apply id="S2.1.p1.18.m18.2.2.1.1.cmml" xref="S2.1.p1.18.m18.2.2.1.1"><csymbol cd="ambiguous" id="S2.1.p1.18.m18.2.2.1.1.1.cmml" xref="S2.1.p1.18.m18.2.2.1.1">superscript</csymbol><ci id="S2.1.p1.18.m18.2.2.1.1.2.cmml" xref="S2.1.p1.18.m18.2.2.1.1.2">cls</ci><ci id="S2.1.p1.18.m18.2.2.1.1.3.cmml" xref="S2.1.p1.18.m18.2.2.1.1.3">πΎ</ci></apply><ci id="S2.1.p1.18.m18.1.1.cmml" xref="S2.1.p1.18.m18.1.1">π΄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.18.m18.2c">\operatorname{cls}^{\gamma}(A)</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.18.m18.2d">roman_cls start_POSTSUPERSCRIPT italic_Ξ³ end_POSTSUPERSCRIPT ( italic_A )</annotation></semantics></math>. Since <math alttext="S" class="ltx_Math" display="inline" id="S2.1.p1.19.m19.1"><semantics id="S2.1.p1.19.m19.1a"><mi id="S2.1.p1.19.m19.1.1" xref="S2.1.p1.19.m19.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.1.p1.19.m19.1b"><ci id="S2.1.p1.19.m19.1.1.cmml" xref="S2.1.p1.19.m19.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.19.m19.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.19.m19.1d">italic_S</annotation></semantics></math> is a topological semigroup, the sequence <math alttext="\{x_{n}y_{n}:n\in\omega\}" class="ltx_Math" display="inline" id="S2.1.p1.20.m20.2"><semantics id="S2.1.p1.20.m20.2a"><mrow id="S2.1.p1.20.m20.2.2.2" xref="S2.1.p1.20.m20.2.2.3.cmml"><mo id="S2.1.p1.20.m20.2.2.2.3" stretchy="false" xref="S2.1.p1.20.m20.2.2.3.1.cmml">{</mo><mrow id="S2.1.p1.20.m20.1.1.1.1" xref="S2.1.p1.20.m20.1.1.1.1.cmml"><msub id="S2.1.p1.20.m20.1.1.1.1.2" xref="S2.1.p1.20.m20.1.1.1.1.2.cmml"><mi id="S2.1.p1.20.m20.1.1.1.1.2.2" xref="S2.1.p1.20.m20.1.1.1.1.2.2.cmml">x</mi><mi id="S2.1.p1.20.m20.1.1.1.1.2.3" xref="S2.1.p1.20.m20.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S2.1.p1.20.m20.1.1.1.1.1" xref="S2.1.p1.20.m20.1.1.1.1.1.cmml">β’</mo><msub id="S2.1.p1.20.m20.1.1.1.1.3" xref="S2.1.p1.20.m20.1.1.1.1.3.cmml"><mi id="S2.1.p1.20.m20.1.1.1.1.3.2" xref="S2.1.p1.20.m20.1.1.1.1.3.2.cmml">y</mi><mi id="S2.1.p1.20.m20.1.1.1.1.3.3" xref="S2.1.p1.20.m20.1.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S2.1.p1.20.m20.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.1.p1.20.m20.2.2.3.1.cmml">:</mo><mrow id="S2.1.p1.20.m20.2.2.2.2" xref="S2.1.p1.20.m20.2.2.2.2.cmml"><mi id="S2.1.p1.20.m20.2.2.2.2.2" xref="S2.1.p1.20.m20.2.2.2.2.2.cmml">n</mi><mo id="S2.1.p1.20.m20.2.2.2.2.1" xref="S2.1.p1.20.m20.2.2.2.2.1.cmml">β</mo><mi id="S2.1.p1.20.m20.2.2.2.2.3" xref="S2.1.p1.20.m20.2.2.2.2.3.cmml">Ο</mi></mrow><mo id="S2.1.p1.20.m20.2.2.2.5" stretchy="false" xref="S2.1.p1.20.m20.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.20.m20.2b"><apply id="S2.1.p1.20.m20.2.2.3.cmml" xref="S2.1.p1.20.m20.2.2.2"><csymbol cd="latexml" id="S2.1.p1.20.m20.2.2.3.1.cmml" xref="S2.1.p1.20.m20.2.2.2.3">conditional-set</csymbol><apply id="S2.1.p1.20.m20.1.1.1.1.cmml" xref="S2.1.p1.20.m20.1.1.1.1"><times id="S2.1.p1.20.m20.1.1.1.1.1.cmml" xref="S2.1.p1.20.m20.1.1.1.1.1"></times><apply id="S2.1.p1.20.m20.1.1.1.1.2.cmml" xref="S2.1.p1.20.m20.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.1.p1.20.m20.1.1.1.1.2.1.cmml" xref="S2.1.p1.20.m20.1.1.1.1.2">subscript</csymbol><ci id="S2.1.p1.20.m20.1.1.1.1.2.2.cmml" xref="S2.1.p1.20.m20.1.1.1.1.2.2">π₯</ci><ci id="S2.1.p1.20.m20.1.1.1.1.2.3.cmml" xref="S2.1.p1.20.m20.1.1.1.1.2.3">π</ci></apply><apply id="S2.1.p1.20.m20.1.1.1.1.3.cmml" xref="S2.1.p1.20.m20.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.1.p1.20.m20.1.1.1.1.3.1.cmml" xref="S2.1.p1.20.m20.1.1.1.1.3">subscript</csymbol><ci id="S2.1.p1.20.m20.1.1.1.1.3.2.cmml" xref="S2.1.p1.20.m20.1.1.1.1.3.2">π¦</ci><ci id="S2.1.p1.20.m20.1.1.1.1.3.3.cmml" xref="S2.1.p1.20.m20.1.1.1.1.3.3">π</ci></apply></apply><apply id="S2.1.p1.20.m20.2.2.2.2.cmml" xref="S2.1.p1.20.m20.2.2.2.2"><in id="S2.1.p1.20.m20.2.2.2.2.1.cmml" xref="S2.1.p1.20.m20.2.2.2.2.1"></in><ci id="S2.1.p1.20.m20.2.2.2.2.2.cmml" xref="S2.1.p1.20.m20.2.2.2.2.2">π</ci><ci id="S2.1.p1.20.m20.2.2.2.2.3.cmml" xref="S2.1.p1.20.m20.2.2.2.2.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.20.m20.2c">\{x_{n}y_{n}:n\in\omega\}</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.20.m20.2d">{ italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n β italic_Ο }</annotation></semantics></math> converges to <math alttext="xy" class="ltx_Math" display="inline" id="S2.1.p1.21.m21.1"><semantics id="S2.1.p1.21.m21.1a"><mrow id="S2.1.p1.21.m21.1.1" xref="S2.1.p1.21.m21.1.1.cmml"><mi id="S2.1.p1.21.m21.1.1.2" xref="S2.1.p1.21.m21.1.1.2.cmml">x</mi><mo id="S2.1.p1.21.m21.1.1.1" xref="S2.1.p1.21.m21.1.1.1.cmml">β’</mo><mi id="S2.1.p1.21.m21.1.1.3" xref="S2.1.p1.21.m21.1.1.3.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.21.m21.1b"><apply id="S2.1.p1.21.m21.1.1.cmml" xref="S2.1.p1.21.m21.1.1"><times id="S2.1.p1.21.m21.1.1.1.cmml" xref="S2.1.p1.21.m21.1.1.1"></times><ci id="S2.1.p1.21.m21.1.1.2.cmml" xref="S2.1.p1.21.m21.1.1.2">π₯</ci><ci id="S2.1.p1.21.m21.1.1.3.cmml" xref="S2.1.p1.21.m21.1.1.3">π¦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.21.m21.1c">xy</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.21.m21.1d">italic_x italic_y</annotation></semantics></math>. Thus <math alttext="xy\in\operatorname{cls}^{\delta}(A)" class="ltx_Math" display="inline" id="S2.1.p1.22.m22.2"><semantics id="S2.1.p1.22.m22.2a"><mrow id="S2.1.p1.22.m22.2.2" xref="S2.1.p1.22.m22.2.2.cmml"><mrow id="S2.1.p1.22.m22.2.2.3" xref="S2.1.p1.22.m22.2.2.3.cmml"><mi id="S2.1.p1.22.m22.2.2.3.2" xref="S2.1.p1.22.m22.2.2.3.2.cmml">x</mi><mo id="S2.1.p1.22.m22.2.2.3.1" xref="S2.1.p1.22.m22.2.2.3.1.cmml">β’</mo><mi id="S2.1.p1.22.m22.2.2.3.3" xref="S2.1.p1.22.m22.2.2.3.3.cmml">y</mi></mrow><mo id="S2.1.p1.22.m22.2.2.2" xref="S2.1.p1.22.m22.2.2.2.cmml">β</mo><mrow id="S2.1.p1.22.m22.2.2.1.1" xref="S2.1.p1.22.m22.2.2.1.2.cmml"><msup id="S2.1.p1.22.m22.2.2.1.1.1" xref="S2.1.p1.22.m22.2.2.1.1.1.cmml"><mi id="S2.1.p1.22.m22.2.2.1.1.1.2" xref="S2.1.p1.22.m22.2.2.1.1.1.2.cmml">cls</mi><mi id="S2.1.p1.22.m22.2.2.1.1.1.3" xref="S2.1.p1.22.m22.2.2.1.1.1.3.cmml">Ξ΄</mi></msup><mo id="S2.1.p1.22.m22.2.2.1.1a" xref="S2.1.p1.22.m22.2.2.1.2.cmml">β‘</mo><mrow id="S2.1.p1.22.m22.2.2.1.1.2" xref="S2.1.p1.22.m22.2.2.1.2.cmml"><mo id="S2.1.p1.22.m22.2.2.1.1.2.1" stretchy="false" xref="S2.1.p1.22.m22.2.2.1.2.cmml">(</mo><mi id="S2.1.p1.22.m22.1.1" xref="S2.1.p1.22.m22.1.1.cmml">A</mi><mo id="S2.1.p1.22.m22.2.2.1.1.2.2" stretchy="false" xref="S2.1.p1.22.m22.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.22.m22.2b"><apply id="S2.1.p1.22.m22.2.2.cmml" xref="S2.1.p1.22.m22.2.2"><in id="S2.1.p1.22.m22.2.2.2.cmml" xref="S2.1.p1.22.m22.2.2.2"></in><apply id="S2.1.p1.22.m22.2.2.3.cmml" xref="S2.1.p1.22.m22.2.2.3"><times id="S2.1.p1.22.m22.2.2.3.1.cmml" xref="S2.1.p1.22.m22.2.2.3.1"></times><ci id="S2.1.p1.22.m22.2.2.3.2.cmml" xref="S2.1.p1.22.m22.2.2.3.2">π₯</ci><ci id="S2.1.p1.22.m22.2.2.3.3.cmml" xref="S2.1.p1.22.m22.2.2.3.3">π¦</ci></apply><apply id="S2.1.p1.22.m22.2.2.1.2.cmml" xref="S2.1.p1.22.m22.2.2.1.1"><apply id="S2.1.p1.22.m22.2.2.1.1.1.cmml" xref="S2.1.p1.22.m22.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.1.p1.22.m22.2.2.1.1.1.1.cmml" xref="S2.1.p1.22.m22.2.2.1.1.1">superscript</csymbol><ci id="S2.1.p1.22.m22.2.2.1.1.1.2.cmml" xref="S2.1.p1.22.m22.2.2.1.1.1.2">cls</ci><ci id="S2.1.p1.22.m22.2.2.1.1.1.3.cmml" xref="S2.1.p1.22.m22.2.2.1.1.1.3">πΏ</ci></apply><ci id="S2.1.p1.22.m22.1.1.cmml" xref="S2.1.p1.22.m22.1.1">π΄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.22.m22.2c">xy\in\operatorname{cls}^{\delta}(A)</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.22.m22.2d">italic_x italic_y β roman_cls start_POSTSUPERSCRIPT italic_Ξ΄ end_POSTSUPERSCRIPT ( italic_A )</annotation></semantics></math>, witnessing that <math alttext="\operatorname{cls}^{\delta}(A)" class="ltx_Math" display="inline" id="S2.1.p1.23.m23.2"><semantics id="S2.1.p1.23.m23.2a"><mrow id="S2.1.p1.23.m23.2.2.1" xref="S2.1.p1.23.m23.2.2.2.cmml"><msup id="S2.1.p1.23.m23.2.2.1.1" xref="S2.1.p1.23.m23.2.2.1.1.cmml"><mi id="S2.1.p1.23.m23.2.2.1.1.2" xref="S2.1.p1.23.m23.2.2.1.1.2.cmml">cls</mi><mi id="S2.1.p1.23.m23.2.2.1.1.3" xref="S2.1.p1.23.m23.2.2.1.1.3.cmml">Ξ΄</mi></msup><mo id="S2.1.p1.23.m23.2.2.1a" xref="S2.1.p1.23.m23.2.2.2.cmml">β‘</mo><mrow id="S2.1.p1.23.m23.2.2.1.2" xref="S2.1.p1.23.m23.2.2.2.cmml"><mo id="S2.1.p1.23.m23.2.2.1.2.1" stretchy="false" xref="S2.1.p1.23.m23.2.2.2.cmml">(</mo><mi id="S2.1.p1.23.m23.1.1" xref="S2.1.p1.23.m23.1.1.cmml">A</mi><mo id="S2.1.p1.23.m23.2.2.1.2.2" stretchy="false" xref="S2.1.p1.23.m23.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.23.m23.2b"><apply id="S2.1.p1.23.m23.2.2.2.cmml" xref="S2.1.p1.23.m23.2.2.1"><apply id="S2.1.p1.23.m23.2.2.1.1.cmml" xref="S2.1.p1.23.m23.2.2.1.1"><csymbol cd="ambiguous" id="S2.1.p1.23.m23.2.2.1.1.1.cmml" xref="S2.1.p1.23.m23.2.2.1.1">superscript</csymbol><ci id="S2.1.p1.23.m23.2.2.1.1.2.cmml" xref="S2.1.p1.23.m23.2.2.1.1.2">cls</ci><ci id="S2.1.p1.23.m23.2.2.1.1.3.cmml" xref="S2.1.p1.23.m23.2.2.1.1.3">πΏ</ci></apply><ci id="S2.1.p1.23.m23.1.1.cmml" xref="S2.1.p1.23.m23.1.1">π΄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.23.m23.2c">\operatorname{cls}^{\delta}(A)</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.23.m23.2d">roman_cls start_POSTSUPERSCRIPT italic_Ξ΄ end_POSTSUPERSCRIPT ( italic_A )</annotation></semantics></math> is a subsemigroup of <math alttext="S" class="ltx_Math" display="inline" id="S2.1.p1.24.m24.1"><semantics id="S2.1.p1.24.m24.1a"><mi id="S2.1.p1.24.m24.1.1" xref="S2.1.p1.24.m24.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.1.p1.24.m24.1b"><ci id="S2.1.p1.24.m24.1.1.cmml" xref="S2.1.p1.24.m24.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.24.m24.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.24.m24.1d">italic_S</annotation></semantics></math>. Hence <math alttext="\operatorname{cls}^{\alpha}(A)" class="ltx_Math" display="inline" id="S2.1.p1.25.m25.2"><semantics id="S2.1.p1.25.m25.2a"><mrow id="S2.1.p1.25.m25.2.2.1" xref="S2.1.p1.25.m25.2.2.2.cmml"><msup id="S2.1.p1.25.m25.2.2.1.1" xref="S2.1.p1.25.m25.2.2.1.1.cmml"><mi id="S2.1.p1.25.m25.2.2.1.1.2" xref="S2.1.p1.25.m25.2.2.1.1.2.cmml">cls</mi><mi id="S2.1.p1.25.m25.2.2.1.1.3" xref="S2.1.p1.25.m25.2.2.1.1.3.cmml">Ξ±</mi></msup><mo id="S2.1.p1.25.m25.2.2.1a" xref="S2.1.p1.25.m25.2.2.2.cmml">β‘</mo><mrow id="S2.1.p1.25.m25.2.2.1.2" xref="S2.1.p1.25.m25.2.2.2.cmml"><mo id="S2.1.p1.25.m25.2.2.1.2.1" stretchy="false" xref="S2.1.p1.25.m25.2.2.2.cmml">(</mo><mi id="S2.1.p1.25.m25.1.1" xref="S2.1.p1.25.m25.1.1.cmml">A</mi><mo id="S2.1.p1.25.m25.2.2.1.2.2" stretchy="false" xref="S2.1.p1.25.m25.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.25.m25.2b"><apply id="S2.1.p1.25.m25.2.2.2.cmml" xref="S2.1.p1.25.m25.2.2.1"><apply id="S2.1.p1.25.m25.2.2.1.1.cmml" xref="S2.1.p1.25.m25.2.2.1.1"><csymbol cd="ambiguous" id="S2.1.p1.25.m25.2.2.1.1.1.cmml" xref="S2.1.p1.25.m25.2.2.1.1">superscript</csymbol><ci id="S2.1.p1.25.m25.2.2.1.1.2.cmml" xref="S2.1.p1.25.m25.2.2.1.1.2">cls</ci><ci id="S2.1.p1.25.m25.2.2.1.1.3.cmml" xref="S2.1.p1.25.m25.2.2.1.1.3">πΌ</ci></apply><ci id="S2.1.p1.25.m25.1.1.cmml" xref="S2.1.p1.25.m25.1.1">π΄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.25.m25.2c">\operatorname{cls}^{\alpha}(A)</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.25.m25.2d">roman_cls start_POSTSUPERSCRIPT italic_Ξ± end_POSTSUPERSCRIPT ( italic_A )</annotation></semantics></math> is a subsemigroup of <math alttext="S" class="ltx_Math" display="inline" id="S2.1.p1.26.m26.1"><semantics id="S2.1.p1.26.m26.1a"><mi id="S2.1.p1.26.m26.1.1" xref="S2.1.p1.26.m26.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.1.p1.26.m26.1b"><ci id="S2.1.p1.26.m26.1.1.cmml" xref="S2.1.p1.26.m26.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.26.m26.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.26.m26.1d">italic_S</annotation></semantics></math> for each <math alttext="\alpha\leq\omega_{1}" class="ltx_Math" display="inline" id="S2.1.p1.27.m27.1"><semantics id="S2.1.p1.27.m27.1a"><mrow id="S2.1.p1.27.m27.1.1" xref="S2.1.p1.27.m27.1.1.cmml"><mi id="S2.1.p1.27.m27.1.1.2" xref="S2.1.p1.27.m27.1.1.2.cmml">Ξ±</mi><mo id="S2.1.p1.27.m27.1.1.1" xref="S2.1.p1.27.m27.1.1.1.cmml">β€</mo><msub id="S2.1.p1.27.m27.1.1.3" xref="S2.1.p1.27.m27.1.1.3.cmml"><mi id="S2.1.p1.27.m27.1.1.3.2" xref="S2.1.p1.27.m27.1.1.3.2.cmml">Ο</mi><mn id="S2.1.p1.27.m27.1.1.3.3" xref="S2.1.p1.27.m27.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.1.p1.27.m27.1b"><apply id="S2.1.p1.27.m27.1.1.cmml" xref="S2.1.p1.27.m27.1.1"><leq id="S2.1.p1.27.m27.1.1.1.cmml" xref="S2.1.p1.27.m27.1.1.1"></leq><ci id="S2.1.p1.27.m27.1.1.2.cmml" xref="S2.1.p1.27.m27.1.1.2">πΌ</ci><apply id="S2.1.p1.27.m27.1.1.3.cmml" xref="S2.1.p1.27.m27.1.1.3"><csymbol cd="ambiguous" id="S2.1.p1.27.m27.1.1.3.1.cmml" xref="S2.1.p1.27.m27.1.1.3">subscript</csymbol><ci id="S2.1.p1.27.m27.1.1.3.2.cmml" xref="S2.1.p1.27.m27.1.1.3.2">π</ci><cn id="S2.1.p1.27.m27.1.1.3.3.cmml" type="integer" xref="S2.1.p1.27.m27.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.p1.27.m27.1c">\alpha\leq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.1.p1.27.m27.1d">italic_Ξ± β€ italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, as required. β</p> </div> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.1">The following problem appears in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib5" title="">5</a>, Problem 2.2]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_problem" id="S2.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem2.1.1.1">Problem 2.2</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem2.2.2"> </span>(Banakh, Pastukhova)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem2.3.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem2.p1"> <p class="ltx_p" id="S2.Thmtheorem2.p1.2"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem2.p1.2.2">Assume that <math alttext="S" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.1.1.m1.1"><semantics id="S2.Thmtheorem2.p1.1.1.m1.1a"><mi id="S2.Thmtheorem2.p1.1.1.m1.1.1" xref="S2.Thmtheorem2.p1.1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.1.1.m1.1b"><ci id="S2.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.1.1.m1.1d">italic_S</annotation></semantics></math> is a compact topological semigroup that contains a dense inverse subsemigroup. Is <math alttext="S" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.2.2.m2.1"><semantics id="S2.Thmtheorem2.p1.2.2.m2.1a"><mi id="S2.Thmtheorem2.p1.2.2.m2.1.1" xref="S2.Thmtheorem2.p1.2.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.2.2.m2.1b"><ci id="S2.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem2.p1.2.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.2.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.2.2.m2.1d">italic_S</annotation></semantics></math> an inverse semigroup?</span></p> </div> </div> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.16">For a semigroup <math alttext="S" class="ltx_Math" display="inline" id="S2.p3.1.m1.1"><semantics id="S2.p3.1.m1.1a"><mi id="S2.p3.1.m1.1.1" xref="S2.p3.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p3.1.m1.1b"><ci id="S2.p3.1.m1.1.1.cmml" xref="S2.p3.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p3.1.m1.1d">italic_S</annotation></semantics></math> by <math alttext="E(S)" class="ltx_Math" display="inline" id="S2.p3.2.m2.1"><semantics id="S2.p3.2.m2.1a"><mrow id="S2.p3.2.m2.1.2" xref="S2.p3.2.m2.1.2.cmml"><mi id="S2.p3.2.m2.1.2.2" xref="S2.p3.2.m2.1.2.2.cmml">E</mi><mo id="S2.p3.2.m2.1.2.1" xref="S2.p3.2.m2.1.2.1.cmml">β’</mo><mrow id="S2.p3.2.m2.1.2.3.2" xref="S2.p3.2.m2.1.2.cmml"><mo id="S2.p3.2.m2.1.2.3.2.1" stretchy="false" xref="S2.p3.2.m2.1.2.cmml">(</mo><mi id="S2.p3.2.m2.1.1" xref="S2.p3.2.m2.1.1.cmml">S</mi><mo id="S2.p3.2.m2.1.2.3.2.2" stretchy="false" xref="S2.p3.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.2.m2.1b"><apply id="S2.p3.2.m2.1.2.cmml" xref="S2.p3.2.m2.1.2"><times id="S2.p3.2.m2.1.2.1.cmml" xref="S2.p3.2.m2.1.2.1"></times><ci id="S2.p3.2.m2.1.2.2.cmml" xref="S2.p3.2.m2.1.2.2">πΈ</ci><ci id="S2.p3.2.m2.1.1.cmml" xref="S2.p3.2.m2.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.2.m2.1c">E(S)</annotation><annotation encoding="application/x-llamapun" id="S2.p3.2.m2.1d">italic_E ( italic_S )</annotation></semantics></math> we denote the set of all idempotents of <math alttext="S" class="ltx_Math" display="inline" id="S2.p3.3.m3.1"><semantics id="S2.p3.3.m3.1a"><mi id="S2.p3.3.m3.1.1" xref="S2.p3.3.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p3.3.m3.1b"><ci id="S2.p3.3.m3.1.1.cmml" xref="S2.p3.3.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.3.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p3.3.m3.1d">italic_S</annotation></semantics></math>. If <math alttext="S" class="ltx_Math" display="inline" id="S2.p3.4.m4.1"><semantics id="S2.p3.4.m4.1a"><mi id="S2.p3.4.m4.1.1" xref="S2.p3.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p3.4.m4.1b"><ci id="S2.p3.4.m4.1.1.cmml" xref="S2.p3.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p3.4.m4.1d">italic_S</annotation></semantics></math> is an inverse semigroup, then <math alttext="E(S)" class="ltx_Math" display="inline" id="S2.p3.5.m5.1"><semantics id="S2.p3.5.m5.1a"><mrow id="S2.p3.5.m5.1.2" xref="S2.p3.5.m5.1.2.cmml"><mi id="S2.p3.5.m5.1.2.2" xref="S2.p3.5.m5.1.2.2.cmml">E</mi><mo id="S2.p3.5.m5.1.2.1" xref="S2.p3.5.m5.1.2.1.cmml">β’</mo><mrow id="S2.p3.5.m5.1.2.3.2" xref="S2.p3.5.m5.1.2.cmml"><mo id="S2.p3.5.m5.1.2.3.2.1" stretchy="false" xref="S2.p3.5.m5.1.2.cmml">(</mo><mi id="S2.p3.5.m5.1.1" xref="S2.p3.5.m5.1.1.cmml">S</mi><mo id="S2.p3.5.m5.1.2.3.2.2" stretchy="false" xref="S2.p3.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.5.m5.1b"><apply id="S2.p3.5.m5.1.2.cmml" xref="S2.p3.5.m5.1.2"><times id="S2.p3.5.m5.1.2.1.cmml" xref="S2.p3.5.m5.1.2.1"></times><ci id="S2.p3.5.m5.1.2.2.cmml" xref="S2.p3.5.m5.1.2.2">πΈ</ci><ci id="S2.p3.5.m5.1.1.cmml" xref="S2.p3.5.m5.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.5.m5.1c">E(S)</annotation><annotation encoding="application/x-llamapun" id="S2.p3.5.m5.1d">italic_E ( italic_S )</annotation></semantics></math> is a semilattice. A semigroup <math alttext="S" class="ltx_Math" display="inline" id="S2.p3.6.m6.1"><semantics id="S2.p3.6.m6.1a"><mi id="S2.p3.6.m6.1.1" xref="S2.p3.6.m6.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p3.6.m6.1b"><ci id="S2.p3.6.m6.1.1.cmml" xref="S2.p3.6.m6.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.6.m6.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p3.6.m6.1d">italic_S</annotation></semantics></math> is called <span class="ltx_text ltx_font_italic" id="S2.p3.16.1">regular</span> if for each <math alttext="x\in S" class="ltx_Math" display="inline" id="S2.p3.7.m7.1"><semantics id="S2.p3.7.m7.1a"><mrow id="S2.p3.7.m7.1.1" xref="S2.p3.7.m7.1.1.cmml"><mi id="S2.p3.7.m7.1.1.2" xref="S2.p3.7.m7.1.1.2.cmml">x</mi><mo id="S2.p3.7.m7.1.1.1" xref="S2.p3.7.m7.1.1.1.cmml">β</mo><mi id="S2.p3.7.m7.1.1.3" xref="S2.p3.7.m7.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.7.m7.1b"><apply id="S2.p3.7.m7.1.1.cmml" xref="S2.p3.7.m7.1.1"><in id="S2.p3.7.m7.1.1.1.cmml" xref="S2.p3.7.m7.1.1.1"></in><ci id="S2.p3.7.m7.1.1.2.cmml" xref="S2.p3.7.m7.1.1.2">π₯</ci><ci id="S2.p3.7.m7.1.1.3.cmml" xref="S2.p3.7.m7.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.7.m7.1c">x\in S</annotation><annotation encoding="application/x-llamapun" id="S2.p3.7.m7.1d">italic_x β italic_S</annotation></semantics></math> there exists <math alttext="y\in S" class="ltx_Math" display="inline" id="S2.p3.8.m8.1"><semantics id="S2.p3.8.m8.1a"><mrow id="S2.p3.8.m8.1.1" xref="S2.p3.8.m8.1.1.cmml"><mi id="S2.p3.8.m8.1.1.2" xref="S2.p3.8.m8.1.1.2.cmml">y</mi><mo id="S2.p3.8.m8.1.1.1" xref="S2.p3.8.m8.1.1.1.cmml">β</mo><mi id="S2.p3.8.m8.1.1.3" xref="S2.p3.8.m8.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.8.m8.1b"><apply id="S2.p3.8.m8.1.1.cmml" xref="S2.p3.8.m8.1.1"><in id="S2.p3.8.m8.1.1.1.cmml" xref="S2.p3.8.m8.1.1.1"></in><ci id="S2.p3.8.m8.1.1.2.cmml" xref="S2.p3.8.m8.1.1.2">π¦</ci><ci id="S2.p3.8.m8.1.1.3.cmml" xref="S2.p3.8.m8.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.8.m8.1c">y\in S</annotation><annotation encoding="application/x-llamapun" id="S2.p3.8.m8.1d">italic_y β italic_S</annotation></semantics></math> such that <math alttext="xyx=x" class="ltx_Math" display="inline" id="S2.p3.9.m9.1"><semantics id="S2.p3.9.m9.1a"><mrow id="S2.p3.9.m9.1.1" xref="S2.p3.9.m9.1.1.cmml"><mrow id="S2.p3.9.m9.1.1.2" xref="S2.p3.9.m9.1.1.2.cmml"><mi id="S2.p3.9.m9.1.1.2.2" xref="S2.p3.9.m9.1.1.2.2.cmml">x</mi><mo id="S2.p3.9.m9.1.1.2.1" xref="S2.p3.9.m9.1.1.2.1.cmml">β’</mo><mi id="S2.p3.9.m9.1.1.2.3" xref="S2.p3.9.m9.1.1.2.3.cmml">y</mi><mo id="S2.p3.9.m9.1.1.2.1a" xref="S2.p3.9.m9.1.1.2.1.cmml">β’</mo><mi id="S2.p3.9.m9.1.1.2.4" xref="S2.p3.9.m9.1.1.2.4.cmml">x</mi></mrow><mo id="S2.p3.9.m9.1.1.1" xref="S2.p3.9.m9.1.1.1.cmml">=</mo><mi id="S2.p3.9.m9.1.1.3" xref="S2.p3.9.m9.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.9.m9.1b"><apply id="S2.p3.9.m9.1.1.cmml" xref="S2.p3.9.m9.1.1"><eq id="S2.p3.9.m9.1.1.1.cmml" xref="S2.p3.9.m9.1.1.1"></eq><apply id="S2.p3.9.m9.1.1.2.cmml" xref="S2.p3.9.m9.1.1.2"><times id="S2.p3.9.m9.1.1.2.1.cmml" xref="S2.p3.9.m9.1.1.2.1"></times><ci id="S2.p3.9.m9.1.1.2.2.cmml" xref="S2.p3.9.m9.1.1.2.2">π₯</ci><ci id="S2.p3.9.m9.1.1.2.3.cmml" xref="S2.p3.9.m9.1.1.2.3">π¦</ci><ci id="S2.p3.9.m9.1.1.2.4.cmml" xref="S2.p3.9.m9.1.1.2.4">π₯</ci></apply><ci id="S2.p3.9.m9.1.1.3.cmml" xref="S2.p3.9.m9.1.1.3">π₯</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.9.m9.1c">xyx=x</annotation><annotation encoding="application/x-llamapun" id="S2.p3.9.m9.1d">italic_x italic_y italic_x = italic_x</annotation></semantics></math>. Note that every inverse semigroup is regular. A regular semigroup <math alttext="S" class="ltx_Math" display="inline" id="S2.p3.10.m10.1"><semantics id="S2.p3.10.m10.1a"><mi id="S2.p3.10.m10.1.1" xref="S2.p3.10.m10.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p3.10.m10.1b"><ci id="S2.p3.10.m10.1.1.cmml" xref="S2.p3.10.m10.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.10.m10.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p3.10.m10.1d">italic_S</annotation></semantics></math> is inverse if an only if <math alttext="E(S)" class="ltx_Math" display="inline" id="S2.p3.11.m11.1"><semantics id="S2.p3.11.m11.1a"><mrow id="S2.p3.11.m11.1.2" xref="S2.p3.11.m11.1.2.cmml"><mi id="S2.p3.11.m11.1.2.2" xref="S2.p3.11.m11.1.2.2.cmml">E</mi><mo id="S2.p3.11.m11.1.2.1" xref="S2.p3.11.m11.1.2.1.cmml">β’</mo><mrow id="S2.p3.11.m11.1.2.3.2" xref="S2.p3.11.m11.1.2.cmml"><mo id="S2.p3.11.m11.1.2.3.2.1" stretchy="false" xref="S2.p3.11.m11.1.2.cmml">(</mo><mi id="S2.p3.11.m11.1.1" xref="S2.p3.11.m11.1.1.cmml">S</mi><mo id="S2.p3.11.m11.1.2.3.2.2" stretchy="false" xref="S2.p3.11.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.11.m11.1b"><apply id="S2.p3.11.m11.1.2.cmml" xref="S2.p3.11.m11.1.2"><times id="S2.p3.11.m11.1.2.1.cmml" xref="S2.p3.11.m11.1.2.1"></times><ci id="S2.p3.11.m11.1.2.2.cmml" xref="S2.p3.11.m11.1.2.2">πΈ</ci><ci id="S2.p3.11.m11.1.1.cmml" xref="S2.p3.11.m11.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.11.m11.1c">E(S)</annotation><annotation encoding="application/x-llamapun" id="S2.p3.11.m11.1d">italic_E ( italic_S )</annotation></semantics></math> is a semilattice (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib31" title="">31</a>, Theorem 5.1.1]</cite>). A filter <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S2.p3.12.m12.1"><semantics id="S2.p3.12.m12.1a"><mi class="ltx_font_mathcaligraphic" id="S2.p3.12.m12.1.1" xref="S2.p3.12.m12.1.1.cmml">β±</mi><annotation-xml encoding="MathML-Content" id="S2.p3.12.m12.1b"><ci id="S2.p3.12.m12.1.1.cmml" xref="S2.p3.12.m12.1.1">β±</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.12.m12.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.12.m12.1d">caligraphic_F</annotation></semantics></math> on a space <math alttext="X" class="ltx_Math" display="inline" id="S2.p3.13.m13.1"><semantics id="S2.p3.13.m13.1a"><mi id="S2.p3.13.m13.1.1" xref="S2.p3.13.m13.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.p3.13.m13.1b"><ci id="S2.p3.13.m13.1.1.cmml" xref="S2.p3.13.m13.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.13.m13.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.p3.13.m13.1d">italic_X</annotation></semantics></math> <span class="ltx_text ltx_font_italic" id="S2.p3.16.2">converges</span> to a point <math alttext="x\in X" class="ltx_Math" display="inline" id="S2.p3.14.m14.1"><semantics id="S2.p3.14.m14.1a"><mrow id="S2.p3.14.m14.1.1" xref="S2.p3.14.m14.1.1.cmml"><mi id="S2.p3.14.m14.1.1.2" xref="S2.p3.14.m14.1.1.2.cmml">x</mi><mo id="S2.p3.14.m14.1.1.1" xref="S2.p3.14.m14.1.1.1.cmml">β</mo><mi id="S2.p3.14.m14.1.1.3" xref="S2.p3.14.m14.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.14.m14.1b"><apply id="S2.p3.14.m14.1.1.cmml" xref="S2.p3.14.m14.1.1"><in id="S2.p3.14.m14.1.1.1.cmml" xref="S2.p3.14.m14.1.1.1"></in><ci id="S2.p3.14.m14.1.1.2.cmml" xref="S2.p3.14.m14.1.1.2">π₯</ci><ci id="S2.p3.14.m14.1.1.3.cmml" xref="S2.p3.14.m14.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.14.m14.1c">x\in X</annotation><annotation encoding="application/x-llamapun" id="S2.p3.14.m14.1d">italic_x β italic_X</annotation></semantics></math> if <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S2.p3.15.m15.1"><semantics id="S2.p3.15.m15.1a"><mi class="ltx_font_mathcaligraphic" id="S2.p3.15.m15.1.1" xref="S2.p3.15.m15.1.1.cmml">β±</mi><annotation-xml encoding="MathML-Content" id="S2.p3.15.m15.1b"><ci id="S2.p3.15.m15.1.1.cmml" xref="S2.p3.15.m15.1.1">β±</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.15.m15.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.15.m15.1d">caligraphic_F</annotation></semantics></math> contains all open neighborhoods of <math alttext="x" class="ltx_Math" display="inline" id="S2.p3.16.m16.1"><semantics id="S2.p3.16.m16.1a"><mi id="S2.p3.16.m16.1.1" xref="S2.p3.16.m16.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.p3.16.m16.1b"><ci id="S2.p3.16.m16.1.1.cmml" xref="S2.p3.16.m16.1.1">π₯</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.16.m16.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.p3.16.m16.1d">italic_x</annotation></semantics></math>. The following proposition is useful for detecting inverse semigroups among topological semigroups with a dense inverse subsemigroup.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S2.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem3.1.1.1">Proposition 2.3</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem3.p1"> <p class="ltx_p" id="S2.Thmtheorem3.p1.10"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem3.p1.10.10">Let <math alttext="S" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.1.1.m1.1"><semantics id="S2.Thmtheorem3.p1.1.1.m1.1a"><mi id="S2.Thmtheorem3.p1.1.1.m1.1.1" xref="S2.Thmtheorem3.p1.1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.1.1.m1.1b"><ci id="S2.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.1.1.m1.1d">italic_S</annotation></semantics></math> be a topological semigroup with a dense inverse subsemigroup <math alttext="X" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.2.2.m2.1"><semantics id="S2.Thmtheorem3.p1.2.2.m2.1a"><mi id="S2.Thmtheorem3.p1.2.2.m2.1.1" xref="S2.Thmtheorem3.p1.2.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.2.2.m2.1b"><ci id="S2.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem3.p1.2.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.2.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.2.2.m2.1d">italic_X</annotation></semantics></math> and for every <math alttext="y\in S\setminus X" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.3.3.m3.1"><semantics id="S2.Thmtheorem3.p1.3.3.m3.1a"><mrow id="S2.Thmtheorem3.p1.3.3.m3.1.1" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.cmml"><mi id="S2.Thmtheorem3.p1.3.3.m3.1.1.2" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.2.cmml">y</mi><mo id="S2.Thmtheorem3.p1.3.3.m3.1.1.1" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.1.cmml">β</mo><mrow id="S2.Thmtheorem3.p1.3.3.m3.1.1.3" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.3.cmml"><mi id="S2.Thmtheorem3.p1.3.3.m3.1.1.3.2" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.3.2.cmml">S</mi><mo id="S2.Thmtheorem3.p1.3.3.m3.1.1.3.1" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.3.1.cmml">β</mo><mi id="S2.Thmtheorem3.p1.3.3.m3.1.1.3.3" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.3.3.m3.1b"><apply id="S2.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1"><in id="S2.Thmtheorem3.p1.3.3.m3.1.1.1.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.1"></in><ci id="S2.Thmtheorem3.p1.3.3.m3.1.1.2.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.2">π¦</ci><apply id="S2.Thmtheorem3.p1.3.3.m3.1.1.3.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.3"><setdiff id="S2.Thmtheorem3.p1.3.3.m3.1.1.3.1.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.3.1"></setdiff><ci id="S2.Thmtheorem3.p1.3.3.m3.1.1.3.2.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.3.2">π</ci><ci id="S2.Thmtheorem3.p1.3.3.m3.1.1.3.3.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.3.3.m3.1c">y\in S\setminus X</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.3.3.m3.1d">italic_y β italic_S β italic_X</annotation></semantics></math> there exists an ultrafilter <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.4.4.m4.1"><semantics id="S2.Thmtheorem3.p1.4.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem3.p1.4.4.m4.1.1" xref="S2.Thmtheorem3.p1.4.4.m4.1.1.cmml">β±</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.4.4.m4.1b"><ci id="S2.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem3.p1.4.4.m4.1.1">β±</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.4.4.m4.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.4.4.m4.1d">caligraphic_F</annotation></semantics></math> convergent to <math alttext="y" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.5.5.m5.1"><semantics id="S2.Thmtheorem3.p1.5.5.m5.1a"><mi id="S2.Thmtheorem3.p1.5.5.m5.1.1" xref="S2.Thmtheorem3.p1.5.5.m5.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.5.5.m5.1b"><ci id="S2.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.1.1">π¦</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.5.5.m5.1c">y</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.5.5.m5.1d">italic_y</annotation></semantics></math> such that <math alttext="X\in\mathcal{F}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.6.6.m6.1"><semantics id="S2.Thmtheorem3.p1.6.6.m6.1a"><mrow id="S2.Thmtheorem3.p1.6.6.m6.1.1" xref="S2.Thmtheorem3.p1.6.6.m6.1.1.cmml"><mi id="S2.Thmtheorem3.p1.6.6.m6.1.1.2" xref="S2.Thmtheorem3.p1.6.6.m6.1.1.2.cmml">X</mi><mo id="S2.Thmtheorem3.p1.6.6.m6.1.1.1" xref="S2.Thmtheorem3.p1.6.6.m6.1.1.1.cmml">β</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem3.p1.6.6.m6.1.1.3" xref="S2.Thmtheorem3.p1.6.6.m6.1.1.3.cmml">β±</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.6.6.m6.1b"><apply id="S2.Thmtheorem3.p1.6.6.m6.1.1.cmml" xref="S2.Thmtheorem3.p1.6.6.m6.1.1"><in id="S2.Thmtheorem3.p1.6.6.m6.1.1.1.cmml" xref="S2.Thmtheorem3.p1.6.6.m6.1.1.1"></in><ci id="S2.Thmtheorem3.p1.6.6.m6.1.1.2.cmml" xref="S2.Thmtheorem3.p1.6.6.m6.1.1.2">π</ci><ci id="S2.Thmtheorem3.p1.6.6.m6.1.1.3.cmml" xref="S2.Thmtheorem3.p1.6.6.m6.1.1.3">β±</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.6.6.m6.1c">X\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.6.6.m6.1d">italic_X β caligraphic_F</annotation></semantics></math> and the ultrafilter <math alttext="\mathcal{F}^{-1}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.7.7.m7.1"><semantics id="S2.Thmtheorem3.p1.7.7.m7.1a"><msup id="S2.Thmtheorem3.p1.7.7.m7.1.1" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem3.p1.7.7.m7.1.1.2" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.2.cmml">β±</mi><mrow id="S2.Thmtheorem3.p1.7.7.m7.1.1.3" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.3.cmml"><mo id="S2.Thmtheorem3.p1.7.7.m7.1.1.3a" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.3.cmml">β</mo><mn id="S2.Thmtheorem3.p1.7.7.m7.1.1.3.2" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.3.2.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.7.7.m7.1b"><apply id="S2.Thmtheorem3.p1.7.7.m7.1.1.cmml" xref="S2.Thmtheorem3.p1.7.7.m7.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.7.7.m7.1.1.1.cmml" xref="S2.Thmtheorem3.p1.7.7.m7.1.1">superscript</csymbol><ci id="S2.Thmtheorem3.p1.7.7.m7.1.1.2.cmml" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.2">β±</ci><apply id="S2.Thmtheorem3.p1.7.7.m7.1.1.3.cmml" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.3"><minus id="S2.Thmtheorem3.p1.7.7.m7.1.1.3.1.cmml" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.3"></minus><cn id="S2.Thmtheorem3.p1.7.7.m7.1.1.3.2.cmml" type="integer" xref="S2.Thmtheorem3.p1.7.7.m7.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.7.7.m7.1c">\mathcal{F}^{-1}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.7.7.m7.1d">caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> generated by the family <math alttext="\{F^{-1}:X\supseteq F\in\mathcal{F}\}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.8.8.m8.2"><semantics id="S2.Thmtheorem3.p1.8.8.m8.2a"><mrow id="S2.Thmtheorem3.p1.8.8.m8.2.2.2" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.3.cmml"><mo id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.3" stretchy="false" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.3.1.cmml">{</mo><msup id="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1" xref="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.cmml"><mi id="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.2" xref="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.2.cmml">F</mi><mrow id="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.3" xref="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.3.cmml"><mo id="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.3a" xref="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.3.cmml">β</mo><mn id="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.3.2" xref="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.3.2.cmml">1</mn></mrow></msup><mo id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.3.1.cmml">:</mo><mrow id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.cmml"><mi id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.2" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.2.cmml">X</mi><mo id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.3" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.3.cmml">β</mo><mi id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.4" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.4.cmml">F</mi><mo id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.5" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.5.cmml">β</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.6" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.6.cmml">β±</mi></mrow><mo id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.5" stretchy="false" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.8.8.m8.2b"><apply id="S2.Thmtheorem3.p1.8.8.m8.2.2.3.cmml" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.2"><csymbol cd="latexml" id="S2.Thmtheorem3.p1.8.8.m8.2.2.3.1.cmml" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.2.3">conditional-set</csymbol><apply id="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1">superscript</csymbol><ci id="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.2.cmml" xref="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.2">πΉ</ci><apply id="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.3.cmml" xref="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.3"><minus id="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.3.1.cmml" xref="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.3"></minus><cn id="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.3.2.cmml" type="integer" xref="S2.Thmtheorem3.p1.8.8.m8.1.1.1.1.3.2">1</cn></apply></apply><apply id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.cmml" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2"><and id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2a.cmml" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2"></and><apply id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2b.cmml" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2"><csymbol cd="latexml" id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.3.cmml" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.3">superset-of-or-equals</csymbol><ci id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.2.cmml" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.2">π</ci><ci id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.4.cmml" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.4">πΉ</ci></apply><apply id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2c.cmml" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2"><in id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.5.cmml" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.5"></in><share href="https://arxiv.org/html/2503.13666v1#S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.4.cmml" id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2d.cmml" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2"></share><ci id="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.6.cmml" xref="S2.Thmtheorem3.p1.8.8.m8.2.2.2.2.6">β±</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.8.8.m8.2c">\{F^{-1}:X\supseteq F\in\mathcal{F}\}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.8.8.m8.2d">{ italic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT : italic_X β italic_F β caligraphic_F }</annotation></semantics></math> converges to some element of <math alttext="S" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.9.9.m9.1"><semantics id="S2.Thmtheorem3.p1.9.9.m9.1a"><mi id="S2.Thmtheorem3.p1.9.9.m9.1.1" xref="S2.Thmtheorem3.p1.9.9.m9.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.9.9.m9.1b"><ci id="S2.Thmtheorem3.p1.9.9.m9.1.1.cmml" xref="S2.Thmtheorem3.p1.9.9.m9.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.9.9.m9.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.9.9.m9.1d">italic_S</annotation></semantics></math>. Then <math alttext="S" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.10.10.m10.1"><semantics id="S2.Thmtheorem3.p1.10.10.m10.1a"><mi id="S2.Thmtheorem3.p1.10.10.m10.1.1" xref="S2.Thmtheorem3.p1.10.10.m10.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.10.10.m10.1b"><ci id="S2.Thmtheorem3.p1.10.10.m10.1.1.cmml" xref="S2.Thmtheorem3.p1.10.10.m10.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.10.10.m10.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.10.10.m10.1d">italic_S</annotation></semantics></math> is an inverse semigroup.</span></p> </div> </div> <div class="ltx_proof" id="S2.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.2.p1"> <p class="ltx_p" id="S2.2.p1.30">First let us show that the semigroup <math alttext="S" class="ltx_Math" display="inline" id="S2.2.p1.1.m1.1"><semantics id="S2.2.p1.1.m1.1a"><mi id="S2.2.p1.1.m1.1.1" xref="S2.2.p1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.2.p1.1.m1.1b"><ci id="S2.2.p1.1.m1.1.1.cmml" xref="S2.2.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.1.m1.1d">italic_S</annotation></semantics></math> is regular. Fix any <math alttext="y\in S\setminus X" class="ltx_Math" display="inline" id="S2.2.p1.2.m2.1"><semantics id="S2.2.p1.2.m2.1a"><mrow id="S2.2.p1.2.m2.1.1" xref="S2.2.p1.2.m2.1.1.cmml"><mi id="S2.2.p1.2.m2.1.1.2" xref="S2.2.p1.2.m2.1.1.2.cmml">y</mi><mo id="S2.2.p1.2.m2.1.1.1" xref="S2.2.p1.2.m2.1.1.1.cmml">β</mo><mrow id="S2.2.p1.2.m2.1.1.3" xref="S2.2.p1.2.m2.1.1.3.cmml"><mi id="S2.2.p1.2.m2.1.1.3.2" xref="S2.2.p1.2.m2.1.1.3.2.cmml">S</mi><mo id="S2.2.p1.2.m2.1.1.3.1" xref="S2.2.p1.2.m2.1.1.3.1.cmml">β</mo><mi id="S2.2.p1.2.m2.1.1.3.3" xref="S2.2.p1.2.m2.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p1.2.m2.1b"><apply id="S2.2.p1.2.m2.1.1.cmml" xref="S2.2.p1.2.m2.1.1"><in id="S2.2.p1.2.m2.1.1.1.cmml" xref="S2.2.p1.2.m2.1.1.1"></in><ci id="S2.2.p1.2.m2.1.1.2.cmml" xref="S2.2.p1.2.m2.1.1.2">π¦</ci><apply id="S2.2.p1.2.m2.1.1.3.cmml" xref="S2.2.p1.2.m2.1.1.3"><setdiff id="S2.2.p1.2.m2.1.1.3.1.cmml" xref="S2.2.p1.2.m2.1.1.3.1"></setdiff><ci id="S2.2.p1.2.m2.1.1.3.2.cmml" xref="S2.2.p1.2.m2.1.1.3.2">π</ci><ci id="S2.2.p1.2.m2.1.1.3.3.cmml" xref="S2.2.p1.2.m2.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.2.m2.1c">y\in S\setminus X</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.2.m2.1d">italic_y β italic_S β italic_X</annotation></semantics></math>. By the assumption, there exists an ultrafilter <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S2.2.p1.3.m3.1"><semantics id="S2.2.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S2.2.p1.3.m3.1.1" xref="S2.2.p1.3.m3.1.1.cmml">β±</mi><annotation-xml encoding="MathML-Content" id="S2.2.p1.3.m3.1b"><ci id="S2.2.p1.3.m3.1.1.cmml" xref="S2.2.p1.3.m3.1.1">β±</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.3.m3.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.3.m3.1d">caligraphic_F</annotation></semantics></math> on <math alttext="S" class="ltx_Math" display="inline" id="S2.2.p1.4.m4.1"><semantics id="S2.2.p1.4.m4.1a"><mi id="S2.2.p1.4.m4.1.1" xref="S2.2.p1.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.2.p1.4.m4.1b"><ci id="S2.2.p1.4.m4.1.1.cmml" xref="S2.2.p1.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.4.m4.1d">italic_S</annotation></semantics></math> convergent to <math alttext="y" class="ltx_Math" display="inline" id="S2.2.p1.5.m5.1"><semantics id="S2.2.p1.5.m5.1a"><mi id="S2.2.p1.5.m5.1.1" xref="S2.2.p1.5.m5.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S2.2.p1.5.m5.1b"><ci id="S2.2.p1.5.m5.1.1.cmml" xref="S2.2.p1.5.m5.1.1">π¦</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.5.m5.1c">y</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.5.m5.1d">italic_y</annotation></semantics></math> such that <math alttext="X\in\mathcal{F}" class="ltx_Math" display="inline" id="S2.2.p1.6.m6.1"><semantics id="S2.2.p1.6.m6.1a"><mrow id="S2.2.p1.6.m6.1.1" xref="S2.2.p1.6.m6.1.1.cmml"><mi id="S2.2.p1.6.m6.1.1.2" xref="S2.2.p1.6.m6.1.1.2.cmml">X</mi><mo id="S2.2.p1.6.m6.1.1.1" xref="S2.2.p1.6.m6.1.1.1.cmml">β</mo><mi class="ltx_font_mathcaligraphic" id="S2.2.p1.6.m6.1.1.3" xref="S2.2.p1.6.m6.1.1.3.cmml">β±</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p1.6.m6.1b"><apply id="S2.2.p1.6.m6.1.1.cmml" xref="S2.2.p1.6.m6.1.1"><in id="S2.2.p1.6.m6.1.1.1.cmml" xref="S2.2.p1.6.m6.1.1.1"></in><ci id="S2.2.p1.6.m6.1.1.2.cmml" xref="S2.2.p1.6.m6.1.1.2">π</ci><ci id="S2.2.p1.6.m6.1.1.3.cmml" xref="S2.2.p1.6.m6.1.1.3">β±</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.6.m6.1c">X\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.6.m6.1d">italic_X β caligraphic_F</annotation></semantics></math> and the ultrafilter <math alttext="\mathcal{F}^{-1}" class="ltx_Math" display="inline" id="S2.2.p1.7.m7.1"><semantics id="S2.2.p1.7.m7.1a"><msup id="S2.2.p1.7.m7.1.1" xref="S2.2.p1.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.2.p1.7.m7.1.1.2" xref="S2.2.p1.7.m7.1.1.2.cmml">β±</mi><mrow id="S2.2.p1.7.m7.1.1.3" xref="S2.2.p1.7.m7.1.1.3.cmml"><mo id="S2.2.p1.7.m7.1.1.3a" xref="S2.2.p1.7.m7.1.1.3.cmml">β</mo><mn id="S2.2.p1.7.m7.1.1.3.2" xref="S2.2.p1.7.m7.1.1.3.2.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.2.p1.7.m7.1b"><apply id="S2.2.p1.7.m7.1.1.cmml" xref="S2.2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.2.p1.7.m7.1.1.1.cmml" xref="S2.2.p1.7.m7.1.1">superscript</csymbol><ci id="S2.2.p1.7.m7.1.1.2.cmml" xref="S2.2.p1.7.m7.1.1.2">β±</ci><apply id="S2.2.p1.7.m7.1.1.3.cmml" xref="S2.2.p1.7.m7.1.1.3"><minus id="S2.2.p1.7.m7.1.1.3.1.cmml" xref="S2.2.p1.7.m7.1.1.3"></minus><cn id="S2.2.p1.7.m7.1.1.3.2.cmml" type="integer" xref="S2.2.p1.7.m7.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.7.m7.1c">\mathcal{F}^{-1}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.7.m7.1d">caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> converges to some element <math alttext="z\in S" class="ltx_Math" display="inline" id="S2.2.p1.8.m8.1"><semantics id="S2.2.p1.8.m8.1a"><mrow id="S2.2.p1.8.m8.1.1" xref="S2.2.p1.8.m8.1.1.cmml"><mi id="S2.2.p1.8.m8.1.1.2" xref="S2.2.p1.8.m8.1.1.2.cmml">z</mi><mo id="S2.2.p1.8.m8.1.1.1" xref="S2.2.p1.8.m8.1.1.1.cmml">β</mo><mi id="S2.2.p1.8.m8.1.1.3" xref="S2.2.p1.8.m8.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p1.8.m8.1b"><apply id="S2.2.p1.8.m8.1.1.cmml" xref="S2.2.p1.8.m8.1.1"><in id="S2.2.p1.8.m8.1.1.1.cmml" xref="S2.2.p1.8.m8.1.1.1"></in><ci id="S2.2.p1.8.m8.1.1.2.cmml" xref="S2.2.p1.8.m8.1.1.2">π§</ci><ci id="S2.2.p1.8.m8.1.1.3.cmml" xref="S2.2.p1.8.m8.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.8.m8.1c">z\in S</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.8.m8.1d">italic_z β italic_S</annotation></semantics></math>. Consider an open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S2.2.p1.9.m9.1"><semantics id="S2.2.p1.9.m9.1a"><mi id="S2.2.p1.9.m9.1.1" xref="S2.2.p1.9.m9.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S2.2.p1.9.m9.1b"><ci id="S2.2.p1.9.m9.1.1.cmml" xref="S2.2.p1.9.m9.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.9.m9.1c">U</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.9.m9.1d">italic_U</annotation></semantics></math> of the point <math alttext="w=yzy" class="ltx_Math" display="inline" id="S2.2.p1.10.m10.1"><semantics id="S2.2.p1.10.m10.1a"><mrow id="S2.2.p1.10.m10.1.1" xref="S2.2.p1.10.m10.1.1.cmml"><mi id="S2.2.p1.10.m10.1.1.2" xref="S2.2.p1.10.m10.1.1.2.cmml">w</mi><mo id="S2.2.p1.10.m10.1.1.1" xref="S2.2.p1.10.m10.1.1.1.cmml">=</mo><mrow id="S2.2.p1.10.m10.1.1.3" xref="S2.2.p1.10.m10.1.1.3.cmml"><mi id="S2.2.p1.10.m10.1.1.3.2" xref="S2.2.p1.10.m10.1.1.3.2.cmml">y</mi><mo id="S2.2.p1.10.m10.1.1.3.1" xref="S2.2.p1.10.m10.1.1.3.1.cmml">β’</mo><mi id="S2.2.p1.10.m10.1.1.3.3" xref="S2.2.p1.10.m10.1.1.3.3.cmml">z</mi><mo id="S2.2.p1.10.m10.1.1.3.1a" xref="S2.2.p1.10.m10.1.1.3.1.cmml">β’</mo><mi id="S2.2.p1.10.m10.1.1.3.4" xref="S2.2.p1.10.m10.1.1.3.4.cmml">y</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p1.10.m10.1b"><apply id="S2.2.p1.10.m10.1.1.cmml" xref="S2.2.p1.10.m10.1.1"><eq id="S2.2.p1.10.m10.1.1.1.cmml" xref="S2.2.p1.10.m10.1.1.1"></eq><ci id="S2.2.p1.10.m10.1.1.2.cmml" xref="S2.2.p1.10.m10.1.1.2">π€</ci><apply id="S2.2.p1.10.m10.1.1.3.cmml" xref="S2.2.p1.10.m10.1.1.3"><times id="S2.2.p1.10.m10.1.1.3.1.cmml" xref="S2.2.p1.10.m10.1.1.3.1"></times><ci id="S2.2.p1.10.m10.1.1.3.2.cmml" xref="S2.2.p1.10.m10.1.1.3.2">π¦</ci><ci id="S2.2.p1.10.m10.1.1.3.3.cmml" xref="S2.2.p1.10.m10.1.1.3.3">π§</ci><ci id="S2.2.p1.10.m10.1.1.3.4.cmml" xref="S2.2.p1.10.m10.1.1.3.4">π¦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.10.m10.1c">w=yzy</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.10.m10.1d">italic_w = italic_y italic_z italic_y</annotation></semantics></math>. By the continuity of the semigroup operation in <math alttext="S" class="ltx_Math" display="inline" id="S2.2.p1.11.m11.1"><semantics id="S2.2.p1.11.m11.1a"><mi id="S2.2.p1.11.m11.1.1" xref="S2.2.p1.11.m11.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.2.p1.11.m11.1b"><ci id="S2.2.p1.11.m11.1.1.cmml" xref="S2.2.p1.11.m11.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.11.m11.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.11.m11.1d">italic_S</annotation></semantics></math>, there exist open neighborhoods <math alttext="V(y)" class="ltx_Math" display="inline" id="S2.2.p1.12.m12.1"><semantics id="S2.2.p1.12.m12.1a"><mrow id="S2.2.p1.12.m12.1.2" xref="S2.2.p1.12.m12.1.2.cmml"><mi id="S2.2.p1.12.m12.1.2.2" xref="S2.2.p1.12.m12.1.2.2.cmml">V</mi><mo id="S2.2.p1.12.m12.1.2.1" xref="S2.2.p1.12.m12.1.2.1.cmml">β’</mo><mrow id="S2.2.p1.12.m12.1.2.3.2" xref="S2.2.p1.12.m12.1.2.cmml"><mo id="S2.2.p1.12.m12.1.2.3.2.1" stretchy="false" xref="S2.2.p1.12.m12.1.2.cmml">(</mo><mi id="S2.2.p1.12.m12.1.1" xref="S2.2.p1.12.m12.1.1.cmml">y</mi><mo id="S2.2.p1.12.m12.1.2.3.2.2" stretchy="false" xref="S2.2.p1.12.m12.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p1.12.m12.1b"><apply id="S2.2.p1.12.m12.1.2.cmml" xref="S2.2.p1.12.m12.1.2"><times id="S2.2.p1.12.m12.1.2.1.cmml" xref="S2.2.p1.12.m12.1.2.1"></times><ci id="S2.2.p1.12.m12.1.2.2.cmml" xref="S2.2.p1.12.m12.1.2.2">π</ci><ci id="S2.2.p1.12.m12.1.1.cmml" xref="S2.2.p1.12.m12.1.1">π¦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.12.m12.1c">V(y)</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.12.m12.1d">italic_V ( italic_y )</annotation></semantics></math> and <math alttext="V(z)" class="ltx_Math" display="inline" id="S2.2.p1.13.m13.1"><semantics id="S2.2.p1.13.m13.1a"><mrow id="S2.2.p1.13.m13.1.2" xref="S2.2.p1.13.m13.1.2.cmml"><mi id="S2.2.p1.13.m13.1.2.2" xref="S2.2.p1.13.m13.1.2.2.cmml">V</mi><mo id="S2.2.p1.13.m13.1.2.1" xref="S2.2.p1.13.m13.1.2.1.cmml">β’</mo><mrow id="S2.2.p1.13.m13.1.2.3.2" xref="S2.2.p1.13.m13.1.2.cmml"><mo id="S2.2.p1.13.m13.1.2.3.2.1" stretchy="false" xref="S2.2.p1.13.m13.1.2.cmml">(</mo><mi id="S2.2.p1.13.m13.1.1" xref="S2.2.p1.13.m13.1.1.cmml">z</mi><mo id="S2.2.p1.13.m13.1.2.3.2.2" stretchy="false" xref="S2.2.p1.13.m13.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p1.13.m13.1b"><apply id="S2.2.p1.13.m13.1.2.cmml" xref="S2.2.p1.13.m13.1.2"><times id="S2.2.p1.13.m13.1.2.1.cmml" xref="S2.2.p1.13.m13.1.2.1"></times><ci id="S2.2.p1.13.m13.1.2.2.cmml" xref="S2.2.p1.13.m13.1.2.2">π</ci><ci id="S2.2.p1.13.m13.1.1.cmml" xref="S2.2.p1.13.m13.1.1">π§</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.13.m13.1c">V(z)</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.13.m13.1d">italic_V ( italic_z )</annotation></semantics></math> of <math alttext="y" class="ltx_Math" display="inline" id="S2.2.p1.14.m14.1"><semantics id="S2.2.p1.14.m14.1a"><mi id="S2.2.p1.14.m14.1.1" xref="S2.2.p1.14.m14.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S2.2.p1.14.m14.1b"><ci id="S2.2.p1.14.m14.1.1.cmml" xref="S2.2.p1.14.m14.1.1">π¦</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.14.m14.1c">y</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.14.m14.1d">italic_y</annotation></semantics></math> and <math alttext="z" class="ltx_Math" display="inline" id="S2.2.p1.15.m15.1"><semantics id="S2.2.p1.15.m15.1a"><mi id="S2.2.p1.15.m15.1.1" xref="S2.2.p1.15.m15.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S2.2.p1.15.m15.1b"><ci id="S2.2.p1.15.m15.1.1.cmml" xref="S2.2.p1.15.m15.1.1">π§</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.15.m15.1c">z</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.15.m15.1d">italic_z</annotation></semantics></math>, respectively, such that <math alttext="V(y)V(z)V(y)\subseteq U" class="ltx_Math" display="inline" id="S2.2.p1.16.m16.3"><semantics id="S2.2.p1.16.m16.3a"><mrow id="S2.2.p1.16.m16.3.4" xref="S2.2.p1.16.m16.3.4.cmml"><mrow id="S2.2.p1.16.m16.3.4.2" xref="S2.2.p1.16.m16.3.4.2.cmml"><mi id="S2.2.p1.16.m16.3.4.2.2" xref="S2.2.p1.16.m16.3.4.2.2.cmml">V</mi><mo id="S2.2.p1.16.m16.3.4.2.1" xref="S2.2.p1.16.m16.3.4.2.1.cmml">β’</mo><mrow id="S2.2.p1.16.m16.3.4.2.3.2" xref="S2.2.p1.16.m16.3.4.2.cmml"><mo id="S2.2.p1.16.m16.3.4.2.3.2.1" stretchy="false" xref="S2.2.p1.16.m16.3.4.2.cmml">(</mo><mi id="S2.2.p1.16.m16.1.1" xref="S2.2.p1.16.m16.1.1.cmml">y</mi><mo id="S2.2.p1.16.m16.3.4.2.3.2.2" stretchy="false" xref="S2.2.p1.16.m16.3.4.2.cmml">)</mo></mrow><mo id="S2.2.p1.16.m16.3.4.2.1a" xref="S2.2.p1.16.m16.3.4.2.1.cmml">β’</mo><mi id="S2.2.p1.16.m16.3.4.2.4" xref="S2.2.p1.16.m16.3.4.2.4.cmml">V</mi><mo id="S2.2.p1.16.m16.3.4.2.1b" xref="S2.2.p1.16.m16.3.4.2.1.cmml">β’</mo><mrow id="S2.2.p1.16.m16.3.4.2.5.2" xref="S2.2.p1.16.m16.3.4.2.cmml"><mo id="S2.2.p1.16.m16.3.4.2.5.2.1" stretchy="false" xref="S2.2.p1.16.m16.3.4.2.cmml">(</mo><mi id="S2.2.p1.16.m16.2.2" xref="S2.2.p1.16.m16.2.2.cmml">z</mi><mo id="S2.2.p1.16.m16.3.4.2.5.2.2" stretchy="false" xref="S2.2.p1.16.m16.3.4.2.cmml">)</mo></mrow><mo id="S2.2.p1.16.m16.3.4.2.1c" xref="S2.2.p1.16.m16.3.4.2.1.cmml">β’</mo><mi id="S2.2.p1.16.m16.3.4.2.6" xref="S2.2.p1.16.m16.3.4.2.6.cmml">V</mi><mo id="S2.2.p1.16.m16.3.4.2.1d" xref="S2.2.p1.16.m16.3.4.2.1.cmml">β’</mo><mrow id="S2.2.p1.16.m16.3.4.2.7.2" xref="S2.2.p1.16.m16.3.4.2.cmml"><mo id="S2.2.p1.16.m16.3.4.2.7.2.1" stretchy="false" xref="S2.2.p1.16.m16.3.4.2.cmml">(</mo><mi id="S2.2.p1.16.m16.3.3" xref="S2.2.p1.16.m16.3.3.cmml">y</mi><mo id="S2.2.p1.16.m16.3.4.2.7.2.2" stretchy="false" xref="S2.2.p1.16.m16.3.4.2.cmml">)</mo></mrow></mrow><mo id="S2.2.p1.16.m16.3.4.1" xref="S2.2.p1.16.m16.3.4.1.cmml">β</mo><mi id="S2.2.p1.16.m16.3.4.3" xref="S2.2.p1.16.m16.3.4.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p1.16.m16.3b"><apply id="S2.2.p1.16.m16.3.4.cmml" xref="S2.2.p1.16.m16.3.4"><subset id="S2.2.p1.16.m16.3.4.1.cmml" xref="S2.2.p1.16.m16.3.4.1"></subset><apply id="S2.2.p1.16.m16.3.4.2.cmml" xref="S2.2.p1.16.m16.3.4.2"><times id="S2.2.p1.16.m16.3.4.2.1.cmml" xref="S2.2.p1.16.m16.3.4.2.1"></times><ci id="S2.2.p1.16.m16.3.4.2.2.cmml" xref="S2.2.p1.16.m16.3.4.2.2">π</ci><ci id="S2.2.p1.16.m16.1.1.cmml" xref="S2.2.p1.16.m16.1.1">π¦</ci><ci id="S2.2.p1.16.m16.3.4.2.4.cmml" xref="S2.2.p1.16.m16.3.4.2.4">π</ci><ci id="S2.2.p1.16.m16.2.2.cmml" xref="S2.2.p1.16.m16.2.2">π§</ci><ci id="S2.2.p1.16.m16.3.4.2.6.cmml" xref="S2.2.p1.16.m16.3.4.2.6">π</ci><ci id="S2.2.p1.16.m16.3.3.cmml" xref="S2.2.p1.16.m16.3.3">π¦</ci></apply><ci id="S2.2.p1.16.m16.3.4.3.cmml" xref="S2.2.p1.16.m16.3.4.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.16.m16.3c">V(y)V(z)V(y)\subseteq U</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.16.m16.3d">italic_V ( italic_y ) italic_V ( italic_z ) italic_V ( italic_y ) β italic_U</annotation></semantics></math>. Then there exist <math alttext="X\supseteq F_{1}\in\mathcal{F}" class="ltx_Math" display="inline" id="S2.2.p1.17.m17.1"><semantics id="S2.2.p1.17.m17.1a"><mrow id="S2.2.p1.17.m17.1.1" xref="S2.2.p1.17.m17.1.1.cmml"><mi id="S2.2.p1.17.m17.1.1.2" xref="S2.2.p1.17.m17.1.1.2.cmml">X</mi><mo id="S2.2.p1.17.m17.1.1.3" xref="S2.2.p1.17.m17.1.1.3.cmml">β</mo><msub id="S2.2.p1.17.m17.1.1.4" xref="S2.2.p1.17.m17.1.1.4.cmml"><mi id="S2.2.p1.17.m17.1.1.4.2" xref="S2.2.p1.17.m17.1.1.4.2.cmml">F</mi><mn id="S2.2.p1.17.m17.1.1.4.3" xref="S2.2.p1.17.m17.1.1.4.3.cmml">1</mn></msub><mo id="S2.2.p1.17.m17.1.1.5" xref="S2.2.p1.17.m17.1.1.5.cmml">β</mo><mi class="ltx_font_mathcaligraphic" id="S2.2.p1.17.m17.1.1.6" xref="S2.2.p1.17.m17.1.1.6.cmml">β±</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p1.17.m17.1b"><apply id="S2.2.p1.17.m17.1.1.cmml" xref="S2.2.p1.17.m17.1.1"><and id="S2.2.p1.17.m17.1.1a.cmml" xref="S2.2.p1.17.m17.1.1"></and><apply id="S2.2.p1.17.m17.1.1b.cmml" xref="S2.2.p1.17.m17.1.1"><csymbol cd="latexml" id="S2.2.p1.17.m17.1.1.3.cmml" xref="S2.2.p1.17.m17.1.1.3">superset-of-or-equals</csymbol><ci id="S2.2.p1.17.m17.1.1.2.cmml" xref="S2.2.p1.17.m17.1.1.2">π</ci><apply id="S2.2.p1.17.m17.1.1.4.cmml" xref="S2.2.p1.17.m17.1.1.4"><csymbol cd="ambiguous" id="S2.2.p1.17.m17.1.1.4.1.cmml" xref="S2.2.p1.17.m17.1.1.4">subscript</csymbol><ci id="S2.2.p1.17.m17.1.1.4.2.cmml" xref="S2.2.p1.17.m17.1.1.4.2">πΉ</ci><cn id="S2.2.p1.17.m17.1.1.4.3.cmml" type="integer" xref="S2.2.p1.17.m17.1.1.4.3">1</cn></apply></apply><apply id="S2.2.p1.17.m17.1.1c.cmml" xref="S2.2.p1.17.m17.1.1"><in id="S2.2.p1.17.m17.1.1.5.cmml" xref="S2.2.p1.17.m17.1.1.5"></in><share href="https://arxiv.org/html/2503.13666v1#S2.2.p1.17.m17.1.1.4.cmml" id="S2.2.p1.17.m17.1.1d.cmml" xref="S2.2.p1.17.m17.1.1"></share><ci id="S2.2.p1.17.m17.1.1.6.cmml" xref="S2.2.p1.17.m17.1.1.6">β±</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.17.m17.1c">X\supseteq F_{1}\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.17.m17.1d">italic_X β italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT β caligraphic_F</annotation></semantics></math> and <math alttext="X\supseteq F_{2}\in\mathcal{F}" class="ltx_Math" display="inline" id="S2.2.p1.18.m18.1"><semantics id="S2.2.p1.18.m18.1a"><mrow id="S2.2.p1.18.m18.1.1" xref="S2.2.p1.18.m18.1.1.cmml"><mi id="S2.2.p1.18.m18.1.1.2" xref="S2.2.p1.18.m18.1.1.2.cmml">X</mi><mo id="S2.2.p1.18.m18.1.1.3" xref="S2.2.p1.18.m18.1.1.3.cmml">β</mo><msub id="S2.2.p1.18.m18.1.1.4" xref="S2.2.p1.18.m18.1.1.4.cmml"><mi id="S2.2.p1.18.m18.1.1.4.2" xref="S2.2.p1.18.m18.1.1.4.2.cmml">F</mi><mn id="S2.2.p1.18.m18.1.1.4.3" xref="S2.2.p1.18.m18.1.1.4.3.cmml">2</mn></msub><mo id="S2.2.p1.18.m18.1.1.5" xref="S2.2.p1.18.m18.1.1.5.cmml">β</mo><mi class="ltx_font_mathcaligraphic" id="S2.2.p1.18.m18.1.1.6" xref="S2.2.p1.18.m18.1.1.6.cmml">β±</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p1.18.m18.1b"><apply id="S2.2.p1.18.m18.1.1.cmml" xref="S2.2.p1.18.m18.1.1"><and id="S2.2.p1.18.m18.1.1a.cmml" xref="S2.2.p1.18.m18.1.1"></and><apply id="S2.2.p1.18.m18.1.1b.cmml" xref="S2.2.p1.18.m18.1.1"><csymbol cd="latexml" id="S2.2.p1.18.m18.1.1.3.cmml" xref="S2.2.p1.18.m18.1.1.3">superset-of-or-equals</csymbol><ci id="S2.2.p1.18.m18.1.1.2.cmml" xref="S2.2.p1.18.m18.1.1.2">π</ci><apply id="S2.2.p1.18.m18.1.1.4.cmml" xref="S2.2.p1.18.m18.1.1.4"><csymbol cd="ambiguous" id="S2.2.p1.18.m18.1.1.4.1.cmml" xref="S2.2.p1.18.m18.1.1.4">subscript</csymbol><ci id="S2.2.p1.18.m18.1.1.4.2.cmml" xref="S2.2.p1.18.m18.1.1.4.2">πΉ</ci><cn id="S2.2.p1.18.m18.1.1.4.3.cmml" type="integer" xref="S2.2.p1.18.m18.1.1.4.3">2</cn></apply></apply><apply id="S2.2.p1.18.m18.1.1c.cmml" xref="S2.2.p1.18.m18.1.1"><in id="S2.2.p1.18.m18.1.1.5.cmml" xref="S2.2.p1.18.m18.1.1.5"></in><share href="https://arxiv.org/html/2503.13666v1#S2.2.p1.18.m18.1.1.4.cmml" id="S2.2.p1.18.m18.1.1d.cmml" xref="S2.2.p1.18.m18.1.1"></share><ci id="S2.2.p1.18.m18.1.1.6.cmml" xref="S2.2.p1.18.m18.1.1.6">β±</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.18.m18.1c">X\supseteq F_{2}\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.18.m18.1d">italic_X β italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT β caligraphic_F</annotation></semantics></math> such that <math alttext="F_{1}\subset V(y)" class="ltx_Math" display="inline" id="S2.2.p1.19.m19.1"><semantics id="S2.2.p1.19.m19.1a"><mrow id="S2.2.p1.19.m19.1.2" xref="S2.2.p1.19.m19.1.2.cmml"><msub id="S2.2.p1.19.m19.1.2.2" xref="S2.2.p1.19.m19.1.2.2.cmml"><mi id="S2.2.p1.19.m19.1.2.2.2" xref="S2.2.p1.19.m19.1.2.2.2.cmml">F</mi><mn id="S2.2.p1.19.m19.1.2.2.3" xref="S2.2.p1.19.m19.1.2.2.3.cmml">1</mn></msub><mo id="S2.2.p1.19.m19.1.2.1" xref="S2.2.p1.19.m19.1.2.1.cmml">β</mo><mrow id="S2.2.p1.19.m19.1.2.3" xref="S2.2.p1.19.m19.1.2.3.cmml"><mi id="S2.2.p1.19.m19.1.2.3.2" xref="S2.2.p1.19.m19.1.2.3.2.cmml">V</mi><mo id="S2.2.p1.19.m19.1.2.3.1" xref="S2.2.p1.19.m19.1.2.3.1.cmml">β’</mo><mrow id="S2.2.p1.19.m19.1.2.3.3.2" xref="S2.2.p1.19.m19.1.2.3.cmml"><mo id="S2.2.p1.19.m19.1.2.3.3.2.1" stretchy="false" xref="S2.2.p1.19.m19.1.2.3.cmml">(</mo><mi id="S2.2.p1.19.m19.1.1" xref="S2.2.p1.19.m19.1.1.cmml">y</mi><mo id="S2.2.p1.19.m19.1.2.3.3.2.2" stretchy="false" xref="S2.2.p1.19.m19.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p1.19.m19.1b"><apply id="S2.2.p1.19.m19.1.2.cmml" xref="S2.2.p1.19.m19.1.2"><subset id="S2.2.p1.19.m19.1.2.1.cmml" xref="S2.2.p1.19.m19.1.2.1"></subset><apply id="S2.2.p1.19.m19.1.2.2.cmml" xref="S2.2.p1.19.m19.1.2.2"><csymbol cd="ambiguous" id="S2.2.p1.19.m19.1.2.2.1.cmml" xref="S2.2.p1.19.m19.1.2.2">subscript</csymbol><ci id="S2.2.p1.19.m19.1.2.2.2.cmml" xref="S2.2.p1.19.m19.1.2.2.2">πΉ</ci><cn id="S2.2.p1.19.m19.1.2.2.3.cmml" type="integer" xref="S2.2.p1.19.m19.1.2.2.3">1</cn></apply><apply id="S2.2.p1.19.m19.1.2.3.cmml" xref="S2.2.p1.19.m19.1.2.3"><times id="S2.2.p1.19.m19.1.2.3.1.cmml" xref="S2.2.p1.19.m19.1.2.3.1"></times><ci id="S2.2.p1.19.m19.1.2.3.2.cmml" xref="S2.2.p1.19.m19.1.2.3.2">π</ci><ci id="S2.2.p1.19.m19.1.1.cmml" xref="S2.2.p1.19.m19.1.1">π¦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.19.m19.1c">F_{1}\subset V(y)</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.19.m19.1d">italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT β italic_V ( italic_y )</annotation></semantics></math> and <math alttext="F_{2}^{-1}\subseteq V(z)" class="ltx_Math" display="inline" id="S2.2.p1.20.m20.1"><semantics id="S2.2.p1.20.m20.1a"><mrow id="S2.2.p1.20.m20.1.2" xref="S2.2.p1.20.m20.1.2.cmml"><msubsup id="S2.2.p1.20.m20.1.2.2" xref="S2.2.p1.20.m20.1.2.2.cmml"><mi id="S2.2.p1.20.m20.1.2.2.2.2" xref="S2.2.p1.20.m20.1.2.2.2.2.cmml">F</mi><mn id="S2.2.p1.20.m20.1.2.2.2.3" xref="S2.2.p1.20.m20.1.2.2.2.3.cmml">2</mn><mrow id="S2.2.p1.20.m20.1.2.2.3" xref="S2.2.p1.20.m20.1.2.2.3.cmml"><mo id="S2.2.p1.20.m20.1.2.2.3a" xref="S2.2.p1.20.m20.1.2.2.3.cmml">β</mo><mn id="S2.2.p1.20.m20.1.2.2.3.2" xref="S2.2.p1.20.m20.1.2.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S2.2.p1.20.m20.1.2.1" xref="S2.2.p1.20.m20.1.2.1.cmml">β</mo><mrow id="S2.2.p1.20.m20.1.2.3" xref="S2.2.p1.20.m20.1.2.3.cmml"><mi id="S2.2.p1.20.m20.1.2.3.2" xref="S2.2.p1.20.m20.1.2.3.2.cmml">V</mi><mo id="S2.2.p1.20.m20.1.2.3.1" xref="S2.2.p1.20.m20.1.2.3.1.cmml">β’</mo><mrow id="S2.2.p1.20.m20.1.2.3.3.2" xref="S2.2.p1.20.m20.1.2.3.cmml"><mo id="S2.2.p1.20.m20.1.2.3.3.2.1" stretchy="false" xref="S2.2.p1.20.m20.1.2.3.cmml">(</mo><mi id="S2.2.p1.20.m20.1.1" xref="S2.2.p1.20.m20.1.1.cmml">z</mi><mo id="S2.2.p1.20.m20.1.2.3.3.2.2" stretchy="false" xref="S2.2.p1.20.m20.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p1.20.m20.1b"><apply id="S2.2.p1.20.m20.1.2.cmml" xref="S2.2.p1.20.m20.1.2"><subset id="S2.2.p1.20.m20.1.2.1.cmml" xref="S2.2.p1.20.m20.1.2.1"></subset><apply id="S2.2.p1.20.m20.1.2.2.cmml" xref="S2.2.p1.20.m20.1.2.2"><csymbol cd="ambiguous" id="S2.2.p1.20.m20.1.2.2.1.cmml" xref="S2.2.p1.20.m20.1.2.2">superscript</csymbol><apply id="S2.2.p1.20.m20.1.2.2.2.cmml" xref="S2.2.p1.20.m20.1.2.2"><csymbol cd="ambiguous" id="S2.2.p1.20.m20.1.2.2.2.1.cmml" xref="S2.2.p1.20.m20.1.2.2">subscript</csymbol><ci id="S2.2.p1.20.m20.1.2.2.2.2.cmml" xref="S2.2.p1.20.m20.1.2.2.2.2">πΉ</ci><cn id="S2.2.p1.20.m20.1.2.2.2.3.cmml" type="integer" xref="S2.2.p1.20.m20.1.2.2.2.3">2</cn></apply><apply id="S2.2.p1.20.m20.1.2.2.3.cmml" xref="S2.2.p1.20.m20.1.2.2.3"><minus id="S2.2.p1.20.m20.1.2.2.3.1.cmml" xref="S2.2.p1.20.m20.1.2.2.3"></minus><cn id="S2.2.p1.20.m20.1.2.2.3.2.cmml" type="integer" xref="S2.2.p1.20.m20.1.2.2.3.2">1</cn></apply></apply><apply id="S2.2.p1.20.m20.1.2.3.cmml" xref="S2.2.p1.20.m20.1.2.3"><times id="S2.2.p1.20.m20.1.2.3.1.cmml" xref="S2.2.p1.20.m20.1.2.3.1"></times><ci id="S2.2.p1.20.m20.1.2.3.2.cmml" xref="S2.2.p1.20.m20.1.2.3.2">π</ci><ci id="S2.2.p1.20.m20.1.1.cmml" xref="S2.2.p1.20.m20.1.1">π§</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.20.m20.1c">F_{2}^{-1}\subseteq V(z)</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.20.m20.1d">italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT β italic_V ( italic_z )</annotation></semantics></math>. Then for <math alttext="F=F_{1}\cap F_{2}" class="ltx_Math" display="inline" id="S2.2.p1.21.m21.1"><semantics id="S2.2.p1.21.m21.1a"><mrow id="S2.2.p1.21.m21.1.1" xref="S2.2.p1.21.m21.1.1.cmml"><mi id="S2.2.p1.21.m21.1.1.2" xref="S2.2.p1.21.m21.1.1.2.cmml">F</mi><mo id="S2.2.p1.21.m21.1.1.1" xref="S2.2.p1.21.m21.1.1.1.cmml">=</mo><mrow id="S2.2.p1.21.m21.1.1.3" xref="S2.2.p1.21.m21.1.1.3.cmml"><msub id="S2.2.p1.21.m21.1.1.3.2" xref="S2.2.p1.21.m21.1.1.3.2.cmml"><mi id="S2.2.p1.21.m21.1.1.3.2.2" xref="S2.2.p1.21.m21.1.1.3.2.2.cmml">F</mi><mn id="S2.2.p1.21.m21.1.1.3.2.3" xref="S2.2.p1.21.m21.1.1.3.2.3.cmml">1</mn></msub><mo id="S2.2.p1.21.m21.1.1.3.1" xref="S2.2.p1.21.m21.1.1.3.1.cmml">β©</mo><msub id="S2.2.p1.21.m21.1.1.3.3" xref="S2.2.p1.21.m21.1.1.3.3.cmml"><mi id="S2.2.p1.21.m21.1.1.3.3.2" xref="S2.2.p1.21.m21.1.1.3.3.2.cmml">F</mi><mn id="S2.2.p1.21.m21.1.1.3.3.3" xref="S2.2.p1.21.m21.1.1.3.3.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p1.21.m21.1b"><apply id="S2.2.p1.21.m21.1.1.cmml" xref="S2.2.p1.21.m21.1.1"><eq id="S2.2.p1.21.m21.1.1.1.cmml" xref="S2.2.p1.21.m21.1.1.1"></eq><ci id="S2.2.p1.21.m21.1.1.2.cmml" xref="S2.2.p1.21.m21.1.1.2">πΉ</ci><apply id="S2.2.p1.21.m21.1.1.3.cmml" xref="S2.2.p1.21.m21.1.1.3"><intersect id="S2.2.p1.21.m21.1.1.3.1.cmml" xref="S2.2.p1.21.m21.1.1.3.1"></intersect><apply id="S2.2.p1.21.m21.1.1.3.2.cmml" xref="S2.2.p1.21.m21.1.1.3.2"><csymbol cd="ambiguous" id="S2.2.p1.21.m21.1.1.3.2.1.cmml" xref="S2.2.p1.21.m21.1.1.3.2">subscript</csymbol><ci id="S2.2.p1.21.m21.1.1.3.2.2.cmml" xref="S2.2.p1.21.m21.1.1.3.2.2">πΉ</ci><cn id="S2.2.p1.21.m21.1.1.3.2.3.cmml" type="integer" xref="S2.2.p1.21.m21.1.1.3.2.3">1</cn></apply><apply id="S2.2.p1.21.m21.1.1.3.3.cmml" xref="S2.2.p1.21.m21.1.1.3.3"><csymbol cd="ambiguous" id="S2.2.p1.21.m21.1.1.3.3.1.cmml" xref="S2.2.p1.21.m21.1.1.3.3">subscript</csymbol><ci id="S2.2.p1.21.m21.1.1.3.3.2.cmml" xref="S2.2.p1.21.m21.1.1.3.3.2">πΉ</ci><cn id="S2.2.p1.21.m21.1.1.3.3.3.cmml" type="integer" xref="S2.2.p1.21.m21.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.21.m21.1c">F=F_{1}\cap F_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.21.m21.1d">italic_F = italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT β© italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> we get <math alttext="F\subseteq FF^{-1}F\subseteq U" class="ltx_Math" display="inline" id="S2.2.p1.22.m22.1"><semantics id="S2.2.p1.22.m22.1a"><mrow id="S2.2.p1.22.m22.1.1" xref="S2.2.p1.22.m22.1.1.cmml"><mi id="S2.2.p1.22.m22.1.1.2" xref="S2.2.p1.22.m22.1.1.2.cmml">F</mi><mo id="S2.2.p1.22.m22.1.1.3" xref="S2.2.p1.22.m22.1.1.3.cmml">β</mo><mrow id="S2.2.p1.22.m22.1.1.4" xref="S2.2.p1.22.m22.1.1.4.cmml"><mi id="S2.2.p1.22.m22.1.1.4.2" xref="S2.2.p1.22.m22.1.1.4.2.cmml">F</mi><mo id="S2.2.p1.22.m22.1.1.4.1" xref="S2.2.p1.22.m22.1.1.4.1.cmml">β’</mo><msup id="S2.2.p1.22.m22.1.1.4.3" xref="S2.2.p1.22.m22.1.1.4.3.cmml"><mi id="S2.2.p1.22.m22.1.1.4.3.2" xref="S2.2.p1.22.m22.1.1.4.3.2.cmml">F</mi><mrow id="S2.2.p1.22.m22.1.1.4.3.3" xref="S2.2.p1.22.m22.1.1.4.3.3.cmml"><mo id="S2.2.p1.22.m22.1.1.4.3.3a" xref="S2.2.p1.22.m22.1.1.4.3.3.cmml">β</mo><mn id="S2.2.p1.22.m22.1.1.4.3.3.2" xref="S2.2.p1.22.m22.1.1.4.3.3.2.cmml">1</mn></mrow></msup><mo id="S2.2.p1.22.m22.1.1.4.1a" xref="S2.2.p1.22.m22.1.1.4.1.cmml">β’</mo><mi id="S2.2.p1.22.m22.1.1.4.4" xref="S2.2.p1.22.m22.1.1.4.4.cmml">F</mi></mrow><mo id="S2.2.p1.22.m22.1.1.5" xref="S2.2.p1.22.m22.1.1.5.cmml">β</mo><mi id="S2.2.p1.22.m22.1.1.6" xref="S2.2.p1.22.m22.1.1.6.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p1.22.m22.1b"><apply id="S2.2.p1.22.m22.1.1.cmml" xref="S2.2.p1.22.m22.1.1"><and id="S2.2.p1.22.m22.1.1a.cmml" xref="S2.2.p1.22.m22.1.1"></and><apply id="S2.2.p1.22.m22.1.1b.cmml" xref="S2.2.p1.22.m22.1.1"><subset id="S2.2.p1.22.m22.1.1.3.cmml" xref="S2.2.p1.22.m22.1.1.3"></subset><ci id="S2.2.p1.22.m22.1.1.2.cmml" xref="S2.2.p1.22.m22.1.1.2">πΉ</ci><apply id="S2.2.p1.22.m22.1.1.4.cmml" xref="S2.2.p1.22.m22.1.1.4"><times id="S2.2.p1.22.m22.1.1.4.1.cmml" xref="S2.2.p1.22.m22.1.1.4.1"></times><ci id="S2.2.p1.22.m22.1.1.4.2.cmml" xref="S2.2.p1.22.m22.1.1.4.2">πΉ</ci><apply id="S2.2.p1.22.m22.1.1.4.3.cmml" xref="S2.2.p1.22.m22.1.1.4.3"><csymbol cd="ambiguous" id="S2.2.p1.22.m22.1.1.4.3.1.cmml" xref="S2.2.p1.22.m22.1.1.4.3">superscript</csymbol><ci id="S2.2.p1.22.m22.1.1.4.3.2.cmml" xref="S2.2.p1.22.m22.1.1.4.3.2">πΉ</ci><apply id="S2.2.p1.22.m22.1.1.4.3.3.cmml" xref="S2.2.p1.22.m22.1.1.4.3.3"><minus id="S2.2.p1.22.m22.1.1.4.3.3.1.cmml" xref="S2.2.p1.22.m22.1.1.4.3.3"></minus><cn id="S2.2.p1.22.m22.1.1.4.3.3.2.cmml" type="integer" xref="S2.2.p1.22.m22.1.1.4.3.3.2">1</cn></apply></apply><ci id="S2.2.p1.22.m22.1.1.4.4.cmml" xref="S2.2.p1.22.m22.1.1.4.4">πΉ</ci></apply></apply><apply id="S2.2.p1.22.m22.1.1c.cmml" xref="S2.2.p1.22.m22.1.1"><subset id="S2.2.p1.22.m22.1.1.5.cmml" xref="S2.2.p1.22.m22.1.1.5"></subset><share href="https://arxiv.org/html/2503.13666v1#S2.2.p1.22.m22.1.1.4.cmml" id="S2.2.p1.22.m22.1.1d.cmml" xref="S2.2.p1.22.m22.1.1"></share><ci id="S2.2.p1.22.m22.1.1.6.cmml" xref="S2.2.p1.22.m22.1.1.6">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.22.m22.1c">F\subseteq FF^{-1}F\subseteq U</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.22.m22.1d">italic_F β italic_F italic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_F β italic_U</annotation></semantics></math>. Since the open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S2.2.p1.23.m23.1"><semantics id="S2.2.p1.23.m23.1a"><mi id="S2.2.p1.23.m23.1.1" xref="S2.2.p1.23.m23.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S2.2.p1.23.m23.1b"><ci id="S2.2.p1.23.m23.1.1.cmml" xref="S2.2.p1.23.m23.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.23.m23.1c">U</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.23.m23.1d">italic_U</annotation></semantics></math> of <math alttext="w" class="ltx_Math" display="inline" id="S2.2.p1.24.m24.1"><semantics id="S2.2.p1.24.m24.1a"><mi id="S2.2.p1.24.m24.1.1" xref="S2.2.p1.24.m24.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S2.2.p1.24.m24.1b"><ci id="S2.2.p1.24.m24.1.1.cmml" xref="S2.2.p1.24.m24.1.1">π€</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.24.m24.1c">w</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.24.m24.1d">italic_w</annotation></semantics></math> is arbitrarily chosen, the filter <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S2.2.p1.25.m25.1"><semantics id="S2.2.p1.25.m25.1a"><mi class="ltx_font_mathcaligraphic" id="S2.2.p1.25.m25.1.1" xref="S2.2.p1.25.m25.1.1.cmml">β±</mi><annotation-xml encoding="MathML-Content" id="S2.2.p1.25.m25.1b"><ci id="S2.2.p1.25.m25.1.1.cmml" xref="S2.2.p1.25.m25.1.1">β±</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.25.m25.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.25.m25.1d">caligraphic_F</annotation></semantics></math> converges to <math alttext="w" class="ltx_Math" display="inline" id="S2.2.p1.26.m26.1"><semantics id="S2.2.p1.26.m26.1a"><mi id="S2.2.p1.26.m26.1.1" xref="S2.2.p1.26.m26.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S2.2.p1.26.m26.1b"><ci id="S2.2.p1.26.m26.1.1.cmml" xref="S2.2.p1.26.m26.1.1">π€</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.26.m26.1c">w</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.26.m26.1d">italic_w</annotation></semantics></math>. Since the space <math alttext="S" class="ltx_Math" display="inline" id="S2.2.p1.27.m27.1"><semantics id="S2.2.p1.27.m27.1a"><mi id="S2.2.p1.27.m27.1.1" xref="S2.2.p1.27.m27.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.2.p1.27.m27.1b"><ci id="S2.2.p1.27.m27.1.1.cmml" xref="S2.2.p1.27.m27.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.27.m27.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.27.m27.1d">italic_S</annotation></semantics></math> is Hausdorff, any filter on <math alttext="S" class="ltx_Math" display="inline" id="S2.2.p1.28.m28.1"><semantics id="S2.2.p1.28.m28.1a"><mi id="S2.2.p1.28.m28.1.1" xref="S2.2.p1.28.m28.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.2.p1.28.m28.1b"><ci id="S2.2.p1.28.m28.1.1.cmml" xref="S2.2.p1.28.m28.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.28.m28.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.28.m28.1d">italic_S</annotation></semantics></math> converges to at most one point, implying that <math alttext="y=w=yzy" class="ltx_Math" display="inline" id="S2.2.p1.29.m29.1"><semantics id="S2.2.p1.29.m29.1a"><mrow id="S2.2.p1.29.m29.1.1" xref="S2.2.p1.29.m29.1.1.cmml"><mi id="S2.2.p1.29.m29.1.1.2" xref="S2.2.p1.29.m29.1.1.2.cmml">y</mi><mo id="S2.2.p1.29.m29.1.1.3" xref="S2.2.p1.29.m29.1.1.3.cmml">=</mo><mi id="S2.2.p1.29.m29.1.1.4" xref="S2.2.p1.29.m29.1.1.4.cmml">w</mi><mo id="S2.2.p1.29.m29.1.1.5" xref="S2.2.p1.29.m29.1.1.5.cmml">=</mo><mrow id="S2.2.p1.29.m29.1.1.6" xref="S2.2.p1.29.m29.1.1.6.cmml"><mi id="S2.2.p1.29.m29.1.1.6.2" xref="S2.2.p1.29.m29.1.1.6.2.cmml">y</mi><mo id="S2.2.p1.29.m29.1.1.6.1" xref="S2.2.p1.29.m29.1.1.6.1.cmml">β’</mo><mi id="S2.2.p1.29.m29.1.1.6.3" xref="S2.2.p1.29.m29.1.1.6.3.cmml">z</mi><mo id="S2.2.p1.29.m29.1.1.6.1a" xref="S2.2.p1.29.m29.1.1.6.1.cmml">β’</mo><mi id="S2.2.p1.29.m29.1.1.6.4" xref="S2.2.p1.29.m29.1.1.6.4.cmml">y</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.2.p1.29.m29.1b"><apply id="S2.2.p1.29.m29.1.1.cmml" xref="S2.2.p1.29.m29.1.1"><and id="S2.2.p1.29.m29.1.1a.cmml" xref="S2.2.p1.29.m29.1.1"></and><apply id="S2.2.p1.29.m29.1.1b.cmml" xref="S2.2.p1.29.m29.1.1"><eq id="S2.2.p1.29.m29.1.1.3.cmml" xref="S2.2.p1.29.m29.1.1.3"></eq><ci id="S2.2.p1.29.m29.1.1.2.cmml" xref="S2.2.p1.29.m29.1.1.2">π¦</ci><ci id="S2.2.p1.29.m29.1.1.4.cmml" xref="S2.2.p1.29.m29.1.1.4">π€</ci></apply><apply id="S2.2.p1.29.m29.1.1c.cmml" xref="S2.2.p1.29.m29.1.1"><eq id="S2.2.p1.29.m29.1.1.5.cmml" xref="S2.2.p1.29.m29.1.1.5"></eq><share href="https://arxiv.org/html/2503.13666v1#S2.2.p1.29.m29.1.1.4.cmml" id="S2.2.p1.29.m29.1.1d.cmml" xref="S2.2.p1.29.m29.1.1"></share><apply id="S2.2.p1.29.m29.1.1.6.cmml" xref="S2.2.p1.29.m29.1.1.6"><times id="S2.2.p1.29.m29.1.1.6.1.cmml" xref="S2.2.p1.29.m29.1.1.6.1"></times><ci id="S2.2.p1.29.m29.1.1.6.2.cmml" xref="S2.2.p1.29.m29.1.1.6.2">π¦</ci><ci id="S2.2.p1.29.m29.1.1.6.3.cmml" xref="S2.2.p1.29.m29.1.1.6.3">π§</ci><ci id="S2.2.p1.29.m29.1.1.6.4.cmml" xref="S2.2.p1.29.m29.1.1.6.4">π¦</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.29.m29.1c">y=w=yzy</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.29.m29.1d">italic_y = italic_w = italic_y italic_z italic_y</annotation></semantics></math>. Hence <math alttext="S" class="ltx_Math" display="inline" id="S2.2.p1.30.m30.1"><semantics id="S2.2.p1.30.m30.1a"><mi id="S2.2.p1.30.m30.1.1" xref="S2.2.p1.30.m30.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.2.p1.30.m30.1b"><ci id="S2.2.p1.30.m30.1.1.cmml" xref="S2.2.p1.30.m30.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.2.p1.30.m30.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.2.p1.30.m30.1d">italic_S</annotation></semantics></math> is a regular semigroup.</p> </div> <div class="ltx_para" id="S2.3.p2"> <p class="ltx_p" id="S2.3.p2.42">To prove that <math alttext="S" class="ltx_Math" display="inline" id="S2.3.p2.1.m1.1"><semantics id="S2.3.p2.1.m1.1a"><mi id="S2.3.p2.1.m1.1.1" xref="S2.3.p2.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.1.m1.1b"><ci id="S2.3.p2.1.m1.1.1.cmml" xref="S2.3.p2.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.1.m1.1d">italic_S</annotation></semantics></math> is an inverse semigroup it suffices to show that idempotents of <math alttext="S" class="ltx_Math" display="inline" id="S2.3.p2.2.m2.1"><semantics id="S2.3.p2.2.m2.1a"><mi id="S2.3.p2.2.m2.1.1" xref="S2.3.p2.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.2.m2.1b"><ci id="S2.3.p2.2.m2.1.1.cmml" xref="S2.3.p2.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.2.m2.1d">italic_S</annotation></semantics></math> commute. For this it is enough to show that <math alttext="E(S)=\overline{E(X)}" class="ltx_Math" display="inline" id="S2.3.p2.3.m3.2"><semantics id="S2.3.p2.3.m3.2a"><mrow id="S2.3.p2.3.m3.2.3" xref="S2.3.p2.3.m3.2.3.cmml"><mrow id="S2.3.p2.3.m3.2.3.2" xref="S2.3.p2.3.m3.2.3.2.cmml"><mi id="S2.3.p2.3.m3.2.3.2.2" xref="S2.3.p2.3.m3.2.3.2.2.cmml">E</mi><mo id="S2.3.p2.3.m3.2.3.2.1" xref="S2.3.p2.3.m3.2.3.2.1.cmml">β’</mo><mrow id="S2.3.p2.3.m3.2.3.2.3.2" xref="S2.3.p2.3.m3.2.3.2.cmml"><mo id="S2.3.p2.3.m3.2.3.2.3.2.1" stretchy="false" xref="S2.3.p2.3.m3.2.3.2.cmml">(</mo><mi id="S2.3.p2.3.m3.2.2" xref="S2.3.p2.3.m3.2.2.cmml">S</mi><mo id="S2.3.p2.3.m3.2.3.2.3.2.2" stretchy="false" xref="S2.3.p2.3.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.3.p2.3.m3.2.3.1" xref="S2.3.p2.3.m3.2.3.1.cmml">=</mo><mover accent="true" id="S2.3.p2.3.m3.1.1" xref="S2.3.p2.3.m3.1.1.cmml"><mrow id="S2.3.p2.3.m3.1.1.1" xref="S2.3.p2.3.m3.1.1.1.cmml"><mi id="S2.3.p2.3.m3.1.1.1.3" xref="S2.3.p2.3.m3.1.1.1.3.cmml">E</mi><mo id="S2.3.p2.3.m3.1.1.1.2" xref="S2.3.p2.3.m3.1.1.1.2.cmml">β’</mo><mrow id="S2.3.p2.3.m3.1.1.1.4.2" xref="S2.3.p2.3.m3.1.1.1.cmml"><mo id="S2.3.p2.3.m3.1.1.1.4.2.1" stretchy="false" xref="S2.3.p2.3.m3.1.1.1.cmml">(</mo><mi id="S2.3.p2.3.m3.1.1.1.1" xref="S2.3.p2.3.m3.1.1.1.1.cmml">X</mi><mo id="S2.3.p2.3.m3.1.1.1.4.2.2" stretchy="false" xref="S2.3.p2.3.m3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.3.p2.3.m3.1.1.2" xref="S2.3.p2.3.m3.1.1.2.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.3.m3.2b"><apply id="S2.3.p2.3.m3.2.3.cmml" xref="S2.3.p2.3.m3.2.3"><eq id="S2.3.p2.3.m3.2.3.1.cmml" xref="S2.3.p2.3.m3.2.3.1"></eq><apply id="S2.3.p2.3.m3.2.3.2.cmml" xref="S2.3.p2.3.m3.2.3.2"><times id="S2.3.p2.3.m3.2.3.2.1.cmml" xref="S2.3.p2.3.m3.2.3.2.1"></times><ci id="S2.3.p2.3.m3.2.3.2.2.cmml" xref="S2.3.p2.3.m3.2.3.2.2">πΈ</ci><ci id="S2.3.p2.3.m3.2.2.cmml" xref="S2.3.p2.3.m3.2.2">π</ci></apply><apply id="S2.3.p2.3.m3.1.1.cmml" xref="S2.3.p2.3.m3.1.1"><ci id="S2.3.p2.3.m3.1.1.2.cmml" xref="S2.3.p2.3.m3.1.1.2">Β―</ci><apply id="S2.3.p2.3.m3.1.1.1.cmml" xref="S2.3.p2.3.m3.1.1.1"><times id="S2.3.p2.3.m3.1.1.1.2.cmml" xref="S2.3.p2.3.m3.1.1.1.2"></times><ci id="S2.3.p2.3.m3.1.1.1.3.cmml" xref="S2.3.p2.3.m3.1.1.1.3">πΈ</ci><ci id="S2.3.p2.3.m3.1.1.1.1.cmml" xref="S2.3.p2.3.m3.1.1.1.1">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.3.m3.2c">E(S)=\overline{E(X)}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.3.m3.2d">italic_E ( italic_S ) = overΒ― start_ARG italic_E ( italic_X ) end_ARG</annotation></semantics></math>, as <math alttext="E(X)" class="ltx_Math" display="inline" id="S2.3.p2.4.m4.1"><semantics id="S2.3.p2.4.m4.1a"><mrow id="S2.3.p2.4.m4.1.2" xref="S2.3.p2.4.m4.1.2.cmml"><mi id="S2.3.p2.4.m4.1.2.2" xref="S2.3.p2.4.m4.1.2.2.cmml">E</mi><mo id="S2.3.p2.4.m4.1.2.1" xref="S2.3.p2.4.m4.1.2.1.cmml">β’</mo><mrow id="S2.3.p2.4.m4.1.2.3.2" xref="S2.3.p2.4.m4.1.2.cmml"><mo id="S2.3.p2.4.m4.1.2.3.2.1" stretchy="false" xref="S2.3.p2.4.m4.1.2.cmml">(</mo><mi id="S2.3.p2.4.m4.1.1" xref="S2.3.p2.4.m4.1.1.cmml">X</mi><mo id="S2.3.p2.4.m4.1.2.3.2.2" stretchy="false" xref="S2.3.p2.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.4.m4.1b"><apply id="S2.3.p2.4.m4.1.2.cmml" xref="S2.3.p2.4.m4.1.2"><times id="S2.3.p2.4.m4.1.2.1.cmml" xref="S2.3.p2.4.m4.1.2.1"></times><ci id="S2.3.p2.4.m4.1.2.2.cmml" xref="S2.3.p2.4.m4.1.2.2">πΈ</ci><ci id="S2.3.p2.4.m4.1.1.cmml" xref="S2.3.p2.4.m4.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.4.m4.1c">E(X)</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.4.m4.1d">italic_E ( italic_X )</annotation></semantics></math> is a commutative semigroup. Fix an idempotent <math alttext="e\in S\setminus X" class="ltx_Math" display="inline" id="S2.3.p2.5.m5.1"><semantics id="S2.3.p2.5.m5.1a"><mrow id="S2.3.p2.5.m5.1.1" xref="S2.3.p2.5.m5.1.1.cmml"><mi id="S2.3.p2.5.m5.1.1.2" xref="S2.3.p2.5.m5.1.1.2.cmml">e</mi><mo id="S2.3.p2.5.m5.1.1.1" xref="S2.3.p2.5.m5.1.1.1.cmml">β</mo><mrow id="S2.3.p2.5.m5.1.1.3" xref="S2.3.p2.5.m5.1.1.3.cmml"><mi id="S2.3.p2.5.m5.1.1.3.2" xref="S2.3.p2.5.m5.1.1.3.2.cmml">S</mi><mo id="S2.3.p2.5.m5.1.1.3.1" xref="S2.3.p2.5.m5.1.1.3.1.cmml">β</mo><mi id="S2.3.p2.5.m5.1.1.3.3" xref="S2.3.p2.5.m5.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.5.m5.1b"><apply id="S2.3.p2.5.m5.1.1.cmml" xref="S2.3.p2.5.m5.1.1"><in id="S2.3.p2.5.m5.1.1.1.cmml" xref="S2.3.p2.5.m5.1.1.1"></in><ci id="S2.3.p2.5.m5.1.1.2.cmml" xref="S2.3.p2.5.m5.1.1.2">π</ci><apply id="S2.3.p2.5.m5.1.1.3.cmml" xref="S2.3.p2.5.m5.1.1.3"><setdiff id="S2.3.p2.5.m5.1.1.3.1.cmml" xref="S2.3.p2.5.m5.1.1.3.1"></setdiff><ci id="S2.3.p2.5.m5.1.1.3.2.cmml" xref="S2.3.p2.5.m5.1.1.3.2">π</ci><ci id="S2.3.p2.5.m5.1.1.3.3.cmml" xref="S2.3.p2.5.m5.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.5.m5.1c">e\in S\setminus X</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.5.m5.1d">italic_e β italic_S β italic_X</annotation></semantics></math>. By the assumption, there exists an ultrafilter <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S2.3.p2.6.m6.1"><semantics id="S2.3.p2.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S2.3.p2.6.m6.1.1" xref="S2.3.p2.6.m6.1.1.cmml">β±</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.6.m6.1b"><ci id="S2.3.p2.6.m6.1.1.cmml" xref="S2.3.p2.6.m6.1.1">β±</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.6.m6.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.6.m6.1d">caligraphic_F</annotation></semantics></math> convergent to <math alttext="e" class="ltx_Math" display="inline" id="S2.3.p2.7.m7.1"><semantics id="S2.3.p2.7.m7.1a"><mi id="S2.3.p2.7.m7.1.1" xref="S2.3.p2.7.m7.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.7.m7.1b"><ci id="S2.3.p2.7.m7.1.1.cmml" xref="S2.3.p2.7.m7.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.7.m7.1c">e</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.7.m7.1d">italic_e</annotation></semantics></math> such that <math alttext="X\in\mathcal{F}" class="ltx_Math" display="inline" id="S2.3.p2.8.m8.1"><semantics id="S2.3.p2.8.m8.1a"><mrow id="S2.3.p2.8.m8.1.1" xref="S2.3.p2.8.m8.1.1.cmml"><mi id="S2.3.p2.8.m8.1.1.2" xref="S2.3.p2.8.m8.1.1.2.cmml">X</mi><mo id="S2.3.p2.8.m8.1.1.1" xref="S2.3.p2.8.m8.1.1.1.cmml">β</mo><mi class="ltx_font_mathcaligraphic" id="S2.3.p2.8.m8.1.1.3" xref="S2.3.p2.8.m8.1.1.3.cmml">β±</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.8.m8.1b"><apply id="S2.3.p2.8.m8.1.1.cmml" xref="S2.3.p2.8.m8.1.1"><in id="S2.3.p2.8.m8.1.1.1.cmml" xref="S2.3.p2.8.m8.1.1.1"></in><ci id="S2.3.p2.8.m8.1.1.2.cmml" xref="S2.3.p2.8.m8.1.1.2">π</ci><ci id="S2.3.p2.8.m8.1.1.3.cmml" xref="S2.3.p2.8.m8.1.1.3">β±</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.8.m8.1c">X\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.8.m8.1d">italic_X β caligraphic_F</annotation></semantics></math> and the ultrafilter <math alttext="\mathcal{F}^{-1}" class="ltx_Math" display="inline" id="S2.3.p2.9.m9.1"><semantics id="S2.3.p2.9.m9.1a"><msup id="S2.3.p2.9.m9.1.1" xref="S2.3.p2.9.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.3.p2.9.m9.1.1.2" xref="S2.3.p2.9.m9.1.1.2.cmml">β±</mi><mrow id="S2.3.p2.9.m9.1.1.3" xref="S2.3.p2.9.m9.1.1.3.cmml"><mo id="S2.3.p2.9.m9.1.1.3a" xref="S2.3.p2.9.m9.1.1.3.cmml">β</mo><mn id="S2.3.p2.9.m9.1.1.3.2" xref="S2.3.p2.9.m9.1.1.3.2.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.3.p2.9.m9.1b"><apply id="S2.3.p2.9.m9.1.1.cmml" xref="S2.3.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S2.3.p2.9.m9.1.1.1.cmml" xref="S2.3.p2.9.m9.1.1">superscript</csymbol><ci id="S2.3.p2.9.m9.1.1.2.cmml" xref="S2.3.p2.9.m9.1.1.2">β±</ci><apply id="S2.3.p2.9.m9.1.1.3.cmml" xref="S2.3.p2.9.m9.1.1.3"><minus id="S2.3.p2.9.m9.1.1.3.1.cmml" xref="S2.3.p2.9.m9.1.1.3"></minus><cn id="S2.3.p2.9.m9.1.1.3.2.cmml" type="integer" xref="S2.3.p2.9.m9.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.9.m9.1c">\mathcal{F}^{-1}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.9.m9.1d">caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> converges to some point <math alttext="f\in S" class="ltx_Math" display="inline" id="S2.3.p2.10.m10.1"><semantics id="S2.3.p2.10.m10.1a"><mrow id="S2.3.p2.10.m10.1.1" xref="S2.3.p2.10.m10.1.1.cmml"><mi id="S2.3.p2.10.m10.1.1.2" xref="S2.3.p2.10.m10.1.1.2.cmml">f</mi><mo id="S2.3.p2.10.m10.1.1.1" xref="S2.3.p2.10.m10.1.1.1.cmml">β</mo><mi id="S2.3.p2.10.m10.1.1.3" xref="S2.3.p2.10.m10.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.10.m10.1b"><apply id="S2.3.p2.10.m10.1.1.cmml" xref="S2.3.p2.10.m10.1.1"><in id="S2.3.p2.10.m10.1.1.1.cmml" xref="S2.3.p2.10.m10.1.1.1"></in><ci id="S2.3.p2.10.m10.1.1.2.cmml" xref="S2.3.p2.10.m10.1.1.2">π</ci><ci id="S2.3.p2.10.m10.1.1.3.cmml" xref="S2.3.p2.10.m10.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.10.m10.1c">f\in S</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.10.m10.1d">italic_f β italic_S</annotation></semantics></math>. Similarly as above one can check that <math alttext="fef=f" class="ltx_Math" display="inline" id="S2.3.p2.11.m11.1"><semantics id="S2.3.p2.11.m11.1a"><mrow id="S2.3.p2.11.m11.1.1" xref="S2.3.p2.11.m11.1.1.cmml"><mrow id="S2.3.p2.11.m11.1.1.2" xref="S2.3.p2.11.m11.1.1.2.cmml"><mi id="S2.3.p2.11.m11.1.1.2.2" xref="S2.3.p2.11.m11.1.1.2.2.cmml">f</mi><mo id="S2.3.p2.11.m11.1.1.2.1" xref="S2.3.p2.11.m11.1.1.2.1.cmml">β’</mo><mi id="S2.3.p2.11.m11.1.1.2.3" xref="S2.3.p2.11.m11.1.1.2.3.cmml">e</mi><mo id="S2.3.p2.11.m11.1.1.2.1a" xref="S2.3.p2.11.m11.1.1.2.1.cmml">β’</mo><mi id="S2.3.p2.11.m11.1.1.2.4" xref="S2.3.p2.11.m11.1.1.2.4.cmml">f</mi></mrow><mo id="S2.3.p2.11.m11.1.1.1" xref="S2.3.p2.11.m11.1.1.1.cmml">=</mo><mi id="S2.3.p2.11.m11.1.1.3" xref="S2.3.p2.11.m11.1.1.3.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.11.m11.1b"><apply id="S2.3.p2.11.m11.1.1.cmml" xref="S2.3.p2.11.m11.1.1"><eq id="S2.3.p2.11.m11.1.1.1.cmml" xref="S2.3.p2.11.m11.1.1.1"></eq><apply id="S2.3.p2.11.m11.1.1.2.cmml" xref="S2.3.p2.11.m11.1.1.2"><times id="S2.3.p2.11.m11.1.1.2.1.cmml" xref="S2.3.p2.11.m11.1.1.2.1"></times><ci id="S2.3.p2.11.m11.1.1.2.2.cmml" xref="S2.3.p2.11.m11.1.1.2.2">π</ci><ci id="S2.3.p2.11.m11.1.1.2.3.cmml" xref="S2.3.p2.11.m11.1.1.2.3">π</ci><ci id="S2.3.p2.11.m11.1.1.2.4.cmml" xref="S2.3.p2.11.m11.1.1.2.4">π</ci></apply><ci id="S2.3.p2.11.m11.1.1.3.cmml" xref="S2.3.p2.11.m11.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.11.m11.1c">fef=f</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.11.m11.1d">italic_f italic_e italic_f = italic_f</annotation></semantics></math> and <math alttext="e=efe" class="ltx_Math" display="inline" id="S2.3.p2.12.m12.1"><semantics id="S2.3.p2.12.m12.1a"><mrow id="S2.3.p2.12.m12.1.1" xref="S2.3.p2.12.m12.1.1.cmml"><mi id="S2.3.p2.12.m12.1.1.2" xref="S2.3.p2.12.m12.1.1.2.cmml">e</mi><mo id="S2.3.p2.12.m12.1.1.1" xref="S2.3.p2.12.m12.1.1.1.cmml">=</mo><mrow id="S2.3.p2.12.m12.1.1.3" xref="S2.3.p2.12.m12.1.1.3.cmml"><mi id="S2.3.p2.12.m12.1.1.3.2" xref="S2.3.p2.12.m12.1.1.3.2.cmml">e</mi><mo id="S2.3.p2.12.m12.1.1.3.1" xref="S2.3.p2.12.m12.1.1.3.1.cmml">β’</mo><mi id="S2.3.p2.12.m12.1.1.3.3" xref="S2.3.p2.12.m12.1.1.3.3.cmml">f</mi><mo id="S2.3.p2.12.m12.1.1.3.1a" xref="S2.3.p2.12.m12.1.1.3.1.cmml">β’</mo><mi id="S2.3.p2.12.m12.1.1.3.4" xref="S2.3.p2.12.m12.1.1.3.4.cmml">e</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.12.m12.1b"><apply id="S2.3.p2.12.m12.1.1.cmml" xref="S2.3.p2.12.m12.1.1"><eq id="S2.3.p2.12.m12.1.1.1.cmml" xref="S2.3.p2.12.m12.1.1.1"></eq><ci id="S2.3.p2.12.m12.1.1.2.cmml" xref="S2.3.p2.12.m12.1.1.2">π</ci><apply id="S2.3.p2.12.m12.1.1.3.cmml" xref="S2.3.p2.12.m12.1.1.3"><times id="S2.3.p2.12.m12.1.1.3.1.cmml" xref="S2.3.p2.12.m12.1.1.3.1"></times><ci id="S2.3.p2.12.m12.1.1.3.2.cmml" xref="S2.3.p2.12.m12.1.1.3.2">π</ci><ci id="S2.3.p2.12.m12.1.1.3.3.cmml" xref="S2.3.p2.12.m12.1.1.3.3">π</ci><ci id="S2.3.p2.12.m12.1.1.3.4.cmml" xref="S2.3.p2.12.m12.1.1.3.4">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.12.m12.1c">e=efe</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.12.m12.1d">italic_e = italic_e italic_f italic_e</annotation></semantics></math>. Since <math alttext="ee=e" class="ltx_Math" display="inline" id="S2.3.p2.13.m13.1"><semantics id="S2.3.p2.13.m13.1a"><mrow id="S2.3.p2.13.m13.1.1" xref="S2.3.p2.13.m13.1.1.cmml"><mrow id="S2.3.p2.13.m13.1.1.2" xref="S2.3.p2.13.m13.1.1.2.cmml"><mi id="S2.3.p2.13.m13.1.1.2.2" xref="S2.3.p2.13.m13.1.1.2.2.cmml">e</mi><mo id="S2.3.p2.13.m13.1.1.2.1" xref="S2.3.p2.13.m13.1.1.2.1.cmml">β’</mo><mi id="S2.3.p2.13.m13.1.1.2.3" xref="S2.3.p2.13.m13.1.1.2.3.cmml">e</mi></mrow><mo id="S2.3.p2.13.m13.1.1.1" xref="S2.3.p2.13.m13.1.1.1.cmml">=</mo><mi id="S2.3.p2.13.m13.1.1.3" xref="S2.3.p2.13.m13.1.1.3.cmml">e</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.13.m13.1b"><apply id="S2.3.p2.13.m13.1.1.cmml" xref="S2.3.p2.13.m13.1.1"><eq id="S2.3.p2.13.m13.1.1.1.cmml" xref="S2.3.p2.13.m13.1.1.1"></eq><apply id="S2.3.p2.13.m13.1.1.2.cmml" xref="S2.3.p2.13.m13.1.1.2"><times id="S2.3.p2.13.m13.1.1.2.1.cmml" xref="S2.3.p2.13.m13.1.1.2.1"></times><ci id="S2.3.p2.13.m13.1.1.2.2.cmml" xref="S2.3.p2.13.m13.1.1.2.2">π</ci><ci id="S2.3.p2.13.m13.1.1.2.3.cmml" xref="S2.3.p2.13.m13.1.1.2.3">π</ci></apply><ci id="S2.3.p2.13.m13.1.1.3.cmml" xref="S2.3.p2.13.m13.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.13.m13.1c">ee=e</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.13.m13.1d">italic_e italic_e = italic_e</annotation></semantics></math>, the continuity of the semigroup operation implies that for every open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S2.3.p2.14.m14.1"><semantics id="S2.3.p2.14.m14.1a"><mi id="S2.3.p2.14.m14.1.1" xref="S2.3.p2.14.m14.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.14.m14.1b"><ci id="S2.3.p2.14.m14.1.1.cmml" xref="S2.3.p2.14.m14.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.14.m14.1c">U</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.14.m14.1d">italic_U</annotation></semantics></math> of <math alttext="e" class="ltx_Math" display="inline" id="S2.3.p2.15.m15.1"><semantics id="S2.3.p2.15.m15.1a"><mi id="S2.3.p2.15.m15.1.1" xref="S2.3.p2.15.m15.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.15.m15.1b"><ci id="S2.3.p2.15.m15.1.1.cmml" xref="S2.3.p2.15.m15.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.15.m15.1c">e</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.15.m15.1d">italic_e</annotation></semantics></math> there exists an open neighborhood <math alttext="V" class="ltx_Math" display="inline" id="S2.3.p2.16.m16.1"><semantics id="S2.3.p2.16.m16.1a"><mi id="S2.3.p2.16.m16.1.1" xref="S2.3.p2.16.m16.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.16.m16.1b"><ci id="S2.3.p2.16.m16.1.1.cmml" xref="S2.3.p2.16.m16.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.16.m16.1c">V</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.16.m16.1d">italic_V</annotation></semantics></math> of <math alttext="e" class="ltx_Math" display="inline" id="S2.3.p2.17.m17.1"><semantics id="S2.3.p2.17.m17.1a"><mi id="S2.3.p2.17.m17.1.1" xref="S2.3.p2.17.m17.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.17.m17.1b"><ci id="S2.3.p2.17.m17.1.1.cmml" xref="S2.3.p2.17.m17.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.17.m17.1c">e</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.17.m17.1d">italic_e</annotation></semantics></math> such that <math alttext="VV\subset U" class="ltx_Math" display="inline" id="S2.3.p2.18.m18.1"><semantics id="S2.3.p2.18.m18.1a"><mrow id="S2.3.p2.18.m18.1.1" xref="S2.3.p2.18.m18.1.1.cmml"><mrow id="S2.3.p2.18.m18.1.1.2" xref="S2.3.p2.18.m18.1.1.2.cmml"><mi id="S2.3.p2.18.m18.1.1.2.2" xref="S2.3.p2.18.m18.1.1.2.2.cmml">V</mi><mo id="S2.3.p2.18.m18.1.1.2.1" xref="S2.3.p2.18.m18.1.1.2.1.cmml">β’</mo><mi id="S2.3.p2.18.m18.1.1.2.3" xref="S2.3.p2.18.m18.1.1.2.3.cmml">V</mi></mrow><mo id="S2.3.p2.18.m18.1.1.1" xref="S2.3.p2.18.m18.1.1.1.cmml">β</mo><mi id="S2.3.p2.18.m18.1.1.3" xref="S2.3.p2.18.m18.1.1.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.18.m18.1b"><apply id="S2.3.p2.18.m18.1.1.cmml" xref="S2.3.p2.18.m18.1.1"><subset id="S2.3.p2.18.m18.1.1.1.cmml" xref="S2.3.p2.18.m18.1.1.1"></subset><apply id="S2.3.p2.18.m18.1.1.2.cmml" xref="S2.3.p2.18.m18.1.1.2"><times id="S2.3.p2.18.m18.1.1.2.1.cmml" xref="S2.3.p2.18.m18.1.1.2.1"></times><ci id="S2.3.p2.18.m18.1.1.2.2.cmml" xref="S2.3.p2.18.m18.1.1.2.2">π</ci><ci id="S2.3.p2.18.m18.1.1.2.3.cmml" xref="S2.3.p2.18.m18.1.1.2.3">π</ci></apply><ci id="S2.3.p2.18.m18.1.1.3.cmml" xref="S2.3.p2.18.m18.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.18.m18.1c">VV\subset U</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.18.m18.1d">italic_V italic_V β italic_U</annotation></semantics></math>. Pick any <math alttext="F\in\mathcal{F}" class="ltx_Math" display="inline" id="S2.3.p2.19.m19.1"><semantics id="S2.3.p2.19.m19.1a"><mrow id="S2.3.p2.19.m19.1.1" xref="S2.3.p2.19.m19.1.1.cmml"><mi id="S2.3.p2.19.m19.1.1.2" xref="S2.3.p2.19.m19.1.1.2.cmml">F</mi><mo id="S2.3.p2.19.m19.1.1.1" xref="S2.3.p2.19.m19.1.1.1.cmml">β</mo><mi class="ltx_font_mathcaligraphic" id="S2.3.p2.19.m19.1.1.3" xref="S2.3.p2.19.m19.1.1.3.cmml">β±</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.19.m19.1b"><apply id="S2.3.p2.19.m19.1.1.cmml" xref="S2.3.p2.19.m19.1.1"><in id="S2.3.p2.19.m19.1.1.1.cmml" xref="S2.3.p2.19.m19.1.1.1"></in><ci id="S2.3.p2.19.m19.1.1.2.cmml" xref="S2.3.p2.19.m19.1.1.2">πΉ</ci><ci id="S2.3.p2.19.m19.1.1.3.cmml" xref="S2.3.p2.19.m19.1.1.3">β±</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.19.m19.1c">F\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.19.m19.1d">italic_F β caligraphic_F</annotation></semantics></math> such that <math alttext="F\subset V" class="ltx_Math" display="inline" id="S2.3.p2.20.m20.1"><semantics id="S2.3.p2.20.m20.1a"><mrow id="S2.3.p2.20.m20.1.1" xref="S2.3.p2.20.m20.1.1.cmml"><mi id="S2.3.p2.20.m20.1.1.2" xref="S2.3.p2.20.m20.1.1.2.cmml">F</mi><mo id="S2.3.p2.20.m20.1.1.1" xref="S2.3.p2.20.m20.1.1.1.cmml">β</mo><mi id="S2.3.p2.20.m20.1.1.3" xref="S2.3.p2.20.m20.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.20.m20.1b"><apply id="S2.3.p2.20.m20.1.1.cmml" xref="S2.3.p2.20.m20.1.1"><subset id="S2.3.p2.20.m20.1.1.1.cmml" xref="S2.3.p2.20.m20.1.1.1"></subset><ci id="S2.3.p2.20.m20.1.1.2.cmml" xref="S2.3.p2.20.m20.1.1.2">πΉ</ci><ci id="S2.3.p2.20.m20.1.1.3.cmml" xref="S2.3.p2.20.m20.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.20.m20.1c">F\subset V</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.20.m20.1d">italic_F β italic_V</annotation></semantics></math> and observe that <math alttext="FF\subset U" class="ltx_Math" display="inline" id="S2.3.p2.21.m21.1"><semantics id="S2.3.p2.21.m21.1a"><mrow id="S2.3.p2.21.m21.1.1" xref="S2.3.p2.21.m21.1.1.cmml"><mrow id="S2.3.p2.21.m21.1.1.2" xref="S2.3.p2.21.m21.1.1.2.cmml"><mi id="S2.3.p2.21.m21.1.1.2.2" xref="S2.3.p2.21.m21.1.1.2.2.cmml">F</mi><mo id="S2.3.p2.21.m21.1.1.2.1" xref="S2.3.p2.21.m21.1.1.2.1.cmml">β’</mo><mi id="S2.3.p2.21.m21.1.1.2.3" xref="S2.3.p2.21.m21.1.1.2.3.cmml">F</mi></mrow><mo id="S2.3.p2.21.m21.1.1.1" xref="S2.3.p2.21.m21.1.1.1.cmml">β</mo><mi id="S2.3.p2.21.m21.1.1.3" xref="S2.3.p2.21.m21.1.1.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.21.m21.1b"><apply id="S2.3.p2.21.m21.1.1.cmml" xref="S2.3.p2.21.m21.1.1"><subset id="S2.3.p2.21.m21.1.1.1.cmml" xref="S2.3.p2.21.m21.1.1.1"></subset><apply id="S2.3.p2.21.m21.1.1.2.cmml" xref="S2.3.p2.21.m21.1.1.2"><times id="S2.3.p2.21.m21.1.1.2.1.cmml" xref="S2.3.p2.21.m21.1.1.2.1"></times><ci id="S2.3.p2.21.m21.1.1.2.2.cmml" xref="S2.3.p2.21.m21.1.1.2.2">πΉ</ci><ci id="S2.3.p2.21.m21.1.1.2.3.cmml" xref="S2.3.p2.21.m21.1.1.2.3">πΉ</ci></apply><ci id="S2.3.p2.21.m21.1.1.3.cmml" xref="S2.3.p2.21.m21.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.21.m21.1c">FF\subset U</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.21.m21.1d">italic_F italic_F β italic_U</annotation></semantics></math>. Consequently, the filter <math alttext="\mathcal{F}^{2}" class="ltx_Math" display="inline" id="S2.3.p2.22.m22.1"><semantics id="S2.3.p2.22.m22.1a"><msup id="S2.3.p2.22.m22.1.1" xref="S2.3.p2.22.m22.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.3.p2.22.m22.1.1.2" xref="S2.3.p2.22.m22.1.1.2.cmml">β±</mi><mn id="S2.3.p2.22.m22.1.1.3" xref="S2.3.p2.22.m22.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S2.3.p2.22.m22.1b"><apply id="S2.3.p2.22.m22.1.1.cmml" xref="S2.3.p2.22.m22.1.1"><csymbol cd="ambiguous" id="S2.3.p2.22.m22.1.1.1.cmml" xref="S2.3.p2.22.m22.1.1">superscript</csymbol><ci id="S2.3.p2.22.m22.1.1.2.cmml" xref="S2.3.p2.22.m22.1.1.2">β±</ci><cn id="S2.3.p2.22.m22.1.1.3.cmml" type="integer" xref="S2.3.p2.22.m22.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.22.m22.1c">\mathcal{F}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.22.m22.1d">caligraphic_F start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> generated by the family <math alttext="\{FF:F\in\mathcal{F}\}" class="ltx_Math" display="inline" id="S2.3.p2.23.m23.2"><semantics id="S2.3.p2.23.m23.2a"><mrow id="S2.3.p2.23.m23.2.2.2" xref="S2.3.p2.23.m23.2.2.3.cmml"><mo id="S2.3.p2.23.m23.2.2.2.3" stretchy="false" xref="S2.3.p2.23.m23.2.2.3.1.cmml">{</mo><mrow id="S2.3.p2.23.m23.1.1.1.1" xref="S2.3.p2.23.m23.1.1.1.1.cmml"><mi id="S2.3.p2.23.m23.1.1.1.1.2" xref="S2.3.p2.23.m23.1.1.1.1.2.cmml">F</mi><mo id="S2.3.p2.23.m23.1.1.1.1.1" xref="S2.3.p2.23.m23.1.1.1.1.1.cmml">β’</mo><mi id="S2.3.p2.23.m23.1.1.1.1.3" xref="S2.3.p2.23.m23.1.1.1.1.3.cmml">F</mi></mrow><mo id="S2.3.p2.23.m23.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.3.p2.23.m23.2.2.3.1.cmml">:</mo><mrow id="S2.3.p2.23.m23.2.2.2.2" xref="S2.3.p2.23.m23.2.2.2.2.cmml"><mi id="S2.3.p2.23.m23.2.2.2.2.2" xref="S2.3.p2.23.m23.2.2.2.2.2.cmml">F</mi><mo id="S2.3.p2.23.m23.2.2.2.2.1" xref="S2.3.p2.23.m23.2.2.2.2.1.cmml">β</mo><mi class="ltx_font_mathcaligraphic" id="S2.3.p2.23.m23.2.2.2.2.3" xref="S2.3.p2.23.m23.2.2.2.2.3.cmml">β±</mi></mrow><mo id="S2.3.p2.23.m23.2.2.2.5" stretchy="false" xref="S2.3.p2.23.m23.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.23.m23.2b"><apply id="S2.3.p2.23.m23.2.2.3.cmml" xref="S2.3.p2.23.m23.2.2.2"><csymbol cd="latexml" id="S2.3.p2.23.m23.2.2.3.1.cmml" xref="S2.3.p2.23.m23.2.2.2.3">conditional-set</csymbol><apply id="S2.3.p2.23.m23.1.1.1.1.cmml" xref="S2.3.p2.23.m23.1.1.1.1"><times id="S2.3.p2.23.m23.1.1.1.1.1.cmml" xref="S2.3.p2.23.m23.1.1.1.1.1"></times><ci id="S2.3.p2.23.m23.1.1.1.1.2.cmml" xref="S2.3.p2.23.m23.1.1.1.1.2">πΉ</ci><ci id="S2.3.p2.23.m23.1.1.1.1.3.cmml" xref="S2.3.p2.23.m23.1.1.1.1.3">πΉ</ci></apply><apply id="S2.3.p2.23.m23.2.2.2.2.cmml" xref="S2.3.p2.23.m23.2.2.2.2"><in id="S2.3.p2.23.m23.2.2.2.2.1.cmml" xref="S2.3.p2.23.m23.2.2.2.2.1"></in><ci id="S2.3.p2.23.m23.2.2.2.2.2.cmml" xref="S2.3.p2.23.m23.2.2.2.2.2">πΉ</ci><ci id="S2.3.p2.23.m23.2.2.2.2.3.cmml" xref="S2.3.p2.23.m23.2.2.2.2.3">β±</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.23.m23.2c">\{FF:F\in\mathcal{F}\}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.23.m23.2d">{ italic_F italic_F : italic_F β caligraphic_F }</annotation></semantics></math> converges to <math alttext="e" class="ltx_Math" display="inline" id="S2.3.p2.24.m24.1"><semantics id="S2.3.p2.24.m24.1a"><mi id="S2.3.p2.24.m24.1.1" xref="S2.3.p2.24.m24.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.24.m24.1b"><ci id="S2.3.p2.24.m24.1.1.cmml" xref="S2.3.p2.24.m24.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.24.m24.1c">e</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.24.m24.1d">italic_e</annotation></semantics></math>. Fix an open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S2.3.p2.25.m25.1"><semantics id="S2.3.p2.25.m25.1a"><mi id="S2.3.p2.25.m25.1.1" xref="S2.3.p2.25.m25.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.25.m25.1b"><ci id="S2.3.p2.25.m25.1.1.cmml" xref="S2.3.p2.25.m25.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.25.m25.1c">U</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.25.m25.1d">italic_U</annotation></semantics></math> of <math alttext="f" class="ltx_Math" display="inline" id="S2.3.p2.26.m26.1"><semantics id="S2.3.p2.26.m26.1a"><mi id="S2.3.p2.26.m26.1.1" xref="S2.3.p2.26.m26.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.26.m26.1b"><ci id="S2.3.p2.26.m26.1.1.cmml" xref="S2.3.p2.26.m26.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.26.m26.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.26.m26.1d">italic_f</annotation></semantics></math>. Since <math alttext="f=fef" class="ltx_Math" display="inline" id="S2.3.p2.27.m27.1"><semantics id="S2.3.p2.27.m27.1a"><mrow id="S2.3.p2.27.m27.1.1" xref="S2.3.p2.27.m27.1.1.cmml"><mi id="S2.3.p2.27.m27.1.1.2" xref="S2.3.p2.27.m27.1.1.2.cmml">f</mi><mo id="S2.3.p2.27.m27.1.1.1" xref="S2.3.p2.27.m27.1.1.1.cmml">=</mo><mrow id="S2.3.p2.27.m27.1.1.3" xref="S2.3.p2.27.m27.1.1.3.cmml"><mi id="S2.3.p2.27.m27.1.1.3.2" xref="S2.3.p2.27.m27.1.1.3.2.cmml">f</mi><mo id="S2.3.p2.27.m27.1.1.3.1" xref="S2.3.p2.27.m27.1.1.3.1.cmml">β’</mo><mi id="S2.3.p2.27.m27.1.1.3.3" xref="S2.3.p2.27.m27.1.1.3.3.cmml">e</mi><mo id="S2.3.p2.27.m27.1.1.3.1a" xref="S2.3.p2.27.m27.1.1.3.1.cmml">β’</mo><mi id="S2.3.p2.27.m27.1.1.3.4" xref="S2.3.p2.27.m27.1.1.3.4.cmml">f</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.27.m27.1b"><apply id="S2.3.p2.27.m27.1.1.cmml" xref="S2.3.p2.27.m27.1.1"><eq id="S2.3.p2.27.m27.1.1.1.cmml" xref="S2.3.p2.27.m27.1.1.1"></eq><ci id="S2.3.p2.27.m27.1.1.2.cmml" xref="S2.3.p2.27.m27.1.1.2">π</ci><apply id="S2.3.p2.27.m27.1.1.3.cmml" xref="S2.3.p2.27.m27.1.1.3"><times id="S2.3.p2.27.m27.1.1.3.1.cmml" xref="S2.3.p2.27.m27.1.1.3.1"></times><ci id="S2.3.p2.27.m27.1.1.3.2.cmml" xref="S2.3.p2.27.m27.1.1.3.2">π</ci><ci id="S2.3.p2.27.m27.1.1.3.3.cmml" xref="S2.3.p2.27.m27.1.1.3.3">π</ci><ci id="S2.3.p2.27.m27.1.1.3.4.cmml" xref="S2.3.p2.27.m27.1.1.3.4">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.27.m27.1c">f=fef</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.27.m27.1d">italic_f = italic_f italic_e italic_f</annotation></semantics></math> and <math alttext="S" class="ltx_Math" display="inline" id="S2.3.p2.28.m28.1"><semantics id="S2.3.p2.28.m28.1a"><mi id="S2.3.p2.28.m28.1.1" xref="S2.3.p2.28.m28.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.28.m28.1b"><ci id="S2.3.p2.28.m28.1.1.cmml" xref="S2.3.p2.28.m28.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.28.m28.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.28.m28.1d">italic_S</annotation></semantics></math> is a topological semigroup, there exist open neighborhoods <math alttext="V(f)" class="ltx_Math" display="inline" id="S2.3.p2.29.m29.1"><semantics id="S2.3.p2.29.m29.1a"><mrow id="S2.3.p2.29.m29.1.2" xref="S2.3.p2.29.m29.1.2.cmml"><mi id="S2.3.p2.29.m29.1.2.2" xref="S2.3.p2.29.m29.1.2.2.cmml">V</mi><mo id="S2.3.p2.29.m29.1.2.1" xref="S2.3.p2.29.m29.1.2.1.cmml">β’</mo><mrow id="S2.3.p2.29.m29.1.2.3.2" xref="S2.3.p2.29.m29.1.2.cmml"><mo id="S2.3.p2.29.m29.1.2.3.2.1" stretchy="false" xref="S2.3.p2.29.m29.1.2.cmml">(</mo><mi id="S2.3.p2.29.m29.1.1" xref="S2.3.p2.29.m29.1.1.cmml">f</mi><mo id="S2.3.p2.29.m29.1.2.3.2.2" stretchy="false" xref="S2.3.p2.29.m29.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.29.m29.1b"><apply id="S2.3.p2.29.m29.1.2.cmml" xref="S2.3.p2.29.m29.1.2"><times id="S2.3.p2.29.m29.1.2.1.cmml" xref="S2.3.p2.29.m29.1.2.1"></times><ci id="S2.3.p2.29.m29.1.2.2.cmml" xref="S2.3.p2.29.m29.1.2.2">π</ci><ci id="S2.3.p2.29.m29.1.1.cmml" xref="S2.3.p2.29.m29.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.29.m29.1c">V(f)</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.29.m29.1d">italic_V ( italic_f )</annotation></semantics></math> and <math alttext="V(e)" class="ltx_Math" display="inline" id="S2.3.p2.30.m30.1"><semantics id="S2.3.p2.30.m30.1a"><mrow id="S2.3.p2.30.m30.1.2" xref="S2.3.p2.30.m30.1.2.cmml"><mi id="S2.3.p2.30.m30.1.2.2" xref="S2.3.p2.30.m30.1.2.2.cmml">V</mi><mo id="S2.3.p2.30.m30.1.2.1" xref="S2.3.p2.30.m30.1.2.1.cmml">β’</mo><mrow id="S2.3.p2.30.m30.1.2.3.2" xref="S2.3.p2.30.m30.1.2.cmml"><mo id="S2.3.p2.30.m30.1.2.3.2.1" stretchy="false" xref="S2.3.p2.30.m30.1.2.cmml">(</mo><mi id="S2.3.p2.30.m30.1.1" xref="S2.3.p2.30.m30.1.1.cmml">e</mi><mo id="S2.3.p2.30.m30.1.2.3.2.2" stretchy="false" xref="S2.3.p2.30.m30.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.30.m30.1b"><apply id="S2.3.p2.30.m30.1.2.cmml" xref="S2.3.p2.30.m30.1.2"><times id="S2.3.p2.30.m30.1.2.1.cmml" xref="S2.3.p2.30.m30.1.2.1"></times><ci id="S2.3.p2.30.m30.1.2.2.cmml" xref="S2.3.p2.30.m30.1.2.2">π</ci><ci id="S2.3.p2.30.m30.1.1.cmml" xref="S2.3.p2.30.m30.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.30.m30.1c">V(e)</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.30.m30.1d">italic_V ( italic_e )</annotation></semantics></math> of <math alttext="f" class="ltx_Math" display="inline" id="S2.3.p2.31.m31.1"><semantics id="S2.3.p2.31.m31.1a"><mi id="S2.3.p2.31.m31.1.1" xref="S2.3.p2.31.m31.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.31.m31.1b"><ci id="S2.3.p2.31.m31.1.1.cmml" xref="S2.3.p2.31.m31.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.31.m31.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.31.m31.1d">italic_f</annotation></semantics></math> and <math alttext="e" class="ltx_Math" display="inline" id="S2.3.p2.32.m32.1"><semantics id="S2.3.p2.32.m32.1a"><mi id="S2.3.p2.32.m32.1.1" xref="S2.3.p2.32.m32.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.32.m32.1b"><ci id="S2.3.p2.32.m32.1.1.cmml" xref="S2.3.p2.32.m32.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.32.m32.1c">e</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.32.m32.1d">italic_e</annotation></semantics></math>, respectively, such that <math alttext="V(f)V(e)V(f)\subset U" class="ltx_Math" display="inline" id="S2.3.p2.33.m33.3"><semantics id="S2.3.p2.33.m33.3a"><mrow id="S2.3.p2.33.m33.3.4" xref="S2.3.p2.33.m33.3.4.cmml"><mrow id="S2.3.p2.33.m33.3.4.2" xref="S2.3.p2.33.m33.3.4.2.cmml"><mi id="S2.3.p2.33.m33.3.4.2.2" xref="S2.3.p2.33.m33.3.4.2.2.cmml">V</mi><mo id="S2.3.p2.33.m33.3.4.2.1" xref="S2.3.p2.33.m33.3.4.2.1.cmml">β’</mo><mrow id="S2.3.p2.33.m33.3.4.2.3.2" xref="S2.3.p2.33.m33.3.4.2.cmml"><mo id="S2.3.p2.33.m33.3.4.2.3.2.1" stretchy="false" xref="S2.3.p2.33.m33.3.4.2.cmml">(</mo><mi id="S2.3.p2.33.m33.1.1" xref="S2.3.p2.33.m33.1.1.cmml">f</mi><mo id="S2.3.p2.33.m33.3.4.2.3.2.2" stretchy="false" xref="S2.3.p2.33.m33.3.4.2.cmml">)</mo></mrow><mo id="S2.3.p2.33.m33.3.4.2.1a" xref="S2.3.p2.33.m33.3.4.2.1.cmml">β’</mo><mi id="S2.3.p2.33.m33.3.4.2.4" xref="S2.3.p2.33.m33.3.4.2.4.cmml">V</mi><mo id="S2.3.p2.33.m33.3.4.2.1b" xref="S2.3.p2.33.m33.3.4.2.1.cmml">β’</mo><mrow id="S2.3.p2.33.m33.3.4.2.5.2" xref="S2.3.p2.33.m33.3.4.2.cmml"><mo id="S2.3.p2.33.m33.3.4.2.5.2.1" stretchy="false" xref="S2.3.p2.33.m33.3.4.2.cmml">(</mo><mi id="S2.3.p2.33.m33.2.2" xref="S2.3.p2.33.m33.2.2.cmml">e</mi><mo id="S2.3.p2.33.m33.3.4.2.5.2.2" stretchy="false" xref="S2.3.p2.33.m33.3.4.2.cmml">)</mo></mrow><mo id="S2.3.p2.33.m33.3.4.2.1c" xref="S2.3.p2.33.m33.3.4.2.1.cmml">β’</mo><mi id="S2.3.p2.33.m33.3.4.2.6" xref="S2.3.p2.33.m33.3.4.2.6.cmml">V</mi><mo id="S2.3.p2.33.m33.3.4.2.1d" xref="S2.3.p2.33.m33.3.4.2.1.cmml">β’</mo><mrow id="S2.3.p2.33.m33.3.4.2.7.2" xref="S2.3.p2.33.m33.3.4.2.cmml"><mo id="S2.3.p2.33.m33.3.4.2.7.2.1" stretchy="false" xref="S2.3.p2.33.m33.3.4.2.cmml">(</mo><mi id="S2.3.p2.33.m33.3.3" xref="S2.3.p2.33.m33.3.3.cmml">f</mi><mo id="S2.3.p2.33.m33.3.4.2.7.2.2" stretchy="false" xref="S2.3.p2.33.m33.3.4.2.cmml">)</mo></mrow></mrow><mo id="S2.3.p2.33.m33.3.4.1" xref="S2.3.p2.33.m33.3.4.1.cmml">β</mo><mi id="S2.3.p2.33.m33.3.4.3" xref="S2.3.p2.33.m33.3.4.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.33.m33.3b"><apply id="S2.3.p2.33.m33.3.4.cmml" xref="S2.3.p2.33.m33.3.4"><subset id="S2.3.p2.33.m33.3.4.1.cmml" xref="S2.3.p2.33.m33.3.4.1"></subset><apply id="S2.3.p2.33.m33.3.4.2.cmml" xref="S2.3.p2.33.m33.3.4.2"><times id="S2.3.p2.33.m33.3.4.2.1.cmml" xref="S2.3.p2.33.m33.3.4.2.1"></times><ci id="S2.3.p2.33.m33.3.4.2.2.cmml" xref="S2.3.p2.33.m33.3.4.2.2">π</ci><ci id="S2.3.p2.33.m33.1.1.cmml" xref="S2.3.p2.33.m33.1.1">π</ci><ci id="S2.3.p2.33.m33.3.4.2.4.cmml" xref="S2.3.p2.33.m33.3.4.2.4">π</ci><ci id="S2.3.p2.33.m33.2.2.cmml" xref="S2.3.p2.33.m33.2.2">π</ci><ci id="S2.3.p2.33.m33.3.4.2.6.cmml" xref="S2.3.p2.33.m33.3.4.2.6">π</ci><ci id="S2.3.p2.33.m33.3.3.cmml" xref="S2.3.p2.33.m33.3.3">π</ci></apply><ci id="S2.3.p2.33.m33.3.4.3.cmml" xref="S2.3.p2.33.m33.3.4.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.33.m33.3c">V(f)V(e)V(f)\subset U</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.33.m33.3d">italic_V ( italic_f ) italic_V ( italic_e ) italic_V ( italic_f ) β italic_U</annotation></semantics></math>. Since the filter <math alttext="\mathcal{F}^{-1}" class="ltx_Math" display="inline" id="S2.3.p2.34.m34.1"><semantics id="S2.3.p2.34.m34.1a"><msup id="S2.3.p2.34.m34.1.1" xref="S2.3.p2.34.m34.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.3.p2.34.m34.1.1.2" xref="S2.3.p2.34.m34.1.1.2.cmml">β±</mi><mrow id="S2.3.p2.34.m34.1.1.3" xref="S2.3.p2.34.m34.1.1.3.cmml"><mo id="S2.3.p2.34.m34.1.1.3a" xref="S2.3.p2.34.m34.1.1.3.cmml">β</mo><mn id="S2.3.p2.34.m34.1.1.3.2" xref="S2.3.p2.34.m34.1.1.3.2.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.3.p2.34.m34.1b"><apply id="S2.3.p2.34.m34.1.1.cmml" xref="S2.3.p2.34.m34.1.1"><csymbol cd="ambiguous" id="S2.3.p2.34.m34.1.1.1.cmml" xref="S2.3.p2.34.m34.1.1">superscript</csymbol><ci id="S2.3.p2.34.m34.1.1.2.cmml" xref="S2.3.p2.34.m34.1.1.2">β±</ci><apply id="S2.3.p2.34.m34.1.1.3.cmml" xref="S2.3.p2.34.m34.1.1.3"><minus id="S2.3.p2.34.m34.1.1.3.1.cmml" xref="S2.3.p2.34.m34.1.1.3"></minus><cn id="S2.3.p2.34.m34.1.1.3.2.cmml" type="integer" xref="S2.3.p2.34.m34.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.34.m34.1c">\mathcal{F}^{-1}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.34.m34.1d">caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> converges to <math alttext="f" class="ltx_Math" display="inline" id="S2.3.p2.35.m35.1"><semantics id="S2.3.p2.35.m35.1a"><mi id="S2.3.p2.35.m35.1.1" xref="S2.3.p2.35.m35.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.35.m35.1b"><ci id="S2.3.p2.35.m35.1.1.cmml" xref="S2.3.p2.35.m35.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.35.m35.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.35.m35.1d">italic_f</annotation></semantics></math> and the filter <math alttext="\mathcal{F}^{2}" class="ltx_Math" display="inline" id="S2.3.p2.36.m36.1"><semantics id="S2.3.p2.36.m36.1a"><msup id="S2.3.p2.36.m36.1.1" xref="S2.3.p2.36.m36.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.3.p2.36.m36.1.1.2" xref="S2.3.p2.36.m36.1.1.2.cmml">β±</mi><mn id="S2.3.p2.36.m36.1.1.3" xref="S2.3.p2.36.m36.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S2.3.p2.36.m36.1b"><apply id="S2.3.p2.36.m36.1.1.cmml" xref="S2.3.p2.36.m36.1.1"><csymbol cd="ambiguous" id="S2.3.p2.36.m36.1.1.1.cmml" xref="S2.3.p2.36.m36.1.1">superscript</csymbol><ci id="S2.3.p2.36.m36.1.1.2.cmml" xref="S2.3.p2.36.m36.1.1.2">β±</ci><cn id="S2.3.p2.36.m36.1.1.3.cmml" type="integer" xref="S2.3.p2.36.m36.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.36.m36.1c">\mathcal{F}^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.36.m36.1d">caligraphic_F start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> converges to <math alttext="e" class="ltx_Math" display="inline" id="S2.3.p2.37.m37.1"><semantics id="S2.3.p2.37.m37.1a"><mi id="S2.3.p2.37.m37.1.1" xref="S2.3.p2.37.m37.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.37.m37.1b"><ci id="S2.3.p2.37.m37.1.1.cmml" xref="S2.3.p2.37.m37.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.37.m37.1c">e</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.37.m37.1d">italic_e</annotation></semantics></math>, there exists <math alttext="F\in\mathcal{F}" class="ltx_Math" display="inline" id="S2.3.p2.38.m38.1"><semantics id="S2.3.p2.38.m38.1a"><mrow id="S2.3.p2.38.m38.1.1" xref="S2.3.p2.38.m38.1.1.cmml"><mi id="S2.3.p2.38.m38.1.1.2" xref="S2.3.p2.38.m38.1.1.2.cmml">F</mi><mo id="S2.3.p2.38.m38.1.1.1" xref="S2.3.p2.38.m38.1.1.1.cmml">β</mo><mi class="ltx_font_mathcaligraphic" id="S2.3.p2.38.m38.1.1.3" xref="S2.3.p2.38.m38.1.1.3.cmml">β±</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.38.m38.1b"><apply id="S2.3.p2.38.m38.1.1.cmml" xref="S2.3.p2.38.m38.1.1"><in id="S2.3.p2.38.m38.1.1.1.cmml" xref="S2.3.p2.38.m38.1.1.1"></in><ci id="S2.3.p2.38.m38.1.1.2.cmml" xref="S2.3.p2.38.m38.1.1.2">πΉ</ci><ci id="S2.3.p2.38.m38.1.1.3.cmml" xref="S2.3.p2.38.m38.1.1.3">β±</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.38.m38.1c">F\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.38.m38.1d">italic_F β caligraphic_F</annotation></semantics></math> such that <math alttext="F\subseteq X" class="ltx_Math" display="inline" id="S2.3.p2.39.m39.1"><semantics id="S2.3.p2.39.m39.1a"><mrow id="S2.3.p2.39.m39.1.1" xref="S2.3.p2.39.m39.1.1.cmml"><mi id="S2.3.p2.39.m39.1.1.2" xref="S2.3.p2.39.m39.1.1.2.cmml">F</mi><mo id="S2.3.p2.39.m39.1.1.1" xref="S2.3.p2.39.m39.1.1.1.cmml">β</mo><mi id="S2.3.p2.39.m39.1.1.3" xref="S2.3.p2.39.m39.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.39.m39.1b"><apply id="S2.3.p2.39.m39.1.1.cmml" xref="S2.3.p2.39.m39.1.1"><subset id="S2.3.p2.39.m39.1.1.1.cmml" xref="S2.3.p2.39.m39.1.1.1"></subset><ci id="S2.3.p2.39.m39.1.1.2.cmml" xref="S2.3.p2.39.m39.1.1.2">πΉ</ci><ci id="S2.3.p2.39.m39.1.1.3.cmml" xref="S2.3.p2.39.m39.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.39.m39.1c">F\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.39.m39.1d">italic_F β italic_X</annotation></semantics></math>, <math alttext="F^{-1}\subset V(f)" class="ltx_Math" display="inline" id="S2.3.p2.40.m40.1"><semantics id="S2.3.p2.40.m40.1a"><mrow id="S2.3.p2.40.m40.1.2" xref="S2.3.p2.40.m40.1.2.cmml"><msup id="S2.3.p2.40.m40.1.2.2" xref="S2.3.p2.40.m40.1.2.2.cmml"><mi id="S2.3.p2.40.m40.1.2.2.2" xref="S2.3.p2.40.m40.1.2.2.2.cmml">F</mi><mrow id="S2.3.p2.40.m40.1.2.2.3" xref="S2.3.p2.40.m40.1.2.2.3.cmml"><mo id="S2.3.p2.40.m40.1.2.2.3a" xref="S2.3.p2.40.m40.1.2.2.3.cmml">β</mo><mn id="S2.3.p2.40.m40.1.2.2.3.2" xref="S2.3.p2.40.m40.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S2.3.p2.40.m40.1.2.1" xref="S2.3.p2.40.m40.1.2.1.cmml">β</mo><mrow id="S2.3.p2.40.m40.1.2.3" xref="S2.3.p2.40.m40.1.2.3.cmml"><mi id="S2.3.p2.40.m40.1.2.3.2" xref="S2.3.p2.40.m40.1.2.3.2.cmml">V</mi><mo id="S2.3.p2.40.m40.1.2.3.1" xref="S2.3.p2.40.m40.1.2.3.1.cmml">β’</mo><mrow id="S2.3.p2.40.m40.1.2.3.3.2" xref="S2.3.p2.40.m40.1.2.3.cmml"><mo id="S2.3.p2.40.m40.1.2.3.3.2.1" stretchy="false" xref="S2.3.p2.40.m40.1.2.3.cmml">(</mo><mi id="S2.3.p2.40.m40.1.1" xref="S2.3.p2.40.m40.1.1.cmml">f</mi><mo id="S2.3.p2.40.m40.1.2.3.3.2.2" stretchy="false" xref="S2.3.p2.40.m40.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.40.m40.1b"><apply id="S2.3.p2.40.m40.1.2.cmml" xref="S2.3.p2.40.m40.1.2"><subset id="S2.3.p2.40.m40.1.2.1.cmml" xref="S2.3.p2.40.m40.1.2.1"></subset><apply id="S2.3.p2.40.m40.1.2.2.cmml" xref="S2.3.p2.40.m40.1.2.2"><csymbol cd="ambiguous" id="S2.3.p2.40.m40.1.2.2.1.cmml" xref="S2.3.p2.40.m40.1.2.2">superscript</csymbol><ci id="S2.3.p2.40.m40.1.2.2.2.cmml" xref="S2.3.p2.40.m40.1.2.2.2">πΉ</ci><apply id="S2.3.p2.40.m40.1.2.2.3.cmml" xref="S2.3.p2.40.m40.1.2.2.3"><minus id="S2.3.p2.40.m40.1.2.2.3.1.cmml" xref="S2.3.p2.40.m40.1.2.2.3"></minus><cn id="S2.3.p2.40.m40.1.2.2.3.2.cmml" type="integer" xref="S2.3.p2.40.m40.1.2.2.3.2">1</cn></apply></apply><apply id="S2.3.p2.40.m40.1.2.3.cmml" xref="S2.3.p2.40.m40.1.2.3"><times id="S2.3.p2.40.m40.1.2.3.1.cmml" xref="S2.3.p2.40.m40.1.2.3.1"></times><ci id="S2.3.p2.40.m40.1.2.3.2.cmml" xref="S2.3.p2.40.m40.1.2.3.2">π</ci><ci id="S2.3.p2.40.m40.1.1.cmml" xref="S2.3.p2.40.m40.1.1">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.40.m40.1c">F^{-1}\subset V(f)</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.40.m40.1d">italic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT β italic_V ( italic_f )</annotation></semantics></math> and <math alttext="FF\subset V(e)" class="ltx_Math" display="inline" id="S2.3.p2.41.m41.1"><semantics id="S2.3.p2.41.m41.1a"><mrow id="S2.3.p2.41.m41.1.2" xref="S2.3.p2.41.m41.1.2.cmml"><mrow id="S2.3.p2.41.m41.1.2.2" xref="S2.3.p2.41.m41.1.2.2.cmml"><mi id="S2.3.p2.41.m41.1.2.2.2" xref="S2.3.p2.41.m41.1.2.2.2.cmml">F</mi><mo id="S2.3.p2.41.m41.1.2.2.1" xref="S2.3.p2.41.m41.1.2.2.1.cmml">β’</mo><mi id="S2.3.p2.41.m41.1.2.2.3" xref="S2.3.p2.41.m41.1.2.2.3.cmml">F</mi></mrow><mo id="S2.3.p2.41.m41.1.2.1" xref="S2.3.p2.41.m41.1.2.1.cmml">β</mo><mrow id="S2.3.p2.41.m41.1.2.3" xref="S2.3.p2.41.m41.1.2.3.cmml"><mi id="S2.3.p2.41.m41.1.2.3.2" xref="S2.3.p2.41.m41.1.2.3.2.cmml">V</mi><mo id="S2.3.p2.41.m41.1.2.3.1" xref="S2.3.p2.41.m41.1.2.3.1.cmml">β’</mo><mrow id="S2.3.p2.41.m41.1.2.3.3.2" xref="S2.3.p2.41.m41.1.2.3.cmml"><mo id="S2.3.p2.41.m41.1.2.3.3.2.1" stretchy="false" xref="S2.3.p2.41.m41.1.2.3.cmml">(</mo><mi id="S2.3.p2.41.m41.1.1" xref="S2.3.p2.41.m41.1.1.cmml">e</mi><mo id="S2.3.p2.41.m41.1.2.3.3.2.2" stretchy="false" xref="S2.3.p2.41.m41.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.41.m41.1b"><apply id="S2.3.p2.41.m41.1.2.cmml" xref="S2.3.p2.41.m41.1.2"><subset id="S2.3.p2.41.m41.1.2.1.cmml" xref="S2.3.p2.41.m41.1.2.1"></subset><apply id="S2.3.p2.41.m41.1.2.2.cmml" xref="S2.3.p2.41.m41.1.2.2"><times id="S2.3.p2.41.m41.1.2.2.1.cmml" xref="S2.3.p2.41.m41.1.2.2.1"></times><ci id="S2.3.p2.41.m41.1.2.2.2.cmml" xref="S2.3.p2.41.m41.1.2.2.2">πΉ</ci><ci id="S2.3.p2.41.m41.1.2.2.3.cmml" xref="S2.3.p2.41.m41.1.2.2.3">πΉ</ci></apply><apply id="S2.3.p2.41.m41.1.2.3.cmml" xref="S2.3.p2.41.m41.1.2.3"><times id="S2.3.p2.41.m41.1.2.3.1.cmml" xref="S2.3.p2.41.m41.1.2.3.1"></times><ci id="S2.3.p2.41.m41.1.2.3.2.cmml" xref="S2.3.p2.41.m41.1.2.3.2">π</ci><ci id="S2.3.p2.41.m41.1.1.cmml" xref="S2.3.p2.41.m41.1.1">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.41.m41.1c">FF\subset V(e)</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.41.m41.1d">italic_F italic_F β italic_V ( italic_e )</annotation></semantics></math>. Fix any <math alttext="x\in F" class="ltx_Math" display="inline" id="S2.3.p2.42.m42.1"><semantics id="S2.3.p2.42.m42.1a"><mrow id="S2.3.p2.42.m42.1.1" xref="S2.3.p2.42.m42.1.1.cmml"><mi id="S2.3.p2.42.m42.1.1.2" xref="S2.3.p2.42.m42.1.1.2.cmml">x</mi><mo id="S2.3.p2.42.m42.1.1.1" xref="S2.3.p2.42.m42.1.1.1.cmml">β</mo><mi id="S2.3.p2.42.m42.1.1.3" xref="S2.3.p2.42.m42.1.1.3.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.42.m42.1b"><apply id="S2.3.p2.42.m42.1.1.cmml" xref="S2.3.p2.42.m42.1.1"><in id="S2.3.p2.42.m42.1.1.1.cmml" xref="S2.3.p2.42.m42.1.1.1"></in><ci id="S2.3.p2.42.m42.1.1.2.cmml" xref="S2.3.p2.42.m42.1.1.2">π₯</ci><ci id="S2.3.p2.42.m42.1.1.3.cmml" xref="S2.3.p2.42.m42.1.1.3">πΉ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.42.m42.1c">x\in F</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.42.m42.1d">italic_x β italic_F</annotation></semantics></math>. Observe that</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="x^{-1}xxx^{-1}\in F^{-1}FFF^{-1}\subset V(f)V(e)V(f)\subseteq U." class="ltx_Math" display="block" id="S2.Ex2.m1.4"><semantics id="S2.Ex2.m1.4a"><mrow id="S2.Ex2.m1.4.4.1" xref="S2.Ex2.m1.4.4.1.1.cmml"><mrow id="S2.Ex2.m1.4.4.1.1" xref="S2.Ex2.m1.4.4.1.1.cmml"><mrow id="S2.Ex2.m1.4.4.1.1.2" xref="S2.Ex2.m1.4.4.1.1.2.cmml"><msup id="S2.Ex2.m1.4.4.1.1.2.2" xref="S2.Ex2.m1.4.4.1.1.2.2.cmml"><mi id="S2.Ex2.m1.4.4.1.1.2.2.2" xref="S2.Ex2.m1.4.4.1.1.2.2.2.cmml">x</mi><mrow id="S2.Ex2.m1.4.4.1.1.2.2.3" xref="S2.Ex2.m1.4.4.1.1.2.2.3.cmml"><mo id="S2.Ex2.m1.4.4.1.1.2.2.3a" xref="S2.Ex2.m1.4.4.1.1.2.2.3.cmml">β</mo><mn id="S2.Ex2.m1.4.4.1.1.2.2.3.2" xref="S2.Ex2.m1.4.4.1.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S2.Ex2.m1.4.4.1.1.2.1" xref="S2.Ex2.m1.4.4.1.1.2.1.cmml">β’</mo><mi id="S2.Ex2.m1.4.4.1.1.2.3" xref="S2.Ex2.m1.4.4.1.1.2.3.cmml">x</mi><mo id="S2.Ex2.m1.4.4.1.1.2.1a" xref="S2.Ex2.m1.4.4.1.1.2.1.cmml">β’</mo><mi id="S2.Ex2.m1.4.4.1.1.2.4" xref="S2.Ex2.m1.4.4.1.1.2.4.cmml">x</mi><mo id="S2.Ex2.m1.4.4.1.1.2.1b" xref="S2.Ex2.m1.4.4.1.1.2.1.cmml">β’</mo><msup id="S2.Ex2.m1.4.4.1.1.2.5" xref="S2.Ex2.m1.4.4.1.1.2.5.cmml"><mi id="S2.Ex2.m1.4.4.1.1.2.5.2" xref="S2.Ex2.m1.4.4.1.1.2.5.2.cmml">x</mi><mrow id="S2.Ex2.m1.4.4.1.1.2.5.3" xref="S2.Ex2.m1.4.4.1.1.2.5.3.cmml"><mo id="S2.Ex2.m1.4.4.1.1.2.5.3a" xref="S2.Ex2.m1.4.4.1.1.2.5.3.cmml">β</mo><mn id="S2.Ex2.m1.4.4.1.1.2.5.3.2" xref="S2.Ex2.m1.4.4.1.1.2.5.3.2.cmml">1</mn></mrow></msup></mrow><mo id="S2.Ex2.m1.4.4.1.1.3" xref="S2.Ex2.m1.4.4.1.1.3.cmml">β</mo><mrow id="S2.Ex2.m1.4.4.1.1.4" xref="S2.Ex2.m1.4.4.1.1.4.cmml"><msup id="S2.Ex2.m1.4.4.1.1.4.2" xref="S2.Ex2.m1.4.4.1.1.4.2.cmml"><mi id="S2.Ex2.m1.4.4.1.1.4.2.2" xref="S2.Ex2.m1.4.4.1.1.4.2.2.cmml">F</mi><mrow id="S2.Ex2.m1.4.4.1.1.4.2.3" xref="S2.Ex2.m1.4.4.1.1.4.2.3.cmml"><mo id="S2.Ex2.m1.4.4.1.1.4.2.3a" xref="S2.Ex2.m1.4.4.1.1.4.2.3.cmml">β</mo><mn id="S2.Ex2.m1.4.4.1.1.4.2.3.2" xref="S2.Ex2.m1.4.4.1.1.4.2.3.2.cmml">1</mn></mrow></msup><mo id="S2.Ex2.m1.4.4.1.1.4.1" xref="S2.Ex2.m1.4.4.1.1.4.1.cmml">β’</mo><mi id="S2.Ex2.m1.4.4.1.1.4.3" xref="S2.Ex2.m1.4.4.1.1.4.3.cmml">F</mi><mo id="S2.Ex2.m1.4.4.1.1.4.1a" xref="S2.Ex2.m1.4.4.1.1.4.1.cmml">β’</mo><mi id="S2.Ex2.m1.4.4.1.1.4.4" xref="S2.Ex2.m1.4.4.1.1.4.4.cmml">F</mi><mo id="S2.Ex2.m1.4.4.1.1.4.1b" xref="S2.Ex2.m1.4.4.1.1.4.1.cmml">β’</mo><msup id="S2.Ex2.m1.4.4.1.1.4.5" xref="S2.Ex2.m1.4.4.1.1.4.5.cmml"><mi id="S2.Ex2.m1.4.4.1.1.4.5.2" xref="S2.Ex2.m1.4.4.1.1.4.5.2.cmml">F</mi><mrow id="S2.Ex2.m1.4.4.1.1.4.5.3" xref="S2.Ex2.m1.4.4.1.1.4.5.3.cmml"><mo id="S2.Ex2.m1.4.4.1.1.4.5.3a" xref="S2.Ex2.m1.4.4.1.1.4.5.3.cmml">β</mo><mn id="S2.Ex2.m1.4.4.1.1.4.5.3.2" xref="S2.Ex2.m1.4.4.1.1.4.5.3.2.cmml">1</mn></mrow></msup></mrow><mo id="S2.Ex2.m1.4.4.1.1.5" xref="S2.Ex2.m1.4.4.1.1.5.cmml">β</mo><mrow id="S2.Ex2.m1.4.4.1.1.6" xref="S2.Ex2.m1.4.4.1.1.6.cmml"><mi id="S2.Ex2.m1.4.4.1.1.6.2" xref="S2.Ex2.m1.4.4.1.1.6.2.cmml">V</mi><mo id="S2.Ex2.m1.4.4.1.1.6.1" xref="S2.Ex2.m1.4.4.1.1.6.1.cmml">β’</mo><mrow id="S2.Ex2.m1.4.4.1.1.6.3.2" xref="S2.Ex2.m1.4.4.1.1.6.cmml"><mo id="S2.Ex2.m1.4.4.1.1.6.3.2.1" stretchy="false" xref="S2.Ex2.m1.4.4.1.1.6.cmml">(</mo><mi id="S2.Ex2.m1.1.1" xref="S2.Ex2.m1.1.1.cmml">f</mi><mo id="S2.Ex2.m1.4.4.1.1.6.3.2.2" stretchy="false" xref="S2.Ex2.m1.4.4.1.1.6.cmml">)</mo></mrow><mo id="S2.Ex2.m1.4.4.1.1.6.1a" xref="S2.Ex2.m1.4.4.1.1.6.1.cmml">β’</mo><mi id="S2.Ex2.m1.4.4.1.1.6.4" xref="S2.Ex2.m1.4.4.1.1.6.4.cmml">V</mi><mo id="S2.Ex2.m1.4.4.1.1.6.1b" xref="S2.Ex2.m1.4.4.1.1.6.1.cmml">β’</mo><mrow id="S2.Ex2.m1.4.4.1.1.6.5.2" xref="S2.Ex2.m1.4.4.1.1.6.cmml"><mo id="S2.Ex2.m1.4.4.1.1.6.5.2.1" stretchy="false" xref="S2.Ex2.m1.4.4.1.1.6.cmml">(</mo><mi id="S2.Ex2.m1.2.2" xref="S2.Ex2.m1.2.2.cmml">e</mi><mo id="S2.Ex2.m1.4.4.1.1.6.5.2.2" stretchy="false" xref="S2.Ex2.m1.4.4.1.1.6.cmml">)</mo></mrow><mo id="S2.Ex2.m1.4.4.1.1.6.1c" xref="S2.Ex2.m1.4.4.1.1.6.1.cmml">β’</mo><mi id="S2.Ex2.m1.4.4.1.1.6.6" xref="S2.Ex2.m1.4.4.1.1.6.6.cmml">V</mi><mo id="S2.Ex2.m1.4.4.1.1.6.1d" xref="S2.Ex2.m1.4.4.1.1.6.1.cmml">β’</mo><mrow id="S2.Ex2.m1.4.4.1.1.6.7.2" xref="S2.Ex2.m1.4.4.1.1.6.cmml"><mo id="S2.Ex2.m1.4.4.1.1.6.7.2.1" stretchy="false" xref="S2.Ex2.m1.4.4.1.1.6.cmml">(</mo><mi id="S2.Ex2.m1.3.3" xref="S2.Ex2.m1.3.3.cmml">f</mi><mo id="S2.Ex2.m1.4.4.1.1.6.7.2.2" stretchy="false" xref="S2.Ex2.m1.4.4.1.1.6.cmml">)</mo></mrow></mrow><mo id="S2.Ex2.m1.4.4.1.1.7" xref="S2.Ex2.m1.4.4.1.1.7.cmml">β</mo><mi id="S2.Ex2.m1.4.4.1.1.8" xref="S2.Ex2.m1.4.4.1.1.8.cmml">U</mi></mrow><mo id="S2.Ex2.m1.4.4.1.2" lspace="0em" xref="S2.Ex2.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex2.m1.4b"><apply id="S2.Ex2.m1.4.4.1.1.cmml" xref="S2.Ex2.m1.4.4.1"><and id="S2.Ex2.m1.4.4.1.1a.cmml" xref="S2.Ex2.m1.4.4.1"></and><apply id="S2.Ex2.m1.4.4.1.1b.cmml" xref="S2.Ex2.m1.4.4.1"><in id="S2.Ex2.m1.4.4.1.1.3.cmml" xref="S2.Ex2.m1.4.4.1.1.3"></in><apply id="S2.Ex2.m1.4.4.1.1.2.cmml" xref="S2.Ex2.m1.4.4.1.1.2"><times id="S2.Ex2.m1.4.4.1.1.2.1.cmml" xref="S2.Ex2.m1.4.4.1.1.2.1"></times><apply id="S2.Ex2.m1.4.4.1.1.2.2.cmml" xref="S2.Ex2.m1.4.4.1.1.2.2"><csymbol cd="ambiguous" id="S2.Ex2.m1.4.4.1.1.2.2.1.cmml" xref="S2.Ex2.m1.4.4.1.1.2.2">superscript</csymbol><ci id="S2.Ex2.m1.4.4.1.1.2.2.2.cmml" xref="S2.Ex2.m1.4.4.1.1.2.2.2">π₯</ci><apply id="S2.Ex2.m1.4.4.1.1.2.2.3.cmml" xref="S2.Ex2.m1.4.4.1.1.2.2.3"><minus id="S2.Ex2.m1.4.4.1.1.2.2.3.1.cmml" xref="S2.Ex2.m1.4.4.1.1.2.2.3"></minus><cn id="S2.Ex2.m1.4.4.1.1.2.2.3.2.cmml" type="integer" xref="S2.Ex2.m1.4.4.1.1.2.2.3.2">1</cn></apply></apply><ci id="S2.Ex2.m1.4.4.1.1.2.3.cmml" xref="S2.Ex2.m1.4.4.1.1.2.3">π₯</ci><ci id="S2.Ex2.m1.4.4.1.1.2.4.cmml" 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xref="S2.Ex2.m1.4.4.1.1.5"></subset><share href="https://arxiv.org/html/2503.13666v1#S2.Ex2.m1.4.4.1.1.4.cmml" id="S2.Ex2.m1.4.4.1.1d.cmml" xref="S2.Ex2.m1.4.4.1"></share><apply id="S2.Ex2.m1.4.4.1.1.6.cmml" xref="S2.Ex2.m1.4.4.1.1.6"><times id="S2.Ex2.m1.4.4.1.1.6.1.cmml" xref="S2.Ex2.m1.4.4.1.1.6.1"></times><ci id="S2.Ex2.m1.4.4.1.1.6.2.cmml" xref="S2.Ex2.m1.4.4.1.1.6.2">π</ci><ci id="S2.Ex2.m1.1.1.cmml" xref="S2.Ex2.m1.1.1">π</ci><ci id="S2.Ex2.m1.4.4.1.1.6.4.cmml" xref="S2.Ex2.m1.4.4.1.1.6.4">π</ci><ci id="S2.Ex2.m1.2.2.cmml" xref="S2.Ex2.m1.2.2">π</ci><ci id="S2.Ex2.m1.4.4.1.1.6.6.cmml" xref="S2.Ex2.m1.4.4.1.1.6.6">π</ci><ci id="S2.Ex2.m1.3.3.cmml" xref="S2.Ex2.m1.3.3">π</ci></apply></apply><apply id="S2.Ex2.m1.4.4.1.1e.cmml" xref="S2.Ex2.m1.4.4.1"><subset id="S2.Ex2.m1.4.4.1.1.7.cmml" xref="S2.Ex2.m1.4.4.1.1.7"></subset><share href="https://arxiv.org/html/2503.13666v1#S2.Ex2.m1.4.4.1.1.6.cmml" id="S2.Ex2.m1.4.4.1.1f.cmml" xref="S2.Ex2.m1.4.4.1"></share><ci id="S2.Ex2.m1.4.4.1.1.8.cmml" xref="S2.Ex2.m1.4.4.1.1.8">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m1.4c">x^{-1}xxx^{-1}\in F^{-1}FFF^{-1}\subset V(f)V(e)V(f)\subseteq U.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m1.4d">italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT β italic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_F italic_F italic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT β italic_V ( italic_f ) italic_V ( italic_e ) italic_V ( italic_f ) β italic_U .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.3.p2.70">Since <math alttext="X" class="ltx_Math" display="inline" id="S2.3.p2.43.m1.1"><semantics id="S2.3.p2.43.m1.1a"><mi id="S2.3.p2.43.m1.1.1" xref="S2.3.p2.43.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.43.m1.1b"><ci id="S2.3.p2.43.m1.1.1.cmml" xref="S2.3.p2.43.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.43.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.43.m1.1d">italic_X</annotation></semantics></math> is an inverse semigroup, the element <math alttext="x^{-1}xxx^{-1}" class="ltx_Math" display="inline" id="S2.3.p2.44.m2.1"><semantics id="S2.3.p2.44.m2.1a"><mrow id="S2.3.p2.44.m2.1.1" xref="S2.3.p2.44.m2.1.1.cmml"><msup id="S2.3.p2.44.m2.1.1.2" xref="S2.3.p2.44.m2.1.1.2.cmml"><mi id="S2.3.p2.44.m2.1.1.2.2" xref="S2.3.p2.44.m2.1.1.2.2.cmml">x</mi><mrow id="S2.3.p2.44.m2.1.1.2.3" xref="S2.3.p2.44.m2.1.1.2.3.cmml"><mo id="S2.3.p2.44.m2.1.1.2.3a" xref="S2.3.p2.44.m2.1.1.2.3.cmml">β</mo><mn id="S2.3.p2.44.m2.1.1.2.3.2" xref="S2.3.p2.44.m2.1.1.2.3.2.cmml">1</mn></mrow></msup><mo id="S2.3.p2.44.m2.1.1.1" xref="S2.3.p2.44.m2.1.1.1.cmml">β’</mo><mi id="S2.3.p2.44.m2.1.1.3" xref="S2.3.p2.44.m2.1.1.3.cmml">x</mi><mo id="S2.3.p2.44.m2.1.1.1a" xref="S2.3.p2.44.m2.1.1.1.cmml">β’</mo><mi id="S2.3.p2.44.m2.1.1.4" xref="S2.3.p2.44.m2.1.1.4.cmml">x</mi><mo id="S2.3.p2.44.m2.1.1.1b" xref="S2.3.p2.44.m2.1.1.1.cmml">β’</mo><msup id="S2.3.p2.44.m2.1.1.5" xref="S2.3.p2.44.m2.1.1.5.cmml"><mi id="S2.3.p2.44.m2.1.1.5.2" xref="S2.3.p2.44.m2.1.1.5.2.cmml">x</mi><mrow id="S2.3.p2.44.m2.1.1.5.3" xref="S2.3.p2.44.m2.1.1.5.3.cmml"><mo id="S2.3.p2.44.m2.1.1.5.3a" xref="S2.3.p2.44.m2.1.1.5.3.cmml">β</mo><mn id="S2.3.p2.44.m2.1.1.5.3.2" xref="S2.3.p2.44.m2.1.1.5.3.2.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.44.m2.1b"><apply id="S2.3.p2.44.m2.1.1.cmml" xref="S2.3.p2.44.m2.1.1"><times id="S2.3.p2.44.m2.1.1.1.cmml" xref="S2.3.p2.44.m2.1.1.1"></times><apply id="S2.3.p2.44.m2.1.1.2.cmml" xref="S2.3.p2.44.m2.1.1.2"><csymbol cd="ambiguous" id="S2.3.p2.44.m2.1.1.2.1.cmml" xref="S2.3.p2.44.m2.1.1.2">superscript</csymbol><ci id="S2.3.p2.44.m2.1.1.2.2.cmml" xref="S2.3.p2.44.m2.1.1.2.2">π₯</ci><apply id="S2.3.p2.44.m2.1.1.2.3.cmml" xref="S2.3.p2.44.m2.1.1.2.3"><minus id="S2.3.p2.44.m2.1.1.2.3.1.cmml" xref="S2.3.p2.44.m2.1.1.2.3"></minus><cn id="S2.3.p2.44.m2.1.1.2.3.2.cmml" type="integer" xref="S2.3.p2.44.m2.1.1.2.3.2">1</cn></apply></apply><ci id="S2.3.p2.44.m2.1.1.3.cmml" xref="S2.3.p2.44.m2.1.1.3">π₯</ci><ci id="S2.3.p2.44.m2.1.1.4.cmml" xref="S2.3.p2.44.m2.1.1.4">π₯</ci><apply id="S2.3.p2.44.m2.1.1.5.cmml" xref="S2.3.p2.44.m2.1.1.5"><csymbol cd="ambiguous" id="S2.3.p2.44.m2.1.1.5.1.cmml" xref="S2.3.p2.44.m2.1.1.5">superscript</csymbol><ci id="S2.3.p2.44.m2.1.1.5.2.cmml" xref="S2.3.p2.44.m2.1.1.5.2">π₯</ci><apply id="S2.3.p2.44.m2.1.1.5.3.cmml" xref="S2.3.p2.44.m2.1.1.5.3"><minus id="S2.3.p2.44.m2.1.1.5.3.1.cmml" xref="S2.3.p2.44.m2.1.1.5.3"></minus><cn id="S2.3.p2.44.m2.1.1.5.3.2.cmml" type="integer" xref="S2.3.p2.44.m2.1.1.5.3.2">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.44.m2.1c">x^{-1}xxx^{-1}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.44.m2.1d">italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> is an idempotent. As the open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S2.3.p2.45.m3.1"><semantics id="S2.3.p2.45.m3.1a"><mi id="S2.3.p2.45.m3.1.1" xref="S2.3.p2.45.m3.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.45.m3.1b"><ci id="S2.3.p2.45.m3.1.1.cmml" xref="S2.3.p2.45.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.45.m3.1c">U</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.45.m3.1d">italic_U</annotation></semantics></math> of <math alttext="f" class="ltx_Math" display="inline" id="S2.3.p2.46.m4.1"><semantics id="S2.3.p2.46.m4.1a"><mi id="S2.3.p2.46.m4.1.1" xref="S2.3.p2.46.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.46.m4.1b"><ci id="S2.3.p2.46.m4.1.1.cmml" xref="S2.3.p2.46.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.46.m4.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.46.m4.1d">italic_f</annotation></semantics></math> is arbitrarily chosen, we get <math alttext="f\in\overline{E(X)}" class="ltx_Math" display="inline" id="S2.3.p2.47.m5.1"><semantics id="S2.3.p2.47.m5.1a"><mrow id="S2.3.p2.47.m5.1.2" xref="S2.3.p2.47.m5.1.2.cmml"><mi id="S2.3.p2.47.m5.1.2.2" xref="S2.3.p2.47.m5.1.2.2.cmml">f</mi><mo id="S2.3.p2.47.m5.1.2.1" xref="S2.3.p2.47.m5.1.2.1.cmml">β</mo><mover accent="true" id="S2.3.p2.47.m5.1.1" xref="S2.3.p2.47.m5.1.1.cmml"><mrow id="S2.3.p2.47.m5.1.1.1" xref="S2.3.p2.47.m5.1.1.1.cmml"><mi id="S2.3.p2.47.m5.1.1.1.3" xref="S2.3.p2.47.m5.1.1.1.3.cmml">E</mi><mo id="S2.3.p2.47.m5.1.1.1.2" xref="S2.3.p2.47.m5.1.1.1.2.cmml">β’</mo><mrow id="S2.3.p2.47.m5.1.1.1.4.2" xref="S2.3.p2.47.m5.1.1.1.cmml"><mo id="S2.3.p2.47.m5.1.1.1.4.2.1" stretchy="false" xref="S2.3.p2.47.m5.1.1.1.cmml">(</mo><mi id="S2.3.p2.47.m5.1.1.1.1" xref="S2.3.p2.47.m5.1.1.1.1.cmml">X</mi><mo id="S2.3.p2.47.m5.1.1.1.4.2.2" stretchy="false" xref="S2.3.p2.47.m5.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.3.p2.47.m5.1.1.2" xref="S2.3.p2.47.m5.1.1.2.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.47.m5.1b"><apply id="S2.3.p2.47.m5.1.2.cmml" xref="S2.3.p2.47.m5.1.2"><in id="S2.3.p2.47.m5.1.2.1.cmml" xref="S2.3.p2.47.m5.1.2.1"></in><ci id="S2.3.p2.47.m5.1.2.2.cmml" xref="S2.3.p2.47.m5.1.2.2">π</ci><apply id="S2.3.p2.47.m5.1.1.cmml" xref="S2.3.p2.47.m5.1.1"><ci id="S2.3.p2.47.m5.1.1.2.cmml" xref="S2.3.p2.47.m5.1.1.2">Β―</ci><apply id="S2.3.p2.47.m5.1.1.1.cmml" xref="S2.3.p2.47.m5.1.1.1"><times id="S2.3.p2.47.m5.1.1.1.2.cmml" xref="S2.3.p2.47.m5.1.1.1.2"></times><ci id="S2.3.p2.47.m5.1.1.1.3.cmml" xref="S2.3.p2.47.m5.1.1.1.3">πΈ</ci><ci id="S2.3.p2.47.m5.1.1.1.1.cmml" xref="S2.3.p2.47.m5.1.1.1.1">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.47.m5.1c">f\in\overline{E(X)}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.47.m5.1d">italic_f β overΒ― start_ARG italic_E ( italic_X ) end_ARG</annotation></semantics></math>. Since <math alttext="S" class="ltx_Math" display="inline" id="S2.3.p2.48.m6.1"><semantics id="S2.3.p2.48.m6.1a"><mi id="S2.3.p2.48.m6.1.1" xref="S2.3.p2.48.m6.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.48.m6.1b"><ci id="S2.3.p2.48.m6.1.1.cmml" xref="S2.3.p2.48.m6.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.48.m6.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.48.m6.1d">italic_S</annotation></semantics></math> is a Hausdorff topological semigroup, <math alttext="f" class="ltx_Math" display="inline" id="S2.3.p2.49.m7.1"><semantics id="S2.3.p2.49.m7.1a"><mi id="S2.3.p2.49.m7.1.1" xref="S2.3.p2.49.m7.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.49.m7.1b"><ci id="S2.3.p2.49.m7.1.1.cmml" xref="S2.3.p2.49.m7.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.49.m7.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.49.m7.1d">italic_f</annotation></semantics></math> is an idempotent. Let <math alttext="\mathcal{F}^{-2}" class="ltx_Math" display="inline" id="S2.3.p2.50.m8.1"><semantics id="S2.3.p2.50.m8.1a"><msup id="S2.3.p2.50.m8.1.1" xref="S2.3.p2.50.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.3.p2.50.m8.1.1.2" xref="S2.3.p2.50.m8.1.1.2.cmml">β±</mi><mrow id="S2.3.p2.50.m8.1.1.3" xref="S2.3.p2.50.m8.1.1.3.cmml"><mo id="S2.3.p2.50.m8.1.1.3a" xref="S2.3.p2.50.m8.1.1.3.cmml">β</mo><mn id="S2.3.p2.50.m8.1.1.3.2" xref="S2.3.p2.50.m8.1.1.3.2.cmml">2</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.3.p2.50.m8.1b"><apply id="S2.3.p2.50.m8.1.1.cmml" xref="S2.3.p2.50.m8.1.1"><csymbol cd="ambiguous" id="S2.3.p2.50.m8.1.1.1.cmml" xref="S2.3.p2.50.m8.1.1">superscript</csymbol><ci id="S2.3.p2.50.m8.1.1.2.cmml" xref="S2.3.p2.50.m8.1.1.2">β±</ci><apply id="S2.3.p2.50.m8.1.1.3.cmml" xref="S2.3.p2.50.m8.1.1.3"><minus id="S2.3.p2.50.m8.1.1.3.1.cmml" xref="S2.3.p2.50.m8.1.1.3"></minus><cn id="S2.3.p2.50.m8.1.1.3.2.cmml" type="integer" xref="S2.3.p2.50.m8.1.1.3.2">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.50.m8.1c">\mathcal{F}^{-2}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.50.m8.1d">caligraphic_F start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math> be the filter generated by the family <math alttext="\{FF:F\in\mathcal{F}^{-1}\}" class="ltx_Math" display="inline" id="S2.3.p2.51.m9.2"><semantics id="S2.3.p2.51.m9.2a"><mrow id="S2.3.p2.51.m9.2.2.2" xref="S2.3.p2.51.m9.2.2.3.cmml"><mo id="S2.3.p2.51.m9.2.2.2.3" stretchy="false" xref="S2.3.p2.51.m9.2.2.3.1.cmml">{</mo><mrow id="S2.3.p2.51.m9.1.1.1.1" xref="S2.3.p2.51.m9.1.1.1.1.cmml"><mi id="S2.3.p2.51.m9.1.1.1.1.2" xref="S2.3.p2.51.m9.1.1.1.1.2.cmml">F</mi><mo id="S2.3.p2.51.m9.1.1.1.1.1" xref="S2.3.p2.51.m9.1.1.1.1.1.cmml">β’</mo><mi id="S2.3.p2.51.m9.1.1.1.1.3" xref="S2.3.p2.51.m9.1.1.1.1.3.cmml">F</mi></mrow><mo id="S2.3.p2.51.m9.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.3.p2.51.m9.2.2.3.1.cmml">:</mo><mrow id="S2.3.p2.51.m9.2.2.2.2" xref="S2.3.p2.51.m9.2.2.2.2.cmml"><mi id="S2.3.p2.51.m9.2.2.2.2.2" xref="S2.3.p2.51.m9.2.2.2.2.2.cmml">F</mi><mo id="S2.3.p2.51.m9.2.2.2.2.1" xref="S2.3.p2.51.m9.2.2.2.2.1.cmml">β</mo><msup id="S2.3.p2.51.m9.2.2.2.2.3" xref="S2.3.p2.51.m9.2.2.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.3.p2.51.m9.2.2.2.2.3.2" xref="S2.3.p2.51.m9.2.2.2.2.3.2.cmml">β±</mi><mrow id="S2.3.p2.51.m9.2.2.2.2.3.3" xref="S2.3.p2.51.m9.2.2.2.2.3.3.cmml"><mo id="S2.3.p2.51.m9.2.2.2.2.3.3a" xref="S2.3.p2.51.m9.2.2.2.2.3.3.cmml">β</mo><mn id="S2.3.p2.51.m9.2.2.2.2.3.3.2" xref="S2.3.p2.51.m9.2.2.2.2.3.3.2.cmml">1</mn></mrow></msup></mrow><mo id="S2.3.p2.51.m9.2.2.2.5" stretchy="false" xref="S2.3.p2.51.m9.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.51.m9.2b"><apply id="S2.3.p2.51.m9.2.2.3.cmml" xref="S2.3.p2.51.m9.2.2.2"><csymbol cd="latexml" id="S2.3.p2.51.m9.2.2.3.1.cmml" xref="S2.3.p2.51.m9.2.2.2.3">conditional-set</csymbol><apply id="S2.3.p2.51.m9.1.1.1.1.cmml" xref="S2.3.p2.51.m9.1.1.1.1"><times id="S2.3.p2.51.m9.1.1.1.1.1.cmml" xref="S2.3.p2.51.m9.1.1.1.1.1"></times><ci id="S2.3.p2.51.m9.1.1.1.1.2.cmml" xref="S2.3.p2.51.m9.1.1.1.1.2">πΉ</ci><ci id="S2.3.p2.51.m9.1.1.1.1.3.cmml" xref="S2.3.p2.51.m9.1.1.1.1.3">πΉ</ci></apply><apply id="S2.3.p2.51.m9.2.2.2.2.cmml" xref="S2.3.p2.51.m9.2.2.2.2"><in id="S2.3.p2.51.m9.2.2.2.2.1.cmml" xref="S2.3.p2.51.m9.2.2.2.2.1"></in><ci id="S2.3.p2.51.m9.2.2.2.2.2.cmml" xref="S2.3.p2.51.m9.2.2.2.2.2">πΉ</ci><apply id="S2.3.p2.51.m9.2.2.2.2.3.cmml" xref="S2.3.p2.51.m9.2.2.2.2.3"><csymbol cd="ambiguous" id="S2.3.p2.51.m9.2.2.2.2.3.1.cmml" xref="S2.3.p2.51.m9.2.2.2.2.3">superscript</csymbol><ci id="S2.3.p2.51.m9.2.2.2.2.3.2.cmml" xref="S2.3.p2.51.m9.2.2.2.2.3.2">β±</ci><apply id="S2.3.p2.51.m9.2.2.2.2.3.3.cmml" xref="S2.3.p2.51.m9.2.2.2.2.3.3"><minus id="S2.3.p2.51.m9.2.2.2.2.3.3.1.cmml" xref="S2.3.p2.51.m9.2.2.2.2.3.3"></minus><cn id="S2.3.p2.51.m9.2.2.2.2.3.3.2.cmml" type="integer" xref="S2.3.p2.51.m9.2.2.2.2.3.3.2">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.51.m9.2c">\{FF:F\in\mathcal{F}^{-1}\}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.51.m9.2d">{ italic_F italic_F : italic_F β caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT }</annotation></semantics></math>. Similarly as above one can check that <math alttext="\mathcal{F}^{-2}" class="ltx_Math" display="inline" id="S2.3.p2.52.m10.1"><semantics id="S2.3.p2.52.m10.1a"><msup id="S2.3.p2.52.m10.1.1" xref="S2.3.p2.52.m10.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.3.p2.52.m10.1.1.2" xref="S2.3.p2.52.m10.1.1.2.cmml">β±</mi><mrow id="S2.3.p2.52.m10.1.1.3" xref="S2.3.p2.52.m10.1.1.3.cmml"><mo id="S2.3.p2.52.m10.1.1.3a" xref="S2.3.p2.52.m10.1.1.3.cmml">β</mo><mn id="S2.3.p2.52.m10.1.1.3.2" xref="S2.3.p2.52.m10.1.1.3.2.cmml">2</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.3.p2.52.m10.1b"><apply id="S2.3.p2.52.m10.1.1.cmml" xref="S2.3.p2.52.m10.1.1"><csymbol cd="ambiguous" id="S2.3.p2.52.m10.1.1.1.cmml" xref="S2.3.p2.52.m10.1.1">superscript</csymbol><ci id="S2.3.p2.52.m10.1.1.2.cmml" xref="S2.3.p2.52.m10.1.1.2">β±</ci><apply id="S2.3.p2.52.m10.1.1.3.cmml" xref="S2.3.p2.52.m10.1.1.3"><minus id="S2.3.p2.52.m10.1.1.3.1.cmml" xref="S2.3.p2.52.m10.1.1.3"></minus><cn id="S2.3.p2.52.m10.1.1.3.2.cmml" type="integer" xref="S2.3.p2.52.m10.1.1.3.2">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.52.m10.1c">\mathcal{F}^{-2}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.52.m10.1d">caligraphic_F start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math> converges to <math alttext="f" class="ltx_Math" display="inline" id="S2.3.p2.53.m11.1"><semantics id="S2.3.p2.53.m11.1a"><mi id="S2.3.p2.53.m11.1.1" xref="S2.3.p2.53.m11.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.53.m11.1b"><ci id="S2.3.p2.53.m11.1.1.cmml" xref="S2.3.p2.53.m11.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.53.m11.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.53.m11.1d">italic_f</annotation></semantics></math>. Since <math alttext="S" class="ltx_Math" display="inline" id="S2.3.p2.54.m12.1"><semantics id="S2.3.p2.54.m12.1a"><mi id="S2.3.p2.54.m12.1.1" xref="S2.3.p2.54.m12.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.54.m12.1b"><ci id="S2.3.p2.54.m12.1.1.cmml" xref="S2.3.p2.54.m12.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.54.m12.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.54.m12.1d">italic_S</annotation></semantics></math> is a topological semigroup and <math alttext="e=efe" class="ltx_Math" display="inline" id="S2.3.p2.55.m13.1"><semantics id="S2.3.p2.55.m13.1a"><mrow id="S2.3.p2.55.m13.1.1" xref="S2.3.p2.55.m13.1.1.cmml"><mi id="S2.3.p2.55.m13.1.1.2" xref="S2.3.p2.55.m13.1.1.2.cmml">e</mi><mo id="S2.3.p2.55.m13.1.1.1" xref="S2.3.p2.55.m13.1.1.1.cmml">=</mo><mrow id="S2.3.p2.55.m13.1.1.3" xref="S2.3.p2.55.m13.1.1.3.cmml"><mi id="S2.3.p2.55.m13.1.1.3.2" xref="S2.3.p2.55.m13.1.1.3.2.cmml">e</mi><mo id="S2.3.p2.55.m13.1.1.3.1" xref="S2.3.p2.55.m13.1.1.3.1.cmml">β’</mo><mi id="S2.3.p2.55.m13.1.1.3.3" xref="S2.3.p2.55.m13.1.1.3.3.cmml">f</mi><mo id="S2.3.p2.55.m13.1.1.3.1a" xref="S2.3.p2.55.m13.1.1.3.1.cmml">β’</mo><mi id="S2.3.p2.55.m13.1.1.3.4" xref="S2.3.p2.55.m13.1.1.3.4.cmml">e</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.55.m13.1b"><apply id="S2.3.p2.55.m13.1.1.cmml" xref="S2.3.p2.55.m13.1.1"><eq id="S2.3.p2.55.m13.1.1.1.cmml" xref="S2.3.p2.55.m13.1.1.1"></eq><ci id="S2.3.p2.55.m13.1.1.2.cmml" xref="S2.3.p2.55.m13.1.1.2">π</ci><apply id="S2.3.p2.55.m13.1.1.3.cmml" xref="S2.3.p2.55.m13.1.1.3"><times id="S2.3.p2.55.m13.1.1.3.1.cmml" xref="S2.3.p2.55.m13.1.1.3.1"></times><ci id="S2.3.p2.55.m13.1.1.3.2.cmml" xref="S2.3.p2.55.m13.1.1.3.2">π</ci><ci id="S2.3.p2.55.m13.1.1.3.3.cmml" xref="S2.3.p2.55.m13.1.1.3.3">π</ci><ci id="S2.3.p2.55.m13.1.1.3.4.cmml" xref="S2.3.p2.55.m13.1.1.3.4">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.55.m13.1c">e=efe</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.55.m13.1d">italic_e = italic_e italic_f italic_e</annotation></semantics></math>, for any open neighborhood <math alttext="W" class="ltx_Math" display="inline" id="S2.3.p2.56.m14.1"><semantics id="S2.3.p2.56.m14.1a"><mi id="S2.3.p2.56.m14.1.1" xref="S2.3.p2.56.m14.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.56.m14.1b"><ci id="S2.3.p2.56.m14.1.1.cmml" xref="S2.3.p2.56.m14.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.56.m14.1c">W</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.56.m14.1d">italic_W</annotation></semantics></math> of <math alttext="e" class="ltx_Math" display="inline" id="S2.3.p2.57.m15.1"><semantics id="S2.3.p2.57.m15.1a"><mi id="S2.3.p2.57.m15.1.1" xref="S2.3.p2.57.m15.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.57.m15.1b"><ci id="S2.3.p2.57.m15.1.1.cmml" xref="S2.3.p2.57.m15.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.57.m15.1c">e</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.57.m15.1d">italic_e</annotation></semantics></math> there exist open neighborhoods <math alttext="V(f)" class="ltx_Math" display="inline" id="S2.3.p2.58.m16.1"><semantics id="S2.3.p2.58.m16.1a"><mrow id="S2.3.p2.58.m16.1.2" xref="S2.3.p2.58.m16.1.2.cmml"><mi id="S2.3.p2.58.m16.1.2.2" xref="S2.3.p2.58.m16.1.2.2.cmml">V</mi><mo id="S2.3.p2.58.m16.1.2.1" xref="S2.3.p2.58.m16.1.2.1.cmml">β’</mo><mrow id="S2.3.p2.58.m16.1.2.3.2" xref="S2.3.p2.58.m16.1.2.cmml"><mo id="S2.3.p2.58.m16.1.2.3.2.1" stretchy="false" xref="S2.3.p2.58.m16.1.2.cmml">(</mo><mi id="S2.3.p2.58.m16.1.1" xref="S2.3.p2.58.m16.1.1.cmml">f</mi><mo id="S2.3.p2.58.m16.1.2.3.2.2" stretchy="false" xref="S2.3.p2.58.m16.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.58.m16.1b"><apply id="S2.3.p2.58.m16.1.2.cmml" xref="S2.3.p2.58.m16.1.2"><times id="S2.3.p2.58.m16.1.2.1.cmml" xref="S2.3.p2.58.m16.1.2.1"></times><ci id="S2.3.p2.58.m16.1.2.2.cmml" xref="S2.3.p2.58.m16.1.2.2">π</ci><ci id="S2.3.p2.58.m16.1.1.cmml" xref="S2.3.p2.58.m16.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.58.m16.1c">V(f)</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.58.m16.1d">italic_V ( italic_f )</annotation></semantics></math> and <math alttext="V(e)" class="ltx_Math" display="inline" id="S2.3.p2.59.m17.1"><semantics id="S2.3.p2.59.m17.1a"><mrow id="S2.3.p2.59.m17.1.2" xref="S2.3.p2.59.m17.1.2.cmml"><mi id="S2.3.p2.59.m17.1.2.2" xref="S2.3.p2.59.m17.1.2.2.cmml">V</mi><mo id="S2.3.p2.59.m17.1.2.1" xref="S2.3.p2.59.m17.1.2.1.cmml">β’</mo><mrow id="S2.3.p2.59.m17.1.2.3.2" xref="S2.3.p2.59.m17.1.2.cmml"><mo id="S2.3.p2.59.m17.1.2.3.2.1" stretchy="false" xref="S2.3.p2.59.m17.1.2.cmml">(</mo><mi id="S2.3.p2.59.m17.1.1" xref="S2.3.p2.59.m17.1.1.cmml">e</mi><mo id="S2.3.p2.59.m17.1.2.3.2.2" stretchy="false" xref="S2.3.p2.59.m17.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.59.m17.1b"><apply id="S2.3.p2.59.m17.1.2.cmml" xref="S2.3.p2.59.m17.1.2"><times id="S2.3.p2.59.m17.1.2.1.cmml" xref="S2.3.p2.59.m17.1.2.1"></times><ci id="S2.3.p2.59.m17.1.2.2.cmml" xref="S2.3.p2.59.m17.1.2.2">π</ci><ci id="S2.3.p2.59.m17.1.1.cmml" xref="S2.3.p2.59.m17.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.59.m17.1c">V(e)</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.59.m17.1d">italic_V ( italic_e )</annotation></semantics></math> of <math alttext="f" class="ltx_Math" display="inline" id="S2.3.p2.60.m18.1"><semantics id="S2.3.p2.60.m18.1a"><mi id="S2.3.p2.60.m18.1.1" xref="S2.3.p2.60.m18.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.60.m18.1b"><ci id="S2.3.p2.60.m18.1.1.cmml" xref="S2.3.p2.60.m18.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.60.m18.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.60.m18.1d">italic_f</annotation></semantics></math> and <math alttext="e" class="ltx_Math" display="inline" id="S2.3.p2.61.m19.1"><semantics id="S2.3.p2.61.m19.1a"><mi id="S2.3.p2.61.m19.1.1" xref="S2.3.p2.61.m19.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.61.m19.1b"><ci id="S2.3.p2.61.m19.1.1.cmml" xref="S2.3.p2.61.m19.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.61.m19.1c">e</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.61.m19.1d">italic_e</annotation></semantics></math>, respectively, such that <math alttext="V(e)V(f)V(e)\subseteq W" class="ltx_Math" display="inline" id="S2.3.p2.62.m20.3"><semantics id="S2.3.p2.62.m20.3a"><mrow id="S2.3.p2.62.m20.3.4" xref="S2.3.p2.62.m20.3.4.cmml"><mrow id="S2.3.p2.62.m20.3.4.2" xref="S2.3.p2.62.m20.3.4.2.cmml"><mi id="S2.3.p2.62.m20.3.4.2.2" xref="S2.3.p2.62.m20.3.4.2.2.cmml">V</mi><mo id="S2.3.p2.62.m20.3.4.2.1" xref="S2.3.p2.62.m20.3.4.2.1.cmml">β’</mo><mrow id="S2.3.p2.62.m20.3.4.2.3.2" xref="S2.3.p2.62.m20.3.4.2.cmml"><mo id="S2.3.p2.62.m20.3.4.2.3.2.1" stretchy="false" xref="S2.3.p2.62.m20.3.4.2.cmml">(</mo><mi id="S2.3.p2.62.m20.1.1" xref="S2.3.p2.62.m20.1.1.cmml">e</mi><mo id="S2.3.p2.62.m20.3.4.2.3.2.2" stretchy="false" xref="S2.3.p2.62.m20.3.4.2.cmml">)</mo></mrow><mo id="S2.3.p2.62.m20.3.4.2.1a" xref="S2.3.p2.62.m20.3.4.2.1.cmml">β’</mo><mi id="S2.3.p2.62.m20.3.4.2.4" xref="S2.3.p2.62.m20.3.4.2.4.cmml">V</mi><mo id="S2.3.p2.62.m20.3.4.2.1b" xref="S2.3.p2.62.m20.3.4.2.1.cmml">β’</mo><mrow id="S2.3.p2.62.m20.3.4.2.5.2" xref="S2.3.p2.62.m20.3.4.2.cmml"><mo id="S2.3.p2.62.m20.3.4.2.5.2.1" stretchy="false" xref="S2.3.p2.62.m20.3.4.2.cmml">(</mo><mi id="S2.3.p2.62.m20.2.2" xref="S2.3.p2.62.m20.2.2.cmml">f</mi><mo id="S2.3.p2.62.m20.3.4.2.5.2.2" stretchy="false" xref="S2.3.p2.62.m20.3.4.2.cmml">)</mo></mrow><mo id="S2.3.p2.62.m20.3.4.2.1c" xref="S2.3.p2.62.m20.3.4.2.1.cmml">β’</mo><mi id="S2.3.p2.62.m20.3.4.2.6" xref="S2.3.p2.62.m20.3.4.2.6.cmml">V</mi><mo id="S2.3.p2.62.m20.3.4.2.1d" xref="S2.3.p2.62.m20.3.4.2.1.cmml">β’</mo><mrow id="S2.3.p2.62.m20.3.4.2.7.2" xref="S2.3.p2.62.m20.3.4.2.cmml"><mo id="S2.3.p2.62.m20.3.4.2.7.2.1" stretchy="false" xref="S2.3.p2.62.m20.3.4.2.cmml">(</mo><mi id="S2.3.p2.62.m20.3.3" xref="S2.3.p2.62.m20.3.3.cmml">e</mi><mo id="S2.3.p2.62.m20.3.4.2.7.2.2" stretchy="false" xref="S2.3.p2.62.m20.3.4.2.cmml">)</mo></mrow></mrow><mo id="S2.3.p2.62.m20.3.4.1" xref="S2.3.p2.62.m20.3.4.1.cmml">β</mo><mi id="S2.3.p2.62.m20.3.4.3" xref="S2.3.p2.62.m20.3.4.3.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.62.m20.3b"><apply id="S2.3.p2.62.m20.3.4.cmml" xref="S2.3.p2.62.m20.3.4"><subset id="S2.3.p2.62.m20.3.4.1.cmml" xref="S2.3.p2.62.m20.3.4.1"></subset><apply id="S2.3.p2.62.m20.3.4.2.cmml" xref="S2.3.p2.62.m20.3.4.2"><times id="S2.3.p2.62.m20.3.4.2.1.cmml" xref="S2.3.p2.62.m20.3.4.2.1"></times><ci id="S2.3.p2.62.m20.3.4.2.2.cmml" xref="S2.3.p2.62.m20.3.4.2.2">π</ci><ci id="S2.3.p2.62.m20.1.1.cmml" xref="S2.3.p2.62.m20.1.1">π</ci><ci id="S2.3.p2.62.m20.3.4.2.4.cmml" xref="S2.3.p2.62.m20.3.4.2.4">π</ci><ci id="S2.3.p2.62.m20.2.2.cmml" xref="S2.3.p2.62.m20.2.2">π</ci><ci id="S2.3.p2.62.m20.3.4.2.6.cmml" xref="S2.3.p2.62.m20.3.4.2.6">π</ci><ci id="S2.3.p2.62.m20.3.3.cmml" xref="S2.3.p2.62.m20.3.3">π</ci></apply><ci id="S2.3.p2.62.m20.3.4.3.cmml" xref="S2.3.p2.62.m20.3.4.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.62.m20.3c">V(e)V(f)V(e)\subseteq W</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.62.m20.3d">italic_V ( italic_e ) italic_V ( italic_f ) italic_V ( italic_e ) β italic_W</annotation></semantics></math>. As the filter <math alttext="\mathcal{F}^{-2}" class="ltx_Math" display="inline" id="S2.3.p2.63.m21.1"><semantics id="S2.3.p2.63.m21.1a"><msup id="S2.3.p2.63.m21.1.1" xref="S2.3.p2.63.m21.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.3.p2.63.m21.1.1.2" xref="S2.3.p2.63.m21.1.1.2.cmml">β±</mi><mrow id="S2.3.p2.63.m21.1.1.3" xref="S2.3.p2.63.m21.1.1.3.cmml"><mo id="S2.3.p2.63.m21.1.1.3a" xref="S2.3.p2.63.m21.1.1.3.cmml">β</mo><mn id="S2.3.p2.63.m21.1.1.3.2" xref="S2.3.p2.63.m21.1.1.3.2.cmml">2</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.3.p2.63.m21.1b"><apply id="S2.3.p2.63.m21.1.1.cmml" xref="S2.3.p2.63.m21.1.1"><csymbol cd="ambiguous" id="S2.3.p2.63.m21.1.1.1.cmml" xref="S2.3.p2.63.m21.1.1">superscript</csymbol><ci id="S2.3.p2.63.m21.1.1.2.cmml" xref="S2.3.p2.63.m21.1.1.2">β±</ci><apply id="S2.3.p2.63.m21.1.1.3.cmml" xref="S2.3.p2.63.m21.1.1.3"><minus id="S2.3.p2.63.m21.1.1.3.1.cmml" xref="S2.3.p2.63.m21.1.1.3"></minus><cn id="S2.3.p2.63.m21.1.1.3.2.cmml" type="integer" xref="S2.3.p2.63.m21.1.1.3.2">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.63.m21.1c">\mathcal{F}^{-2}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.63.m21.1d">caligraphic_F start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math> converges to <math alttext="f" class="ltx_Math" display="inline" id="S2.3.p2.64.m22.1"><semantics id="S2.3.p2.64.m22.1a"><mi id="S2.3.p2.64.m22.1.1" xref="S2.3.p2.64.m22.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.64.m22.1b"><ci id="S2.3.p2.64.m22.1.1.cmml" xref="S2.3.p2.64.m22.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.64.m22.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.64.m22.1d">italic_f</annotation></semantics></math> and the filter <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S2.3.p2.65.m23.1"><semantics id="S2.3.p2.65.m23.1a"><mi class="ltx_font_mathcaligraphic" id="S2.3.p2.65.m23.1.1" xref="S2.3.p2.65.m23.1.1.cmml">β±</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.65.m23.1b"><ci id="S2.3.p2.65.m23.1.1.cmml" xref="S2.3.p2.65.m23.1.1">β±</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.65.m23.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.65.m23.1d">caligraphic_F</annotation></semantics></math> converges to <math alttext="e" class="ltx_Math" display="inline" id="S2.3.p2.66.m24.1"><semantics id="S2.3.p2.66.m24.1a"><mi id="S2.3.p2.66.m24.1.1" xref="S2.3.p2.66.m24.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.66.m24.1b"><ci id="S2.3.p2.66.m24.1.1.cmml" xref="S2.3.p2.66.m24.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.66.m24.1c">e</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.66.m24.1d">italic_e</annotation></semantics></math>, there exists <math alttext="F\in\mathcal{F}" class="ltx_Math" display="inline" id="S2.3.p2.67.m25.1"><semantics id="S2.3.p2.67.m25.1a"><mrow id="S2.3.p2.67.m25.1.1" xref="S2.3.p2.67.m25.1.1.cmml"><mi id="S2.3.p2.67.m25.1.1.2" xref="S2.3.p2.67.m25.1.1.2.cmml">F</mi><mo id="S2.3.p2.67.m25.1.1.1" xref="S2.3.p2.67.m25.1.1.1.cmml">β</mo><mi class="ltx_font_mathcaligraphic" id="S2.3.p2.67.m25.1.1.3" xref="S2.3.p2.67.m25.1.1.3.cmml">β±</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.67.m25.1b"><apply id="S2.3.p2.67.m25.1.1.cmml" xref="S2.3.p2.67.m25.1.1"><in id="S2.3.p2.67.m25.1.1.1.cmml" xref="S2.3.p2.67.m25.1.1.1"></in><ci id="S2.3.p2.67.m25.1.1.2.cmml" xref="S2.3.p2.67.m25.1.1.2">πΉ</ci><ci id="S2.3.p2.67.m25.1.1.3.cmml" xref="S2.3.p2.67.m25.1.1.3">β±</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.67.m25.1c">F\in\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.67.m25.1d">italic_F β caligraphic_F</annotation></semantics></math> such that <math alttext="F\subseteq X\cap V(e)" class="ltx_Math" display="inline" id="S2.3.p2.68.m26.1"><semantics id="S2.3.p2.68.m26.1a"><mrow id="S2.3.p2.68.m26.1.2" xref="S2.3.p2.68.m26.1.2.cmml"><mi id="S2.3.p2.68.m26.1.2.2" xref="S2.3.p2.68.m26.1.2.2.cmml">F</mi><mo id="S2.3.p2.68.m26.1.2.1" xref="S2.3.p2.68.m26.1.2.1.cmml">β</mo><mrow id="S2.3.p2.68.m26.1.2.3" xref="S2.3.p2.68.m26.1.2.3.cmml"><mi id="S2.3.p2.68.m26.1.2.3.2" xref="S2.3.p2.68.m26.1.2.3.2.cmml">X</mi><mo id="S2.3.p2.68.m26.1.2.3.1" xref="S2.3.p2.68.m26.1.2.3.1.cmml">β©</mo><mrow id="S2.3.p2.68.m26.1.2.3.3" xref="S2.3.p2.68.m26.1.2.3.3.cmml"><mi id="S2.3.p2.68.m26.1.2.3.3.2" xref="S2.3.p2.68.m26.1.2.3.3.2.cmml">V</mi><mo id="S2.3.p2.68.m26.1.2.3.3.1" xref="S2.3.p2.68.m26.1.2.3.3.1.cmml">β’</mo><mrow id="S2.3.p2.68.m26.1.2.3.3.3.2" xref="S2.3.p2.68.m26.1.2.3.3.cmml"><mo id="S2.3.p2.68.m26.1.2.3.3.3.2.1" stretchy="false" xref="S2.3.p2.68.m26.1.2.3.3.cmml">(</mo><mi id="S2.3.p2.68.m26.1.1" xref="S2.3.p2.68.m26.1.1.cmml">e</mi><mo id="S2.3.p2.68.m26.1.2.3.3.3.2.2" stretchy="false" xref="S2.3.p2.68.m26.1.2.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.68.m26.1b"><apply id="S2.3.p2.68.m26.1.2.cmml" xref="S2.3.p2.68.m26.1.2"><subset id="S2.3.p2.68.m26.1.2.1.cmml" xref="S2.3.p2.68.m26.1.2.1"></subset><ci id="S2.3.p2.68.m26.1.2.2.cmml" xref="S2.3.p2.68.m26.1.2.2">πΉ</ci><apply id="S2.3.p2.68.m26.1.2.3.cmml" xref="S2.3.p2.68.m26.1.2.3"><intersect id="S2.3.p2.68.m26.1.2.3.1.cmml" xref="S2.3.p2.68.m26.1.2.3.1"></intersect><ci id="S2.3.p2.68.m26.1.2.3.2.cmml" xref="S2.3.p2.68.m26.1.2.3.2">π</ci><apply id="S2.3.p2.68.m26.1.2.3.3.cmml" xref="S2.3.p2.68.m26.1.2.3.3"><times id="S2.3.p2.68.m26.1.2.3.3.1.cmml" xref="S2.3.p2.68.m26.1.2.3.3.1"></times><ci id="S2.3.p2.68.m26.1.2.3.3.2.cmml" xref="S2.3.p2.68.m26.1.2.3.3.2">π</ci><ci id="S2.3.p2.68.m26.1.1.cmml" xref="S2.3.p2.68.m26.1.1">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.68.m26.1c">F\subseteq X\cap V(e)</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.68.m26.1d">italic_F β italic_X β© italic_V ( italic_e )</annotation></semantics></math> and <math alttext="(FF)^{-1}=F^{-1}F^{-1}\subset V(f)" class="ltx_Math" display="inline" id="S2.3.p2.69.m27.2"><semantics id="S2.3.p2.69.m27.2a"><mrow id="S2.3.p2.69.m27.2.2" xref="S2.3.p2.69.m27.2.2.cmml"><msup id="S2.3.p2.69.m27.2.2.1" xref="S2.3.p2.69.m27.2.2.1.cmml"><mrow id="S2.3.p2.69.m27.2.2.1.1.1" xref="S2.3.p2.69.m27.2.2.1.1.1.1.cmml"><mo id="S2.3.p2.69.m27.2.2.1.1.1.2" stretchy="false" xref="S2.3.p2.69.m27.2.2.1.1.1.1.cmml">(</mo><mrow id="S2.3.p2.69.m27.2.2.1.1.1.1" xref="S2.3.p2.69.m27.2.2.1.1.1.1.cmml"><mi id="S2.3.p2.69.m27.2.2.1.1.1.1.2" xref="S2.3.p2.69.m27.2.2.1.1.1.1.2.cmml">F</mi><mo id="S2.3.p2.69.m27.2.2.1.1.1.1.1" xref="S2.3.p2.69.m27.2.2.1.1.1.1.1.cmml">β’</mo><mi id="S2.3.p2.69.m27.2.2.1.1.1.1.3" xref="S2.3.p2.69.m27.2.2.1.1.1.1.3.cmml">F</mi></mrow><mo id="S2.3.p2.69.m27.2.2.1.1.1.3" stretchy="false" xref="S2.3.p2.69.m27.2.2.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.3.p2.69.m27.2.2.1.3" xref="S2.3.p2.69.m27.2.2.1.3.cmml"><mo id="S2.3.p2.69.m27.2.2.1.3a" xref="S2.3.p2.69.m27.2.2.1.3.cmml">β</mo><mn id="S2.3.p2.69.m27.2.2.1.3.2" xref="S2.3.p2.69.m27.2.2.1.3.2.cmml">1</mn></mrow></msup><mo id="S2.3.p2.69.m27.2.2.3" xref="S2.3.p2.69.m27.2.2.3.cmml">=</mo><mrow id="S2.3.p2.69.m27.2.2.4" xref="S2.3.p2.69.m27.2.2.4.cmml"><msup id="S2.3.p2.69.m27.2.2.4.2" xref="S2.3.p2.69.m27.2.2.4.2.cmml"><mi id="S2.3.p2.69.m27.2.2.4.2.2" xref="S2.3.p2.69.m27.2.2.4.2.2.cmml">F</mi><mrow id="S2.3.p2.69.m27.2.2.4.2.3" xref="S2.3.p2.69.m27.2.2.4.2.3.cmml"><mo id="S2.3.p2.69.m27.2.2.4.2.3a" xref="S2.3.p2.69.m27.2.2.4.2.3.cmml">β</mo><mn id="S2.3.p2.69.m27.2.2.4.2.3.2" xref="S2.3.p2.69.m27.2.2.4.2.3.2.cmml">1</mn></mrow></msup><mo id="S2.3.p2.69.m27.2.2.4.1" xref="S2.3.p2.69.m27.2.2.4.1.cmml">β’</mo><msup id="S2.3.p2.69.m27.2.2.4.3" xref="S2.3.p2.69.m27.2.2.4.3.cmml"><mi id="S2.3.p2.69.m27.2.2.4.3.2" xref="S2.3.p2.69.m27.2.2.4.3.2.cmml">F</mi><mrow id="S2.3.p2.69.m27.2.2.4.3.3" xref="S2.3.p2.69.m27.2.2.4.3.3.cmml"><mo id="S2.3.p2.69.m27.2.2.4.3.3a" xref="S2.3.p2.69.m27.2.2.4.3.3.cmml">β</mo><mn id="S2.3.p2.69.m27.2.2.4.3.3.2" xref="S2.3.p2.69.m27.2.2.4.3.3.2.cmml">1</mn></mrow></msup></mrow><mo id="S2.3.p2.69.m27.2.2.5" xref="S2.3.p2.69.m27.2.2.5.cmml">β</mo><mrow id="S2.3.p2.69.m27.2.2.6" xref="S2.3.p2.69.m27.2.2.6.cmml"><mi id="S2.3.p2.69.m27.2.2.6.2" xref="S2.3.p2.69.m27.2.2.6.2.cmml">V</mi><mo id="S2.3.p2.69.m27.2.2.6.1" xref="S2.3.p2.69.m27.2.2.6.1.cmml">β’</mo><mrow id="S2.3.p2.69.m27.2.2.6.3.2" xref="S2.3.p2.69.m27.2.2.6.cmml"><mo id="S2.3.p2.69.m27.2.2.6.3.2.1" stretchy="false" xref="S2.3.p2.69.m27.2.2.6.cmml">(</mo><mi id="S2.3.p2.69.m27.1.1" xref="S2.3.p2.69.m27.1.1.cmml">f</mi><mo id="S2.3.p2.69.m27.2.2.6.3.2.2" stretchy="false" xref="S2.3.p2.69.m27.2.2.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.69.m27.2b"><apply id="S2.3.p2.69.m27.2.2.cmml" xref="S2.3.p2.69.m27.2.2"><and id="S2.3.p2.69.m27.2.2a.cmml" xref="S2.3.p2.69.m27.2.2"></and><apply id="S2.3.p2.69.m27.2.2b.cmml" xref="S2.3.p2.69.m27.2.2"><eq id="S2.3.p2.69.m27.2.2.3.cmml" xref="S2.3.p2.69.m27.2.2.3"></eq><apply id="S2.3.p2.69.m27.2.2.1.cmml" xref="S2.3.p2.69.m27.2.2.1"><csymbol cd="ambiguous" id="S2.3.p2.69.m27.2.2.1.2.cmml" xref="S2.3.p2.69.m27.2.2.1">superscript</csymbol><apply id="S2.3.p2.69.m27.2.2.1.1.1.1.cmml" xref="S2.3.p2.69.m27.2.2.1.1.1"><times id="S2.3.p2.69.m27.2.2.1.1.1.1.1.cmml" xref="S2.3.p2.69.m27.2.2.1.1.1.1.1"></times><ci id="S2.3.p2.69.m27.2.2.1.1.1.1.2.cmml" xref="S2.3.p2.69.m27.2.2.1.1.1.1.2">πΉ</ci><ci id="S2.3.p2.69.m27.2.2.1.1.1.1.3.cmml" xref="S2.3.p2.69.m27.2.2.1.1.1.1.3">πΉ</ci></apply><apply id="S2.3.p2.69.m27.2.2.1.3.cmml" xref="S2.3.p2.69.m27.2.2.1.3"><minus id="S2.3.p2.69.m27.2.2.1.3.1.cmml" xref="S2.3.p2.69.m27.2.2.1.3"></minus><cn id="S2.3.p2.69.m27.2.2.1.3.2.cmml" type="integer" xref="S2.3.p2.69.m27.2.2.1.3.2">1</cn></apply></apply><apply id="S2.3.p2.69.m27.2.2.4.cmml" xref="S2.3.p2.69.m27.2.2.4"><times id="S2.3.p2.69.m27.2.2.4.1.cmml" xref="S2.3.p2.69.m27.2.2.4.1"></times><apply id="S2.3.p2.69.m27.2.2.4.2.cmml" xref="S2.3.p2.69.m27.2.2.4.2"><csymbol cd="ambiguous" id="S2.3.p2.69.m27.2.2.4.2.1.cmml" xref="S2.3.p2.69.m27.2.2.4.2">superscript</csymbol><ci id="S2.3.p2.69.m27.2.2.4.2.2.cmml" xref="S2.3.p2.69.m27.2.2.4.2.2">πΉ</ci><apply id="S2.3.p2.69.m27.2.2.4.2.3.cmml" xref="S2.3.p2.69.m27.2.2.4.2.3"><minus id="S2.3.p2.69.m27.2.2.4.2.3.1.cmml" xref="S2.3.p2.69.m27.2.2.4.2.3"></minus><cn id="S2.3.p2.69.m27.2.2.4.2.3.2.cmml" type="integer" xref="S2.3.p2.69.m27.2.2.4.2.3.2">1</cn></apply></apply><apply id="S2.3.p2.69.m27.2.2.4.3.cmml" xref="S2.3.p2.69.m27.2.2.4.3"><csymbol cd="ambiguous" id="S2.3.p2.69.m27.2.2.4.3.1.cmml" xref="S2.3.p2.69.m27.2.2.4.3">superscript</csymbol><ci id="S2.3.p2.69.m27.2.2.4.3.2.cmml" xref="S2.3.p2.69.m27.2.2.4.3.2">πΉ</ci><apply id="S2.3.p2.69.m27.2.2.4.3.3.cmml" xref="S2.3.p2.69.m27.2.2.4.3.3"><minus id="S2.3.p2.69.m27.2.2.4.3.3.1.cmml" xref="S2.3.p2.69.m27.2.2.4.3.3"></minus><cn id="S2.3.p2.69.m27.2.2.4.3.3.2.cmml" type="integer" xref="S2.3.p2.69.m27.2.2.4.3.3.2">1</cn></apply></apply></apply></apply><apply id="S2.3.p2.69.m27.2.2c.cmml" xref="S2.3.p2.69.m27.2.2"><subset id="S2.3.p2.69.m27.2.2.5.cmml" xref="S2.3.p2.69.m27.2.2.5"></subset><share href="https://arxiv.org/html/2503.13666v1#S2.3.p2.69.m27.2.2.4.cmml" id="S2.3.p2.69.m27.2.2d.cmml" xref="S2.3.p2.69.m27.2.2"></share><apply id="S2.3.p2.69.m27.2.2.6.cmml" xref="S2.3.p2.69.m27.2.2.6"><times id="S2.3.p2.69.m27.2.2.6.1.cmml" xref="S2.3.p2.69.m27.2.2.6.1"></times><ci id="S2.3.p2.69.m27.2.2.6.2.cmml" xref="S2.3.p2.69.m27.2.2.6.2">π</ci><ci id="S2.3.p2.69.m27.1.1.cmml" xref="S2.3.p2.69.m27.1.1">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.69.m27.2c">(FF)^{-1}=F^{-1}F^{-1}\subset V(f)</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.69.m27.2d">( italic_F italic_F ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT β italic_V ( italic_f )</annotation></semantics></math>. Then for any <math alttext="x\in F" class="ltx_Math" display="inline" id="S2.3.p2.70.m28.1"><semantics id="S2.3.p2.70.m28.1a"><mrow id="S2.3.p2.70.m28.1.1" xref="S2.3.p2.70.m28.1.1.cmml"><mi id="S2.3.p2.70.m28.1.1.2" xref="S2.3.p2.70.m28.1.1.2.cmml">x</mi><mo id="S2.3.p2.70.m28.1.1.1" xref="S2.3.p2.70.m28.1.1.1.cmml">β</mo><mi id="S2.3.p2.70.m28.1.1.3" xref="S2.3.p2.70.m28.1.1.3.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.70.m28.1b"><apply id="S2.3.p2.70.m28.1.1.cmml" xref="S2.3.p2.70.m28.1.1"><in id="S2.3.p2.70.m28.1.1.1.cmml" xref="S2.3.p2.70.m28.1.1.1"></in><ci id="S2.3.p2.70.m28.1.1.2.cmml" xref="S2.3.p2.70.m28.1.1.2">π₯</ci><ci id="S2.3.p2.70.m28.1.1.3.cmml" xref="S2.3.p2.70.m28.1.1.3">πΉ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.70.m28.1c">x\in F</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.70.m28.1d">italic_x β italic_F</annotation></semantics></math> we have</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="E(X)\ni xx^{-1}x^{-1}x\in FF^{-1}F^{-1}F\subset V(e)V(f)V(e)\subseteq W." class="ltx_Math" display="block" id="S2.Ex3.m1.5"><semantics id="S2.Ex3.m1.5a"><mrow id="S2.Ex3.m1.5.5.1" xref="S2.Ex3.m1.5.5.1.1.cmml"><mrow id="S2.Ex3.m1.5.5.1.1" xref="S2.Ex3.m1.5.5.1.1.cmml"><mrow id="S2.Ex3.m1.5.5.1.1.2" xref="S2.Ex3.m1.5.5.1.1.2.cmml"><mi id="S2.Ex3.m1.5.5.1.1.2.2" xref="S2.Ex3.m1.5.5.1.1.2.2.cmml">E</mi><mo id="S2.Ex3.m1.5.5.1.1.2.1" xref="S2.Ex3.m1.5.5.1.1.2.1.cmml">β’</mo><mrow id="S2.Ex3.m1.5.5.1.1.2.3.2" xref="S2.Ex3.m1.5.5.1.1.2.cmml"><mo id="S2.Ex3.m1.5.5.1.1.2.3.2.1" stretchy="false" xref="S2.Ex3.m1.5.5.1.1.2.cmml">(</mo><mi id="S2.Ex3.m1.1.1" xref="S2.Ex3.m1.1.1.cmml">X</mi><mo id="S2.Ex3.m1.5.5.1.1.2.3.2.2" stretchy="false" 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id="S2.Ex3.m1.5.5.1.1.6.5.cmml" xref="S2.Ex3.m1.5.5.1.1.6.5">πΉ</ci></apply></apply><apply id="S2.Ex3.m1.5.5.1.1e.cmml" xref="S2.Ex3.m1.5.5.1"><subset id="S2.Ex3.m1.5.5.1.1.7.cmml" xref="S2.Ex3.m1.5.5.1.1.7"></subset><share href="https://arxiv.org/html/2503.13666v1#S2.Ex3.m1.5.5.1.1.6.cmml" id="S2.Ex3.m1.5.5.1.1f.cmml" xref="S2.Ex3.m1.5.5.1"></share><apply id="S2.Ex3.m1.5.5.1.1.8.cmml" xref="S2.Ex3.m1.5.5.1.1.8"><times id="S2.Ex3.m1.5.5.1.1.8.1.cmml" xref="S2.Ex3.m1.5.5.1.1.8.1"></times><ci id="S2.Ex3.m1.5.5.1.1.8.2.cmml" xref="S2.Ex3.m1.5.5.1.1.8.2">π</ci><ci id="S2.Ex3.m1.2.2.cmml" xref="S2.Ex3.m1.2.2">π</ci><ci id="S2.Ex3.m1.5.5.1.1.8.4.cmml" xref="S2.Ex3.m1.5.5.1.1.8.4">π</ci><ci id="S2.Ex3.m1.3.3.cmml" xref="S2.Ex3.m1.3.3">π</ci><ci id="S2.Ex3.m1.5.5.1.1.8.6.cmml" xref="S2.Ex3.m1.5.5.1.1.8.6">π</ci><ci id="S2.Ex3.m1.4.4.cmml" xref="S2.Ex3.m1.4.4">π</ci></apply></apply><apply id="S2.Ex3.m1.5.5.1.1g.cmml" xref="S2.Ex3.m1.5.5.1"><subset id="S2.Ex3.m1.5.5.1.1.9.cmml" xref="S2.Ex3.m1.5.5.1.1.9"></subset><share href="https://arxiv.org/html/2503.13666v1#S2.Ex3.m1.5.5.1.1.8.cmml" id="S2.Ex3.m1.5.5.1.1h.cmml" xref="S2.Ex3.m1.5.5.1"></share><ci id="S2.Ex3.m1.5.5.1.1.10.cmml" xref="S2.Ex3.m1.5.5.1.1.10">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3.m1.5c">E(X)\ni xx^{-1}x^{-1}x\in FF^{-1}F^{-1}F\subset V(e)V(f)V(e)\subseteq W.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m1.5d">italic_E ( italic_X ) β italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x β italic_F italic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_F β italic_V ( italic_e ) italic_V ( italic_f ) italic_V ( italic_e ) β italic_W .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.3.p2.73">Thus <math alttext="e\in\overline{E(X)}" class="ltx_Math" display="inline" id="S2.3.p2.71.m1.1"><semantics id="S2.3.p2.71.m1.1a"><mrow id="S2.3.p2.71.m1.1.2" xref="S2.3.p2.71.m1.1.2.cmml"><mi id="S2.3.p2.71.m1.1.2.2" xref="S2.3.p2.71.m1.1.2.2.cmml">e</mi><mo id="S2.3.p2.71.m1.1.2.1" xref="S2.3.p2.71.m1.1.2.1.cmml">β</mo><mover accent="true" id="S2.3.p2.71.m1.1.1" xref="S2.3.p2.71.m1.1.1.cmml"><mrow id="S2.3.p2.71.m1.1.1.1" xref="S2.3.p2.71.m1.1.1.1.cmml"><mi id="S2.3.p2.71.m1.1.1.1.3" xref="S2.3.p2.71.m1.1.1.1.3.cmml">E</mi><mo id="S2.3.p2.71.m1.1.1.1.2" xref="S2.3.p2.71.m1.1.1.1.2.cmml">β’</mo><mrow id="S2.3.p2.71.m1.1.1.1.4.2" xref="S2.3.p2.71.m1.1.1.1.cmml"><mo id="S2.3.p2.71.m1.1.1.1.4.2.1" stretchy="false" xref="S2.3.p2.71.m1.1.1.1.cmml">(</mo><mi id="S2.3.p2.71.m1.1.1.1.1" xref="S2.3.p2.71.m1.1.1.1.1.cmml">X</mi><mo id="S2.3.p2.71.m1.1.1.1.4.2.2" stretchy="false" xref="S2.3.p2.71.m1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.3.p2.71.m1.1.1.2" xref="S2.3.p2.71.m1.1.1.2.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.71.m1.1b"><apply id="S2.3.p2.71.m1.1.2.cmml" xref="S2.3.p2.71.m1.1.2"><in id="S2.3.p2.71.m1.1.2.1.cmml" xref="S2.3.p2.71.m1.1.2.1"></in><ci id="S2.3.p2.71.m1.1.2.2.cmml" xref="S2.3.p2.71.m1.1.2.2">π</ci><apply id="S2.3.p2.71.m1.1.1.cmml" xref="S2.3.p2.71.m1.1.1"><ci id="S2.3.p2.71.m1.1.1.2.cmml" xref="S2.3.p2.71.m1.1.1.2">Β―</ci><apply id="S2.3.p2.71.m1.1.1.1.cmml" xref="S2.3.p2.71.m1.1.1.1"><times id="S2.3.p2.71.m1.1.1.1.2.cmml" xref="S2.3.p2.71.m1.1.1.1.2"></times><ci id="S2.3.p2.71.m1.1.1.1.3.cmml" xref="S2.3.p2.71.m1.1.1.1.3">πΈ</ci><ci id="S2.3.p2.71.m1.1.1.1.1.cmml" xref="S2.3.p2.71.m1.1.1.1.1">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.71.m1.1c">e\in\overline{E(X)}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.71.m1.1d">italic_e β overΒ― start_ARG italic_E ( italic_X ) end_ARG</annotation></semantics></math> witnessing that <math alttext="E(S)=\overline{E(X)}" class="ltx_Math" display="inline" id="S2.3.p2.72.m2.2"><semantics id="S2.3.p2.72.m2.2a"><mrow id="S2.3.p2.72.m2.2.3" xref="S2.3.p2.72.m2.2.3.cmml"><mrow id="S2.3.p2.72.m2.2.3.2" xref="S2.3.p2.72.m2.2.3.2.cmml"><mi id="S2.3.p2.72.m2.2.3.2.2" xref="S2.3.p2.72.m2.2.3.2.2.cmml">E</mi><mo id="S2.3.p2.72.m2.2.3.2.1" xref="S2.3.p2.72.m2.2.3.2.1.cmml">β’</mo><mrow id="S2.3.p2.72.m2.2.3.2.3.2" xref="S2.3.p2.72.m2.2.3.2.cmml"><mo id="S2.3.p2.72.m2.2.3.2.3.2.1" stretchy="false" xref="S2.3.p2.72.m2.2.3.2.cmml">(</mo><mi id="S2.3.p2.72.m2.2.2" xref="S2.3.p2.72.m2.2.2.cmml">S</mi><mo id="S2.3.p2.72.m2.2.3.2.3.2.2" stretchy="false" xref="S2.3.p2.72.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.3.p2.72.m2.2.3.1" xref="S2.3.p2.72.m2.2.3.1.cmml">=</mo><mover accent="true" id="S2.3.p2.72.m2.1.1" xref="S2.3.p2.72.m2.1.1.cmml"><mrow id="S2.3.p2.72.m2.1.1.1" xref="S2.3.p2.72.m2.1.1.1.cmml"><mi id="S2.3.p2.72.m2.1.1.1.3" xref="S2.3.p2.72.m2.1.1.1.3.cmml">E</mi><mo id="S2.3.p2.72.m2.1.1.1.2" xref="S2.3.p2.72.m2.1.1.1.2.cmml">β’</mo><mrow id="S2.3.p2.72.m2.1.1.1.4.2" xref="S2.3.p2.72.m2.1.1.1.cmml"><mo id="S2.3.p2.72.m2.1.1.1.4.2.1" stretchy="false" xref="S2.3.p2.72.m2.1.1.1.cmml">(</mo><mi id="S2.3.p2.72.m2.1.1.1.1" xref="S2.3.p2.72.m2.1.1.1.1.cmml">X</mi><mo id="S2.3.p2.72.m2.1.1.1.4.2.2" stretchy="false" xref="S2.3.p2.72.m2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.3.p2.72.m2.1.1.2" xref="S2.3.p2.72.m2.1.1.2.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S2.3.p2.72.m2.2b"><apply id="S2.3.p2.72.m2.2.3.cmml" xref="S2.3.p2.72.m2.2.3"><eq id="S2.3.p2.72.m2.2.3.1.cmml" xref="S2.3.p2.72.m2.2.3.1"></eq><apply id="S2.3.p2.72.m2.2.3.2.cmml" xref="S2.3.p2.72.m2.2.3.2"><times id="S2.3.p2.72.m2.2.3.2.1.cmml" xref="S2.3.p2.72.m2.2.3.2.1"></times><ci id="S2.3.p2.72.m2.2.3.2.2.cmml" xref="S2.3.p2.72.m2.2.3.2.2">πΈ</ci><ci id="S2.3.p2.72.m2.2.2.cmml" xref="S2.3.p2.72.m2.2.2">π</ci></apply><apply id="S2.3.p2.72.m2.1.1.cmml" xref="S2.3.p2.72.m2.1.1"><ci id="S2.3.p2.72.m2.1.1.2.cmml" xref="S2.3.p2.72.m2.1.1.2">Β―</ci><apply id="S2.3.p2.72.m2.1.1.1.cmml" xref="S2.3.p2.72.m2.1.1.1"><times id="S2.3.p2.72.m2.1.1.1.2.cmml" xref="S2.3.p2.72.m2.1.1.1.2"></times><ci id="S2.3.p2.72.m2.1.1.1.3.cmml" xref="S2.3.p2.72.m2.1.1.1.3">πΈ</ci><ci id="S2.3.p2.72.m2.1.1.1.1.cmml" xref="S2.3.p2.72.m2.1.1.1.1">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.72.m2.2c">E(S)=\overline{E(X)}</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.72.m2.2d">italic_E ( italic_S ) = overΒ― start_ARG italic_E ( italic_X ) end_ARG</annotation></semantics></math>, as required. Hence <math alttext="S" class="ltx_Math" display="inline" id="S2.3.p2.73.m3.1"><semantics id="S2.3.p2.73.m3.1a"><mi id="S2.3.p2.73.m3.1.1" xref="S2.3.p2.73.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.3.p2.73.m3.1b"><ci id="S2.3.p2.73.m3.1.1.cmml" xref="S2.3.p2.73.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.3.p2.73.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.3.p2.73.m3.1d">italic_S</annotation></semantics></math> is an inverse semigroup. β</p> </div> </div> <div class="ltx_para" id="S2.p4"> <p class="ltx_p" id="S2.p4.5">A space <math alttext="X" class="ltx_Math" display="inline" id="S2.p4.1.m1.1"><semantics id="S2.p4.1.m1.1a"><mi id="S2.p4.1.m1.1.1" xref="S2.p4.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.p4.1.m1.1b"><ci id="S2.p4.1.m1.1.1.cmml" xref="S2.p4.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.p4.1.m1.1d">italic_X</annotation></semantics></math> is called <span class="ltx_text ltx_font_italic" id="S2.p4.5.1">sequentially compact</span> if each sequence in <math alttext="X" class="ltx_Math" display="inline" id="S2.p4.2.m2.1"><semantics id="S2.p4.2.m2.1a"><mi id="S2.p4.2.m2.1.1" xref="S2.p4.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.p4.2.m2.1b"><ci id="S2.p4.2.m2.1.1.cmml" xref="S2.p4.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.p4.2.m2.1d">italic_X</annotation></semantics></math> contains a convergent subsequence. We are in a position to prove Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem3" title="Theorem C. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">C</span></a> and, in particular, to solve Problem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S2.Thmtheorem2" title="Problem 2.2 (Banakh, Pastukhova). β£ 2. Compact-like semigroups with dense inverse subsemigroups and the automatic continuity of inversion β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">2.2</span></a>. We need to show that a topological semigroup <math alttext="S" class="ltx_Math" display="inline" id="S2.p4.3.m3.1"><semantics id="S2.p4.3.m3.1a"><mi id="S2.p4.3.m3.1.1" xref="S2.p4.3.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p4.3.m3.1b"><ci id="S2.p4.3.m3.1.1.cmml" xref="S2.p4.3.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.3.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p4.3.m3.1d">italic_S</annotation></semantics></math> containing a dense inverse subsemigroup is a topological inverse semigroup provided (i) <math alttext="S" class="ltx_Math" display="inline" id="S2.p4.4.m4.1"><semantics id="S2.p4.4.m4.1a"><mi id="S2.p4.4.m4.1.1" xref="S2.p4.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p4.4.m4.1b"><ci id="S2.p4.4.m4.1.1.cmml" xref="S2.p4.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p4.4.m4.1d">italic_S</annotation></semantics></math> is compact, or (ii) <math alttext="S" class="ltx_Math" display="inline" id="S2.p4.5.m5.1"><semantics id="S2.p4.5.m5.1a"><mi id="S2.p4.5.m5.1.1" xref="S2.p4.5.m5.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p4.5.m5.1b"><ci id="S2.p4.5.m5.1.1.cmml" xref="S2.p4.5.m5.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.5.m5.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p4.5.m5.1d">italic_S</annotation></semantics></math> is countably compact and sequential.</p> </div> <div class="ltx_proof" id="S2.5"> <h6 class="ltx_title ltx_runin ltx_font_bold ltx_title_proof">Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem3" title="Theorem C. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">C</span></a>.</h6> <div class="ltx_para" id="S2.4.p1"> <p class="ltx_p" id="S2.4.p1.15">(i) Let <math alttext="S" class="ltx_Math" display="inline" id="S2.4.p1.1.m1.1"><semantics id="S2.4.p1.1.m1.1a"><mi id="S2.4.p1.1.m1.1.1" xref="S2.4.p1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.4.p1.1.m1.1b"><ci id="S2.4.p1.1.m1.1.1.cmml" xref="S2.4.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.1.m1.1d">italic_S</annotation></semantics></math> be a compact topological semigroup containing a dense inverse subsemigroup <math alttext="X" class="ltx_Math" display="inline" id="S2.4.p1.2.m2.1"><semantics id="S2.4.p1.2.m2.1a"><mi id="S2.4.p1.2.m2.1.1" xref="S2.4.p1.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.4.p1.2.m2.1b"><ci id="S2.4.p1.2.m2.1.1.cmml" xref="S2.4.p1.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.2.m2.1d">italic_X</annotation></semantics></math>. Fix any <math alttext="y\in S\setminus X" class="ltx_Math" display="inline" id="S2.4.p1.3.m3.1"><semantics id="S2.4.p1.3.m3.1a"><mrow id="S2.4.p1.3.m3.1.1" xref="S2.4.p1.3.m3.1.1.cmml"><mi id="S2.4.p1.3.m3.1.1.2" xref="S2.4.p1.3.m3.1.1.2.cmml">y</mi><mo id="S2.4.p1.3.m3.1.1.1" xref="S2.4.p1.3.m3.1.1.1.cmml">β</mo><mrow id="S2.4.p1.3.m3.1.1.3" xref="S2.4.p1.3.m3.1.1.3.cmml"><mi id="S2.4.p1.3.m3.1.1.3.2" xref="S2.4.p1.3.m3.1.1.3.2.cmml">S</mi><mo id="S2.4.p1.3.m3.1.1.3.1" xref="S2.4.p1.3.m3.1.1.3.1.cmml">β</mo><mi id="S2.4.p1.3.m3.1.1.3.3" xref="S2.4.p1.3.m3.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.4.p1.3.m3.1b"><apply id="S2.4.p1.3.m3.1.1.cmml" xref="S2.4.p1.3.m3.1.1"><in id="S2.4.p1.3.m3.1.1.1.cmml" xref="S2.4.p1.3.m3.1.1.1"></in><ci id="S2.4.p1.3.m3.1.1.2.cmml" xref="S2.4.p1.3.m3.1.1.2">π¦</ci><apply id="S2.4.p1.3.m3.1.1.3.cmml" xref="S2.4.p1.3.m3.1.1.3"><setdiff id="S2.4.p1.3.m3.1.1.3.1.cmml" xref="S2.4.p1.3.m3.1.1.3.1"></setdiff><ci id="S2.4.p1.3.m3.1.1.3.2.cmml" xref="S2.4.p1.3.m3.1.1.3.2">π</ci><ci id="S2.4.p1.3.m3.1.1.3.3.cmml" xref="S2.4.p1.3.m3.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.3.m3.1c">y\in S\setminus X</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.3.m3.1d">italic_y β italic_S β italic_X</annotation></semantics></math> and consider an ultrafilter <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S2.4.p1.4.m4.1"><semantics id="S2.4.p1.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S2.4.p1.4.m4.1.1" xref="S2.4.p1.4.m4.1.1.cmml">β±</mi><annotation-xml encoding="MathML-Content" id="S2.4.p1.4.m4.1b"><ci id="S2.4.p1.4.m4.1.1.cmml" xref="S2.4.p1.4.m4.1.1">β±</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.4.m4.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.4.m4.1d">caligraphic_F</annotation></semantics></math> on <math alttext="S" class="ltx_Math" display="inline" id="S2.4.p1.5.m5.1"><semantics id="S2.4.p1.5.m5.1a"><mi id="S2.4.p1.5.m5.1.1" xref="S2.4.p1.5.m5.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.4.p1.5.m5.1b"><ci id="S2.4.p1.5.m5.1.1.cmml" xref="S2.4.p1.5.m5.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.5.m5.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.5.m5.1d">italic_S</annotation></semantics></math> that contains the family <math alttext="\{U\cap X:U" class="ltx_math_unparsed" display="inline" id="S2.4.p1.6.m6.1"><semantics id="S2.4.p1.6.m6.1a"><mrow id="S2.4.p1.6.m6.1b"><mo id="S2.4.p1.6.m6.1.1" stretchy="false">{</mo><mi id="S2.4.p1.6.m6.1.2">U</mi><mo id="S2.4.p1.6.m6.1.3">β©</mo><mi id="S2.4.p1.6.m6.1.4">X</mi><mo id="S2.4.p1.6.m6.1.5" lspace="0.278em" rspace="0.278em">:</mo><mi id="S2.4.p1.6.m6.1.6">U</mi></mrow><annotation encoding="application/x-tex" id="S2.4.p1.6.m6.1c">\{U\cap X:U</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.6.m6.1d">{ italic_U β© italic_X : italic_U</annotation></semantics></math> is an open neighborhood of <math alttext="y\}" class="ltx_math_unparsed" display="inline" id="S2.4.p1.7.m7.1"><semantics id="S2.4.p1.7.m7.1a"><mrow id="S2.4.p1.7.m7.1b"><mi id="S2.4.p1.7.m7.1.1">y</mi><mo id="S2.4.p1.7.m7.1.2" stretchy="false">}</mo></mrow><annotation encoding="application/x-tex" id="S2.4.p1.7.m7.1c">y\}</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.7.m7.1d">italic_y }</annotation></semantics></math>. It is clear that <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S2.4.p1.8.m8.1"><semantics id="S2.4.p1.8.m8.1a"><mi class="ltx_font_mathcaligraphic" id="S2.4.p1.8.m8.1.1" xref="S2.4.p1.8.m8.1.1.cmml">β±</mi><annotation-xml encoding="MathML-Content" id="S2.4.p1.8.m8.1b"><ci id="S2.4.p1.8.m8.1.1.cmml" xref="S2.4.p1.8.m8.1.1">β±</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.8.m8.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.8.m8.1d">caligraphic_F</annotation></semantics></math> converges to <math alttext="y" class="ltx_Math" display="inline" id="S2.4.p1.9.m9.1"><semantics id="S2.4.p1.9.m9.1a"><mi id="S2.4.p1.9.m9.1.1" xref="S2.4.p1.9.m9.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S2.4.p1.9.m9.1b"><ci id="S2.4.p1.9.m9.1.1.cmml" xref="S2.4.p1.9.m9.1.1">π¦</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.9.m9.1c">y</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.9.m9.1d">italic_y</annotation></semantics></math>. By the compactness of <math alttext="S" class="ltx_Math" display="inline" id="S2.4.p1.10.m10.1"><semantics id="S2.4.p1.10.m10.1a"><mi id="S2.4.p1.10.m10.1.1" xref="S2.4.p1.10.m10.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.4.p1.10.m10.1b"><ci id="S2.4.p1.10.m10.1.1.cmml" xref="S2.4.p1.10.m10.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.10.m10.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.10.m10.1d">italic_S</annotation></semantics></math>, the ultrafilter <math alttext="\mathcal{F}^{-1}" class="ltx_Math" display="inline" id="S2.4.p1.11.m11.1"><semantics id="S2.4.p1.11.m11.1a"><msup id="S2.4.p1.11.m11.1.1" xref="S2.4.p1.11.m11.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.4.p1.11.m11.1.1.2" xref="S2.4.p1.11.m11.1.1.2.cmml">β±</mi><mrow id="S2.4.p1.11.m11.1.1.3" xref="S2.4.p1.11.m11.1.1.3.cmml"><mo id="S2.4.p1.11.m11.1.1.3a" xref="S2.4.p1.11.m11.1.1.3.cmml">β</mo><mn id="S2.4.p1.11.m11.1.1.3.2" xref="S2.4.p1.11.m11.1.1.3.2.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.4.p1.11.m11.1b"><apply id="S2.4.p1.11.m11.1.1.cmml" xref="S2.4.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S2.4.p1.11.m11.1.1.1.cmml" xref="S2.4.p1.11.m11.1.1">superscript</csymbol><ci id="S2.4.p1.11.m11.1.1.2.cmml" xref="S2.4.p1.11.m11.1.1.2">β±</ci><apply id="S2.4.p1.11.m11.1.1.3.cmml" xref="S2.4.p1.11.m11.1.1.3"><minus id="S2.4.p1.11.m11.1.1.3.1.cmml" xref="S2.4.p1.11.m11.1.1.3"></minus><cn id="S2.4.p1.11.m11.1.1.3.2.cmml" type="integer" xref="S2.4.p1.11.m11.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.11.m11.1c">\mathcal{F}^{-1}</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.11.m11.1d">caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> generated by the family <math alttext="\{F^{-1}:X\supseteq F\in\mathcal{F}\}" class="ltx_Math" display="inline" id="S2.4.p1.12.m12.2"><semantics id="S2.4.p1.12.m12.2a"><mrow id="S2.4.p1.12.m12.2.2.2" xref="S2.4.p1.12.m12.2.2.3.cmml"><mo id="S2.4.p1.12.m12.2.2.2.3" stretchy="false" xref="S2.4.p1.12.m12.2.2.3.1.cmml">{</mo><msup id="S2.4.p1.12.m12.1.1.1.1" xref="S2.4.p1.12.m12.1.1.1.1.cmml"><mi id="S2.4.p1.12.m12.1.1.1.1.2" xref="S2.4.p1.12.m12.1.1.1.1.2.cmml">F</mi><mrow id="S2.4.p1.12.m12.1.1.1.1.3" xref="S2.4.p1.12.m12.1.1.1.1.3.cmml"><mo id="S2.4.p1.12.m12.1.1.1.1.3a" xref="S2.4.p1.12.m12.1.1.1.1.3.cmml">β</mo><mn id="S2.4.p1.12.m12.1.1.1.1.3.2" xref="S2.4.p1.12.m12.1.1.1.1.3.2.cmml">1</mn></mrow></msup><mo id="S2.4.p1.12.m12.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.4.p1.12.m12.2.2.3.1.cmml">:</mo><mrow id="S2.4.p1.12.m12.2.2.2.2" xref="S2.4.p1.12.m12.2.2.2.2.cmml"><mi id="S2.4.p1.12.m12.2.2.2.2.2" xref="S2.4.p1.12.m12.2.2.2.2.2.cmml">X</mi><mo id="S2.4.p1.12.m12.2.2.2.2.3" xref="S2.4.p1.12.m12.2.2.2.2.3.cmml">β</mo><mi id="S2.4.p1.12.m12.2.2.2.2.4" xref="S2.4.p1.12.m12.2.2.2.2.4.cmml">F</mi><mo id="S2.4.p1.12.m12.2.2.2.2.5" xref="S2.4.p1.12.m12.2.2.2.2.5.cmml">β</mo><mi class="ltx_font_mathcaligraphic" id="S2.4.p1.12.m12.2.2.2.2.6" xref="S2.4.p1.12.m12.2.2.2.2.6.cmml">β±</mi></mrow><mo id="S2.4.p1.12.m12.2.2.2.5" stretchy="false" xref="S2.4.p1.12.m12.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.4.p1.12.m12.2b"><apply id="S2.4.p1.12.m12.2.2.3.cmml" xref="S2.4.p1.12.m12.2.2.2"><csymbol cd="latexml" id="S2.4.p1.12.m12.2.2.3.1.cmml" xref="S2.4.p1.12.m12.2.2.2.3">conditional-set</csymbol><apply id="S2.4.p1.12.m12.1.1.1.1.cmml" xref="S2.4.p1.12.m12.1.1.1.1"><csymbol cd="ambiguous" id="S2.4.p1.12.m12.1.1.1.1.1.cmml" xref="S2.4.p1.12.m12.1.1.1.1">superscript</csymbol><ci id="S2.4.p1.12.m12.1.1.1.1.2.cmml" xref="S2.4.p1.12.m12.1.1.1.1.2">πΉ</ci><apply id="S2.4.p1.12.m12.1.1.1.1.3.cmml" xref="S2.4.p1.12.m12.1.1.1.1.3"><minus id="S2.4.p1.12.m12.1.1.1.1.3.1.cmml" xref="S2.4.p1.12.m12.1.1.1.1.3"></minus><cn id="S2.4.p1.12.m12.1.1.1.1.3.2.cmml" type="integer" xref="S2.4.p1.12.m12.1.1.1.1.3.2">1</cn></apply></apply><apply id="S2.4.p1.12.m12.2.2.2.2.cmml" xref="S2.4.p1.12.m12.2.2.2.2"><and id="S2.4.p1.12.m12.2.2.2.2a.cmml" xref="S2.4.p1.12.m12.2.2.2.2"></and><apply id="S2.4.p1.12.m12.2.2.2.2b.cmml" xref="S2.4.p1.12.m12.2.2.2.2"><csymbol cd="latexml" id="S2.4.p1.12.m12.2.2.2.2.3.cmml" xref="S2.4.p1.12.m12.2.2.2.2.3">superset-of-or-equals</csymbol><ci id="S2.4.p1.12.m12.2.2.2.2.2.cmml" xref="S2.4.p1.12.m12.2.2.2.2.2">π</ci><ci id="S2.4.p1.12.m12.2.2.2.2.4.cmml" xref="S2.4.p1.12.m12.2.2.2.2.4">πΉ</ci></apply><apply id="S2.4.p1.12.m12.2.2.2.2c.cmml" xref="S2.4.p1.12.m12.2.2.2.2"><in id="S2.4.p1.12.m12.2.2.2.2.5.cmml" xref="S2.4.p1.12.m12.2.2.2.2.5"></in><share href="https://arxiv.org/html/2503.13666v1#S2.4.p1.12.m12.2.2.2.2.4.cmml" id="S2.4.p1.12.m12.2.2.2.2d.cmml" xref="S2.4.p1.12.m12.2.2.2.2"></share><ci id="S2.4.p1.12.m12.2.2.2.2.6.cmml" xref="S2.4.p1.12.m12.2.2.2.2.6">β±</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.12.m12.2c">\{F^{-1}:X\supseteq F\in\mathcal{F}\}</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.12.m12.2d">{ italic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT : italic_X β italic_F β caligraphic_F }</annotation></semantics></math> converges to some point of <math alttext="S" class="ltx_Math" display="inline" id="S2.4.p1.13.m13.1"><semantics id="S2.4.p1.13.m13.1a"><mi id="S2.4.p1.13.m13.1.1" xref="S2.4.p1.13.m13.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.4.p1.13.m13.1b"><ci id="S2.4.p1.13.m13.1.1.cmml" xref="S2.4.p1.13.m13.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.13.m13.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.13.m13.1d">italic_S</annotation></semantics></math>. Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S2.Thmtheorem3" title="Proposition 2.3. β£ 2. Compact-like semigroups with dense inverse subsemigroups and the automatic continuity of inversion β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">2.3</span></a> implies that <math alttext="S" class="ltx_Math" display="inline" id="S2.4.p1.14.m14.1"><semantics id="S2.4.p1.14.m14.1a"><mi id="S2.4.p1.14.m14.1.1" xref="S2.4.p1.14.m14.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.4.p1.14.m14.1b"><ci id="S2.4.p1.14.m14.1.1.cmml" xref="S2.4.p1.14.m14.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.14.m14.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.14.m14.1d">italic_S</annotation></semantics></math> is an inverse topological semigroup. By <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib28" title="">28</a>, Proposition 1.6.7]</cite>, inversion is automatically continuous in compact inverse topological semigroups. Hence <math alttext="S" class="ltx_Math" display="inline" id="S2.4.p1.15.m15.1"><semantics id="S2.4.p1.15.m15.1a"><mi id="S2.4.p1.15.m15.1.1" xref="S2.4.p1.15.m15.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.4.p1.15.m15.1b"><ci id="S2.4.p1.15.m15.1.1.cmml" xref="S2.4.p1.15.m15.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.4.p1.15.m15.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.4.p1.15.m15.1d">italic_S</annotation></semantics></math> is a topological inverse semigroup.</p> </div> <div class="ltx_para" id="S2.5.p2"> <p class="ltx_p" id="S2.5.p2.1">(ii) Since the space <math alttext="S" class="ltx_Math" display="inline" id="S2.5.p2.1.m1.1"><semantics id="S2.5.p2.1.m1.1a"><mi id="S2.5.p2.1.m1.1.1" xref="S2.5.p2.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.1.m1.1b"><ci id="S2.5.p2.1.m1.1.1.cmml" xref="S2.5.p2.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.1.m1.1d">italic_S</annotation></semantics></math> is sequential we get that</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="S=\overline{X}=\operatorname{cls}(X)=\operatorname{cls}^{\omega_{1}}(X)=% \bigcup_{\xi<\omega_{1}}\operatorname{cls}^{\xi}(X)." class="ltx_Math" display="block" id="S2.Ex4.m1.5"><semantics id="S2.Ex4.m1.5a"><mrow id="S2.Ex4.m1.5.5.1" xref="S2.Ex4.m1.5.5.1.1.cmml"><mrow id="S2.Ex4.m1.5.5.1.1" xref="S2.Ex4.m1.5.5.1.1.cmml"><mi id="S2.Ex4.m1.5.5.1.1.4" xref="S2.Ex4.m1.5.5.1.1.4.cmml">S</mi><mo id="S2.Ex4.m1.5.5.1.1.5" xref="S2.Ex4.m1.5.5.1.1.5.cmml">=</mo><mover accent="true" id="S2.Ex4.m1.5.5.1.1.6" xref="S2.Ex4.m1.5.5.1.1.6.cmml"><mi id="S2.Ex4.m1.5.5.1.1.6.2" xref="S2.Ex4.m1.5.5.1.1.6.2.cmml">X</mi><mo id="S2.Ex4.m1.5.5.1.1.6.1" xref="S2.Ex4.m1.5.5.1.1.6.1.cmml">Β―</mo></mover><mo id="S2.Ex4.m1.5.5.1.1.7" xref="S2.Ex4.m1.5.5.1.1.7.cmml">=</mo><mrow id="S2.Ex4.m1.5.5.1.1.8.2" xref="S2.Ex4.m1.5.5.1.1.8.1.cmml"><mi id="S2.Ex4.m1.1.1" xref="S2.Ex4.m1.1.1.cmml">cls</mi><mo id="S2.Ex4.m1.5.5.1.1.8.2a" xref="S2.Ex4.m1.5.5.1.1.8.1.cmml">β‘</mo><mrow id="S2.Ex4.m1.5.5.1.1.8.2.1" xref="S2.Ex4.m1.5.5.1.1.8.1.cmml"><mo id="S2.Ex4.m1.5.5.1.1.8.2.1.1" stretchy="false" xref="S2.Ex4.m1.5.5.1.1.8.1.cmml">(</mo><mi id="S2.Ex4.m1.2.2" xref="S2.Ex4.m1.2.2.cmml">X</mi><mo id="S2.Ex4.m1.5.5.1.1.8.2.1.2" stretchy="false" xref="S2.Ex4.m1.5.5.1.1.8.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex4.m1.5.5.1.1.9" xref="S2.Ex4.m1.5.5.1.1.9.cmml">=</mo><mrow id="S2.Ex4.m1.5.5.1.1.1.1" xref="S2.Ex4.m1.5.5.1.1.1.2.cmml"><msup id="S2.Ex4.m1.5.5.1.1.1.1.1" xref="S2.Ex4.m1.5.5.1.1.1.1.1.cmml"><mi id="S2.Ex4.m1.5.5.1.1.1.1.1.2" xref="S2.Ex4.m1.5.5.1.1.1.1.1.2.cmml">cls</mi><msub 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xref="S2.Ex4.m1.5.5.1.1.2.1.1.1.2">cls</ci><ci id="S2.Ex4.m1.5.5.1.1.2.1.1.1.3.cmml" xref="S2.Ex4.m1.5.5.1.1.2.1.1.1.3">π</ci></apply><ci id="S2.Ex4.m1.4.4.cmml" xref="S2.Ex4.m1.4.4">π</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m1.5c">S=\overline{X}=\operatorname{cls}(X)=\operatorname{cls}^{\omega_{1}}(X)=% \bigcup_{\xi<\omega_{1}}\operatorname{cls}^{\xi}(X).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m1.5d">italic_S = overΒ― start_ARG italic_X end_ARG = roman_cls ( italic_X ) = roman_cls start_POSTSUPERSCRIPT italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( italic_X ) = β start_POSTSUBSCRIPT italic_ΞΎ < italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_cls start_POSTSUPERSCRIPT italic_ΞΎ end_POSTSUPERSCRIPT ( italic_X ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.5.p2.33">First let us show that <math alttext="S" class="ltx_Math" display="inline" id="S2.5.p2.2.m1.1"><semantics id="S2.5.p2.2.m1.1a"><mi id="S2.5.p2.2.m1.1.1" xref="S2.5.p2.2.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.2.m1.1b"><ci id="S2.5.p2.2.m1.1.1.cmml" xref="S2.5.p2.2.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.2.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.2.m1.1d">italic_S</annotation></semantics></math> is an inverse semigroup. Note that <math alttext="\operatorname{cls}^{0}(X)=X" class="ltx_Math" display="inline" id="S2.5.p2.3.m2.2"><semantics id="S2.5.p2.3.m2.2a"><mrow id="S2.5.p2.3.m2.2.2" xref="S2.5.p2.3.m2.2.2.cmml"><mrow id="S2.5.p2.3.m2.2.2.1.1" xref="S2.5.p2.3.m2.2.2.1.2.cmml"><msup id="S2.5.p2.3.m2.2.2.1.1.1" xref="S2.5.p2.3.m2.2.2.1.1.1.cmml"><mi id="S2.5.p2.3.m2.2.2.1.1.1.2" xref="S2.5.p2.3.m2.2.2.1.1.1.2.cmml">cls</mi><mn id="S2.5.p2.3.m2.2.2.1.1.1.3" xref="S2.5.p2.3.m2.2.2.1.1.1.3.cmml">0</mn></msup><mo id="S2.5.p2.3.m2.2.2.1.1a" xref="S2.5.p2.3.m2.2.2.1.2.cmml">β‘</mo><mrow id="S2.5.p2.3.m2.2.2.1.1.2" xref="S2.5.p2.3.m2.2.2.1.2.cmml"><mo id="S2.5.p2.3.m2.2.2.1.1.2.1" stretchy="false" xref="S2.5.p2.3.m2.2.2.1.2.cmml">(</mo><mi id="S2.5.p2.3.m2.1.1" xref="S2.5.p2.3.m2.1.1.cmml">X</mi><mo id="S2.5.p2.3.m2.2.2.1.1.2.2" stretchy="false" xref="S2.5.p2.3.m2.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S2.5.p2.3.m2.2.2.2" xref="S2.5.p2.3.m2.2.2.2.cmml">=</mo><mi id="S2.5.p2.3.m2.2.2.3" xref="S2.5.p2.3.m2.2.2.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.3.m2.2b"><apply id="S2.5.p2.3.m2.2.2.cmml" xref="S2.5.p2.3.m2.2.2"><eq id="S2.5.p2.3.m2.2.2.2.cmml" xref="S2.5.p2.3.m2.2.2.2"></eq><apply id="S2.5.p2.3.m2.2.2.1.2.cmml" xref="S2.5.p2.3.m2.2.2.1.1"><apply id="S2.5.p2.3.m2.2.2.1.1.1.cmml" xref="S2.5.p2.3.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.3.m2.2.2.1.1.1.1.cmml" xref="S2.5.p2.3.m2.2.2.1.1.1">superscript</csymbol><ci id="S2.5.p2.3.m2.2.2.1.1.1.2.cmml" xref="S2.5.p2.3.m2.2.2.1.1.1.2">cls</ci><cn id="S2.5.p2.3.m2.2.2.1.1.1.3.cmml" type="integer" xref="S2.5.p2.3.m2.2.2.1.1.1.3">0</cn></apply><ci id="S2.5.p2.3.m2.1.1.cmml" xref="S2.5.p2.3.m2.1.1">π</ci></apply><ci id="S2.5.p2.3.m2.2.2.3.cmml" xref="S2.5.p2.3.m2.2.2.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.3.m2.2c">\operatorname{cls}^{0}(X)=X</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.3.m2.2d">roman_cls start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_X ) = italic_X</annotation></semantics></math> is an inverse subsemigroup of <math alttext="S" class="ltx_Math" display="inline" id="S2.5.p2.4.m3.1"><semantics id="S2.5.p2.4.m3.1a"><mi id="S2.5.p2.4.m3.1.1" xref="S2.5.p2.4.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.4.m3.1b"><ci id="S2.5.p2.4.m3.1.1.cmml" xref="S2.5.p2.4.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.4.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.4.m3.1d">italic_S</annotation></semantics></math>. Assume that <math alttext="\operatorname{cls}^{\xi}(X)" class="ltx_Math" display="inline" id="S2.5.p2.5.m4.2"><semantics id="S2.5.p2.5.m4.2a"><mrow id="S2.5.p2.5.m4.2.2.1" xref="S2.5.p2.5.m4.2.2.2.cmml"><msup id="S2.5.p2.5.m4.2.2.1.1" xref="S2.5.p2.5.m4.2.2.1.1.cmml"><mi id="S2.5.p2.5.m4.2.2.1.1.2" xref="S2.5.p2.5.m4.2.2.1.1.2.cmml">cls</mi><mi id="S2.5.p2.5.m4.2.2.1.1.3" xref="S2.5.p2.5.m4.2.2.1.1.3.cmml">ΞΎ</mi></msup><mo id="S2.5.p2.5.m4.2.2.1a" xref="S2.5.p2.5.m4.2.2.2.cmml">β‘</mo><mrow id="S2.5.p2.5.m4.2.2.1.2" xref="S2.5.p2.5.m4.2.2.2.cmml"><mo id="S2.5.p2.5.m4.2.2.1.2.1" stretchy="false" xref="S2.5.p2.5.m4.2.2.2.cmml">(</mo><mi id="S2.5.p2.5.m4.1.1" xref="S2.5.p2.5.m4.1.1.cmml">X</mi><mo id="S2.5.p2.5.m4.2.2.1.2.2" stretchy="false" xref="S2.5.p2.5.m4.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.5.m4.2b"><apply id="S2.5.p2.5.m4.2.2.2.cmml" xref="S2.5.p2.5.m4.2.2.1"><apply id="S2.5.p2.5.m4.2.2.1.1.cmml" xref="S2.5.p2.5.m4.2.2.1.1"><csymbol cd="ambiguous" id="S2.5.p2.5.m4.2.2.1.1.1.cmml" xref="S2.5.p2.5.m4.2.2.1.1">superscript</csymbol><ci id="S2.5.p2.5.m4.2.2.1.1.2.cmml" xref="S2.5.p2.5.m4.2.2.1.1.2">cls</ci><ci id="S2.5.p2.5.m4.2.2.1.1.3.cmml" xref="S2.5.p2.5.m4.2.2.1.1.3">π</ci></apply><ci id="S2.5.p2.5.m4.1.1.cmml" xref="S2.5.p2.5.m4.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.5.m4.2c">\operatorname{cls}^{\xi}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.5.m4.2d">roman_cls start_POSTSUPERSCRIPT italic_ΞΎ end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math> is an inverse subsemigroup of <math alttext="S" class="ltx_Math" display="inline" id="S2.5.p2.6.m5.1"><semantics id="S2.5.p2.6.m5.1a"><mi id="S2.5.p2.6.m5.1.1" xref="S2.5.p2.6.m5.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.6.m5.1b"><ci id="S2.5.p2.6.m5.1.1.cmml" xref="S2.5.p2.6.m5.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.6.m5.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.6.m5.1d">italic_S</annotation></semantics></math> for every <math alttext="\xi<\alpha\leq\omega_{1}" class="ltx_Math" display="inline" id="S2.5.p2.7.m6.1"><semantics id="S2.5.p2.7.m6.1a"><mrow id="S2.5.p2.7.m6.1.1" xref="S2.5.p2.7.m6.1.1.cmml"><mi id="S2.5.p2.7.m6.1.1.2" xref="S2.5.p2.7.m6.1.1.2.cmml">ΞΎ</mi><mo id="S2.5.p2.7.m6.1.1.3" xref="S2.5.p2.7.m6.1.1.3.cmml"><</mo><mi id="S2.5.p2.7.m6.1.1.4" xref="S2.5.p2.7.m6.1.1.4.cmml">Ξ±</mi><mo id="S2.5.p2.7.m6.1.1.5" xref="S2.5.p2.7.m6.1.1.5.cmml">β€</mo><msub id="S2.5.p2.7.m6.1.1.6" xref="S2.5.p2.7.m6.1.1.6.cmml"><mi id="S2.5.p2.7.m6.1.1.6.2" xref="S2.5.p2.7.m6.1.1.6.2.cmml">Ο</mi><mn id="S2.5.p2.7.m6.1.1.6.3" xref="S2.5.p2.7.m6.1.1.6.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.7.m6.1b"><apply id="S2.5.p2.7.m6.1.1.cmml" xref="S2.5.p2.7.m6.1.1"><and id="S2.5.p2.7.m6.1.1a.cmml" xref="S2.5.p2.7.m6.1.1"></and><apply id="S2.5.p2.7.m6.1.1b.cmml" xref="S2.5.p2.7.m6.1.1"><lt id="S2.5.p2.7.m6.1.1.3.cmml" xref="S2.5.p2.7.m6.1.1.3"></lt><ci id="S2.5.p2.7.m6.1.1.2.cmml" xref="S2.5.p2.7.m6.1.1.2">π</ci><ci id="S2.5.p2.7.m6.1.1.4.cmml" xref="S2.5.p2.7.m6.1.1.4">πΌ</ci></apply><apply id="S2.5.p2.7.m6.1.1c.cmml" xref="S2.5.p2.7.m6.1.1"><leq id="S2.5.p2.7.m6.1.1.5.cmml" xref="S2.5.p2.7.m6.1.1.5"></leq><share href="https://arxiv.org/html/2503.13666v1#S2.5.p2.7.m6.1.1.4.cmml" id="S2.5.p2.7.m6.1.1d.cmml" xref="S2.5.p2.7.m6.1.1"></share><apply id="S2.5.p2.7.m6.1.1.6.cmml" xref="S2.5.p2.7.m6.1.1.6"><csymbol cd="ambiguous" id="S2.5.p2.7.m6.1.1.6.1.cmml" xref="S2.5.p2.7.m6.1.1.6">subscript</csymbol><ci id="S2.5.p2.7.m6.1.1.6.2.cmml" xref="S2.5.p2.7.m6.1.1.6.2">π</ci><cn id="S2.5.p2.7.m6.1.1.6.3.cmml" type="integer" xref="S2.5.p2.7.m6.1.1.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.7.m6.1c">\xi<\alpha\leq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.7.m6.1d">italic_ΞΎ < italic_Ξ± β€ italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. If the ordinal <math alttext="\alpha" class="ltx_Math" display="inline" id="S2.5.p2.8.m7.1"><semantics id="S2.5.p2.8.m7.1a"><mi id="S2.5.p2.8.m7.1.1" xref="S2.5.p2.8.m7.1.1.cmml">Ξ±</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.8.m7.1b"><ci id="S2.5.p2.8.m7.1.1.cmml" xref="S2.5.p2.8.m7.1.1">πΌ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.8.m7.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.8.m7.1d">italic_Ξ±</annotation></semantics></math> is limit, then <math alttext="\operatorname{cls}^{\alpha}(X)=\bigcup_{\xi<\alpha}\operatorname{cls}^{\xi}(X)" class="ltx_Math" display="inline" id="S2.5.p2.9.m8.4"><semantics id="S2.5.p2.9.m8.4a"><mrow id="S2.5.p2.9.m8.4.4" xref="S2.5.p2.9.m8.4.4.cmml"><mrow id="S2.5.p2.9.m8.3.3.1.1" xref="S2.5.p2.9.m8.3.3.1.2.cmml"><msup id="S2.5.p2.9.m8.3.3.1.1.1" xref="S2.5.p2.9.m8.3.3.1.1.1.cmml"><mi id="S2.5.p2.9.m8.3.3.1.1.1.2" xref="S2.5.p2.9.m8.3.3.1.1.1.2.cmml">cls</mi><mi id="S2.5.p2.9.m8.3.3.1.1.1.3" xref="S2.5.p2.9.m8.3.3.1.1.1.3.cmml">Ξ±</mi></msup><mo id="S2.5.p2.9.m8.3.3.1.1a" xref="S2.5.p2.9.m8.3.3.1.2.cmml">β‘</mo><mrow id="S2.5.p2.9.m8.3.3.1.1.2" xref="S2.5.p2.9.m8.3.3.1.2.cmml"><mo id="S2.5.p2.9.m8.3.3.1.1.2.1" stretchy="false" xref="S2.5.p2.9.m8.3.3.1.2.cmml">(</mo><mi id="S2.5.p2.9.m8.1.1" xref="S2.5.p2.9.m8.1.1.cmml">X</mi><mo id="S2.5.p2.9.m8.3.3.1.1.2.2" stretchy="false" xref="S2.5.p2.9.m8.3.3.1.2.cmml">)</mo></mrow></mrow><mo id="S2.5.p2.9.m8.4.4.3" rspace="0.111em" xref="S2.5.p2.9.m8.4.4.3.cmml">=</mo><mrow id="S2.5.p2.9.m8.4.4.2" xref="S2.5.p2.9.m8.4.4.2.cmml"><msub id="S2.5.p2.9.m8.4.4.2.2" xref="S2.5.p2.9.m8.4.4.2.2.cmml"><mo id="S2.5.p2.9.m8.4.4.2.2.2" xref="S2.5.p2.9.m8.4.4.2.2.2.cmml">β</mo><mrow id="S2.5.p2.9.m8.4.4.2.2.3" xref="S2.5.p2.9.m8.4.4.2.2.3.cmml"><mi id="S2.5.p2.9.m8.4.4.2.2.3.2" xref="S2.5.p2.9.m8.4.4.2.2.3.2.cmml">ΞΎ</mi><mo id="S2.5.p2.9.m8.4.4.2.2.3.1" xref="S2.5.p2.9.m8.4.4.2.2.3.1.cmml"><</mo><mi id="S2.5.p2.9.m8.4.4.2.2.3.3" xref="S2.5.p2.9.m8.4.4.2.2.3.3.cmml">Ξ±</mi></mrow></msub><mrow id="S2.5.p2.9.m8.4.4.2.1.1" xref="S2.5.p2.9.m8.4.4.2.1.2.cmml"><msup id="S2.5.p2.9.m8.4.4.2.1.1.1" xref="S2.5.p2.9.m8.4.4.2.1.1.1.cmml"><mi id="S2.5.p2.9.m8.4.4.2.1.1.1.2" xref="S2.5.p2.9.m8.4.4.2.1.1.1.2.cmml">cls</mi><mi id="S2.5.p2.9.m8.4.4.2.1.1.1.3" xref="S2.5.p2.9.m8.4.4.2.1.1.1.3.cmml">ΞΎ</mi></msup><mo id="S2.5.p2.9.m8.4.4.2.1.1a" xref="S2.5.p2.9.m8.4.4.2.1.2.cmml">β‘</mo><mrow id="S2.5.p2.9.m8.4.4.2.1.1.2" xref="S2.5.p2.9.m8.4.4.2.1.2.cmml"><mo id="S2.5.p2.9.m8.4.4.2.1.1.2.1" stretchy="false" xref="S2.5.p2.9.m8.4.4.2.1.2.cmml">(</mo><mi id="S2.5.p2.9.m8.2.2" xref="S2.5.p2.9.m8.2.2.cmml">X</mi><mo id="S2.5.p2.9.m8.4.4.2.1.1.2.2" stretchy="false" xref="S2.5.p2.9.m8.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.9.m8.4b"><apply id="S2.5.p2.9.m8.4.4.cmml" xref="S2.5.p2.9.m8.4.4"><eq id="S2.5.p2.9.m8.4.4.3.cmml" xref="S2.5.p2.9.m8.4.4.3"></eq><apply id="S2.5.p2.9.m8.3.3.1.2.cmml" xref="S2.5.p2.9.m8.3.3.1.1"><apply id="S2.5.p2.9.m8.3.3.1.1.1.cmml" xref="S2.5.p2.9.m8.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.9.m8.3.3.1.1.1.1.cmml" xref="S2.5.p2.9.m8.3.3.1.1.1">superscript</csymbol><ci id="S2.5.p2.9.m8.3.3.1.1.1.2.cmml" xref="S2.5.p2.9.m8.3.3.1.1.1.2">cls</ci><ci id="S2.5.p2.9.m8.3.3.1.1.1.3.cmml" xref="S2.5.p2.9.m8.3.3.1.1.1.3">πΌ</ci></apply><ci id="S2.5.p2.9.m8.1.1.cmml" xref="S2.5.p2.9.m8.1.1">π</ci></apply><apply id="S2.5.p2.9.m8.4.4.2.cmml" xref="S2.5.p2.9.m8.4.4.2"><apply id="S2.5.p2.9.m8.4.4.2.2.cmml" xref="S2.5.p2.9.m8.4.4.2.2"><csymbol cd="ambiguous" id="S2.5.p2.9.m8.4.4.2.2.1.cmml" xref="S2.5.p2.9.m8.4.4.2.2">subscript</csymbol><union id="S2.5.p2.9.m8.4.4.2.2.2.cmml" xref="S2.5.p2.9.m8.4.4.2.2.2"></union><apply id="S2.5.p2.9.m8.4.4.2.2.3.cmml" xref="S2.5.p2.9.m8.4.4.2.2.3"><lt id="S2.5.p2.9.m8.4.4.2.2.3.1.cmml" xref="S2.5.p2.9.m8.4.4.2.2.3.1"></lt><ci id="S2.5.p2.9.m8.4.4.2.2.3.2.cmml" xref="S2.5.p2.9.m8.4.4.2.2.3.2">π</ci><ci id="S2.5.p2.9.m8.4.4.2.2.3.3.cmml" xref="S2.5.p2.9.m8.4.4.2.2.3.3">πΌ</ci></apply></apply><apply id="S2.5.p2.9.m8.4.4.2.1.2.cmml" xref="S2.5.p2.9.m8.4.4.2.1.1"><apply id="S2.5.p2.9.m8.4.4.2.1.1.1.cmml" xref="S2.5.p2.9.m8.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.9.m8.4.4.2.1.1.1.1.cmml" xref="S2.5.p2.9.m8.4.4.2.1.1.1">superscript</csymbol><ci id="S2.5.p2.9.m8.4.4.2.1.1.1.2.cmml" xref="S2.5.p2.9.m8.4.4.2.1.1.1.2">cls</ci><ci id="S2.5.p2.9.m8.4.4.2.1.1.1.3.cmml" xref="S2.5.p2.9.m8.4.4.2.1.1.1.3">π</ci></apply><ci id="S2.5.p2.9.m8.2.2.cmml" xref="S2.5.p2.9.m8.2.2">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.9.m8.4c">\operatorname{cls}^{\alpha}(X)=\bigcup_{\xi<\alpha}\operatorname{cls}^{\xi}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.9.m8.4d">roman_cls start_POSTSUPERSCRIPT italic_Ξ± end_POSTSUPERSCRIPT ( italic_X ) = β start_POSTSUBSCRIPT italic_ΞΎ < italic_Ξ± end_POSTSUBSCRIPT roman_cls start_POSTSUPERSCRIPT italic_ΞΎ end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math> is an inverse semigroup, being an ascending union of inverse semigroups. Suppose that <math alttext="\alpha=\gamma+1" class="ltx_Math" display="inline" id="S2.5.p2.10.m9.1"><semantics id="S2.5.p2.10.m9.1a"><mrow id="S2.5.p2.10.m9.1.1" xref="S2.5.p2.10.m9.1.1.cmml"><mi id="S2.5.p2.10.m9.1.1.2" xref="S2.5.p2.10.m9.1.1.2.cmml">Ξ±</mi><mo id="S2.5.p2.10.m9.1.1.1" xref="S2.5.p2.10.m9.1.1.1.cmml">=</mo><mrow id="S2.5.p2.10.m9.1.1.3" xref="S2.5.p2.10.m9.1.1.3.cmml"><mi id="S2.5.p2.10.m9.1.1.3.2" xref="S2.5.p2.10.m9.1.1.3.2.cmml">Ξ³</mi><mo id="S2.5.p2.10.m9.1.1.3.1" xref="S2.5.p2.10.m9.1.1.3.1.cmml">+</mo><mn id="S2.5.p2.10.m9.1.1.3.3" xref="S2.5.p2.10.m9.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.10.m9.1b"><apply id="S2.5.p2.10.m9.1.1.cmml" xref="S2.5.p2.10.m9.1.1"><eq id="S2.5.p2.10.m9.1.1.1.cmml" xref="S2.5.p2.10.m9.1.1.1"></eq><ci id="S2.5.p2.10.m9.1.1.2.cmml" xref="S2.5.p2.10.m9.1.1.2">πΌ</ci><apply id="S2.5.p2.10.m9.1.1.3.cmml" xref="S2.5.p2.10.m9.1.1.3"><plus id="S2.5.p2.10.m9.1.1.3.1.cmml" xref="S2.5.p2.10.m9.1.1.3.1"></plus><ci id="S2.5.p2.10.m9.1.1.3.2.cmml" xref="S2.5.p2.10.m9.1.1.3.2">πΎ</ci><cn id="S2.5.p2.10.m9.1.1.3.3.cmml" type="integer" xref="S2.5.p2.10.m9.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.10.m9.1c">\alpha=\gamma+1</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.10.m9.1d">italic_Ξ± = italic_Ξ³ + 1</annotation></semantics></math> for some <math alttext="\gamma<\omega_{1}" class="ltx_Math" display="inline" id="S2.5.p2.11.m10.1"><semantics id="S2.5.p2.11.m10.1a"><mrow id="S2.5.p2.11.m10.1.1" xref="S2.5.p2.11.m10.1.1.cmml"><mi id="S2.5.p2.11.m10.1.1.2" xref="S2.5.p2.11.m10.1.1.2.cmml">Ξ³</mi><mo id="S2.5.p2.11.m10.1.1.1" xref="S2.5.p2.11.m10.1.1.1.cmml"><</mo><msub id="S2.5.p2.11.m10.1.1.3" xref="S2.5.p2.11.m10.1.1.3.cmml"><mi id="S2.5.p2.11.m10.1.1.3.2" xref="S2.5.p2.11.m10.1.1.3.2.cmml">Ο</mi><mn id="S2.5.p2.11.m10.1.1.3.3" xref="S2.5.p2.11.m10.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.11.m10.1b"><apply id="S2.5.p2.11.m10.1.1.cmml" xref="S2.5.p2.11.m10.1.1"><lt id="S2.5.p2.11.m10.1.1.1.cmml" xref="S2.5.p2.11.m10.1.1.1"></lt><ci id="S2.5.p2.11.m10.1.1.2.cmml" xref="S2.5.p2.11.m10.1.1.2">πΎ</ci><apply id="S2.5.p2.11.m10.1.1.3.cmml" xref="S2.5.p2.11.m10.1.1.3"><csymbol cd="ambiguous" id="S2.5.p2.11.m10.1.1.3.1.cmml" xref="S2.5.p2.11.m10.1.1.3">subscript</csymbol><ci id="S2.5.p2.11.m10.1.1.3.2.cmml" xref="S2.5.p2.11.m10.1.1.3.2">π</ci><cn id="S2.5.p2.11.m10.1.1.3.3.cmml" type="integer" xref="S2.5.p2.11.m10.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.11.m10.1c">\gamma<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.11.m10.1d">italic_Ξ³ < italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S2.Thmtheorem1" title="Lemma 2.1. β£ 2. Compact-like semigroups with dense inverse subsemigroups and the automatic continuity of inversion β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">2.1</span></a> implies that <math alttext="\operatorname{cls}^{\alpha}(X)=s(\operatorname{cls}^{\gamma}(X))" class="ltx_Math" display="inline" id="S2.5.p2.12.m11.4"><semantics id="S2.5.p2.12.m11.4a"><mrow id="S2.5.p2.12.m11.4.4" xref="S2.5.p2.12.m11.4.4.cmml"><mrow id="S2.5.p2.12.m11.3.3.1.1" xref="S2.5.p2.12.m11.3.3.1.2.cmml"><msup id="S2.5.p2.12.m11.3.3.1.1.1" xref="S2.5.p2.12.m11.3.3.1.1.1.cmml"><mi id="S2.5.p2.12.m11.3.3.1.1.1.2" xref="S2.5.p2.12.m11.3.3.1.1.1.2.cmml">cls</mi><mi id="S2.5.p2.12.m11.3.3.1.1.1.3" xref="S2.5.p2.12.m11.3.3.1.1.1.3.cmml">Ξ±</mi></msup><mo id="S2.5.p2.12.m11.3.3.1.1a" xref="S2.5.p2.12.m11.3.3.1.2.cmml">β‘</mo><mrow id="S2.5.p2.12.m11.3.3.1.1.2" xref="S2.5.p2.12.m11.3.3.1.2.cmml"><mo id="S2.5.p2.12.m11.3.3.1.1.2.1" stretchy="false" xref="S2.5.p2.12.m11.3.3.1.2.cmml">(</mo><mi id="S2.5.p2.12.m11.1.1" xref="S2.5.p2.12.m11.1.1.cmml">X</mi><mo id="S2.5.p2.12.m11.3.3.1.1.2.2" stretchy="false" xref="S2.5.p2.12.m11.3.3.1.2.cmml">)</mo></mrow></mrow><mo id="S2.5.p2.12.m11.4.4.3" xref="S2.5.p2.12.m11.4.4.3.cmml">=</mo><mrow id="S2.5.p2.12.m11.4.4.2" xref="S2.5.p2.12.m11.4.4.2.cmml"><mi id="S2.5.p2.12.m11.4.4.2.3" xref="S2.5.p2.12.m11.4.4.2.3.cmml">s</mi><mo id="S2.5.p2.12.m11.4.4.2.2" xref="S2.5.p2.12.m11.4.4.2.2.cmml">β’</mo><mrow id="S2.5.p2.12.m11.4.4.2.1.1" xref="S2.5.p2.12.m11.4.4.2.cmml"><mo id="S2.5.p2.12.m11.4.4.2.1.1.2" stretchy="false" xref="S2.5.p2.12.m11.4.4.2.cmml">(</mo><mrow id="S2.5.p2.12.m11.4.4.2.1.1.1.1" xref="S2.5.p2.12.m11.4.4.2.1.1.1.2.cmml"><msup id="S2.5.p2.12.m11.4.4.2.1.1.1.1.1" xref="S2.5.p2.12.m11.4.4.2.1.1.1.1.1.cmml"><mi id="S2.5.p2.12.m11.4.4.2.1.1.1.1.1.2" xref="S2.5.p2.12.m11.4.4.2.1.1.1.1.1.2.cmml">cls</mi><mi id="S2.5.p2.12.m11.4.4.2.1.1.1.1.1.3" xref="S2.5.p2.12.m11.4.4.2.1.1.1.1.1.3.cmml">Ξ³</mi></msup><mo id="S2.5.p2.12.m11.4.4.2.1.1.1.1a" xref="S2.5.p2.12.m11.4.4.2.1.1.1.2.cmml">β‘</mo><mrow id="S2.5.p2.12.m11.4.4.2.1.1.1.1.2" xref="S2.5.p2.12.m11.4.4.2.1.1.1.2.cmml"><mo id="S2.5.p2.12.m11.4.4.2.1.1.1.1.2.1" stretchy="false" xref="S2.5.p2.12.m11.4.4.2.1.1.1.2.cmml">(</mo><mi id="S2.5.p2.12.m11.2.2" xref="S2.5.p2.12.m11.2.2.cmml">X</mi><mo id="S2.5.p2.12.m11.4.4.2.1.1.1.1.2.2" stretchy="false" xref="S2.5.p2.12.m11.4.4.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.5.p2.12.m11.4.4.2.1.1.3" stretchy="false" xref="S2.5.p2.12.m11.4.4.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.12.m11.4b"><apply id="S2.5.p2.12.m11.4.4.cmml" xref="S2.5.p2.12.m11.4.4"><eq id="S2.5.p2.12.m11.4.4.3.cmml" xref="S2.5.p2.12.m11.4.4.3"></eq><apply id="S2.5.p2.12.m11.3.3.1.2.cmml" xref="S2.5.p2.12.m11.3.3.1.1"><apply id="S2.5.p2.12.m11.3.3.1.1.1.cmml" xref="S2.5.p2.12.m11.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.12.m11.3.3.1.1.1.1.cmml" xref="S2.5.p2.12.m11.3.3.1.1.1">superscript</csymbol><ci id="S2.5.p2.12.m11.3.3.1.1.1.2.cmml" xref="S2.5.p2.12.m11.3.3.1.1.1.2">cls</ci><ci id="S2.5.p2.12.m11.3.3.1.1.1.3.cmml" xref="S2.5.p2.12.m11.3.3.1.1.1.3">πΌ</ci></apply><ci id="S2.5.p2.12.m11.1.1.cmml" xref="S2.5.p2.12.m11.1.1">π</ci></apply><apply id="S2.5.p2.12.m11.4.4.2.cmml" xref="S2.5.p2.12.m11.4.4.2"><times id="S2.5.p2.12.m11.4.4.2.2.cmml" xref="S2.5.p2.12.m11.4.4.2.2"></times><ci id="S2.5.p2.12.m11.4.4.2.3.cmml" xref="S2.5.p2.12.m11.4.4.2.3">π </ci><apply id="S2.5.p2.12.m11.4.4.2.1.1.1.2.cmml" xref="S2.5.p2.12.m11.4.4.2.1.1.1.1"><apply id="S2.5.p2.12.m11.4.4.2.1.1.1.1.1.cmml" xref="S2.5.p2.12.m11.4.4.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.12.m11.4.4.2.1.1.1.1.1.1.cmml" xref="S2.5.p2.12.m11.4.4.2.1.1.1.1.1">superscript</csymbol><ci id="S2.5.p2.12.m11.4.4.2.1.1.1.1.1.2.cmml" xref="S2.5.p2.12.m11.4.4.2.1.1.1.1.1.2">cls</ci><ci id="S2.5.p2.12.m11.4.4.2.1.1.1.1.1.3.cmml" xref="S2.5.p2.12.m11.4.4.2.1.1.1.1.1.3">πΎ</ci></apply><ci id="S2.5.p2.12.m11.2.2.cmml" xref="S2.5.p2.12.m11.2.2">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.12.m11.4c">\operatorname{cls}^{\alpha}(X)=s(\operatorname{cls}^{\gamma}(X))</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.12.m11.4d">roman_cls start_POSTSUPERSCRIPT italic_Ξ± end_POSTSUPERSCRIPT ( italic_X ) = italic_s ( roman_cls start_POSTSUPERSCRIPT italic_Ξ³ end_POSTSUPERSCRIPT ( italic_X ) )</annotation></semantics></math> is a subsemigroup of <math alttext="S" class="ltx_Math" display="inline" id="S2.5.p2.13.m12.1"><semantics id="S2.5.p2.13.m12.1a"><mi id="S2.5.p2.13.m12.1.1" xref="S2.5.p2.13.m12.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.13.m12.1b"><ci id="S2.5.p2.13.m12.1.1.cmml" xref="S2.5.p2.13.m12.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.13.m12.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.13.m12.1d">italic_S</annotation></semantics></math> that contains densely the inverse semigroup <math alttext="\operatorname{cls}^{\gamma}(X)" class="ltx_Math" display="inline" id="S2.5.p2.14.m13.2"><semantics id="S2.5.p2.14.m13.2a"><mrow id="S2.5.p2.14.m13.2.2.1" xref="S2.5.p2.14.m13.2.2.2.cmml"><msup id="S2.5.p2.14.m13.2.2.1.1" xref="S2.5.p2.14.m13.2.2.1.1.cmml"><mi id="S2.5.p2.14.m13.2.2.1.1.2" xref="S2.5.p2.14.m13.2.2.1.1.2.cmml">cls</mi><mi id="S2.5.p2.14.m13.2.2.1.1.3" xref="S2.5.p2.14.m13.2.2.1.1.3.cmml">Ξ³</mi></msup><mo id="S2.5.p2.14.m13.2.2.1a" xref="S2.5.p2.14.m13.2.2.2.cmml">β‘</mo><mrow id="S2.5.p2.14.m13.2.2.1.2" xref="S2.5.p2.14.m13.2.2.2.cmml"><mo id="S2.5.p2.14.m13.2.2.1.2.1" stretchy="false" xref="S2.5.p2.14.m13.2.2.2.cmml">(</mo><mi id="S2.5.p2.14.m13.1.1" xref="S2.5.p2.14.m13.1.1.cmml">X</mi><mo id="S2.5.p2.14.m13.2.2.1.2.2" stretchy="false" xref="S2.5.p2.14.m13.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.14.m13.2b"><apply id="S2.5.p2.14.m13.2.2.2.cmml" xref="S2.5.p2.14.m13.2.2.1"><apply id="S2.5.p2.14.m13.2.2.1.1.cmml" xref="S2.5.p2.14.m13.2.2.1.1"><csymbol cd="ambiguous" id="S2.5.p2.14.m13.2.2.1.1.1.cmml" xref="S2.5.p2.14.m13.2.2.1.1">superscript</csymbol><ci id="S2.5.p2.14.m13.2.2.1.1.2.cmml" xref="S2.5.p2.14.m13.2.2.1.1.2">cls</ci><ci id="S2.5.p2.14.m13.2.2.1.1.3.cmml" xref="S2.5.p2.14.m13.2.2.1.1.3">πΎ</ci></apply><ci id="S2.5.p2.14.m13.1.1.cmml" xref="S2.5.p2.14.m13.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.14.m13.2c">\operatorname{cls}^{\gamma}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.14.m13.2d">roman_cls start_POSTSUPERSCRIPT italic_Ξ³ end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math>. Fix any <math alttext="x\in\operatorname{cls}^{\alpha}(X)\setminus\operatorname{cls}^{\gamma}(X)" class="ltx_Math" display="inline" id="S2.5.p2.15.m14.4"><semantics id="S2.5.p2.15.m14.4a"><mrow id="S2.5.p2.15.m14.4.4" xref="S2.5.p2.15.m14.4.4.cmml"><mi id="S2.5.p2.15.m14.4.4.4" xref="S2.5.p2.15.m14.4.4.4.cmml">x</mi><mo id="S2.5.p2.15.m14.4.4.3" xref="S2.5.p2.15.m14.4.4.3.cmml">β</mo><mrow id="S2.5.p2.15.m14.4.4.2" xref="S2.5.p2.15.m14.4.4.2.cmml"><mrow id="S2.5.p2.15.m14.3.3.1.1.1" xref="S2.5.p2.15.m14.3.3.1.1.2.cmml"><msup id="S2.5.p2.15.m14.3.3.1.1.1.1" xref="S2.5.p2.15.m14.3.3.1.1.1.1.cmml"><mi id="S2.5.p2.15.m14.3.3.1.1.1.1.2" xref="S2.5.p2.15.m14.3.3.1.1.1.1.2.cmml">cls</mi><mi id="S2.5.p2.15.m14.3.3.1.1.1.1.3" xref="S2.5.p2.15.m14.3.3.1.1.1.1.3.cmml">Ξ±</mi></msup><mo id="S2.5.p2.15.m14.3.3.1.1.1a" xref="S2.5.p2.15.m14.3.3.1.1.2.cmml">β‘</mo><mrow id="S2.5.p2.15.m14.3.3.1.1.1.2" xref="S2.5.p2.15.m14.3.3.1.1.2.cmml"><mo id="S2.5.p2.15.m14.3.3.1.1.1.2.1" stretchy="false" xref="S2.5.p2.15.m14.3.3.1.1.2.cmml">(</mo><mi id="S2.5.p2.15.m14.1.1" xref="S2.5.p2.15.m14.1.1.cmml">X</mi><mo id="S2.5.p2.15.m14.3.3.1.1.1.2.2" stretchy="false" xref="S2.5.p2.15.m14.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.5.p2.15.m14.4.4.2.3" xref="S2.5.p2.15.m14.4.4.2.3.cmml">β</mo><mrow id="S2.5.p2.15.m14.4.4.2.2.1" xref="S2.5.p2.15.m14.4.4.2.2.2.cmml"><msup id="S2.5.p2.15.m14.4.4.2.2.1.1" xref="S2.5.p2.15.m14.4.4.2.2.1.1.cmml"><mi id="S2.5.p2.15.m14.4.4.2.2.1.1.2" xref="S2.5.p2.15.m14.4.4.2.2.1.1.2.cmml">cls</mi><mi id="S2.5.p2.15.m14.4.4.2.2.1.1.3" xref="S2.5.p2.15.m14.4.4.2.2.1.1.3.cmml">Ξ³</mi></msup><mo id="S2.5.p2.15.m14.4.4.2.2.1a" xref="S2.5.p2.15.m14.4.4.2.2.2.cmml">β‘</mo><mrow id="S2.5.p2.15.m14.4.4.2.2.1.2" xref="S2.5.p2.15.m14.4.4.2.2.2.cmml"><mo id="S2.5.p2.15.m14.4.4.2.2.1.2.1" stretchy="false" xref="S2.5.p2.15.m14.4.4.2.2.2.cmml">(</mo><mi id="S2.5.p2.15.m14.2.2" xref="S2.5.p2.15.m14.2.2.cmml">X</mi><mo id="S2.5.p2.15.m14.4.4.2.2.1.2.2" stretchy="false" xref="S2.5.p2.15.m14.4.4.2.2.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.15.m14.4b"><apply id="S2.5.p2.15.m14.4.4.cmml" xref="S2.5.p2.15.m14.4.4"><in id="S2.5.p2.15.m14.4.4.3.cmml" xref="S2.5.p2.15.m14.4.4.3"></in><ci id="S2.5.p2.15.m14.4.4.4.cmml" xref="S2.5.p2.15.m14.4.4.4">π₯</ci><apply id="S2.5.p2.15.m14.4.4.2.cmml" xref="S2.5.p2.15.m14.4.4.2"><setdiff id="S2.5.p2.15.m14.4.4.2.3.cmml" xref="S2.5.p2.15.m14.4.4.2.3"></setdiff><apply id="S2.5.p2.15.m14.3.3.1.1.2.cmml" xref="S2.5.p2.15.m14.3.3.1.1.1"><apply id="S2.5.p2.15.m14.3.3.1.1.1.1.cmml" xref="S2.5.p2.15.m14.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.15.m14.3.3.1.1.1.1.1.cmml" xref="S2.5.p2.15.m14.3.3.1.1.1.1">superscript</csymbol><ci id="S2.5.p2.15.m14.3.3.1.1.1.1.2.cmml" xref="S2.5.p2.15.m14.3.3.1.1.1.1.2">cls</ci><ci id="S2.5.p2.15.m14.3.3.1.1.1.1.3.cmml" xref="S2.5.p2.15.m14.3.3.1.1.1.1.3">πΌ</ci></apply><ci id="S2.5.p2.15.m14.1.1.cmml" xref="S2.5.p2.15.m14.1.1">π</ci></apply><apply id="S2.5.p2.15.m14.4.4.2.2.2.cmml" xref="S2.5.p2.15.m14.4.4.2.2.1"><apply id="S2.5.p2.15.m14.4.4.2.2.1.1.cmml" xref="S2.5.p2.15.m14.4.4.2.2.1.1"><csymbol cd="ambiguous" id="S2.5.p2.15.m14.4.4.2.2.1.1.1.cmml" xref="S2.5.p2.15.m14.4.4.2.2.1.1">superscript</csymbol><ci id="S2.5.p2.15.m14.4.4.2.2.1.1.2.cmml" xref="S2.5.p2.15.m14.4.4.2.2.1.1.2">cls</ci><ci id="S2.5.p2.15.m14.4.4.2.2.1.1.3.cmml" xref="S2.5.p2.15.m14.4.4.2.2.1.1.3">πΎ</ci></apply><ci id="S2.5.p2.15.m14.2.2.cmml" xref="S2.5.p2.15.m14.2.2">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.15.m14.4c">x\in\operatorname{cls}^{\alpha}(X)\setminus\operatorname{cls}^{\gamma}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.15.m14.4d">italic_x β roman_cls start_POSTSUPERSCRIPT italic_Ξ± end_POSTSUPERSCRIPT ( italic_X ) β roman_cls start_POSTSUPERSCRIPT italic_Ξ³ end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math> and a sequence <math alttext="\{x_{n}:n\in\omega\}\subset\operatorname{cls}^{\gamma}(X)" class="ltx_Math" display="inline" id="S2.5.p2.16.m15.4"><semantics id="S2.5.p2.16.m15.4a"><mrow id="S2.5.p2.16.m15.4.4" xref="S2.5.p2.16.m15.4.4.cmml"><mrow id="S2.5.p2.16.m15.3.3.2.2" xref="S2.5.p2.16.m15.3.3.2.3.cmml"><mo id="S2.5.p2.16.m15.3.3.2.2.3" stretchy="false" xref="S2.5.p2.16.m15.3.3.2.3.1.cmml">{</mo><msub id="S2.5.p2.16.m15.2.2.1.1.1" xref="S2.5.p2.16.m15.2.2.1.1.1.cmml"><mi id="S2.5.p2.16.m15.2.2.1.1.1.2" xref="S2.5.p2.16.m15.2.2.1.1.1.2.cmml">x</mi><mi id="S2.5.p2.16.m15.2.2.1.1.1.3" xref="S2.5.p2.16.m15.2.2.1.1.1.3.cmml">n</mi></msub><mo id="S2.5.p2.16.m15.3.3.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.5.p2.16.m15.3.3.2.3.1.cmml">:</mo><mrow id="S2.5.p2.16.m15.3.3.2.2.2" xref="S2.5.p2.16.m15.3.3.2.2.2.cmml"><mi id="S2.5.p2.16.m15.3.3.2.2.2.2" xref="S2.5.p2.16.m15.3.3.2.2.2.2.cmml">n</mi><mo id="S2.5.p2.16.m15.3.3.2.2.2.1" xref="S2.5.p2.16.m15.3.3.2.2.2.1.cmml">β</mo><mi id="S2.5.p2.16.m15.3.3.2.2.2.3" xref="S2.5.p2.16.m15.3.3.2.2.2.3.cmml">Ο</mi></mrow><mo id="S2.5.p2.16.m15.3.3.2.2.5" stretchy="false" xref="S2.5.p2.16.m15.3.3.2.3.1.cmml">}</mo></mrow><mo id="S2.5.p2.16.m15.4.4.4" xref="S2.5.p2.16.m15.4.4.4.cmml">β</mo><mrow id="S2.5.p2.16.m15.4.4.3.1" xref="S2.5.p2.16.m15.4.4.3.2.cmml"><msup id="S2.5.p2.16.m15.4.4.3.1.1" xref="S2.5.p2.16.m15.4.4.3.1.1.cmml"><mi id="S2.5.p2.16.m15.4.4.3.1.1.2" xref="S2.5.p2.16.m15.4.4.3.1.1.2.cmml">cls</mi><mi id="S2.5.p2.16.m15.4.4.3.1.1.3" xref="S2.5.p2.16.m15.4.4.3.1.1.3.cmml">Ξ³</mi></msup><mo id="S2.5.p2.16.m15.4.4.3.1a" xref="S2.5.p2.16.m15.4.4.3.2.cmml">β‘</mo><mrow id="S2.5.p2.16.m15.4.4.3.1.2" xref="S2.5.p2.16.m15.4.4.3.2.cmml"><mo id="S2.5.p2.16.m15.4.4.3.1.2.1" stretchy="false" xref="S2.5.p2.16.m15.4.4.3.2.cmml">(</mo><mi id="S2.5.p2.16.m15.1.1" xref="S2.5.p2.16.m15.1.1.cmml">X</mi><mo id="S2.5.p2.16.m15.4.4.3.1.2.2" stretchy="false" xref="S2.5.p2.16.m15.4.4.3.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.16.m15.4b"><apply id="S2.5.p2.16.m15.4.4.cmml" xref="S2.5.p2.16.m15.4.4"><subset id="S2.5.p2.16.m15.4.4.4.cmml" xref="S2.5.p2.16.m15.4.4.4"></subset><apply id="S2.5.p2.16.m15.3.3.2.3.cmml" xref="S2.5.p2.16.m15.3.3.2.2"><csymbol cd="latexml" id="S2.5.p2.16.m15.3.3.2.3.1.cmml" xref="S2.5.p2.16.m15.3.3.2.2.3">conditional-set</csymbol><apply id="S2.5.p2.16.m15.2.2.1.1.1.cmml" xref="S2.5.p2.16.m15.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.16.m15.2.2.1.1.1.1.cmml" xref="S2.5.p2.16.m15.2.2.1.1.1">subscript</csymbol><ci id="S2.5.p2.16.m15.2.2.1.1.1.2.cmml" xref="S2.5.p2.16.m15.2.2.1.1.1.2">π₯</ci><ci id="S2.5.p2.16.m15.2.2.1.1.1.3.cmml" xref="S2.5.p2.16.m15.2.2.1.1.1.3">π</ci></apply><apply id="S2.5.p2.16.m15.3.3.2.2.2.cmml" xref="S2.5.p2.16.m15.3.3.2.2.2"><in id="S2.5.p2.16.m15.3.3.2.2.2.1.cmml" xref="S2.5.p2.16.m15.3.3.2.2.2.1"></in><ci id="S2.5.p2.16.m15.3.3.2.2.2.2.cmml" xref="S2.5.p2.16.m15.3.3.2.2.2.2">π</ci><ci id="S2.5.p2.16.m15.3.3.2.2.2.3.cmml" xref="S2.5.p2.16.m15.3.3.2.2.2.3">π</ci></apply></apply><apply id="S2.5.p2.16.m15.4.4.3.2.cmml" xref="S2.5.p2.16.m15.4.4.3.1"><apply id="S2.5.p2.16.m15.4.4.3.1.1.cmml" xref="S2.5.p2.16.m15.4.4.3.1.1"><csymbol cd="ambiguous" id="S2.5.p2.16.m15.4.4.3.1.1.1.cmml" xref="S2.5.p2.16.m15.4.4.3.1.1">superscript</csymbol><ci id="S2.5.p2.16.m15.4.4.3.1.1.2.cmml" xref="S2.5.p2.16.m15.4.4.3.1.1.2">cls</ci><ci id="S2.5.p2.16.m15.4.4.3.1.1.3.cmml" xref="S2.5.p2.16.m15.4.4.3.1.1.3">πΎ</ci></apply><ci id="S2.5.p2.16.m15.1.1.cmml" xref="S2.5.p2.16.m15.1.1">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.16.m15.4c">\{x_{n}:n\in\omega\}\subset\operatorname{cls}^{\gamma}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.16.m15.4d">{ italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n β italic_Ο } β roman_cls start_POSTSUPERSCRIPT italic_Ξ³ end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math> convergent to <math alttext="x" class="ltx_Math" display="inline" id="S2.5.p2.17.m16.1"><semantics id="S2.5.p2.17.m16.1a"><mi id="S2.5.p2.17.m16.1.1" xref="S2.5.p2.17.m16.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.17.m16.1b"><ci id="S2.5.p2.17.m16.1.1.cmml" xref="S2.5.p2.17.m16.1.1">π₯</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.17.m16.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.17.m16.1d">italic_x</annotation></semantics></math>. By <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib22" title="">22</a>, Theorem 3.10.31]</cite>, each Hausdorff countably compact sequential space is sequentially compact. Thus the sequence <math alttext="\{x_{n}^{-1}:n\in\omega\}\subseteq\operatorname{cls}^{\gamma}(X)" class="ltx_Math" display="inline" id="S2.5.p2.18.m17.4"><semantics id="S2.5.p2.18.m17.4a"><mrow id="S2.5.p2.18.m17.4.4" xref="S2.5.p2.18.m17.4.4.cmml"><mrow id="S2.5.p2.18.m17.3.3.2.2" xref="S2.5.p2.18.m17.3.3.2.3.cmml"><mo id="S2.5.p2.18.m17.3.3.2.2.3" stretchy="false" xref="S2.5.p2.18.m17.3.3.2.3.1.cmml">{</mo><msubsup id="S2.5.p2.18.m17.2.2.1.1.1" xref="S2.5.p2.18.m17.2.2.1.1.1.cmml"><mi id="S2.5.p2.18.m17.2.2.1.1.1.2.2" xref="S2.5.p2.18.m17.2.2.1.1.1.2.2.cmml">x</mi><mi id="S2.5.p2.18.m17.2.2.1.1.1.2.3" xref="S2.5.p2.18.m17.2.2.1.1.1.2.3.cmml">n</mi><mrow id="S2.5.p2.18.m17.2.2.1.1.1.3" xref="S2.5.p2.18.m17.2.2.1.1.1.3.cmml"><mo id="S2.5.p2.18.m17.2.2.1.1.1.3a" xref="S2.5.p2.18.m17.2.2.1.1.1.3.cmml">β</mo><mn id="S2.5.p2.18.m17.2.2.1.1.1.3.2" xref="S2.5.p2.18.m17.2.2.1.1.1.3.2.cmml">1</mn></mrow></msubsup><mo id="S2.5.p2.18.m17.3.3.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.5.p2.18.m17.3.3.2.3.1.cmml">:</mo><mrow id="S2.5.p2.18.m17.3.3.2.2.2" xref="S2.5.p2.18.m17.3.3.2.2.2.cmml"><mi id="S2.5.p2.18.m17.3.3.2.2.2.2" xref="S2.5.p2.18.m17.3.3.2.2.2.2.cmml">n</mi><mo id="S2.5.p2.18.m17.3.3.2.2.2.1" xref="S2.5.p2.18.m17.3.3.2.2.2.1.cmml">β</mo><mi id="S2.5.p2.18.m17.3.3.2.2.2.3" xref="S2.5.p2.18.m17.3.3.2.2.2.3.cmml">Ο</mi></mrow><mo id="S2.5.p2.18.m17.3.3.2.2.5" stretchy="false" xref="S2.5.p2.18.m17.3.3.2.3.1.cmml">}</mo></mrow><mo id="S2.5.p2.18.m17.4.4.4" xref="S2.5.p2.18.m17.4.4.4.cmml">β</mo><mrow id="S2.5.p2.18.m17.4.4.3.1" xref="S2.5.p2.18.m17.4.4.3.2.cmml"><msup id="S2.5.p2.18.m17.4.4.3.1.1" xref="S2.5.p2.18.m17.4.4.3.1.1.cmml"><mi id="S2.5.p2.18.m17.4.4.3.1.1.2" xref="S2.5.p2.18.m17.4.4.3.1.1.2.cmml">cls</mi><mi id="S2.5.p2.18.m17.4.4.3.1.1.3" xref="S2.5.p2.18.m17.4.4.3.1.1.3.cmml">Ξ³</mi></msup><mo id="S2.5.p2.18.m17.4.4.3.1a" xref="S2.5.p2.18.m17.4.4.3.2.cmml">β‘</mo><mrow id="S2.5.p2.18.m17.4.4.3.1.2" xref="S2.5.p2.18.m17.4.4.3.2.cmml"><mo id="S2.5.p2.18.m17.4.4.3.1.2.1" stretchy="false" xref="S2.5.p2.18.m17.4.4.3.2.cmml">(</mo><mi id="S2.5.p2.18.m17.1.1" xref="S2.5.p2.18.m17.1.1.cmml">X</mi><mo id="S2.5.p2.18.m17.4.4.3.1.2.2" stretchy="false" xref="S2.5.p2.18.m17.4.4.3.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.18.m17.4b"><apply id="S2.5.p2.18.m17.4.4.cmml" xref="S2.5.p2.18.m17.4.4"><subset id="S2.5.p2.18.m17.4.4.4.cmml" xref="S2.5.p2.18.m17.4.4.4"></subset><apply id="S2.5.p2.18.m17.3.3.2.3.cmml" xref="S2.5.p2.18.m17.3.3.2.2"><csymbol cd="latexml" id="S2.5.p2.18.m17.3.3.2.3.1.cmml" xref="S2.5.p2.18.m17.3.3.2.2.3">conditional-set</csymbol><apply id="S2.5.p2.18.m17.2.2.1.1.1.cmml" xref="S2.5.p2.18.m17.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.18.m17.2.2.1.1.1.1.cmml" xref="S2.5.p2.18.m17.2.2.1.1.1">superscript</csymbol><apply id="S2.5.p2.18.m17.2.2.1.1.1.2.cmml" xref="S2.5.p2.18.m17.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.18.m17.2.2.1.1.1.2.1.cmml" xref="S2.5.p2.18.m17.2.2.1.1.1">subscript</csymbol><ci id="S2.5.p2.18.m17.2.2.1.1.1.2.2.cmml" xref="S2.5.p2.18.m17.2.2.1.1.1.2.2">π₯</ci><ci id="S2.5.p2.18.m17.2.2.1.1.1.2.3.cmml" xref="S2.5.p2.18.m17.2.2.1.1.1.2.3">π</ci></apply><apply id="S2.5.p2.18.m17.2.2.1.1.1.3.cmml" xref="S2.5.p2.18.m17.2.2.1.1.1.3"><minus id="S2.5.p2.18.m17.2.2.1.1.1.3.1.cmml" xref="S2.5.p2.18.m17.2.2.1.1.1.3"></minus><cn id="S2.5.p2.18.m17.2.2.1.1.1.3.2.cmml" type="integer" xref="S2.5.p2.18.m17.2.2.1.1.1.3.2">1</cn></apply></apply><apply id="S2.5.p2.18.m17.3.3.2.2.2.cmml" xref="S2.5.p2.18.m17.3.3.2.2.2"><in id="S2.5.p2.18.m17.3.3.2.2.2.1.cmml" xref="S2.5.p2.18.m17.3.3.2.2.2.1"></in><ci id="S2.5.p2.18.m17.3.3.2.2.2.2.cmml" xref="S2.5.p2.18.m17.3.3.2.2.2.2">π</ci><ci id="S2.5.p2.18.m17.3.3.2.2.2.3.cmml" xref="S2.5.p2.18.m17.3.3.2.2.2.3">π</ci></apply></apply><apply id="S2.5.p2.18.m17.4.4.3.2.cmml" xref="S2.5.p2.18.m17.4.4.3.1"><apply id="S2.5.p2.18.m17.4.4.3.1.1.cmml" xref="S2.5.p2.18.m17.4.4.3.1.1"><csymbol cd="ambiguous" id="S2.5.p2.18.m17.4.4.3.1.1.1.cmml" xref="S2.5.p2.18.m17.4.4.3.1.1">superscript</csymbol><ci id="S2.5.p2.18.m17.4.4.3.1.1.2.cmml" xref="S2.5.p2.18.m17.4.4.3.1.1.2">cls</ci><ci id="S2.5.p2.18.m17.4.4.3.1.1.3.cmml" xref="S2.5.p2.18.m17.4.4.3.1.1.3">πΎ</ci></apply><ci id="S2.5.p2.18.m17.1.1.cmml" xref="S2.5.p2.18.m17.1.1">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.18.m17.4c">\{x_{n}^{-1}:n\in\omega\}\subseteq\operatorname{cls}^{\gamma}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.18.m17.4d">{ italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT : italic_n β italic_Ο } β roman_cls start_POSTSUPERSCRIPT italic_Ξ³ end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math> contains a convergent in <math alttext="S" class="ltx_Math" display="inline" id="S2.5.p2.19.m18.1"><semantics id="S2.5.p2.19.m18.1a"><mi id="S2.5.p2.19.m18.1.1" xref="S2.5.p2.19.m18.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.19.m18.1b"><ci id="S2.5.p2.19.m18.1.1.cmml" xref="S2.5.p2.19.m18.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.19.m18.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.19.m18.1d">italic_S</annotation></semantics></math> subsequence <math alttext="\{x^{-1}_{n_{k}}:k\in\omega\}" class="ltx_Math" display="inline" id="S2.5.p2.20.m19.2"><semantics id="S2.5.p2.20.m19.2a"><mrow id="S2.5.p2.20.m19.2.2.2" xref="S2.5.p2.20.m19.2.2.3.cmml"><mo id="S2.5.p2.20.m19.2.2.2.3" stretchy="false" xref="S2.5.p2.20.m19.2.2.3.1.cmml">{</mo><msubsup id="S2.5.p2.20.m19.1.1.1.1" xref="S2.5.p2.20.m19.1.1.1.1.cmml"><mi id="S2.5.p2.20.m19.1.1.1.1.2.2" xref="S2.5.p2.20.m19.1.1.1.1.2.2.cmml">x</mi><msub id="S2.5.p2.20.m19.1.1.1.1.3" xref="S2.5.p2.20.m19.1.1.1.1.3.cmml"><mi id="S2.5.p2.20.m19.1.1.1.1.3.2" xref="S2.5.p2.20.m19.1.1.1.1.3.2.cmml">n</mi><mi id="S2.5.p2.20.m19.1.1.1.1.3.3" xref="S2.5.p2.20.m19.1.1.1.1.3.3.cmml">k</mi></msub><mrow id="S2.5.p2.20.m19.1.1.1.1.2.3" xref="S2.5.p2.20.m19.1.1.1.1.2.3.cmml"><mo id="S2.5.p2.20.m19.1.1.1.1.2.3a" xref="S2.5.p2.20.m19.1.1.1.1.2.3.cmml">β</mo><mn id="S2.5.p2.20.m19.1.1.1.1.2.3.2" xref="S2.5.p2.20.m19.1.1.1.1.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S2.5.p2.20.m19.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.5.p2.20.m19.2.2.3.1.cmml">:</mo><mrow id="S2.5.p2.20.m19.2.2.2.2" xref="S2.5.p2.20.m19.2.2.2.2.cmml"><mi id="S2.5.p2.20.m19.2.2.2.2.2" xref="S2.5.p2.20.m19.2.2.2.2.2.cmml">k</mi><mo id="S2.5.p2.20.m19.2.2.2.2.1" xref="S2.5.p2.20.m19.2.2.2.2.1.cmml">β</mo><mi id="S2.5.p2.20.m19.2.2.2.2.3" xref="S2.5.p2.20.m19.2.2.2.2.3.cmml">Ο</mi></mrow><mo id="S2.5.p2.20.m19.2.2.2.5" stretchy="false" xref="S2.5.p2.20.m19.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.20.m19.2b"><apply id="S2.5.p2.20.m19.2.2.3.cmml" xref="S2.5.p2.20.m19.2.2.2"><csymbol cd="latexml" id="S2.5.p2.20.m19.2.2.3.1.cmml" xref="S2.5.p2.20.m19.2.2.2.3">conditional-set</csymbol><apply id="S2.5.p2.20.m19.1.1.1.1.cmml" xref="S2.5.p2.20.m19.1.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.20.m19.1.1.1.1.1.cmml" xref="S2.5.p2.20.m19.1.1.1.1">subscript</csymbol><apply id="S2.5.p2.20.m19.1.1.1.1.2.cmml" xref="S2.5.p2.20.m19.1.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.20.m19.1.1.1.1.2.1.cmml" xref="S2.5.p2.20.m19.1.1.1.1">superscript</csymbol><ci id="S2.5.p2.20.m19.1.1.1.1.2.2.cmml" xref="S2.5.p2.20.m19.1.1.1.1.2.2">π₯</ci><apply id="S2.5.p2.20.m19.1.1.1.1.2.3.cmml" xref="S2.5.p2.20.m19.1.1.1.1.2.3"><minus id="S2.5.p2.20.m19.1.1.1.1.2.3.1.cmml" xref="S2.5.p2.20.m19.1.1.1.1.2.3"></minus><cn id="S2.5.p2.20.m19.1.1.1.1.2.3.2.cmml" type="integer" xref="S2.5.p2.20.m19.1.1.1.1.2.3.2">1</cn></apply></apply><apply id="S2.5.p2.20.m19.1.1.1.1.3.cmml" xref="S2.5.p2.20.m19.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.5.p2.20.m19.1.1.1.1.3.1.cmml" xref="S2.5.p2.20.m19.1.1.1.1.3">subscript</csymbol><ci id="S2.5.p2.20.m19.1.1.1.1.3.2.cmml" xref="S2.5.p2.20.m19.1.1.1.1.3.2">π</ci><ci id="S2.5.p2.20.m19.1.1.1.1.3.3.cmml" xref="S2.5.p2.20.m19.1.1.1.1.3.3">π</ci></apply></apply><apply id="S2.5.p2.20.m19.2.2.2.2.cmml" xref="S2.5.p2.20.m19.2.2.2.2"><in id="S2.5.p2.20.m19.2.2.2.2.1.cmml" xref="S2.5.p2.20.m19.2.2.2.2.1"></in><ci id="S2.5.p2.20.m19.2.2.2.2.2.cmml" xref="S2.5.p2.20.m19.2.2.2.2.2">π</ci><ci id="S2.5.p2.20.m19.2.2.2.2.3.cmml" xref="S2.5.p2.20.m19.2.2.2.2.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.20.m19.2c">\{x^{-1}_{n_{k}}:k\in\omega\}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.20.m19.2d">{ italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT : italic_k β italic_Ο }</annotation></semantics></math>. Set <math alttext="y=\lim_{k\in\omega}x^{-1}_{n_{k}}" class="ltx_Math" display="inline" id="S2.5.p2.21.m20.1"><semantics id="S2.5.p2.21.m20.1a"><mrow id="S2.5.p2.21.m20.1.1" xref="S2.5.p2.21.m20.1.1.cmml"><mi id="S2.5.p2.21.m20.1.1.2" xref="S2.5.p2.21.m20.1.1.2.cmml">y</mi><mo id="S2.5.p2.21.m20.1.1.1" rspace="0.1389em" xref="S2.5.p2.21.m20.1.1.1.cmml">=</mo><mrow id="S2.5.p2.21.m20.1.1.3" xref="S2.5.p2.21.m20.1.1.3.cmml"><msub id="S2.5.p2.21.m20.1.1.3.1" xref="S2.5.p2.21.m20.1.1.3.1.cmml"><mo id="S2.5.p2.21.m20.1.1.3.1.2" lspace="0.1389em" rspace="0.167em" xref="S2.5.p2.21.m20.1.1.3.1.2.cmml">lim</mo><mrow id="S2.5.p2.21.m20.1.1.3.1.3" xref="S2.5.p2.21.m20.1.1.3.1.3.cmml"><mi id="S2.5.p2.21.m20.1.1.3.1.3.2" xref="S2.5.p2.21.m20.1.1.3.1.3.2.cmml">k</mi><mo id="S2.5.p2.21.m20.1.1.3.1.3.1" xref="S2.5.p2.21.m20.1.1.3.1.3.1.cmml">β</mo><mi id="S2.5.p2.21.m20.1.1.3.1.3.3" xref="S2.5.p2.21.m20.1.1.3.1.3.3.cmml">Ο</mi></mrow></msub><msubsup id="S2.5.p2.21.m20.1.1.3.2" xref="S2.5.p2.21.m20.1.1.3.2.cmml"><mi id="S2.5.p2.21.m20.1.1.3.2.2.2" xref="S2.5.p2.21.m20.1.1.3.2.2.2.cmml">x</mi><msub id="S2.5.p2.21.m20.1.1.3.2.3" xref="S2.5.p2.21.m20.1.1.3.2.3.cmml"><mi id="S2.5.p2.21.m20.1.1.3.2.3.2" xref="S2.5.p2.21.m20.1.1.3.2.3.2.cmml">n</mi><mi id="S2.5.p2.21.m20.1.1.3.2.3.3" xref="S2.5.p2.21.m20.1.1.3.2.3.3.cmml">k</mi></msub><mrow id="S2.5.p2.21.m20.1.1.3.2.2.3" xref="S2.5.p2.21.m20.1.1.3.2.2.3.cmml"><mo id="S2.5.p2.21.m20.1.1.3.2.2.3a" xref="S2.5.p2.21.m20.1.1.3.2.2.3.cmml">β</mo><mn id="S2.5.p2.21.m20.1.1.3.2.2.3.2" xref="S2.5.p2.21.m20.1.1.3.2.2.3.2.cmml">1</mn></mrow></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.21.m20.1b"><apply id="S2.5.p2.21.m20.1.1.cmml" xref="S2.5.p2.21.m20.1.1"><eq id="S2.5.p2.21.m20.1.1.1.cmml" xref="S2.5.p2.21.m20.1.1.1"></eq><ci id="S2.5.p2.21.m20.1.1.2.cmml" xref="S2.5.p2.21.m20.1.1.2">π¦</ci><apply id="S2.5.p2.21.m20.1.1.3.cmml" xref="S2.5.p2.21.m20.1.1.3"><apply id="S2.5.p2.21.m20.1.1.3.1.cmml" xref="S2.5.p2.21.m20.1.1.3.1"><csymbol cd="ambiguous" id="S2.5.p2.21.m20.1.1.3.1.1.cmml" xref="S2.5.p2.21.m20.1.1.3.1">subscript</csymbol><limit id="S2.5.p2.21.m20.1.1.3.1.2.cmml" xref="S2.5.p2.21.m20.1.1.3.1.2"></limit><apply id="S2.5.p2.21.m20.1.1.3.1.3.cmml" xref="S2.5.p2.21.m20.1.1.3.1.3"><in id="S2.5.p2.21.m20.1.1.3.1.3.1.cmml" xref="S2.5.p2.21.m20.1.1.3.1.3.1"></in><ci id="S2.5.p2.21.m20.1.1.3.1.3.2.cmml" xref="S2.5.p2.21.m20.1.1.3.1.3.2">π</ci><ci id="S2.5.p2.21.m20.1.1.3.1.3.3.cmml" xref="S2.5.p2.21.m20.1.1.3.1.3.3">π</ci></apply></apply><apply id="S2.5.p2.21.m20.1.1.3.2.cmml" xref="S2.5.p2.21.m20.1.1.3.2"><csymbol cd="ambiguous" id="S2.5.p2.21.m20.1.1.3.2.1.cmml" xref="S2.5.p2.21.m20.1.1.3.2">subscript</csymbol><apply id="S2.5.p2.21.m20.1.1.3.2.2.cmml" xref="S2.5.p2.21.m20.1.1.3.2"><csymbol cd="ambiguous" id="S2.5.p2.21.m20.1.1.3.2.2.1.cmml" xref="S2.5.p2.21.m20.1.1.3.2">superscript</csymbol><ci id="S2.5.p2.21.m20.1.1.3.2.2.2.cmml" xref="S2.5.p2.21.m20.1.1.3.2.2.2">π₯</ci><apply id="S2.5.p2.21.m20.1.1.3.2.2.3.cmml" xref="S2.5.p2.21.m20.1.1.3.2.2.3"><minus id="S2.5.p2.21.m20.1.1.3.2.2.3.1.cmml" xref="S2.5.p2.21.m20.1.1.3.2.2.3"></minus><cn id="S2.5.p2.21.m20.1.1.3.2.2.3.2.cmml" type="integer" xref="S2.5.p2.21.m20.1.1.3.2.2.3.2">1</cn></apply></apply><apply id="S2.5.p2.21.m20.1.1.3.2.3.cmml" xref="S2.5.p2.21.m20.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.5.p2.21.m20.1.1.3.2.3.1.cmml" xref="S2.5.p2.21.m20.1.1.3.2.3">subscript</csymbol><ci id="S2.5.p2.21.m20.1.1.3.2.3.2.cmml" xref="S2.5.p2.21.m20.1.1.3.2.3.2">π</ci><ci id="S2.5.p2.21.m20.1.1.3.2.3.3.cmml" xref="S2.5.p2.21.m20.1.1.3.2.3.3">π</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.21.m20.1c">y=\lim_{k\in\omega}x^{-1}_{n_{k}}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.21.m20.1d">italic_y = roman_lim start_POSTSUBSCRIPT italic_k β italic_Ο end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and note that <math alttext="y\in s(\operatorname{cls}^{\gamma}(X))=\operatorname{cls}^{\alpha}(X)" class="ltx_Math" display="inline" id="S2.5.p2.22.m21.4"><semantics id="S2.5.p2.22.m21.4a"><mrow id="S2.5.p2.22.m21.4.4" xref="S2.5.p2.22.m21.4.4.cmml"><mi id="S2.5.p2.22.m21.4.4.4" xref="S2.5.p2.22.m21.4.4.4.cmml">y</mi><mo id="S2.5.p2.22.m21.4.4.5" xref="S2.5.p2.22.m21.4.4.5.cmml">β</mo><mrow id="S2.5.p2.22.m21.3.3.1" xref="S2.5.p2.22.m21.3.3.1.cmml"><mi id="S2.5.p2.22.m21.3.3.1.3" xref="S2.5.p2.22.m21.3.3.1.3.cmml">s</mi><mo id="S2.5.p2.22.m21.3.3.1.2" xref="S2.5.p2.22.m21.3.3.1.2.cmml">β’</mo><mrow id="S2.5.p2.22.m21.3.3.1.1.1" xref="S2.5.p2.22.m21.3.3.1.cmml"><mo id="S2.5.p2.22.m21.3.3.1.1.1.2" stretchy="false" xref="S2.5.p2.22.m21.3.3.1.cmml">(</mo><mrow id="S2.5.p2.22.m21.3.3.1.1.1.1.1" xref="S2.5.p2.22.m21.3.3.1.1.1.1.2.cmml"><msup id="S2.5.p2.22.m21.3.3.1.1.1.1.1.1" xref="S2.5.p2.22.m21.3.3.1.1.1.1.1.1.cmml"><mi id="S2.5.p2.22.m21.3.3.1.1.1.1.1.1.2" xref="S2.5.p2.22.m21.3.3.1.1.1.1.1.1.2.cmml">cls</mi><mi id="S2.5.p2.22.m21.3.3.1.1.1.1.1.1.3" xref="S2.5.p2.22.m21.3.3.1.1.1.1.1.1.3.cmml">Ξ³</mi></msup><mo id="S2.5.p2.22.m21.3.3.1.1.1.1.1a" xref="S2.5.p2.22.m21.3.3.1.1.1.1.2.cmml">β‘</mo><mrow id="S2.5.p2.22.m21.3.3.1.1.1.1.1.2" xref="S2.5.p2.22.m21.3.3.1.1.1.1.2.cmml"><mo id="S2.5.p2.22.m21.3.3.1.1.1.1.1.2.1" stretchy="false" xref="S2.5.p2.22.m21.3.3.1.1.1.1.2.cmml">(</mo><mi id="S2.5.p2.22.m21.1.1" xref="S2.5.p2.22.m21.1.1.cmml">X</mi><mo id="S2.5.p2.22.m21.3.3.1.1.1.1.1.2.2" stretchy="false" xref="S2.5.p2.22.m21.3.3.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.5.p2.22.m21.3.3.1.1.1.3" stretchy="false" xref="S2.5.p2.22.m21.3.3.1.cmml">)</mo></mrow></mrow><mo id="S2.5.p2.22.m21.4.4.6" xref="S2.5.p2.22.m21.4.4.6.cmml">=</mo><mrow id="S2.5.p2.22.m21.4.4.2.1" xref="S2.5.p2.22.m21.4.4.2.2.cmml"><msup id="S2.5.p2.22.m21.4.4.2.1.1" xref="S2.5.p2.22.m21.4.4.2.1.1.cmml"><mi id="S2.5.p2.22.m21.4.4.2.1.1.2" xref="S2.5.p2.22.m21.4.4.2.1.1.2.cmml">cls</mi><mi id="S2.5.p2.22.m21.4.4.2.1.1.3" xref="S2.5.p2.22.m21.4.4.2.1.1.3.cmml">Ξ±</mi></msup><mo id="S2.5.p2.22.m21.4.4.2.1a" xref="S2.5.p2.22.m21.4.4.2.2.cmml">β‘</mo><mrow id="S2.5.p2.22.m21.4.4.2.1.2" xref="S2.5.p2.22.m21.4.4.2.2.cmml"><mo id="S2.5.p2.22.m21.4.4.2.1.2.1" stretchy="false" xref="S2.5.p2.22.m21.4.4.2.2.cmml">(</mo><mi id="S2.5.p2.22.m21.2.2" xref="S2.5.p2.22.m21.2.2.cmml">X</mi><mo id="S2.5.p2.22.m21.4.4.2.1.2.2" stretchy="false" xref="S2.5.p2.22.m21.4.4.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.22.m21.4b"><apply id="S2.5.p2.22.m21.4.4.cmml" xref="S2.5.p2.22.m21.4.4"><and id="S2.5.p2.22.m21.4.4a.cmml" xref="S2.5.p2.22.m21.4.4"></and><apply id="S2.5.p2.22.m21.4.4b.cmml" xref="S2.5.p2.22.m21.4.4"><in id="S2.5.p2.22.m21.4.4.5.cmml" xref="S2.5.p2.22.m21.4.4.5"></in><ci id="S2.5.p2.22.m21.4.4.4.cmml" xref="S2.5.p2.22.m21.4.4.4">π¦</ci><apply id="S2.5.p2.22.m21.3.3.1.cmml" xref="S2.5.p2.22.m21.3.3.1"><times id="S2.5.p2.22.m21.3.3.1.2.cmml" xref="S2.5.p2.22.m21.3.3.1.2"></times><ci id="S2.5.p2.22.m21.3.3.1.3.cmml" xref="S2.5.p2.22.m21.3.3.1.3">π </ci><apply id="S2.5.p2.22.m21.3.3.1.1.1.1.2.cmml" xref="S2.5.p2.22.m21.3.3.1.1.1.1.1"><apply id="S2.5.p2.22.m21.3.3.1.1.1.1.1.1.cmml" xref="S2.5.p2.22.m21.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.22.m21.3.3.1.1.1.1.1.1.1.cmml" xref="S2.5.p2.22.m21.3.3.1.1.1.1.1.1">superscript</csymbol><ci id="S2.5.p2.22.m21.3.3.1.1.1.1.1.1.2.cmml" xref="S2.5.p2.22.m21.3.3.1.1.1.1.1.1.2">cls</ci><ci id="S2.5.p2.22.m21.3.3.1.1.1.1.1.1.3.cmml" xref="S2.5.p2.22.m21.3.3.1.1.1.1.1.1.3">πΎ</ci></apply><ci id="S2.5.p2.22.m21.1.1.cmml" xref="S2.5.p2.22.m21.1.1">π</ci></apply></apply></apply><apply id="S2.5.p2.22.m21.4.4c.cmml" xref="S2.5.p2.22.m21.4.4"><eq id="S2.5.p2.22.m21.4.4.6.cmml" xref="S2.5.p2.22.m21.4.4.6"></eq><share href="https://arxiv.org/html/2503.13666v1#S2.5.p2.22.m21.3.3.1.cmml" id="S2.5.p2.22.m21.4.4d.cmml" xref="S2.5.p2.22.m21.4.4"></share><apply id="S2.5.p2.22.m21.4.4.2.2.cmml" xref="S2.5.p2.22.m21.4.4.2.1"><apply id="S2.5.p2.22.m21.4.4.2.1.1.cmml" xref="S2.5.p2.22.m21.4.4.2.1.1"><csymbol cd="ambiguous" id="S2.5.p2.22.m21.4.4.2.1.1.1.cmml" xref="S2.5.p2.22.m21.4.4.2.1.1">superscript</csymbol><ci id="S2.5.p2.22.m21.4.4.2.1.1.2.cmml" xref="S2.5.p2.22.m21.4.4.2.1.1.2">cls</ci><ci id="S2.5.p2.22.m21.4.4.2.1.1.3.cmml" xref="S2.5.p2.22.m21.4.4.2.1.1.3">πΌ</ci></apply><ci id="S2.5.p2.22.m21.2.2.cmml" xref="S2.5.p2.22.m21.2.2">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.22.m21.4c">y\in s(\operatorname{cls}^{\gamma}(X))=\operatorname{cls}^{\alpha}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.22.m21.4d">italic_y β italic_s ( roman_cls start_POSTSUPERSCRIPT italic_Ξ³ end_POSTSUPERSCRIPT ( italic_X ) ) = roman_cls start_POSTSUPERSCRIPT italic_Ξ± end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math>. Fix any free ultrafilter <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S2.5.p2.23.m22.1"><semantics id="S2.5.p2.23.m22.1a"><mi class="ltx_font_mathcaligraphic" id="S2.5.p2.23.m22.1.1" xref="S2.5.p2.23.m22.1.1.cmml">β±</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.23.m22.1b"><ci id="S2.5.p2.23.m22.1.1.cmml" xref="S2.5.p2.23.m22.1.1">β±</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.23.m22.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.23.m22.1d">caligraphic_F</annotation></semantics></math> on <math alttext="\operatorname{cls}^{\alpha}(X)" class="ltx_Math" display="inline" id="S2.5.p2.24.m23.2"><semantics id="S2.5.p2.24.m23.2a"><mrow id="S2.5.p2.24.m23.2.2.1" xref="S2.5.p2.24.m23.2.2.2.cmml"><msup id="S2.5.p2.24.m23.2.2.1.1" xref="S2.5.p2.24.m23.2.2.1.1.cmml"><mi id="S2.5.p2.24.m23.2.2.1.1.2" xref="S2.5.p2.24.m23.2.2.1.1.2.cmml">cls</mi><mi id="S2.5.p2.24.m23.2.2.1.1.3" xref="S2.5.p2.24.m23.2.2.1.1.3.cmml">Ξ±</mi></msup><mo id="S2.5.p2.24.m23.2.2.1a" xref="S2.5.p2.24.m23.2.2.2.cmml">β‘</mo><mrow id="S2.5.p2.24.m23.2.2.1.2" xref="S2.5.p2.24.m23.2.2.2.cmml"><mo id="S2.5.p2.24.m23.2.2.1.2.1" stretchy="false" xref="S2.5.p2.24.m23.2.2.2.cmml">(</mo><mi id="S2.5.p2.24.m23.1.1" xref="S2.5.p2.24.m23.1.1.cmml">X</mi><mo id="S2.5.p2.24.m23.2.2.1.2.2" stretchy="false" xref="S2.5.p2.24.m23.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.24.m23.2b"><apply id="S2.5.p2.24.m23.2.2.2.cmml" xref="S2.5.p2.24.m23.2.2.1"><apply id="S2.5.p2.24.m23.2.2.1.1.cmml" xref="S2.5.p2.24.m23.2.2.1.1"><csymbol cd="ambiguous" id="S2.5.p2.24.m23.2.2.1.1.1.cmml" xref="S2.5.p2.24.m23.2.2.1.1">superscript</csymbol><ci id="S2.5.p2.24.m23.2.2.1.1.2.cmml" xref="S2.5.p2.24.m23.2.2.1.1.2">cls</ci><ci id="S2.5.p2.24.m23.2.2.1.1.3.cmml" xref="S2.5.p2.24.m23.2.2.1.1.3">πΌ</ci></apply><ci id="S2.5.p2.24.m23.1.1.cmml" xref="S2.5.p2.24.m23.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.24.m23.2c">\operatorname{cls}^{\alpha}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.24.m23.2d">roman_cls start_POSTSUPERSCRIPT italic_Ξ± end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math> which contains the set <math alttext="\{x_{n_{k}}:k\in\omega\}" class="ltx_Math" display="inline" id="S2.5.p2.25.m24.2"><semantics id="S2.5.p2.25.m24.2a"><mrow id="S2.5.p2.25.m24.2.2.2" xref="S2.5.p2.25.m24.2.2.3.cmml"><mo id="S2.5.p2.25.m24.2.2.2.3" stretchy="false" xref="S2.5.p2.25.m24.2.2.3.1.cmml">{</mo><msub id="S2.5.p2.25.m24.1.1.1.1" xref="S2.5.p2.25.m24.1.1.1.1.cmml"><mi id="S2.5.p2.25.m24.1.1.1.1.2" xref="S2.5.p2.25.m24.1.1.1.1.2.cmml">x</mi><msub id="S2.5.p2.25.m24.1.1.1.1.3" xref="S2.5.p2.25.m24.1.1.1.1.3.cmml"><mi id="S2.5.p2.25.m24.1.1.1.1.3.2" xref="S2.5.p2.25.m24.1.1.1.1.3.2.cmml">n</mi><mi id="S2.5.p2.25.m24.1.1.1.1.3.3" xref="S2.5.p2.25.m24.1.1.1.1.3.3.cmml">k</mi></msub></msub><mo id="S2.5.p2.25.m24.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.5.p2.25.m24.2.2.3.1.cmml">:</mo><mrow id="S2.5.p2.25.m24.2.2.2.2" xref="S2.5.p2.25.m24.2.2.2.2.cmml"><mi id="S2.5.p2.25.m24.2.2.2.2.2" xref="S2.5.p2.25.m24.2.2.2.2.2.cmml">k</mi><mo id="S2.5.p2.25.m24.2.2.2.2.1" xref="S2.5.p2.25.m24.2.2.2.2.1.cmml">β</mo><mi id="S2.5.p2.25.m24.2.2.2.2.3" xref="S2.5.p2.25.m24.2.2.2.2.3.cmml">Ο</mi></mrow><mo id="S2.5.p2.25.m24.2.2.2.5" stretchy="false" xref="S2.5.p2.25.m24.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.25.m24.2b"><apply id="S2.5.p2.25.m24.2.2.3.cmml" xref="S2.5.p2.25.m24.2.2.2"><csymbol cd="latexml" id="S2.5.p2.25.m24.2.2.3.1.cmml" xref="S2.5.p2.25.m24.2.2.2.3">conditional-set</csymbol><apply id="S2.5.p2.25.m24.1.1.1.1.cmml" xref="S2.5.p2.25.m24.1.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.25.m24.1.1.1.1.1.cmml" xref="S2.5.p2.25.m24.1.1.1.1">subscript</csymbol><ci id="S2.5.p2.25.m24.1.1.1.1.2.cmml" xref="S2.5.p2.25.m24.1.1.1.1.2">π₯</ci><apply id="S2.5.p2.25.m24.1.1.1.1.3.cmml" xref="S2.5.p2.25.m24.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.5.p2.25.m24.1.1.1.1.3.1.cmml" xref="S2.5.p2.25.m24.1.1.1.1.3">subscript</csymbol><ci id="S2.5.p2.25.m24.1.1.1.1.3.2.cmml" xref="S2.5.p2.25.m24.1.1.1.1.3.2">π</ci><ci id="S2.5.p2.25.m24.1.1.1.1.3.3.cmml" xref="S2.5.p2.25.m24.1.1.1.1.3.3">π</ci></apply></apply><apply id="S2.5.p2.25.m24.2.2.2.2.cmml" xref="S2.5.p2.25.m24.2.2.2.2"><in id="S2.5.p2.25.m24.2.2.2.2.1.cmml" xref="S2.5.p2.25.m24.2.2.2.2.1"></in><ci id="S2.5.p2.25.m24.2.2.2.2.2.cmml" xref="S2.5.p2.25.m24.2.2.2.2.2">π</ci><ci id="S2.5.p2.25.m24.2.2.2.2.3.cmml" xref="S2.5.p2.25.m24.2.2.2.2.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.25.m24.2c">\{x_{n_{k}}:k\in\omega\}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.25.m24.2d">{ italic_x start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT : italic_k β italic_Ο }</annotation></semantics></math>. It is easy to see that <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S2.5.p2.26.m25.1"><semantics id="S2.5.p2.26.m25.1a"><mi class="ltx_font_mathcaligraphic" id="S2.5.p2.26.m25.1.1" xref="S2.5.p2.26.m25.1.1.cmml">β±</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.26.m25.1b"><ci id="S2.5.p2.26.m25.1.1.cmml" xref="S2.5.p2.26.m25.1.1">β±</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.26.m25.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.26.m25.1d">caligraphic_F</annotation></semantics></math> converges to <math alttext="x" class="ltx_Math" display="inline" id="S2.5.p2.27.m26.1"><semantics id="S2.5.p2.27.m26.1a"><mi id="S2.5.p2.27.m26.1.1" xref="S2.5.p2.27.m26.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.27.m26.1b"><ci id="S2.5.p2.27.m26.1.1.cmml" xref="S2.5.p2.27.m26.1.1">π₯</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.27.m26.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.27.m26.1d">italic_x</annotation></semantics></math>, and the ultrafilter <math alttext="\mathcal{F}^{-1}" class="ltx_Math" display="inline" id="S2.5.p2.28.m27.1"><semantics id="S2.5.p2.28.m27.1a"><msup id="S2.5.p2.28.m27.1.1" xref="S2.5.p2.28.m27.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.5.p2.28.m27.1.1.2" xref="S2.5.p2.28.m27.1.1.2.cmml">β±</mi><mrow id="S2.5.p2.28.m27.1.1.3" xref="S2.5.p2.28.m27.1.1.3.cmml"><mo id="S2.5.p2.28.m27.1.1.3a" xref="S2.5.p2.28.m27.1.1.3.cmml">β</mo><mn id="S2.5.p2.28.m27.1.1.3.2" xref="S2.5.p2.28.m27.1.1.3.2.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.5.p2.28.m27.1b"><apply id="S2.5.p2.28.m27.1.1.cmml" xref="S2.5.p2.28.m27.1.1"><csymbol cd="ambiguous" id="S2.5.p2.28.m27.1.1.1.cmml" xref="S2.5.p2.28.m27.1.1">superscript</csymbol><ci id="S2.5.p2.28.m27.1.1.2.cmml" xref="S2.5.p2.28.m27.1.1.2">β±</ci><apply id="S2.5.p2.28.m27.1.1.3.cmml" xref="S2.5.p2.28.m27.1.1.3"><minus id="S2.5.p2.28.m27.1.1.3.1.cmml" xref="S2.5.p2.28.m27.1.1.3"></minus><cn id="S2.5.p2.28.m27.1.1.3.2.cmml" type="integer" xref="S2.5.p2.28.m27.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.28.m27.1c">\mathcal{F}^{-1}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.28.m27.1d">caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> generated by the family <math alttext="\{F^{-1}\colon\operatorname{cls}^{\gamma}(X)\supseteq F\in\mathcal{F}\}" class="ltx_Math" display="inline" id="S2.5.p2.29.m28.3"><semantics id="S2.5.p2.29.m28.3a"><mrow id="S2.5.p2.29.m28.3.3.2" xref="S2.5.p2.29.m28.3.3.3.cmml"><mo id="S2.5.p2.29.m28.3.3.2.3" stretchy="false" xref="S2.5.p2.29.m28.3.3.3.1.cmml">{</mo><msup id="S2.5.p2.29.m28.2.2.1.1" xref="S2.5.p2.29.m28.2.2.1.1.cmml"><mi id="S2.5.p2.29.m28.2.2.1.1.2" xref="S2.5.p2.29.m28.2.2.1.1.2.cmml">F</mi><mrow id="S2.5.p2.29.m28.2.2.1.1.3" xref="S2.5.p2.29.m28.2.2.1.1.3.cmml"><mo id="S2.5.p2.29.m28.2.2.1.1.3a" xref="S2.5.p2.29.m28.2.2.1.1.3.cmml">β</mo><mn id="S2.5.p2.29.m28.2.2.1.1.3.2" xref="S2.5.p2.29.m28.2.2.1.1.3.2.cmml">1</mn></mrow></msup><mo id="S2.5.p2.29.m28.3.3.2.4" lspace="0.278em" rspace="0.278em" xref="S2.5.p2.29.m28.3.3.3.1.cmml">:</mo><mrow id="S2.5.p2.29.m28.3.3.2.2" xref="S2.5.p2.29.m28.3.3.2.2.cmml"><mrow id="S2.5.p2.29.m28.3.3.2.2.1.1" xref="S2.5.p2.29.m28.3.3.2.2.1.2.cmml"><msup id="S2.5.p2.29.m28.3.3.2.2.1.1.1" xref="S2.5.p2.29.m28.3.3.2.2.1.1.1.cmml"><mi id="S2.5.p2.29.m28.3.3.2.2.1.1.1.2" xref="S2.5.p2.29.m28.3.3.2.2.1.1.1.2.cmml">cls</mi><mi id="S2.5.p2.29.m28.3.3.2.2.1.1.1.3" xref="S2.5.p2.29.m28.3.3.2.2.1.1.1.3.cmml">Ξ³</mi></msup><mo id="S2.5.p2.29.m28.3.3.2.2.1.1a" xref="S2.5.p2.29.m28.3.3.2.2.1.2.cmml">β‘</mo><mrow id="S2.5.p2.29.m28.3.3.2.2.1.1.2" xref="S2.5.p2.29.m28.3.3.2.2.1.2.cmml"><mo id="S2.5.p2.29.m28.3.3.2.2.1.1.2.1" stretchy="false" xref="S2.5.p2.29.m28.3.3.2.2.1.2.cmml">(</mo><mi id="S2.5.p2.29.m28.1.1" xref="S2.5.p2.29.m28.1.1.cmml">X</mi><mo id="S2.5.p2.29.m28.3.3.2.2.1.1.2.2" stretchy="false" xref="S2.5.p2.29.m28.3.3.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S2.5.p2.29.m28.3.3.2.2.3" xref="S2.5.p2.29.m28.3.3.2.2.3.cmml">β</mo><mi id="S2.5.p2.29.m28.3.3.2.2.4" xref="S2.5.p2.29.m28.3.3.2.2.4.cmml">F</mi><mo id="S2.5.p2.29.m28.3.3.2.2.5" xref="S2.5.p2.29.m28.3.3.2.2.5.cmml">β</mo><mi class="ltx_font_mathcaligraphic" id="S2.5.p2.29.m28.3.3.2.2.6" xref="S2.5.p2.29.m28.3.3.2.2.6.cmml">β±</mi></mrow><mo id="S2.5.p2.29.m28.3.3.2.5" stretchy="false" xref="S2.5.p2.29.m28.3.3.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.29.m28.3b"><apply id="S2.5.p2.29.m28.3.3.3.cmml" xref="S2.5.p2.29.m28.3.3.2"><csymbol cd="latexml" id="S2.5.p2.29.m28.3.3.3.1.cmml" xref="S2.5.p2.29.m28.3.3.2.3">conditional-set</csymbol><apply id="S2.5.p2.29.m28.2.2.1.1.cmml" xref="S2.5.p2.29.m28.2.2.1.1"><csymbol cd="ambiguous" id="S2.5.p2.29.m28.2.2.1.1.1.cmml" xref="S2.5.p2.29.m28.2.2.1.1">superscript</csymbol><ci id="S2.5.p2.29.m28.2.2.1.1.2.cmml" xref="S2.5.p2.29.m28.2.2.1.1.2">πΉ</ci><apply id="S2.5.p2.29.m28.2.2.1.1.3.cmml" xref="S2.5.p2.29.m28.2.2.1.1.3"><minus id="S2.5.p2.29.m28.2.2.1.1.3.1.cmml" xref="S2.5.p2.29.m28.2.2.1.1.3"></minus><cn id="S2.5.p2.29.m28.2.2.1.1.3.2.cmml" type="integer" xref="S2.5.p2.29.m28.2.2.1.1.3.2">1</cn></apply></apply><apply id="S2.5.p2.29.m28.3.3.2.2.cmml" xref="S2.5.p2.29.m28.3.3.2.2"><and id="S2.5.p2.29.m28.3.3.2.2a.cmml" xref="S2.5.p2.29.m28.3.3.2.2"></and><apply id="S2.5.p2.29.m28.3.3.2.2b.cmml" xref="S2.5.p2.29.m28.3.3.2.2"><csymbol cd="latexml" id="S2.5.p2.29.m28.3.3.2.2.3.cmml" xref="S2.5.p2.29.m28.3.3.2.2.3">superset-of-or-equals</csymbol><apply id="S2.5.p2.29.m28.3.3.2.2.1.2.cmml" xref="S2.5.p2.29.m28.3.3.2.2.1.1"><apply id="S2.5.p2.29.m28.3.3.2.2.1.1.1.cmml" xref="S2.5.p2.29.m28.3.3.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.29.m28.3.3.2.2.1.1.1.1.cmml" xref="S2.5.p2.29.m28.3.3.2.2.1.1.1">superscript</csymbol><ci id="S2.5.p2.29.m28.3.3.2.2.1.1.1.2.cmml" xref="S2.5.p2.29.m28.3.3.2.2.1.1.1.2">cls</ci><ci id="S2.5.p2.29.m28.3.3.2.2.1.1.1.3.cmml" xref="S2.5.p2.29.m28.3.3.2.2.1.1.1.3">πΎ</ci></apply><ci id="S2.5.p2.29.m28.1.1.cmml" xref="S2.5.p2.29.m28.1.1">π</ci></apply><ci id="S2.5.p2.29.m28.3.3.2.2.4.cmml" xref="S2.5.p2.29.m28.3.3.2.2.4">πΉ</ci></apply><apply id="S2.5.p2.29.m28.3.3.2.2c.cmml" xref="S2.5.p2.29.m28.3.3.2.2"><in id="S2.5.p2.29.m28.3.3.2.2.5.cmml" xref="S2.5.p2.29.m28.3.3.2.2.5"></in><share href="https://arxiv.org/html/2503.13666v1#S2.5.p2.29.m28.3.3.2.2.4.cmml" id="S2.5.p2.29.m28.3.3.2.2d.cmml" xref="S2.5.p2.29.m28.3.3.2.2"></share><ci id="S2.5.p2.29.m28.3.3.2.2.6.cmml" xref="S2.5.p2.29.m28.3.3.2.2.6">β±</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.29.m28.3c">\{F^{-1}\colon\operatorname{cls}^{\gamma}(X)\supseteq F\in\mathcal{F}\}</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.29.m28.3d">{ italic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT : roman_cls start_POSTSUPERSCRIPT italic_Ξ³ end_POSTSUPERSCRIPT ( italic_X ) β italic_F β caligraphic_F }</annotation></semantics></math> converges to <math alttext="y" class="ltx_Math" display="inline" id="S2.5.p2.30.m29.1"><semantics id="S2.5.p2.30.m29.1a"><mi id="S2.5.p2.30.m29.1.1" xref="S2.5.p2.30.m29.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.30.m29.1b"><ci id="S2.5.p2.30.m29.1.1.cmml" xref="S2.5.p2.30.m29.1.1">π¦</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.30.m29.1c">y</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.30.m29.1d">italic_y</annotation></semantics></math>. Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S2.Thmtheorem3" title="Proposition 2.3. β£ 2. Compact-like semigroups with dense inverse subsemigroups and the automatic continuity of inversion β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">2.3</span></a> implies that <math alttext="\operatorname{cls}^{\alpha}(X)" class="ltx_Math" display="inline" id="S2.5.p2.31.m30.2"><semantics id="S2.5.p2.31.m30.2a"><mrow id="S2.5.p2.31.m30.2.2.1" xref="S2.5.p2.31.m30.2.2.2.cmml"><msup id="S2.5.p2.31.m30.2.2.1.1" xref="S2.5.p2.31.m30.2.2.1.1.cmml"><mi id="S2.5.p2.31.m30.2.2.1.1.2" xref="S2.5.p2.31.m30.2.2.1.1.2.cmml">cls</mi><mi id="S2.5.p2.31.m30.2.2.1.1.3" xref="S2.5.p2.31.m30.2.2.1.1.3.cmml">Ξ±</mi></msup><mo id="S2.5.p2.31.m30.2.2.1a" xref="S2.5.p2.31.m30.2.2.2.cmml">β‘</mo><mrow id="S2.5.p2.31.m30.2.2.1.2" xref="S2.5.p2.31.m30.2.2.2.cmml"><mo id="S2.5.p2.31.m30.2.2.1.2.1" stretchy="false" xref="S2.5.p2.31.m30.2.2.2.cmml">(</mo><mi id="S2.5.p2.31.m30.1.1" xref="S2.5.p2.31.m30.1.1.cmml">X</mi><mo id="S2.5.p2.31.m30.2.2.1.2.2" stretchy="false" xref="S2.5.p2.31.m30.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.31.m30.2b"><apply id="S2.5.p2.31.m30.2.2.2.cmml" xref="S2.5.p2.31.m30.2.2.1"><apply id="S2.5.p2.31.m30.2.2.1.1.cmml" xref="S2.5.p2.31.m30.2.2.1.1"><csymbol cd="ambiguous" id="S2.5.p2.31.m30.2.2.1.1.1.cmml" xref="S2.5.p2.31.m30.2.2.1.1">superscript</csymbol><ci id="S2.5.p2.31.m30.2.2.1.1.2.cmml" xref="S2.5.p2.31.m30.2.2.1.1.2">cls</ci><ci id="S2.5.p2.31.m30.2.2.1.1.3.cmml" xref="S2.5.p2.31.m30.2.2.1.1.3">πΌ</ci></apply><ci id="S2.5.p2.31.m30.1.1.cmml" xref="S2.5.p2.31.m30.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.31.m30.2c">\operatorname{cls}^{\alpha}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.31.m30.2d">roman_cls start_POSTSUPERSCRIPT italic_Ξ± end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math> is an inverse semigroup. Hence <math alttext="S=\operatorname{cls}^{\omega_{1}}(X)" class="ltx_Math" display="inline" id="S2.5.p2.32.m31.2"><semantics id="S2.5.p2.32.m31.2a"><mrow id="S2.5.p2.32.m31.2.2" xref="S2.5.p2.32.m31.2.2.cmml"><mi id="S2.5.p2.32.m31.2.2.3" xref="S2.5.p2.32.m31.2.2.3.cmml">S</mi><mo id="S2.5.p2.32.m31.2.2.2" xref="S2.5.p2.32.m31.2.2.2.cmml">=</mo><mrow id="S2.5.p2.32.m31.2.2.1.1" xref="S2.5.p2.32.m31.2.2.1.2.cmml"><msup id="S2.5.p2.32.m31.2.2.1.1.1" xref="S2.5.p2.32.m31.2.2.1.1.1.cmml"><mi id="S2.5.p2.32.m31.2.2.1.1.1.2" xref="S2.5.p2.32.m31.2.2.1.1.1.2.cmml">cls</mi><msub id="S2.5.p2.32.m31.2.2.1.1.1.3" xref="S2.5.p2.32.m31.2.2.1.1.1.3.cmml"><mi id="S2.5.p2.32.m31.2.2.1.1.1.3.2" xref="S2.5.p2.32.m31.2.2.1.1.1.3.2.cmml">Ο</mi><mn id="S2.5.p2.32.m31.2.2.1.1.1.3.3" xref="S2.5.p2.32.m31.2.2.1.1.1.3.3.cmml">1</mn></msub></msup><mo id="S2.5.p2.32.m31.2.2.1.1a" xref="S2.5.p2.32.m31.2.2.1.2.cmml">β‘</mo><mrow id="S2.5.p2.32.m31.2.2.1.1.2" xref="S2.5.p2.32.m31.2.2.1.2.cmml"><mo id="S2.5.p2.32.m31.2.2.1.1.2.1" stretchy="false" xref="S2.5.p2.32.m31.2.2.1.2.cmml">(</mo><mi id="S2.5.p2.32.m31.1.1" xref="S2.5.p2.32.m31.1.1.cmml">X</mi><mo id="S2.5.p2.32.m31.2.2.1.1.2.2" stretchy="false" xref="S2.5.p2.32.m31.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.5.p2.32.m31.2b"><apply id="S2.5.p2.32.m31.2.2.cmml" xref="S2.5.p2.32.m31.2.2"><eq id="S2.5.p2.32.m31.2.2.2.cmml" xref="S2.5.p2.32.m31.2.2.2"></eq><ci id="S2.5.p2.32.m31.2.2.3.cmml" xref="S2.5.p2.32.m31.2.2.3">π</ci><apply id="S2.5.p2.32.m31.2.2.1.2.cmml" xref="S2.5.p2.32.m31.2.2.1.1"><apply id="S2.5.p2.32.m31.2.2.1.1.1.cmml" xref="S2.5.p2.32.m31.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.5.p2.32.m31.2.2.1.1.1.1.cmml" xref="S2.5.p2.32.m31.2.2.1.1.1">superscript</csymbol><ci id="S2.5.p2.32.m31.2.2.1.1.1.2.cmml" xref="S2.5.p2.32.m31.2.2.1.1.1.2">cls</ci><apply id="S2.5.p2.32.m31.2.2.1.1.1.3.cmml" xref="S2.5.p2.32.m31.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.5.p2.32.m31.2.2.1.1.1.3.1.cmml" xref="S2.5.p2.32.m31.2.2.1.1.1.3">subscript</csymbol><ci id="S2.5.p2.32.m31.2.2.1.1.1.3.2.cmml" xref="S2.5.p2.32.m31.2.2.1.1.1.3.2">π</ci><cn id="S2.5.p2.32.m31.2.2.1.1.1.3.3.cmml" type="integer" xref="S2.5.p2.32.m31.2.2.1.1.1.3.3">1</cn></apply></apply><ci id="S2.5.p2.32.m31.1.1.cmml" xref="S2.5.p2.32.m31.1.1">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.32.m31.2c">S=\operatorname{cls}^{\omega_{1}}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.32.m31.2d">italic_S = roman_cls start_POSTSUPERSCRIPT italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math> is an inverse semigroup. By <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib4" title="">4</a>, Corollary 1]</cite>, inversion is automatically continuous on every countably compact sequential inverse topological semigroup. Hence, <math alttext="S" class="ltx_Math" display="inline" id="S2.5.p2.33.m32.1"><semantics id="S2.5.p2.33.m32.1a"><mi id="S2.5.p2.33.m32.1.1" xref="S2.5.p2.33.m32.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.5.p2.33.m32.1b"><ci id="S2.5.p2.33.m32.1.1.cmml" xref="S2.5.p2.33.m32.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.5.p2.33.m32.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.5.p2.33.m32.1d">italic_S</annotation></semantics></math> is a topological inverse semigroup. β</p> </div> </div> <div class="ltx_para" id="S2.p5"> <p class="ltx_p" id="S2.p5.2">The <span class="ltx_text ltx_font_italic" id="S2.p5.2.1">bicyclic monoid</span> is generated by two elements <math alttext="p,q" class="ltx_Math" display="inline" id="S2.p5.1.m1.2"><semantics id="S2.p5.1.m1.2a"><mrow id="S2.p5.1.m1.2.3.2" xref="S2.p5.1.m1.2.3.1.cmml"><mi id="S2.p5.1.m1.1.1" xref="S2.p5.1.m1.1.1.cmml">p</mi><mo id="S2.p5.1.m1.2.3.2.1" xref="S2.p5.1.m1.2.3.1.cmml">,</mo><mi id="S2.p5.1.m1.2.2" xref="S2.p5.1.m1.2.2.cmml">q</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.1.m1.2b"><list id="S2.p5.1.m1.2.3.1.cmml" xref="S2.p5.1.m1.2.3.2"><ci id="S2.p5.1.m1.1.1.cmml" xref="S2.p5.1.m1.1.1">π</ci><ci id="S2.p5.1.m1.2.2.cmml" xref="S2.p5.1.m1.2.2">π</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.1.m1.2c">p,q</annotation><annotation encoding="application/x-llamapun" id="S2.p5.1.m1.2d">italic_p , italic_q</annotation></semantics></math> subject to the one condition <math alttext="qp=1" class="ltx_Math" display="inline" id="S2.p5.2.m2.1"><semantics id="S2.p5.2.m2.1a"><mrow id="S2.p5.2.m2.1.1" xref="S2.p5.2.m2.1.1.cmml"><mrow id="S2.p5.2.m2.1.1.2" xref="S2.p5.2.m2.1.1.2.cmml"><mi id="S2.p5.2.m2.1.1.2.2" xref="S2.p5.2.m2.1.1.2.2.cmml">q</mi><mo id="S2.p5.2.m2.1.1.2.1" xref="S2.p5.2.m2.1.1.2.1.cmml">β’</mo><mi id="S2.p5.2.m2.1.1.2.3" xref="S2.p5.2.m2.1.1.2.3.cmml">p</mi></mrow><mo id="S2.p5.2.m2.1.1.1" xref="S2.p5.2.m2.1.1.1.cmml">=</mo><mn id="S2.p5.2.m2.1.1.3" xref="S2.p5.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.2.m2.1b"><apply id="S2.p5.2.m2.1.1.cmml" xref="S2.p5.2.m2.1.1"><eq id="S2.p5.2.m2.1.1.1.cmml" xref="S2.p5.2.m2.1.1.1"></eq><apply id="S2.p5.2.m2.1.1.2.cmml" xref="S2.p5.2.m2.1.1.2"><times id="S2.p5.2.m2.1.1.2.1.cmml" xref="S2.p5.2.m2.1.1.2.1"></times><ci id="S2.p5.2.m2.1.1.2.2.cmml" xref="S2.p5.2.m2.1.1.2.2">π</ci><ci id="S2.p5.2.m2.1.1.2.3.cmml" xref="S2.p5.2.m2.1.1.2.3">π</ci></apply><cn id="S2.p5.2.m2.1.1.3.cmml" type="integer" xref="S2.p5.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.2.m2.1c">qp=1</annotation><annotation encoding="application/x-llamapun" id="S2.p5.2.m2.1d">italic_q italic_p = 1</annotation></semantics></math>. It is well known that bicyclic monoid is an inverse semigroup (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib36" title="">36</a>, Chapter 3.4]</cite>). The following two results that appear in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib3" title="">3</a>, Theorem 6.1 and Theorem 6.6]</cite> establish some sharpness of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem3" title="Theorem C. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">C</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S2.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem4.1.1.1">Theorem 2.4</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem4.2.2"> </span>(Banakh, Dimitrova, Gutik)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem4.3.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem4.p1"> <p class="ltx_p" id="S2.Thmtheorem4.p1.1"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem4.p1.1.1">There exists a pseudocompact topological semigroup which is not an inverse semigroup but contains a dense copy of the bicyclic monoid.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S2.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem5.1.1.1">Theorem 2.5</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem5.2.2"> </span>(Banakh, Dimitrova, Gutik)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem5.3.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem5.p1"> <p class="ltx_p" id="S2.Thmtheorem5.p1.1"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem5.p1.1.1">If there exists a torsion-free abelian countably compact topological group without non-trivial convergent sequences, then there exists a Tychonoff countably compact topological semigroup which is not an inverse semigroup but contains a dense copy of the bicyclic monoid.</span></p> </div> </div> <div class="ltx_para" id="S2.p6"> <p class="ltx_p" id="S2.p6.1">Noteworthy, the existence of a torsion-free abelian countably compact topological group without non-trivial convergent sequences is still not established in ZFC. However, various consistent examples of such groups were constructed in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib13" title="">13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib15" title="">15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib16" title="">16</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib24" title="">24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib30" title="">30</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib34" title="">34</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib51" title="">51</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib52" title="">52</a>]</cite>.</p> </div> <div class="ltx_para" id="S2.p7"> <p class="ltx_p" id="S2.p7.3">Recall that an inverse semigroup <math alttext="S" class="ltx_Math" display="inline" id="S2.p7.1.m1.1"><semantics id="S2.p7.1.m1.1a"><mi id="S2.p7.1.m1.1.1" xref="S2.p7.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p7.1.m1.1b"><ci id="S2.p7.1.m1.1.1.cmml" xref="S2.p7.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p7.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p7.1.m1.1d">italic_S</annotation></semantics></math> is called <span class="ltx_text ltx_font_italic" id="S2.p7.3.1">Clifford</span> if <math alttext="xx^{-1}=x^{-1}x" class="ltx_Math" display="inline" id="S2.p7.2.m2.1"><semantics id="S2.p7.2.m2.1a"><mrow id="S2.p7.2.m2.1.1" xref="S2.p7.2.m2.1.1.cmml"><mrow id="S2.p7.2.m2.1.1.2" xref="S2.p7.2.m2.1.1.2.cmml"><mi id="S2.p7.2.m2.1.1.2.2" xref="S2.p7.2.m2.1.1.2.2.cmml">x</mi><mo id="S2.p7.2.m2.1.1.2.1" xref="S2.p7.2.m2.1.1.2.1.cmml">β’</mo><msup id="S2.p7.2.m2.1.1.2.3" xref="S2.p7.2.m2.1.1.2.3.cmml"><mi id="S2.p7.2.m2.1.1.2.3.2" xref="S2.p7.2.m2.1.1.2.3.2.cmml">x</mi><mrow id="S2.p7.2.m2.1.1.2.3.3" xref="S2.p7.2.m2.1.1.2.3.3.cmml"><mo id="S2.p7.2.m2.1.1.2.3.3a" xref="S2.p7.2.m2.1.1.2.3.3.cmml">β</mo><mn id="S2.p7.2.m2.1.1.2.3.3.2" xref="S2.p7.2.m2.1.1.2.3.3.2.cmml">1</mn></mrow></msup></mrow><mo id="S2.p7.2.m2.1.1.1" xref="S2.p7.2.m2.1.1.1.cmml">=</mo><mrow id="S2.p7.2.m2.1.1.3" xref="S2.p7.2.m2.1.1.3.cmml"><msup id="S2.p7.2.m2.1.1.3.2" xref="S2.p7.2.m2.1.1.3.2.cmml"><mi id="S2.p7.2.m2.1.1.3.2.2" xref="S2.p7.2.m2.1.1.3.2.2.cmml">x</mi><mrow id="S2.p7.2.m2.1.1.3.2.3" xref="S2.p7.2.m2.1.1.3.2.3.cmml"><mo id="S2.p7.2.m2.1.1.3.2.3a" xref="S2.p7.2.m2.1.1.3.2.3.cmml">β</mo><mn id="S2.p7.2.m2.1.1.3.2.3.2" xref="S2.p7.2.m2.1.1.3.2.3.2.cmml">1</mn></mrow></msup><mo id="S2.p7.2.m2.1.1.3.1" xref="S2.p7.2.m2.1.1.3.1.cmml">β’</mo><mi id="S2.p7.2.m2.1.1.3.3" xref="S2.p7.2.m2.1.1.3.3.cmml">x</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p7.2.m2.1b"><apply id="S2.p7.2.m2.1.1.cmml" xref="S2.p7.2.m2.1.1"><eq id="S2.p7.2.m2.1.1.1.cmml" xref="S2.p7.2.m2.1.1.1"></eq><apply id="S2.p7.2.m2.1.1.2.cmml" xref="S2.p7.2.m2.1.1.2"><times id="S2.p7.2.m2.1.1.2.1.cmml" xref="S2.p7.2.m2.1.1.2.1"></times><ci id="S2.p7.2.m2.1.1.2.2.cmml" xref="S2.p7.2.m2.1.1.2.2">π₯</ci><apply id="S2.p7.2.m2.1.1.2.3.cmml" xref="S2.p7.2.m2.1.1.2.3"><csymbol cd="ambiguous" id="S2.p7.2.m2.1.1.2.3.1.cmml" xref="S2.p7.2.m2.1.1.2.3">superscript</csymbol><ci id="S2.p7.2.m2.1.1.2.3.2.cmml" xref="S2.p7.2.m2.1.1.2.3.2">π₯</ci><apply id="S2.p7.2.m2.1.1.2.3.3.cmml" xref="S2.p7.2.m2.1.1.2.3.3"><minus id="S2.p7.2.m2.1.1.2.3.3.1.cmml" xref="S2.p7.2.m2.1.1.2.3.3"></minus><cn id="S2.p7.2.m2.1.1.2.3.3.2.cmml" type="integer" xref="S2.p7.2.m2.1.1.2.3.3.2">1</cn></apply></apply></apply><apply id="S2.p7.2.m2.1.1.3.cmml" xref="S2.p7.2.m2.1.1.3"><times id="S2.p7.2.m2.1.1.3.1.cmml" xref="S2.p7.2.m2.1.1.3.1"></times><apply id="S2.p7.2.m2.1.1.3.2.cmml" xref="S2.p7.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S2.p7.2.m2.1.1.3.2.1.cmml" xref="S2.p7.2.m2.1.1.3.2">superscript</csymbol><ci id="S2.p7.2.m2.1.1.3.2.2.cmml" xref="S2.p7.2.m2.1.1.3.2.2">π₯</ci><apply id="S2.p7.2.m2.1.1.3.2.3.cmml" xref="S2.p7.2.m2.1.1.3.2.3"><minus id="S2.p7.2.m2.1.1.3.2.3.1.cmml" xref="S2.p7.2.m2.1.1.3.2.3"></minus><cn id="S2.p7.2.m2.1.1.3.2.3.2.cmml" type="integer" xref="S2.p7.2.m2.1.1.3.2.3.2">1</cn></apply></apply><ci id="S2.p7.2.m2.1.1.3.3.cmml" xref="S2.p7.2.m2.1.1.3.3">π₯</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p7.2.m2.1c">xx^{-1}=x^{-1}x</annotation><annotation encoding="application/x-llamapun" id="S2.p7.2.m2.1d">italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x</annotation></semantics></math> for all <math alttext="x\in S" class="ltx_Math" display="inline" id="S2.p7.3.m3.1"><semantics id="S2.p7.3.m3.1a"><mrow id="S2.p7.3.m3.1.1" xref="S2.p7.3.m3.1.1.cmml"><mi id="S2.p7.3.m3.1.1.2" xref="S2.p7.3.m3.1.1.2.cmml">x</mi><mo id="S2.p7.3.m3.1.1.1" xref="S2.p7.3.m3.1.1.1.cmml">β</mo><mi id="S2.p7.3.m3.1.1.3" xref="S2.p7.3.m3.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p7.3.m3.1b"><apply id="S2.p7.3.m3.1.1.cmml" xref="S2.p7.3.m3.1.1"><in id="S2.p7.3.m3.1.1.1.cmml" xref="S2.p7.3.m3.1.1.1"></in><ci id="S2.p7.3.m3.1.1.2.cmml" xref="S2.p7.3.m3.1.1.2">π₯</ci><ci id="S2.p7.3.m3.1.1.3.cmml" xref="S2.p7.3.m3.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p7.3.m3.1c">x\in S</annotation><annotation encoding="application/x-llamapun" id="S2.p7.3.m3.1d">italic_x β italic_S</annotation></semantics></math>. The automatic continuity of inversion in groups, Clifford semigroups, and inverse semigroups is widely studied (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib6" title="">6</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib14" title="">14</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib21" title="">21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib46" title="">46</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib48" title="">48</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib50" title="">50</a>]</cite>). In most cases the continuity of inversion follows from the continuity of multiplication and some compact-like property. For instance, the following two results appear in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib27" title="">27</a>, Corollary 2]</cite> and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib4" title="">4</a>, Theorem 2]</cite>, respectively.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S2.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem6.1.1.1">Theorem 2.6</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem6.2.2"> </span>(Gutik, Pagon, RepovΕ‘)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem6.3.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem6.p1"> <p class="ltx_p" id="S2.Thmtheorem6.p1.3"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem6.p1.3.3">Let <math alttext="S" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.1.1.m1.1"><semantics id="S2.Thmtheorem6.p1.1.1.m1.1a"><mi id="S2.Thmtheorem6.p1.1.1.m1.1.1" xref="S2.Thmtheorem6.p1.1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.1.1.m1.1b"><ci id="S2.Thmtheorem6.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem6.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.1.1.m1.1d">italic_S</annotation></semantics></math> be a Tychonoff Clifford topological semigroup such that <math alttext="S{\times}S" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.2.2.m2.1"><semantics id="S2.Thmtheorem6.p1.2.2.m2.1a"><mrow id="S2.Thmtheorem6.p1.2.2.m2.1.1" xref="S2.Thmtheorem6.p1.2.2.m2.1.1.cmml"><mi id="S2.Thmtheorem6.p1.2.2.m2.1.1.2" xref="S2.Thmtheorem6.p1.2.2.m2.1.1.2.cmml">S</mi><mo id="S2.Thmtheorem6.p1.2.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.Thmtheorem6.p1.2.2.m2.1.1.1.cmml">Γ</mo><mi id="S2.Thmtheorem6.p1.2.2.m2.1.1.3" xref="S2.Thmtheorem6.p1.2.2.m2.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.2.2.m2.1b"><apply id="S2.Thmtheorem6.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem6.p1.2.2.m2.1.1"><times id="S2.Thmtheorem6.p1.2.2.m2.1.1.1.cmml" xref="S2.Thmtheorem6.p1.2.2.m2.1.1.1"></times><ci id="S2.Thmtheorem6.p1.2.2.m2.1.1.2.cmml" xref="S2.Thmtheorem6.p1.2.2.m2.1.1.2">π</ci><ci id="S2.Thmtheorem6.p1.2.2.m2.1.1.3.cmml" xref="S2.Thmtheorem6.p1.2.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.2.2.m2.1c">S{\times}S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.2.2.m2.1d">italic_S Γ italic_S</annotation></semantics></math> is pseudocompact. Then inversion is continuous in <math alttext="S" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.3.3.m3.1"><semantics id="S2.Thmtheorem6.p1.3.3.m3.1a"><mi id="S2.Thmtheorem6.p1.3.3.m3.1.1" xref="S2.Thmtheorem6.p1.3.3.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.3.3.m3.1b"><ci id="S2.Thmtheorem6.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem6.p1.3.3.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.3.3.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.3.3.m3.1d">italic_S</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S2.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem7.1.1.1">Theorem 2.7</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem7.2.2"> </span>(Banakh, Gutik)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem7.3.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem7.p1"> <p class="ltx_p" id="S2.Thmtheorem7.p1.3"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem7.p1.3.3">Let <math alttext="S" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.1.1.m1.1"><semantics id="S2.Thmtheorem7.p1.1.1.m1.1a"><mi id="S2.Thmtheorem7.p1.1.1.m1.1.1" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.1.1.m1.1b"><ci id="S2.Thmtheorem7.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.1.1.m1.1d">italic_S</annotation></semantics></math> be a regular topologically periodic Clifford topological semigroup such that <math alttext="S{\times}S" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.2.2.m2.1"><semantics id="S2.Thmtheorem7.p1.2.2.m2.1a"><mrow id="S2.Thmtheorem7.p1.2.2.m2.1.1" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.cmml"><mi id="S2.Thmtheorem7.p1.2.2.m2.1.1.2" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.2.cmml">S</mi><mo id="S2.Thmtheorem7.p1.2.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.1.cmml">Γ</mo><mi id="S2.Thmtheorem7.p1.2.2.m2.1.1.3" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.2.2.m2.1b"><apply id="S2.Thmtheorem7.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem7.p1.2.2.m2.1.1"><times id="S2.Thmtheorem7.p1.2.2.m2.1.1.1.cmml" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.1"></times><ci id="S2.Thmtheorem7.p1.2.2.m2.1.1.2.cmml" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.2">π</ci><ci id="S2.Thmtheorem7.p1.2.2.m2.1.1.3.cmml" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.2.2.m2.1c">S{\times}S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.2.2.m2.1d">italic_S Γ italic_S</annotation></semantics></math> is countably compact. Then inversion is continuous in <math alttext="S" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.3.3.m3.1"><semantics id="S2.Thmtheorem7.p1.3.3.m3.1a"><mi id="S2.Thmtheorem7.p1.3.3.m3.1.1" xref="S2.Thmtheorem7.p1.3.3.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.3.3.m3.1b"><ci id="S2.Thmtheorem7.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem7.p1.3.3.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.3.3.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.3.3.m3.1d">italic_S</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.p8"> <p class="ltx_p" id="S2.p8.5">The rest of this section is devoted to the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem4" title="Theorem D. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">D</span></a>, which in fact generalizes Theorems <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S2.Thmtheorem6" title="Theorem 2.6 (Gutik, Pagon, RepovΕ‘). β£ 2. Compact-like semigroups with dense inverse subsemigroups and the automatic continuity of inversion β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">2.6</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S2.Thmtheorem7" title="Theorem 2.7 (Banakh, Gutik). β£ 2. Compact-like semigroups with dense inverse subsemigroups and the automatic continuity of inversion β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">2.7</span></a>. A semigroup <math alttext="S" class="ltx_Math" display="inline" id="S2.p8.1.m1.1"><semantics id="S2.p8.1.m1.1a"><mi id="S2.p8.1.m1.1.1" xref="S2.p8.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p8.1.m1.1b"><ci id="S2.p8.1.m1.1.1.cmml" xref="S2.p8.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p8.1.m1.1d">italic_S</annotation></semantics></math> endowed with a topology is called <span class="ltx_text ltx_font_italic" id="S2.p8.5.1">semitopological</span> if for every <math alttext="s\in S" class="ltx_Math" display="inline" id="S2.p8.2.m2.1"><semantics id="S2.p8.2.m2.1a"><mrow id="S2.p8.2.m2.1.1" xref="S2.p8.2.m2.1.1.cmml"><mi id="S2.p8.2.m2.1.1.2" xref="S2.p8.2.m2.1.1.2.cmml">s</mi><mo id="S2.p8.2.m2.1.1.1" xref="S2.p8.2.m2.1.1.1.cmml">β</mo><mi id="S2.p8.2.m2.1.1.3" xref="S2.p8.2.m2.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p8.2.m2.1b"><apply id="S2.p8.2.m2.1.1.cmml" xref="S2.p8.2.m2.1.1"><in id="S2.p8.2.m2.1.1.1.cmml" xref="S2.p8.2.m2.1.1.1"></in><ci id="S2.p8.2.m2.1.1.2.cmml" xref="S2.p8.2.m2.1.1.2">π </ci><ci id="S2.p8.2.m2.1.1.3.cmml" xref="S2.p8.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.2.m2.1c">s\in S</annotation><annotation encoding="application/x-llamapun" id="S2.p8.2.m2.1d">italic_s β italic_S</annotation></semantics></math> the shifts <math alttext="l_{s}:x\mapsto sx" class="ltx_Math" display="inline" id="S2.p8.3.m3.1"><semantics id="S2.p8.3.m3.1a"><mrow id="S2.p8.3.m3.1.1" xref="S2.p8.3.m3.1.1.cmml"><msub id="S2.p8.3.m3.1.1.2" xref="S2.p8.3.m3.1.1.2.cmml"><mi id="S2.p8.3.m3.1.1.2.2" xref="S2.p8.3.m3.1.1.2.2.cmml">l</mi><mi id="S2.p8.3.m3.1.1.2.3" xref="S2.p8.3.m3.1.1.2.3.cmml">s</mi></msub><mo id="S2.p8.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.p8.3.m3.1.1.1.cmml">:</mo><mrow id="S2.p8.3.m3.1.1.3" xref="S2.p8.3.m3.1.1.3.cmml"><mi id="S2.p8.3.m3.1.1.3.2" xref="S2.p8.3.m3.1.1.3.2.cmml">x</mi><mo id="S2.p8.3.m3.1.1.3.1" stretchy="false" xref="S2.p8.3.m3.1.1.3.1.cmml">β¦</mo><mrow id="S2.p8.3.m3.1.1.3.3" xref="S2.p8.3.m3.1.1.3.3.cmml"><mi id="S2.p8.3.m3.1.1.3.3.2" xref="S2.p8.3.m3.1.1.3.3.2.cmml">s</mi><mo id="S2.p8.3.m3.1.1.3.3.1" xref="S2.p8.3.m3.1.1.3.3.1.cmml">β’</mo><mi id="S2.p8.3.m3.1.1.3.3.3" xref="S2.p8.3.m3.1.1.3.3.3.cmml">x</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p8.3.m3.1b"><apply id="S2.p8.3.m3.1.1.cmml" xref="S2.p8.3.m3.1.1"><ci id="S2.p8.3.m3.1.1.1.cmml" xref="S2.p8.3.m3.1.1.1">:</ci><apply id="S2.p8.3.m3.1.1.2.cmml" xref="S2.p8.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.p8.3.m3.1.1.2.1.cmml" xref="S2.p8.3.m3.1.1.2">subscript</csymbol><ci id="S2.p8.3.m3.1.1.2.2.cmml" xref="S2.p8.3.m3.1.1.2.2">π</ci><ci id="S2.p8.3.m3.1.1.2.3.cmml" xref="S2.p8.3.m3.1.1.2.3">π </ci></apply><apply id="S2.p8.3.m3.1.1.3.cmml" xref="S2.p8.3.m3.1.1.3"><csymbol cd="latexml" id="S2.p8.3.m3.1.1.3.1.cmml" xref="S2.p8.3.m3.1.1.3.1">maps-to</csymbol><ci id="S2.p8.3.m3.1.1.3.2.cmml" xref="S2.p8.3.m3.1.1.3.2">π₯</ci><apply id="S2.p8.3.m3.1.1.3.3.cmml" xref="S2.p8.3.m3.1.1.3.3"><times id="S2.p8.3.m3.1.1.3.3.1.cmml" xref="S2.p8.3.m3.1.1.3.3.1"></times><ci id="S2.p8.3.m3.1.1.3.3.2.cmml" xref="S2.p8.3.m3.1.1.3.3.2">π </ci><ci id="S2.p8.3.m3.1.1.3.3.3.cmml" xref="S2.p8.3.m3.1.1.3.3.3">π₯</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.3.m3.1c">l_{s}:x\mapsto sx</annotation><annotation encoding="application/x-llamapun" id="S2.p8.3.m3.1d">italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT : italic_x β¦ italic_s italic_x</annotation></semantics></math> and <math alttext="r_{s}:x\mapsto xs" class="ltx_Math" display="inline" id="S2.p8.4.m4.1"><semantics id="S2.p8.4.m4.1a"><mrow id="S2.p8.4.m4.1.1" xref="S2.p8.4.m4.1.1.cmml"><msub id="S2.p8.4.m4.1.1.2" xref="S2.p8.4.m4.1.1.2.cmml"><mi id="S2.p8.4.m4.1.1.2.2" xref="S2.p8.4.m4.1.1.2.2.cmml">r</mi><mi id="S2.p8.4.m4.1.1.2.3" xref="S2.p8.4.m4.1.1.2.3.cmml">s</mi></msub><mo id="S2.p8.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.p8.4.m4.1.1.1.cmml">:</mo><mrow id="S2.p8.4.m4.1.1.3" xref="S2.p8.4.m4.1.1.3.cmml"><mi id="S2.p8.4.m4.1.1.3.2" xref="S2.p8.4.m4.1.1.3.2.cmml">x</mi><mo id="S2.p8.4.m4.1.1.3.1" stretchy="false" xref="S2.p8.4.m4.1.1.3.1.cmml">β¦</mo><mrow id="S2.p8.4.m4.1.1.3.3" xref="S2.p8.4.m4.1.1.3.3.cmml"><mi id="S2.p8.4.m4.1.1.3.3.2" xref="S2.p8.4.m4.1.1.3.3.2.cmml">x</mi><mo id="S2.p8.4.m4.1.1.3.3.1" xref="S2.p8.4.m4.1.1.3.3.1.cmml">β’</mo><mi id="S2.p8.4.m4.1.1.3.3.3" xref="S2.p8.4.m4.1.1.3.3.3.cmml">s</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p8.4.m4.1b"><apply id="S2.p8.4.m4.1.1.cmml" xref="S2.p8.4.m4.1.1"><ci id="S2.p8.4.m4.1.1.1.cmml" xref="S2.p8.4.m4.1.1.1">:</ci><apply id="S2.p8.4.m4.1.1.2.cmml" xref="S2.p8.4.m4.1.1.2"><csymbol cd="ambiguous" id="S2.p8.4.m4.1.1.2.1.cmml" xref="S2.p8.4.m4.1.1.2">subscript</csymbol><ci id="S2.p8.4.m4.1.1.2.2.cmml" xref="S2.p8.4.m4.1.1.2.2">π</ci><ci id="S2.p8.4.m4.1.1.2.3.cmml" xref="S2.p8.4.m4.1.1.2.3">π </ci></apply><apply id="S2.p8.4.m4.1.1.3.cmml" xref="S2.p8.4.m4.1.1.3"><csymbol cd="latexml" id="S2.p8.4.m4.1.1.3.1.cmml" xref="S2.p8.4.m4.1.1.3.1">maps-to</csymbol><ci id="S2.p8.4.m4.1.1.3.2.cmml" xref="S2.p8.4.m4.1.1.3.2">π₯</ci><apply id="S2.p8.4.m4.1.1.3.3.cmml" xref="S2.p8.4.m4.1.1.3.3"><times id="S2.p8.4.m4.1.1.3.3.1.cmml" xref="S2.p8.4.m4.1.1.3.3.1"></times><ci id="S2.p8.4.m4.1.1.3.3.2.cmml" xref="S2.p8.4.m4.1.1.3.3.2">π₯</ci><ci id="S2.p8.4.m4.1.1.3.3.3.cmml" xref="S2.p8.4.m4.1.1.3.3.3">π </ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.4.m4.1c">r_{s}:x\mapsto xs</annotation><annotation encoding="application/x-llamapun" id="S2.p8.4.m4.1d">italic_r start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT : italic_x β¦ italic_x italic_s</annotation></semantics></math> are continuous in <math alttext="S" class="ltx_Math" display="inline" id="S2.p8.5.m5.1"><semantics id="S2.p8.5.m5.1a"><mi id="S2.p8.5.m5.1.1" xref="S2.p8.5.m5.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p8.5.m5.1b"><ci id="S2.p8.5.m5.1.1.cmml" xref="S2.p8.5.m5.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p8.5.m5.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p8.5.m5.1d">italic_S</annotation></semantics></math>. It is clear that every topological semigroup is semitopological.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S2.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem8.1.1.1">Proposition 2.8</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem8.p1"> <p class="ltx_p" id="S2.Thmtheorem8.p1.1"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem8.p1.1.1">Each inverse topologically periodic semitopological semigroup <math alttext="S" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.1.1.m1.1"><semantics id="S2.Thmtheorem8.p1.1.1.m1.1a"><mi id="S2.Thmtheorem8.p1.1.1.m1.1.1" xref="S2.Thmtheorem8.p1.1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.1.1.m1.1b"><ci id="S2.Thmtheorem8.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem8.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.1.1.m1.1d">italic_S</annotation></semantics></math> is Clifford.</span></p> </div> </div> <div class="ltx_proof" id="S2.6"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.6.p1"> <p class="ltx_p" id="S2.6.p1.6">Fix an element <math alttext="x\in S" class="ltx_Math" display="inline" id="S2.6.p1.1.m1.1"><semantics id="S2.6.p1.1.m1.1a"><mrow id="S2.6.p1.1.m1.1.1" xref="S2.6.p1.1.m1.1.1.cmml"><mi id="S2.6.p1.1.m1.1.1.2" xref="S2.6.p1.1.m1.1.1.2.cmml">x</mi><mo id="S2.6.p1.1.m1.1.1.1" xref="S2.6.p1.1.m1.1.1.1.cmml">β</mo><mi id="S2.6.p1.1.m1.1.1.3" xref="S2.6.p1.1.m1.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p1.1.m1.1b"><apply id="S2.6.p1.1.m1.1.1.cmml" xref="S2.6.p1.1.m1.1.1"><in id="S2.6.p1.1.m1.1.1.1.cmml" xref="S2.6.p1.1.m1.1.1.1"></in><ci id="S2.6.p1.1.m1.1.1.2.cmml" xref="S2.6.p1.1.m1.1.1.2">π₯</ci><ci id="S2.6.p1.1.m1.1.1.3.cmml" xref="S2.6.p1.1.m1.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p1.1.m1.1c">x\in S</annotation><annotation encoding="application/x-llamapun" id="S2.6.p1.1.m1.1d">italic_x β italic_S</annotation></semantics></math> and an open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S2.6.p1.2.m2.1"><semantics id="S2.6.p1.2.m2.1a"><mi id="S2.6.p1.2.m2.1.1" xref="S2.6.p1.2.m2.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S2.6.p1.2.m2.1b"><ci id="S2.6.p1.2.m2.1.1.cmml" xref="S2.6.p1.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p1.2.m2.1c">U</annotation><annotation encoding="application/x-llamapun" id="S2.6.p1.2.m2.1d">italic_U</annotation></semantics></math> of <math alttext="x" class="ltx_Math" display="inline" id="S2.6.p1.3.m3.1"><semantics id="S2.6.p1.3.m3.1a"><mi id="S2.6.p1.3.m3.1.1" xref="S2.6.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.6.p1.3.m3.1b"><ci id="S2.6.p1.3.m3.1.1.cmml" xref="S2.6.p1.3.m3.1.1">π₯</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.6.p1.3.m3.1d">italic_x</annotation></semantics></math>. Since <math alttext="X" class="ltx_Math" display="inline" id="S2.6.p1.4.m4.1"><semantics id="S2.6.p1.4.m4.1a"><mi id="S2.6.p1.4.m4.1.1" xref="S2.6.p1.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.6.p1.4.m4.1b"><ci id="S2.6.p1.4.m4.1.1.cmml" xref="S2.6.p1.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p1.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.6.p1.4.m4.1d">italic_X</annotation></semantics></math> is topologically periodic, there exists a positive integer <math alttext="n\geq 2" class="ltx_Math" display="inline" id="S2.6.p1.5.m5.1"><semantics id="S2.6.p1.5.m5.1a"><mrow id="S2.6.p1.5.m5.1.1" xref="S2.6.p1.5.m5.1.1.cmml"><mi id="S2.6.p1.5.m5.1.1.2" xref="S2.6.p1.5.m5.1.1.2.cmml">n</mi><mo id="S2.6.p1.5.m5.1.1.1" xref="S2.6.p1.5.m5.1.1.1.cmml">β₯</mo><mn id="S2.6.p1.5.m5.1.1.3" xref="S2.6.p1.5.m5.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p1.5.m5.1b"><apply id="S2.6.p1.5.m5.1.1.cmml" xref="S2.6.p1.5.m5.1.1"><geq id="S2.6.p1.5.m5.1.1.1.cmml" xref="S2.6.p1.5.m5.1.1.1"></geq><ci id="S2.6.p1.5.m5.1.1.2.cmml" xref="S2.6.p1.5.m5.1.1.2">π</ci><cn id="S2.6.p1.5.m5.1.1.3.cmml" type="integer" xref="S2.6.p1.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p1.5.m5.1c">n\geq 2</annotation><annotation encoding="application/x-llamapun" id="S2.6.p1.5.m5.1d">italic_n β₯ 2</annotation></semantics></math> such that <math alttext="x^{n}\in U" class="ltx_Math" display="inline" id="S2.6.p1.6.m6.1"><semantics id="S2.6.p1.6.m6.1a"><mrow id="S2.6.p1.6.m6.1.1" xref="S2.6.p1.6.m6.1.1.cmml"><msup id="S2.6.p1.6.m6.1.1.2" xref="S2.6.p1.6.m6.1.1.2.cmml"><mi id="S2.6.p1.6.m6.1.1.2.2" xref="S2.6.p1.6.m6.1.1.2.2.cmml">x</mi><mi id="S2.6.p1.6.m6.1.1.2.3" xref="S2.6.p1.6.m6.1.1.2.3.cmml">n</mi></msup><mo id="S2.6.p1.6.m6.1.1.1" xref="S2.6.p1.6.m6.1.1.1.cmml">β</mo><mi id="S2.6.p1.6.m6.1.1.3" xref="S2.6.p1.6.m6.1.1.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p1.6.m6.1b"><apply id="S2.6.p1.6.m6.1.1.cmml" xref="S2.6.p1.6.m6.1.1"><in id="S2.6.p1.6.m6.1.1.1.cmml" xref="S2.6.p1.6.m6.1.1.1"></in><apply id="S2.6.p1.6.m6.1.1.2.cmml" xref="S2.6.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S2.6.p1.6.m6.1.1.2.1.cmml" xref="S2.6.p1.6.m6.1.1.2">superscript</csymbol><ci id="S2.6.p1.6.m6.1.1.2.2.cmml" xref="S2.6.p1.6.m6.1.1.2.2">π₯</ci><ci id="S2.6.p1.6.m6.1.1.2.3.cmml" xref="S2.6.p1.6.m6.1.1.2.3">π</ci></apply><ci id="S2.6.p1.6.m6.1.1.3.cmml" xref="S2.6.p1.6.m6.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p1.6.m6.1c">x^{n}\in U</annotation><annotation encoding="application/x-llamapun" id="S2.6.p1.6.m6.1d">italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT β italic_U</annotation></semantics></math>. Then</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="x^{n}=x^{2}x^{-2}x^{n}\in U\cap(x^{2}x^{-2})U," class="ltx_Math" display="block" id="S2.Ex5.m1.1"><semantics id="S2.Ex5.m1.1a"><mrow id="S2.Ex5.m1.1.1.1" xref="S2.Ex5.m1.1.1.1.1.cmml"><mrow id="S2.Ex5.m1.1.1.1.1" xref="S2.Ex5.m1.1.1.1.1.cmml"><msup id="S2.Ex5.m1.1.1.1.1.3" xref="S2.Ex5.m1.1.1.1.1.3.cmml"><mi id="S2.Ex5.m1.1.1.1.1.3.2" xref="S2.Ex5.m1.1.1.1.1.3.2.cmml">x</mi><mi id="S2.Ex5.m1.1.1.1.1.3.3" xref="S2.Ex5.m1.1.1.1.1.3.3.cmml">n</mi></msup><mo id="S2.Ex5.m1.1.1.1.1.4" xref="S2.Ex5.m1.1.1.1.1.4.cmml">=</mo><mrow id="S2.Ex5.m1.1.1.1.1.5" xref="S2.Ex5.m1.1.1.1.1.5.cmml"><msup id="S2.Ex5.m1.1.1.1.1.5.2" xref="S2.Ex5.m1.1.1.1.1.5.2.cmml"><mi id="S2.Ex5.m1.1.1.1.1.5.2.2" xref="S2.Ex5.m1.1.1.1.1.5.2.2.cmml">x</mi><mn id="S2.Ex5.m1.1.1.1.1.5.2.3" xref="S2.Ex5.m1.1.1.1.1.5.2.3.cmml">2</mn></msup><mo id="S2.Ex5.m1.1.1.1.1.5.1" xref="S2.Ex5.m1.1.1.1.1.5.1.cmml">β’</mo><msup id="S2.Ex5.m1.1.1.1.1.5.3" xref="S2.Ex5.m1.1.1.1.1.5.3.cmml"><mi id="S2.Ex5.m1.1.1.1.1.5.3.2" xref="S2.Ex5.m1.1.1.1.1.5.3.2.cmml">x</mi><mrow id="S2.Ex5.m1.1.1.1.1.5.3.3" xref="S2.Ex5.m1.1.1.1.1.5.3.3.cmml"><mo id="S2.Ex5.m1.1.1.1.1.5.3.3a" xref="S2.Ex5.m1.1.1.1.1.5.3.3.cmml">β</mo><mn id="S2.Ex5.m1.1.1.1.1.5.3.3.2" xref="S2.Ex5.m1.1.1.1.1.5.3.3.2.cmml">2</mn></mrow></msup><mo id="S2.Ex5.m1.1.1.1.1.5.1a" xref="S2.Ex5.m1.1.1.1.1.5.1.cmml">β’</mo><msup id="S2.Ex5.m1.1.1.1.1.5.4" xref="S2.Ex5.m1.1.1.1.1.5.4.cmml"><mi id="S2.Ex5.m1.1.1.1.1.5.4.2" xref="S2.Ex5.m1.1.1.1.1.5.4.2.cmml">x</mi><mi id="S2.Ex5.m1.1.1.1.1.5.4.3" xref="S2.Ex5.m1.1.1.1.1.5.4.3.cmml">n</mi></msup></mrow><mo id="S2.Ex5.m1.1.1.1.1.6" xref="S2.Ex5.m1.1.1.1.1.6.cmml">β</mo><mrow id="S2.Ex5.m1.1.1.1.1.1" xref="S2.Ex5.m1.1.1.1.1.1.cmml"><mi id="S2.Ex5.m1.1.1.1.1.1.3" xref="S2.Ex5.m1.1.1.1.1.1.3.cmml">U</mi><mo id="S2.Ex5.m1.1.1.1.1.1.2" xref="S2.Ex5.m1.1.1.1.1.1.2.cmml">β©</mo><mrow id="S2.Ex5.m1.1.1.1.1.1.1" xref="S2.Ex5.m1.1.1.1.1.1.1.cmml"><mrow id="S2.Ex5.m1.1.1.1.1.1.1.1.1" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex5.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex5.m1.1.1.1.1.1.1.1.1.1" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.cmml"><msup id="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.2" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.2.2" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.2.2.cmml">x</mi><mn id="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.2.3" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.2.3.cmml">2</mn></msup><mo id="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.1.cmml">β’</mo><msup id="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.3" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.3.2" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.3.2.cmml">x</mi><mrow id="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.3.3" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.3.3.cmml"><mo id="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.3.3a" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.3.3.cmml">β</mo><mn id="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.3.3.2" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.3.3.2.cmml">2</mn></mrow></msup></mrow><mo id="S2.Ex5.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.Ex5.m1.1.1.1.1.1.1.2" xref="S2.Ex5.m1.1.1.1.1.1.1.2.cmml">β’</mo><mi id="S2.Ex5.m1.1.1.1.1.1.1.3" xref="S2.Ex5.m1.1.1.1.1.1.1.3.cmml">U</mi></mrow></mrow></mrow><mo id="S2.Ex5.m1.1.1.1.2" xref="S2.Ex5.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex5.m1.1b"><apply id="S2.Ex5.m1.1.1.1.1.cmml" xref="S2.Ex5.m1.1.1.1"><and id="S2.Ex5.m1.1.1.1.1a.cmml" xref="S2.Ex5.m1.1.1.1"></and><apply id="S2.Ex5.m1.1.1.1.1b.cmml" xref="S2.Ex5.m1.1.1.1"><eq id="S2.Ex5.m1.1.1.1.1.4.cmml" xref="S2.Ex5.m1.1.1.1.1.4"></eq><apply id="S2.Ex5.m1.1.1.1.1.3.cmml" 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id="S2.Ex6.m1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S2.Ex6.m1.1.1.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S2.Ex6.m1.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S2.Ex6.m1.1.1.1.1.1.1.1.1.1.2.2">π₯</ci><apply id="S2.Ex6.m1.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S2.Ex6.m1.1.1.1.1.1.1.1.1.1.2.3"><minus id="S2.Ex6.m1.1.1.1.1.1.1.1.1.1.2.3.1.cmml" xref="S2.Ex6.m1.1.1.1.1.1.1.1.1.1.2.3"></minus><cn id="S2.Ex6.m1.1.1.1.1.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S2.Ex6.m1.1.1.1.1.1.1.1.1.1.2.3.2">2</cn></apply></apply><apply id="S2.Ex6.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S2.Ex6.m1.1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex6.m1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S2.Ex6.m1.1.1.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="S2.Ex6.m1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S2.Ex6.m1.1.1.1.1.1.1.1.1.1.3.2">π₯</ci><cn id="S2.Ex6.m1.1.1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.Ex6.m1.1.1.1.1.1.1.1.1.1.3.3">2</cn></apply></apply></apply><ci id="S2.Ex6.m1.1.1.1.1.1.3.cmml" xref="S2.Ex6.m1.1.1.1.1.1.3">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex6.m1.1c">x^{n}=x^{n}x^{-2}x^{2}\in U(x^{-2}x^{2})\cap U.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex6.m1.1d">italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT = italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT β italic_U ( italic_x start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) β© italic_U .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.6.p1.13">It follows that <math alttext="V\cap(x^{2}x^{-2})V\neq\varnothing" class="ltx_Math" display="inline" id="S2.6.p1.7.m1.1"><semantics id="S2.6.p1.7.m1.1a"><mrow id="S2.6.p1.7.m1.1.1" xref="S2.6.p1.7.m1.1.1.cmml"><mrow id="S2.6.p1.7.m1.1.1.1" xref="S2.6.p1.7.m1.1.1.1.cmml"><mi id="S2.6.p1.7.m1.1.1.1.3" xref="S2.6.p1.7.m1.1.1.1.3.cmml">V</mi><mo id="S2.6.p1.7.m1.1.1.1.2" xref="S2.6.p1.7.m1.1.1.1.2.cmml">β©</mo><mrow id="S2.6.p1.7.m1.1.1.1.1" xref="S2.6.p1.7.m1.1.1.1.1.cmml"><mrow id="S2.6.p1.7.m1.1.1.1.1.1.1" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.cmml"><mo id="S2.6.p1.7.m1.1.1.1.1.1.1.2" stretchy="false" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.6.p1.7.m1.1.1.1.1.1.1.1" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.cmml"><msup id="S2.6.p1.7.m1.1.1.1.1.1.1.1.2" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.2.cmml"><mi id="S2.6.p1.7.m1.1.1.1.1.1.1.1.2.2" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.2.2.cmml">x</mi><mn id="S2.6.p1.7.m1.1.1.1.1.1.1.1.2.3" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.2.3.cmml">2</mn></msup><mo id="S2.6.p1.7.m1.1.1.1.1.1.1.1.1" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.1.cmml">β’</mo><msup id="S2.6.p1.7.m1.1.1.1.1.1.1.1.3" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.cmml"><mi id="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.2" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.2.cmml">x</mi><mrow id="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.3" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.3.cmml"><mo id="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.3a" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.3.cmml">β</mo><mn id="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.3.2" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.3.2.cmml">2</mn></mrow></msup></mrow><mo id="S2.6.p1.7.m1.1.1.1.1.1.1.3" stretchy="false" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.6.p1.7.m1.1.1.1.1.2" xref="S2.6.p1.7.m1.1.1.1.1.2.cmml">β’</mo><mi id="S2.6.p1.7.m1.1.1.1.1.3" xref="S2.6.p1.7.m1.1.1.1.1.3.cmml">V</mi></mrow></mrow><mo id="S2.6.p1.7.m1.1.1.2" xref="S2.6.p1.7.m1.1.1.2.cmml">β </mo><mi id="S2.6.p1.7.m1.1.1.3" mathvariant="normal" xref="S2.6.p1.7.m1.1.1.3.cmml">β </mi></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p1.7.m1.1b"><apply id="S2.6.p1.7.m1.1.1.cmml" xref="S2.6.p1.7.m1.1.1"><neq id="S2.6.p1.7.m1.1.1.2.cmml" xref="S2.6.p1.7.m1.1.1.2"></neq><apply id="S2.6.p1.7.m1.1.1.1.cmml" xref="S2.6.p1.7.m1.1.1.1"><intersect id="S2.6.p1.7.m1.1.1.1.2.cmml" xref="S2.6.p1.7.m1.1.1.1.2"></intersect><ci id="S2.6.p1.7.m1.1.1.1.3.cmml" xref="S2.6.p1.7.m1.1.1.1.3">π</ci><apply id="S2.6.p1.7.m1.1.1.1.1.cmml" xref="S2.6.p1.7.m1.1.1.1.1"><times id="S2.6.p1.7.m1.1.1.1.1.2.cmml" xref="S2.6.p1.7.m1.1.1.1.1.2"></times><apply id="S2.6.p1.7.m1.1.1.1.1.1.1.1.cmml" xref="S2.6.p1.7.m1.1.1.1.1.1.1"><times id="S2.6.p1.7.m1.1.1.1.1.1.1.1.1.cmml" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.1"></times><apply id="S2.6.p1.7.m1.1.1.1.1.1.1.1.2.cmml" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.6.p1.7.m1.1.1.1.1.1.1.1.2.1.cmml" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S2.6.p1.7.m1.1.1.1.1.1.1.1.2.2.cmml" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.2.2">π₯</ci><cn id="S2.6.p1.7.m1.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.2.3">2</cn></apply><apply id="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.cmml" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.1.cmml" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.2.cmml" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.2">π₯</ci><apply id="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.3.cmml" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.3"><minus id="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.3.1.cmml" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.3"></minus><cn id="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S2.6.p1.7.m1.1.1.1.1.1.1.1.3.3.2">2</cn></apply></apply></apply><ci id="S2.6.p1.7.m1.1.1.1.1.3.cmml" xref="S2.6.p1.7.m1.1.1.1.1.3">π</ci></apply></apply><emptyset id="S2.6.p1.7.m1.1.1.3.cmml" xref="S2.6.p1.7.m1.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p1.7.m1.1c">V\cap(x^{2}x^{-2})V\neq\varnothing</annotation><annotation encoding="application/x-llamapun" id="S2.6.p1.7.m1.1d">italic_V β© ( italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ) italic_V β β </annotation></semantics></math> and <math alttext="V(x^{-2}x^{2})\cap V\neq\varnothing" class="ltx_Math" display="inline" id="S2.6.p1.8.m2.1"><semantics id="S2.6.p1.8.m2.1a"><mrow id="S2.6.p1.8.m2.1.1" xref="S2.6.p1.8.m2.1.1.cmml"><mrow id="S2.6.p1.8.m2.1.1.1" xref="S2.6.p1.8.m2.1.1.1.cmml"><mrow id="S2.6.p1.8.m2.1.1.1.1" xref="S2.6.p1.8.m2.1.1.1.1.cmml"><mi id="S2.6.p1.8.m2.1.1.1.1.3" xref="S2.6.p1.8.m2.1.1.1.1.3.cmml">V</mi><mo id="S2.6.p1.8.m2.1.1.1.1.2" xref="S2.6.p1.8.m2.1.1.1.1.2.cmml">β’</mo><mrow id="S2.6.p1.8.m2.1.1.1.1.1.1" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.cmml"><mo id="S2.6.p1.8.m2.1.1.1.1.1.1.2" stretchy="false" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.6.p1.8.m2.1.1.1.1.1.1.1" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.cmml"><msup id="S2.6.p1.8.m2.1.1.1.1.1.1.1.2" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.cmml"><mi id="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.2" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.2.cmml">x</mi><mrow id="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.3" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.3.cmml"><mo id="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.3a" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.3.cmml">β</mo><mn id="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.3.2" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.3.2.cmml">2</mn></mrow></msup><mo id="S2.6.p1.8.m2.1.1.1.1.1.1.1.1" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.1.cmml">β’</mo><msup id="S2.6.p1.8.m2.1.1.1.1.1.1.1.3" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.3.cmml"><mi id="S2.6.p1.8.m2.1.1.1.1.1.1.1.3.2" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.3.2.cmml">x</mi><mn id="S2.6.p1.8.m2.1.1.1.1.1.1.1.3.3" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.3.3.cmml">2</mn></msup></mrow><mo id="S2.6.p1.8.m2.1.1.1.1.1.1.3" stretchy="false" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.6.p1.8.m2.1.1.1.2" xref="S2.6.p1.8.m2.1.1.1.2.cmml">β©</mo><mi id="S2.6.p1.8.m2.1.1.1.3" xref="S2.6.p1.8.m2.1.1.1.3.cmml">V</mi></mrow><mo id="S2.6.p1.8.m2.1.1.2" xref="S2.6.p1.8.m2.1.1.2.cmml">β </mo><mi id="S2.6.p1.8.m2.1.1.3" mathvariant="normal" xref="S2.6.p1.8.m2.1.1.3.cmml">β </mi></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p1.8.m2.1b"><apply id="S2.6.p1.8.m2.1.1.cmml" xref="S2.6.p1.8.m2.1.1"><neq id="S2.6.p1.8.m2.1.1.2.cmml" xref="S2.6.p1.8.m2.1.1.2"></neq><apply id="S2.6.p1.8.m2.1.1.1.cmml" xref="S2.6.p1.8.m2.1.1.1"><intersect id="S2.6.p1.8.m2.1.1.1.2.cmml" xref="S2.6.p1.8.m2.1.1.1.2"></intersect><apply id="S2.6.p1.8.m2.1.1.1.1.cmml" xref="S2.6.p1.8.m2.1.1.1.1"><times id="S2.6.p1.8.m2.1.1.1.1.2.cmml" xref="S2.6.p1.8.m2.1.1.1.1.2"></times><ci id="S2.6.p1.8.m2.1.1.1.1.3.cmml" xref="S2.6.p1.8.m2.1.1.1.1.3">π</ci><apply id="S2.6.p1.8.m2.1.1.1.1.1.1.1.cmml" xref="S2.6.p1.8.m2.1.1.1.1.1.1"><times id="S2.6.p1.8.m2.1.1.1.1.1.1.1.1.cmml" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.1"></times><apply id="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.cmml" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.1.cmml" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.2.cmml" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.2">π₯</ci><apply id="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.3.cmml" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.3"><minus id="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.3.1.cmml" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.3"></minus><cn id="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.2.3.2">2</cn></apply></apply><apply id="S2.6.p1.8.m2.1.1.1.1.1.1.1.3.cmml" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.6.p1.8.m2.1.1.1.1.1.1.1.3.1.cmml" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="S2.6.p1.8.m2.1.1.1.1.1.1.1.3.2.cmml" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.3.2">π₯</ci><cn id="S2.6.p1.8.m2.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.6.p1.8.m2.1.1.1.1.1.1.1.3.3">2</cn></apply></apply></apply><ci id="S2.6.p1.8.m2.1.1.1.3.cmml" xref="S2.6.p1.8.m2.1.1.1.3">π</ci></apply><emptyset id="S2.6.p1.8.m2.1.1.3.cmml" xref="S2.6.p1.8.m2.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p1.8.m2.1c">V(x^{-2}x^{2})\cap V\neq\varnothing</annotation><annotation encoding="application/x-llamapun" id="S2.6.p1.8.m2.1d">italic_V ( italic_x start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) β© italic_V β β </annotation></semantics></math> for each open neighborhood <math alttext="V" class="ltx_Math" display="inline" id="S2.6.p1.9.m3.1"><semantics id="S2.6.p1.9.m3.1a"><mi id="S2.6.p1.9.m3.1.1" xref="S2.6.p1.9.m3.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S2.6.p1.9.m3.1b"><ci id="S2.6.p1.9.m3.1.1.cmml" xref="S2.6.p1.9.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p1.9.m3.1c">V</annotation><annotation encoding="application/x-llamapun" id="S2.6.p1.9.m3.1d">italic_V</annotation></semantics></math> of <math alttext="x" class="ltx_Math" display="inline" id="S2.6.p1.10.m4.1"><semantics id="S2.6.p1.10.m4.1a"><mi id="S2.6.p1.10.m4.1.1" xref="S2.6.p1.10.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.6.p1.10.m4.1b"><ci id="S2.6.p1.10.m4.1.1.cmml" xref="S2.6.p1.10.m4.1.1">π₯</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p1.10.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.6.p1.10.m4.1d">italic_x</annotation></semantics></math>. Since <math alttext="X" class="ltx_Math" display="inline" id="S2.6.p1.11.m5.1"><semantics id="S2.6.p1.11.m5.1a"><mi id="S2.6.p1.11.m5.1.1" xref="S2.6.p1.11.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.6.p1.11.m5.1b"><ci id="S2.6.p1.11.m5.1.1.cmml" xref="S2.6.p1.11.m5.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p1.11.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.6.p1.11.m5.1d">italic_X</annotation></semantics></math> is a Hausdorff semitopological semigroup, we obtain <math alttext="x^{2}x^{-2}x=x=xx^{-2}x^{2}" class="ltx_Math" display="inline" id="S2.6.p1.12.m6.1"><semantics id="S2.6.p1.12.m6.1a"><mrow id="S2.6.p1.12.m6.1.1" xref="S2.6.p1.12.m6.1.1.cmml"><mrow id="S2.6.p1.12.m6.1.1.2" xref="S2.6.p1.12.m6.1.1.2.cmml"><msup id="S2.6.p1.12.m6.1.1.2.2" xref="S2.6.p1.12.m6.1.1.2.2.cmml"><mi id="S2.6.p1.12.m6.1.1.2.2.2" xref="S2.6.p1.12.m6.1.1.2.2.2.cmml">x</mi><mn id="S2.6.p1.12.m6.1.1.2.2.3" xref="S2.6.p1.12.m6.1.1.2.2.3.cmml">2</mn></msup><mo id="S2.6.p1.12.m6.1.1.2.1" xref="S2.6.p1.12.m6.1.1.2.1.cmml">β’</mo><msup id="S2.6.p1.12.m6.1.1.2.3" xref="S2.6.p1.12.m6.1.1.2.3.cmml"><mi id="S2.6.p1.12.m6.1.1.2.3.2" xref="S2.6.p1.12.m6.1.1.2.3.2.cmml">x</mi><mrow id="S2.6.p1.12.m6.1.1.2.3.3" xref="S2.6.p1.12.m6.1.1.2.3.3.cmml"><mo id="S2.6.p1.12.m6.1.1.2.3.3a" xref="S2.6.p1.12.m6.1.1.2.3.3.cmml">β</mo><mn id="S2.6.p1.12.m6.1.1.2.3.3.2" xref="S2.6.p1.12.m6.1.1.2.3.3.2.cmml">2</mn></mrow></msup><mo id="S2.6.p1.12.m6.1.1.2.1a" xref="S2.6.p1.12.m6.1.1.2.1.cmml">β’</mo><mi id="S2.6.p1.12.m6.1.1.2.4" xref="S2.6.p1.12.m6.1.1.2.4.cmml">x</mi></mrow><mo id="S2.6.p1.12.m6.1.1.3" xref="S2.6.p1.12.m6.1.1.3.cmml">=</mo><mi id="S2.6.p1.12.m6.1.1.4" xref="S2.6.p1.12.m6.1.1.4.cmml">x</mi><mo id="S2.6.p1.12.m6.1.1.5" xref="S2.6.p1.12.m6.1.1.5.cmml">=</mo><mrow id="S2.6.p1.12.m6.1.1.6" xref="S2.6.p1.12.m6.1.1.6.cmml"><mi id="S2.6.p1.12.m6.1.1.6.2" xref="S2.6.p1.12.m6.1.1.6.2.cmml">x</mi><mo id="S2.6.p1.12.m6.1.1.6.1" xref="S2.6.p1.12.m6.1.1.6.1.cmml">β’</mo><msup id="S2.6.p1.12.m6.1.1.6.3" xref="S2.6.p1.12.m6.1.1.6.3.cmml"><mi id="S2.6.p1.12.m6.1.1.6.3.2" xref="S2.6.p1.12.m6.1.1.6.3.2.cmml">x</mi><mrow id="S2.6.p1.12.m6.1.1.6.3.3" xref="S2.6.p1.12.m6.1.1.6.3.3.cmml"><mo id="S2.6.p1.12.m6.1.1.6.3.3a" xref="S2.6.p1.12.m6.1.1.6.3.3.cmml">β</mo><mn id="S2.6.p1.12.m6.1.1.6.3.3.2" xref="S2.6.p1.12.m6.1.1.6.3.3.2.cmml">2</mn></mrow></msup><mo id="S2.6.p1.12.m6.1.1.6.1a" xref="S2.6.p1.12.m6.1.1.6.1.cmml">β’</mo><msup id="S2.6.p1.12.m6.1.1.6.4" xref="S2.6.p1.12.m6.1.1.6.4.cmml"><mi id="S2.6.p1.12.m6.1.1.6.4.2" xref="S2.6.p1.12.m6.1.1.6.4.2.cmml">x</mi><mn id="S2.6.p1.12.m6.1.1.6.4.3" xref="S2.6.p1.12.m6.1.1.6.4.3.cmml">2</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p1.12.m6.1b"><apply id="S2.6.p1.12.m6.1.1.cmml" xref="S2.6.p1.12.m6.1.1"><and id="S2.6.p1.12.m6.1.1a.cmml" xref="S2.6.p1.12.m6.1.1"></and><apply id="S2.6.p1.12.m6.1.1b.cmml" xref="S2.6.p1.12.m6.1.1"><eq id="S2.6.p1.12.m6.1.1.3.cmml" xref="S2.6.p1.12.m6.1.1.3"></eq><apply id="S2.6.p1.12.m6.1.1.2.cmml" xref="S2.6.p1.12.m6.1.1.2"><times id="S2.6.p1.12.m6.1.1.2.1.cmml" xref="S2.6.p1.12.m6.1.1.2.1"></times><apply id="S2.6.p1.12.m6.1.1.2.2.cmml" xref="S2.6.p1.12.m6.1.1.2.2"><csymbol cd="ambiguous" id="S2.6.p1.12.m6.1.1.2.2.1.cmml" xref="S2.6.p1.12.m6.1.1.2.2">superscript</csymbol><ci id="S2.6.p1.12.m6.1.1.2.2.2.cmml" xref="S2.6.p1.12.m6.1.1.2.2.2">π₯</ci><cn id="S2.6.p1.12.m6.1.1.2.2.3.cmml" type="integer" xref="S2.6.p1.12.m6.1.1.2.2.3">2</cn></apply><apply id="S2.6.p1.12.m6.1.1.2.3.cmml" xref="S2.6.p1.12.m6.1.1.2.3"><csymbol cd="ambiguous" id="S2.6.p1.12.m6.1.1.2.3.1.cmml" xref="S2.6.p1.12.m6.1.1.2.3">superscript</csymbol><ci id="S2.6.p1.12.m6.1.1.2.3.2.cmml" xref="S2.6.p1.12.m6.1.1.2.3.2">π₯</ci><apply id="S2.6.p1.12.m6.1.1.2.3.3.cmml" xref="S2.6.p1.12.m6.1.1.2.3.3"><minus id="S2.6.p1.12.m6.1.1.2.3.3.1.cmml" xref="S2.6.p1.12.m6.1.1.2.3.3"></minus><cn id="S2.6.p1.12.m6.1.1.2.3.3.2.cmml" type="integer" xref="S2.6.p1.12.m6.1.1.2.3.3.2">2</cn></apply></apply><ci id="S2.6.p1.12.m6.1.1.2.4.cmml" xref="S2.6.p1.12.m6.1.1.2.4">π₯</ci></apply><ci id="S2.6.p1.12.m6.1.1.4.cmml" xref="S2.6.p1.12.m6.1.1.4">π₯</ci></apply><apply id="S2.6.p1.12.m6.1.1c.cmml" xref="S2.6.p1.12.m6.1.1"><eq id="S2.6.p1.12.m6.1.1.5.cmml" xref="S2.6.p1.12.m6.1.1.5"></eq><share href="https://arxiv.org/html/2503.13666v1#S2.6.p1.12.m6.1.1.4.cmml" id="S2.6.p1.12.m6.1.1d.cmml" xref="S2.6.p1.12.m6.1.1"></share><apply id="S2.6.p1.12.m6.1.1.6.cmml" xref="S2.6.p1.12.m6.1.1.6"><times id="S2.6.p1.12.m6.1.1.6.1.cmml" xref="S2.6.p1.12.m6.1.1.6.1"></times><ci id="S2.6.p1.12.m6.1.1.6.2.cmml" xref="S2.6.p1.12.m6.1.1.6.2">π₯</ci><apply id="S2.6.p1.12.m6.1.1.6.3.cmml" xref="S2.6.p1.12.m6.1.1.6.3"><csymbol cd="ambiguous" id="S2.6.p1.12.m6.1.1.6.3.1.cmml" xref="S2.6.p1.12.m6.1.1.6.3">superscript</csymbol><ci id="S2.6.p1.12.m6.1.1.6.3.2.cmml" xref="S2.6.p1.12.m6.1.1.6.3.2">π₯</ci><apply id="S2.6.p1.12.m6.1.1.6.3.3.cmml" xref="S2.6.p1.12.m6.1.1.6.3.3"><minus id="S2.6.p1.12.m6.1.1.6.3.3.1.cmml" xref="S2.6.p1.12.m6.1.1.6.3.3"></minus><cn id="S2.6.p1.12.m6.1.1.6.3.3.2.cmml" type="integer" xref="S2.6.p1.12.m6.1.1.6.3.3.2">2</cn></apply></apply><apply id="S2.6.p1.12.m6.1.1.6.4.cmml" xref="S2.6.p1.12.m6.1.1.6.4"><csymbol cd="ambiguous" id="S2.6.p1.12.m6.1.1.6.4.1.cmml" xref="S2.6.p1.12.m6.1.1.6.4">superscript</csymbol><ci id="S2.6.p1.12.m6.1.1.6.4.2.cmml" xref="S2.6.p1.12.m6.1.1.6.4.2">π₯</ci><cn id="S2.6.p1.12.m6.1.1.6.4.3.cmml" type="integer" xref="S2.6.p1.12.m6.1.1.6.4.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p1.12.m6.1c">x^{2}x^{-2}x=x=xx^{-2}x^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.6.p1.12.m6.1d">italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT italic_x = italic_x = italic_x italic_x start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. It follows that <math alttext="x^{-1}x^{2}x^{-2}=x^{-1}=x^{-2}x^{2}x^{-1}" class="ltx_Math" display="inline" id="S2.6.p1.13.m7.1"><semantics id="S2.6.p1.13.m7.1a"><mrow id="S2.6.p1.13.m7.1.1" xref="S2.6.p1.13.m7.1.1.cmml"><mrow id="S2.6.p1.13.m7.1.1.2" xref="S2.6.p1.13.m7.1.1.2.cmml"><msup id="S2.6.p1.13.m7.1.1.2.2" xref="S2.6.p1.13.m7.1.1.2.2.cmml"><mi id="S2.6.p1.13.m7.1.1.2.2.2" xref="S2.6.p1.13.m7.1.1.2.2.2.cmml">x</mi><mrow id="S2.6.p1.13.m7.1.1.2.2.3" xref="S2.6.p1.13.m7.1.1.2.2.3.cmml"><mo id="S2.6.p1.13.m7.1.1.2.2.3a" xref="S2.6.p1.13.m7.1.1.2.2.3.cmml">β</mo><mn id="S2.6.p1.13.m7.1.1.2.2.3.2" xref="S2.6.p1.13.m7.1.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S2.6.p1.13.m7.1.1.2.1" xref="S2.6.p1.13.m7.1.1.2.1.cmml">β’</mo><msup id="S2.6.p1.13.m7.1.1.2.3" xref="S2.6.p1.13.m7.1.1.2.3.cmml"><mi id="S2.6.p1.13.m7.1.1.2.3.2" xref="S2.6.p1.13.m7.1.1.2.3.2.cmml">x</mi><mn id="S2.6.p1.13.m7.1.1.2.3.3" xref="S2.6.p1.13.m7.1.1.2.3.3.cmml">2</mn></msup><mo id="S2.6.p1.13.m7.1.1.2.1a" xref="S2.6.p1.13.m7.1.1.2.1.cmml">β’</mo><msup id="S2.6.p1.13.m7.1.1.2.4" xref="S2.6.p1.13.m7.1.1.2.4.cmml"><mi id="S2.6.p1.13.m7.1.1.2.4.2" xref="S2.6.p1.13.m7.1.1.2.4.2.cmml">x</mi><mrow id="S2.6.p1.13.m7.1.1.2.4.3" xref="S2.6.p1.13.m7.1.1.2.4.3.cmml"><mo id="S2.6.p1.13.m7.1.1.2.4.3a" xref="S2.6.p1.13.m7.1.1.2.4.3.cmml">β</mo><mn id="S2.6.p1.13.m7.1.1.2.4.3.2" xref="S2.6.p1.13.m7.1.1.2.4.3.2.cmml">2</mn></mrow></msup></mrow><mo id="S2.6.p1.13.m7.1.1.3" xref="S2.6.p1.13.m7.1.1.3.cmml">=</mo><msup id="S2.6.p1.13.m7.1.1.4" xref="S2.6.p1.13.m7.1.1.4.cmml"><mi id="S2.6.p1.13.m7.1.1.4.2" xref="S2.6.p1.13.m7.1.1.4.2.cmml">x</mi><mrow id="S2.6.p1.13.m7.1.1.4.3" xref="S2.6.p1.13.m7.1.1.4.3.cmml"><mo id="S2.6.p1.13.m7.1.1.4.3a" xref="S2.6.p1.13.m7.1.1.4.3.cmml">β</mo><mn id="S2.6.p1.13.m7.1.1.4.3.2" xref="S2.6.p1.13.m7.1.1.4.3.2.cmml">1</mn></mrow></msup><mo id="S2.6.p1.13.m7.1.1.5" xref="S2.6.p1.13.m7.1.1.5.cmml">=</mo><mrow id="S2.6.p1.13.m7.1.1.6" xref="S2.6.p1.13.m7.1.1.6.cmml"><msup id="S2.6.p1.13.m7.1.1.6.2" xref="S2.6.p1.13.m7.1.1.6.2.cmml"><mi id="S2.6.p1.13.m7.1.1.6.2.2" xref="S2.6.p1.13.m7.1.1.6.2.2.cmml">x</mi><mrow id="S2.6.p1.13.m7.1.1.6.2.3" xref="S2.6.p1.13.m7.1.1.6.2.3.cmml"><mo id="S2.6.p1.13.m7.1.1.6.2.3a" xref="S2.6.p1.13.m7.1.1.6.2.3.cmml">β</mo><mn id="S2.6.p1.13.m7.1.1.6.2.3.2" xref="S2.6.p1.13.m7.1.1.6.2.3.2.cmml">2</mn></mrow></msup><mo id="S2.6.p1.13.m7.1.1.6.1" xref="S2.6.p1.13.m7.1.1.6.1.cmml">β’</mo><msup id="S2.6.p1.13.m7.1.1.6.3" xref="S2.6.p1.13.m7.1.1.6.3.cmml"><mi id="S2.6.p1.13.m7.1.1.6.3.2" xref="S2.6.p1.13.m7.1.1.6.3.2.cmml">x</mi><mn id="S2.6.p1.13.m7.1.1.6.3.3" xref="S2.6.p1.13.m7.1.1.6.3.3.cmml">2</mn></msup><mo id="S2.6.p1.13.m7.1.1.6.1a" xref="S2.6.p1.13.m7.1.1.6.1.cmml">β’</mo><msup id="S2.6.p1.13.m7.1.1.6.4" xref="S2.6.p1.13.m7.1.1.6.4.cmml"><mi id="S2.6.p1.13.m7.1.1.6.4.2" xref="S2.6.p1.13.m7.1.1.6.4.2.cmml">x</mi><mrow id="S2.6.p1.13.m7.1.1.6.4.3" xref="S2.6.p1.13.m7.1.1.6.4.3.cmml"><mo id="S2.6.p1.13.m7.1.1.6.4.3a" xref="S2.6.p1.13.m7.1.1.6.4.3.cmml">β</mo><mn id="S2.6.p1.13.m7.1.1.6.4.3.2" xref="S2.6.p1.13.m7.1.1.6.4.3.2.cmml">1</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p1.13.m7.1b"><apply id="S2.6.p1.13.m7.1.1.cmml" xref="S2.6.p1.13.m7.1.1"><and id="S2.6.p1.13.m7.1.1a.cmml" xref="S2.6.p1.13.m7.1.1"></and><apply id="S2.6.p1.13.m7.1.1b.cmml" xref="S2.6.p1.13.m7.1.1"><eq id="S2.6.p1.13.m7.1.1.3.cmml" xref="S2.6.p1.13.m7.1.1.3"></eq><apply id="S2.6.p1.13.m7.1.1.2.cmml" xref="S2.6.p1.13.m7.1.1.2"><times id="S2.6.p1.13.m7.1.1.2.1.cmml" xref="S2.6.p1.13.m7.1.1.2.1"></times><apply id="S2.6.p1.13.m7.1.1.2.2.cmml" xref="S2.6.p1.13.m7.1.1.2.2"><csymbol cd="ambiguous" id="S2.6.p1.13.m7.1.1.2.2.1.cmml" xref="S2.6.p1.13.m7.1.1.2.2">superscript</csymbol><ci id="S2.6.p1.13.m7.1.1.2.2.2.cmml" xref="S2.6.p1.13.m7.1.1.2.2.2">π₯</ci><apply id="S2.6.p1.13.m7.1.1.2.2.3.cmml" xref="S2.6.p1.13.m7.1.1.2.2.3"><minus id="S2.6.p1.13.m7.1.1.2.2.3.1.cmml" xref="S2.6.p1.13.m7.1.1.2.2.3"></minus><cn id="S2.6.p1.13.m7.1.1.2.2.3.2.cmml" type="integer" xref="S2.6.p1.13.m7.1.1.2.2.3.2">1</cn></apply></apply><apply id="S2.6.p1.13.m7.1.1.2.3.cmml" xref="S2.6.p1.13.m7.1.1.2.3"><csymbol cd="ambiguous" id="S2.6.p1.13.m7.1.1.2.3.1.cmml" xref="S2.6.p1.13.m7.1.1.2.3">superscript</csymbol><ci id="S2.6.p1.13.m7.1.1.2.3.2.cmml" xref="S2.6.p1.13.m7.1.1.2.3.2">π₯</ci><cn id="S2.6.p1.13.m7.1.1.2.3.3.cmml" type="integer" xref="S2.6.p1.13.m7.1.1.2.3.3">2</cn></apply><apply id="S2.6.p1.13.m7.1.1.2.4.cmml" xref="S2.6.p1.13.m7.1.1.2.4"><csymbol cd="ambiguous" id="S2.6.p1.13.m7.1.1.2.4.1.cmml" xref="S2.6.p1.13.m7.1.1.2.4">superscript</csymbol><ci id="S2.6.p1.13.m7.1.1.2.4.2.cmml" xref="S2.6.p1.13.m7.1.1.2.4.2">π₯</ci><apply id="S2.6.p1.13.m7.1.1.2.4.3.cmml" xref="S2.6.p1.13.m7.1.1.2.4.3"><minus id="S2.6.p1.13.m7.1.1.2.4.3.1.cmml" xref="S2.6.p1.13.m7.1.1.2.4.3"></minus><cn id="S2.6.p1.13.m7.1.1.2.4.3.2.cmml" type="integer" xref="S2.6.p1.13.m7.1.1.2.4.3.2">2</cn></apply></apply></apply><apply id="S2.6.p1.13.m7.1.1.4.cmml" xref="S2.6.p1.13.m7.1.1.4"><csymbol cd="ambiguous" id="S2.6.p1.13.m7.1.1.4.1.cmml" xref="S2.6.p1.13.m7.1.1.4">superscript</csymbol><ci id="S2.6.p1.13.m7.1.1.4.2.cmml" xref="S2.6.p1.13.m7.1.1.4.2">π₯</ci><apply id="S2.6.p1.13.m7.1.1.4.3.cmml" xref="S2.6.p1.13.m7.1.1.4.3"><minus id="S2.6.p1.13.m7.1.1.4.3.1.cmml" xref="S2.6.p1.13.m7.1.1.4.3"></minus><cn id="S2.6.p1.13.m7.1.1.4.3.2.cmml" type="integer" xref="S2.6.p1.13.m7.1.1.4.3.2">1</cn></apply></apply></apply><apply id="S2.6.p1.13.m7.1.1c.cmml" xref="S2.6.p1.13.m7.1.1"><eq id="S2.6.p1.13.m7.1.1.5.cmml" xref="S2.6.p1.13.m7.1.1.5"></eq><share href="https://arxiv.org/html/2503.13666v1#S2.6.p1.13.m7.1.1.4.cmml" id="S2.6.p1.13.m7.1.1d.cmml" xref="S2.6.p1.13.m7.1.1"></share><apply id="S2.6.p1.13.m7.1.1.6.cmml" xref="S2.6.p1.13.m7.1.1.6"><times id="S2.6.p1.13.m7.1.1.6.1.cmml" xref="S2.6.p1.13.m7.1.1.6.1"></times><apply id="S2.6.p1.13.m7.1.1.6.2.cmml" xref="S2.6.p1.13.m7.1.1.6.2"><csymbol cd="ambiguous" id="S2.6.p1.13.m7.1.1.6.2.1.cmml" xref="S2.6.p1.13.m7.1.1.6.2">superscript</csymbol><ci id="S2.6.p1.13.m7.1.1.6.2.2.cmml" xref="S2.6.p1.13.m7.1.1.6.2.2">π₯</ci><apply id="S2.6.p1.13.m7.1.1.6.2.3.cmml" xref="S2.6.p1.13.m7.1.1.6.2.3"><minus id="S2.6.p1.13.m7.1.1.6.2.3.1.cmml" xref="S2.6.p1.13.m7.1.1.6.2.3"></minus><cn id="S2.6.p1.13.m7.1.1.6.2.3.2.cmml" type="integer" xref="S2.6.p1.13.m7.1.1.6.2.3.2">2</cn></apply></apply><apply id="S2.6.p1.13.m7.1.1.6.3.cmml" xref="S2.6.p1.13.m7.1.1.6.3"><csymbol cd="ambiguous" id="S2.6.p1.13.m7.1.1.6.3.1.cmml" xref="S2.6.p1.13.m7.1.1.6.3">superscript</csymbol><ci id="S2.6.p1.13.m7.1.1.6.3.2.cmml" xref="S2.6.p1.13.m7.1.1.6.3.2">π₯</ci><cn id="S2.6.p1.13.m7.1.1.6.3.3.cmml" type="integer" xref="S2.6.p1.13.m7.1.1.6.3.3">2</cn></apply><apply id="S2.6.p1.13.m7.1.1.6.4.cmml" xref="S2.6.p1.13.m7.1.1.6.4"><csymbol cd="ambiguous" id="S2.6.p1.13.m7.1.1.6.4.1.cmml" xref="S2.6.p1.13.m7.1.1.6.4">superscript</csymbol><ci id="S2.6.p1.13.m7.1.1.6.4.2.cmml" xref="S2.6.p1.13.m7.1.1.6.4.2">π₯</ci><apply id="S2.6.p1.13.m7.1.1.6.4.3.cmml" xref="S2.6.p1.13.m7.1.1.6.4.3"><minus id="S2.6.p1.13.m7.1.1.6.4.3.1.cmml" xref="S2.6.p1.13.m7.1.1.6.4.3"></minus><cn id="S2.6.p1.13.m7.1.1.6.4.3.2.cmml" type="integer" xref="S2.6.p1.13.m7.1.1.6.4.3.2">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p1.13.m7.1c">x^{-1}x^{2}x^{-2}=x^{-1}=x^{-2}x^{2}x^{-1}</annotation><annotation encoding="application/x-llamapun" id="S2.6.p1.13.m7.1d">italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT = italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_x start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. Then</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="x^{-1}x=x^{-1}x^{2}x^{-2}x^{2}x^{-2}x=x^{-1}x^{2}x^{-2}x=(x^{-1}x)(xx^{-1})(x^% {-1}x)=(xx^{-1})(x^{-1}x)." class="ltx_Math" display="block" id="S2.Ex7.m1.1"><semantics id="S2.Ex7.m1.1a"><mrow id="S2.Ex7.m1.1.1.1" xref="S2.Ex7.m1.1.1.1.1.cmml"><mrow id="S2.Ex7.m1.1.1.1.1" xref="S2.Ex7.m1.1.1.1.1.cmml"><mrow id="S2.Ex7.m1.1.1.1.1.7" xref="S2.Ex7.m1.1.1.1.1.7.cmml"><msup id="S2.Ex7.m1.1.1.1.1.7.2" xref="S2.Ex7.m1.1.1.1.1.7.2.cmml"><mi id="S2.Ex7.m1.1.1.1.1.7.2.2" xref="S2.Ex7.m1.1.1.1.1.7.2.2.cmml">x</mi><mrow id="S2.Ex7.m1.1.1.1.1.7.2.3" xref="S2.Ex7.m1.1.1.1.1.7.2.3.cmml"><mo id="S2.Ex7.m1.1.1.1.1.7.2.3a" xref="S2.Ex7.m1.1.1.1.1.7.2.3.cmml">β</mo><mn id="S2.Ex7.m1.1.1.1.1.7.2.3.2" xref="S2.Ex7.m1.1.1.1.1.7.2.3.2.cmml">1</mn></mrow></msup><mo 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id="S2.Ex7.m1.1.1.1.1.5.2.1.1.3.cmml" xref="S2.Ex7.m1.1.1.1.1.5.2.1.1.3">π₯</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex7.m1.1c">x^{-1}x=x^{-1}x^{2}x^{-2}x^{2}x^{-2}x=x^{-1}x^{2}x^{-2}x=(x^{-1}x)(xx^{-1})(x^% {-1}x)=(xx^{-1})(x^{-1}x).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex7.m1.1d">italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x = italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT italic_x = italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT italic_x = ( italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x ) ( italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) ( italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x ) = ( italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) ( italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.6.p1.15">In the formula above, the second equality follows from the fact that <math alttext="x^{2}x^{-2}" class="ltx_Math" display="inline" id="S2.6.p1.14.m1.1"><semantics id="S2.6.p1.14.m1.1a"><mrow id="S2.6.p1.14.m1.1.1" xref="S2.6.p1.14.m1.1.1.cmml"><msup id="S2.6.p1.14.m1.1.1.2" xref="S2.6.p1.14.m1.1.1.2.cmml"><mi id="S2.6.p1.14.m1.1.1.2.2" xref="S2.6.p1.14.m1.1.1.2.2.cmml">x</mi><mn id="S2.6.p1.14.m1.1.1.2.3" xref="S2.6.p1.14.m1.1.1.2.3.cmml">2</mn></msup><mo id="S2.6.p1.14.m1.1.1.1" xref="S2.6.p1.14.m1.1.1.1.cmml">β’</mo><msup id="S2.6.p1.14.m1.1.1.3" xref="S2.6.p1.14.m1.1.1.3.cmml"><mi id="S2.6.p1.14.m1.1.1.3.2" xref="S2.6.p1.14.m1.1.1.3.2.cmml">x</mi><mrow id="S2.6.p1.14.m1.1.1.3.3" xref="S2.6.p1.14.m1.1.1.3.3.cmml"><mo id="S2.6.p1.14.m1.1.1.3.3a" xref="S2.6.p1.14.m1.1.1.3.3.cmml">β</mo><mn id="S2.6.p1.14.m1.1.1.3.3.2" xref="S2.6.p1.14.m1.1.1.3.3.2.cmml">2</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p1.14.m1.1b"><apply id="S2.6.p1.14.m1.1.1.cmml" xref="S2.6.p1.14.m1.1.1"><times id="S2.6.p1.14.m1.1.1.1.cmml" xref="S2.6.p1.14.m1.1.1.1"></times><apply id="S2.6.p1.14.m1.1.1.2.cmml" xref="S2.6.p1.14.m1.1.1.2"><csymbol cd="ambiguous" id="S2.6.p1.14.m1.1.1.2.1.cmml" xref="S2.6.p1.14.m1.1.1.2">superscript</csymbol><ci id="S2.6.p1.14.m1.1.1.2.2.cmml" xref="S2.6.p1.14.m1.1.1.2.2">π₯</ci><cn id="S2.6.p1.14.m1.1.1.2.3.cmml" type="integer" xref="S2.6.p1.14.m1.1.1.2.3">2</cn></apply><apply id="S2.6.p1.14.m1.1.1.3.cmml" xref="S2.6.p1.14.m1.1.1.3"><csymbol cd="ambiguous" id="S2.6.p1.14.m1.1.1.3.1.cmml" xref="S2.6.p1.14.m1.1.1.3">superscript</csymbol><ci id="S2.6.p1.14.m1.1.1.3.2.cmml" xref="S2.6.p1.14.m1.1.1.3.2">π₯</ci><apply id="S2.6.p1.14.m1.1.1.3.3.cmml" xref="S2.6.p1.14.m1.1.1.3.3"><minus id="S2.6.p1.14.m1.1.1.3.3.1.cmml" xref="S2.6.p1.14.m1.1.1.3.3"></minus><cn id="S2.6.p1.14.m1.1.1.3.3.2.cmml" type="integer" xref="S2.6.p1.14.m1.1.1.3.3.2">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p1.14.m1.1c">x^{2}x^{-2}</annotation><annotation encoding="application/x-llamapun" id="S2.6.p1.14.m1.1d">italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math> is an idempotent, whereas the last equality holds as idempotents commute in <math alttext="S" class="ltx_Math" display="inline" id="S2.6.p1.15.m2.1"><semantics id="S2.6.p1.15.m2.1a"><mi id="S2.6.p1.15.m2.1.1" xref="S2.6.p1.15.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.6.p1.15.m2.1b"><ci id="S2.6.p1.15.m2.1.1.cmml" xref="S2.6.p1.15.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p1.15.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.6.p1.15.m2.1d">italic_S</annotation></semantics></math>. Similarly</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="xx^{-1}=xx^{-2}x^{2}x^{-2}x^{2}x^{-1}=xx^{-2}x^{2}x^{-1}=(xx^{-1})(x^{-1}x)(xx% ^{-1})=(xx^{-1})(x^{-1}x)." class="ltx_Math" display="block" id="S2.Ex8.m1.1"><semantics id="S2.Ex8.m1.1a"><mrow id="S2.Ex8.m1.1.1.1" xref="S2.Ex8.m1.1.1.1.1.cmml"><mrow id="S2.Ex8.m1.1.1.1.1" xref="S2.Ex8.m1.1.1.1.1.cmml"><mrow id="S2.Ex8.m1.1.1.1.1.7" xref="S2.Ex8.m1.1.1.1.1.7.cmml"><mi id="S2.Ex8.m1.1.1.1.1.7.2" xref="S2.Ex8.m1.1.1.1.1.7.2.cmml">x</mi><mo id="S2.Ex8.m1.1.1.1.1.7.1" xref="S2.Ex8.m1.1.1.1.1.7.1.cmml">β’</mo><msup id="S2.Ex8.m1.1.1.1.1.7.3" xref="S2.Ex8.m1.1.1.1.1.7.3.cmml"><mi id="S2.Ex8.m1.1.1.1.1.7.3.2" xref="S2.Ex8.m1.1.1.1.1.7.3.2.cmml">x</mi><mrow id="S2.Ex8.m1.1.1.1.1.7.3.3" xref="S2.Ex8.m1.1.1.1.1.7.3.3.cmml"><mo id="S2.Ex8.m1.1.1.1.1.7.3.3a" 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xref="S2.Ex8.m1.1.1.1.1.5.2.1.1.3">π₯</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex8.m1.1c">xx^{-1}=xx^{-2}x^{2}x^{-2}x^{2}x^{-1}=xx^{-2}x^{2}x^{-1}=(xx^{-1})(x^{-1}x)(xx% ^{-1})=(xx^{-1})(x^{-1}x).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex8.m1.1d">italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_x italic_x start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_x italic_x start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = ( italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) ( italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x ) ( italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) = ( italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) ( italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.6.p1.17">Hence <math alttext="xx^{-1}=x^{-1}x" class="ltx_Math" display="inline" id="S2.6.p1.16.m1.1"><semantics id="S2.6.p1.16.m1.1a"><mrow id="S2.6.p1.16.m1.1.1" xref="S2.6.p1.16.m1.1.1.cmml"><mrow id="S2.6.p1.16.m1.1.1.2" xref="S2.6.p1.16.m1.1.1.2.cmml"><mi id="S2.6.p1.16.m1.1.1.2.2" xref="S2.6.p1.16.m1.1.1.2.2.cmml">x</mi><mo id="S2.6.p1.16.m1.1.1.2.1" xref="S2.6.p1.16.m1.1.1.2.1.cmml">β’</mo><msup id="S2.6.p1.16.m1.1.1.2.3" xref="S2.6.p1.16.m1.1.1.2.3.cmml"><mi id="S2.6.p1.16.m1.1.1.2.3.2" xref="S2.6.p1.16.m1.1.1.2.3.2.cmml">x</mi><mrow id="S2.6.p1.16.m1.1.1.2.3.3" xref="S2.6.p1.16.m1.1.1.2.3.3.cmml"><mo id="S2.6.p1.16.m1.1.1.2.3.3a" xref="S2.6.p1.16.m1.1.1.2.3.3.cmml">β</mo><mn id="S2.6.p1.16.m1.1.1.2.3.3.2" xref="S2.6.p1.16.m1.1.1.2.3.3.2.cmml">1</mn></mrow></msup></mrow><mo id="S2.6.p1.16.m1.1.1.1" xref="S2.6.p1.16.m1.1.1.1.cmml">=</mo><mrow id="S2.6.p1.16.m1.1.1.3" xref="S2.6.p1.16.m1.1.1.3.cmml"><msup id="S2.6.p1.16.m1.1.1.3.2" xref="S2.6.p1.16.m1.1.1.3.2.cmml"><mi id="S2.6.p1.16.m1.1.1.3.2.2" xref="S2.6.p1.16.m1.1.1.3.2.2.cmml">x</mi><mrow id="S2.6.p1.16.m1.1.1.3.2.3" xref="S2.6.p1.16.m1.1.1.3.2.3.cmml"><mo id="S2.6.p1.16.m1.1.1.3.2.3a" xref="S2.6.p1.16.m1.1.1.3.2.3.cmml">β</mo><mn id="S2.6.p1.16.m1.1.1.3.2.3.2" xref="S2.6.p1.16.m1.1.1.3.2.3.2.cmml">1</mn></mrow></msup><mo id="S2.6.p1.16.m1.1.1.3.1" xref="S2.6.p1.16.m1.1.1.3.1.cmml">β’</mo><mi id="S2.6.p1.16.m1.1.1.3.3" xref="S2.6.p1.16.m1.1.1.3.3.cmml">x</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.6.p1.16.m1.1b"><apply id="S2.6.p1.16.m1.1.1.cmml" xref="S2.6.p1.16.m1.1.1"><eq id="S2.6.p1.16.m1.1.1.1.cmml" xref="S2.6.p1.16.m1.1.1.1"></eq><apply id="S2.6.p1.16.m1.1.1.2.cmml" xref="S2.6.p1.16.m1.1.1.2"><times id="S2.6.p1.16.m1.1.1.2.1.cmml" xref="S2.6.p1.16.m1.1.1.2.1"></times><ci id="S2.6.p1.16.m1.1.1.2.2.cmml" xref="S2.6.p1.16.m1.1.1.2.2">π₯</ci><apply id="S2.6.p1.16.m1.1.1.2.3.cmml" xref="S2.6.p1.16.m1.1.1.2.3"><csymbol cd="ambiguous" id="S2.6.p1.16.m1.1.1.2.3.1.cmml" xref="S2.6.p1.16.m1.1.1.2.3">superscript</csymbol><ci id="S2.6.p1.16.m1.1.1.2.3.2.cmml" xref="S2.6.p1.16.m1.1.1.2.3.2">π₯</ci><apply id="S2.6.p1.16.m1.1.1.2.3.3.cmml" xref="S2.6.p1.16.m1.1.1.2.3.3"><minus id="S2.6.p1.16.m1.1.1.2.3.3.1.cmml" xref="S2.6.p1.16.m1.1.1.2.3.3"></minus><cn id="S2.6.p1.16.m1.1.1.2.3.3.2.cmml" type="integer" xref="S2.6.p1.16.m1.1.1.2.3.3.2">1</cn></apply></apply></apply><apply id="S2.6.p1.16.m1.1.1.3.cmml" xref="S2.6.p1.16.m1.1.1.3"><times id="S2.6.p1.16.m1.1.1.3.1.cmml" xref="S2.6.p1.16.m1.1.1.3.1"></times><apply id="S2.6.p1.16.m1.1.1.3.2.cmml" xref="S2.6.p1.16.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.6.p1.16.m1.1.1.3.2.1.cmml" xref="S2.6.p1.16.m1.1.1.3.2">superscript</csymbol><ci id="S2.6.p1.16.m1.1.1.3.2.2.cmml" xref="S2.6.p1.16.m1.1.1.3.2.2">π₯</ci><apply id="S2.6.p1.16.m1.1.1.3.2.3.cmml" xref="S2.6.p1.16.m1.1.1.3.2.3"><minus id="S2.6.p1.16.m1.1.1.3.2.3.1.cmml" xref="S2.6.p1.16.m1.1.1.3.2.3"></minus><cn id="S2.6.p1.16.m1.1.1.3.2.3.2.cmml" type="integer" xref="S2.6.p1.16.m1.1.1.3.2.3.2">1</cn></apply></apply><ci id="S2.6.p1.16.m1.1.1.3.3.cmml" xref="S2.6.p1.16.m1.1.1.3.3">π₯</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p1.16.m1.1c">xx^{-1}=x^{-1}x</annotation><annotation encoding="application/x-llamapun" id="S2.6.p1.16.m1.1d">italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x</annotation></semantics></math> and, as such, the semigroup <math alttext="S" class="ltx_Math" display="inline" id="S2.6.p1.17.m2.1"><semantics id="S2.6.p1.17.m2.1a"><mi id="S2.6.p1.17.m2.1.1" xref="S2.6.p1.17.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.6.p1.17.m2.1b"><ci id="S2.6.p1.17.m2.1.1.cmml" xref="S2.6.p1.17.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.6.p1.17.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.6.p1.17.m2.1d">italic_S</annotation></semantics></math> is Clifford. β</p> </div> </div> <div class="ltx_para" id="S2.p9"> <p class="ltx_p" id="S2.p9.2">By <math alttext="\beta X" class="ltx_Math" display="inline" id="S2.p9.1.m1.1"><semantics id="S2.p9.1.m1.1a"><mrow id="S2.p9.1.m1.1.1" xref="S2.p9.1.m1.1.1.cmml"><mi id="S2.p9.1.m1.1.1.2" xref="S2.p9.1.m1.1.1.2.cmml">Ξ²</mi><mo id="S2.p9.1.m1.1.1.1" xref="S2.p9.1.m1.1.1.1.cmml">β’</mo><mi id="S2.p9.1.m1.1.1.3" xref="S2.p9.1.m1.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p9.1.m1.1b"><apply id="S2.p9.1.m1.1.1.cmml" xref="S2.p9.1.m1.1.1"><times id="S2.p9.1.m1.1.1.1.cmml" xref="S2.p9.1.m1.1.1.1"></times><ci id="S2.p9.1.m1.1.1.2.cmml" xref="S2.p9.1.m1.1.1.2">π½</ci><ci id="S2.p9.1.m1.1.1.3.cmml" xref="S2.p9.1.m1.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p9.1.m1.1c">\beta X</annotation><annotation encoding="application/x-llamapun" id="S2.p9.1.m1.1d">italic_Ξ² italic_X</annotation></semantics></math> we denote the Stone-Δech compactification of a Tychonoff space <math alttext="X" class="ltx_Math" display="inline" id="S2.p9.2.m2.1"><semantics id="S2.p9.2.m2.1a"><mi id="S2.p9.2.m2.1.1" xref="S2.p9.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.p9.2.m2.1b"><ci id="S2.p9.2.m2.1.1.cmml" xref="S2.p9.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p9.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.p9.2.m2.1d">italic_X</annotation></semantics></math> (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib22" title="">22</a>, Chapter 3.6]</cite>). The following result appears in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib3" title="">3</a>, Theorem 2.3]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S2.Thmtheorem9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem9.1.1.1">Theorem 2.9</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem9.2.2"> </span>(Banakh, Dimitrova, Gutik)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem9.3.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem9.p1"> <p class="ltx_p" id="S2.Thmtheorem9.p1.4"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem9.p1.4.4">Let <math alttext="S" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.1.1.m1.1"><semantics id="S2.Thmtheorem9.p1.1.1.m1.1a"><mi id="S2.Thmtheorem9.p1.1.1.m1.1.1" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.1.1.m1.1b"><ci id="S2.Thmtheorem9.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.1.1.m1.1d">italic_S</annotation></semantics></math> be a Tychonoff topological semigroup such that <math alttext="S{\times}S" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.2.2.m2.1"><semantics id="S2.Thmtheorem9.p1.2.2.m2.1a"><mrow id="S2.Thmtheorem9.p1.2.2.m2.1.1" xref="S2.Thmtheorem9.p1.2.2.m2.1.1.cmml"><mi id="S2.Thmtheorem9.p1.2.2.m2.1.1.2" xref="S2.Thmtheorem9.p1.2.2.m2.1.1.2.cmml">S</mi><mo id="S2.Thmtheorem9.p1.2.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.Thmtheorem9.p1.2.2.m2.1.1.1.cmml">Γ</mo><mi id="S2.Thmtheorem9.p1.2.2.m2.1.1.3" xref="S2.Thmtheorem9.p1.2.2.m2.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.2.2.m2.1b"><apply id="S2.Thmtheorem9.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem9.p1.2.2.m2.1.1"><times id="S2.Thmtheorem9.p1.2.2.m2.1.1.1.cmml" xref="S2.Thmtheorem9.p1.2.2.m2.1.1.1"></times><ci id="S2.Thmtheorem9.p1.2.2.m2.1.1.2.cmml" xref="S2.Thmtheorem9.p1.2.2.m2.1.1.2">π</ci><ci id="S2.Thmtheorem9.p1.2.2.m2.1.1.3.cmml" xref="S2.Thmtheorem9.p1.2.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.2.2.m2.1c">S{\times}S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.2.2.m2.1d">italic_S Γ italic_S</annotation></semantics></math> is pseudocompact. Then the semigroup operation of <math alttext="S" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.3.3.m3.1"><semantics id="S2.Thmtheorem9.p1.3.3.m3.1a"><mi id="S2.Thmtheorem9.p1.3.3.m3.1.1" xref="S2.Thmtheorem9.p1.3.3.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.3.3.m3.1b"><ci id="S2.Thmtheorem9.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.3.3.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.3.3.m3.1d">italic_S</annotation></semantics></math> extends to a continuous semigroup operation on <math alttext="\beta S" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.4.4.m4.1"><semantics id="S2.Thmtheorem9.p1.4.4.m4.1a"><mrow id="S2.Thmtheorem9.p1.4.4.m4.1.1" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.cmml"><mi id="S2.Thmtheorem9.p1.4.4.m4.1.1.2" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.2.cmml">Ξ²</mi><mo id="S2.Thmtheorem9.p1.4.4.m4.1.1.1" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.1.cmml">β’</mo><mi id="S2.Thmtheorem9.p1.4.4.m4.1.1.3" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.4.4.m4.1b"><apply id="S2.Thmtheorem9.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1"><times id="S2.Thmtheorem9.p1.4.4.m4.1.1.1.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.1"></times><ci id="S2.Thmtheorem9.p1.4.4.m4.1.1.2.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.2">π½</ci><ci id="S2.Thmtheorem9.p1.4.4.m4.1.1.3.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.4.4.m4.1c">\beta S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.4.4.m4.1d">italic_Ξ² italic_S</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.p10"> <p class="ltx_p" id="S2.p10.4">We are in a position to prove Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem4" title="Theorem D. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">D</span></a>. We need to show that inversion is continuous in any inverse topological semigroup <math alttext="S" class="ltx_Math" display="inline" id="S2.p10.1.m1.1"><semantics id="S2.p10.1.m1.1a"><mi id="S2.p10.1.m1.1.1" xref="S2.p10.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p10.1.m1.1b"><ci id="S2.p10.1.m1.1.1.cmml" xref="S2.p10.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p10.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p10.1.m1.1d">italic_S</annotation></semantics></math> provided (i) <math alttext="S{\times}S" class="ltx_Math" display="inline" id="S2.p10.2.m2.1"><semantics id="S2.p10.2.m2.1a"><mrow id="S2.p10.2.m2.1.1" xref="S2.p10.2.m2.1.1.cmml"><mi id="S2.p10.2.m2.1.1.2" xref="S2.p10.2.m2.1.1.2.cmml">S</mi><mo id="S2.p10.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.p10.2.m2.1.1.1.cmml">Γ</mo><mi id="S2.p10.2.m2.1.1.3" xref="S2.p10.2.m2.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p10.2.m2.1b"><apply id="S2.p10.2.m2.1.1.cmml" xref="S2.p10.2.m2.1.1"><times id="S2.p10.2.m2.1.1.1.cmml" xref="S2.p10.2.m2.1.1.1"></times><ci id="S2.p10.2.m2.1.1.2.cmml" xref="S2.p10.2.m2.1.1.2">π</ci><ci id="S2.p10.2.m2.1.1.3.cmml" xref="S2.p10.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p10.2.m2.1c">S{\times}S</annotation><annotation encoding="application/x-llamapun" id="S2.p10.2.m2.1d">italic_S Γ italic_S</annotation></semantics></math> is pseudocompact, or (ii) <math alttext="S" class="ltx_Math" display="inline" id="S2.p10.3.m3.1"><semantics id="S2.p10.3.m3.1a"><mi id="S2.p10.3.m3.1.1" xref="S2.p10.3.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p10.3.m3.1b"><ci id="S2.p10.3.m3.1.1.cmml" xref="S2.p10.3.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p10.3.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p10.3.m3.1d">italic_S</annotation></semantics></math> is regular, topologically periodic and <math alttext="S{\times}S" class="ltx_Math" display="inline" id="S2.p10.4.m4.1"><semantics id="S2.p10.4.m4.1a"><mrow id="S2.p10.4.m4.1.1" xref="S2.p10.4.m4.1.1.cmml"><mi id="S2.p10.4.m4.1.1.2" xref="S2.p10.4.m4.1.1.2.cmml">S</mi><mo id="S2.p10.4.m4.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.p10.4.m4.1.1.1.cmml">Γ</mo><mi id="S2.p10.4.m4.1.1.3" xref="S2.p10.4.m4.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p10.4.m4.1b"><apply id="S2.p10.4.m4.1.1.cmml" xref="S2.p10.4.m4.1.1"><times id="S2.p10.4.m4.1.1.1.cmml" xref="S2.p10.4.m4.1.1.1"></times><ci id="S2.p10.4.m4.1.1.2.cmml" xref="S2.p10.4.m4.1.1.2">π</ci><ci id="S2.p10.4.m4.1.1.3.cmml" xref="S2.p10.4.m4.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p10.4.m4.1c">S{\times}S</annotation><annotation encoding="application/x-llamapun" id="S2.p10.4.m4.1d">italic_S Γ italic_S</annotation></semantics></math> is countably compact.</p> </div> <div class="ltx_proof" id="S2.8"> <h6 class="ltx_title ltx_runin ltx_font_bold ltx_title_proof">Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem4" title="Theorem D. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">D</span></a>.</h6> <div class="ltx_para" id="S2.7.p1"> <p class="ltx_p" id="S2.7.p1.5">(i) By Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S2.Thmtheorem9" title="Theorem 2.9 (Banakh, Dimitrova, Gutik). β£ 2. Compact-like semigroups with dense inverse subsemigroups and the automatic continuity of inversion β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">2.9</span></a>, <math alttext="\beta S" class="ltx_Math" display="inline" id="S2.7.p1.1.m1.1"><semantics id="S2.7.p1.1.m1.1a"><mrow id="S2.7.p1.1.m1.1.1" xref="S2.7.p1.1.m1.1.1.cmml"><mi id="S2.7.p1.1.m1.1.1.2" xref="S2.7.p1.1.m1.1.1.2.cmml">Ξ²</mi><mo id="S2.7.p1.1.m1.1.1.1" xref="S2.7.p1.1.m1.1.1.1.cmml">β’</mo><mi id="S2.7.p1.1.m1.1.1.3" xref="S2.7.p1.1.m1.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.7.p1.1.m1.1b"><apply id="S2.7.p1.1.m1.1.1.cmml" xref="S2.7.p1.1.m1.1.1"><times id="S2.7.p1.1.m1.1.1.1.cmml" xref="S2.7.p1.1.m1.1.1.1"></times><ci id="S2.7.p1.1.m1.1.1.2.cmml" xref="S2.7.p1.1.m1.1.1.2">π½</ci><ci id="S2.7.p1.1.m1.1.1.3.cmml" xref="S2.7.p1.1.m1.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.7.p1.1.m1.1c">\beta S</annotation><annotation encoding="application/x-llamapun" id="S2.7.p1.1.m1.1d">italic_Ξ² italic_S</annotation></semantics></math> is a compact topological semigroup that contains <math alttext="S" class="ltx_Math" display="inline" id="S2.7.p1.2.m2.1"><semantics id="S2.7.p1.2.m2.1a"><mi id="S2.7.p1.2.m2.1.1" xref="S2.7.p1.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.7.p1.2.m2.1b"><ci id="S2.7.p1.2.m2.1.1.cmml" xref="S2.7.p1.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.7.p1.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.7.p1.2.m2.1d">italic_S</annotation></semantics></math> as a dense inverse subsemigroup. Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem3" title="Theorem C. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">C</span></a> implies that <math alttext="\beta S" class="ltx_Math" display="inline" id="S2.7.p1.3.m3.1"><semantics id="S2.7.p1.3.m3.1a"><mrow id="S2.7.p1.3.m3.1.1" xref="S2.7.p1.3.m3.1.1.cmml"><mi id="S2.7.p1.3.m3.1.1.2" xref="S2.7.p1.3.m3.1.1.2.cmml">Ξ²</mi><mo id="S2.7.p1.3.m3.1.1.1" xref="S2.7.p1.3.m3.1.1.1.cmml">β’</mo><mi id="S2.7.p1.3.m3.1.1.3" xref="S2.7.p1.3.m3.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.7.p1.3.m3.1b"><apply id="S2.7.p1.3.m3.1.1.cmml" xref="S2.7.p1.3.m3.1.1"><times id="S2.7.p1.3.m3.1.1.1.cmml" xref="S2.7.p1.3.m3.1.1.1"></times><ci id="S2.7.p1.3.m3.1.1.2.cmml" xref="S2.7.p1.3.m3.1.1.2">π½</ci><ci id="S2.7.p1.3.m3.1.1.3.cmml" xref="S2.7.p1.3.m3.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.7.p1.3.m3.1c">\beta S</annotation><annotation encoding="application/x-llamapun" id="S2.7.p1.3.m3.1d">italic_Ξ² italic_S</annotation></semantics></math> is a topological inverse semigroup. Then the continuity of inversion in <math alttext="\beta S" class="ltx_Math" display="inline" id="S2.7.p1.4.m4.1"><semantics id="S2.7.p1.4.m4.1a"><mrow id="S2.7.p1.4.m4.1.1" xref="S2.7.p1.4.m4.1.1.cmml"><mi id="S2.7.p1.4.m4.1.1.2" xref="S2.7.p1.4.m4.1.1.2.cmml">Ξ²</mi><mo id="S2.7.p1.4.m4.1.1.1" xref="S2.7.p1.4.m4.1.1.1.cmml">β’</mo><mi id="S2.7.p1.4.m4.1.1.3" xref="S2.7.p1.4.m4.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.7.p1.4.m4.1b"><apply id="S2.7.p1.4.m4.1.1.cmml" xref="S2.7.p1.4.m4.1.1"><times id="S2.7.p1.4.m4.1.1.1.cmml" xref="S2.7.p1.4.m4.1.1.1"></times><ci id="S2.7.p1.4.m4.1.1.2.cmml" xref="S2.7.p1.4.m4.1.1.2">π½</ci><ci id="S2.7.p1.4.m4.1.1.3.cmml" xref="S2.7.p1.4.m4.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.7.p1.4.m4.1c">\beta S</annotation><annotation encoding="application/x-llamapun" id="S2.7.p1.4.m4.1d">italic_Ξ² italic_S</annotation></semantics></math> yields the continuity of inversion in <math alttext="S" class="ltx_Math" display="inline" id="S2.7.p1.5.m5.1"><semantics id="S2.7.p1.5.m5.1a"><mi id="S2.7.p1.5.m5.1.1" xref="S2.7.p1.5.m5.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.7.p1.5.m5.1b"><ci id="S2.7.p1.5.m5.1.1.cmml" xref="S2.7.p1.5.m5.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.7.p1.5.m5.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.7.p1.5.m5.1d">italic_S</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.8.p2"> <p class="ltx_p" id="S2.8.p2.1">(ii) The proof follows from Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S2.Thmtheorem8" title="Proposition 2.8. β£ 2. Compact-like semigroups with dense inverse subsemigroups and the automatic continuity of inversion β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">2.8</span></a> and Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S2.Thmtheorem7" title="Theorem 2.7 (Banakh, Gutik). β£ 2. Compact-like semigroups with dense inverse subsemigroups and the automatic continuity of inversion β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">2.7</span></a>. β</p> </div> </div> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3. </span>Chains in topological semilattices</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.13">Recall that each semilattice possesses the natural partial order <math alttext="\leq" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><mo id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml">β€</mo><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><leq id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">\leq</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">β€</annotation></semantics></math> defined by <math alttext="a\leq b" class="ltx_Math" display="inline" id="S3.p1.2.m2.1"><semantics id="S3.p1.2.m2.1a"><mrow id="S3.p1.2.m2.1.1" xref="S3.p1.2.m2.1.1.cmml"><mi id="S3.p1.2.m2.1.1.2" xref="S3.p1.2.m2.1.1.2.cmml">a</mi><mo id="S3.p1.2.m2.1.1.1" xref="S3.p1.2.m2.1.1.1.cmml">β€</mo><mi id="S3.p1.2.m2.1.1.3" xref="S3.p1.2.m2.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1b"><apply id="S3.p1.2.m2.1.1.cmml" xref="S3.p1.2.m2.1.1"><leq id="S3.p1.2.m2.1.1.1.cmml" xref="S3.p1.2.m2.1.1.1"></leq><ci id="S3.p1.2.m2.1.1.2.cmml" xref="S3.p1.2.m2.1.1.2">π</ci><ci id="S3.p1.2.m2.1.1.3.cmml" xref="S3.p1.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">a\leq b</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">italic_a β€ italic_b</annotation></semantics></math> if and only if <math alttext="ab=ba=a" class="ltx_Math" display="inline" id="S3.p1.3.m3.1"><semantics id="S3.p1.3.m3.1a"><mrow id="S3.p1.3.m3.1.1" xref="S3.p1.3.m3.1.1.cmml"><mrow id="S3.p1.3.m3.1.1.2" xref="S3.p1.3.m3.1.1.2.cmml"><mi id="S3.p1.3.m3.1.1.2.2" xref="S3.p1.3.m3.1.1.2.2.cmml">a</mi><mo id="S3.p1.3.m3.1.1.2.1" xref="S3.p1.3.m3.1.1.2.1.cmml">β’</mo><mi id="S3.p1.3.m3.1.1.2.3" xref="S3.p1.3.m3.1.1.2.3.cmml">b</mi></mrow><mo id="S3.p1.3.m3.1.1.3" xref="S3.p1.3.m3.1.1.3.cmml">=</mo><mrow id="S3.p1.3.m3.1.1.4" xref="S3.p1.3.m3.1.1.4.cmml"><mi id="S3.p1.3.m3.1.1.4.2" xref="S3.p1.3.m3.1.1.4.2.cmml">b</mi><mo id="S3.p1.3.m3.1.1.4.1" xref="S3.p1.3.m3.1.1.4.1.cmml">β’</mo><mi id="S3.p1.3.m3.1.1.4.3" xref="S3.p1.3.m3.1.1.4.3.cmml">a</mi></mrow><mo id="S3.p1.3.m3.1.1.5" xref="S3.p1.3.m3.1.1.5.cmml">=</mo><mi id="S3.p1.3.m3.1.1.6" xref="S3.p1.3.m3.1.1.6.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.3.m3.1b"><apply id="S3.p1.3.m3.1.1.cmml" xref="S3.p1.3.m3.1.1"><and id="S3.p1.3.m3.1.1a.cmml" xref="S3.p1.3.m3.1.1"></and><apply id="S3.p1.3.m3.1.1b.cmml" xref="S3.p1.3.m3.1.1"><eq id="S3.p1.3.m3.1.1.3.cmml" xref="S3.p1.3.m3.1.1.3"></eq><apply id="S3.p1.3.m3.1.1.2.cmml" xref="S3.p1.3.m3.1.1.2"><times id="S3.p1.3.m3.1.1.2.1.cmml" xref="S3.p1.3.m3.1.1.2.1"></times><ci id="S3.p1.3.m3.1.1.2.2.cmml" xref="S3.p1.3.m3.1.1.2.2">π</ci><ci id="S3.p1.3.m3.1.1.2.3.cmml" xref="S3.p1.3.m3.1.1.2.3">π</ci></apply><apply id="S3.p1.3.m3.1.1.4.cmml" xref="S3.p1.3.m3.1.1.4"><times id="S3.p1.3.m3.1.1.4.1.cmml" xref="S3.p1.3.m3.1.1.4.1"></times><ci id="S3.p1.3.m3.1.1.4.2.cmml" xref="S3.p1.3.m3.1.1.4.2">π</ci><ci id="S3.p1.3.m3.1.1.4.3.cmml" xref="S3.p1.3.m3.1.1.4.3">π</ci></apply></apply><apply id="S3.p1.3.m3.1.1c.cmml" xref="S3.p1.3.m3.1.1"><eq id="S3.p1.3.m3.1.1.5.cmml" xref="S3.p1.3.m3.1.1.5"></eq><share href="https://arxiv.org/html/2503.13666v1#S3.p1.3.m3.1.1.4.cmml" id="S3.p1.3.m3.1.1d.cmml" xref="S3.p1.3.m3.1.1"></share><ci id="S3.p1.3.m3.1.1.6.cmml" xref="S3.p1.3.m3.1.1.6">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.3.m3.1c">ab=ba=a</annotation><annotation encoding="application/x-llamapun" id="S3.p1.3.m3.1d">italic_a italic_b = italic_b italic_a = italic_a</annotation></semantics></math>. Further, if a semilattice is treated as a poset, then by default it is assumed to carry the natural partial order. For an element <math alttext="x" class="ltx_Math" display="inline" id="S3.p1.4.m4.1"><semantics id="S3.p1.4.m4.1a"><mi id="S3.p1.4.m4.1.1" xref="S3.p1.4.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.p1.4.m4.1b"><ci id="S3.p1.4.m4.1.1.cmml" xref="S3.p1.4.m4.1.1">π₯</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.4.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.p1.4.m4.1d">italic_x</annotation></semantics></math> of a semilattice <math alttext="X" class="ltx_Math" display="inline" id="S3.p1.5.m5.1"><semantics id="S3.p1.5.m5.1a"><mi id="S3.p1.5.m5.1.1" xref="S3.p1.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.p1.5.m5.1b"><ci id="S3.p1.5.m5.1.1.cmml" xref="S3.p1.5.m5.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.p1.5.m5.1d">italic_X</annotation></semantics></math> put <math alttext="{\downarrow}x=\{y\in X\colon y\leq x\}" class="ltx_Math" display="inline" id="S3.p1.6.m6.2"><semantics id="S3.p1.6.m6.2a"><mrow id="S3.p1.6.m6.2.2" xref="S3.p1.6.m6.2.2.cmml"><mi id="S3.p1.6.m6.2.2.4" xref="S3.p1.6.m6.2.2.4.cmml"></mi><mo id="S3.p1.6.m6.2.2.5" stretchy="false" xref="S3.p1.6.m6.2.2.5.cmml">β</mo><mi id="S3.p1.6.m6.2.2.6" xref="S3.p1.6.m6.2.2.6.cmml">x</mi><mo id="S3.p1.6.m6.2.2.7" xref="S3.p1.6.m6.2.2.7.cmml">=</mo><mrow id="S3.p1.6.m6.2.2.2.2" xref="S3.p1.6.m6.2.2.2.3.cmml"><mo id="S3.p1.6.m6.2.2.2.2.3" stretchy="false" xref="S3.p1.6.m6.2.2.2.3.1.cmml">{</mo><mrow id="S3.p1.6.m6.1.1.1.1.1" xref="S3.p1.6.m6.1.1.1.1.1.cmml"><mi id="S3.p1.6.m6.1.1.1.1.1.2" xref="S3.p1.6.m6.1.1.1.1.1.2.cmml">y</mi><mo id="S3.p1.6.m6.1.1.1.1.1.1" xref="S3.p1.6.m6.1.1.1.1.1.1.cmml">β</mo><mi id="S3.p1.6.m6.1.1.1.1.1.3" xref="S3.p1.6.m6.1.1.1.1.1.3.cmml">X</mi></mrow><mo id="S3.p1.6.m6.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.p1.6.m6.2.2.2.3.1.cmml">:</mo><mrow id="S3.p1.6.m6.2.2.2.2.2" xref="S3.p1.6.m6.2.2.2.2.2.cmml"><mi id="S3.p1.6.m6.2.2.2.2.2.2" xref="S3.p1.6.m6.2.2.2.2.2.2.cmml">y</mi><mo id="S3.p1.6.m6.2.2.2.2.2.1" xref="S3.p1.6.m6.2.2.2.2.2.1.cmml">β€</mo><mi id="S3.p1.6.m6.2.2.2.2.2.3" xref="S3.p1.6.m6.2.2.2.2.2.3.cmml">x</mi></mrow><mo id="S3.p1.6.m6.2.2.2.2.5" stretchy="false" xref="S3.p1.6.m6.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.6.m6.2b"><apply id="S3.p1.6.m6.2.2.cmml" xref="S3.p1.6.m6.2.2"><and id="S3.p1.6.m6.2.2a.cmml" xref="S3.p1.6.m6.2.2"></and><apply id="S3.p1.6.m6.2.2b.cmml" xref="S3.p1.6.m6.2.2"><ci id="S3.p1.6.m6.2.2.5.cmml" xref="S3.p1.6.m6.2.2.5">β</ci><csymbol cd="latexml" id="S3.p1.6.m6.2.2.4.cmml" xref="S3.p1.6.m6.2.2.4">absent</csymbol><ci id="S3.p1.6.m6.2.2.6.cmml" xref="S3.p1.6.m6.2.2.6">π₯</ci></apply><apply id="S3.p1.6.m6.2.2c.cmml" xref="S3.p1.6.m6.2.2"><eq id="S3.p1.6.m6.2.2.7.cmml" xref="S3.p1.6.m6.2.2.7"></eq><share href="https://arxiv.org/html/2503.13666v1#S3.p1.6.m6.2.2.6.cmml" id="S3.p1.6.m6.2.2d.cmml" xref="S3.p1.6.m6.2.2"></share><apply id="S3.p1.6.m6.2.2.2.3.cmml" xref="S3.p1.6.m6.2.2.2.2"><csymbol cd="latexml" id="S3.p1.6.m6.2.2.2.3.1.cmml" xref="S3.p1.6.m6.2.2.2.2.3">conditional-set</csymbol><apply id="S3.p1.6.m6.1.1.1.1.1.cmml" xref="S3.p1.6.m6.1.1.1.1.1"><in id="S3.p1.6.m6.1.1.1.1.1.1.cmml" xref="S3.p1.6.m6.1.1.1.1.1.1"></in><ci id="S3.p1.6.m6.1.1.1.1.1.2.cmml" xref="S3.p1.6.m6.1.1.1.1.1.2">π¦</ci><ci id="S3.p1.6.m6.1.1.1.1.1.3.cmml" xref="S3.p1.6.m6.1.1.1.1.1.3">π</ci></apply><apply id="S3.p1.6.m6.2.2.2.2.2.cmml" xref="S3.p1.6.m6.2.2.2.2.2"><leq id="S3.p1.6.m6.2.2.2.2.2.1.cmml" xref="S3.p1.6.m6.2.2.2.2.2.1"></leq><ci id="S3.p1.6.m6.2.2.2.2.2.2.cmml" xref="S3.p1.6.m6.2.2.2.2.2.2">π¦</ci><ci id="S3.p1.6.m6.2.2.2.2.2.3.cmml" xref="S3.p1.6.m6.2.2.2.2.2.3">π₯</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.6.m6.2c">{\downarrow}x=\{y\in X\colon y\leq x\}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.6.m6.2d">β italic_x = { italic_y β italic_X : italic_y β€ italic_x }</annotation></semantics></math> and <math alttext="{\uparrow}x=\{y\in X\colon x\leq y\}" class="ltx_Math" display="inline" id="S3.p1.7.m7.2"><semantics id="S3.p1.7.m7.2a"><mrow id="S3.p1.7.m7.2.2" xref="S3.p1.7.m7.2.2.cmml"><mi id="S3.p1.7.m7.2.2.4" xref="S3.p1.7.m7.2.2.4.cmml"></mi><mo id="S3.p1.7.m7.2.2.5" stretchy="false" xref="S3.p1.7.m7.2.2.5.cmml">β</mo><mi id="S3.p1.7.m7.2.2.6" xref="S3.p1.7.m7.2.2.6.cmml">x</mi><mo id="S3.p1.7.m7.2.2.7" xref="S3.p1.7.m7.2.2.7.cmml">=</mo><mrow id="S3.p1.7.m7.2.2.2.2" xref="S3.p1.7.m7.2.2.2.3.cmml"><mo id="S3.p1.7.m7.2.2.2.2.3" stretchy="false" xref="S3.p1.7.m7.2.2.2.3.1.cmml">{</mo><mrow id="S3.p1.7.m7.1.1.1.1.1" xref="S3.p1.7.m7.1.1.1.1.1.cmml"><mi id="S3.p1.7.m7.1.1.1.1.1.2" xref="S3.p1.7.m7.1.1.1.1.1.2.cmml">y</mi><mo id="S3.p1.7.m7.1.1.1.1.1.1" xref="S3.p1.7.m7.1.1.1.1.1.1.cmml">β</mo><mi id="S3.p1.7.m7.1.1.1.1.1.3" xref="S3.p1.7.m7.1.1.1.1.1.3.cmml">X</mi></mrow><mo id="S3.p1.7.m7.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.p1.7.m7.2.2.2.3.1.cmml">:</mo><mrow id="S3.p1.7.m7.2.2.2.2.2" xref="S3.p1.7.m7.2.2.2.2.2.cmml"><mi id="S3.p1.7.m7.2.2.2.2.2.2" xref="S3.p1.7.m7.2.2.2.2.2.2.cmml">x</mi><mo id="S3.p1.7.m7.2.2.2.2.2.1" xref="S3.p1.7.m7.2.2.2.2.2.1.cmml">β€</mo><mi id="S3.p1.7.m7.2.2.2.2.2.3" xref="S3.p1.7.m7.2.2.2.2.2.3.cmml">y</mi></mrow><mo id="S3.p1.7.m7.2.2.2.2.5" stretchy="false" xref="S3.p1.7.m7.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.7.m7.2b"><apply id="S3.p1.7.m7.2.2.cmml" xref="S3.p1.7.m7.2.2"><and id="S3.p1.7.m7.2.2a.cmml" xref="S3.p1.7.m7.2.2"></and><apply id="S3.p1.7.m7.2.2b.cmml" xref="S3.p1.7.m7.2.2"><ci id="S3.p1.7.m7.2.2.5.cmml" xref="S3.p1.7.m7.2.2.5">β</ci><csymbol cd="latexml" id="S3.p1.7.m7.2.2.4.cmml" xref="S3.p1.7.m7.2.2.4">absent</csymbol><ci id="S3.p1.7.m7.2.2.6.cmml" xref="S3.p1.7.m7.2.2.6">π₯</ci></apply><apply id="S3.p1.7.m7.2.2c.cmml" xref="S3.p1.7.m7.2.2"><eq id="S3.p1.7.m7.2.2.7.cmml" xref="S3.p1.7.m7.2.2.7"></eq><share href="https://arxiv.org/html/2503.13666v1#S3.p1.7.m7.2.2.6.cmml" id="S3.p1.7.m7.2.2d.cmml" xref="S3.p1.7.m7.2.2"></share><apply id="S3.p1.7.m7.2.2.2.3.cmml" xref="S3.p1.7.m7.2.2.2.2"><csymbol cd="latexml" id="S3.p1.7.m7.2.2.2.3.1.cmml" xref="S3.p1.7.m7.2.2.2.2.3">conditional-set</csymbol><apply id="S3.p1.7.m7.1.1.1.1.1.cmml" xref="S3.p1.7.m7.1.1.1.1.1"><in id="S3.p1.7.m7.1.1.1.1.1.1.cmml" xref="S3.p1.7.m7.1.1.1.1.1.1"></in><ci id="S3.p1.7.m7.1.1.1.1.1.2.cmml" xref="S3.p1.7.m7.1.1.1.1.1.2">π¦</ci><ci id="S3.p1.7.m7.1.1.1.1.1.3.cmml" xref="S3.p1.7.m7.1.1.1.1.1.3">π</ci></apply><apply id="S3.p1.7.m7.2.2.2.2.2.cmml" xref="S3.p1.7.m7.2.2.2.2.2"><leq id="S3.p1.7.m7.2.2.2.2.2.1.cmml" xref="S3.p1.7.m7.2.2.2.2.2.1"></leq><ci id="S3.p1.7.m7.2.2.2.2.2.2.cmml" xref="S3.p1.7.m7.2.2.2.2.2.2">π₯</ci><ci id="S3.p1.7.m7.2.2.2.2.2.3.cmml" xref="S3.p1.7.m7.2.2.2.2.2.3">π¦</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.7.m7.2c">{\uparrow}x=\{y\in X\colon x\leq y\}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.7.m7.2d">β italic_x = { italic_y β italic_X : italic_x β€ italic_y }</annotation></semantics></math>. Note that if <math alttext="X" class="ltx_Math" display="inline" id="S3.p1.8.m8.1"><semantics id="S3.p1.8.m8.1a"><mi id="S3.p1.8.m8.1.1" xref="S3.p1.8.m8.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.p1.8.m8.1b"><ci id="S3.p1.8.m8.1.1.cmml" xref="S3.p1.8.m8.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.8.m8.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.p1.8.m8.1d">italic_X</annotation></semantics></math> is a semitopological semilattice, then for every <math alttext="x\in X" class="ltx_Math" display="inline" id="S3.p1.9.m9.1"><semantics id="S3.p1.9.m9.1a"><mrow id="S3.p1.9.m9.1.1" xref="S3.p1.9.m9.1.1.cmml"><mi id="S3.p1.9.m9.1.1.2" xref="S3.p1.9.m9.1.1.2.cmml">x</mi><mo id="S3.p1.9.m9.1.1.1" xref="S3.p1.9.m9.1.1.1.cmml">β</mo><mi id="S3.p1.9.m9.1.1.3" xref="S3.p1.9.m9.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.9.m9.1b"><apply id="S3.p1.9.m9.1.1.cmml" xref="S3.p1.9.m9.1.1"><in id="S3.p1.9.m9.1.1.1.cmml" xref="S3.p1.9.m9.1.1.1"></in><ci id="S3.p1.9.m9.1.1.2.cmml" xref="S3.p1.9.m9.1.1.2">π₯</ci><ci id="S3.p1.9.m9.1.1.3.cmml" xref="S3.p1.9.m9.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.9.m9.1c">x\in X</annotation><annotation encoding="application/x-llamapun" id="S3.p1.9.m9.1d">italic_x β italic_X</annotation></semantics></math> the sets <math alttext="{\downarrow}x" class="ltx_Math" display="inline" id="S3.p1.10.m10.1"><semantics id="S3.p1.10.m10.1a"><mrow id="S3.p1.10.m10.1.1" xref="S3.p1.10.m10.1.1.cmml"><mi id="S3.p1.10.m10.1.1.2" xref="S3.p1.10.m10.1.1.2.cmml"></mi><mo id="S3.p1.10.m10.1.1.1" stretchy="false" xref="S3.p1.10.m10.1.1.1.cmml">β</mo><mi id="S3.p1.10.m10.1.1.3" xref="S3.p1.10.m10.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.10.m10.1b"><apply id="S3.p1.10.m10.1.1.cmml" xref="S3.p1.10.m10.1.1"><ci id="S3.p1.10.m10.1.1.1.cmml" xref="S3.p1.10.m10.1.1.1">β</ci><csymbol cd="latexml" id="S3.p1.10.m10.1.1.2.cmml" xref="S3.p1.10.m10.1.1.2">absent</csymbol><ci id="S3.p1.10.m10.1.1.3.cmml" xref="S3.p1.10.m10.1.1.3">π₯</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.10.m10.1c">{\downarrow}x</annotation><annotation encoding="application/x-llamapun" id="S3.p1.10.m10.1d">β italic_x</annotation></semantics></math> and <math alttext="{\uparrow}x" class="ltx_Math" display="inline" id="S3.p1.11.m11.1"><semantics id="S3.p1.11.m11.1a"><mrow id="S3.p1.11.m11.1.1" xref="S3.p1.11.m11.1.1.cmml"><mi id="S3.p1.11.m11.1.1.2" xref="S3.p1.11.m11.1.1.2.cmml"></mi><mo id="S3.p1.11.m11.1.1.1" stretchy="false" xref="S3.p1.11.m11.1.1.1.cmml">β</mo><mi id="S3.p1.11.m11.1.1.3" xref="S3.p1.11.m11.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.11.m11.1b"><apply id="S3.p1.11.m11.1.1.cmml" xref="S3.p1.11.m11.1.1"><ci id="S3.p1.11.m11.1.1.1.cmml" xref="S3.p1.11.m11.1.1.1">β</ci><csymbol cd="latexml" id="S3.p1.11.m11.1.1.2.cmml" xref="S3.p1.11.m11.1.1.2">absent</csymbol><ci id="S3.p1.11.m11.1.1.3.cmml" xref="S3.p1.11.m11.1.1.3">π₯</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.11.m11.1c">{\uparrow}x</annotation><annotation encoding="application/x-llamapun" id="S3.p1.11.m11.1d">β italic_x</annotation></semantics></math> are closed. A semitopological semilattice <math alttext="X" class="ltx_Math" display="inline" id="S3.p1.12.m12.1"><semantics id="S3.p1.12.m12.1a"><mi id="S3.p1.12.m12.1.1" xref="S3.p1.12.m12.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.p1.12.m12.1b"><ci id="S3.p1.12.m12.1.1.cmml" xref="S3.p1.12.m12.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.12.m12.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.p1.12.m12.1d">italic_X</annotation></semantics></math> is called <span class="ltx_text ltx_font_italic" id="S3.p1.13.1">chain-compact</span> if all maximal chains in <math alttext="X" class="ltx_Math" display="inline" id="S3.p1.13.m13.1"><semantics id="S3.p1.13.m13.1a"><mi id="S3.p1.13.m13.1.1" xref="S3.p1.13.m13.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.p1.13.m13.1b"><ci id="S3.p1.13.m13.1.1.cmml" xref="S3.p1.13.m13.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.13.m13.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.p1.13.m13.1d">italic_X</annotation></semantics></math> are compact. The following two theorems follow from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib1" title="">1</a>, Theorem 3.1]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.1.1.1">Theorem 3.1</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.2.2"> </span>(Banakh, Bardyla)<span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.3.3">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem1.p1"> <p class="ltx_p" id="S3.Thmtheorem1.p1.1"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem1.p1.1.1">For a semitopological semilattice <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.1.1.m1.1"><semantics id="S3.Thmtheorem1.p1.1.1.m1.1a"><mi id="S3.Thmtheorem1.p1.1.1.m1.1.1" xref="S3.Thmtheorem1.p1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.1.1.m1.1b"><ci id="S3.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.1.1.m1.1d">italic_X</annotation></semantics></math> the following is equivalent:</span></p> <ol class="ltx_enumerate" id="S3.I1"> <li class="ltx_item" id="S3.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S3.I1.i1.p1"> <p class="ltx_p" id="S3.I1.i1.p1.1"><math alttext="X" class="ltx_Math" display="inline" id="S3.I1.i1.p1.1.m1.1"><semantics id="S3.I1.i1.p1.1.m1.1a"><mi id="S3.I1.i1.p1.1.m1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.1.m1.1b"><ci id="S3.I1.i1.p1.1.m1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.1.m1.1d">italic_X</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i1.p1.1.1"> is chain-compact;</span></p> </div> </li> <li class="ltx_item" id="S3.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S3.I1.i2.p1"> <p class="ltx_p" id="S3.I1.i2.p1.3"><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.3.1">each nonempty chain </span><math alttext="L\subseteq X" class="ltx_Math" display="inline" id="S3.I1.i2.p1.1.m1.1"><semantics id="S3.I1.i2.p1.1.m1.1a"><mrow id="S3.I1.i2.p1.1.m1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.cmml"><mi id="S3.I1.i2.p1.1.m1.1.1.2" xref="S3.I1.i2.p1.1.m1.1.1.2.cmml">L</mi><mo id="S3.I1.i2.p1.1.m1.1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.1.cmml">β</mo><mi id="S3.I1.i2.p1.1.m1.1.1.3" xref="S3.I1.i2.p1.1.m1.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.1.m1.1b"><apply id="S3.I1.i2.p1.1.m1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1"><subset id="S3.I1.i2.p1.1.m1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1"></subset><ci id="S3.I1.i2.p1.1.m1.1.1.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.2">πΏ</ci><ci id="S3.I1.i2.p1.1.m1.1.1.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.1.m1.1c">L\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.1.m1.1d">italic_L β italic_X</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.3.2"> has </span><math alttext="\inf L\in\overline{L}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.2.m2.1"><semantics id="S3.I1.i2.p1.2.m2.1a"><mrow id="S3.I1.i2.p1.2.m2.1.1" xref="S3.I1.i2.p1.2.m2.1.1.cmml"><mrow id="S3.I1.i2.p1.2.m2.1.1.2" xref="S3.I1.i2.p1.2.m2.1.1.2.cmml"><mo id="S3.I1.i2.p1.2.m2.1.1.2.1" rspace="0.167em" xref="S3.I1.i2.p1.2.m2.1.1.2.1.cmml">inf</mo><mi id="S3.I1.i2.p1.2.m2.1.1.2.2" xref="S3.I1.i2.p1.2.m2.1.1.2.2.cmml">L</mi></mrow><mo id="S3.I1.i2.p1.2.m2.1.1.1" xref="S3.I1.i2.p1.2.m2.1.1.1.cmml">β</mo><mover accent="true" id="S3.I1.i2.p1.2.m2.1.1.3" xref="S3.I1.i2.p1.2.m2.1.1.3.cmml"><mi id="S3.I1.i2.p1.2.m2.1.1.3.2" xref="S3.I1.i2.p1.2.m2.1.1.3.2.cmml">L</mi><mo id="S3.I1.i2.p1.2.m2.1.1.3.1" xref="S3.I1.i2.p1.2.m2.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.2.m2.1b"><apply id="S3.I1.i2.p1.2.m2.1.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1"><in id="S3.I1.i2.p1.2.m2.1.1.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1.1"></in><apply id="S3.I1.i2.p1.2.m2.1.1.2.cmml" xref="S3.I1.i2.p1.2.m2.1.1.2"><csymbol cd="latexml" id="S3.I1.i2.p1.2.m2.1.1.2.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1.2.1">infimum</csymbol><ci id="S3.I1.i2.p1.2.m2.1.1.2.2.cmml" xref="S3.I1.i2.p1.2.m2.1.1.2.2">πΏ</ci></apply><apply id="S3.I1.i2.p1.2.m2.1.1.3.cmml" xref="S3.I1.i2.p1.2.m2.1.1.3"><ci id="S3.I1.i2.p1.2.m2.1.1.3.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1.3.1">Β―</ci><ci id="S3.I1.i2.p1.2.m2.1.1.3.2.cmml" xref="S3.I1.i2.p1.2.m2.1.1.3.2">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.2.m2.1c">\inf L\in\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.2.m2.1d">roman_inf italic_L β overΒ― start_ARG italic_L end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.3.3"> and </span><math alttext="\sup L\in\overline{L}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.3.m3.1"><semantics id="S3.I1.i2.p1.3.m3.1a"><mrow id="S3.I1.i2.p1.3.m3.1.1" xref="S3.I1.i2.p1.3.m3.1.1.cmml"><mrow id="S3.I1.i2.p1.3.m3.1.1.2" xref="S3.I1.i2.p1.3.m3.1.1.2.cmml"><mo id="S3.I1.i2.p1.3.m3.1.1.2.1" rspace="0.167em" xref="S3.I1.i2.p1.3.m3.1.1.2.1.cmml">sup</mo><mi id="S3.I1.i2.p1.3.m3.1.1.2.2" xref="S3.I1.i2.p1.3.m3.1.1.2.2.cmml">L</mi></mrow><mo id="S3.I1.i2.p1.3.m3.1.1.1" xref="S3.I1.i2.p1.3.m3.1.1.1.cmml">β</mo><mover accent="true" id="S3.I1.i2.p1.3.m3.1.1.3" xref="S3.I1.i2.p1.3.m3.1.1.3.cmml"><mi id="S3.I1.i2.p1.3.m3.1.1.3.2" xref="S3.I1.i2.p1.3.m3.1.1.3.2.cmml">L</mi><mo id="S3.I1.i2.p1.3.m3.1.1.3.1" xref="S3.I1.i2.p1.3.m3.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.3.m3.1b"><apply id="S3.I1.i2.p1.3.m3.1.1.cmml" xref="S3.I1.i2.p1.3.m3.1.1"><in id="S3.I1.i2.p1.3.m3.1.1.1.cmml" xref="S3.I1.i2.p1.3.m3.1.1.1"></in><apply id="S3.I1.i2.p1.3.m3.1.1.2.cmml" xref="S3.I1.i2.p1.3.m3.1.1.2"><csymbol cd="latexml" id="S3.I1.i2.p1.3.m3.1.1.2.1.cmml" xref="S3.I1.i2.p1.3.m3.1.1.2.1">supremum</csymbol><ci id="S3.I1.i2.p1.3.m3.1.1.2.2.cmml" xref="S3.I1.i2.p1.3.m3.1.1.2.2">πΏ</ci></apply><apply id="S3.I1.i2.p1.3.m3.1.1.3.cmml" xref="S3.I1.i2.p1.3.m3.1.1.3"><ci id="S3.I1.i2.p1.3.m3.1.1.3.1.cmml" xref="S3.I1.i2.p1.3.m3.1.1.3.1">Β―</ci><ci id="S3.I1.i2.p1.3.m3.1.1.3.2.cmml" xref="S3.I1.i2.p1.3.m3.1.1.3.2">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.3.m3.1c">\sup L\in\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.3.m3.1d">roman_sup italic_L β overΒ― start_ARG italic_L end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.3.4">;</span></p> </div> </li> <li class="ltx_item" id="S3.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(3)</span> <div class="ltx_para" id="S3.I1.i3.p1"> <p class="ltx_p" id="S3.I1.i3.p1.1"><span class="ltx_text ltx_font_italic" id="S3.I1.i3.p1.1.1">no closed chain in </span><math alttext="X" class="ltx_Math" display="inline" id="S3.I1.i3.p1.1.m1.1"><semantics id="S3.I1.i3.p1.1.m1.1a"><mi id="S3.I1.i3.p1.1.m1.1.1" xref="S3.I1.i3.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.1.m1.1b"><ci id="S3.I1.i3.p1.1.m1.1.1.cmml" xref="S3.I1.i3.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.1.m1.1d">italic_X</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i3.p1.1.2"> is topologically isomorphic to an infinite regular cardinal endowed with the order topology and the semilattice operation of minimum or maximum.</span></p> </div> </li> </ol> </div> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.1.1.1">Theorem 3.2</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.2.2"> </span>(Banakh, Bardyla)<span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.3.3">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem2.p1"> <p class="ltx_p" id="S3.Thmtheorem2.p1.4"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p1.4.4">If <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.1.1.m1.1"><semantics id="S3.Thmtheorem2.p1.1.1.m1.1a"><mi id="S3.Thmtheorem2.p1.1.1.m1.1.1" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.1.1.m1.1b"><ci id="S3.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.1.1.m1.1d">italic_X</annotation></semantics></math> is a chain-compact subsemilattice of a topological semigroup <math alttext="Y" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.2.2.m2.1"><semantics id="S3.Thmtheorem2.p1.2.2.m2.1a"><mi id="S3.Thmtheorem2.p1.2.2.m2.1.1" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.2.2.m2.1b"><ci id="S3.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.2.2.m2.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.2.2.m2.1d">italic_Y</annotation></semantics></math>, then <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.3.3.m3.1"><semantics id="S3.Thmtheorem2.p1.3.3.m3.1a"><mi id="S3.Thmtheorem2.p1.3.3.m3.1.1" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.3.3.m3.1b"><ci id="S3.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.3.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.3.3.m3.1d">italic_X</annotation></semantics></math> is closed in <math alttext="Y" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.4.4.m4.1"><semantics id="S3.Thmtheorem2.p1.4.4.m4.1a"><mi id="S3.Thmtheorem2.p1.4.4.m4.1.1" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.4.4.m4.1b"><ci id="S3.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem2.p1.4.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.4.4.m4.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.4.4.m4.1d">italic_Y</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.1">The following fact is well known.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S3.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem3.1.1.1">Proposition 3.3</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem3.2.2"> </span>(Folklore)<span class="ltx_text ltx_font_bold" id="S3.Thmtheorem3.3.3">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem3.p1"> <p class="ltx_p" id="S3.Thmtheorem3.p1.1"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem3.p1.1.1">The closure of a chain in a semitopological semilattice is a chain.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem4.1.1.1">Lemma 3.4</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem4.p1"> <p class="ltx_p" id="S3.Thmtheorem4.p1.8"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem4.p1.8.8">Let <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.1.1.m1.1"><semantics id="S3.Thmtheorem4.p1.1.1.m1.1a"><mi id="S3.Thmtheorem4.p1.1.1.m1.1.1" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.1.1.m1.1b"><ci id="S3.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.1.1.m1.1d">italic_X</annotation></semantics></math> be a semitopological semilattice and <math alttext="L\subseteq X" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.2.2.m2.1"><semantics id="S3.Thmtheorem4.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem4.p1.2.2.m2.1.1" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem4.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.2.cmml">L</mi><mo id="S3.Thmtheorem4.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.1.cmml">β</mo><mi id="S3.Thmtheorem4.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.2.2.m2.1b"><apply id="S3.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1"><subset id="S3.Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.1"></subset><ci id="S3.Thmtheorem4.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.2">πΏ</ci><ci id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.2.2.m2.1c">L\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.2.2.m2.1d">italic_L β italic_X</annotation></semantics></math> be a chain isomorphic to <math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.3.3.m3.2"><semantics id="S3.Thmtheorem4.p1.3.3.m3.2a"><mrow id="S3.Thmtheorem4.p1.3.3.m3.2.2.1" xref="S3.Thmtheorem4.p1.3.3.m3.2.2.2.cmml"><mo id="S3.Thmtheorem4.p1.3.3.m3.2.2.1.2" stretchy="false" xref="S3.Thmtheorem4.p1.3.3.m3.2.2.2.cmml">(</mo><msub id="S3.Thmtheorem4.p1.3.3.m3.2.2.1.1" xref="S3.Thmtheorem4.p1.3.3.m3.2.2.1.1.cmml"><mi id="S3.Thmtheorem4.p1.3.3.m3.2.2.1.1.2" xref="S3.Thmtheorem4.p1.3.3.m3.2.2.1.1.2.cmml">Ο</mi><mn id="S3.Thmtheorem4.p1.3.3.m3.2.2.1.1.3" xref="S3.Thmtheorem4.p1.3.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.Thmtheorem4.p1.3.3.m3.2.2.1.3" xref="S3.Thmtheorem4.p1.3.3.m3.2.2.2.cmml">,</mo><mi id="S3.Thmtheorem4.p1.3.3.m3.1.1" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.cmml">min</mi><mo id="S3.Thmtheorem4.p1.3.3.m3.2.2.1.4" stretchy="false" xref="S3.Thmtheorem4.p1.3.3.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.3.3.m3.2b"><interval closure="open" id="S3.Thmtheorem4.p1.3.3.m3.2.2.2.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.2.2.1"><apply id="S3.Thmtheorem4.p1.3.3.m3.2.2.1.1.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.3.3.m3.2.2.1.1.1.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.2.2.1.1">subscript</csymbol><ci id="S3.Thmtheorem4.p1.3.3.m3.2.2.1.1.2.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.2.2.1.1.2">π</ci><cn id="S3.Thmtheorem4.p1.3.3.m3.2.2.1.1.3.cmml" type="integer" xref="S3.Thmtheorem4.p1.3.3.m3.2.2.1.1.3">1</cn></apply><min id="S3.Thmtheorem4.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.3.3.m3.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.3.3.m3.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math>. If <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.4.4.m4.1"><semantics id="S3.Thmtheorem4.p1.4.4.m4.1a"><mover accent="true" id="S3.Thmtheorem4.p1.4.4.m4.1.1" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem4.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.2.cmml">L</mi><mo id="S3.Thmtheorem4.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.4.4.m4.1b"><apply id="S3.Thmtheorem4.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1"><ci id="S3.Thmtheorem4.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.1">Β―</ci><ci id="S3.Thmtheorem4.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.4.4.m4.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.4.4.m4.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> is compact, then there exists <math alttext="z=\sup L\in\overline{L}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.5.5.m5.1"><semantics id="S3.Thmtheorem4.p1.5.5.m5.1a"><mrow id="S3.Thmtheorem4.p1.5.5.m5.1.1" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem4.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.2.cmml">z</mi><mo id="S3.Thmtheorem4.p1.5.5.m5.1.1.3" rspace="0.1389em" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.3.cmml">=</mo><mrow id="S3.Thmtheorem4.p1.5.5.m5.1.1.4" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.4.cmml"><mo id="S3.Thmtheorem4.p1.5.5.m5.1.1.4.1" lspace="0.1389em" rspace="0.167em" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.4.1.cmml">sup</mo><mi id="S3.Thmtheorem4.p1.5.5.m5.1.1.4.2" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.4.2.cmml">L</mi></mrow><mo id="S3.Thmtheorem4.p1.5.5.m5.1.1.5" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.5.cmml">β</mo><mover accent="true" id="S3.Thmtheorem4.p1.5.5.m5.1.1.6" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.6.cmml"><mi id="S3.Thmtheorem4.p1.5.5.m5.1.1.6.2" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.6.2.cmml">L</mi><mo id="S3.Thmtheorem4.p1.5.5.m5.1.1.6.1" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.6.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.5.5.m5.1b"><apply id="S3.Thmtheorem4.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1"><and id="S3.Thmtheorem4.p1.5.5.m5.1.1a.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1"></and><apply id="S3.Thmtheorem4.p1.5.5.m5.1.1b.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1"><eq id="S3.Thmtheorem4.p1.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.3"></eq><ci id="S3.Thmtheorem4.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.2">π§</ci><apply id="S3.Thmtheorem4.p1.5.5.m5.1.1.4.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.4"><csymbol cd="latexml" id="S3.Thmtheorem4.p1.5.5.m5.1.1.4.1.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.4.1">supremum</csymbol><ci id="S3.Thmtheorem4.p1.5.5.m5.1.1.4.2.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.4.2">πΏ</ci></apply></apply><apply id="S3.Thmtheorem4.p1.5.5.m5.1.1c.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1"><in id="S3.Thmtheorem4.p1.5.5.m5.1.1.5.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.5"></in><share href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem4.p1.5.5.m5.1.1.4.cmml" id="S3.Thmtheorem4.p1.5.5.m5.1.1d.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1"></share><apply id="S3.Thmtheorem4.p1.5.5.m5.1.1.6.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.6"><ci id="S3.Thmtheorem4.p1.5.5.m5.1.1.6.1.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.6.1">Β―</ci><ci id="S3.Thmtheorem4.p1.5.5.m5.1.1.6.2.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.6.2">πΏ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.5.5.m5.1c">z=\sup L\in\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.5.5.m5.1d">italic_z = roman_sup italic_L β overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> such that for each open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.6.6.m6.1"><semantics id="S3.Thmtheorem4.p1.6.6.m6.1a"><mi id="S3.Thmtheorem4.p1.6.6.m6.1.1" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.6.6.m6.1b"><ci id="S3.Thmtheorem4.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem4.p1.6.6.m6.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.6.6.m6.1c">U</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.6.6.m6.1d">italic_U</annotation></semantics></math> of <math alttext="z" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.7.7.m7.1"><semantics id="S3.Thmtheorem4.p1.7.7.m7.1a"><mi id="S3.Thmtheorem4.p1.7.7.m7.1.1" xref="S3.Thmtheorem4.p1.7.7.m7.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.7.7.m7.1b"><ci id="S3.Thmtheorem4.p1.7.7.m7.1.1.cmml" xref="S3.Thmtheorem4.p1.7.7.m7.1.1">π§</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.7.7.m7.1c">z</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.7.7.m7.1d">italic_z</annotation></semantics></math> the set <math alttext="L\setminus U" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.8.8.m8.1"><semantics id="S3.Thmtheorem4.p1.8.8.m8.1a"><mrow id="S3.Thmtheorem4.p1.8.8.m8.1.1" xref="S3.Thmtheorem4.p1.8.8.m8.1.1.cmml"><mi id="S3.Thmtheorem4.p1.8.8.m8.1.1.2" xref="S3.Thmtheorem4.p1.8.8.m8.1.1.2.cmml">L</mi><mo id="S3.Thmtheorem4.p1.8.8.m8.1.1.1" xref="S3.Thmtheorem4.p1.8.8.m8.1.1.1.cmml">β</mo><mi id="S3.Thmtheorem4.p1.8.8.m8.1.1.3" xref="S3.Thmtheorem4.p1.8.8.m8.1.1.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.8.8.m8.1b"><apply id="S3.Thmtheorem4.p1.8.8.m8.1.1.cmml" xref="S3.Thmtheorem4.p1.8.8.m8.1.1"><setdiff id="S3.Thmtheorem4.p1.8.8.m8.1.1.1.cmml" xref="S3.Thmtheorem4.p1.8.8.m8.1.1.1"></setdiff><ci id="S3.Thmtheorem4.p1.8.8.m8.1.1.2.cmml" xref="S3.Thmtheorem4.p1.8.8.m8.1.1.2">πΏ</ci><ci id="S3.Thmtheorem4.p1.8.8.m8.1.1.3.cmml" xref="S3.Thmtheorem4.p1.8.8.m8.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.8.8.m8.1c">L\setminus U</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.8.8.m8.1d">italic_L β italic_U</annotation></semantics></math> is countable.</span></p> </div> </div> <div class="ltx_proof" id="S3.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.1.p1"> <p class="ltx_p" id="S3.1.p1.14">By Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem3" title="Proposition 3.3 (Folklore). β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.3</span></a>, <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.1.p1.1.m1.1"><semantics id="S3.1.p1.1.m1.1a"><mover accent="true" id="S3.1.p1.1.m1.1.1" xref="S3.1.p1.1.m1.1.1.cmml"><mi id="S3.1.p1.1.m1.1.1.2" xref="S3.1.p1.1.m1.1.1.2.cmml">L</mi><mo id="S3.1.p1.1.m1.1.1.1" xref="S3.1.p1.1.m1.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.1.p1.1.m1.1b"><apply id="S3.1.p1.1.m1.1.1.cmml" xref="S3.1.p1.1.m1.1.1"><ci id="S3.1.p1.1.m1.1.1.1.cmml" xref="S3.1.p1.1.m1.1.1.1">Β―</ci><ci id="S3.1.p1.1.m1.1.1.2.cmml" xref="S3.1.p1.1.m1.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.1.m1.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.1.m1.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> is a compact chain. Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem1" title="Theorem 3.1 (Banakh, Bardyla). β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.1</span></a>(2) applied to <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.1.p1.2.m2.1"><semantics id="S3.1.p1.2.m2.1a"><mover accent="true" id="S3.1.p1.2.m2.1.1" xref="S3.1.p1.2.m2.1.1.cmml"><mi id="S3.1.p1.2.m2.1.1.2" xref="S3.1.p1.2.m2.1.1.2.cmml">L</mi><mo id="S3.1.p1.2.m2.1.1.1" xref="S3.1.p1.2.m2.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.1.p1.2.m2.1b"><apply id="S3.1.p1.2.m2.1.1.cmml" xref="S3.1.p1.2.m2.1.1"><ci id="S3.1.p1.2.m2.1.1.1.cmml" xref="S3.1.p1.2.m2.1.1.1">Β―</ci><ci id="S3.1.p1.2.m2.1.1.2.cmml" xref="S3.1.p1.2.m2.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.2.m2.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.2.m2.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> implies that <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.1.p1.3.m3.1"><semantics id="S3.1.p1.3.m3.1a"><mover accent="true" id="S3.1.p1.3.m3.1.1" xref="S3.1.p1.3.m3.1.1.cmml"><mi id="S3.1.p1.3.m3.1.1.2" xref="S3.1.p1.3.m3.1.1.2.cmml">L</mi><mo id="S3.1.p1.3.m3.1.1.1" xref="S3.1.p1.3.m3.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.1.p1.3.m3.1b"><apply id="S3.1.p1.3.m3.1.1.cmml" xref="S3.1.p1.3.m3.1.1"><ci id="S3.1.p1.3.m3.1.1.1.cmml" xref="S3.1.p1.3.m3.1.1.1">Β―</ci><ci id="S3.1.p1.3.m3.1.1.2.cmml" xref="S3.1.p1.3.m3.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.3.m3.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.3.m3.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> contains the supremum of <math alttext="L" class="ltx_Math" display="inline" id="S3.1.p1.4.m4.1"><semantics id="S3.1.p1.4.m4.1a"><mi id="S3.1.p1.4.m4.1.1" xref="S3.1.p1.4.m4.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.1.p1.4.m4.1b"><ci id="S3.1.p1.4.m4.1.1.cmml" xref="S3.1.p1.4.m4.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.4.m4.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.4.m4.1d">italic_L</annotation></semantics></math> which we denote by <math alttext="z" class="ltx_Math" display="inline" id="S3.1.p1.5.m5.1"><semantics id="S3.1.p1.5.m5.1a"><mi id="S3.1.p1.5.m5.1.1" xref="S3.1.p1.5.m5.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S3.1.p1.5.m5.1b"><ci id="S3.1.p1.5.m5.1.1.cmml" xref="S3.1.p1.5.m5.1.1">π§</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.5.m5.1c">z</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.5.m5.1d">italic_z</annotation></semantics></math>. It is clear that <math alttext="z=\sup L^{\prime}" class="ltx_Math" display="inline" id="S3.1.p1.6.m6.1"><semantics id="S3.1.p1.6.m6.1a"><mrow id="S3.1.p1.6.m6.1.1" xref="S3.1.p1.6.m6.1.1.cmml"><mi id="S3.1.p1.6.m6.1.1.2" xref="S3.1.p1.6.m6.1.1.2.cmml">z</mi><mo id="S3.1.p1.6.m6.1.1.1" rspace="0.1389em" xref="S3.1.p1.6.m6.1.1.1.cmml">=</mo><mrow id="S3.1.p1.6.m6.1.1.3" xref="S3.1.p1.6.m6.1.1.3.cmml"><mo id="S3.1.p1.6.m6.1.1.3.1" lspace="0.1389em" rspace="0.167em" xref="S3.1.p1.6.m6.1.1.3.1.cmml">sup</mo><msup id="S3.1.p1.6.m6.1.1.3.2" xref="S3.1.p1.6.m6.1.1.3.2.cmml"><mi id="S3.1.p1.6.m6.1.1.3.2.2" xref="S3.1.p1.6.m6.1.1.3.2.2.cmml">L</mi><mo id="S3.1.p1.6.m6.1.1.3.2.3" xref="S3.1.p1.6.m6.1.1.3.2.3.cmml">β²</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.6.m6.1b"><apply id="S3.1.p1.6.m6.1.1.cmml" xref="S3.1.p1.6.m6.1.1"><eq id="S3.1.p1.6.m6.1.1.1.cmml" xref="S3.1.p1.6.m6.1.1.1"></eq><ci id="S3.1.p1.6.m6.1.1.2.cmml" xref="S3.1.p1.6.m6.1.1.2">π§</ci><apply id="S3.1.p1.6.m6.1.1.3.cmml" xref="S3.1.p1.6.m6.1.1.3"><csymbol cd="latexml" id="S3.1.p1.6.m6.1.1.3.1.cmml" xref="S3.1.p1.6.m6.1.1.3.1">supremum</csymbol><apply id="S3.1.p1.6.m6.1.1.3.2.cmml" xref="S3.1.p1.6.m6.1.1.3.2"><csymbol cd="ambiguous" id="S3.1.p1.6.m6.1.1.3.2.1.cmml" xref="S3.1.p1.6.m6.1.1.3.2">superscript</csymbol><ci id="S3.1.p1.6.m6.1.1.3.2.2.cmml" xref="S3.1.p1.6.m6.1.1.3.2.2">πΏ</ci><ci id="S3.1.p1.6.m6.1.1.3.2.3.cmml" xref="S3.1.p1.6.m6.1.1.3.2.3">β²</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.6.m6.1c">z=\sup L^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.6.m6.1d">italic_z = roman_sup italic_L start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT</annotation></semantics></math> for each uncountable subset <math alttext="L^{\prime}" class="ltx_Math" display="inline" id="S3.1.p1.7.m7.1"><semantics id="S3.1.p1.7.m7.1a"><msup id="S3.1.p1.7.m7.1.1" xref="S3.1.p1.7.m7.1.1.cmml"><mi id="S3.1.p1.7.m7.1.1.2" xref="S3.1.p1.7.m7.1.1.2.cmml">L</mi><mo id="S3.1.p1.7.m7.1.1.3" xref="S3.1.p1.7.m7.1.1.3.cmml">β²</mo></msup><annotation-xml encoding="MathML-Content" id="S3.1.p1.7.m7.1b"><apply id="S3.1.p1.7.m7.1.1.cmml" xref="S3.1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.1.p1.7.m7.1.1.1.cmml" xref="S3.1.p1.7.m7.1.1">superscript</csymbol><ci id="S3.1.p1.7.m7.1.1.2.cmml" xref="S3.1.p1.7.m7.1.1.2">πΏ</ci><ci id="S3.1.p1.7.m7.1.1.3.cmml" xref="S3.1.p1.7.m7.1.1.3">β²</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.7.m7.1c">L^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.7.m7.1d">italic_L start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT</annotation></semantics></math> of <math alttext="L" class="ltx_Math" display="inline" id="S3.1.p1.8.m8.1"><semantics id="S3.1.p1.8.m8.1a"><mi id="S3.1.p1.8.m8.1.1" xref="S3.1.p1.8.m8.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.1.p1.8.m8.1b"><ci id="S3.1.p1.8.m8.1.1.cmml" xref="S3.1.p1.8.m8.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.8.m8.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.8.m8.1d">italic_L</annotation></semantics></math>. By Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem1" title="Theorem 3.1 (Banakh, Bardyla). β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.1</span></a>(2), <math alttext="z\in\overline{L^{\prime}}" class="ltx_Math" display="inline" id="S3.1.p1.9.m9.1"><semantics id="S3.1.p1.9.m9.1a"><mrow id="S3.1.p1.9.m9.1.1" xref="S3.1.p1.9.m9.1.1.cmml"><mi id="S3.1.p1.9.m9.1.1.2" xref="S3.1.p1.9.m9.1.1.2.cmml">z</mi><mo id="S3.1.p1.9.m9.1.1.1" xref="S3.1.p1.9.m9.1.1.1.cmml">β</mo><mover accent="true" id="S3.1.p1.9.m9.1.1.3" xref="S3.1.p1.9.m9.1.1.3.cmml"><msup id="S3.1.p1.9.m9.1.1.3.2" xref="S3.1.p1.9.m9.1.1.3.2.cmml"><mi id="S3.1.p1.9.m9.1.1.3.2.2" xref="S3.1.p1.9.m9.1.1.3.2.2.cmml">L</mi><mo id="S3.1.p1.9.m9.1.1.3.2.3" xref="S3.1.p1.9.m9.1.1.3.2.3.cmml">β²</mo></msup><mo id="S3.1.p1.9.m9.1.1.3.1" xref="S3.1.p1.9.m9.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.9.m9.1b"><apply id="S3.1.p1.9.m9.1.1.cmml" xref="S3.1.p1.9.m9.1.1"><in id="S3.1.p1.9.m9.1.1.1.cmml" xref="S3.1.p1.9.m9.1.1.1"></in><ci id="S3.1.p1.9.m9.1.1.2.cmml" xref="S3.1.p1.9.m9.1.1.2">π§</ci><apply id="S3.1.p1.9.m9.1.1.3.cmml" xref="S3.1.p1.9.m9.1.1.3"><ci id="S3.1.p1.9.m9.1.1.3.1.cmml" xref="S3.1.p1.9.m9.1.1.3.1">Β―</ci><apply id="S3.1.p1.9.m9.1.1.3.2.cmml" xref="S3.1.p1.9.m9.1.1.3.2"><csymbol cd="ambiguous" id="S3.1.p1.9.m9.1.1.3.2.1.cmml" xref="S3.1.p1.9.m9.1.1.3.2">superscript</csymbol><ci id="S3.1.p1.9.m9.1.1.3.2.2.cmml" xref="S3.1.p1.9.m9.1.1.3.2.2">πΏ</ci><ci id="S3.1.p1.9.m9.1.1.3.2.3.cmml" xref="S3.1.p1.9.m9.1.1.3.2.3">β²</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.9.m9.1c">z\in\overline{L^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.9.m9.1d">italic_z β overΒ― start_ARG italic_L start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math> for each uncountable subset <math alttext="L^{\prime}" class="ltx_Math" display="inline" id="S3.1.p1.10.m10.1"><semantics id="S3.1.p1.10.m10.1a"><msup id="S3.1.p1.10.m10.1.1" xref="S3.1.p1.10.m10.1.1.cmml"><mi id="S3.1.p1.10.m10.1.1.2" xref="S3.1.p1.10.m10.1.1.2.cmml">L</mi><mo id="S3.1.p1.10.m10.1.1.3" xref="S3.1.p1.10.m10.1.1.3.cmml">β²</mo></msup><annotation-xml encoding="MathML-Content" id="S3.1.p1.10.m10.1b"><apply id="S3.1.p1.10.m10.1.1.cmml" xref="S3.1.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S3.1.p1.10.m10.1.1.1.cmml" xref="S3.1.p1.10.m10.1.1">superscript</csymbol><ci id="S3.1.p1.10.m10.1.1.2.cmml" xref="S3.1.p1.10.m10.1.1.2">πΏ</ci><ci id="S3.1.p1.10.m10.1.1.3.cmml" xref="S3.1.p1.10.m10.1.1.3">β²</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.10.m10.1c">L^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.10.m10.1d">italic_L start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT</annotation></semantics></math> of <math alttext="L" class="ltx_Math" display="inline" id="S3.1.p1.11.m11.1"><semantics id="S3.1.p1.11.m11.1a"><mi id="S3.1.p1.11.m11.1.1" xref="S3.1.p1.11.m11.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.1.p1.11.m11.1b"><ci id="S3.1.p1.11.m11.1.1.cmml" xref="S3.1.p1.11.m11.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.11.m11.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.11.m11.1d">italic_L</annotation></semantics></math>. It follows that for each open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S3.1.p1.12.m12.1"><semantics id="S3.1.p1.12.m12.1a"><mi id="S3.1.p1.12.m12.1.1" xref="S3.1.p1.12.m12.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.1.p1.12.m12.1b"><ci id="S3.1.p1.12.m12.1.1.cmml" xref="S3.1.p1.12.m12.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.12.m12.1c">U</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.12.m12.1d">italic_U</annotation></semantics></math> of <math alttext="z" class="ltx_Math" display="inline" id="S3.1.p1.13.m13.1"><semantics id="S3.1.p1.13.m13.1a"><mi id="S3.1.p1.13.m13.1.1" xref="S3.1.p1.13.m13.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S3.1.p1.13.m13.1b"><ci id="S3.1.p1.13.m13.1.1.cmml" xref="S3.1.p1.13.m13.1.1">π§</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.13.m13.1c">z</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.13.m13.1d">italic_z</annotation></semantics></math> the set <math alttext="L\setminus U" class="ltx_Math" display="inline" id="S3.1.p1.14.m14.1"><semantics id="S3.1.p1.14.m14.1a"><mrow id="S3.1.p1.14.m14.1.1" xref="S3.1.p1.14.m14.1.1.cmml"><mi id="S3.1.p1.14.m14.1.1.2" xref="S3.1.p1.14.m14.1.1.2.cmml">L</mi><mo id="S3.1.p1.14.m14.1.1.1" xref="S3.1.p1.14.m14.1.1.1.cmml">β</mo><mi id="S3.1.p1.14.m14.1.1.3" xref="S3.1.p1.14.m14.1.1.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.14.m14.1b"><apply id="S3.1.p1.14.m14.1.1.cmml" xref="S3.1.p1.14.m14.1.1"><setdiff id="S3.1.p1.14.m14.1.1.1.cmml" xref="S3.1.p1.14.m14.1.1.1"></setdiff><ci id="S3.1.p1.14.m14.1.1.2.cmml" xref="S3.1.p1.14.m14.1.1.2">πΏ</ci><ci id="S3.1.p1.14.m14.1.1.3.cmml" xref="S3.1.p1.14.m14.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.14.m14.1c">L\setminus U</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.14.m14.1d">italic_L β italic_U</annotation></semantics></math> is countable, as required. β</p> </div> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.1">Dually, one can prove the following.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem5.1.1.1">Lemma 3.5</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem5.p1"> <p class="ltx_p" id="S3.Thmtheorem5.p1.8"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem5.p1.8.8">Let <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.1.1.m1.1"><semantics id="S3.Thmtheorem5.p1.1.1.m1.1a"><mi id="S3.Thmtheorem5.p1.1.1.m1.1.1" xref="S3.Thmtheorem5.p1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.1.1.m1.1b"><ci id="S3.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem5.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.1.1.m1.1d">italic_X</annotation></semantics></math> be a semitopological semilattice and <math alttext="L\subseteq X" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.2.2.m2.1"><semantics id="S3.Thmtheorem5.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem5.p1.2.2.m2.1.1" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem5.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.2.cmml">L</mi><mo id="S3.Thmtheorem5.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.1.cmml">β</mo><mi id="S3.Thmtheorem5.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.2.2.m2.1b"><apply id="S3.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.1.1"><subset id="S3.Thmtheorem5.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.1"></subset><ci id="S3.Thmtheorem5.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.2">πΏ</ci><ci id="S3.Thmtheorem5.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.2.2.m2.1c">L\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.2.2.m2.1d">italic_L β italic_X</annotation></semantics></math> be a chain isomorphic to <math alttext="(\omega_{1},\max)" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.3.3.m3.2"><semantics id="S3.Thmtheorem5.p1.3.3.m3.2a"><mrow id="S3.Thmtheorem5.p1.3.3.m3.2.2.1" xref="S3.Thmtheorem5.p1.3.3.m3.2.2.2.cmml"><mo id="S3.Thmtheorem5.p1.3.3.m3.2.2.1.2" stretchy="false" xref="S3.Thmtheorem5.p1.3.3.m3.2.2.2.cmml">(</mo><msub id="S3.Thmtheorem5.p1.3.3.m3.2.2.1.1" xref="S3.Thmtheorem5.p1.3.3.m3.2.2.1.1.cmml"><mi id="S3.Thmtheorem5.p1.3.3.m3.2.2.1.1.2" xref="S3.Thmtheorem5.p1.3.3.m3.2.2.1.1.2.cmml">Ο</mi><mn id="S3.Thmtheorem5.p1.3.3.m3.2.2.1.1.3" xref="S3.Thmtheorem5.p1.3.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.Thmtheorem5.p1.3.3.m3.2.2.1.3" xref="S3.Thmtheorem5.p1.3.3.m3.2.2.2.cmml">,</mo><mi id="S3.Thmtheorem5.p1.3.3.m3.1.1" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.cmml">max</mi><mo id="S3.Thmtheorem5.p1.3.3.m3.2.2.1.4" stretchy="false" xref="S3.Thmtheorem5.p1.3.3.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.3.3.m3.2b"><interval closure="open" id="S3.Thmtheorem5.p1.3.3.m3.2.2.2.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.2.2.1"><apply id="S3.Thmtheorem5.p1.3.3.m3.2.2.1.1.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.3.3.m3.2.2.1.1.1.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.2.2.1.1">subscript</csymbol><ci id="S3.Thmtheorem5.p1.3.3.m3.2.2.1.1.2.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.2.2.1.1.2">π</ci><cn id="S3.Thmtheorem5.p1.3.3.m3.2.2.1.1.3.cmml" type="integer" xref="S3.Thmtheorem5.p1.3.3.m3.2.2.1.1.3">1</cn></apply><max id="S3.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.1.1"></max></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.3.3.m3.2c">(\omega_{1},\max)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.3.3.m3.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_max )</annotation></semantics></math>. If <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.4.4.m4.1"><semantics id="S3.Thmtheorem5.p1.4.4.m4.1a"><mover accent="true" id="S3.Thmtheorem5.p1.4.4.m4.1.1" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem5.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.2.cmml">L</mi><mo id="S3.Thmtheorem5.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.4.4.m4.1b"><apply id="S3.Thmtheorem5.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1"><ci id="S3.Thmtheorem5.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.1">Β―</ci><ci id="S3.Thmtheorem5.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.4.4.m4.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.4.4.m4.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> is compact, then there exists <math alttext="z=\inf L\in\overline{L}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.5.5.m5.1"><semantics id="S3.Thmtheorem5.p1.5.5.m5.1a"><mrow id="S3.Thmtheorem5.p1.5.5.m5.1.1" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem5.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.2.cmml">z</mi><mo id="S3.Thmtheorem5.p1.5.5.m5.1.1.3" rspace="0.1389em" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.3.cmml">=</mo><mrow id="S3.Thmtheorem5.p1.5.5.m5.1.1.4" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.4.cmml"><mo id="S3.Thmtheorem5.p1.5.5.m5.1.1.4.1" lspace="0.1389em" rspace="0.167em" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.4.1.cmml">inf</mo><mi id="S3.Thmtheorem5.p1.5.5.m5.1.1.4.2" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.4.2.cmml">L</mi></mrow><mo id="S3.Thmtheorem5.p1.5.5.m5.1.1.5" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.5.cmml">β</mo><mover accent="true" id="S3.Thmtheorem5.p1.5.5.m5.1.1.6" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.6.cmml"><mi id="S3.Thmtheorem5.p1.5.5.m5.1.1.6.2" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.6.2.cmml">L</mi><mo id="S3.Thmtheorem5.p1.5.5.m5.1.1.6.1" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.6.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.5.5.m5.1b"><apply id="S3.Thmtheorem5.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1"><and id="S3.Thmtheorem5.p1.5.5.m5.1.1a.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1"></and><apply id="S3.Thmtheorem5.p1.5.5.m5.1.1b.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1"><eq id="S3.Thmtheorem5.p1.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.3"></eq><ci id="S3.Thmtheorem5.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.2">π§</ci><apply id="S3.Thmtheorem5.p1.5.5.m5.1.1.4.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.4"><csymbol cd="latexml" id="S3.Thmtheorem5.p1.5.5.m5.1.1.4.1.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.4.1">infimum</csymbol><ci id="S3.Thmtheorem5.p1.5.5.m5.1.1.4.2.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.4.2">πΏ</ci></apply></apply><apply id="S3.Thmtheorem5.p1.5.5.m5.1.1c.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1"><in id="S3.Thmtheorem5.p1.5.5.m5.1.1.5.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.5"></in><share href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem5.p1.5.5.m5.1.1.4.cmml" id="S3.Thmtheorem5.p1.5.5.m5.1.1d.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1"></share><apply id="S3.Thmtheorem5.p1.5.5.m5.1.1.6.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.6"><ci id="S3.Thmtheorem5.p1.5.5.m5.1.1.6.1.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.6.1">Β―</ci><ci id="S3.Thmtheorem5.p1.5.5.m5.1.1.6.2.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.6.2">πΏ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.5.5.m5.1c">z=\inf L\in\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.5.5.m5.1d">italic_z = roman_inf italic_L β overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> such that for each open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.6.6.m6.1"><semantics id="S3.Thmtheorem5.p1.6.6.m6.1a"><mi id="S3.Thmtheorem5.p1.6.6.m6.1.1" xref="S3.Thmtheorem5.p1.6.6.m6.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.6.6.m6.1b"><ci id="S3.Thmtheorem5.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem5.p1.6.6.m6.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.6.6.m6.1c">U</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.6.6.m6.1d">italic_U</annotation></semantics></math> of <math alttext="z" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.7.7.m7.1"><semantics id="S3.Thmtheorem5.p1.7.7.m7.1a"><mi id="S3.Thmtheorem5.p1.7.7.m7.1.1" xref="S3.Thmtheorem5.p1.7.7.m7.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.7.7.m7.1b"><ci id="S3.Thmtheorem5.p1.7.7.m7.1.1.cmml" xref="S3.Thmtheorem5.p1.7.7.m7.1.1">π§</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.7.7.m7.1c">z</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.7.7.m7.1d">italic_z</annotation></semantics></math> the set <math alttext="L\setminus U" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.8.8.m8.1"><semantics id="S3.Thmtheorem5.p1.8.8.m8.1a"><mrow id="S3.Thmtheorem5.p1.8.8.m8.1.1" xref="S3.Thmtheorem5.p1.8.8.m8.1.1.cmml"><mi id="S3.Thmtheorem5.p1.8.8.m8.1.1.2" xref="S3.Thmtheorem5.p1.8.8.m8.1.1.2.cmml">L</mi><mo id="S3.Thmtheorem5.p1.8.8.m8.1.1.1" xref="S3.Thmtheorem5.p1.8.8.m8.1.1.1.cmml">β</mo><mi id="S3.Thmtheorem5.p1.8.8.m8.1.1.3" xref="S3.Thmtheorem5.p1.8.8.m8.1.1.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.8.8.m8.1b"><apply id="S3.Thmtheorem5.p1.8.8.m8.1.1.cmml" xref="S3.Thmtheorem5.p1.8.8.m8.1.1"><setdiff id="S3.Thmtheorem5.p1.8.8.m8.1.1.1.cmml" xref="S3.Thmtheorem5.p1.8.8.m8.1.1.1"></setdiff><ci id="S3.Thmtheorem5.p1.8.8.m8.1.1.2.cmml" xref="S3.Thmtheorem5.p1.8.8.m8.1.1.2">πΏ</ci><ci id="S3.Thmtheorem5.p1.8.8.m8.1.1.3.cmml" xref="S3.Thmtheorem5.p1.8.8.m8.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.8.8.m8.1c">L\setminus U</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.8.8.m8.1d">italic_L β italic_U</annotation></semantics></math> is countable.</span></p> </div> </div> <div class="ltx_para" id="S3.p4"> <p class="ltx_p" id="S3.p4.5">Recall that a space <math alttext="X" class="ltx_Math" display="inline" id="S3.p4.1.m1.1"><semantics id="S3.p4.1.m1.1a"><mi id="S3.p4.1.m1.1.1" xref="S3.p4.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.p4.1.m1.1b"><ci id="S3.p4.1.m1.1.1.cmml" xref="S3.p4.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.p4.1.m1.1d">italic_X</annotation></semantics></math> is called <span class="ltx_text ltx_font_italic" id="S3.p4.5.1">countably tight</span> if for every <math alttext="A\subseteq X" class="ltx_Math" display="inline" id="S3.p4.2.m2.1"><semantics id="S3.p4.2.m2.1a"><mrow id="S3.p4.2.m2.1.1" xref="S3.p4.2.m2.1.1.cmml"><mi id="S3.p4.2.m2.1.1.2" xref="S3.p4.2.m2.1.1.2.cmml">A</mi><mo id="S3.p4.2.m2.1.1.1" xref="S3.p4.2.m2.1.1.1.cmml">β</mo><mi id="S3.p4.2.m2.1.1.3" xref="S3.p4.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.2.m2.1b"><apply id="S3.p4.2.m2.1.1.cmml" xref="S3.p4.2.m2.1.1"><subset id="S3.p4.2.m2.1.1.1.cmml" xref="S3.p4.2.m2.1.1.1"></subset><ci id="S3.p4.2.m2.1.1.2.cmml" xref="S3.p4.2.m2.1.1.2">π΄</ci><ci id="S3.p4.2.m2.1.1.3.cmml" xref="S3.p4.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.2.m2.1c">A\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S3.p4.2.m2.1d">italic_A β italic_X</annotation></semantics></math> and <math alttext="x\in\overline{A}" class="ltx_Math" display="inline" id="S3.p4.3.m3.1"><semantics id="S3.p4.3.m3.1a"><mrow id="S3.p4.3.m3.1.1" xref="S3.p4.3.m3.1.1.cmml"><mi id="S3.p4.3.m3.1.1.2" xref="S3.p4.3.m3.1.1.2.cmml">x</mi><mo id="S3.p4.3.m3.1.1.1" xref="S3.p4.3.m3.1.1.1.cmml">β</mo><mover accent="true" id="S3.p4.3.m3.1.1.3" xref="S3.p4.3.m3.1.1.3.cmml"><mi id="S3.p4.3.m3.1.1.3.2" xref="S3.p4.3.m3.1.1.3.2.cmml">A</mi><mo id="S3.p4.3.m3.1.1.3.1" xref="S3.p4.3.m3.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.3.m3.1b"><apply id="S3.p4.3.m3.1.1.cmml" xref="S3.p4.3.m3.1.1"><in id="S3.p4.3.m3.1.1.1.cmml" xref="S3.p4.3.m3.1.1.1"></in><ci id="S3.p4.3.m3.1.1.2.cmml" xref="S3.p4.3.m3.1.1.2">π₯</ci><apply id="S3.p4.3.m3.1.1.3.cmml" xref="S3.p4.3.m3.1.1.3"><ci id="S3.p4.3.m3.1.1.3.1.cmml" xref="S3.p4.3.m3.1.1.3.1">Β―</ci><ci id="S3.p4.3.m3.1.1.3.2.cmml" xref="S3.p4.3.m3.1.1.3.2">π΄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.3.m3.1c">x\in\overline{A}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.3.m3.1d">italic_x β overΒ― start_ARG italic_A end_ARG</annotation></semantics></math> there exists a countable subset <math alttext="B\subseteq A" class="ltx_Math" display="inline" id="S3.p4.4.m4.1"><semantics id="S3.p4.4.m4.1a"><mrow id="S3.p4.4.m4.1.1" xref="S3.p4.4.m4.1.1.cmml"><mi id="S3.p4.4.m4.1.1.2" xref="S3.p4.4.m4.1.1.2.cmml">B</mi><mo id="S3.p4.4.m4.1.1.1" xref="S3.p4.4.m4.1.1.1.cmml">β</mo><mi id="S3.p4.4.m4.1.1.3" xref="S3.p4.4.m4.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.4.m4.1b"><apply id="S3.p4.4.m4.1.1.cmml" xref="S3.p4.4.m4.1.1"><subset id="S3.p4.4.m4.1.1.1.cmml" xref="S3.p4.4.m4.1.1.1"></subset><ci id="S3.p4.4.m4.1.1.2.cmml" xref="S3.p4.4.m4.1.1.2">π΅</ci><ci id="S3.p4.4.m4.1.1.3.cmml" xref="S3.p4.4.m4.1.1.3">π΄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.4.m4.1c">B\subseteq A</annotation><annotation encoding="application/x-llamapun" id="S3.p4.4.m4.1d">italic_B β italic_A</annotation></semantics></math> such that <math alttext="x\in\overline{B}" class="ltx_Math" display="inline" id="S3.p4.5.m5.1"><semantics id="S3.p4.5.m5.1a"><mrow id="S3.p4.5.m5.1.1" xref="S3.p4.5.m5.1.1.cmml"><mi id="S3.p4.5.m5.1.1.2" xref="S3.p4.5.m5.1.1.2.cmml">x</mi><mo id="S3.p4.5.m5.1.1.1" xref="S3.p4.5.m5.1.1.1.cmml">β</mo><mover accent="true" id="S3.p4.5.m5.1.1.3" xref="S3.p4.5.m5.1.1.3.cmml"><mi id="S3.p4.5.m5.1.1.3.2" xref="S3.p4.5.m5.1.1.3.2.cmml">B</mi><mo id="S3.p4.5.m5.1.1.3.1" xref="S3.p4.5.m5.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.5.m5.1b"><apply id="S3.p4.5.m5.1.1.cmml" xref="S3.p4.5.m5.1.1"><in id="S3.p4.5.m5.1.1.1.cmml" xref="S3.p4.5.m5.1.1.1"></in><ci id="S3.p4.5.m5.1.1.2.cmml" xref="S3.p4.5.m5.1.1.2">π₯</ci><apply id="S3.p4.5.m5.1.1.3.cmml" xref="S3.p4.5.m5.1.1.3"><ci id="S3.p4.5.m5.1.1.3.1.cmml" xref="S3.p4.5.m5.1.1.3.1">Β―</ci><ci id="S3.p4.5.m5.1.1.3.2.cmml" xref="S3.p4.5.m5.1.1.3.2">π΅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.5.m5.1c">x\in\overline{B}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.5.m5.1d">italic_x β overΒ― start_ARG italic_B end_ARG</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem6.1.1.1">Lemma 3.6</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem6.p1"> <p class="ltx_p" id="S3.Thmtheorem6.p1.5"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem6.p1.5.5">Let <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.1.1.m1.1"><semantics id="S3.Thmtheorem6.p1.1.1.m1.1a"><mi id="S3.Thmtheorem6.p1.1.1.m1.1.1" xref="S3.Thmtheorem6.p1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.1.1.m1.1b"><ci id="S3.Thmtheorem6.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem6.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.1.1.m1.1d">italic_X</annotation></semantics></math> be a countably tight semitopological semilattice and <math alttext="L" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.2.2.m2.1"><semantics id="S3.Thmtheorem6.p1.2.2.m2.1a"><mi id="S3.Thmtheorem6.p1.2.2.m2.1.1" xref="S3.Thmtheorem6.p1.2.2.m2.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.2.2.m2.1b"><ci id="S3.Thmtheorem6.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem6.p1.2.2.m2.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.2.2.m2.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.2.2.m2.1d">italic_L</annotation></semantics></math> be a chain isomorphic to <math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.3.3.m3.2"><semantics id="S3.Thmtheorem6.p1.3.3.m3.2a"><mrow id="S3.Thmtheorem6.p1.3.3.m3.2.2.1" xref="S3.Thmtheorem6.p1.3.3.m3.2.2.2.cmml"><mo id="S3.Thmtheorem6.p1.3.3.m3.2.2.1.2" stretchy="false" xref="S3.Thmtheorem6.p1.3.3.m3.2.2.2.cmml">(</mo><msub id="S3.Thmtheorem6.p1.3.3.m3.2.2.1.1" xref="S3.Thmtheorem6.p1.3.3.m3.2.2.1.1.cmml"><mi id="S3.Thmtheorem6.p1.3.3.m3.2.2.1.1.2" xref="S3.Thmtheorem6.p1.3.3.m3.2.2.1.1.2.cmml">Ο</mi><mn id="S3.Thmtheorem6.p1.3.3.m3.2.2.1.1.3" xref="S3.Thmtheorem6.p1.3.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.Thmtheorem6.p1.3.3.m3.2.2.1.3" xref="S3.Thmtheorem6.p1.3.3.m3.2.2.2.cmml">,</mo><mi id="S3.Thmtheorem6.p1.3.3.m3.1.1" xref="S3.Thmtheorem6.p1.3.3.m3.1.1.cmml">min</mi><mo id="S3.Thmtheorem6.p1.3.3.m3.2.2.1.4" stretchy="false" xref="S3.Thmtheorem6.p1.3.3.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.3.3.m3.2b"><interval closure="open" id="S3.Thmtheorem6.p1.3.3.m3.2.2.2.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.2.2.1"><apply id="S3.Thmtheorem6.p1.3.3.m3.2.2.1.1.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.2.2.1.1">subscript</csymbol><ci id="S3.Thmtheorem6.p1.3.3.m3.2.2.1.1.2.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.2.2.1.1.2">π</ci><cn id="S3.Thmtheorem6.p1.3.3.m3.2.2.1.1.3.cmml" type="integer" xref="S3.Thmtheorem6.p1.3.3.m3.2.2.1.1.3">1</cn></apply><min id="S3.Thmtheorem6.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.3.3.m3.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.3.3.m3.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math> or <math alttext="(\omega_{1},\max)" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.4.4.m4.2"><semantics id="S3.Thmtheorem6.p1.4.4.m4.2a"><mrow id="S3.Thmtheorem6.p1.4.4.m4.2.2.1" xref="S3.Thmtheorem6.p1.4.4.m4.2.2.2.cmml"><mo id="S3.Thmtheorem6.p1.4.4.m4.2.2.1.2" stretchy="false" xref="S3.Thmtheorem6.p1.4.4.m4.2.2.2.cmml">(</mo><msub id="S3.Thmtheorem6.p1.4.4.m4.2.2.1.1" xref="S3.Thmtheorem6.p1.4.4.m4.2.2.1.1.cmml"><mi id="S3.Thmtheorem6.p1.4.4.m4.2.2.1.1.2" xref="S3.Thmtheorem6.p1.4.4.m4.2.2.1.1.2.cmml">Ο</mi><mn id="S3.Thmtheorem6.p1.4.4.m4.2.2.1.1.3" xref="S3.Thmtheorem6.p1.4.4.m4.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.Thmtheorem6.p1.4.4.m4.2.2.1.3" xref="S3.Thmtheorem6.p1.4.4.m4.2.2.2.cmml">,</mo><mi id="S3.Thmtheorem6.p1.4.4.m4.1.1" xref="S3.Thmtheorem6.p1.4.4.m4.1.1.cmml">max</mi><mo id="S3.Thmtheorem6.p1.4.4.m4.2.2.1.4" stretchy="false" xref="S3.Thmtheorem6.p1.4.4.m4.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.4.4.m4.2b"><interval closure="open" id="S3.Thmtheorem6.p1.4.4.m4.2.2.2.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.2.2.1"><apply id="S3.Thmtheorem6.p1.4.4.m4.2.2.1.1.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.4.4.m4.2.2.1.1.1.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.2.2.1.1">subscript</csymbol><ci id="S3.Thmtheorem6.p1.4.4.m4.2.2.1.1.2.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.2.2.1.1.2">π</ci><cn id="S3.Thmtheorem6.p1.4.4.m4.2.2.1.1.3.cmml" type="integer" xref="S3.Thmtheorem6.p1.4.4.m4.2.2.1.1.3">1</cn></apply><max id="S3.Thmtheorem6.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.1.1"></max></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.4.4.m4.2c">(\omega_{1},\max)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.4.4.m4.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_max )</annotation></semantics></math>. Then <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.5.5.m5.1"><semantics id="S3.Thmtheorem6.p1.5.5.m5.1a"><mover accent="true" id="S3.Thmtheorem6.p1.5.5.m5.1.1" xref="S3.Thmtheorem6.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem6.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem6.p1.5.5.m5.1.1.2.cmml">L</mi><mo id="S3.Thmtheorem6.p1.5.5.m5.1.1.1" xref="S3.Thmtheorem6.p1.5.5.m5.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.5.5.m5.1b"><apply id="S3.Thmtheorem6.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem6.p1.5.5.m5.1.1"><ci id="S3.Thmtheorem6.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem6.p1.5.5.m5.1.1.1">Β―</ci><ci id="S3.Thmtheorem6.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem6.p1.5.5.m5.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.5.5.m5.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.5.5.m5.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> is not compact.</span></p> </div> </div> <div class="ltx_proof" id="S3.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.2.p1"> <p class="ltx_p" id="S3.2.p1.16">Let <math alttext="L" class="ltx_Math" display="inline" id="S3.2.p1.1.m1.1"><semantics id="S3.2.p1.1.m1.1a"><mi id="S3.2.p1.1.m1.1.1" xref="S3.2.p1.1.m1.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.2.p1.1.m1.1b"><ci id="S3.2.p1.1.m1.1.1.cmml" xref="S3.2.p1.1.m1.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p1.1.m1.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.2.p1.1.m1.1d">italic_L</annotation></semantics></math> be a chain isomorphic to <math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S3.2.p1.2.m2.2"><semantics id="S3.2.p1.2.m2.2a"><mrow id="S3.2.p1.2.m2.2.2.1" xref="S3.2.p1.2.m2.2.2.2.cmml"><mo id="S3.2.p1.2.m2.2.2.1.2" stretchy="false" xref="S3.2.p1.2.m2.2.2.2.cmml">(</mo><msub id="S3.2.p1.2.m2.2.2.1.1" xref="S3.2.p1.2.m2.2.2.1.1.cmml"><mi id="S3.2.p1.2.m2.2.2.1.1.2" xref="S3.2.p1.2.m2.2.2.1.1.2.cmml">Ο</mi><mn id="S3.2.p1.2.m2.2.2.1.1.3" xref="S3.2.p1.2.m2.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.2.p1.2.m2.2.2.1.3" xref="S3.2.p1.2.m2.2.2.2.cmml">,</mo><mi id="S3.2.p1.2.m2.1.1" xref="S3.2.p1.2.m2.1.1.cmml">min</mi><mo id="S3.2.p1.2.m2.2.2.1.4" stretchy="false" xref="S3.2.p1.2.m2.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p1.2.m2.2b"><interval closure="open" id="S3.2.p1.2.m2.2.2.2.cmml" xref="S3.2.p1.2.m2.2.2.1"><apply id="S3.2.p1.2.m2.2.2.1.1.cmml" xref="S3.2.p1.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S3.2.p1.2.m2.2.2.1.1.1.cmml" xref="S3.2.p1.2.m2.2.2.1.1">subscript</csymbol><ci id="S3.2.p1.2.m2.2.2.1.1.2.cmml" xref="S3.2.p1.2.m2.2.2.1.1.2">π</ci><cn id="S3.2.p1.2.m2.2.2.1.1.3.cmml" type="integer" xref="S3.2.p1.2.m2.2.2.1.1.3">1</cn></apply><min id="S3.2.p1.2.m2.1.1.cmml" xref="S3.2.p1.2.m2.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p1.2.m2.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S3.2.p1.2.m2.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math>. If <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.2.p1.3.m3.1"><semantics id="S3.2.p1.3.m3.1a"><mover accent="true" id="S3.2.p1.3.m3.1.1" xref="S3.2.p1.3.m3.1.1.cmml"><mi id="S3.2.p1.3.m3.1.1.2" xref="S3.2.p1.3.m3.1.1.2.cmml">L</mi><mo id="S3.2.p1.3.m3.1.1.1" xref="S3.2.p1.3.m3.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.2.p1.3.m3.1b"><apply id="S3.2.p1.3.m3.1.1.cmml" xref="S3.2.p1.3.m3.1.1"><ci id="S3.2.p1.3.m3.1.1.1.cmml" xref="S3.2.p1.3.m3.1.1.1">Β―</ci><ci id="S3.2.p1.3.m3.1.1.2.cmml" xref="S3.2.p1.3.m3.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p1.3.m3.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p1.3.m3.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> is compact, then Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem4" title="Lemma 3.4. β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.4</span></a> yields <math alttext="z=\sup L\in\overline{L}" class="ltx_Math" display="inline" id="S3.2.p1.4.m4.1"><semantics id="S3.2.p1.4.m4.1a"><mrow id="S3.2.p1.4.m4.1.1" xref="S3.2.p1.4.m4.1.1.cmml"><mi id="S3.2.p1.4.m4.1.1.2" xref="S3.2.p1.4.m4.1.1.2.cmml">z</mi><mo id="S3.2.p1.4.m4.1.1.3" rspace="0.1389em" xref="S3.2.p1.4.m4.1.1.3.cmml">=</mo><mrow id="S3.2.p1.4.m4.1.1.4" xref="S3.2.p1.4.m4.1.1.4.cmml"><mo id="S3.2.p1.4.m4.1.1.4.1" lspace="0.1389em" rspace="0.167em" xref="S3.2.p1.4.m4.1.1.4.1.cmml">sup</mo><mi id="S3.2.p1.4.m4.1.1.4.2" xref="S3.2.p1.4.m4.1.1.4.2.cmml">L</mi></mrow><mo id="S3.2.p1.4.m4.1.1.5" xref="S3.2.p1.4.m4.1.1.5.cmml">β</mo><mover accent="true" id="S3.2.p1.4.m4.1.1.6" xref="S3.2.p1.4.m4.1.1.6.cmml"><mi id="S3.2.p1.4.m4.1.1.6.2" xref="S3.2.p1.4.m4.1.1.6.2.cmml">L</mi><mo id="S3.2.p1.4.m4.1.1.6.1" xref="S3.2.p1.4.m4.1.1.6.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p1.4.m4.1b"><apply id="S3.2.p1.4.m4.1.1.cmml" xref="S3.2.p1.4.m4.1.1"><and id="S3.2.p1.4.m4.1.1a.cmml" xref="S3.2.p1.4.m4.1.1"></and><apply id="S3.2.p1.4.m4.1.1b.cmml" xref="S3.2.p1.4.m4.1.1"><eq id="S3.2.p1.4.m4.1.1.3.cmml" xref="S3.2.p1.4.m4.1.1.3"></eq><ci id="S3.2.p1.4.m4.1.1.2.cmml" xref="S3.2.p1.4.m4.1.1.2">π§</ci><apply id="S3.2.p1.4.m4.1.1.4.cmml" xref="S3.2.p1.4.m4.1.1.4"><csymbol cd="latexml" id="S3.2.p1.4.m4.1.1.4.1.cmml" xref="S3.2.p1.4.m4.1.1.4.1">supremum</csymbol><ci id="S3.2.p1.4.m4.1.1.4.2.cmml" xref="S3.2.p1.4.m4.1.1.4.2">πΏ</ci></apply></apply><apply id="S3.2.p1.4.m4.1.1c.cmml" xref="S3.2.p1.4.m4.1.1"><in id="S3.2.p1.4.m4.1.1.5.cmml" xref="S3.2.p1.4.m4.1.1.5"></in><share href="https://arxiv.org/html/2503.13666v1#S3.2.p1.4.m4.1.1.4.cmml" id="S3.2.p1.4.m4.1.1d.cmml" xref="S3.2.p1.4.m4.1.1"></share><apply id="S3.2.p1.4.m4.1.1.6.cmml" xref="S3.2.p1.4.m4.1.1.6"><ci id="S3.2.p1.4.m4.1.1.6.1.cmml" xref="S3.2.p1.4.m4.1.1.6.1">Β―</ci><ci id="S3.2.p1.4.m4.1.1.6.2.cmml" xref="S3.2.p1.4.m4.1.1.6.2">πΏ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p1.4.m4.1c">z=\sup L\in\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p1.4.m4.1d">italic_z = roman_sup italic_L β overΒ― start_ARG italic_L end_ARG</annotation></semantics></math>. The countable tightness of <math alttext="X" class="ltx_Math" display="inline" id="S3.2.p1.5.m5.1"><semantics id="S3.2.p1.5.m5.1a"><mi id="S3.2.p1.5.m5.1.1" xref="S3.2.p1.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.2.p1.5.m5.1b"><ci id="S3.2.p1.5.m5.1.1.cmml" xref="S3.2.p1.5.m5.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p1.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.2.p1.5.m5.1d">italic_X</annotation></semantics></math> implies that there exists a countable subset <math alttext="A" class="ltx_Math" display="inline" id="S3.2.p1.6.m6.1"><semantics id="S3.2.p1.6.m6.1a"><mi id="S3.2.p1.6.m6.1.1" xref="S3.2.p1.6.m6.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.2.p1.6.m6.1b"><ci id="S3.2.p1.6.m6.1.1.cmml" xref="S3.2.p1.6.m6.1.1">π΄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p1.6.m6.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.2.p1.6.m6.1d">italic_A</annotation></semantics></math> of <math alttext="L" class="ltx_Math" display="inline" id="S3.2.p1.7.m7.1"><semantics id="S3.2.p1.7.m7.1a"><mi id="S3.2.p1.7.m7.1.1" xref="S3.2.p1.7.m7.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.2.p1.7.m7.1b"><ci id="S3.2.p1.7.m7.1.1.cmml" xref="S3.2.p1.7.m7.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p1.7.m7.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.2.p1.7.m7.1d">italic_L</annotation></semantics></math> such that <math alttext="z\in\overline{A}" class="ltx_Math" display="inline" id="S3.2.p1.8.m8.1"><semantics id="S3.2.p1.8.m8.1a"><mrow id="S3.2.p1.8.m8.1.1" xref="S3.2.p1.8.m8.1.1.cmml"><mi id="S3.2.p1.8.m8.1.1.2" xref="S3.2.p1.8.m8.1.1.2.cmml">z</mi><mo id="S3.2.p1.8.m8.1.1.1" xref="S3.2.p1.8.m8.1.1.1.cmml">β</mo><mover accent="true" id="S3.2.p1.8.m8.1.1.3" xref="S3.2.p1.8.m8.1.1.3.cmml"><mi id="S3.2.p1.8.m8.1.1.3.2" xref="S3.2.p1.8.m8.1.1.3.2.cmml">A</mi><mo id="S3.2.p1.8.m8.1.1.3.1" xref="S3.2.p1.8.m8.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p1.8.m8.1b"><apply id="S3.2.p1.8.m8.1.1.cmml" xref="S3.2.p1.8.m8.1.1"><in id="S3.2.p1.8.m8.1.1.1.cmml" xref="S3.2.p1.8.m8.1.1.1"></in><ci id="S3.2.p1.8.m8.1.1.2.cmml" xref="S3.2.p1.8.m8.1.1.2">π§</ci><apply id="S3.2.p1.8.m8.1.1.3.cmml" xref="S3.2.p1.8.m8.1.1.3"><ci id="S3.2.p1.8.m8.1.1.3.1.cmml" xref="S3.2.p1.8.m8.1.1.3.1">Β―</ci><ci id="S3.2.p1.8.m8.1.1.3.2.cmml" xref="S3.2.p1.8.m8.1.1.3.2">π΄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p1.8.m8.1c">z\in\overline{A}</annotation><annotation encoding="application/x-llamapun" id="S3.2.p1.8.m8.1d">italic_z β overΒ― start_ARG italic_A end_ARG</annotation></semantics></math>. Then there exists <math alttext="l\in L" class="ltx_Math" display="inline" id="S3.2.p1.9.m9.1"><semantics id="S3.2.p1.9.m9.1a"><mrow id="S3.2.p1.9.m9.1.1" xref="S3.2.p1.9.m9.1.1.cmml"><mi id="S3.2.p1.9.m9.1.1.2" xref="S3.2.p1.9.m9.1.1.2.cmml">l</mi><mo id="S3.2.p1.9.m9.1.1.1" xref="S3.2.p1.9.m9.1.1.1.cmml">β</mo><mi id="S3.2.p1.9.m9.1.1.3" xref="S3.2.p1.9.m9.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p1.9.m9.1b"><apply id="S3.2.p1.9.m9.1.1.cmml" xref="S3.2.p1.9.m9.1.1"><in id="S3.2.p1.9.m9.1.1.1.cmml" xref="S3.2.p1.9.m9.1.1.1"></in><ci id="S3.2.p1.9.m9.1.1.2.cmml" xref="S3.2.p1.9.m9.1.1.2">π</ci><ci id="S3.2.p1.9.m9.1.1.3.cmml" xref="S3.2.p1.9.m9.1.1.3">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p1.9.m9.1c">l\in L</annotation><annotation encoding="application/x-llamapun" id="S3.2.p1.9.m9.1d">italic_l β italic_L</annotation></semantics></math> such that <math alttext="A\subseteq{\downarrow}l" class="ltx_math_unparsed" display="inline" id="S3.2.p1.10.m10.1"><semantics id="S3.2.p1.10.m10.1a"><mrow id="S3.2.p1.10.m10.1b"><mi id="S3.2.p1.10.m10.1.1">A</mi><mo id="S3.2.p1.10.m10.1.2" rspace="0em">β</mo><mo id="S3.2.p1.10.m10.1.3" lspace="0em" stretchy="false">β</mo><mi id="S3.2.p1.10.m10.1.4">l</mi></mrow><annotation encoding="application/x-tex" id="S3.2.p1.10.m10.1c">A\subseteq{\downarrow}l</annotation><annotation encoding="application/x-llamapun" id="S3.2.p1.10.m10.1d">italic_A β β italic_l</annotation></semantics></math>. Since <math alttext="X" class="ltx_Math" display="inline" id="S3.2.p1.11.m11.1"><semantics id="S3.2.p1.11.m11.1a"><mi id="S3.2.p1.11.m11.1.1" xref="S3.2.p1.11.m11.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.2.p1.11.m11.1b"><ci id="S3.2.p1.11.m11.1.1.cmml" xref="S3.2.p1.11.m11.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p1.11.m11.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.2.p1.11.m11.1d">italic_X</annotation></semantics></math> is a semitopological semilattice, <math alttext="X\setminus{\downarrow}l" class="ltx_Math" display="inline" id="S3.2.p1.12.m12.1"><semantics id="S3.2.p1.12.m12.1a"><mrow id="S3.2.p1.12.m12.1.1" xref="S3.2.p1.12.m12.1.1.cmml"><mrow id="S3.2.p1.12.m12.1.1.2" xref="S3.2.p1.12.m12.1.1.2.cmml"><mi id="S3.2.p1.12.m12.1.1.2.2" xref="S3.2.p1.12.m12.1.1.2.2.cmml">X</mi><mo id="S3.2.p1.12.m12.1.1.2.3" rspace="0em" xref="S3.2.p1.12.m12.1.1.2.3.cmml">β</mo></mrow><mo id="S3.2.p1.12.m12.1.1.1" lspace="0em" stretchy="false" xref="S3.2.p1.12.m12.1.1.1.cmml">β</mo><mi id="S3.2.p1.12.m12.1.1.3" xref="S3.2.p1.12.m12.1.1.3.cmml">l</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p1.12.m12.1b"><apply id="S3.2.p1.12.m12.1.1.cmml" xref="S3.2.p1.12.m12.1.1"><ci id="S3.2.p1.12.m12.1.1.1.cmml" xref="S3.2.p1.12.m12.1.1.1">β</ci><apply id="S3.2.p1.12.m12.1.1.2.cmml" xref="S3.2.p1.12.m12.1.1.2"><csymbol cd="latexml" id="S3.2.p1.12.m12.1.1.2.1.cmml" xref="S3.2.p1.12.m12.1.1.2">limit-from</csymbol><ci id="S3.2.p1.12.m12.1.1.2.2.cmml" xref="S3.2.p1.12.m12.1.1.2.2">π</ci><setdiff id="S3.2.p1.12.m12.1.1.2.3.cmml" xref="S3.2.p1.12.m12.1.1.2.3"></setdiff></apply><ci id="S3.2.p1.12.m12.1.1.3.cmml" xref="S3.2.p1.12.m12.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p1.12.m12.1c">X\setminus{\downarrow}l</annotation><annotation encoding="application/x-llamapun" id="S3.2.p1.12.m12.1d">italic_X β β italic_l</annotation></semantics></math> is an open neighborhood of <math alttext="z" class="ltx_Math" display="inline" id="S3.2.p1.13.m13.1"><semantics id="S3.2.p1.13.m13.1a"><mi id="S3.2.p1.13.m13.1.1" xref="S3.2.p1.13.m13.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S3.2.p1.13.m13.1b"><ci id="S3.2.p1.13.m13.1.1.cmml" xref="S3.2.p1.13.m13.1.1">π§</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p1.13.m13.1c">z</annotation><annotation encoding="application/x-llamapun" id="S3.2.p1.13.m13.1d">italic_z</annotation></semantics></math> disjoint with <math alttext="A" class="ltx_Math" display="inline" id="S3.2.p1.14.m14.1"><semantics id="S3.2.p1.14.m14.1a"><mi id="S3.2.p1.14.m14.1.1" xref="S3.2.p1.14.m14.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.2.p1.14.m14.1b"><ci id="S3.2.p1.14.m14.1.1.cmml" xref="S3.2.p1.14.m14.1.1">π΄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p1.14.m14.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.2.p1.14.m14.1d">italic_A</annotation></semantics></math>, a contradiction. In case when <math alttext="L" class="ltx_Math" display="inline" id="S3.2.p1.15.m15.1"><semantics id="S3.2.p1.15.m15.1a"><mi id="S3.2.p1.15.m15.1.1" xref="S3.2.p1.15.m15.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.2.p1.15.m15.1b"><ci id="S3.2.p1.15.m15.1.1.cmml" xref="S3.2.p1.15.m15.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p1.15.m15.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.2.p1.15.m15.1d">italic_L</annotation></semantics></math> is isomorphic to <math alttext="(\omega_{1},\max)" class="ltx_Math" display="inline" id="S3.2.p1.16.m16.2"><semantics id="S3.2.p1.16.m16.2a"><mrow id="S3.2.p1.16.m16.2.2.1" xref="S3.2.p1.16.m16.2.2.2.cmml"><mo id="S3.2.p1.16.m16.2.2.1.2" stretchy="false" xref="S3.2.p1.16.m16.2.2.2.cmml">(</mo><msub id="S3.2.p1.16.m16.2.2.1.1" xref="S3.2.p1.16.m16.2.2.1.1.cmml"><mi id="S3.2.p1.16.m16.2.2.1.1.2" xref="S3.2.p1.16.m16.2.2.1.1.2.cmml">Ο</mi><mn id="S3.2.p1.16.m16.2.2.1.1.3" xref="S3.2.p1.16.m16.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.2.p1.16.m16.2.2.1.3" xref="S3.2.p1.16.m16.2.2.2.cmml">,</mo><mi id="S3.2.p1.16.m16.1.1" xref="S3.2.p1.16.m16.1.1.cmml">max</mi><mo id="S3.2.p1.16.m16.2.2.1.4" stretchy="false" xref="S3.2.p1.16.m16.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.2.p1.16.m16.2b"><interval closure="open" id="S3.2.p1.16.m16.2.2.2.cmml" xref="S3.2.p1.16.m16.2.2.1"><apply id="S3.2.p1.16.m16.2.2.1.1.cmml" xref="S3.2.p1.16.m16.2.2.1.1"><csymbol cd="ambiguous" id="S3.2.p1.16.m16.2.2.1.1.1.cmml" xref="S3.2.p1.16.m16.2.2.1.1">subscript</csymbol><ci id="S3.2.p1.16.m16.2.2.1.1.2.cmml" xref="S3.2.p1.16.m16.2.2.1.1.2">π</ci><cn id="S3.2.p1.16.m16.2.2.1.1.3.cmml" type="integer" xref="S3.2.p1.16.m16.2.2.1.1.3">1</cn></apply><max id="S3.2.p1.16.m16.1.1.cmml" xref="S3.2.p1.16.m16.1.1"></max></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.2.p1.16.m16.2c">(\omega_{1},\max)</annotation><annotation encoding="application/x-llamapun" id="S3.2.p1.16.m16.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_max )</annotation></semantics></math>, the proof is similar. β</p> </div> </div> <div class="ltx_para" id="S3.p5"> <p class="ltx_p" id="S3.p5.8">We are in a position to prove Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem2" title="Theorem B. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">B</span></a>. We need to show that for a Tychonoff Nyikos topological semilattice <math alttext="X" class="ltx_Math" display="inline" id="S3.p5.1.m1.1"><semantics id="S3.p5.1.m1.1a"><mi id="S3.p5.1.m1.1.1" xref="S3.p5.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.p5.1.m1.1b"><ci id="S3.p5.1.m1.1.1.cmml" xref="S3.p5.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.p5.1.m1.1d">italic_X</annotation></semantics></math> the following conditions are equivalent: (i) <math alttext="X" class="ltx_Math" display="inline" id="S3.p5.2.m2.1"><semantics id="S3.p5.2.m2.1a"><mi id="S3.p5.2.m2.1.1" xref="S3.p5.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.p5.2.m2.1b"><ci id="S3.p5.2.m2.1.1.cmml" xref="S3.p5.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.p5.2.m2.1d">italic_X</annotation></semantics></math> is compact; (ii) no chain in <math alttext="X" class="ltx_Math" display="inline" id="S3.p5.3.m3.1"><semantics id="S3.p5.3.m3.1a"><mi id="S3.p5.3.m3.1.1" xref="S3.p5.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.p5.3.m3.1b"><ci id="S3.p5.3.m3.1.1.cmml" xref="S3.p5.3.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.p5.3.m3.1d">italic_X</annotation></semantics></math> is isomorphic to <math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S3.p5.4.m4.2"><semantics id="S3.p5.4.m4.2a"><mrow id="S3.p5.4.m4.2.2.1" xref="S3.p5.4.m4.2.2.2.cmml"><mo id="S3.p5.4.m4.2.2.1.2" stretchy="false" xref="S3.p5.4.m4.2.2.2.cmml">(</mo><msub id="S3.p5.4.m4.2.2.1.1" xref="S3.p5.4.m4.2.2.1.1.cmml"><mi id="S3.p5.4.m4.2.2.1.1.2" xref="S3.p5.4.m4.2.2.1.1.2.cmml">Ο</mi><mn id="S3.p5.4.m4.2.2.1.1.3" xref="S3.p5.4.m4.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.p5.4.m4.2.2.1.3" xref="S3.p5.4.m4.2.2.2.cmml">,</mo><mi id="S3.p5.4.m4.1.1" xref="S3.p5.4.m4.1.1.cmml">min</mi><mo id="S3.p5.4.m4.2.2.1.4" stretchy="false" xref="S3.p5.4.m4.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.4.m4.2b"><interval closure="open" id="S3.p5.4.m4.2.2.2.cmml" xref="S3.p5.4.m4.2.2.1"><apply id="S3.p5.4.m4.2.2.1.1.cmml" xref="S3.p5.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S3.p5.4.m4.2.2.1.1.1.cmml" xref="S3.p5.4.m4.2.2.1.1">subscript</csymbol><ci id="S3.p5.4.m4.2.2.1.1.2.cmml" xref="S3.p5.4.m4.2.2.1.1.2">π</ci><cn id="S3.p5.4.m4.2.2.1.1.3.cmml" type="integer" xref="S3.p5.4.m4.2.2.1.1.3">1</cn></apply><min id="S3.p5.4.m4.1.1.cmml" xref="S3.p5.4.m4.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.4.m4.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.4.m4.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math> or <math alttext="(\omega_{1},\max)" class="ltx_Math" display="inline" id="S3.p5.5.m5.2"><semantics id="S3.p5.5.m5.2a"><mrow id="S3.p5.5.m5.2.2.1" xref="S3.p5.5.m5.2.2.2.cmml"><mo id="S3.p5.5.m5.2.2.1.2" stretchy="false" xref="S3.p5.5.m5.2.2.2.cmml">(</mo><msub id="S3.p5.5.m5.2.2.1.1" xref="S3.p5.5.m5.2.2.1.1.cmml"><mi id="S3.p5.5.m5.2.2.1.1.2" xref="S3.p5.5.m5.2.2.1.1.2.cmml">Ο</mi><mn id="S3.p5.5.m5.2.2.1.1.3" xref="S3.p5.5.m5.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.p5.5.m5.2.2.1.3" xref="S3.p5.5.m5.2.2.2.cmml">,</mo><mi id="S3.p5.5.m5.1.1" xref="S3.p5.5.m5.1.1.cmml">max</mi><mo id="S3.p5.5.m5.2.2.1.4" stretchy="false" xref="S3.p5.5.m5.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.5.m5.2b"><interval closure="open" id="S3.p5.5.m5.2.2.2.cmml" xref="S3.p5.5.m5.2.2.1"><apply id="S3.p5.5.m5.2.2.1.1.cmml" xref="S3.p5.5.m5.2.2.1.1"><csymbol cd="ambiguous" id="S3.p5.5.m5.2.2.1.1.1.cmml" xref="S3.p5.5.m5.2.2.1.1">subscript</csymbol><ci id="S3.p5.5.m5.2.2.1.1.2.cmml" xref="S3.p5.5.m5.2.2.1.1.2">π</ci><cn id="S3.p5.5.m5.2.2.1.1.3.cmml" type="integer" xref="S3.p5.5.m5.2.2.1.1.3">1</cn></apply><max id="S3.p5.5.m5.1.1.cmml" xref="S3.p5.5.m5.1.1"></max></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.5.m5.2c">(\omega_{1},\max)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.5.m5.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_max )</annotation></semantics></math>; (iii) no chain in <math alttext="X" class="ltx_Math" display="inline" id="S3.p5.6.m6.1"><semantics id="S3.p5.6.m6.1a"><mi id="S3.p5.6.m6.1.1" xref="S3.p5.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.p5.6.m6.1b"><ci id="S3.p5.6.m6.1.1.cmml" xref="S3.p5.6.m6.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.p5.6.m6.1d">italic_X</annotation></semantics></math> is topologically isomorphic to <math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S3.p5.7.m7.2"><semantics id="S3.p5.7.m7.2a"><mrow id="S3.p5.7.m7.2.2.1" xref="S3.p5.7.m7.2.2.2.cmml"><mo id="S3.p5.7.m7.2.2.1.2" stretchy="false" xref="S3.p5.7.m7.2.2.2.cmml">(</mo><msub id="S3.p5.7.m7.2.2.1.1" xref="S3.p5.7.m7.2.2.1.1.cmml"><mi id="S3.p5.7.m7.2.2.1.1.2" xref="S3.p5.7.m7.2.2.1.1.2.cmml">Ο</mi><mn id="S3.p5.7.m7.2.2.1.1.3" xref="S3.p5.7.m7.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.p5.7.m7.2.2.1.3" xref="S3.p5.7.m7.2.2.2.cmml">,</mo><mi id="S3.p5.7.m7.1.1" xref="S3.p5.7.m7.1.1.cmml">min</mi><mo id="S3.p5.7.m7.2.2.1.4" stretchy="false" xref="S3.p5.7.m7.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.7.m7.2b"><interval closure="open" id="S3.p5.7.m7.2.2.2.cmml" xref="S3.p5.7.m7.2.2.1"><apply id="S3.p5.7.m7.2.2.1.1.cmml" xref="S3.p5.7.m7.2.2.1.1"><csymbol cd="ambiguous" id="S3.p5.7.m7.2.2.1.1.1.cmml" xref="S3.p5.7.m7.2.2.1.1">subscript</csymbol><ci id="S3.p5.7.m7.2.2.1.1.2.cmml" xref="S3.p5.7.m7.2.2.1.1.2">π</ci><cn id="S3.p5.7.m7.2.2.1.1.3.cmml" type="integer" xref="S3.p5.7.m7.2.2.1.1.3">1</cn></apply><min id="S3.p5.7.m7.1.1.cmml" xref="S3.p5.7.m7.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.7.m7.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.7.m7.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math> or <math alttext="(\omega_{1},\max)" class="ltx_Math" display="inline" id="S3.p5.8.m8.2"><semantics id="S3.p5.8.m8.2a"><mrow id="S3.p5.8.m8.2.2.1" xref="S3.p5.8.m8.2.2.2.cmml"><mo id="S3.p5.8.m8.2.2.1.2" stretchy="false" xref="S3.p5.8.m8.2.2.2.cmml">(</mo><msub id="S3.p5.8.m8.2.2.1.1" xref="S3.p5.8.m8.2.2.1.1.cmml"><mi id="S3.p5.8.m8.2.2.1.1.2" xref="S3.p5.8.m8.2.2.1.1.2.cmml">Ο</mi><mn id="S3.p5.8.m8.2.2.1.1.3" xref="S3.p5.8.m8.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.p5.8.m8.2.2.1.3" xref="S3.p5.8.m8.2.2.2.cmml">,</mo><mi id="S3.p5.8.m8.1.1" xref="S3.p5.8.m8.1.1.cmml">max</mi><mo id="S3.p5.8.m8.2.2.1.4" stretchy="false" xref="S3.p5.8.m8.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.8.m8.2b"><interval closure="open" id="S3.p5.8.m8.2.2.2.cmml" xref="S3.p5.8.m8.2.2.1"><apply id="S3.p5.8.m8.2.2.1.1.cmml" xref="S3.p5.8.m8.2.2.1.1"><csymbol cd="ambiguous" id="S3.p5.8.m8.2.2.1.1.1.cmml" xref="S3.p5.8.m8.2.2.1.1">subscript</csymbol><ci id="S3.p5.8.m8.2.2.1.1.2.cmml" xref="S3.p5.8.m8.2.2.1.1.2">π</ci><cn id="S3.p5.8.m8.2.2.1.1.3.cmml" type="integer" xref="S3.p5.8.m8.2.2.1.1.3">1</cn></apply><max id="S3.p5.8.m8.1.1.cmml" xref="S3.p5.8.m8.1.1"></max></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.8.m8.2c">(\omega_{1},\max)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.8.m8.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_max )</annotation></semantics></math> equipped with the order topology.</p> </div> <div class="ltx_proof" id="S3.4"> <h6 class="ltx_title ltx_runin ltx_font_bold ltx_title_proof">Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem2" title="Theorem B. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">B</span></a>.</h6> <div class="ltx_para" id="S3.3.p1"> <p class="ltx_p" id="S3.3.p1.2">Since each first-countable space is countably tight, the implication (i) <math alttext="\Rightarrow" class="ltx_Math" display="inline" id="S3.3.p1.1.m1.1"><semantics id="S3.3.p1.1.m1.1a"><mo id="S3.3.p1.1.m1.1.1" stretchy="false" xref="S3.3.p1.1.m1.1.1.cmml">β</mo><annotation-xml encoding="MathML-Content" id="S3.3.p1.1.m1.1b"><ci id="S3.3.p1.1.m1.1.1.cmml" xref="S3.3.p1.1.m1.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.3.p1.1.m1.1c">\Rightarrow</annotation><annotation encoding="application/x-llamapun" id="S3.3.p1.1.m1.1d">β</annotation></semantics></math> (ii) follows from Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem6" title="Lemma 3.6. β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.6</span></a>. The implication (ii) <math alttext="\Rightarrow" class="ltx_Math" display="inline" id="S3.3.p1.2.m2.1"><semantics id="S3.3.p1.2.m2.1a"><mo id="S3.3.p1.2.m2.1.1" stretchy="false" xref="S3.3.p1.2.m2.1.1.cmml">β</mo><annotation-xml encoding="MathML-Content" id="S3.3.p1.2.m2.1b"><ci id="S3.3.p1.2.m2.1.1.cmml" xref="S3.3.p1.2.m2.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.3.p1.2.m2.1c">\Rightarrow</annotation><annotation encoding="application/x-llamapun" id="S3.3.p1.2.m2.1d">β</annotation></semantics></math> (iii) is obvious.</p> </div> <div class="ltx_para" id="S3.4.p2"> <p class="ltx_p" id="S3.4.p2.20">(iii) <math alttext="\Rightarrow" class="ltx_Math" display="inline" id="S3.4.p2.1.m1.1"><semantics id="S3.4.p2.1.m1.1a"><mo id="S3.4.p2.1.m1.1.1" stretchy="false" xref="S3.4.p2.1.m1.1.1.cmml">β</mo><annotation-xml encoding="MathML-Content" id="S3.4.p2.1.m1.1b"><ci id="S3.4.p2.1.m1.1.1.cmml" xref="S3.4.p2.1.m1.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.1.m1.1c">\Rightarrow</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.1.m1.1d">β</annotation></semantics></math> (i): Since <math alttext="X" class="ltx_Math" display="inline" id="S3.4.p2.2.m2.1"><semantics id="S3.4.p2.2.m2.1a"><mi id="S3.4.p2.2.m2.1.1" xref="S3.4.p2.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.4.p2.2.m2.1b"><ci id="S3.4.p2.2.m2.1.1.cmml" xref="S3.4.p2.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.2.m2.1d">italic_X</annotation></semantics></math> is first-countable, for any cardinal <math alttext="\kappa>\omega_{1}" class="ltx_Math" display="inline" id="S3.4.p2.3.m3.1"><semantics id="S3.4.p2.3.m3.1a"><mrow id="S3.4.p2.3.m3.1.1" xref="S3.4.p2.3.m3.1.1.cmml"><mi id="S3.4.p2.3.m3.1.1.2" xref="S3.4.p2.3.m3.1.1.2.cmml">ΞΊ</mi><mo id="S3.4.p2.3.m3.1.1.1" xref="S3.4.p2.3.m3.1.1.1.cmml">></mo><msub id="S3.4.p2.3.m3.1.1.3" xref="S3.4.p2.3.m3.1.1.3.cmml"><mi id="S3.4.p2.3.m3.1.1.3.2" xref="S3.4.p2.3.m3.1.1.3.2.cmml">Ο</mi><mn id="S3.4.p2.3.m3.1.1.3.3" xref="S3.4.p2.3.m3.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p2.3.m3.1b"><apply id="S3.4.p2.3.m3.1.1.cmml" xref="S3.4.p2.3.m3.1.1"><gt id="S3.4.p2.3.m3.1.1.1.cmml" xref="S3.4.p2.3.m3.1.1.1"></gt><ci id="S3.4.p2.3.m3.1.1.2.cmml" xref="S3.4.p2.3.m3.1.1.2">π </ci><apply id="S3.4.p2.3.m3.1.1.3.cmml" xref="S3.4.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.4.p2.3.m3.1.1.3.1.cmml" xref="S3.4.p2.3.m3.1.1.3">subscript</csymbol><ci id="S3.4.p2.3.m3.1.1.3.2.cmml" xref="S3.4.p2.3.m3.1.1.3.2">π</ci><cn id="S3.4.p2.3.m3.1.1.3.3.cmml" type="integer" xref="S3.4.p2.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.3.m3.1c">\kappa>\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.3.m3.1d">italic_ΞΊ > italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> there is no chain in <math alttext="X" class="ltx_Math" display="inline" id="S3.4.p2.4.m4.1"><semantics id="S3.4.p2.4.m4.1a"><mi id="S3.4.p2.4.m4.1.1" xref="S3.4.p2.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.4.p2.4.m4.1b"><ci id="S3.4.p2.4.m4.1.1.cmml" xref="S3.4.p2.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.4.m4.1d">italic_X</annotation></semantics></math> that is topologically isomorphic to <math alttext="(\kappa,\min)" class="ltx_Math" display="inline" id="S3.4.p2.5.m5.2"><semantics id="S3.4.p2.5.m5.2a"><mrow id="S3.4.p2.5.m5.2.3.2" xref="S3.4.p2.5.m5.2.3.1.cmml"><mo id="S3.4.p2.5.m5.2.3.2.1" stretchy="false" xref="S3.4.p2.5.m5.2.3.1.cmml">(</mo><mi id="S3.4.p2.5.m5.1.1" xref="S3.4.p2.5.m5.1.1.cmml">ΞΊ</mi><mo id="S3.4.p2.5.m5.2.3.2.2" xref="S3.4.p2.5.m5.2.3.1.cmml">,</mo><mi id="S3.4.p2.5.m5.2.2" xref="S3.4.p2.5.m5.2.2.cmml">min</mi><mo id="S3.4.p2.5.m5.2.3.2.3" stretchy="false" xref="S3.4.p2.5.m5.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p2.5.m5.2b"><interval closure="open" id="S3.4.p2.5.m5.2.3.1.cmml" xref="S3.4.p2.5.m5.2.3.2"><ci id="S3.4.p2.5.m5.1.1.cmml" xref="S3.4.p2.5.m5.1.1">π </ci><min id="S3.4.p2.5.m5.2.2.cmml" xref="S3.4.p2.5.m5.2.2"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.5.m5.2c">(\kappa,\min)</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.5.m5.2d">( italic_ΞΊ , roman_min )</annotation></semantics></math> or <math alttext="(\kappa,\max)" class="ltx_Math" display="inline" id="S3.4.p2.6.m6.2"><semantics id="S3.4.p2.6.m6.2a"><mrow id="S3.4.p2.6.m6.2.3.2" xref="S3.4.p2.6.m6.2.3.1.cmml"><mo id="S3.4.p2.6.m6.2.3.2.1" stretchy="false" xref="S3.4.p2.6.m6.2.3.1.cmml">(</mo><mi id="S3.4.p2.6.m6.1.1" xref="S3.4.p2.6.m6.1.1.cmml">ΞΊ</mi><mo id="S3.4.p2.6.m6.2.3.2.2" xref="S3.4.p2.6.m6.2.3.1.cmml">,</mo><mi id="S3.4.p2.6.m6.2.2" xref="S3.4.p2.6.m6.2.2.cmml">max</mi><mo id="S3.4.p2.6.m6.2.3.2.3" stretchy="false" xref="S3.4.p2.6.m6.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p2.6.m6.2b"><interval closure="open" id="S3.4.p2.6.m6.2.3.1.cmml" xref="S3.4.p2.6.m6.2.3.2"><ci id="S3.4.p2.6.m6.1.1.cmml" xref="S3.4.p2.6.m6.1.1">π </ci><max id="S3.4.p2.6.m6.2.2.cmml" xref="S3.4.p2.6.m6.2.2"></max></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.6.m6.2c">(\kappa,\max)</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.6.m6.2d">( italic_ΞΊ , roman_max )</annotation></semantics></math> endowed with the order topology. Since <math alttext="X" class="ltx_Math" display="inline" id="S3.4.p2.7.m7.1"><semantics id="S3.4.p2.7.m7.1a"><mi id="S3.4.p2.7.m7.1.1" xref="S3.4.p2.7.m7.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.4.p2.7.m7.1b"><ci id="S3.4.p2.7.m7.1.1.cmml" xref="S3.4.p2.7.m7.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.7.m7.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.7.m7.1d">italic_X</annotation></semantics></math> is countably compact, no chain in <math alttext="X" class="ltx_Math" display="inline" id="S3.4.p2.8.m8.1"><semantics id="S3.4.p2.8.m8.1a"><mi id="S3.4.p2.8.m8.1.1" xref="S3.4.p2.8.m8.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.4.p2.8.m8.1b"><ci id="S3.4.p2.8.m8.1.1.cmml" xref="S3.4.p2.8.m8.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.8.m8.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.8.m8.1d">italic_X</annotation></semantics></math> is topologically isomorphic to the discrete semilattices <math alttext="(\omega,\min)" class="ltx_Math" display="inline" id="S3.4.p2.9.m9.2"><semantics id="S3.4.p2.9.m9.2a"><mrow id="S3.4.p2.9.m9.2.3.2" xref="S3.4.p2.9.m9.2.3.1.cmml"><mo id="S3.4.p2.9.m9.2.3.2.1" stretchy="false" xref="S3.4.p2.9.m9.2.3.1.cmml">(</mo><mi id="S3.4.p2.9.m9.1.1" xref="S3.4.p2.9.m9.1.1.cmml">Ο</mi><mo id="S3.4.p2.9.m9.2.3.2.2" xref="S3.4.p2.9.m9.2.3.1.cmml">,</mo><mi id="S3.4.p2.9.m9.2.2" xref="S3.4.p2.9.m9.2.2.cmml">min</mi><mo id="S3.4.p2.9.m9.2.3.2.3" stretchy="false" xref="S3.4.p2.9.m9.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p2.9.m9.2b"><interval closure="open" id="S3.4.p2.9.m9.2.3.1.cmml" xref="S3.4.p2.9.m9.2.3.2"><ci id="S3.4.p2.9.m9.1.1.cmml" xref="S3.4.p2.9.m9.1.1">π</ci><min id="S3.4.p2.9.m9.2.2.cmml" xref="S3.4.p2.9.m9.2.2"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.9.m9.2c">(\omega,\min)</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.9.m9.2d">( italic_Ο , roman_min )</annotation></semantics></math> and <math alttext="(\omega,\max)" class="ltx_Math" display="inline" id="S3.4.p2.10.m10.2"><semantics id="S3.4.p2.10.m10.2a"><mrow id="S3.4.p2.10.m10.2.3.2" xref="S3.4.p2.10.m10.2.3.1.cmml"><mo id="S3.4.p2.10.m10.2.3.2.1" stretchy="false" xref="S3.4.p2.10.m10.2.3.1.cmml">(</mo><mi id="S3.4.p2.10.m10.1.1" xref="S3.4.p2.10.m10.1.1.cmml">Ο</mi><mo id="S3.4.p2.10.m10.2.3.2.2" xref="S3.4.p2.10.m10.2.3.1.cmml">,</mo><mi id="S3.4.p2.10.m10.2.2" xref="S3.4.p2.10.m10.2.2.cmml">max</mi><mo id="S3.4.p2.10.m10.2.3.2.3" stretchy="false" xref="S3.4.p2.10.m10.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p2.10.m10.2b"><interval closure="open" id="S3.4.p2.10.m10.2.3.1.cmml" xref="S3.4.p2.10.m10.2.3.2"><ci id="S3.4.p2.10.m10.1.1.cmml" xref="S3.4.p2.10.m10.1.1">π</ci><max id="S3.4.p2.10.m10.2.2.cmml" xref="S3.4.p2.10.m10.2.2"></max></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.10.m10.2c">(\omega,\max)</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.10.m10.2d">( italic_Ο , roman_max )</annotation></semantics></math>. By the assumption, no chain in <math alttext="X" class="ltx_Math" display="inline" id="S3.4.p2.11.m11.1"><semantics id="S3.4.p2.11.m11.1a"><mi id="S3.4.p2.11.m11.1.1" xref="S3.4.p2.11.m11.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.4.p2.11.m11.1b"><ci id="S3.4.p2.11.m11.1.1.cmml" xref="S3.4.p2.11.m11.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.11.m11.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.11.m11.1d">italic_X</annotation></semantics></math> is topologically isomorphic to <math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S3.4.p2.12.m12.2"><semantics id="S3.4.p2.12.m12.2a"><mrow id="S3.4.p2.12.m12.2.2.1" xref="S3.4.p2.12.m12.2.2.2.cmml"><mo id="S3.4.p2.12.m12.2.2.1.2" stretchy="false" xref="S3.4.p2.12.m12.2.2.2.cmml">(</mo><msub id="S3.4.p2.12.m12.2.2.1.1" xref="S3.4.p2.12.m12.2.2.1.1.cmml"><mi id="S3.4.p2.12.m12.2.2.1.1.2" xref="S3.4.p2.12.m12.2.2.1.1.2.cmml">Ο</mi><mn id="S3.4.p2.12.m12.2.2.1.1.3" xref="S3.4.p2.12.m12.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.4.p2.12.m12.2.2.1.3" xref="S3.4.p2.12.m12.2.2.2.cmml">,</mo><mi id="S3.4.p2.12.m12.1.1" xref="S3.4.p2.12.m12.1.1.cmml">min</mi><mo id="S3.4.p2.12.m12.2.2.1.4" stretchy="false" xref="S3.4.p2.12.m12.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p2.12.m12.2b"><interval closure="open" id="S3.4.p2.12.m12.2.2.2.cmml" xref="S3.4.p2.12.m12.2.2.1"><apply id="S3.4.p2.12.m12.2.2.1.1.cmml" xref="S3.4.p2.12.m12.2.2.1.1"><csymbol cd="ambiguous" id="S3.4.p2.12.m12.2.2.1.1.1.cmml" xref="S3.4.p2.12.m12.2.2.1.1">subscript</csymbol><ci id="S3.4.p2.12.m12.2.2.1.1.2.cmml" xref="S3.4.p2.12.m12.2.2.1.1.2">π</ci><cn id="S3.4.p2.12.m12.2.2.1.1.3.cmml" type="integer" xref="S3.4.p2.12.m12.2.2.1.1.3">1</cn></apply><min id="S3.4.p2.12.m12.1.1.cmml" xref="S3.4.p2.12.m12.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.12.m12.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.12.m12.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math> or <math alttext="(\omega_{1},\max)" class="ltx_Math" display="inline" id="S3.4.p2.13.m13.2"><semantics id="S3.4.p2.13.m13.2a"><mrow id="S3.4.p2.13.m13.2.2.1" xref="S3.4.p2.13.m13.2.2.2.cmml"><mo id="S3.4.p2.13.m13.2.2.1.2" stretchy="false" xref="S3.4.p2.13.m13.2.2.2.cmml">(</mo><msub id="S3.4.p2.13.m13.2.2.1.1" xref="S3.4.p2.13.m13.2.2.1.1.cmml"><mi id="S3.4.p2.13.m13.2.2.1.1.2" xref="S3.4.p2.13.m13.2.2.1.1.2.cmml">Ο</mi><mn id="S3.4.p2.13.m13.2.2.1.1.3" xref="S3.4.p2.13.m13.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.4.p2.13.m13.2.2.1.3" xref="S3.4.p2.13.m13.2.2.2.cmml">,</mo><mi id="S3.4.p2.13.m13.1.1" xref="S3.4.p2.13.m13.1.1.cmml">max</mi><mo id="S3.4.p2.13.m13.2.2.1.4" stretchy="false" xref="S3.4.p2.13.m13.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p2.13.m13.2b"><interval closure="open" id="S3.4.p2.13.m13.2.2.2.cmml" xref="S3.4.p2.13.m13.2.2.1"><apply id="S3.4.p2.13.m13.2.2.1.1.cmml" xref="S3.4.p2.13.m13.2.2.1.1"><csymbol cd="ambiguous" id="S3.4.p2.13.m13.2.2.1.1.1.cmml" xref="S3.4.p2.13.m13.2.2.1.1">subscript</csymbol><ci id="S3.4.p2.13.m13.2.2.1.1.2.cmml" xref="S3.4.p2.13.m13.2.2.1.1.2">π</ci><cn id="S3.4.p2.13.m13.2.2.1.1.3.cmml" type="integer" xref="S3.4.p2.13.m13.2.2.1.1.3">1</cn></apply><max id="S3.4.p2.13.m13.1.1.cmml" xref="S3.4.p2.13.m13.1.1"></max></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.13.m13.2c">(\omega_{1},\max)</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.13.m13.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_max )</annotation></semantics></math> endowed with the order topology. Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem1" title="Theorem 3.1 (Banakh, Bardyla). β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.1</span></a>(3) implies that the semilattice <math alttext="X" class="ltx_Math" display="inline" id="S3.4.p2.14.m14.1"><semantics id="S3.4.p2.14.m14.1a"><mi id="S3.4.p2.14.m14.1.1" xref="S3.4.p2.14.m14.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.4.p2.14.m14.1b"><ci id="S3.4.p2.14.m14.1.1.cmml" xref="S3.4.p2.14.m14.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.14.m14.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.14.m14.1d">italic_X</annotation></semantics></math> is chain-compact. By <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib22" title="">22</a>, Corollary 3.10.15]</cite>, <math alttext="X{\times}X" class="ltx_Math" display="inline" id="S3.4.p2.15.m15.1"><semantics id="S3.4.p2.15.m15.1a"><mrow id="S3.4.p2.15.m15.1.1" xref="S3.4.p2.15.m15.1.1.cmml"><mi id="S3.4.p2.15.m15.1.1.2" xref="S3.4.p2.15.m15.1.1.2.cmml">X</mi><mo id="S3.4.p2.15.m15.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.4.p2.15.m15.1.1.1.cmml">Γ</mo><mi id="S3.4.p2.15.m15.1.1.3" xref="S3.4.p2.15.m15.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p2.15.m15.1b"><apply id="S3.4.p2.15.m15.1.1.cmml" xref="S3.4.p2.15.m15.1.1"><times id="S3.4.p2.15.m15.1.1.1.cmml" xref="S3.4.p2.15.m15.1.1.1"></times><ci id="S3.4.p2.15.m15.1.1.2.cmml" xref="S3.4.p2.15.m15.1.1.2">π</ci><ci id="S3.4.p2.15.m15.1.1.3.cmml" xref="S3.4.p2.15.m15.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.15.m15.1c">X{\times}X</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.15.m15.1d">italic_X Γ italic_X</annotation></semantics></math> is countable compact and thus pseudocompact. By Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S2.Thmtheorem9" title="Theorem 2.9 (Banakh, Dimitrova, Gutik). β£ 2. Compact-like semigroups with dense inverse subsemigroups and the automatic continuity of inversion β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">2.9</span></a>, <math alttext="\beta X" class="ltx_Math" display="inline" id="S3.4.p2.16.m16.1"><semantics id="S3.4.p2.16.m16.1a"><mrow id="S3.4.p2.16.m16.1.1" xref="S3.4.p2.16.m16.1.1.cmml"><mi id="S3.4.p2.16.m16.1.1.2" xref="S3.4.p2.16.m16.1.1.2.cmml">Ξ²</mi><mo id="S3.4.p2.16.m16.1.1.1" xref="S3.4.p2.16.m16.1.1.1.cmml">β’</mo><mi id="S3.4.p2.16.m16.1.1.3" xref="S3.4.p2.16.m16.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p2.16.m16.1b"><apply id="S3.4.p2.16.m16.1.1.cmml" xref="S3.4.p2.16.m16.1.1"><times id="S3.4.p2.16.m16.1.1.1.cmml" xref="S3.4.p2.16.m16.1.1.1"></times><ci id="S3.4.p2.16.m16.1.1.2.cmml" xref="S3.4.p2.16.m16.1.1.2">π½</ci><ci id="S3.4.p2.16.m16.1.1.3.cmml" xref="S3.4.p2.16.m16.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.16.m16.1c">\beta X</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.16.m16.1d">italic_Ξ² italic_X</annotation></semantics></math> is a topological semigroup that contains <math alttext="X" class="ltx_Math" display="inline" id="S3.4.p2.17.m17.1"><semantics id="S3.4.p2.17.m17.1a"><mi id="S3.4.p2.17.m17.1.1" xref="S3.4.p2.17.m17.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.4.p2.17.m17.1b"><ci id="S3.4.p2.17.m17.1.1.cmml" xref="S3.4.p2.17.m17.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.17.m17.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.17.m17.1d">italic_X</annotation></semantics></math> as a dense subsemilattice. Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem2" title="Theorem 3.2 (Banakh, Bardyla). β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.2</span></a> implies that <math alttext="X" class="ltx_Math" display="inline" id="S3.4.p2.18.m18.1"><semantics id="S3.4.p2.18.m18.1a"><mi id="S3.4.p2.18.m18.1.1" xref="S3.4.p2.18.m18.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.4.p2.18.m18.1b"><ci id="S3.4.p2.18.m18.1.1.cmml" xref="S3.4.p2.18.m18.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.18.m18.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.18.m18.1d">italic_X</annotation></semantics></math> is closed in <math alttext="\beta X" class="ltx_Math" display="inline" id="S3.4.p2.19.m19.1"><semantics id="S3.4.p2.19.m19.1a"><mrow id="S3.4.p2.19.m19.1.1" xref="S3.4.p2.19.m19.1.1.cmml"><mi id="S3.4.p2.19.m19.1.1.2" xref="S3.4.p2.19.m19.1.1.2.cmml">Ξ²</mi><mo id="S3.4.p2.19.m19.1.1.1" xref="S3.4.p2.19.m19.1.1.1.cmml">β’</mo><mi id="S3.4.p2.19.m19.1.1.3" xref="S3.4.p2.19.m19.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p2.19.m19.1b"><apply id="S3.4.p2.19.m19.1.1.cmml" xref="S3.4.p2.19.m19.1.1"><times id="S3.4.p2.19.m19.1.1.1.cmml" xref="S3.4.p2.19.m19.1.1.1"></times><ci id="S3.4.p2.19.m19.1.1.2.cmml" xref="S3.4.p2.19.m19.1.1.2">π½</ci><ci id="S3.4.p2.19.m19.1.1.3.cmml" xref="S3.4.p2.19.m19.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.19.m19.1c">\beta X</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.19.m19.1d">italic_Ξ² italic_X</annotation></semantics></math> and thus <math alttext="X=\beta X" class="ltx_Math" display="inline" id="S3.4.p2.20.m20.1"><semantics id="S3.4.p2.20.m20.1a"><mrow id="S3.4.p2.20.m20.1.1" xref="S3.4.p2.20.m20.1.1.cmml"><mi id="S3.4.p2.20.m20.1.1.2" xref="S3.4.p2.20.m20.1.1.2.cmml">X</mi><mo id="S3.4.p2.20.m20.1.1.1" xref="S3.4.p2.20.m20.1.1.1.cmml">=</mo><mrow id="S3.4.p2.20.m20.1.1.3" xref="S3.4.p2.20.m20.1.1.3.cmml"><mi id="S3.4.p2.20.m20.1.1.3.2" xref="S3.4.p2.20.m20.1.1.3.2.cmml">Ξ²</mi><mo id="S3.4.p2.20.m20.1.1.3.1" xref="S3.4.p2.20.m20.1.1.3.1.cmml">β’</mo><mi id="S3.4.p2.20.m20.1.1.3.3" xref="S3.4.p2.20.m20.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.4.p2.20.m20.1b"><apply id="S3.4.p2.20.m20.1.1.cmml" xref="S3.4.p2.20.m20.1.1"><eq id="S3.4.p2.20.m20.1.1.1.cmml" xref="S3.4.p2.20.m20.1.1.1"></eq><ci id="S3.4.p2.20.m20.1.1.2.cmml" xref="S3.4.p2.20.m20.1.1.2">π</ci><apply id="S3.4.p2.20.m20.1.1.3.cmml" xref="S3.4.p2.20.m20.1.1.3"><times id="S3.4.p2.20.m20.1.1.3.1.cmml" xref="S3.4.p2.20.m20.1.1.3.1"></times><ci id="S3.4.p2.20.m20.1.1.3.2.cmml" xref="S3.4.p2.20.m20.1.1.3.2">π½</ci><ci id="S3.4.p2.20.m20.1.1.3.3.cmml" xref="S3.4.p2.20.m20.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.4.p2.20.m20.1c">X=\beta X</annotation><annotation encoding="application/x-llamapun" id="S3.4.p2.20.m20.1d">italic_X = italic_Ξ² italic_X</annotation></semantics></math> is compact. β</p> </div> </div> <div class="ltx_para" id="S3.p6"> <p class="ltx_p" id="S3.p6.1">The rest of this section contains technical results on chains in topological semilattices. They will be used to prove the compactness of locally compact Nyikos topological semilattices (see Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S4.Thmtheorem12" title="Theorem 4.12. β£ 4. Nyikos semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">4.12</span></a>), which is a milestone in the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem1" title="Theorem A. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">A</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem7.1.1.1">Lemma 3.7</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem7.p1"> <p class="ltx_p" id="S3.Thmtheorem7.p1.2"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem7.p1.2.2">Let <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.1.1.m1.1"><semantics id="S3.Thmtheorem7.p1.1.1.m1.1a"><mi id="S3.Thmtheorem7.p1.1.1.m1.1.1" xref="S3.Thmtheorem7.p1.1.1.m1.1.1.cmml">Ξ±</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.1.1.m1.1b"><ci id="S3.Thmtheorem7.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem7.p1.1.1.m1.1.1">πΌ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.1.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.1.1.m1.1d">italic_Ξ±</annotation></semantics></math> be an ordinal and <math alttext="Y" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.2.2.m2.1"><semantics id="S3.Thmtheorem7.p1.2.2.m2.1a"><mi id="S3.Thmtheorem7.p1.2.2.m2.1.1" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.2.2.m2.1b"><ci id="S3.Thmtheorem7.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.2.2.m2.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.2.2.m2.1d">italic_Y</annotation></semantics></math> be a semilattice. Then the following assertions hold:</span></p> <ol class="ltx_enumerate" id="S3.I2"> <li class="ltx_item" id="S3.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S3.I2.i1.p1"> <p class="ltx_p" id="S3.I2.i1.p1.4"><span class="ltx_text ltx_font_italic" id="S3.I2.i1.p1.4.1">if </span><math alttext="h:(\alpha,\min)\rightarrow Y" class="ltx_Math" display="inline" id="S3.I2.i1.p1.1.m1.2"><semantics id="S3.I2.i1.p1.1.m1.2a"><mrow id="S3.I2.i1.p1.1.m1.2.3" xref="S3.I2.i1.p1.1.m1.2.3.cmml"><mi id="S3.I2.i1.p1.1.m1.2.3.2" xref="S3.I2.i1.p1.1.m1.2.3.2.cmml">h</mi><mo id="S3.I2.i1.p1.1.m1.2.3.1" lspace="0.278em" rspace="0.278em" xref="S3.I2.i1.p1.1.m1.2.3.1.cmml">:</mo><mrow id="S3.I2.i1.p1.1.m1.2.3.3" xref="S3.I2.i1.p1.1.m1.2.3.3.cmml"><mrow id="S3.I2.i1.p1.1.m1.2.3.3.2.2" xref="S3.I2.i1.p1.1.m1.2.3.3.2.1.cmml"><mo id="S3.I2.i1.p1.1.m1.2.3.3.2.2.1" stretchy="false" xref="S3.I2.i1.p1.1.m1.2.3.3.2.1.cmml">(</mo><mi id="S3.I2.i1.p1.1.m1.1.1" xref="S3.I2.i1.p1.1.m1.1.1.cmml">Ξ±</mi><mo id="S3.I2.i1.p1.1.m1.2.3.3.2.2.2" xref="S3.I2.i1.p1.1.m1.2.3.3.2.1.cmml">,</mo><mi id="S3.I2.i1.p1.1.m1.2.2" xref="S3.I2.i1.p1.1.m1.2.2.cmml">min</mi><mo id="S3.I2.i1.p1.1.m1.2.3.3.2.2.3" stretchy="false" xref="S3.I2.i1.p1.1.m1.2.3.3.2.1.cmml">)</mo></mrow><mo id="S3.I2.i1.p1.1.m1.2.3.3.1" stretchy="false" xref="S3.I2.i1.p1.1.m1.2.3.3.1.cmml">β</mo><mi id="S3.I2.i1.p1.1.m1.2.3.3.3" xref="S3.I2.i1.p1.1.m1.2.3.3.3.cmml">Y</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.1.m1.2b"><apply id="S3.I2.i1.p1.1.m1.2.3.cmml" xref="S3.I2.i1.p1.1.m1.2.3"><ci id="S3.I2.i1.p1.1.m1.2.3.1.cmml" xref="S3.I2.i1.p1.1.m1.2.3.1">:</ci><ci id="S3.I2.i1.p1.1.m1.2.3.2.cmml" xref="S3.I2.i1.p1.1.m1.2.3.2">β</ci><apply id="S3.I2.i1.p1.1.m1.2.3.3.cmml" xref="S3.I2.i1.p1.1.m1.2.3.3"><ci id="S3.I2.i1.p1.1.m1.2.3.3.1.cmml" xref="S3.I2.i1.p1.1.m1.2.3.3.1">β</ci><interval closure="open" id="S3.I2.i1.p1.1.m1.2.3.3.2.1.cmml" xref="S3.I2.i1.p1.1.m1.2.3.3.2.2"><ci id="S3.I2.i1.p1.1.m1.1.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1">πΌ</ci><min id="S3.I2.i1.p1.1.m1.2.2.cmml" xref="S3.I2.i1.p1.1.m1.2.2"></min></interval><ci id="S3.I2.i1.p1.1.m1.2.3.3.3.cmml" xref="S3.I2.i1.p1.1.m1.2.3.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.1.m1.2c">h:(\alpha,\min)\rightarrow Y</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.1.m1.2d">italic_h : ( italic_Ξ± , roman_min ) β italic_Y</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.i1.p1.4.2"> is a homomorphism, then </span><math alttext="h(\alpha)" class="ltx_Math" display="inline" id="S3.I2.i1.p1.2.m2.1"><semantics id="S3.I2.i1.p1.2.m2.1a"><mrow id="S3.I2.i1.p1.2.m2.1.2" xref="S3.I2.i1.p1.2.m2.1.2.cmml"><mi id="S3.I2.i1.p1.2.m2.1.2.2" xref="S3.I2.i1.p1.2.m2.1.2.2.cmml">h</mi><mo id="S3.I2.i1.p1.2.m2.1.2.1" xref="S3.I2.i1.p1.2.m2.1.2.1.cmml">β’</mo><mrow id="S3.I2.i1.p1.2.m2.1.2.3.2" xref="S3.I2.i1.p1.2.m2.1.2.cmml"><mo id="S3.I2.i1.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S3.I2.i1.p1.2.m2.1.2.cmml">(</mo><mi id="S3.I2.i1.p1.2.m2.1.1" xref="S3.I2.i1.p1.2.m2.1.1.cmml">Ξ±</mi><mo id="S3.I2.i1.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S3.I2.i1.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.2.m2.1b"><apply id="S3.I2.i1.p1.2.m2.1.2.cmml" xref="S3.I2.i1.p1.2.m2.1.2"><times id="S3.I2.i1.p1.2.m2.1.2.1.cmml" xref="S3.I2.i1.p1.2.m2.1.2.1"></times><ci id="S3.I2.i1.p1.2.m2.1.2.2.cmml" xref="S3.I2.i1.p1.2.m2.1.2.2">β</ci><ci id="S3.I2.i1.p1.2.m2.1.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1">πΌ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.2.m2.1c">h(\alpha)</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.2.m2.1d">italic_h ( italic_Ξ± )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.i1.p1.4.3"> is isomorphic to </span><math alttext="(\beta,\min)" class="ltx_Math" display="inline" id="S3.I2.i1.p1.3.m3.2"><semantics id="S3.I2.i1.p1.3.m3.2a"><mrow id="S3.I2.i1.p1.3.m3.2.3.2" xref="S3.I2.i1.p1.3.m3.2.3.1.cmml"><mo id="S3.I2.i1.p1.3.m3.2.3.2.1" stretchy="false" xref="S3.I2.i1.p1.3.m3.2.3.1.cmml">(</mo><mi id="S3.I2.i1.p1.3.m3.1.1" xref="S3.I2.i1.p1.3.m3.1.1.cmml">Ξ²</mi><mo id="S3.I2.i1.p1.3.m3.2.3.2.2" xref="S3.I2.i1.p1.3.m3.2.3.1.cmml">,</mo><mi id="S3.I2.i1.p1.3.m3.2.2" xref="S3.I2.i1.p1.3.m3.2.2.cmml">min</mi><mo id="S3.I2.i1.p1.3.m3.2.3.2.3" stretchy="false" xref="S3.I2.i1.p1.3.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.3.m3.2b"><interval closure="open" id="S3.I2.i1.p1.3.m3.2.3.1.cmml" xref="S3.I2.i1.p1.3.m3.2.3.2"><ci id="S3.I2.i1.p1.3.m3.1.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1">π½</ci><min id="S3.I2.i1.p1.3.m3.2.2.cmml" xref="S3.I2.i1.p1.3.m3.2.2"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.3.m3.2c">(\beta,\min)</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.3.m3.2d">( italic_Ξ² , roman_min )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.i1.p1.4.4"> for some </span><math alttext="\beta\leq\alpha" class="ltx_Math" display="inline" id="S3.I2.i1.p1.4.m4.1"><semantics id="S3.I2.i1.p1.4.m4.1a"><mrow id="S3.I2.i1.p1.4.m4.1.1" xref="S3.I2.i1.p1.4.m4.1.1.cmml"><mi id="S3.I2.i1.p1.4.m4.1.1.2" xref="S3.I2.i1.p1.4.m4.1.1.2.cmml">Ξ²</mi><mo id="S3.I2.i1.p1.4.m4.1.1.1" xref="S3.I2.i1.p1.4.m4.1.1.1.cmml">β€</mo><mi id="S3.I2.i1.p1.4.m4.1.1.3" xref="S3.I2.i1.p1.4.m4.1.1.3.cmml">Ξ±</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.4.m4.1b"><apply id="S3.I2.i1.p1.4.m4.1.1.cmml" xref="S3.I2.i1.p1.4.m4.1.1"><leq id="S3.I2.i1.p1.4.m4.1.1.1.cmml" xref="S3.I2.i1.p1.4.m4.1.1.1"></leq><ci id="S3.I2.i1.p1.4.m4.1.1.2.cmml" xref="S3.I2.i1.p1.4.m4.1.1.2">π½</ci><ci id="S3.I2.i1.p1.4.m4.1.1.3.cmml" xref="S3.I2.i1.p1.4.m4.1.1.3">πΌ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.4.m4.1c">\beta\leq\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.4.m4.1d">italic_Ξ² β€ italic_Ξ±</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.i1.p1.4.5">;</span></p> </div> </li> <li class="ltx_item" id="S3.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S3.I2.i2.p1"> <p class="ltx_p" id="S3.I2.i2.p1.4"><span class="ltx_text ltx_font_italic" id="S3.I2.i2.p1.4.1">if </span><math alttext="h:(\alpha,\max)\rightarrow Y" class="ltx_Math" display="inline" id="S3.I2.i2.p1.1.m1.2"><semantics id="S3.I2.i2.p1.1.m1.2a"><mrow id="S3.I2.i2.p1.1.m1.2.3" xref="S3.I2.i2.p1.1.m1.2.3.cmml"><mi id="S3.I2.i2.p1.1.m1.2.3.2" xref="S3.I2.i2.p1.1.m1.2.3.2.cmml">h</mi><mo id="S3.I2.i2.p1.1.m1.2.3.1" lspace="0.278em" rspace="0.278em" xref="S3.I2.i2.p1.1.m1.2.3.1.cmml">:</mo><mrow id="S3.I2.i2.p1.1.m1.2.3.3" xref="S3.I2.i2.p1.1.m1.2.3.3.cmml"><mrow id="S3.I2.i2.p1.1.m1.2.3.3.2.2" xref="S3.I2.i2.p1.1.m1.2.3.3.2.1.cmml"><mo id="S3.I2.i2.p1.1.m1.2.3.3.2.2.1" stretchy="false" xref="S3.I2.i2.p1.1.m1.2.3.3.2.1.cmml">(</mo><mi id="S3.I2.i2.p1.1.m1.1.1" xref="S3.I2.i2.p1.1.m1.1.1.cmml">Ξ±</mi><mo id="S3.I2.i2.p1.1.m1.2.3.3.2.2.2" xref="S3.I2.i2.p1.1.m1.2.3.3.2.1.cmml">,</mo><mi id="S3.I2.i2.p1.1.m1.2.2" xref="S3.I2.i2.p1.1.m1.2.2.cmml">max</mi><mo id="S3.I2.i2.p1.1.m1.2.3.3.2.2.3" stretchy="false" xref="S3.I2.i2.p1.1.m1.2.3.3.2.1.cmml">)</mo></mrow><mo id="S3.I2.i2.p1.1.m1.2.3.3.1" stretchy="false" xref="S3.I2.i2.p1.1.m1.2.3.3.1.cmml">β</mo><mi id="S3.I2.i2.p1.1.m1.2.3.3.3" xref="S3.I2.i2.p1.1.m1.2.3.3.3.cmml">Y</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.1.m1.2b"><apply id="S3.I2.i2.p1.1.m1.2.3.cmml" xref="S3.I2.i2.p1.1.m1.2.3"><ci id="S3.I2.i2.p1.1.m1.2.3.1.cmml" xref="S3.I2.i2.p1.1.m1.2.3.1">:</ci><ci id="S3.I2.i2.p1.1.m1.2.3.2.cmml" xref="S3.I2.i2.p1.1.m1.2.3.2">β</ci><apply id="S3.I2.i2.p1.1.m1.2.3.3.cmml" xref="S3.I2.i2.p1.1.m1.2.3.3"><ci id="S3.I2.i2.p1.1.m1.2.3.3.1.cmml" xref="S3.I2.i2.p1.1.m1.2.3.3.1">β</ci><interval closure="open" id="S3.I2.i2.p1.1.m1.2.3.3.2.1.cmml" xref="S3.I2.i2.p1.1.m1.2.3.3.2.2"><ci id="S3.I2.i2.p1.1.m1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1">πΌ</ci><max id="S3.I2.i2.p1.1.m1.2.2.cmml" xref="S3.I2.i2.p1.1.m1.2.2"></max></interval><ci id="S3.I2.i2.p1.1.m1.2.3.3.3.cmml" xref="S3.I2.i2.p1.1.m1.2.3.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.1.m1.2c">h:(\alpha,\max)\rightarrow Y</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.1.m1.2d">italic_h : ( italic_Ξ± , roman_max ) β italic_Y</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.i2.p1.4.2"> is a homomorphism, then </span><math alttext="h(\alpha)" class="ltx_Math" display="inline" id="S3.I2.i2.p1.2.m2.1"><semantics id="S3.I2.i2.p1.2.m2.1a"><mrow id="S3.I2.i2.p1.2.m2.1.2" xref="S3.I2.i2.p1.2.m2.1.2.cmml"><mi id="S3.I2.i2.p1.2.m2.1.2.2" xref="S3.I2.i2.p1.2.m2.1.2.2.cmml">h</mi><mo id="S3.I2.i2.p1.2.m2.1.2.1" xref="S3.I2.i2.p1.2.m2.1.2.1.cmml">β’</mo><mrow id="S3.I2.i2.p1.2.m2.1.2.3.2" xref="S3.I2.i2.p1.2.m2.1.2.cmml"><mo id="S3.I2.i2.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S3.I2.i2.p1.2.m2.1.2.cmml">(</mo><mi id="S3.I2.i2.p1.2.m2.1.1" xref="S3.I2.i2.p1.2.m2.1.1.cmml">Ξ±</mi><mo id="S3.I2.i2.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S3.I2.i2.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.2.m2.1b"><apply id="S3.I2.i2.p1.2.m2.1.2.cmml" xref="S3.I2.i2.p1.2.m2.1.2"><times id="S3.I2.i2.p1.2.m2.1.2.1.cmml" xref="S3.I2.i2.p1.2.m2.1.2.1"></times><ci id="S3.I2.i2.p1.2.m2.1.2.2.cmml" xref="S3.I2.i2.p1.2.m2.1.2.2">β</ci><ci id="S3.I2.i2.p1.2.m2.1.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1">πΌ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.2.m2.1c">h(\alpha)</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.2.m2.1d">italic_h ( italic_Ξ± )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.i2.p1.4.3"> is isomorphic to </span><math alttext="(\beta,\max)" class="ltx_Math" display="inline" id="S3.I2.i2.p1.3.m3.2"><semantics id="S3.I2.i2.p1.3.m3.2a"><mrow id="S3.I2.i2.p1.3.m3.2.3.2" xref="S3.I2.i2.p1.3.m3.2.3.1.cmml"><mo id="S3.I2.i2.p1.3.m3.2.3.2.1" stretchy="false" xref="S3.I2.i2.p1.3.m3.2.3.1.cmml">(</mo><mi id="S3.I2.i2.p1.3.m3.1.1" xref="S3.I2.i2.p1.3.m3.1.1.cmml">Ξ²</mi><mo id="S3.I2.i2.p1.3.m3.2.3.2.2" xref="S3.I2.i2.p1.3.m3.2.3.1.cmml">,</mo><mi id="S3.I2.i2.p1.3.m3.2.2" xref="S3.I2.i2.p1.3.m3.2.2.cmml">max</mi><mo id="S3.I2.i2.p1.3.m3.2.3.2.3" stretchy="false" xref="S3.I2.i2.p1.3.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.3.m3.2b"><interval closure="open" id="S3.I2.i2.p1.3.m3.2.3.1.cmml" xref="S3.I2.i2.p1.3.m3.2.3.2"><ci id="S3.I2.i2.p1.3.m3.1.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1">π½</ci><max id="S3.I2.i2.p1.3.m3.2.2.cmml" xref="S3.I2.i2.p1.3.m3.2.2"></max></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.3.m3.2c">(\beta,\max)</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.3.m3.2d">( italic_Ξ² , roman_max )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.i2.p1.4.4"> for some </span><math alttext="\beta\leq\alpha" class="ltx_Math" display="inline" id="S3.I2.i2.p1.4.m4.1"><semantics id="S3.I2.i2.p1.4.m4.1a"><mrow id="S3.I2.i2.p1.4.m4.1.1" xref="S3.I2.i2.p1.4.m4.1.1.cmml"><mi id="S3.I2.i2.p1.4.m4.1.1.2" xref="S3.I2.i2.p1.4.m4.1.1.2.cmml">Ξ²</mi><mo id="S3.I2.i2.p1.4.m4.1.1.1" xref="S3.I2.i2.p1.4.m4.1.1.1.cmml">β€</mo><mi id="S3.I2.i2.p1.4.m4.1.1.3" xref="S3.I2.i2.p1.4.m4.1.1.3.cmml">Ξ±</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.4.m4.1b"><apply id="S3.I2.i2.p1.4.m4.1.1.cmml" xref="S3.I2.i2.p1.4.m4.1.1"><leq id="S3.I2.i2.p1.4.m4.1.1.1.cmml" xref="S3.I2.i2.p1.4.m4.1.1.1"></leq><ci id="S3.I2.i2.p1.4.m4.1.1.2.cmml" xref="S3.I2.i2.p1.4.m4.1.1.2">π½</ci><ci id="S3.I2.i2.p1.4.m4.1.1.3.cmml" xref="S3.I2.i2.p1.4.m4.1.1.3">πΌ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.4.m4.1c">\beta\leq\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.4.m4.1d">italic_Ξ² β€ italic_Ξ±</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.i2.p1.4.5">.</span></p> </div> </li> </ol> </div> </div> <div class="ltx_proof" id="S3.6"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.5.p1"> <p class="ltx_p" id="S3.5.p1.9">(1) Consider the subsemilattice <math alttext="A=\{\min(h^{-1}(x)):x\in h(\alpha)\}" class="ltx_Math" display="inline" id="S3.5.p1.1.m1.5"><semantics id="S3.5.p1.1.m1.5a"><mrow id="S3.5.p1.1.m1.5.5" xref="S3.5.p1.1.m1.5.5.cmml"><mi id="S3.5.p1.1.m1.5.5.4" xref="S3.5.p1.1.m1.5.5.4.cmml">A</mi><mo id="S3.5.p1.1.m1.5.5.3" xref="S3.5.p1.1.m1.5.5.3.cmml">=</mo><mrow id="S3.5.p1.1.m1.5.5.2.2" xref="S3.5.p1.1.m1.5.5.2.3.cmml"><mo id="S3.5.p1.1.m1.5.5.2.2.3" stretchy="false" xref="S3.5.p1.1.m1.5.5.2.3.1.cmml">{</mo><mrow id="S3.5.p1.1.m1.4.4.1.1.1.1" xref="S3.5.p1.1.m1.4.4.1.1.1.2.cmml"><mi id="S3.5.p1.1.m1.2.2" xref="S3.5.p1.1.m1.2.2.cmml">min</mi><mo id="S3.5.p1.1.m1.4.4.1.1.1.1a" xref="S3.5.p1.1.m1.4.4.1.1.1.2.cmml">β‘</mo><mrow id="S3.5.p1.1.m1.4.4.1.1.1.1.1" xref="S3.5.p1.1.m1.4.4.1.1.1.2.cmml"><mo id="S3.5.p1.1.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="S3.5.p1.1.m1.4.4.1.1.1.2.cmml">(</mo><mrow id="S3.5.p1.1.m1.4.4.1.1.1.1.1.1" xref="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.cmml"><msup id="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2" xref="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.cmml"><mi id="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.2" xref="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.2.cmml">h</mi><mrow id="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.3" xref="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.3.cmml"><mo id="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.3a" xref="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.3.cmml">β</mo><mn id="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.3.2" xref="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.3.2.cmml">1</mn></mrow></msup><mo id="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.1" xref="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.1.cmml">β’</mo><mrow id="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.3.2" xref="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.cmml"><mo id="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.cmml">(</mo><mi id="S3.5.p1.1.m1.1.1" xref="S3.5.p1.1.m1.1.1.cmml">x</mi><mo id="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.5.p1.1.m1.4.4.1.1.1.1.1.3" rspace="0.278em" stretchy="false" xref="S3.5.p1.1.m1.4.4.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.5.p1.1.m1.5.5.2.2.4" rspace="0.278em" xref="S3.5.p1.1.m1.5.5.2.3.1.cmml">:</mo><mrow id="S3.5.p1.1.m1.5.5.2.2.2" xref="S3.5.p1.1.m1.5.5.2.2.2.cmml"><mi id="S3.5.p1.1.m1.5.5.2.2.2.2" xref="S3.5.p1.1.m1.5.5.2.2.2.2.cmml">x</mi><mo id="S3.5.p1.1.m1.5.5.2.2.2.1" xref="S3.5.p1.1.m1.5.5.2.2.2.1.cmml">β</mo><mrow id="S3.5.p1.1.m1.5.5.2.2.2.3" xref="S3.5.p1.1.m1.5.5.2.2.2.3.cmml"><mi id="S3.5.p1.1.m1.5.5.2.2.2.3.2" xref="S3.5.p1.1.m1.5.5.2.2.2.3.2.cmml">h</mi><mo id="S3.5.p1.1.m1.5.5.2.2.2.3.1" xref="S3.5.p1.1.m1.5.5.2.2.2.3.1.cmml">β’</mo><mrow id="S3.5.p1.1.m1.5.5.2.2.2.3.3.2" xref="S3.5.p1.1.m1.5.5.2.2.2.3.cmml"><mo id="S3.5.p1.1.m1.5.5.2.2.2.3.3.2.1" stretchy="false" xref="S3.5.p1.1.m1.5.5.2.2.2.3.cmml">(</mo><mi id="S3.5.p1.1.m1.3.3" xref="S3.5.p1.1.m1.3.3.cmml">Ξ±</mi><mo id="S3.5.p1.1.m1.5.5.2.2.2.3.3.2.2" stretchy="false" xref="S3.5.p1.1.m1.5.5.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.5.p1.1.m1.5.5.2.2.5" stretchy="false" xref="S3.5.p1.1.m1.5.5.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p1.1.m1.5b"><apply id="S3.5.p1.1.m1.5.5.cmml" xref="S3.5.p1.1.m1.5.5"><eq id="S3.5.p1.1.m1.5.5.3.cmml" xref="S3.5.p1.1.m1.5.5.3"></eq><ci id="S3.5.p1.1.m1.5.5.4.cmml" xref="S3.5.p1.1.m1.5.5.4">π΄</ci><apply id="S3.5.p1.1.m1.5.5.2.3.cmml" xref="S3.5.p1.1.m1.5.5.2.2"><csymbol cd="latexml" id="S3.5.p1.1.m1.5.5.2.3.1.cmml" xref="S3.5.p1.1.m1.5.5.2.2.3">conditional-set</csymbol><apply id="S3.5.p1.1.m1.4.4.1.1.1.2.cmml" xref="S3.5.p1.1.m1.4.4.1.1.1.1"><min id="S3.5.p1.1.m1.2.2.cmml" xref="S3.5.p1.1.m1.2.2"></min><apply id="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.cmml" xref="S3.5.p1.1.m1.4.4.1.1.1.1.1.1"><times id="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.1.cmml" xref="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.1"></times><apply id="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.cmml" xref="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.1.cmml" xref="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2">superscript</csymbol><ci id="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.2.cmml" xref="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.2">β</ci><apply id="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.3.cmml" xref="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.3"><minus id="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.3.1.cmml" xref="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.3"></minus><cn id="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S3.5.p1.1.m1.4.4.1.1.1.1.1.1.2.3.2">1</cn></apply></apply><ci id="S3.5.p1.1.m1.1.1.cmml" xref="S3.5.p1.1.m1.1.1">π₯</ci></apply></apply><apply id="S3.5.p1.1.m1.5.5.2.2.2.cmml" xref="S3.5.p1.1.m1.5.5.2.2.2"><in id="S3.5.p1.1.m1.5.5.2.2.2.1.cmml" xref="S3.5.p1.1.m1.5.5.2.2.2.1"></in><ci id="S3.5.p1.1.m1.5.5.2.2.2.2.cmml" xref="S3.5.p1.1.m1.5.5.2.2.2.2">π₯</ci><apply id="S3.5.p1.1.m1.5.5.2.2.2.3.cmml" xref="S3.5.p1.1.m1.5.5.2.2.2.3"><times id="S3.5.p1.1.m1.5.5.2.2.2.3.1.cmml" xref="S3.5.p1.1.m1.5.5.2.2.2.3.1"></times><ci id="S3.5.p1.1.m1.5.5.2.2.2.3.2.cmml" xref="S3.5.p1.1.m1.5.5.2.2.2.3.2">β</ci><ci id="S3.5.p1.1.m1.3.3.cmml" xref="S3.5.p1.1.m1.3.3">πΌ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.1.m1.5c">A=\{\min(h^{-1}(x)):x\in h(\alpha)\}</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.1.m1.5d">italic_A = { roman_min ( italic_h start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_x ) ) : italic_x β italic_h ( italic_Ξ± ) }</annotation></semantics></math> of <math alttext="(\alpha,\min)" class="ltx_Math" display="inline" id="S3.5.p1.2.m2.2"><semantics id="S3.5.p1.2.m2.2a"><mrow id="S3.5.p1.2.m2.2.3.2" xref="S3.5.p1.2.m2.2.3.1.cmml"><mo id="S3.5.p1.2.m2.2.3.2.1" stretchy="false" xref="S3.5.p1.2.m2.2.3.1.cmml">(</mo><mi id="S3.5.p1.2.m2.1.1" xref="S3.5.p1.2.m2.1.1.cmml">Ξ±</mi><mo id="S3.5.p1.2.m2.2.3.2.2" xref="S3.5.p1.2.m2.2.3.1.cmml">,</mo><mi id="S3.5.p1.2.m2.2.2" xref="S3.5.p1.2.m2.2.2.cmml">min</mi><mo id="S3.5.p1.2.m2.2.3.2.3" stretchy="false" xref="S3.5.p1.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p1.2.m2.2b"><interval closure="open" id="S3.5.p1.2.m2.2.3.1.cmml" xref="S3.5.p1.2.m2.2.3.2"><ci id="S3.5.p1.2.m2.1.1.cmml" xref="S3.5.p1.2.m2.1.1">πΌ</ci><min id="S3.5.p1.2.m2.2.2.cmml" xref="S3.5.p1.2.m2.2.2"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.2.m2.2c">(\alpha,\min)</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.2.m2.2d">( italic_Ξ± , roman_min )</annotation></semantics></math>. Since <math alttext="A\subseteq\alpha" class="ltx_Math" display="inline" id="S3.5.p1.3.m3.1"><semantics id="S3.5.p1.3.m3.1a"><mrow id="S3.5.p1.3.m3.1.1" xref="S3.5.p1.3.m3.1.1.cmml"><mi id="S3.5.p1.3.m3.1.1.2" xref="S3.5.p1.3.m3.1.1.2.cmml">A</mi><mo id="S3.5.p1.3.m3.1.1.1" xref="S3.5.p1.3.m3.1.1.1.cmml">β</mo><mi id="S3.5.p1.3.m3.1.1.3" xref="S3.5.p1.3.m3.1.1.3.cmml">Ξ±</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p1.3.m3.1b"><apply id="S3.5.p1.3.m3.1.1.cmml" xref="S3.5.p1.3.m3.1.1"><subset id="S3.5.p1.3.m3.1.1.1.cmml" xref="S3.5.p1.3.m3.1.1.1"></subset><ci id="S3.5.p1.3.m3.1.1.2.cmml" xref="S3.5.p1.3.m3.1.1.2">π΄</ci><ci id="S3.5.p1.3.m3.1.1.3.cmml" xref="S3.5.p1.3.m3.1.1.3">πΌ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.3.m3.1c">A\subseteq\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.3.m3.1d">italic_A β italic_Ξ±</annotation></semantics></math>, there exists an ordinal <math alttext="\beta\leq\alpha" class="ltx_Math" display="inline" id="S3.5.p1.4.m4.1"><semantics id="S3.5.p1.4.m4.1a"><mrow id="S3.5.p1.4.m4.1.1" xref="S3.5.p1.4.m4.1.1.cmml"><mi id="S3.5.p1.4.m4.1.1.2" xref="S3.5.p1.4.m4.1.1.2.cmml">Ξ²</mi><mo id="S3.5.p1.4.m4.1.1.1" xref="S3.5.p1.4.m4.1.1.1.cmml">β€</mo><mi id="S3.5.p1.4.m4.1.1.3" xref="S3.5.p1.4.m4.1.1.3.cmml">Ξ±</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p1.4.m4.1b"><apply id="S3.5.p1.4.m4.1.1.cmml" xref="S3.5.p1.4.m4.1.1"><leq id="S3.5.p1.4.m4.1.1.1.cmml" xref="S3.5.p1.4.m4.1.1.1"></leq><ci id="S3.5.p1.4.m4.1.1.2.cmml" xref="S3.5.p1.4.m4.1.1.2">π½</ci><ci id="S3.5.p1.4.m4.1.1.3.cmml" xref="S3.5.p1.4.m4.1.1.3">πΌ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.4.m4.1c">\beta\leq\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.4.m4.1d">italic_Ξ² β€ italic_Ξ±</annotation></semantics></math> such that <math alttext="A" class="ltx_Math" display="inline" id="S3.5.p1.5.m5.1"><semantics id="S3.5.p1.5.m5.1a"><mi id="S3.5.p1.5.m5.1.1" xref="S3.5.p1.5.m5.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.5.p1.5.m5.1b"><ci id="S3.5.p1.5.m5.1.1.cmml" xref="S3.5.p1.5.m5.1.1">π΄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.5.m5.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.5.m5.1d">italic_A</annotation></semantics></math> is isomorphic to <math alttext="(\beta,\min)" class="ltx_Math" display="inline" id="S3.5.p1.6.m6.2"><semantics id="S3.5.p1.6.m6.2a"><mrow id="S3.5.p1.6.m6.2.3.2" xref="S3.5.p1.6.m6.2.3.1.cmml"><mo id="S3.5.p1.6.m6.2.3.2.1" stretchy="false" xref="S3.5.p1.6.m6.2.3.1.cmml">(</mo><mi id="S3.5.p1.6.m6.1.1" xref="S3.5.p1.6.m6.1.1.cmml">Ξ²</mi><mo id="S3.5.p1.6.m6.2.3.2.2" xref="S3.5.p1.6.m6.2.3.1.cmml">,</mo><mi id="S3.5.p1.6.m6.2.2" xref="S3.5.p1.6.m6.2.2.cmml">min</mi><mo id="S3.5.p1.6.m6.2.3.2.3" stretchy="false" xref="S3.5.p1.6.m6.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p1.6.m6.2b"><interval closure="open" id="S3.5.p1.6.m6.2.3.1.cmml" xref="S3.5.p1.6.m6.2.3.2"><ci id="S3.5.p1.6.m6.1.1.cmml" xref="S3.5.p1.6.m6.1.1">π½</ci><min id="S3.5.p1.6.m6.2.2.cmml" xref="S3.5.p1.6.m6.2.2"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.6.m6.2c">(\beta,\min)</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.6.m6.2d">( italic_Ξ² , roman_min )</annotation></semantics></math>. It is straightforward to check that <math alttext="h(\alpha)" class="ltx_Math" display="inline" id="S3.5.p1.7.m7.1"><semantics id="S3.5.p1.7.m7.1a"><mrow id="S3.5.p1.7.m7.1.2" xref="S3.5.p1.7.m7.1.2.cmml"><mi id="S3.5.p1.7.m7.1.2.2" xref="S3.5.p1.7.m7.1.2.2.cmml">h</mi><mo id="S3.5.p1.7.m7.1.2.1" xref="S3.5.p1.7.m7.1.2.1.cmml">β’</mo><mrow id="S3.5.p1.7.m7.1.2.3.2" xref="S3.5.p1.7.m7.1.2.cmml"><mo id="S3.5.p1.7.m7.1.2.3.2.1" stretchy="false" xref="S3.5.p1.7.m7.1.2.cmml">(</mo><mi id="S3.5.p1.7.m7.1.1" xref="S3.5.p1.7.m7.1.1.cmml">Ξ±</mi><mo id="S3.5.p1.7.m7.1.2.3.2.2" stretchy="false" xref="S3.5.p1.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p1.7.m7.1b"><apply id="S3.5.p1.7.m7.1.2.cmml" xref="S3.5.p1.7.m7.1.2"><times id="S3.5.p1.7.m7.1.2.1.cmml" xref="S3.5.p1.7.m7.1.2.1"></times><ci id="S3.5.p1.7.m7.1.2.2.cmml" xref="S3.5.p1.7.m7.1.2.2">β</ci><ci id="S3.5.p1.7.m7.1.1.cmml" xref="S3.5.p1.7.m7.1.1">πΌ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.7.m7.1c">h(\alpha)</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.7.m7.1d">italic_h ( italic_Ξ± )</annotation></semantics></math> is isomorphic to <math alttext="A" class="ltx_Math" display="inline" id="S3.5.p1.8.m8.1"><semantics id="S3.5.p1.8.m8.1a"><mi id="S3.5.p1.8.m8.1.1" xref="S3.5.p1.8.m8.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.5.p1.8.m8.1b"><ci id="S3.5.p1.8.m8.1.1.cmml" xref="S3.5.p1.8.m8.1.1">π΄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.8.m8.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.8.m8.1d">italic_A</annotation></semantics></math> and thus to <math alttext="(\beta,\min)" class="ltx_Math" display="inline" id="S3.5.p1.9.m9.2"><semantics id="S3.5.p1.9.m9.2a"><mrow id="S3.5.p1.9.m9.2.3.2" xref="S3.5.p1.9.m9.2.3.1.cmml"><mo id="S3.5.p1.9.m9.2.3.2.1" stretchy="false" xref="S3.5.p1.9.m9.2.3.1.cmml">(</mo><mi id="S3.5.p1.9.m9.1.1" xref="S3.5.p1.9.m9.1.1.cmml">Ξ²</mi><mo id="S3.5.p1.9.m9.2.3.2.2" xref="S3.5.p1.9.m9.2.3.1.cmml">,</mo><mi id="S3.5.p1.9.m9.2.2" xref="S3.5.p1.9.m9.2.2.cmml">min</mi><mo id="S3.5.p1.9.m9.2.3.2.3" stretchy="false" xref="S3.5.p1.9.m9.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.5.p1.9.m9.2b"><interval closure="open" id="S3.5.p1.9.m9.2.3.1.cmml" xref="S3.5.p1.9.m9.2.3.2"><ci id="S3.5.p1.9.m9.1.1.cmml" xref="S3.5.p1.9.m9.1.1">π½</ci><min id="S3.5.p1.9.m9.2.2.cmml" xref="S3.5.p1.9.m9.2.2"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.5.p1.9.m9.2c">(\beta,\min)</annotation><annotation encoding="application/x-llamapun" id="S3.5.p1.9.m9.2d">( italic_Ξ² , roman_min )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.6.p2"> <p class="ltx_p" id="S3.6.p2.1">The proof of (2) is analogous. β</p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem8.1.1.1">Lemma 3.8</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem8.p1"> <p class="ltx_p" id="S3.Thmtheorem8.p1.4"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem8.p1.4.4">Let <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.1.1.m1.1"><semantics id="S3.Thmtheorem8.p1.1.1.m1.1a"><mi id="S3.Thmtheorem8.p1.1.1.m1.1.1" xref="S3.Thmtheorem8.p1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.1.1.m1.1b"><ci id="S3.Thmtheorem8.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem8.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.1.1.m1.1d">italic_X</annotation></semantics></math> be a countably compact semitopological semilattice, <math alttext="x\in X" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.2.2.m2.1"><semantics id="S3.Thmtheorem8.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem8.p1.2.2.m2.1.1" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem8.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.2.cmml">x</mi><mo id="S3.Thmtheorem8.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.1.cmml">β</mo><mi id="S3.Thmtheorem8.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.2.2.m2.1b"><apply id="S3.Thmtheorem8.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.1"><in id="S3.Thmtheorem8.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.1"></in><ci id="S3.Thmtheorem8.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.2">π₯</ci><ci id="S3.Thmtheorem8.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.2.2.m2.1c">x\in X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.2.2.m2.1d">italic_x β italic_X</annotation></semantics></math>, and <math alttext="C" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.3.3.m3.1"><semantics id="S3.Thmtheorem8.p1.3.3.m3.1a"><mi id="S3.Thmtheorem8.p1.3.3.m3.1.1" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.3.3.m3.1b"><ci id="S3.Thmtheorem8.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1">πΆ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.3.3.m3.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.3.3.m3.1d">italic_C</annotation></semantics></math> be a chain isomorphic to <math alttext="(\omega,\min)" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.4.4.m4.2"><semantics id="S3.Thmtheorem8.p1.4.4.m4.2a"><mrow id="S3.Thmtheorem8.p1.4.4.m4.2.3.2" xref="S3.Thmtheorem8.p1.4.4.m4.2.3.1.cmml"><mo id="S3.Thmtheorem8.p1.4.4.m4.2.3.2.1" stretchy="false" xref="S3.Thmtheorem8.p1.4.4.m4.2.3.1.cmml">(</mo><mi id="S3.Thmtheorem8.p1.4.4.m4.1.1" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.cmml">Ο</mi><mo id="S3.Thmtheorem8.p1.4.4.m4.2.3.2.2" xref="S3.Thmtheorem8.p1.4.4.m4.2.3.1.cmml">,</mo><mi id="S3.Thmtheorem8.p1.4.4.m4.2.2" xref="S3.Thmtheorem8.p1.4.4.m4.2.2.cmml">min</mi><mo id="S3.Thmtheorem8.p1.4.4.m4.2.3.2.3" stretchy="false" xref="S3.Thmtheorem8.p1.4.4.m4.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.4.4.m4.2b"><interval closure="open" id="S3.Thmtheorem8.p1.4.4.m4.2.3.1.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.2.3.2"><ci id="S3.Thmtheorem8.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.1.1">π</ci><min id="S3.Thmtheorem8.p1.4.4.m4.2.2.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.2.2"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.4.4.m4.2c">(\omega,\min)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.4.4.m4.2d">( italic_Ο , roman_min )</annotation></semantics></math>. Then the following conditions are equivalent:</span></p> <ol class="ltx_enumerate" id="S3.I3"> <li class="ltx_item" id="S3.I3.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S3.I3.i1.p1"> <p class="ltx_p" id="S3.I3.i1.p1.3"><span class="ltx_text ltx_font_italic" id="S3.I3.i1.p1.3.1">For each open neighborhood </span><math alttext="U" class="ltx_Math" display="inline" id="S3.I3.i1.p1.1.m1.1"><semantics id="S3.I3.i1.p1.1.m1.1a"><mi id="S3.I3.i1.p1.1.m1.1.1" xref="S3.I3.i1.p1.1.m1.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.I3.i1.p1.1.m1.1b"><ci id="S3.I3.i1.p1.1.m1.1.1.cmml" xref="S3.I3.i1.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.1.m1.1c">U</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.1.m1.1d">italic_U</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i1.p1.3.2"> of </span><math alttext="x" class="ltx_Math" display="inline" id="S3.I3.i1.p1.2.m2.1"><semantics id="S3.I3.i1.p1.2.m2.1a"><mi id="S3.I3.i1.p1.2.m2.1.1" xref="S3.I3.i1.p1.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.I3.i1.p1.2.m2.1b"><ci id="S3.I3.i1.p1.2.m2.1.1.cmml" xref="S3.I3.i1.p1.2.m2.1.1">π₯</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.2.m2.1d">italic_x</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i1.p1.3.3"> the set </span><math alttext="C\setminus U" class="ltx_Math" display="inline" id="S3.I3.i1.p1.3.m3.1"><semantics id="S3.I3.i1.p1.3.m3.1a"><mrow id="S3.I3.i1.p1.3.m3.1.1" xref="S3.I3.i1.p1.3.m3.1.1.cmml"><mi id="S3.I3.i1.p1.3.m3.1.1.2" xref="S3.I3.i1.p1.3.m3.1.1.2.cmml">C</mi><mo id="S3.I3.i1.p1.3.m3.1.1.1" xref="S3.I3.i1.p1.3.m3.1.1.1.cmml">β</mo><mi id="S3.I3.i1.p1.3.m3.1.1.3" xref="S3.I3.i1.p1.3.m3.1.1.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i1.p1.3.m3.1b"><apply id="S3.I3.i1.p1.3.m3.1.1.cmml" xref="S3.I3.i1.p1.3.m3.1.1"><setdiff id="S3.I3.i1.p1.3.m3.1.1.1.cmml" xref="S3.I3.i1.p1.3.m3.1.1.1"></setdiff><ci id="S3.I3.i1.p1.3.m3.1.1.2.cmml" xref="S3.I3.i1.p1.3.m3.1.1.2">πΆ</ci><ci id="S3.I3.i1.p1.3.m3.1.1.3.cmml" xref="S3.I3.i1.p1.3.m3.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.3.m3.1c">C\setminus U</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.3.m3.1d">italic_C β italic_U</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i1.p1.3.4"> is finite;</span></p> </div> </li> <li class="ltx_item" id="S3.I3.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S3.I3.i2.p1"> <p class="ltx_p" id="S3.I3.i2.p1.1"><math alttext="x\in\overline{C}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.1.m1.1"><semantics id="S3.I3.i2.p1.1.m1.1a"><mrow id="S3.I3.i2.p1.1.m1.1.1" xref="S3.I3.i2.p1.1.m1.1.1.cmml"><mi id="S3.I3.i2.p1.1.m1.1.1.2" xref="S3.I3.i2.p1.1.m1.1.1.2.cmml">x</mi><mo id="S3.I3.i2.p1.1.m1.1.1.1" xref="S3.I3.i2.p1.1.m1.1.1.1.cmml">β</mo><mover accent="true" id="S3.I3.i2.p1.1.m1.1.1.3" xref="S3.I3.i2.p1.1.m1.1.1.3.cmml"><mi id="S3.I3.i2.p1.1.m1.1.1.3.2" xref="S3.I3.i2.p1.1.m1.1.1.3.2.cmml">C</mi><mo id="S3.I3.i2.p1.1.m1.1.1.3.1" xref="S3.I3.i2.p1.1.m1.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.1.m1.1b"><apply id="S3.I3.i2.p1.1.m1.1.1.cmml" xref="S3.I3.i2.p1.1.m1.1.1"><in id="S3.I3.i2.p1.1.m1.1.1.1.cmml" xref="S3.I3.i2.p1.1.m1.1.1.1"></in><ci id="S3.I3.i2.p1.1.m1.1.1.2.cmml" xref="S3.I3.i2.p1.1.m1.1.1.2">π₯</ci><apply id="S3.I3.i2.p1.1.m1.1.1.3.cmml" xref="S3.I3.i2.p1.1.m1.1.1.3"><ci id="S3.I3.i2.p1.1.m1.1.1.3.1.cmml" xref="S3.I3.i2.p1.1.m1.1.1.3.1">Β―</ci><ci id="S3.I3.i2.p1.1.m1.1.1.3.2.cmml" xref="S3.I3.i2.p1.1.m1.1.1.3.2">πΆ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.1.m1.1c">x\in\overline{C}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.1.m1.1d">italic_x β overΒ― start_ARG italic_C end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i2.p1.1.1">;</span></p> </div> </li> <li class="ltx_item" id="S3.I3.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(3)</span> <div class="ltx_para" id="S3.I3.i3.p1"> <p class="ltx_p" id="S3.I3.i3.p1.1"><math alttext="x=\sup C" class="ltx_Math" display="inline" id="S3.I3.i3.p1.1.m1.1"><semantics id="S3.I3.i3.p1.1.m1.1a"><mrow id="S3.I3.i3.p1.1.m1.1.1" xref="S3.I3.i3.p1.1.m1.1.1.cmml"><mi id="S3.I3.i3.p1.1.m1.1.1.2" xref="S3.I3.i3.p1.1.m1.1.1.2.cmml">x</mi><mo id="S3.I3.i3.p1.1.m1.1.1.1" rspace="0.1389em" xref="S3.I3.i3.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S3.I3.i3.p1.1.m1.1.1.3" xref="S3.I3.i3.p1.1.m1.1.1.3.cmml"><mo id="S3.I3.i3.p1.1.m1.1.1.3.1" lspace="0.1389em" rspace="0.167em" xref="S3.I3.i3.p1.1.m1.1.1.3.1.cmml">sup</mo><mi id="S3.I3.i3.p1.1.m1.1.1.3.2" xref="S3.I3.i3.p1.1.m1.1.1.3.2.cmml">C</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i3.p1.1.m1.1b"><apply id="S3.I3.i3.p1.1.m1.1.1.cmml" xref="S3.I3.i3.p1.1.m1.1.1"><eq id="S3.I3.i3.p1.1.m1.1.1.1.cmml" xref="S3.I3.i3.p1.1.m1.1.1.1"></eq><ci id="S3.I3.i3.p1.1.m1.1.1.2.cmml" xref="S3.I3.i3.p1.1.m1.1.1.2">π₯</ci><apply id="S3.I3.i3.p1.1.m1.1.1.3.cmml" xref="S3.I3.i3.p1.1.m1.1.1.3"><csymbol cd="latexml" id="S3.I3.i3.p1.1.m1.1.1.3.1.cmml" xref="S3.I3.i3.p1.1.m1.1.1.3.1">supremum</csymbol><ci id="S3.I3.i3.p1.1.m1.1.1.3.2.cmml" xref="S3.I3.i3.p1.1.m1.1.1.3.2">πΆ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i3.p1.1.m1.1c">x=\sup C</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i3.p1.1.m1.1d">italic_x = roman_sup italic_C</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I3.i3.p1.1.1">.</span></p> </div> </li> </ol> </div> </div> <div class="ltx_proof" id="S3.9"> <h6 class="ltx_title ltx_runin ltx_title_proof"><span class="ltx_text ltx_font_italic" id="S3.9.1.1">Proof.</span></h6> <div class="ltx_para" id="S3.7.p1"> <p class="ltx_p" id="S3.7.p1.1">The implication (1) <math alttext="\Rightarrow" class="ltx_Math" display="inline" id="S3.7.p1.1.m1.1"><semantics id="S3.7.p1.1.m1.1a"><mo id="S3.7.p1.1.m1.1.1" stretchy="false" xref="S3.7.p1.1.m1.1.1.cmml">β</mo><annotation-xml encoding="MathML-Content" id="S3.7.p1.1.m1.1b"><ci id="S3.7.p1.1.m1.1.1.cmml" xref="S3.7.p1.1.m1.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.7.p1.1.m1.1c">\Rightarrow</annotation><annotation encoding="application/x-llamapun" id="S3.7.p1.1.m1.1d">β</annotation></semantics></math> (2) is obvious.</p> </div> <div class="ltx_para" id="S3.8.p2"> <p class="ltx_p" id="S3.8.p2.24">(2) <math alttext="\Rightarrow" class="ltx_Math" display="inline" id="S3.8.p2.1.m1.1"><semantics id="S3.8.p2.1.m1.1a"><mo id="S3.8.p2.1.m1.1.1" stretchy="false" xref="S3.8.p2.1.m1.1.1.cmml">β</mo><annotation-xml encoding="MathML-Content" id="S3.8.p2.1.m1.1b"><ci id="S3.8.p2.1.m1.1.1.cmml" xref="S3.8.p2.1.m1.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.1.m1.1c">\Rightarrow</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.1.m1.1d">β</annotation></semantics></math> (3): Assume that <math alttext="x\in\overline{C}" class="ltx_Math" display="inline" id="S3.8.p2.2.m2.1"><semantics id="S3.8.p2.2.m2.1a"><mrow id="S3.8.p2.2.m2.1.1" xref="S3.8.p2.2.m2.1.1.cmml"><mi id="S3.8.p2.2.m2.1.1.2" xref="S3.8.p2.2.m2.1.1.2.cmml">x</mi><mo id="S3.8.p2.2.m2.1.1.1" xref="S3.8.p2.2.m2.1.1.1.cmml">β</mo><mover accent="true" id="S3.8.p2.2.m2.1.1.3" xref="S3.8.p2.2.m2.1.1.3.cmml"><mi id="S3.8.p2.2.m2.1.1.3.2" xref="S3.8.p2.2.m2.1.1.3.2.cmml">C</mi><mo id="S3.8.p2.2.m2.1.1.3.1" xref="S3.8.p2.2.m2.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.8.p2.2.m2.1b"><apply id="S3.8.p2.2.m2.1.1.cmml" xref="S3.8.p2.2.m2.1.1"><in id="S3.8.p2.2.m2.1.1.1.cmml" xref="S3.8.p2.2.m2.1.1.1"></in><ci id="S3.8.p2.2.m2.1.1.2.cmml" xref="S3.8.p2.2.m2.1.1.2">π₯</ci><apply id="S3.8.p2.2.m2.1.1.3.cmml" xref="S3.8.p2.2.m2.1.1.3"><ci id="S3.8.p2.2.m2.1.1.3.1.cmml" xref="S3.8.p2.2.m2.1.1.3.1">Β―</ci><ci id="S3.8.p2.2.m2.1.1.3.2.cmml" xref="S3.8.p2.2.m2.1.1.3.2">πΆ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.2.m2.1c">x\in\overline{C}</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.2.m2.1d">italic_x β overΒ― start_ARG italic_C end_ARG</annotation></semantics></math>. Fix any <math alttext="c\in C" class="ltx_Math" display="inline" id="S3.8.p2.3.m3.1"><semantics id="S3.8.p2.3.m3.1a"><mrow id="S3.8.p2.3.m3.1.1" xref="S3.8.p2.3.m3.1.1.cmml"><mi id="S3.8.p2.3.m3.1.1.2" xref="S3.8.p2.3.m3.1.1.2.cmml">c</mi><mo id="S3.8.p2.3.m3.1.1.1" xref="S3.8.p2.3.m3.1.1.1.cmml">β</mo><mi id="S3.8.p2.3.m3.1.1.3" xref="S3.8.p2.3.m3.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.8.p2.3.m3.1b"><apply id="S3.8.p2.3.m3.1.1.cmml" xref="S3.8.p2.3.m3.1.1"><in id="S3.8.p2.3.m3.1.1.1.cmml" xref="S3.8.p2.3.m3.1.1.1"></in><ci id="S3.8.p2.3.m3.1.1.2.cmml" xref="S3.8.p2.3.m3.1.1.2">π</ci><ci id="S3.8.p2.3.m3.1.1.3.cmml" xref="S3.8.p2.3.m3.1.1.3">πΆ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.3.m3.1c">c\in C</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.3.m3.1d">italic_c β italic_C</annotation></semantics></math> and open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S3.8.p2.4.m4.1"><semantics id="S3.8.p2.4.m4.1a"><mi id="S3.8.p2.4.m4.1.1" xref="S3.8.p2.4.m4.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.8.p2.4.m4.1b"><ci id="S3.8.p2.4.m4.1.1.cmml" xref="S3.8.p2.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.4.m4.1c">U</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.4.m4.1d">italic_U</annotation></semantics></math> of <math alttext="x" class="ltx_Math" display="inline" id="S3.8.p2.5.m5.1"><semantics id="S3.8.p2.5.m5.1a"><mi id="S3.8.p2.5.m5.1.1" xref="S3.8.p2.5.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.8.p2.5.m5.1b"><ci id="S3.8.p2.5.m5.1.1.cmml" xref="S3.8.p2.5.m5.1.1">π₯</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.5.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.5.m5.1d">italic_x</annotation></semantics></math>. Note that the set <math alttext="{\uparrow}c\cap U" class="ltx_Math" display="inline" id="S3.8.p2.6.m6.1"><semantics id="S3.8.p2.6.m6.1a"><mrow id="S3.8.p2.6.m6.1.1" xref="S3.8.p2.6.m6.1.1.cmml"><mi id="S3.8.p2.6.m6.1.1.2" xref="S3.8.p2.6.m6.1.1.2.cmml"></mi><mo id="S3.8.p2.6.m6.1.1.1" stretchy="false" xref="S3.8.p2.6.m6.1.1.1.cmml">β</mo><mrow id="S3.8.p2.6.m6.1.1.3" xref="S3.8.p2.6.m6.1.1.3.cmml"><mi id="S3.8.p2.6.m6.1.1.3.2" xref="S3.8.p2.6.m6.1.1.3.2.cmml">c</mi><mo id="S3.8.p2.6.m6.1.1.3.1" xref="S3.8.p2.6.m6.1.1.3.1.cmml">β©</mo><mi id="S3.8.p2.6.m6.1.1.3.3" xref="S3.8.p2.6.m6.1.1.3.3.cmml">U</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.8.p2.6.m6.1b"><apply id="S3.8.p2.6.m6.1.1.cmml" xref="S3.8.p2.6.m6.1.1"><ci id="S3.8.p2.6.m6.1.1.1.cmml" xref="S3.8.p2.6.m6.1.1.1">β</ci><csymbol cd="latexml" id="S3.8.p2.6.m6.1.1.2.cmml" xref="S3.8.p2.6.m6.1.1.2">absent</csymbol><apply id="S3.8.p2.6.m6.1.1.3.cmml" xref="S3.8.p2.6.m6.1.1.3"><intersect id="S3.8.p2.6.m6.1.1.3.1.cmml" xref="S3.8.p2.6.m6.1.1.3.1"></intersect><ci id="S3.8.p2.6.m6.1.1.3.2.cmml" xref="S3.8.p2.6.m6.1.1.3.2">π</ci><ci id="S3.8.p2.6.m6.1.1.3.3.cmml" xref="S3.8.p2.6.m6.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.6.m6.1c">{\uparrow}c\cap U</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.6.m6.1d">β italic_c β© italic_U</annotation></semantics></math> is nonempty, witnessing that <math alttext="c\in cU" class="ltx_Math" display="inline" id="S3.8.p2.7.m7.1"><semantics id="S3.8.p2.7.m7.1a"><mrow id="S3.8.p2.7.m7.1.1" xref="S3.8.p2.7.m7.1.1.cmml"><mi id="S3.8.p2.7.m7.1.1.2" xref="S3.8.p2.7.m7.1.1.2.cmml">c</mi><mo id="S3.8.p2.7.m7.1.1.1" xref="S3.8.p2.7.m7.1.1.1.cmml">β</mo><mrow id="S3.8.p2.7.m7.1.1.3" xref="S3.8.p2.7.m7.1.1.3.cmml"><mi id="S3.8.p2.7.m7.1.1.3.2" xref="S3.8.p2.7.m7.1.1.3.2.cmml">c</mi><mo id="S3.8.p2.7.m7.1.1.3.1" xref="S3.8.p2.7.m7.1.1.3.1.cmml">β’</mo><mi id="S3.8.p2.7.m7.1.1.3.3" xref="S3.8.p2.7.m7.1.1.3.3.cmml">U</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.8.p2.7.m7.1b"><apply id="S3.8.p2.7.m7.1.1.cmml" xref="S3.8.p2.7.m7.1.1"><in id="S3.8.p2.7.m7.1.1.1.cmml" xref="S3.8.p2.7.m7.1.1.1"></in><ci id="S3.8.p2.7.m7.1.1.2.cmml" xref="S3.8.p2.7.m7.1.1.2">π</ci><apply id="S3.8.p2.7.m7.1.1.3.cmml" xref="S3.8.p2.7.m7.1.1.3"><times id="S3.8.p2.7.m7.1.1.3.1.cmml" xref="S3.8.p2.7.m7.1.1.3.1"></times><ci id="S3.8.p2.7.m7.1.1.3.2.cmml" xref="S3.8.p2.7.m7.1.1.3.2">π</ci><ci id="S3.8.p2.7.m7.1.1.3.3.cmml" xref="S3.8.p2.7.m7.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.7.m7.1c">c\in cU</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.7.m7.1d">italic_c β italic_c italic_U</annotation></semantics></math>. Since <math alttext="U" class="ltx_Math" display="inline" id="S3.8.p2.8.m8.1"><semantics id="S3.8.p2.8.m8.1a"><mi id="S3.8.p2.8.m8.1.1" xref="S3.8.p2.8.m8.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.8.p2.8.m8.1b"><ci id="S3.8.p2.8.m8.1.1.cmml" xref="S3.8.p2.8.m8.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.8.m8.1c">U</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.8.m8.1d">italic_U</annotation></semantics></math> was arbitrarily chosen and <math alttext="X" class="ltx_Math" display="inline" id="S3.8.p2.9.m9.1"><semantics id="S3.8.p2.9.m9.1a"><mi id="S3.8.p2.9.m9.1.1" xref="S3.8.p2.9.m9.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.8.p2.9.m9.1b"><ci id="S3.8.p2.9.m9.1.1.cmml" xref="S3.8.p2.9.m9.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.9.m9.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.9.m9.1d">italic_X</annotation></semantics></math> is a Hausdorff semitopological semilattice, we get that <math alttext="cx=c" class="ltx_Math" display="inline" id="S3.8.p2.10.m10.1"><semantics id="S3.8.p2.10.m10.1a"><mrow id="S3.8.p2.10.m10.1.1" xref="S3.8.p2.10.m10.1.1.cmml"><mrow id="S3.8.p2.10.m10.1.1.2" xref="S3.8.p2.10.m10.1.1.2.cmml"><mi id="S3.8.p2.10.m10.1.1.2.2" xref="S3.8.p2.10.m10.1.1.2.2.cmml">c</mi><mo id="S3.8.p2.10.m10.1.1.2.1" xref="S3.8.p2.10.m10.1.1.2.1.cmml">β’</mo><mi id="S3.8.p2.10.m10.1.1.2.3" xref="S3.8.p2.10.m10.1.1.2.3.cmml">x</mi></mrow><mo id="S3.8.p2.10.m10.1.1.1" xref="S3.8.p2.10.m10.1.1.1.cmml">=</mo><mi id="S3.8.p2.10.m10.1.1.3" xref="S3.8.p2.10.m10.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.8.p2.10.m10.1b"><apply id="S3.8.p2.10.m10.1.1.cmml" xref="S3.8.p2.10.m10.1.1"><eq id="S3.8.p2.10.m10.1.1.1.cmml" xref="S3.8.p2.10.m10.1.1.1"></eq><apply id="S3.8.p2.10.m10.1.1.2.cmml" xref="S3.8.p2.10.m10.1.1.2"><times id="S3.8.p2.10.m10.1.1.2.1.cmml" xref="S3.8.p2.10.m10.1.1.2.1"></times><ci id="S3.8.p2.10.m10.1.1.2.2.cmml" xref="S3.8.p2.10.m10.1.1.2.2">π</ci><ci id="S3.8.p2.10.m10.1.1.2.3.cmml" xref="S3.8.p2.10.m10.1.1.2.3">π₯</ci></apply><ci id="S3.8.p2.10.m10.1.1.3.cmml" xref="S3.8.p2.10.m10.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.10.m10.1c">cx=c</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.10.m10.1d">italic_c italic_x = italic_c</annotation></semantics></math>. Since the point <math alttext="c" class="ltx_Math" display="inline" id="S3.8.p2.11.m11.1"><semantics id="S3.8.p2.11.m11.1a"><mi id="S3.8.p2.11.m11.1.1" xref="S3.8.p2.11.m11.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.8.p2.11.m11.1b"><ci id="S3.8.p2.11.m11.1.1.cmml" xref="S3.8.p2.11.m11.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.11.m11.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.11.m11.1d">italic_c</annotation></semantics></math> was chosen arbitrarily, we obtain that <math alttext="c\leq x" class="ltx_Math" display="inline" id="S3.8.p2.12.m12.1"><semantics id="S3.8.p2.12.m12.1a"><mrow id="S3.8.p2.12.m12.1.1" xref="S3.8.p2.12.m12.1.1.cmml"><mi id="S3.8.p2.12.m12.1.1.2" xref="S3.8.p2.12.m12.1.1.2.cmml">c</mi><mo id="S3.8.p2.12.m12.1.1.1" xref="S3.8.p2.12.m12.1.1.1.cmml">β€</mo><mi id="S3.8.p2.12.m12.1.1.3" xref="S3.8.p2.12.m12.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.8.p2.12.m12.1b"><apply id="S3.8.p2.12.m12.1.1.cmml" xref="S3.8.p2.12.m12.1.1"><leq id="S3.8.p2.12.m12.1.1.1.cmml" xref="S3.8.p2.12.m12.1.1.1"></leq><ci id="S3.8.p2.12.m12.1.1.2.cmml" xref="S3.8.p2.12.m12.1.1.2">π</ci><ci id="S3.8.p2.12.m12.1.1.3.cmml" xref="S3.8.p2.12.m12.1.1.3">π₯</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.12.m12.1c">c\leq x</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.12.m12.1d">italic_c β€ italic_x</annotation></semantics></math> for all <math alttext="c\in C" class="ltx_Math" display="inline" id="S3.8.p2.13.m13.1"><semantics id="S3.8.p2.13.m13.1a"><mrow id="S3.8.p2.13.m13.1.1" xref="S3.8.p2.13.m13.1.1.cmml"><mi id="S3.8.p2.13.m13.1.1.2" xref="S3.8.p2.13.m13.1.1.2.cmml">c</mi><mo id="S3.8.p2.13.m13.1.1.1" xref="S3.8.p2.13.m13.1.1.1.cmml">β</mo><mi id="S3.8.p2.13.m13.1.1.3" xref="S3.8.p2.13.m13.1.1.3.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.8.p2.13.m13.1b"><apply id="S3.8.p2.13.m13.1.1.cmml" xref="S3.8.p2.13.m13.1.1"><in id="S3.8.p2.13.m13.1.1.1.cmml" xref="S3.8.p2.13.m13.1.1.1"></in><ci id="S3.8.p2.13.m13.1.1.2.cmml" xref="S3.8.p2.13.m13.1.1.2">π</ci><ci id="S3.8.p2.13.m13.1.1.3.cmml" xref="S3.8.p2.13.m13.1.1.3">πΆ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.13.m13.1c">c\in C</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.13.m13.1d">italic_c β italic_C</annotation></semantics></math>, i.e. <math alttext="x" class="ltx_Math" display="inline" id="S3.8.p2.14.m14.1"><semantics id="S3.8.p2.14.m14.1a"><mi id="S3.8.p2.14.m14.1.1" xref="S3.8.p2.14.m14.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.8.p2.14.m14.1b"><ci id="S3.8.p2.14.m14.1.1.cmml" xref="S3.8.p2.14.m14.1.1">π₯</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.14.m14.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.14.m14.1d">italic_x</annotation></semantics></math> is an upper bound of <math alttext="C" class="ltx_Math" display="inline" id="S3.8.p2.15.m15.1"><semantics id="S3.8.p2.15.m15.1a"><mi id="S3.8.p2.15.m15.1.1" xref="S3.8.p2.15.m15.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.8.p2.15.m15.1b"><ci id="S3.8.p2.15.m15.1.1.cmml" xref="S3.8.p2.15.m15.1.1">πΆ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.15.m15.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.15.m15.1d">italic_C</annotation></semantics></math>. Assume that <math alttext="y" class="ltx_Math" display="inline" id="S3.8.p2.16.m16.1"><semantics id="S3.8.p2.16.m16.1a"><mi id="S3.8.p2.16.m16.1.1" xref="S3.8.p2.16.m16.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S3.8.p2.16.m16.1b"><ci id="S3.8.p2.16.m16.1.1.cmml" xref="S3.8.p2.16.m16.1.1">π¦</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.16.m16.1c">y</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.16.m16.1d">italic_y</annotation></semantics></math> is another upper bound of <math alttext="C" class="ltx_Math" display="inline" id="S3.8.p2.17.m17.1"><semantics id="S3.8.p2.17.m17.1a"><mi id="S3.8.p2.17.m17.1.1" xref="S3.8.p2.17.m17.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.8.p2.17.m17.1b"><ci id="S3.8.p2.17.m17.1.1.cmml" xref="S3.8.p2.17.m17.1.1">πΆ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.17.m17.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.17.m17.1d">italic_C</annotation></semantics></math>. Taking into account that <math alttext="x\in\overline{C}" class="ltx_Math" display="inline" id="S3.8.p2.18.m18.1"><semantics id="S3.8.p2.18.m18.1a"><mrow id="S3.8.p2.18.m18.1.1" xref="S3.8.p2.18.m18.1.1.cmml"><mi id="S3.8.p2.18.m18.1.1.2" xref="S3.8.p2.18.m18.1.1.2.cmml">x</mi><mo id="S3.8.p2.18.m18.1.1.1" xref="S3.8.p2.18.m18.1.1.1.cmml">β</mo><mover accent="true" id="S3.8.p2.18.m18.1.1.3" xref="S3.8.p2.18.m18.1.1.3.cmml"><mi id="S3.8.p2.18.m18.1.1.3.2" xref="S3.8.p2.18.m18.1.1.3.2.cmml">C</mi><mo id="S3.8.p2.18.m18.1.1.3.1" xref="S3.8.p2.18.m18.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.8.p2.18.m18.1b"><apply id="S3.8.p2.18.m18.1.1.cmml" xref="S3.8.p2.18.m18.1.1"><in id="S3.8.p2.18.m18.1.1.1.cmml" xref="S3.8.p2.18.m18.1.1.1"></in><ci id="S3.8.p2.18.m18.1.1.2.cmml" xref="S3.8.p2.18.m18.1.1.2">π₯</ci><apply id="S3.8.p2.18.m18.1.1.3.cmml" xref="S3.8.p2.18.m18.1.1.3"><ci id="S3.8.p2.18.m18.1.1.3.1.cmml" xref="S3.8.p2.18.m18.1.1.3.1">Β―</ci><ci id="S3.8.p2.18.m18.1.1.3.2.cmml" xref="S3.8.p2.18.m18.1.1.3.2">πΆ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.18.m18.1c">x\in\overline{C}</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.18.m18.1d">italic_x β overΒ― start_ARG italic_C end_ARG</annotation></semantics></math>, we have <math alttext="U\cap yU\neq\varnothing" class="ltx_Math" display="inline" id="S3.8.p2.19.m19.1"><semantics id="S3.8.p2.19.m19.1a"><mrow id="S3.8.p2.19.m19.1.1" xref="S3.8.p2.19.m19.1.1.cmml"><mrow id="S3.8.p2.19.m19.1.1.2" xref="S3.8.p2.19.m19.1.1.2.cmml"><mi id="S3.8.p2.19.m19.1.1.2.2" xref="S3.8.p2.19.m19.1.1.2.2.cmml">U</mi><mo id="S3.8.p2.19.m19.1.1.2.1" xref="S3.8.p2.19.m19.1.1.2.1.cmml">β©</mo><mrow id="S3.8.p2.19.m19.1.1.2.3" xref="S3.8.p2.19.m19.1.1.2.3.cmml"><mi id="S3.8.p2.19.m19.1.1.2.3.2" xref="S3.8.p2.19.m19.1.1.2.3.2.cmml">y</mi><mo id="S3.8.p2.19.m19.1.1.2.3.1" xref="S3.8.p2.19.m19.1.1.2.3.1.cmml">β’</mo><mi id="S3.8.p2.19.m19.1.1.2.3.3" xref="S3.8.p2.19.m19.1.1.2.3.3.cmml">U</mi></mrow></mrow><mo id="S3.8.p2.19.m19.1.1.1" xref="S3.8.p2.19.m19.1.1.1.cmml">β </mo><mi id="S3.8.p2.19.m19.1.1.3" mathvariant="normal" xref="S3.8.p2.19.m19.1.1.3.cmml">β </mi></mrow><annotation-xml encoding="MathML-Content" id="S3.8.p2.19.m19.1b"><apply id="S3.8.p2.19.m19.1.1.cmml" xref="S3.8.p2.19.m19.1.1"><neq id="S3.8.p2.19.m19.1.1.1.cmml" xref="S3.8.p2.19.m19.1.1.1"></neq><apply id="S3.8.p2.19.m19.1.1.2.cmml" xref="S3.8.p2.19.m19.1.1.2"><intersect id="S3.8.p2.19.m19.1.1.2.1.cmml" xref="S3.8.p2.19.m19.1.1.2.1"></intersect><ci id="S3.8.p2.19.m19.1.1.2.2.cmml" xref="S3.8.p2.19.m19.1.1.2.2">π</ci><apply id="S3.8.p2.19.m19.1.1.2.3.cmml" xref="S3.8.p2.19.m19.1.1.2.3"><times id="S3.8.p2.19.m19.1.1.2.3.1.cmml" xref="S3.8.p2.19.m19.1.1.2.3.1"></times><ci id="S3.8.p2.19.m19.1.1.2.3.2.cmml" xref="S3.8.p2.19.m19.1.1.2.3.2">π¦</ci><ci id="S3.8.p2.19.m19.1.1.2.3.3.cmml" xref="S3.8.p2.19.m19.1.1.2.3.3">π</ci></apply></apply><emptyset id="S3.8.p2.19.m19.1.1.3.cmml" xref="S3.8.p2.19.m19.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.19.m19.1c">U\cap yU\neq\varnothing</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.19.m19.1d">italic_U β© italic_y italic_U β β </annotation></semantics></math> for each open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S3.8.p2.20.m20.1"><semantics id="S3.8.p2.20.m20.1a"><mi id="S3.8.p2.20.m20.1.1" xref="S3.8.p2.20.m20.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.8.p2.20.m20.1b"><ci id="S3.8.p2.20.m20.1.1.cmml" xref="S3.8.p2.20.m20.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.20.m20.1c">U</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.20.m20.1d">italic_U</annotation></semantics></math> of <math alttext="x" class="ltx_Math" display="inline" id="S3.8.p2.21.m21.1"><semantics id="S3.8.p2.21.m21.1a"><mi id="S3.8.p2.21.m21.1.1" xref="S3.8.p2.21.m21.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.8.p2.21.m21.1b"><ci id="S3.8.p2.21.m21.1.1.cmml" xref="S3.8.p2.21.m21.1.1">π₯</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.21.m21.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.21.m21.1d">italic_x</annotation></semantics></math>. Since <math alttext="X" class="ltx_Math" display="inline" id="S3.8.p2.22.m22.1"><semantics id="S3.8.p2.22.m22.1a"><mi id="S3.8.p2.22.m22.1.1" xref="S3.8.p2.22.m22.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.8.p2.22.m22.1b"><ci id="S3.8.p2.22.m22.1.1.cmml" xref="S3.8.p2.22.m22.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.22.m22.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.22.m22.1d">italic_X</annotation></semantics></math> is a Hausdorff semitopological semilattice, <math alttext="x=yx" class="ltx_Math" display="inline" id="S3.8.p2.23.m23.1"><semantics id="S3.8.p2.23.m23.1a"><mrow id="S3.8.p2.23.m23.1.1" xref="S3.8.p2.23.m23.1.1.cmml"><mi id="S3.8.p2.23.m23.1.1.2" xref="S3.8.p2.23.m23.1.1.2.cmml">x</mi><mo id="S3.8.p2.23.m23.1.1.1" xref="S3.8.p2.23.m23.1.1.1.cmml">=</mo><mrow id="S3.8.p2.23.m23.1.1.3" xref="S3.8.p2.23.m23.1.1.3.cmml"><mi id="S3.8.p2.23.m23.1.1.3.2" xref="S3.8.p2.23.m23.1.1.3.2.cmml">y</mi><mo id="S3.8.p2.23.m23.1.1.3.1" xref="S3.8.p2.23.m23.1.1.3.1.cmml">β’</mo><mi id="S3.8.p2.23.m23.1.1.3.3" xref="S3.8.p2.23.m23.1.1.3.3.cmml">x</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.8.p2.23.m23.1b"><apply id="S3.8.p2.23.m23.1.1.cmml" xref="S3.8.p2.23.m23.1.1"><eq id="S3.8.p2.23.m23.1.1.1.cmml" xref="S3.8.p2.23.m23.1.1.1"></eq><ci id="S3.8.p2.23.m23.1.1.2.cmml" xref="S3.8.p2.23.m23.1.1.2">π₯</ci><apply id="S3.8.p2.23.m23.1.1.3.cmml" xref="S3.8.p2.23.m23.1.1.3"><times id="S3.8.p2.23.m23.1.1.3.1.cmml" xref="S3.8.p2.23.m23.1.1.3.1"></times><ci id="S3.8.p2.23.m23.1.1.3.2.cmml" xref="S3.8.p2.23.m23.1.1.3.2">π¦</ci><ci id="S3.8.p2.23.m23.1.1.3.3.cmml" xref="S3.8.p2.23.m23.1.1.3.3">π₯</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.23.m23.1c">x=yx</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.23.m23.1d">italic_x = italic_y italic_x</annotation></semantics></math>. Hence <math alttext="x=\sup C" class="ltx_Math" display="inline" id="S3.8.p2.24.m24.1"><semantics id="S3.8.p2.24.m24.1a"><mrow id="S3.8.p2.24.m24.1.1" xref="S3.8.p2.24.m24.1.1.cmml"><mi id="S3.8.p2.24.m24.1.1.2" xref="S3.8.p2.24.m24.1.1.2.cmml">x</mi><mo id="S3.8.p2.24.m24.1.1.1" rspace="0.1389em" xref="S3.8.p2.24.m24.1.1.1.cmml">=</mo><mrow id="S3.8.p2.24.m24.1.1.3" xref="S3.8.p2.24.m24.1.1.3.cmml"><mo id="S3.8.p2.24.m24.1.1.3.1" lspace="0.1389em" rspace="0.167em" xref="S3.8.p2.24.m24.1.1.3.1.cmml">sup</mo><mi id="S3.8.p2.24.m24.1.1.3.2" xref="S3.8.p2.24.m24.1.1.3.2.cmml">C</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.8.p2.24.m24.1b"><apply id="S3.8.p2.24.m24.1.1.cmml" xref="S3.8.p2.24.m24.1.1"><eq id="S3.8.p2.24.m24.1.1.1.cmml" xref="S3.8.p2.24.m24.1.1.1"></eq><ci id="S3.8.p2.24.m24.1.1.2.cmml" xref="S3.8.p2.24.m24.1.1.2">π₯</ci><apply id="S3.8.p2.24.m24.1.1.3.cmml" xref="S3.8.p2.24.m24.1.1.3"><csymbol cd="latexml" id="S3.8.p2.24.m24.1.1.3.1.cmml" xref="S3.8.p2.24.m24.1.1.3.1">supremum</csymbol><ci id="S3.8.p2.24.m24.1.1.3.2.cmml" xref="S3.8.p2.24.m24.1.1.3.2">πΆ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.8.p2.24.m24.1c">x=\sup C</annotation><annotation encoding="application/x-llamapun" id="S3.8.p2.24.m24.1d">italic_x = roman_sup italic_C</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.9.p3"> <p class="ltx_p" id="S3.9.p3.13">(3) <math alttext="\Rightarrow" class="ltx_Math" display="inline" id="S3.9.p3.1.m1.1"><semantics id="S3.9.p3.1.m1.1a"><mo id="S3.9.p3.1.m1.1.1" stretchy="false" xref="S3.9.p3.1.m1.1.1.cmml">β</mo><annotation-xml encoding="MathML-Content" id="S3.9.p3.1.m1.1b"><ci id="S3.9.p3.1.m1.1.1.cmml" xref="S3.9.p3.1.m1.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p3.1.m1.1c">\Rightarrow</annotation><annotation encoding="application/x-llamapun" id="S3.9.p3.1.m1.1d">β</annotation></semantics></math> (1): To derive a contradiction, assume that there exists an open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S3.9.p3.2.m2.1"><semantics id="S3.9.p3.2.m2.1a"><mi id="S3.9.p3.2.m2.1.1" xref="S3.9.p3.2.m2.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.9.p3.2.m2.1b"><ci id="S3.9.p3.2.m2.1.1.cmml" xref="S3.9.p3.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p3.2.m2.1c">U</annotation><annotation encoding="application/x-llamapun" id="S3.9.p3.2.m2.1d">italic_U</annotation></semantics></math> of <math alttext="x" class="ltx_Math" display="inline" id="S3.9.p3.3.m3.1"><semantics id="S3.9.p3.3.m3.1a"><mi id="S3.9.p3.3.m3.1.1" xref="S3.9.p3.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.9.p3.3.m3.1b"><ci id="S3.9.p3.3.m3.1.1.cmml" xref="S3.9.p3.3.m3.1.1">π₯</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p3.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.9.p3.3.m3.1d">italic_x</annotation></semantics></math> such that the set <math alttext="D=C\setminus U" class="ltx_Math" display="inline" id="S3.9.p3.4.m4.1"><semantics id="S3.9.p3.4.m4.1a"><mrow id="S3.9.p3.4.m4.1.1" xref="S3.9.p3.4.m4.1.1.cmml"><mi id="S3.9.p3.4.m4.1.1.2" xref="S3.9.p3.4.m4.1.1.2.cmml">D</mi><mo id="S3.9.p3.4.m4.1.1.1" xref="S3.9.p3.4.m4.1.1.1.cmml">=</mo><mrow id="S3.9.p3.4.m4.1.1.3" xref="S3.9.p3.4.m4.1.1.3.cmml"><mi id="S3.9.p3.4.m4.1.1.3.2" xref="S3.9.p3.4.m4.1.1.3.2.cmml">C</mi><mo id="S3.9.p3.4.m4.1.1.3.1" xref="S3.9.p3.4.m4.1.1.3.1.cmml">β</mo><mi id="S3.9.p3.4.m4.1.1.3.3" xref="S3.9.p3.4.m4.1.1.3.3.cmml">U</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p3.4.m4.1b"><apply id="S3.9.p3.4.m4.1.1.cmml" xref="S3.9.p3.4.m4.1.1"><eq id="S3.9.p3.4.m4.1.1.1.cmml" xref="S3.9.p3.4.m4.1.1.1"></eq><ci id="S3.9.p3.4.m4.1.1.2.cmml" xref="S3.9.p3.4.m4.1.1.2">π·</ci><apply id="S3.9.p3.4.m4.1.1.3.cmml" xref="S3.9.p3.4.m4.1.1.3"><setdiff id="S3.9.p3.4.m4.1.1.3.1.cmml" xref="S3.9.p3.4.m4.1.1.3.1"></setdiff><ci id="S3.9.p3.4.m4.1.1.3.2.cmml" xref="S3.9.p3.4.m4.1.1.3.2">πΆ</ci><ci id="S3.9.p3.4.m4.1.1.3.3.cmml" xref="S3.9.p3.4.m4.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p3.4.m4.1c">D=C\setminus U</annotation><annotation encoding="application/x-llamapun" id="S3.9.p3.4.m4.1d">italic_D = italic_C β italic_U</annotation></semantics></math> is infinite. It is clear that <math alttext="D" class="ltx_Math" display="inline" id="S3.9.p3.5.m5.1"><semantics id="S3.9.p3.5.m5.1a"><mi id="S3.9.p3.5.m5.1.1" xref="S3.9.p3.5.m5.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.9.p3.5.m5.1b"><ci id="S3.9.p3.5.m5.1.1.cmml" xref="S3.9.p3.5.m5.1.1">π·</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p3.5.m5.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.9.p3.5.m5.1d">italic_D</annotation></semantics></math> is a cofinal subset of <math alttext="C" class="ltx_Math" display="inline" id="S3.9.p3.6.m6.1"><semantics id="S3.9.p3.6.m6.1a"><mi id="S3.9.p3.6.m6.1.1" xref="S3.9.p3.6.m6.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.9.p3.6.m6.1b"><ci id="S3.9.p3.6.m6.1.1.cmml" xref="S3.9.p3.6.m6.1.1">πΆ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p3.6.m6.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.9.p3.6.m6.1d">italic_C</annotation></semantics></math> isomorphic to <math alttext="(\omega,\min)" class="ltx_Math" display="inline" id="S3.9.p3.7.m7.2"><semantics id="S3.9.p3.7.m7.2a"><mrow id="S3.9.p3.7.m7.2.3.2" xref="S3.9.p3.7.m7.2.3.1.cmml"><mo id="S3.9.p3.7.m7.2.3.2.1" stretchy="false" xref="S3.9.p3.7.m7.2.3.1.cmml">(</mo><mi id="S3.9.p3.7.m7.1.1" xref="S3.9.p3.7.m7.1.1.cmml">Ο</mi><mo id="S3.9.p3.7.m7.2.3.2.2" xref="S3.9.p3.7.m7.2.3.1.cmml">,</mo><mi id="S3.9.p3.7.m7.2.2" xref="S3.9.p3.7.m7.2.2.cmml">min</mi><mo id="S3.9.p3.7.m7.2.3.2.3" stretchy="false" xref="S3.9.p3.7.m7.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p3.7.m7.2b"><interval closure="open" id="S3.9.p3.7.m7.2.3.1.cmml" xref="S3.9.p3.7.m7.2.3.2"><ci id="S3.9.p3.7.m7.1.1.cmml" xref="S3.9.p3.7.m7.1.1">π</ci><min id="S3.9.p3.7.m7.2.2.cmml" xref="S3.9.p3.7.m7.2.2"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p3.7.m7.2c">(\omega,\min)</annotation><annotation encoding="application/x-llamapun" id="S3.9.p3.7.m7.2d">( italic_Ο , roman_min )</annotation></semantics></math>. In particular, we have <math alttext="\sup D=x" class="ltx_Math" display="inline" id="S3.9.p3.8.m8.1"><semantics id="S3.9.p3.8.m8.1a"><mrow id="S3.9.p3.8.m8.1.1" xref="S3.9.p3.8.m8.1.1.cmml"><mrow id="S3.9.p3.8.m8.1.1.2" xref="S3.9.p3.8.m8.1.1.2.cmml"><mo id="S3.9.p3.8.m8.1.1.2.1" rspace="0.167em" xref="S3.9.p3.8.m8.1.1.2.1.cmml">sup</mo><mi id="S3.9.p3.8.m8.1.1.2.2" xref="S3.9.p3.8.m8.1.1.2.2.cmml">D</mi></mrow><mo id="S3.9.p3.8.m8.1.1.1" xref="S3.9.p3.8.m8.1.1.1.cmml">=</mo><mi id="S3.9.p3.8.m8.1.1.3" xref="S3.9.p3.8.m8.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p3.8.m8.1b"><apply id="S3.9.p3.8.m8.1.1.cmml" xref="S3.9.p3.8.m8.1.1"><eq id="S3.9.p3.8.m8.1.1.1.cmml" xref="S3.9.p3.8.m8.1.1.1"></eq><apply id="S3.9.p3.8.m8.1.1.2.cmml" xref="S3.9.p3.8.m8.1.1.2"><csymbol cd="latexml" id="S3.9.p3.8.m8.1.1.2.1.cmml" xref="S3.9.p3.8.m8.1.1.2.1">supremum</csymbol><ci id="S3.9.p3.8.m8.1.1.2.2.cmml" xref="S3.9.p3.8.m8.1.1.2.2">π·</ci></apply><ci id="S3.9.p3.8.m8.1.1.3.cmml" xref="S3.9.p3.8.m8.1.1.3">π₯</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p3.8.m8.1c">\sup D=x</annotation><annotation encoding="application/x-llamapun" id="S3.9.p3.8.m8.1d">roman_sup italic_D = italic_x</annotation></semantics></math>. By the countable compactness of <math alttext="X" class="ltx_Math" display="inline" id="S3.9.p3.9.m9.1"><semantics id="S3.9.p3.9.m9.1a"><mi id="S3.9.p3.9.m9.1.1" xref="S3.9.p3.9.m9.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.9.p3.9.m9.1b"><ci id="S3.9.p3.9.m9.1.1.cmml" xref="S3.9.p3.9.m9.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p3.9.m9.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.9.p3.9.m9.1d">italic_X</annotation></semantics></math>, there exists <math alttext="y\in\overline{D}" class="ltx_Math" display="inline" id="S3.9.p3.10.m10.1"><semantics id="S3.9.p3.10.m10.1a"><mrow id="S3.9.p3.10.m10.1.1" xref="S3.9.p3.10.m10.1.1.cmml"><mi id="S3.9.p3.10.m10.1.1.2" xref="S3.9.p3.10.m10.1.1.2.cmml">y</mi><mo id="S3.9.p3.10.m10.1.1.1" xref="S3.9.p3.10.m10.1.1.1.cmml">β</mo><mover accent="true" id="S3.9.p3.10.m10.1.1.3" xref="S3.9.p3.10.m10.1.1.3.cmml"><mi id="S3.9.p3.10.m10.1.1.3.2" xref="S3.9.p3.10.m10.1.1.3.2.cmml">D</mi><mo id="S3.9.p3.10.m10.1.1.3.1" xref="S3.9.p3.10.m10.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p3.10.m10.1b"><apply id="S3.9.p3.10.m10.1.1.cmml" xref="S3.9.p3.10.m10.1.1"><in id="S3.9.p3.10.m10.1.1.1.cmml" xref="S3.9.p3.10.m10.1.1.1"></in><ci id="S3.9.p3.10.m10.1.1.2.cmml" xref="S3.9.p3.10.m10.1.1.2">π¦</ci><apply id="S3.9.p3.10.m10.1.1.3.cmml" xref="S3.9.p3.10.m10.1.1.3"><ci id="S3.9.p3.10.m10.1.1.3.1.cmml" xref="S3.9.p3.10.m10.1.1.3.1">Β―</ci><ci id="S3.9.p3.10.m10.1.1.3.2.cmml" xref="S3.9.p3.10.m10.1.1.3.2">π·</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p3.10.m10.1c">y\in\overline{D}</annotation><annotation encoding="application/x-llamapun" id="S3.9.p3.10.m10.1d">italic_y β overΒ― start_ARG italic_D end_ARG</annotation></semantics></math>. Repeating the arguments above, we obtain <math alttext="y=\sup D" class="ltx_Math" display="inline" id="S3.9.p3.11.m11.1"><semantics id="S3.9.p3.11.m11.1a"><mrow id="S3.9.p3.11.m11.1.1" xref="S3.9.p3.11.m11.1.1.cmml"><mi id="S3.9.p3.11.m11.1.1.2" xref="S3.9.p3.11.m11.1.1.2.cmml">y</mi><mo id="S3.9.p3.11.m11.1.1.1" rspace="0.1389em" xref="S3.9.p3.11.m11.1.1.1.cmml">=</mo><mrow id="S3.9.p3.11.m11.1.1.3" xref="S3.9.p3.11.m11.1.1.3.cmml"><mo id="S3.9.p3.11.m11.1.1.3.1" lspace="0.1389em" rspace="0.167em" xref="S3.9.p3.11.m11.1.1.3.1.cmml">sup</mo><mi id="S3.9.p3.11.m11.1.1.3.2" xref="S3.9.p3.11.m11.1.1.3.2.cmml">D</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p3.11.m11.1b"><apply id="S3.9.p3.11.m11.1.1.cmml" xref="S3.9.p3.11.m11.1.1"><eq id="S3.9.p3.11.m11.1.1.1.cmml" xref="S3.9.p3.11.m11.1.1.1"></eq><ci id="S3.9.p3.11.m11.1.1.2.cmml" xref="S3.9.p3.11.m11.1.1.2">π¦</ci><apply id="S3.9.p3.11.m11.1.1.3.cmml" xref="S3.9.p3.11.m11.1.1.3"><csymbol cd="latexml" id="S3.9.p3.11.m11.1.1.3.1.cmml" xref="S3.9.p3.11.m11.1.1.3.1">supremum</csymbol><ci id="S3.9.p3.11.m11.1.1.3.2.cmml" xref="S3.9.p3.11.m11.1.1.3.2">π·</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p3.11.m11.1c">y=\sup D</annotation><annotation encoding="application/x-llamapun" id="S3.9.p3.11.m11.1d">italic_y = roman_sup italic_D</annotation></semantics></math>. Hence <math alttext="y=x" class="ltx_Math" display="inline" id="S3.9.p3.12.m12.1"><semantics id="S3.9.p3.12.m12.1a"><mrow id="S3.9.p3.12.m12.1.1" xref="S3.9.p3.12.m12.1.1.cmml"><mi id="S3.9.p3.12.m12.1.1.2" xref="S3.9.p3.12.m12.1.1.2.cmml">y</mi><mo id="S3.9.p3.12.m12.1.1.1" xref="S3.9.p3.12.m12.1.1.1.cmml">=</mo><mi id="S3.9.p3.12.m12.1.1.3" xref="S3.9.p3.12.m12.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p3.12.m12.1b"><apply id="S3.9.p3.12.m12.1.1.cmml" xref="S3.9.p3.12.m12.1.1"><eq id="S3.9.p3.12.m12.1.1.1.cmml" xref="S3.9.p3.12.m12.1.1.1"></eq><ci id="S3.9.p3.12.m12.1.1.2.cmml" xref="S3.9.p3.12.m12.1.1.2">π¦</ci><ci id="S3.9.p3.12.m12.1.1.3.cmml" xref="S3.9.p3.12.m12.1.1.3">π₯</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p3.12.m12.1c">y=x</annotation><annotation encoding="application/x-llamapun" id="S3.9.p3.12.m12.1d">italic_y = italic_x</annotation></semantics></math>, but <math alttext="x\notin\overline{D}" class="ltx_Math" display="inline" id="S3.9.p3.13.m13.1"><semantics id="S3.9.p3.13.m13.1a"><mrow id="S3.9.p3.13.m13.1.1" xref="S3.9.p3.13.m13.1.1.cmml"><mi id="S3.9.p3.13.m13.1.1.2" xref="S3.9.p3.13.m13.1.1.2.cmml">x</mi><mo id="S3.9.p3.13.m13.1.1.1" xref="S3.9.p3.13.m13.1.1.1.cmml">β</mo><mover accent="true" id="S3.9.p3.13.m13.1.1.3" xref="S3.9.p3.13.m13.1.1.3.cmml"><mi id="S3.9.p3.13.m13.1.1.3.2" xref="S3.9.p3.13.m13.1.1.3.2.cmml">D</mi><mo id="S3.9.p3.13.m13.1.1.3.1" xref="S3.9.p3.13.m13.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.9.p3.13.m13.1b"><apply id="S3.9.p3.13.m13.1.1.cmml" xref="S3.9.p3.13.m13.1.1"><notin id="S3.9.p3.13.m13.1.1.1.cmml" xref="S3.9.p3.13.m13.1.1.1"></notin><ci id="S3.9.p3.13.m13.1.1.2.cmml" xref="S3.9.p3.13.m13.1.1.2">π₯</ci><apply id="S3.9.p3.13.m13.1.1.3.cmml" xref="S3.9.p3.13.m13.1.1.3"><ci id="S3.9.p3.13.m13.1.1.3.1.cmml" xref="S3.9.p3.13.m13.1.1.3.1">Β―</ci><ci id="S3.9.p3.13.m13.1.1.3.2.cmml" xref="S3.9.p3.13.m13.1.1.3.2">π·</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.9.p3.13.m13.1c">x\notin\overline{D}</annotation><annotation encoding="application/x-llamapun" id="S3.9.p3.13.m13.1d">italic_x β overΒ― start_ARG italic_D end_ARG</annotation></semantics></math>, a contradiction. β</p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem9.1.1.1">Lemma 3.9</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem9.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem9.p1"> <p class="ltx_p" id="S3.Thmtheorem9.p1.9"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem9.p1.9.9">Let <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.1.1.m1.1"><semantics id="S3.Thmtheorem9.p1.1.1.m1.1a"><mi id="S3.Thmtheorem9.p1.1.1.m1.1.1" xref="S3.Thmtheorem9.p1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.1.1.m1.1b"><ci id="S3.Thmtheorem9.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem9.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.1.1.m1.1d">italic_X</annotation></semantics></math> be a semitopological semilattice and <math alttext="L\subset X" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.2.2.m2.1"><semantics id="S3.Thmtheorem9.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem9.p1.2.2.m2.1.1" xref="S3.Thmtheorem9.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem9.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem9.p1.2.2.m2.1.1.2.cmml">L</mi><mo id="S3.Thmtheorem9.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem9.p1.2.2.m2.1.1.1.cmml">β</mo><mi id="S3.Thmtheorem9.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem9.p1.2.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.2.2.m2.1b"><apply id="S3.Thmtheorem9.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.1.1"><subset id="S3.Thmtheorem9.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.1.1.1"></subset><ci id="S3.Thmtheorem9.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.1.1.2">πΏ</ci><ci id="S3.Thmtheorem9.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.2.2.m2.1c">L\subset X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.2.2.m2.1d">italic_L β italic_X</annotation></semantics></math> be a chain isomorphic to <math alttext="(\alpha,\min)" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.3.3.m3.2"><semantics id="S3.Thmtheorem9.p1.3.3.m3.2a"><mrow id="S3.Thmtheorem9.p1.3.3.m3.2.3.2" xref="S3.Thmtheorem9.p1.3.3.m3.2.3.1.cmml"><mo id="S3.Thmtheorem9.p1.3.3.m3.2.3.2.1" stretchy="false" xref="S3.Thmtheorem9.p1.3.3.m3.2.3.1.cmml">(</mo><mi id="S3.Thmtheorem9.p1.3.3.m3.1.1" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.cmml">Ξ±</mi><mo id="S3.Thmtheorem9.p1.3.3.m3.2.3.2.2" xref="S3.Thmtheorem9.p1.3.3.m3.2.3.1.cmml">,</mo><mi id="S3.Thmtheorem9.p1.3.3.m3.2.2" xref="S3.Thmtheorem9.p1.3.3.m3.2.2.cmml">min</mi><mo id="S3.Thmtheorem9.p1.3.3.m3.2.3.2.3" stretchy="false" xref="S3.Thmtheorem9.p1.3.3.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.3.3.m3.2b"><interval closure="open" id="S3.Thmtheorem9.p1.3.3.m3.2.3.1.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.2.3.2"><ci id="S3.Thmtheorem9.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.1.1">πΌ</ci><min id="S3.Thmtheorem9.p1.3.3.m3.2.2.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.2.2"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.3.3.m3.2c">(\alpha,\min)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.3.3.m3.2d">( italic_Ξ± , roman_min )</annotation></semantics></math> for some ordinal <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.4.4.m4.1"><semantics id="S3.Thmtheorem9.p1.4.4.m4.1a"><mi id="S3.Thmtheorem9.p1.4.4.m4.1.1" xref="S3.Thmtheorem9.p1.4.4.m4.1.1.cmml">Ξ±</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.4.4.m4.1b"><ci id="S3.Thmtheorem9.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem9.p1.4.4.m4.1.1">πΌ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.4.4.m4.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.4.4.m4.1d">italic_Ξ±</annotation></semantics></math>. If <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.5.5.m5.1"><semantics id="S3.Thmtheorem9.p1.5.5.m5.1a"><mover accent="true" id="S3.Thmtheorem9.p1.5.5.m5.1.1" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem9.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.2.cmml">L</mi><mo id="S3.Thmtheorem9.p1.5.5.m5.1.1.1" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.5.5.m5.1b"><apply id="S3.Thmtheorem9.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1"><ci id="S3.Thmtheorem9.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.1">Β―</ci><ci id="S3.Thmtheorem9.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.5.5.m5.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.5.5.m5.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> does not contain maximum, then <math alttext="L" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.6.6.m6.1"><semantics id="S3.Thmtheorem9.p1.6.6.m6.1a"><mi id="S3.Thmtheorem9.p1.6.6.m6.1.1" xref="S3.Thmtheorem9.p1.6.6.m6.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.6.6.m6.1b"><ci id="S3.Thmtheorem9.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.6.6.m6.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.6.6.m6.1d">italic_L</annotation></semantics></math> is cofinal in <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.7.7.m7.1"><semantics id="S3.Thmtheorem9.p1.7.7.m7.1a"><mover accent="true" id="S3.Thmtheorem9.p1.7.7.m7.1.1" xref="S3.Thmtheorem9.p1.7.7.m7.1.1.cmml"><mi id="S3.Thmtheorem9.p1.7.7.m7.1.1.2" xref="S3.Thmtheorem9.p1.7.7.m7.1.1.2.cmml">L</mi><mo id="S3.Thmtheorem9.p1.7.7.m7.1.1.1" xref="S3.Thmtheorem9.p1.7.7.m7.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.7.7.m7.1b"><apply id="S3.Thmtheorem9.p1.7.7.m7.1.1.cmml" xref="S3.Thmtheorem9.p1.7.7.m7.1.1"><ci id="S3.Thmtheorem9.p1.7.7.m7.1.1.1.cmml" xref="S3.Thmtheorem9.p1.7.7.m7.1.1.1">Β―</ci><ci id="S3.Thmtheorem9.p1.7.7.m7.1.1.2.cmml" xref="S3.Thmtheorem9.p1.7.7.m7.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.7.7.m7.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.7.7.m7.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> and <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.8.8.m8.1"><semantics id="S3.Thmtheorem9.p1.8.8.m8.1a"><mover accent="true" id="S3.Thmtheorem9.p1.8.8.m8.1.1" xref="S3.Thmtheorem9.p1.8.8.m8.1.1.cmml"><mi id="S3.Thmtheorem9.p1.8.8.m8.1.1.2" xref="S3.Thmtheorem9.p1.8.8.m8.1.1.2.cmml">L</mi><mo id="S3.Thmtheorem9.p1.8.8.m8.1.1.1" xref="S3.Thmtheorem9.p1.8.8.m8.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.8.8.m8.1b"><apply id="S3.Thmtheorem9.p1.8.8.m8.1.1.cmml" xref="S3.Thmtheorem9.p1.8.8.m8.1.1"><ci id="S3.Thmtheorem9.p1.8.8.m8.1.1.1.cmml" xref="S3.Thmtheorem9.p1.8.8.m8.1.1.1">Β―</ci><ci id="S3.Thmtheorem9.p1.8.8.m8.1.1.2.cmml" xref="S3.Thmtheorem9.p1.8.8.m8.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.8.8.m8.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.8.8.m8.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> is isomorphic to <math alttext="(\alpha,\min)" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.9.9.m9.2"><semantics id="S3.Thmtheorem9.p1.9.9.m9.2a"><mrow id="S3.Thmtheorem9.p1.9.9.m9.2.3.2" xref="S3.Thmtheorem9.p1.9.9.m9.2.3.1.cmml"><mo id="S3.Thmtheorem9.p1.9.9.m9.2.3.2.1" stretchy="false" xref="S3.Thmtheorem9.p1.9.9.m9.2.3.1.cmml">(</mo><mi id="S3.Thmtheorem9.p1.9.9.m9.1.1" xref="S3.Thmtheorem9.p1.9.9.m9.1.1.cmml">Ξ±</mi><mo id="S3.Thmtheorem9.p1.9.9.m9.2.3.2.2" xref="S3.Thmtheorem9.p1.9.9.m9.2.3.1.cmml">,</mo><mi id="S3.Thmtheorem9.p1.9.9.m9.2.2" xref="S3.Thmtheorem9.p1.9.9.m9.2.2.cmml">min</mi><mo id="S3.Thmtheorem9.p1.9.9.m9.2.3.2.3" stretchy="false" xref="S3.Thmtheorem9.p1.9.9.m9.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.9.9.m9.2b"><interval closure="open" id="S3.Thmtheorem9.p1.9.9.m9.2.3.1.cmml" xref="S3.Thmtheorem9.p1.9.9.m9.2.3.2"><ci id="S3.Thmtheorem9.p1.9.9.m9.1.1.cmml" xref="S3.Thmtheorem9.p1.9.9.m9.1.1">πΌ</ci><min id="S3.Thmtheorem9.p1.9.9.m9.2.2.cmml" xref="S3.Thmtheorem9.p1.9.9.m9.2.2"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.9.9.m9.2c">(\alpha,\min)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.9.9.m9.2d">( italic_Ξ± , roman_min )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S3.16"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.10.p1"> <p class="ltx_p" id="S3.10.p1.15">Let <math alttext="L=\{l_{\xi}:\xi\in\alpha\}" class="ltx_Math" display="inline" id="S3.10.p1.1.m1.2"><semantics id="S3.10.p1.1.m1.2a"><mrow id="S3.10.p1.1.m1.2.2" xref="S3.10.p1.1.m1.2.2.cmml"><mi id="S3.10.p1.1.m1.2.2.4" xref="S3.10.p1.1.m1.2.2.4.cmml">L</mi><mo id="S3.10.p1.1.m1.2.2.3" xref="S3.10.p1.1.m1.2.2.3.cmml">=</mo><mrow id="S3.10.p1.1.m1.2.2.2.2" xref="S3.10.p1.1.m1.2.2.2.3.cmml"><mo id="S3.10.p1.1.m1.2.2.2.2.3" stretchy="false" xref="S3.10.p1.1.m1.2.2.2.3.1.cmml">{</mo><msub id="S3.10.p1.1.m1.1.1.1.1.1" xref="S3.10.p1.1.m1.1.1.1.1.1.cmml"><mi id="S3.10.p1.1.m1.1.1.1.1.1.2" xref="S3.10.p1.1.m1.1.1.1.1.1.2.cmml">l</mi><mi id="S3.10.p1.1.m1.1.1.1.1.1.3" xref="S3.10.p1.1.m1.1.1.1.1.1.3.cmml">ΞΎ</mi></msub><mo id="S3.10.p1.1.m1.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.10.p1.1.m1.2.2.2.3.1.cmml">:</mo><mrow id="S3.10.p1.1.m1.2.2.2.2.2" xref="S3.10.p1.1.m1.2.2.2.2.2.cmml"><mi id="S3.10.p1.1.m1.2.2.2.2.2.2" xref="S3.10.p1.1.m1.2.2.2.2.2.2.cmml">ΞΎ</mi><mo id="S3.10.p1.1.m1.2.2.2.2.2.1" xref="S3.10.p1.1.m1.2.2.2.2.2.1.cmml">β</mo><mi id="S3.10.p1.1.m1.2.2.2.2.2.3" xref="S3.10.p1.1.m1.2.2.2.2.2.3.cmml">Ξ±</mi></mrow><mo id="S3.10.p1.1.m1.2.2.2.2.5" stretchy="false" xref="S3.10.p1.1.m1.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p1.1.m1.2b"><apply id="S3.10.p1.1.m1.2.2.cmml" xref="S3.10.p1.1.m1.2.2"><eq id="S3.10.p1.1.m1.2.2.3.cmml" xref="S3.10.p1.1.m1.2.2.3"></eq><ci id="S3.10.p1.1.m1.2.2.4.cmml" xref="S3.10.p1.1.m1.2.2.4">πΏ</ci><apply id="S3.10.p1.1.m1.2.2.2.3.cmml" xref="S3.10.p1.1.m1.2.2.2.2"><csymbol cd="latexml" id="S3.10.p1.1.m1.2.2.2.3.1.cmml" xref="S3.10.p1.1.m1.2.2.2.2.3">conditional-set</csymbol><apply id="S3.10.p1.1.m1.1.1.1.1.1.cmml" xref="S3.10.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.10.p1.1.m1.1.1.1.1.1.1.cmml" xref="S3.10.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S3.10.p1.1.m1.1.1.1.1.1.2.cmml" xref="S3.10.p1.1.m1.1.1.1.1.1.2">π</ci><ci id="S3.10.p1.1.m1.1.1.1.1.1.3.cmml" xref="S3.10.p1.1.m1.1.1.1.1.1.3">π</ci></apply><apply id="S3.10.p1.1.m1.2.2.2.2.2.cmml" xref="S3.10.p1.1.m1.2.2.2.2.2"><in id="S3.10.p1.1.m1.2.2.2.2.2.1.cmml" xref="S3.10.p1.1.m1.2.2.2.2.2.1"></in><ci id="S3.10.p1.1.m1.2.2.2.2.2.2.cmml" xref="S3.10.p1.1.m1.2.2.2.2.2.2">π</ci><ci id="S3.10.p1.1.m1.2.2.2.2.2.3.cmml" xref="S3.10.p1.1.m1.2.2.2.2.2.3">πΌ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.1.m1.2c">L=\{l_{\xi}:\xi\in\alpha\}</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.1.m1.2d">italic_L = { italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT : italic_ΞΎ β italic_Ξ± }</annotation></semantics></math>, where <math alttext="l_{\xi}\leq l_{\eta}" class="ltx_Math" display="inline" id="S3.10.p1.2.m2.1"><semantics id="S3.10.p1.2.m2.1a"><mrow id="S3.10.p1.2.m2.1.1" xref="S3.10.p1.2.m2.1.1.cmml"><msub id="S3.10.p1.2.m2.1.1.2" xref="S3.10.p1.2.m2.1.1.2.cmml"><mi id="S3.10.p1.2.m2.1.1.2.2" xref="S3.10.p1.2.m2.1.1.2.2.cmml">l</mi><mi id="S3.10.p1.2.m2.1.1.2.3" xref="S3.10.p1.2.m2.1.1.2.3.cmml">ΞΎ</mi></msub><mo id="S3.10.p1.2.m2.1.1.1" xref="S3.10.p1.2.m2.1.1.1.cmml">β€</mo><msub id="S3.10.p1.2.m2.1.1.3" xref="S3.10.p1.2.m2.1.1.3.cmml"><mi id="S3.10.p1.2.m2.1.1.3.2" xref="S3.10.p1.2.m2.1.1.3.2.cmml">l</mi><mi id="S3.10.p1.2.m2.1.1.3.3" xref="S3.10.p1.2.m2.1.1.3.3.cmml">Ξ·</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p1.2.m2.1b"><apply id="S3.10.p1.2.m2.1.1.cmml" xref="S3.10.p1.2.m2.1.1"><leq id="S3.10.p1.2.m2.1.1.1.cmml" xref="S3.10.p1.2.m2.1.1.1"></leq><apply id="S3.10.p1.2.m2.1.1.2.cmml" xref="S3.10.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.10.p1.2.m2.1.1.2.1.cmml" xref="S3.10.p1.2.m2.1.1.2">subscript</csymbol><ci id="S3.10.p1.2.m2.1.1.2.2.cmml" xref="S3.10.p1.2.m2.1.1.2.2">π</ci><ci id="S3.10.p1.2.m2.1.1.2.3.cmml" xref="S3.10.p1.2.m2.1.1.2.3">π</ci></apply><apply id="S3.10.p1.2.m2.1.1.3.cmml" xref="S3.10.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.10.p1.2.m2.1.1.3.1.cmml" xref="S3.10.p1.2.m2.1.1.3">subscript</csymbol><ci id="S3.10.p1.2.m2.1.1.3.2.cmml" xref="S3.10.p1.2.m2.1.1.3.2">π</ci><ci id="S3.10.p1.2.m2.1.1.3.3.cmml" xref="S3.10.p1.2.m2.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.2.m2.1c">l_{\xi}\leq l_{\eta}</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.2.m2.1d">italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT β€ italic_l start_POSTSUBSCRIPT italic_Ξ· end_POSTSUBSCRIPT</annotation></semantics></math> if and only if <math alttext="\xi\leq\eta" class="ltx_Math" display="inline" id="S3.10.p1.3.m3.1"><semantics id="S3.10.p1.3.m3.1a"><mrow id="S3.10.p1.3.m3.1.1" xref="S3.10.p1.3.m3.1.1.cmml"><mi id="S3.10.p1.3.m3.1.1.2" xref="S3.10.p1.3.m3.1.1.2.cmml">ΞΎ</mi><mo id="S3.10.p1.3.m3.1.1.1" xref="S3.10.p1.3.m3.1.1.1.cmml">β€</mo><mi id="S3.10.p1.3.m3.1.1.3" xref="S3.10.p1.3.m3.1.1.3.cmml">Ξ·</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p1.3.m3.1b"><apply id="S3.10.p1.3.m3.1.1.cmml" xref="S3.10.p1.3.m3.1.1"><leq id="S3.10.p1.3.m3.1.1.1.cmml" xref="S3.10.p1.3.m3.1.1.1"></leq><ci id="S3.10.p1.3.m3.1.1.2.cmml" xref="S3.10.p1.3.m3.1.1.2">π</ci><ci id="S3.10.p1.3.m3.1.1.3.cmml" xref="S3.10.p1.3.m3.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.3.m3.1c">\xi\leq\eta</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.3.m3.1d">italic_ΞΎ β€ italic_Ξ·</annotation></semantics></math>. Fix any <math alttext="x\in\overline{L}" class="ltx_Math" display="inline" id="S3.10.p1.4.m4.1"><semantics id="S3.10.p1.4.m4.1a"><mrow id="S3.10.p1.4.m4.1.1" xref="S3.10.p1.4.m4.1.1.cmml"><mi id="S3.10.p1.4.m4.1.1.2" xref="S3.10.p1.4.m4.1.1.2.cmml">x</mi><mo id="S3.10.p1.4.m4.1.1.1" xref="S3.10.p1.4.m4.1.1.1.cmml">β</mo><mover accent="true" id="S3.10.p1.4.m4.1.1.3" xref="S3.10.p1.4.m4.1.1.3.cmml"><mi id="S3.10.p1.4.m4.1.1.3.2" xref="S3.10.p1.4.m4.1.1.3.2.cmml">L</mi><mo id="S3.10.p1.4.m4.1.1.3.1" xref="S3.10.p1.4.m4.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p1.4.m4.1b"><apply id="S3.10.p1.4.m4.1.1.cmml" xref="S3.10.p1.4.m4.1.1"><in id="S3.10.p1.4.m4.1.1.1.cmml" xref="S3.10.p1.4.m4.1.1.1"></in><ci id="S3.10.p1.4.m4.1.1.2.cmml" xref="S3.10.p1.4.m4.1.1.2">π₯</ci><apply id="S3.10.p1.4.m4.1.1.3.cmml" xref="S3.10.p1.4.m4.1.1.3"><ci id="S3.10.p1.4.m4.1.1.3.1.cmml" xref="S3.10.p1.4.m4.1.1.3.1">Β―</ci><ci id="S3.10.p1.4.m4.1.1.3.2.cmml" xref="S3.10.p1.4.m4.1.1.3.2">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.4.m4.1c">x\in\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.4.m4.1d">italic_x β overΒ― start_ARG italic_L end_ARG</annotation></semantics></math>. If <math alttext="L\subseteq{\downarrow}x" class="ltx_math_unparsed" display="inline" id="S3.10.p1.5.m5.1"><semantics id="S3.10.p1.5.m5.1a"><mrow id="S3.10.p1.5.m5.1b"><mi id="S3.10.p1.5.m5.1.1">L</mi><mo id="S3.10.p1.5.m5.1.2" rspace="0em">β</mo><mo id="S3.10.p1.5.m5.1.3" lspace="0em" stretchy="false">β</mo><mi id="S3.10.p1.5.m5.1.4">x</mi></mrow><annotation encoding="application/x-tex" id="S3.10.p1.5.m5.1c">L\subseteq{\downarrow}x</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.5.m5.1d">italic_L β β italic_x</annotation></semantics></math>, then <math alttext="\overline{L}\subseteq{\downarrow}x" class="ltx_math_unparsed" display="inline" id="S3.10.p1.6.m6.1"><semantics id="S3.10.p1.6.m6.1a"><mrow id="S3.10.p1.6.m6.1b"><mover accent="true" id="S3.10.p1.6.m6.1.1"><mi id="S3.10.p1.6.m6.1.1.2">L</mi><mo id="S3.10.p1.6.m6.1.1.1">Β―</mo></mover><mo id="S3.10.p1.6.m6.1.2" rspace="0em">β</mo><mo id="S3.10.p1.6.m6.1.3" lspace="0em" stretchy="false">β</mo><mi id="S3.10.p1.6.m6.1.4">x</mi></mrow><annotation encoding="application/x-tex" id="S3.10.p1.6.m6.1c">\overline{L}\subseteq{\downarrow}x</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.6.m6.1d">overΒ― start_ARG italic_L end_ARG β β italic_x</annotation></semantics></math>, as <math alttext="{\downarrow}x" class="ltx_Math" display="inline" id="S3.10.p1.7.m7.1"><semantics id="S3.10.p1.7.m7.1a"><mrow id="S3.10.p1.7.m7.1.1" xref="S3.10.p1.7.m7.1.1.cmml"><mi id="S3.10.p1.7.m7.1.1.2" xref="S3.10.p1.7.m7.1.1.2.cmml"></mi><mo id="S3.10.p1.7.m7.1.1.1" stretchy="false" xref="S3.10.p1.7.m7.1.1.1.cmml">β</mo><mi id="S3.10.p1.7.m7.1.1.3" xref="S3.10.p1.7.m7.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p1.7.m7.1b"><apply id="S3.10.p1.7.m7.1.1.cmml" xref="S3.10.p1.7.m7.1.1"><ci id="S3.10.p1.7.m7.1.1.1.cmml" xref="S3.10.p1.7.m7.1.1.1">β</ci><csymbol cd="latexml" id="S3.10.p1.7.m7.1.1.2.cmml" xref="S3.10.p1.7.m7.1.1.2">absent</csymbol><ci id="S3.10.p1.7.m7.1.1.3.cmml" xref="S3.10.p1.7.m7.1.1.3">π₯</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.7.m7.1c">{\downarrow}x</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.7.m7.1d">β italic_x</annotation></semantics></math> is closed in <math alttext="X" class="ltx_Math" display="inline" id="S3.10.p1.8.m8.1"><semantics id="S3.10.p1.8.m8.1a"><mi id="S3.10.p1.8.m8.1.1" xref="S3.10.p1.8.m8.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.10.p1.8.m8.1b"><ci id="S3.10.p1.8.m8.1.1.cmml" xref="S3.10.p1.8.m8.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.8.m8.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.8.m8.1d">italic_X</annotation></semantics></math>. But then <math alttext="x" class="ltx_Math" display="inline" id="S3.10.p1.9.m9.1"><semantics id="S3.10.p1.9.m9.1a"><mi id="S3.10.p1.9.m9.1.1" xref="S3.10.p1.9.m9.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.10.p1.9.m9.1b"><ci id="S3.10.p1.9.m9.1.1.cmml" xref="S3.10.p1.9.m9.1.1">π₯</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.9.m9.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.9.m9.1d">italic_x</annotation></semantics></math> is the maximum of <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.10.p1.10.m10.1"><semantics id="S3.10.p1.10.m10.1a"><mover accent="true" id="S3.10.p1.10.m10.1.1" xref="S3.10.p1.10.m10.1.1.cmml"><mi id="S3.10.p1.10.m10.1.1.2" xref="S3.10.p1.10.m10.1.1.2.cmml">L</mi><mo id="S3.10.p1.10.m10.1.1.1" xref="S3.10.p1.10.m10.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.10.p1.10.m10.1b"><apply id="S3.10.p1.10.m10.1.1.cmml" xref="S3.10.p1.10.m10.1.1"><ci id="S3.10.p1.10.m10.1.1.1.cmml" xref="S3.10.p1.10.m10.1.1.1">Β―</ci><ci id="S3.10.p1.10.m10.1.1.2.cmml" xref="S3.10.p1.10.m10.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.10.m10.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.10.m10.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math>, a contradiction. So, for each <math alttext="x\in\overline{L}" class="ltx_Math" display="inline" id="S3.10.p1.11.m11.1"><semantics id="S3.10.p1.11.m11.1a"><mrow id="S3.10.p1.11.m11.1.1" xref="S3.10.p1.11.m11.1.1.cmml"><mi id="S3.10.p1.11.m11.1.1.2" xref="S3.10.p1.11.m11.1.1.2.cmml">x</mi><mo id="S3.10.p1.11.m11.1.1.1" xref="S3.10.p1.11.m11.1.1.1.cmml">β</mo><mover accent="true" id="S3.10.p1.11.m11.1.1.3" xref="S3.10.p1.11.m11.1.1.3.cmml"><mi id="S3.10.p1.11.m11.1.1.3.2" xref="S3.10.p1.11.m11.1.1.3.2.cmml">L</mi><mo id="S3.10.p1.11.m11.1.1.3.1" xref="S3.10.p1.11.m11.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p1.11.m11.1b"><apply id="S3.10.p1.11.m11.1.1.cmml" xref="S3.10.p1.11.m11.1.1"><in id="S3.10.p1.11.m11.1.1.1.cmml" xref="S3.10.p1.11.m11.1.1.1"></in><ci id="S3.10.p1.11.m11.1.1.2.cmml" xref="S3.10.p1.11.m11.1.1.2">π₯</ci><apply id="S3.10.p1.11.m11.1.1.3.cmml" xref="S3.10.p1.11.m11.1.1.3"><ci id="S3.10.p1.11.m11.1.1.3.1.cmml" xref="S3.10.p1.11.m11.1.1.3.1">Β―</ci><ci id="S3.10.p1.11.m11.1.1.3.2.cmml" xref="S3.10.p1.11.m11.1.1.3.2">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.11.m11.1c">x\in\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.11.m11.1d">italic_x β overΒ― start_ARG italic_L end_ARG</annotation></semantics></math>, the set <math alttext="L\setminus{\downarrow}x" class="ltx_Math" display="inline" id="S3.10.p1.12.m12.1"><semantics id="S3.10.p1.12.m12.1a"><mrow id="S3.10.p1.12.m12.1.1" xref="S3.10.p1.12.m12.1.1.cmml"><mrow id="S3.10.p1.12.m12.1.1.2" xref="S3.10.p1.12.m12.1.1.2.cmml"><mi id="S3.10.p1.12.m12.1.1.2.2" xref="S3.10.p1.12.m12.1.1.2.2.cmml">L</mi><mo id="S3.10.p1.12.m12.1.1.2.3" rspace="0em" xref="S3.10.p1.12.m12.1.1.2.3.cmml">β</mo></mrow><mo id="S3.10.p1.12.m12.1.1.1" lspace="0em" stretchy="false" xref="S3.10.p1.12.m12.1.1.1.cmml">β</mo><mi id="S3.10.p1.12.m12.1.1.3" xref="S3.10.p1.12.m12.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.10.p1.12.m12.1b"><apply id="S3.10.p1.12.m12.1.1.cmml" xref="S3.10.p1.12.m12.1.1"><ci id="S3.10.p1.12.m12.1.1.1.cmml" xref="S3.10.p1.12.m12.1.1.1">β</ci><apply id="S3.10.p1.12.m12.1.1.2.cmml" xref="S3.10.p1.12.m12.1.1.2"><csymbol cd="latexml" id="S3.10.p1.12.m12.1.1.2.1.cmml" xref="S3.10.p1.12.m12.1.1.2">limit-from</csymbol><ci id="S3.10.p1.12.m12.1.1.2.2.cmml" xref="S3.10.p1.12.m12.1.1.2.2">πΏ</ci><setdiff id="S3.10.p1.12.m12.1.1.2.3.cmml" xref="S3.10.p1.12.m12.1.1.2.3"></setdiff></apply><ci id="S3.10.p1.12.m12.1.1.3.cmml" xref="S3.10.p1.12.m12.1.1.3">π₯</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.12.m12.1c">L\setminus{\downarrow}x</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.12.m12.1d">italic_L β β italic_x</annotation></semantics></math> is nonempty. It follows that the ordinal <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.10.p1.13.m13.1"><semantics id="S3.10.p1.13.m13.1a"><mi id="S3.10.p1.13.m13.1.1" xref="S3.10.p1.13.m13.1.1.cmml">Ξ±</mi><annotation-xml encoding="MathML-Content" id="S3.10.p1.13.m13.1b"><ci id="S3.10.p1.13.m13.1.1.cmml" xref="S3.10.p1.13.m13.1.1">πΌ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.13.m13.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.13.m13.1d">italic_Ξ±</annotation></semantics></math> is limit and <math alttext="L" class="ltx_Math" display="inline" id="S3.10.p1.14.m14.1"><semantics id="S3.10.p1.14.m14.1a"><mi id="S3.10.p1.14.m14.1.1" xref="S3.10.p1.14.m14.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.10.p1.14.m14.1b"><ci id="S3.10.p1.14.m14.1.1.cmml" xref="S3.10.p1.14.m14.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.14.m14.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.14.m14.1d">italic_L</annotation></semantics></math> is cofinal in <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.10.p1.15.m15.1"><semantics id="S3.10.p1.15.m15.1a"><mover accent="true" id="S3.10.p1.15.m15.1.1" xref="S3.10.p1.15.m15.1.1.cmml"><mi id="S3.10.p1.15.m15.1.1.2" xref="S3.10.p1.15.m15.1.1.2.cmml">L</mi><mo id="S3.10.p1.15.m15.1.1.1" xref="S3.10.p1.15.m15.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.10.p1.15.m15.1b"><apply id="S3.10.p1.15.m15.1.1.cmml" xref="S3.10.p1.15.m15.1.1"><ci id="S3.10.p1.15.m15.1.1.1.cmml" xref="S3.10.p1.15.m15.1.1.1">Β―</ci><ci id="S3.10.p1.15.m15.1.1.2.cmml" xref="S3.10.p1.15.m15.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.10.p1.15.m15.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.10.p1.15.m15.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.11.p2"> <p class="ltx_p" id="S3.11.p2.26">For every <math alttext="x\in\overline{L}\setminus L" class="ltx_Math" display="inline" id="S3.11.p2.1.m1.1"><semantics id="S3.11.p2.1.m1.1a"><mrow id="S3.11.p2.1.m1.1.1" xref="S3.11.p2.1.m1.1.1.cmml"><mi id="S3.11.p2.1.m1.1.1.2" xref="S3.11.p2.1.m1.1.1.2.cmml">x</mi><mo id="S3.11.p2.1.m1.1.1.1" xref="S3.11.p2.1.m1.1.1.1.cmml">β</mo><mrow id="S3.11.p2.1.m1.1.1.3" xref="S3.11.p2.1.m1.1.1.3.cmml"><mover accent="true" id="S3.11.p2.1.m1.1.1.3.2" xref="S3.11.p2.1.m1.1.1.3.2.cmml"><mi id="S3.11.p2.1.m1.1.1.3.2.2" xref="S3.11.p2.1.m1.1.1.3.2.2.cmml">L</mi><mo id="S3.11.p2.1.m1.1.1.3.2.1" xref="S3.11.p2.1.m1.1.1.3.2.1.cmml">Β―</mo></mover><mo id="S3.11.p2.1.m1.1.1.3.1" xref="S3.11.p2.1.m1.1.1.3.1.cmml">β</mo><mi id="S3.11.p2.1.m1.1.1.3.3" xref="S3.11.p2.1.m1.1.1.3.3.cmml">L</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.1.m1.1b"><apply id="S3.11.p2.1.m1.1.1.cmml" xref="S3.11.p2.1.m1.1.1"><in id="S3.11.p2.1.m1.1.1.1.cmml" xref="S3.11.p2.1.m1.1.1.1"></in><ci id="S3.11.p2.1.m1.1.1.2.cmml" xref="S3.11.p2.1.m1.1.1.2">π₯</ci><apply id="S3.11.p2.1.m1.1.1.3.cmml" xref="S3.11.p2.1.m1.1.1.3"><setdiff id="S3.11.p2.1.m1.1.1.3.1.cmml" xref="S3.11.p2.1.m1.1.1.3.1"></setdiff><apply id="S3.11.p2.1.m1.1.1.3.2.cmml" xref="S3.11.p2.1.m1.1.1.3.2"><ci id="S3.11.p2.1.m1.1.1.3.2.1.cmml" xref="S3.11.p2.1.m1.1.1.3.2.1">Β―</ci><ci id="S3.11.p2.1.m1.1.1.3.2.2.cmml" xref="S3.11.p2.1.m1.1.1.3.2.2">πΏ</ci></apply><ci id="S3.11.p2.1.m1.1.1.3.3.cmml" xref="S3.11.p2.1.m1.1.1.3.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.1.m1.1c">x\in\overline{L}\setminus L</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.1.m1.1d">italic_x β overΒ― start_ARG italic_L end_ARG β italic_L</annotation></semantics></math> let <math alttext="\delta(x)=\min\{\xi\in\alpha:x\leq l_{\xi}\}" class="ltx_Math" display="inline" id="S3.11.p2.2.m2.3"><semantics id="S3.11.p2.2.m2.3a"><mrow id="S3.11.p2.2.m2.3.3" xref="S3.11.p2.2.m2.3.3.cmml"><mrow id="S3.11.p2.2.m2.3.3.3" xref="S3.11.p2.2.m2.3.3.3.cmml"><mi id="S3.11.p2.2.m2.3.3.3.2" xref="S3.11.p2.2.m2.3.3.3.2.cmml">Ξ΄</mi><mo id="S3.11.p2.2.m2.3.3.3.1" xref="S3.11.p2.2.m2.3.3.3.1.cmml">β’</mo><mrow id="S3.11.p2.2.m2.3.3.3.3.2" xref="S3.11.p2.2.m2.3.3.3.cmml"><mo id="S3.11.p2.2.m2.3.3.3.3.2.1" stretchy="false" xref="S3.11.p2.2.m2.3.3.3.cmml">(</mo><mi id="S3.11.p2.2.m2.1.1" xref="S3.11.p2.2.m2.1.1.cmml">x</mi><mo id="S3.11.p2.2.m2.3.3.3.3.2.2" stretchy="false" xref="S3.11.p2.2.m2.3.3.3.cmml">)</mo></mrow></mrow><mo id="S3.11.p2.2.m2.3.3.2" xref="S3.11.p2.2.m2.3.3.2.cmml">=</mo><mrow id="S3.11.p2.2.m2.3.3.1.1" xref="S3.11.p2.2.m2.3.3.1.2.cmml"><mi id="S3.11.p2.2.m2.2.2" xref="S3.11.p2.2.m2.2.2.cmml">min</mi><mo id="S3.11.p2.2.m2.3.3.1.1a" xref="S3.11.p2.2.m2.3.3.1.2.cmml">β‘</mo><mrow id="S3.11.p2.2.m2.3.3.1.1.1" xref="S3.11.p2.2.m2.3.3.1.2.cmml"><mo id="S3.11.p2.2.m2.3.3.1.1.1.2" stretchy="false" xref="S3.11.p2.2.m2.3.3.1.2.cmml">{</mo><mrow id="S3.11.p2.2.m2.3.3.1.1.1.1" xref="S3.11.p2.2.m2.3.3.1.1.1.1.cmml"><mrow id="S3.11.p2.2.m2.3.3.1.1.1.1.2" xref="S3.11.p2.2.m2.3.3.1.1.1.1.2.cmml"><mi id="S3.11.p2.2.m2.3.3.1.1.1.1.2.2" xref="S3.11.p2.2.m2.3.3.1.1.1.1.2.2.cmml">ΞΎ</mi><mo id="S3.11.p2.2.m2.3.3.1.1.1.1.2.1" xref="S3.11.p2.2.m2.3.3.1.1.1.1.2.1.cmml">β</mo><mi id="S3.11.p2.2.m2.3.3.1.1.1.1.2.3" xref="S3.11.p2.2.m2.3.3.1.1.1.1.2.3.cmml">Ξ±</mi></mrow><mo id="S3.11.p2.2.m2.3.3.1.1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.11.p2.2.m2.3.3.1.1.1.1.1.cmml">:</mo><mrow id="S3.11.p2.2.m2.3.3.1.1.1.1.3" xref="S3.11.p2.2.m2.3.3.1.1.1.1.3.cmml"><mi id="S3.11.p2.2.m2.3.3.1.1.1.1.3.2" xref="S3.11.p2.2.m2.3.3.1.1.1.1.3.2.cmml">x</mi><mo id="S3.11.p2.2.m2.3.3.1.1.1.1.3.1" xref="S3.11.p2.2.m2.3.3.1.1.1.1.3.1.cmml">β€</mo><msub id="S3.11.p2.2.m2.3.3.1.1.1.1.3.3" xref="S3.11.p2.2.m2.3.3.1.1.1.1.3.3.cmml"><mi id="S3.11.p2.2.m2.3.3.1.1.1.1.3.3.2" xref="S3.11.p2.2.m2.3.3.1.1.1.1.3.3.2.cmml">l</mi><mi id="S3.11.p2.2.m2.3.3.1.1.1.1.3.3.3" xref="S3.11.p2.2.m2.3.3.1.1.1.1.3.3.3.cmml">ΞΎ</mi></msub></mrow></mrow><mo id="S3.11.p2.2.m2.3.3.1.1.1.3" stretchy="false" xref="S3.11.p2.2.m2.3.3.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.2.m2.3b"><apply id="S3.11.p2.2.m2.3.3.cmml" xref="S3.11.p2.2.m2.3.3"><eq id="S3.11.p2.2.m2.3.3.2.cmml" xref="S3.11.p2.2.m2.3.3.2"></eq><apply id="S3.11.p2.2.m2.3.3.3.cmml" xref="S3.11.p2.2.m2.3.3.3"><times id="S3.11.p2.2.m2.3.3.3.1.cmml" xref="S3.11.p2.2.m2.3.3.3.1"></times><ci id="S3.11.p2.2.m2.3.3.3.2.cmml" xref="S3.11.p2.2.m2.3.3.3.2">πΏ</ci><ci id="S3.11.p2.2.m2.1.1.cmml" xref="S3.11.p2.2.m2.1.1">π₯</ci></apply><apply id="S3.11.p2.2.m2.3.3.1.2.cmml" xref="S3.11.p2.2.m2.3.3.1.1"><min id="S3.11.p2.2.m2.2.2.cmml" xref="S3.11.p2.2.m2.2.2"></min><apply id="S3.11.p2.2.m2.3.3.1.1.1.1.cmml" xref="S3.11.p2.2.m2.3.3.1.1.1.1"><ci id="S3.11.p2.2.m2.3.3.1.1.1.1.1.cmml" xref="S3.11.p2.2.m2.3.3.1.1.1.1.1">:</ci><apply id="S3.11.p2.2.m2.3.3.1.1.1.1.2.cmml" xref="S3.11.p2.2.m2.3.3.1.1.1.1.2"><in id="S3.11.p2.2.m2.3.3.1.1.1.1.2.1.cmml" xref="S3.11.p2.2.m2.3.3.1.1.1.1.2.1"></in><ci id="S3.11.p2.2.m2.3.3.1.1.1.1.2.2.cmml" xref="S3.11.p2.2.m2.3.3.1.1.1.1.2.2">π</ci><ci id="S3.11.p2.2.m2.3.3.1.1.1.1.2.3.cmml" xref="S3.11.p2.2.m2.3.3.1.1.1.1.2.3">πΌ</ci></apply><apply id="S3.11.p2.2.m2.3.3.1.1.1.1.3.cmml" xref="S3.11.p2.2.m2.3.3.1.1.1.1.3"><leq id="S3.11.p2.2.m2.3.3.1.1.1.1.3.1.cmml" xref="S3.11.p2.2.m2.3.3.1.1.1.1.3.1"></leq><ci id="S3.11.p2.2.m2.3.3.1.1.1.1.3.2.cmml" xref="S3.11.p2.2.m2.3.3.1.1.1.1.3.2">π₯</ci><apply id="S3.11.p2.2.m2.3.3.1.1.1.1.3.3.cmml" xref="S3.11.p2.2.m2.3.3.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.11.p2.2.m2.3.3.1.1.1.1.3.3.1.cmml" xref="S3.11.p2.2.m2.3.3.1.1.1.1.3.3">subscript</csymbol><ci id="S3.11.p2.2.m2.3.3.1.1.1.1.3.3.2.cmml" xref="S3.11.p2.2.m2.3.3.1.1.1.1.3.3.2">π</ci><ci id="S3.11.p2.2.m2.3.3.1.1.1.1.3.3.3.cmml" xref="S3.11.p2.2.m2.3.3.1.1.1.1.3.3.3">π</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.2.m2.3c">\delta(x)=\min\{\xi\in\alpha:x\leq l_{\xi}\}</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.2.m2.3d">italic_Ξ΄ ( italic_x ) = roman_min { italic_ΞΎ β italic_Ξ± : italic_x β€ italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT }</annotation></semantics></math>. Since <math alttext="L" class="ltx_Math" display="inline" id="S3.11.p2.3.m3.1"><semantics id="S3.11.p2.3.m3.1a"><mi id="S3.11.p2.3.m3.1.1" xref="S3.11.p2.3.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.11.p2.3.m3.1b"><ci id="S3.11.p2.3.m3.1.1.cmml" xref="S3.11.p2.3.m3.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.3.m3.1d">italic_L</annotation></semantics></math> is cofinal in <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.11.p2.4.m4.1"><semantics id="S3.11.p2.4.m4.1a"><mover accent="true" id="S3.11.p2.4.m4.1.1" xref="S3.11.p2.4.m4.1.1.cmml"><mi id="S3.11.p2.4.m4.1.1.2" xref="S3.11.p2.4.m4.1.1.2.cmml">L</mi><mo id="S3.11.p2.4.m4.1.1.1" xref="S3.11.p2.4.m4.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.11.p2.4.m4.1b"><apply id="S3.11.p2.4.m4.1.1.cmml" xref="S3.11.p2.4.m4.1.1"><ci id="S3.11.p2.4.m4.1.1.1.cmml" xref="S3.11.p2.4.m4.1.1.1">Β―</ci><ci id="S3.11.p2.4.m4.1.1.2.cmml" xref="S3.11.p2.4.m4.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.4.m4.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.4.m4.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math>, the ordinals <math alttext="\delta(x)" class="ltx_Math" display="inline" id="S3.11.p2.5.m5.1"><semantics id="S3.11.p2.5.m5.1a"><mrow id="S3.11.p2.5.m5.1.2" xref="S3.11.p2.5.m5.1.2.cmml"><mi id="S3.11.p2.5.m5.1.2.2" xref="S3.11.p2.5.m5.1.2.2.cmml">Ξ΄</mi><mo id="S3.11.p2.5.m5.1.2.1" xref="S3.11.p2.5.m5.1.2.1.cmml">β’</mo><mrow id="S3.11.p2.5.m5.1.2.3.2" xref="S3.11.p2.5.m5.1.2.cmml"><mo id="S3.11.p2.5.m5.1.2.3.2.1" stretchy="false" xref="S3.11.p2.5.m5.1.2.cmml">(</mo><mi id="S3.11.p2.5.m5.1.1" xref="S3.11.p2.5.m5.1.1.cmml">x</mi><mo id="S3.11.p2.5.m5.1.2.3.2.2" stretchy="false" xref="S3.11.p2.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.5.m5.1b"><apply id="S3.11.p2.5.m5.1.2.cmml" xref="S3.11.p2.5.m5.1.2"><times id="S3.11.p2.5.m5.1.2.1.cmml" xref="S3.11.p2.5.m5.1.2.1"></times><ci id="S3.11.p2.5.m5.1.2.2.cmml" xref="S3.11.p2.5.m5.1.2.2">πΏ</ci><ci id="S3.11.p2.5.m5.1.1.cmml" xref="S3.11.p2.5.m5.1.1">π₯</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.5.m5.1c">\delta(x)</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.5.m5.1d">italic_Ξ΄ ( italic_x )</annotation></semantics></math>, <math alttext="x\in\overline{L}\setminus L" class="ltx_Math" display="inline" id="S3.11.p2.6.m6.1"><semantics id="S3.11.p2.6.m6.1a"><mrow id="S3.11.p2.6.m6.1.1" xref="S3.11.p2.6.m6.1.1.cmml"><mi id="S3.11.p2.6.m6.1.1.2" xref="S3.11.p2.6.m6.1.1.2.cmml">x</mi><mo id="S3.11.p2.6.m6.1.1.1" xref="S3.11.p2.6.m6.1.1.1.cmml">β</mo><mrow id="S3.11.p2.6.m6.1.1.3" xref="S3.11.p2.6.m6.1.1.3.cmml"><mover accent="true" id="S3.11.p2.6.m6.1.1.3.2" xref="S3.11.p2.6.m6.1.1.3.2.cmml"><mi id="S3.11.p2.6.m6.1.1.3.2.2" xref="S3.11.p2.6.m6.1.1.3.2.2.cmml">L</mi><mo id="S3.11.p2.6.m6.1.1.3.2.1" xref="S3.11.p2.6.m6.1.1.3.2.1.cmml">Β―</mo></mover><mo id="S3.11.p2.6.m6.1.1.3.1" xref="S3.11.p2.6.m6.1.1.3.1.cmml">β</mo><mi id="S3.11.p2.6.m6.1.1.3.3" xref="S3.11.p2.6.m6.1.1.3.3.cmml">L</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.6.m6.1b"><apply id="S3.11.p2.6.m6.1.1.cmml" xref="S3.11.p2.6.m6.1.1"><in id="S3.11.p2.6.m6.1.1.1.cmml" xref="S3.11.p2.6.m6.1.1.1"></in><ci id="S3.11.p2.6.m6.1.1.2.cmml" xref="S3.11.p2.6.m6.1.1.2">π₯</ci><apply id="S3.11.p2.6.m6.1.1.3.cmml" xref="S3.11.p2.6.m6.1.1.3"><setdiff id="S3.11.p2.6.m6.1.1.3.1.cmml" xref="S3.11.p2.6.m6.1.1.3.1"></setdiff><apply id="S3.11.p2.6.m6.1.1.3.2.cmml" xref="S3.11.p2.6.m6.1.1.3.2"><ci id="S3.11.p2.6.m6.1.1.3.2.1.cmml" xref="S3.11.p2.6.m6.1.1.3.2.1">Β―</ci><ci id="S3.11.p2.6.m6.1.1.3.2.2.cmml" xref="S3.11.p2.6.m6.1.1.3.2.2">πΏ</ci></apply><ci id="S3.11.p2.6.m6.1.1.3.3.cmml" xref="S3.11.p2.6.m6.1.1.3.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.6.m6.1c">x\in\overline{L}\setminus L</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.6.m6.1d">italic_x β overΒ― start_ARG italic_L end_ARG β italic_L</annotation></semantics></math>, are well defined. By Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem3" title="Proposition 3.3 (Folklore). β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.3</span></a>, <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.11.p2.7.m7.1"><semantics id="S3.11.p2.7.m7.1a"><mover accent="true" id="S3.11.p2.7.m7.1.1" xref="S3.11.p2.7.m7.1.1.cmml"><mi id="S3.11.p2.7.m7.1.1.2" xref="S3.11.p2.7.m7.1.1.2.cmml">L</mi><mo id="S3.11.p2.7.m7.1.1.1" xref="S3.11.p2.7.m7.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.11.p2.7.m7.1b"><apply id="S3.11.p2.7.m7.1.1.cmml" xref="S3.11.p2.7.m7.1.1"><ci id="S3.11.p2.7.m7.1.1.1.cmml" xref="S3.11.p2.7.m7.1.1.1">Β―</ci><ci id="S3.11.p2.7.m7.1.1.2.cmml" xref="S3.11.p2.7.m7.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.7.m7.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.7.m7.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> is a chain. Seeking a contradiction, assume that there exists an infinite decreasing chain <math alttext="\{x_{n}:n\in\omega\}\subset\overline{L}" class="ltx_Math" display="inline" id="S3.11.p2.8.m8.2"><semantics id="S3.11.p2.8.m8.2a"><mrow id="S3.11.p2.8.m8.2.2" xref="S3.11.p2.8.m8.2.2.cmml"><mrow id="S3.11.p2.8.m8.2.2.2.2" xref="S3.11.p2.8.m8.2.2.2.3.cmml"><mo id="S3.11.p2.8.m8.2.2.2.2.3" stretchy="false" xref="S3.11.p2.8.m8.2.2.2.3.1.cmml">{</mo><msub id="S3.11.p2.8.m8.1.1.1.1.1" xref="S3.11.p2.8.m8.1.1.1.1.1.cmml"><mi id="S3.11.p2.8.m8.1.1.1.1.1.2" xref="S3.11.p2.8.m8.1.1.1.1.1.2.cmml">x</mi><mi id="S3.11.p2.8.m8.1.1.1.1.1.3" xref="S3.11.p2.8.m8.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S3.11.p2.8.m8.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.11.p2.8.m8.2.2.2.3.1.cmml">:</mo><mrow id="S3.11.p2.8.m8.2.2.2.2.2" xref="S3.11.p2.8.m8.2.2.2.2.2.cmml"><mi id="S3.11.p2.8.m8.2.2.2.2.2.2" xref="S3.11.p2.8.m8.2.2.2.2.2.2.cmml">n</mi><mo id="S3.11.p2.8.m8.2.2.2.2.2.1" xref="S3.11.p2.8.m8.2.2.2.2.2.1.cmml">β</mo><mi id="S3.11.p2.8.m8.2.2.2.2.2.3" xref="S3.11.p2.8.m8.2.2.2.2.2.3.cmml">Ο</mi></mrow><mo id="S3.11.p2.8.m8.2.2.2.2.5" stretchy="false" xref="S3.11.p2.8.m8.2.2.2.3.1.cmml">}</mo></mrow><mo id="S3.11.p2.8.m8.2.2.3" xref="S3.11.p2.8.m8.2.2.3.cmml">β</mo><mover accent="true" id="S3.11.p2.8.m8.2.2.4" xref="S3.11.p2.8.m8.2.2.4.cmml"><mi id="S3.11.p2.8.m8.2.2.4.2" xref="S3.11.p2.8.m8.2.2.4.2.cmml">L</mi><mo id="S3.11.p2.8.m8.2.2.4.1" xref="S3.11.p2.8.m8.2.2.4.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.8.m8.2b"><apply id="S3.11.p2.8.m8.2.2.cmml" xref="S3.11.p2.8.m8.2.2"><subset id="S3.11.p2.8.m8.2.2.3.cmml" xref="S3.11.p2.8.m8.2.2.3"></subset><apply id="S3.11.p2.8.m8.2.2.2.3.cmml" xref="S3.11.p2.8.m8.2.2.2.2"><csymbol cd="latexml" id="S3.11.p2.8.m8.2.2.2.3.1.cmml" xref="S3.11.p2.8.m8.2.2.2.2.3">conditional-set</csymbol><apply id="S3.11.p2.8.m8.1.1.1.1.1.cmml" xref="S3.11.p2.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.11.p2.8.m8.1.1.1.1.1.1.cmml" xref="S3.11.p2.8.m8.1.1.1.1.1">subscript</csymbol><ci id="S3.11.p2.8.m8.1.1.1.1.1.2.cmml" xref="S3.11.p2.8.m8.1.1.1.1.1.2">π₯</ci><ci id="S3.11.p2.8.m8.1.1.1.1.1.3.cmml" xref="S3.11.p2.8.m8.1.1.1.1.1.3">π</ci></apply><apply id="S3.11.p2.8.m8.2.2.2.2.2.cmml" xref="S3.11.p2.8.m8.2.2.2.2.2"><in id="S3.11.p2.8.m8.2.2.2.2.2.1.cmml" xref="S3.11.p2.8.m8.2.2.2.2.2.1"></in><ci id="S3.11.p2.8.m8.2.2.2.2.2.2.cmml" xref="S3.11.p2.8.m8.2.2.2.2.2.2">π</ci><ci id="S3.11.p2.8.m8.2.2.2.2.2.3.cmml" xref="S3.11.p2.8.m8.2.2.2.2.2.3">π</ci></apply></apply><apply id="S3.11.p2.8.m8.2.2.4.cmml" xref="S3.11.p2.8.m8.2.2.4"><ci id="S3.11.p2.8.m8.2.2.4.1.cmml" xref="S3.11.p2.8.m8.2.2.4.1">Β―</ci><ci id="S3.11.p2.8.m8.2.2.4.2.cmml" xref="S3.11.p2.8.m8.2.2.4.2">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.8.m8.2c">\{x_{n}:n\in\omega\}\subset\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.8.m8.2d">{ italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n β italic_Ο } β overΒ― start_ARG italic_L end_ARG</annotation></semantics></math>. The chain <math alttext="L" class="ltx_Math" display="inline" id="S3.11.p2.9.m9.1"><semantics id="S3.11.p2.9.m9.1a"><mi id="S3.11.p2.9.m9.1.1" xref="S3.11.p2.9.m9.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.11.p2.9.m9.1b"><ci id="S3.11.p2.9.m9.1.1.cmml" xref="S3.11.p2.9.m9.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.9.m9.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.9.m9.1d">italic_L</annotation></semantics></math>, being isomorphic to an ordinal, contains only finite decreasing chains. Hence we lose no generality assuming that <math alttext="\{x_{n}:n\in\omega\}\subset\overline{L}\setminus L" class="ltx_Math" display="inline" id="S3.11.p2.10.m10.2"><semantics id="S3.11.p2.10.m10.2a"><mrow id="S3.11.p2.10.m10.2.2" xref="S3.11.p2.10.m10.2.2.cmml"><mrow id="S3.11.p2.10.m10.2.2.2.2" xref="S3.11.p2.10.m10.2.2.2.3.cmml"><mo id="S3.11.p2.10.m10.2.2.2.2.3" stretchy="false" xref="S3.11.p2.10.m10.2.2.2.3.1.cmml">{</mo><msub id="S3.11.p2.10.m10.1.1.1.1.1" xref="S3.11.p2.10.m10.1.1.1.1.1.cmml"><mi id="S3.11.p2.10.m10.1.1.1.1.1.2" xref="S3.11.p2.10.m10.1.1.1.1.1.2.cmml">x</mi><mi id="S3.11.p2.10.m10.1.1.1.1.1.3" xref="S3.11.p2.10.m10.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S3.11.p2.10.m10.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.11.p2.10.m10.2.2.2.3.1.cmml">:</mo><mrow id="S3.11.p2.10.m10.2.2.2.2.2" xref="S3.11.p2.10.m10.2.2.2.2.2.cmml"><mi id="S3.11.p2.10.m10.2.2.2.2.2.2" xref="S3.11.p2.10.m10.2.2.2.2.2.2.cmml">n</mi><mo id="S3.11.p2.10.m10.2.2.2.2.2.1" xref="S3.11.p2.10.m10.2.2.2.2.2.1.cmml">β</mo><mi id="S3.11.p2.10.m10.2.2.2.2.2.3" xref="S3.11.p2.10.m10.2.2.2.2.2.3.cmml">Ο</mi></mrow><mo id="S3.11.p2.10.m10.2.2.2.2.5" stretchy="false" xref="S3.11.p2.10.m10.2.2.2.3.1.cmml">}</mo></mrow><mo id="S3.11.p2.10.m10.2.2.3" xref="S3.11.p2.10.m10.2.2.3.cmml">β</mo><mrow id="S3.11.p2.10.m10.2.2.4" xref="S3.11.p2.10.m10.2.2.4.cmml"><mover accent="true" id="S3.11.p2.10.m10.2.2.4.2" xref="S3.11.p2.10.m10.2.2.4.2.cmml"><mi id="S3.11.p2.10.m10.2.2.4.2.2" xref="S3.11.p2.10.m10.2.2.4.2.2.cmml">L</mi><mo id="S3.11.p2.10.m10.2.2.4.2.1" xref="S3.11.p2.10.m10.2.2.4.2.1.cmml">Β―</mo></mover><mo id="S3.11.p2.10.m10.2.2.4.1" xref="S3.11.p2.10.m10.2.2.4.1.cmml">β</mo><mi id="S3.11.p2.10.m10.2.2.4.3" xref="S3.11.p2.10.m10.2.2.4.3.cmml">L</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.10.m10.2b"><apply id="S3.11.p2.10.m10.2.2.cmml" xref="S3.11.p2.10.m10.2.2"><subset id="S3.11.p2.10.m10.2.2.3.cmml" xref="S3.11.p2.10.m10.2.2.3"></subset><apply id="S3.11.p2.10.m10.2.2.2.3.cmml" xref="S3.11.p2.10.m10.2.2.2.2"><csymbol cd="latexml" id="S3.11.p2.10.m10.2.2.2.3.1.cmml" xref="S3.11.p2.10.m10.2.2.2.2.3">conditional-set</csymbol><apply id="S3.11.p2.10.m10.1.1.1.1.1.cmml" xref="S3.11.p2.10.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.11.p2.10.m10.1.1.1.1.1.1.cmml" xref="S3.11.p2.10.m10.1.1.1.1.1">subscript</csymbol><ci id="S3.11.p2.10.m10.1.1.1.1.1.2.cmml" xref="S3.11.p2.10.m10.1.1.1.1.1.2">π₯</ci><ci id="S3.11.p2.10.m10.1.1.1.1.1.3.cmml" xref="S3.11.p2.10.m10.1.1.1.1.1.3">π</ci></apply><apply id="S3.11.p2.10.m10.2.2.2.2.2.cmml" xref="S3.11.p2.10.m10.2.2.2.2.2"><in id="S3.11.p2.10.m10.2.2.2.2.2.1.cmml" xref="S3.11.p2.10.m10.2.2.2.2.2.1"></in><ci id="S3.11.p2.10.m10.2.2.2.2.2.2.cmml" xref="S3.11.p2.10.m10.2.2.2.2.2.2">π</ci><ci id="S3.11.p2.10.m10.2.2.2.2.2.3.cmml" xref="S3.11.p2.10.m10.2.2.2.2.2.3">π</ci></apply></apply><apply id="S3.11.p2.10.m10.2.2.4.cmml" xref="S3.11.p2.10.m10.2.2.4"><setdiff id="S3.11.p2.10.m10.2.2.4.1.cmml" xref="S3.11.p2.10.m10.2.2.4.1"></setdiff><apply id="S3.11.p2.10.m10.2.2.4.2.cmml" xref="S3.11.p2.10.m10.2.2.4.2"><ci id="S3.11.p2.10.m10.2.2.4.2.1.cmml" xref="S3.11.p2.10.m10.2.2.4.2.1">Β―</ci><ci id="S3.11.p2.10.m10.2.2.4.2.2.cmml" xref="S3.11.p2.10.m10.2.2.4.2.2">πΏ</ci></apply><ci id="S3.11.p2.10.m10.2.2.4.3.cmml" xref="S3.11.p2.10.m10.2.2.4.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.10.m10.2c">\{x_{n}:n\in\omega\}\subset\overline{L}\setminus L</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.10.m10.2d">{ italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n β italic_Ο } β overΒ― start_ARG italic_L end_ARG β italic_L</annotation></semantics></math>. Observe that <math alttext="\{\delta(x_{n}):n\in\omega\}" class="ltx_Math" display="inline" id="S3.11.p2.11.m11.2"><semantics id="S3.11.p2.11.m11.2a"><mrow id="S3.11.p2.11.m11.2.2.2" xref="S3.11.p2.11.m11.2.2.3.cmml"><mo id="S3.11.p2.11.m11.2.2.2.3" stretchy="false" xref="S3.11.p2.11.m11.2.2.3.1.cmml">{</mo><mrow id="S3.11.p2.11.m11.1.1.1.1" xref="S3.11.p2.11.m11.1.1.1.1.cmml"><mi id="S3.11.p2.11.m11.1.1.1.1.3" xref="S3.11.p2.11.m11.1.1.1.1.3.cmml">Ξ΄</mi><mo id="S3.11.p2.11.m11.1.1.1.1.2" xref="S3.11.p2.11.m11.1.1.1.1.2.cmml">β’</mo><mrow id="S3.11.p2.11.m11.1.1.1.1.1.1" xref="S3.11.p2.11.m11.1.1.1.1.1.1.1.cmml"><mo id="S3.11.p2.11.m11.1.1.1.1.1.1.2" stretchy="false" xref="S3.11.p2.11.m11.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.11.p2.11.m11.1.1.1.1.1.1.1" xref="S3.11.p2.11.m11.1.1.1.1.1.1.1.cmml"><mi id="S3.11.p2.11.m11.1.1.1.1.1.1.1.2" xref="S3.11.p2.11.m11.1.1.1.1.1.1.1.2.cmml">x</mi><mi id="S3.11.p2.11.m11.1.1.1.1.1.1.1.3" xref="S3.11.p2.11.m11.1.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S3.11.p2.11.m11.1.1.1.1.1.1.3" rspace="0.278em" stretchy="false" xref="S3.11.p2.11.m11.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.11.p2.11.m11.2.2.2.4" rspace="0.278em" xref="S3.11.p2.11.m11.2.2.3.1.cmml">:</mo><mrow id="S3.11.p2.11.m11.2.2.2.2" xref="S3.11.p2.11.m11.2.2.2.2.cmml"><mi id="S3.11.p2.11.m11.2.2.2.2.2" xref="S3.11.p2.11.m11.2.2.2.2.2.cmml">n</mi><mo id="S3.11.p2.11.m11.2.2.2.2.1" xref="S3.11.p2.11.m11.2.2.2.2.1.cmml">β</mo><mi id="S3.11.p2.11.m11.2.2.2.2.3" xref="S3.11.p2.11.m11.2.2.2.2.3.cmml">Ο</mi></mrow><mo id="S3.11.p2.11.m11.2.2.2.5" stretchy="false" xref="S3.11.p2.11.m11.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.11.m11.2b"><apply id="S3.11.p2.11.m11.2.2.3.cmml" xref="S3.11.p2.11.m11.2.2.2"><csymbol cd="latexml" id="S3.11.p2.11.m11.2.2.3.1.cmml" xref="S3.11.p2.11.m11.2.2.2.3">conditional-set</csymbol><apply id="S3.11.p2.11.m11.1.1.1.1.cmml" xref="S3.11.p2.11.m11.1.1.1.1"><times id="S3.11.p2.11.m11.1.1.1.1.2.cmml" xref="S3.11.p2.11.m11.1.1.1.1.2"></times><ci id="S3.11.p2.11.m11.1.1.1.1.3.cmml" xref="S3.11.p2.11.m11.1.1.1.1.3">πΏ</ci><apply id="S3.11.p2.11.m11.1.1.1.1.1.1.1.cmml" xref="S3.11.p2.11.m11.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.11.p2.11.m11.1.1.1.1.1.1.1.1.cmml" xref="S3.11.p2.11.m11.1.1.1.1.1.1">subscript</csymbol><ci id="S3.11.p2.11.m11.1.1.1.1.1.1.1.2.cmml" xref="S3.11.p2.11.m11.1.1.1.1.1.1.1.2">π₯</ci><ci id="S3.11.p2.11.m11.1.1.1.1.1.1.1.3.cmml" xref="S3.11.p2.11.m11.1.1.1.1.1.1.1.3">π</ci></apply></apply><apply id="S3.11.p2.11.m11.2.2.2.2.cmml" xref="S3.11.p2.11.m11.2.2.2.2"><in id="S3.11.p2.11.m11.2.2.2.2.1.cmml" xref="S3.11.p2.11.m11.2.2.2.2.1"></in><ci id="S3.11.p2.11.m11.2.2.2.2.2.cmml" xref="S3.11.p2.11.m11.2.2.2.2.2">π</ci><ci id="S3.11.p2.11.m11.2.2.2.2.3.cmml" xref="S3.11.p2.11.m11.2.2.2.2.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.11.m11.2c">\{\delta(x_{n}):n\in\omega\}</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.11.m11.2d">{ italic_Ξ΄ ( italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) : italic_n β italic_Ο }</annotation></semantics></math> is a nonincreasing chain in <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.11.p2.12.m12.1"><semantics id="S3.11.p2.12.m12.1a"><mi id="S3.11.p2.12.m12.1.1" xref="S3.11.p2.12.m12.1.1.cmml">Ξ±</mi><annotation-xml encoding="MathML-Content" id="S3.11.p2.12.m12.1b"><ci id="S3.11.p2.12.m12.1.1.cmml" xref="S3.11.p2.12.m12.1.1">πΌ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.12.m12.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.12.m12.1d">italic_Ξ±</annotation></semantics></math>. Thus there exists <math alttext="k\in\omega" class="ltx_Math" display="inline" id="S3.11.p2.13.m13.1"><semantics id="S3.11.p2.13.m13.1a"><mrow id="S3.11.p2.13.m13.1.1" xref="S3.11.p2.13.m13.1.1.cmml"><mi id="S3.11.p2.13.m13.1.1.2" xref="S3.11.p2.13.m13.1.1.2.cmml">k</mi><mo id="S3.11.p2.13.m13.1.1.1" xref="S3.11.p2.13.m13.1.1.1.cmml">β</mo><mi id="S3.11.p2.13.m13.1.1.3" xref="S3.11.p2.13.m13.1.1.3.cmml">Ο</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.13.m13.1b"><apply id="S3.11.p2.13.m13.1.1.cmml" xref="S3.11.p2.13.m13.1.1"><in id="S3.11.p2.13.m13.1.1.1.cmml" xref="S3.11.p2.13.m13.1.1.1"></in><ci id="S3.11.p2.13.m13.1.1.2.cmml" xref="S3.11.p2.13.m13.1.1.2">π</ci><ci id="S3.11.p2.13.m13.1.1.3.cmml" xref="S3.11.p2.13.m13.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.13.m13.1c">k\in\omega</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.13.m13.1d">italic_k β italic_Ο</annotation></semantics></math> such that <math alttext="\delta(x_{n})=\delta(x_{m})" class="ltx_Math" display="inline" id="S3.11.p2.14.m14.2"><semantics id="S3.11.p2.14.m14.2a"><mrow id="S3.11.p2.14.m14.2.2" xref="S3.11.p2.14.m14.2.2.cmml"><mrow id="S3.11.p2.14.m14.1.1.1" xref="S3.11.p2.14.m14.1.1.1.cmml"><mi id="S3.11.p2.14.m14.1.1.1.3" xref="S3.11.p2.14.m14.1.1.1.3.cmml">Ξ΄</mi><mo id="S3.11.p2.14.m14.1.1.1.2" xref="S3.11.p2.14.m14.1.1.1.2.cmml">β’</mo><mrow id="S3.11.p2.14.m14.1.1.1.1.1" xref="S3.11.p2.14.m14.1.1.1.1.1.1.cmml"><mo id="S3.11.p2.14.m14.1.1.1.1.1.2" stretchy="false" xref="S3.11.p2.14.m14.1.1.1.1.1.1.cmml">(</mo><msub id="S3.11.p2.14.m14.1.1.1.1.1.1" xref="S3.11.p2.14.m14.1.1.1.1.1.1.cmml"><mi id="S3.11.p2.14.m14.1.1.1.1.1.1.2" xref="S3.11.p2.14.m14.1.1.1.1.1.1.2.cmml">x</mi><mi id="S3.11.p2.14.m14.1.1.1.1.1.1.3" xref="S3.11.p2.14.m14.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S3.11.p2.14.m14.1.1.1.1.1.3" stretchy="false" xref="S3.11.p2.14.m14.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.11.p2.14.m14.2.2.3" xref="S3.11.p2.14.m14.2.2.3.cmml">=</mo><mrow id="S3.11.p2.14.m14.2.2.2" xref="S3.11.p2.14.m14.2.2.2.cmml"><mi id="S3.11.p2.14.m14.2.2.2.3" xref="S3.11.p2.14.m14.2.2.2.3.cmml">Ξ΄</mi><mo id="S3.11.p2.14.m14.2.2.2.2" xref="S3.11.p2.14.m14.2.2.2.2.cmml">β’</mo><mrow id="S3.11.p2.14.m14.2.2.2.1.1" xref="S3.11.p2.14.m14.2.2.2.1.1.1.cmml"><mo id="S3.11.p2.14.m14.2.2.2.1.1.2" stretchy="false" xref="S3.11.p2.14.m14.2.2.2.1.1.1.cmml">(</mo><msub id="S3.11.p2.14.m14.2.2.2.1.1.1" xref="S3.11.p2.14.m14.2.2.2.1.1.1.cmml"><mi id="S3.11.p2.14.m14.2.2.2.1.1.1.2" xref="S3.11.p2.14.m14.2.2.2.1.1.1.2.cmml">x</mi><mi id="S3.11.p2.14.m14.2.2.2.1.1.1.3" xref="S3.11.p2.14.m14.2.2.2.1.1.1.3.cmml">m</mi></msub><mo id="S3.11.p2.14.m14.2.2.2.1.1.3" stretchy="false" xref="S3.11.p2.14.m14.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.14.m14.2b"><apply id="S3.11.p2.14.m14.2.2.cmml" xref="S3.11.p2.14.m14.2.2"><eq id="S3.11.p2.14.m14.2.2.3.cmml" xref="S3.11.p2.14.m14.2.2.3"></eq><apply id="S3.11.p2.14.m14.1.1.1.cmml" xref="S3.11.p2.14.m14.1.1.1"><times id="S3.11.p2.14.m14.1.1.1.2.cmml" xref="S3.11.p2.14.m14.1.1.1.2"></times><ci id="S3.11.p2.14.m14.1.1.1.3.cmml" xref="S3.11.p2.14.m14.1.1.1.3">πΏ</ci><apply id="S3.11.p2.14.m14.1.1.1.1.1.1.cmml" xref="S3.11.p2.14.m14.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.11.p2.14.m14.1.1.1.1.1.1.1.cmml" xref="S3.11.p2.14.m14.1.1.1.1.1">subscript</csymbol><ci id="S3.11.p2.14.m14.1.1.1.1.1.1.2.cmml" xref="S3.11.p2.14.m14.1.1.1.1.1.1.2">π₯</ci><ci id="S3.11.p2.14.m14.1.1.1.1.1.1.3.cmml" xref="S3.11.p2.14.m14.1.1.1.1.1.1.3">π</ci></apply></apply><apply id="S3.11.p2.14.m14.2.2.2.cmml" xref="S3.11.p2.14.m14.2.2.2"><times id="S3.11.p2.14.m14.2.2.2.2.cmml" xref="S3.11.p2.14.m14.2.2.2.2"></times><ci id="S3.11.p2.14.m14.2.2.2.3.cmml" xref="S3.11.p2.14.m14.2.2.2.3">πΏ</ci><apply id="S3.11.p2.14.m14.2.2.2.1.1.1.cmml" xref="S3.11.p2.14.m14.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.11.p2.14.m14.2.2.2.1.1.1.1.cmml" xref="S3.11.p2.14.m14.2.2.2.1.1">subscript</csymbol><ci id="S3.11.p2.14.m14.2.2.2.1.1.1.2.cmml" xref="S3.11.p2.14.m14.2.2.2.1.1.1.2">π₯</ci><ci id="S3.11.p2.14.m14.2.2.2.1.1.1.3.cmml" xref="S3.11.p2.14.m14.2.2.2.1.1.1.3">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.14.m14.2c">\delta(x_{n})=\delta(x_{m})</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.14.m14.2d">italic_Ξ΄ ( italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = italic_Ξ΄ ( italic_x start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT )</annotation></semantics></math> for each <math alttext="n,m\geq k" class="ltx_Math" display="inline" id="S3.11.p2.15.m15.2"><semantics id="S3.11.p2.15.m15.2a"><mrow id="S3.11.p2.15.m15.2.3" xref="S3.11.p2.15.m15.2.3.cmml"><mrow id="S3.11.p2.15.m15.2.3.2.2" xref="S3.11.p2.15.m15.2.3.2.1.cmml"><mi id="S3.11.p2.15.m15.1.1" xref="S3.11.p2.15.m15.1.1.cmml">n</mi><mo id="S3.11.p2.15.m15.2.3.2.2.1" xref="S3.11.p2.15.m15.2.3.2.1.cmml">,</mo><mi id="S3.11.p2.15.m15.2.2" xref="S3.11.p2.15.m15.2.2.cmml">m</mi></mrow><mo id="S3.11.p2.15.m15.2.3.1" xref="S3.11.p2.15.m15.2.3.1.cmml">β₯</mo><mi id="S3.11.p2.15.m15.2.3.3" xref="S3.11.p2.15.m15.2.3.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.15.m15.2b"><apply id="S3.11.p2.15.m15.2.3.cmml" xref="S3.11.p2.15.m15.2.3"><geq id="S3.11.p2.15.m15.2.3.1.cmml" xref="S3.11.p2.15.m15.2.3.1"></geq><list id="S3.11.p2.15.m15.2.3.2.1.cmml" xref="S3.11.p2.15.m15.2.3.2.2"><ci id="S3.11.p2.15.m15.1.1.cmml" xref="S3.11.p2.15.m15.1.1">π</ci><ci id="S3.11.p2.15.m15.2.2.cmml" xref="S3.11.p2.15.m15.2.2">π</ci></list><ci id="S3.11.p2.15.m15.2.3.3.cmml" xref="S3.11.p2.15.m15.2.3.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.15.m15.2c">n,m\geq k</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.15.m15.2d">italic_n , italic_m β₯ italic_k</annotation></semantics></math>. Then the set <math alttext="W=X\setminus({\uparrow}x_{k}\cup{\downarrow}x_{k+2})" class="ltx_math_unparsed" display="inline" id="S3.11.p2.16.m16.1"><semantics id="S3.11.p2.16.m16.1a"><mrow id="S3.11.p2.16.m16.1b"><mi id="S3.11.p2.16.m16.1.1">W</mi><mo id="S3.11.p2.16.m16.1.2">=</mo><mi id="S3.11.p2.16.m16.1.3">X</mi><mo id="S3.11.p2.16.m16.1.4">β</mo><mrow id="S3.11.p2.16.m16.1.5"><mo id="S3.11.p2.16.m16.1.5.1" stretchy="false">(</mo><mo id="S3.11.p2.16.m16.1.5.2" lspace="0em" stretchy="false">β</mo><msub id="S3.11.p2.16.m16.1.5.3"><mi id="S3.11.p2.16.m16.1.5.3.2">x</mi><mi id="S3.11.p2.16.m16.1.5.3.3">k</mi></msub><mo id="S3.11.p2.16.m16.1.5.4" rspace="0em">βͺ</mo><mo id="S3.11.p2.16.m16.1.5.5" lspace="0em" stretchy="false">β</mo><msub id="S3.11.p2.16.m16.1.5.6"><mi id="S3.11.p2.16.m16.1.5.6.2">x</mi><mrow id="S3.11.p2.16.m16.1.5.6.3"><mi id="S3.11.p2.16.m16.1.5.6.3.2">k</mi><mo id="S3.11.p2.16.m16.1.5.6.3.1">+</mo><mn id="S3.11.p2.16.m16.1.5.6.3.3">2</mn></mrow></msub><mo id="S3.11.p2.16.m16.1.5.7" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.11.p2.16.m16.1c">W=X\setminus({\uparrow}x_{k}\cup{\downarrow}x_{k+2})</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.16.m16.1d">italic_W = italic_X β ( β italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT βͺ β italic_x start_POSTSUBSCRIPT italic_k + 2 end_POSTSUBSCRIPT )</annotation></semantics></math> is an open neighborhood of <math alttext="x_{k+1}" class="ltx_Math" display="inline" id="S3.11.p2.17.m17.1"><semantics id="S3.11.p2.17.m17.1a"><msub id="S3.11.p2.17.m17.1.1" xref="S3.11.p2.17.m17.1.1.cmml"><mi id="S3.11.p2.17.m17.1.1.2" xref="S3.11.p2.17.m17.1.1.2.cmml">x</mi><mrow id="S3.11.p2.17.m17.1.1.3" xref="S3.11.p2.17.m17.1.1.3.cmml"><mi id="S3.11.p2.17.m17.1.1.3.2" xref="S3.11.p2.17.m17.1.1.3.2.cmml">k</mi><mo id="S3.11.p2.17.m17.1.1.3.1" xref="S3.11.p2.17.m17.1.1.3.1.cmml">+</mo><mn id="S3.11.p2.17.m17.1.1.3.3" xref="S3.11.p2.17.m17.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.11.p2.17.m17.1b"><apply id="S3.11.p2.17.m17.1.1.cmml" xref="S3.11.p2.17.m17.1.1"><csymbol cd="ambiguous" id="S3.11.p2.17.m17.1.1.1.cmml" xref="S3.11.p2.17.m17.1.1">subscript</csymbol><ci id="S3.11.p2.17.m17.1.1.2.cmml" xref="S3.11.p2.17.m17.1.1.2">π₯</ci><apply id="S3.11.p2.17.m17.1.1.3.cmml" xref="S3.11.p2.17.m17.1.1.3"><plus id="S3.11.p2.17.m17.1.1.3.1.cmml" xref="S3.11.p2.17.m17.1.1.3.1"></plus><ci id="S3.11.p2.17.m17.1.1.3.2.cmml" xref="S3.11.p2.17.m17.1.1.3.2">π</ci><cn id="S3.11.p2.17.m17.1.1.3.3.cmml" type="integer" xref="S3.11.p2.17.m17.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.17.m17.1c">x_{k+1}</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.17.m17.1d">italic_x start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT</annotation></semantics></math> which is disjoint with <math alttext="L" class="ltx_Math" display="inline" id="S3.11.p2.18.m18.1"><semantics id="S3.11.p2.18.m18.1a"><mi id="S3.11.p2.18.m18.1.1" xref="S3.11.p2.18.m18.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.11.p2.18.m18.1b"><ci id="S3.11.p2.18.m18.1.1.cmml" xref="S3.11.p2.18.m18.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.18.m18.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.18.m18.1d">italic_L</annotation></semantics></math>. The obtained contradiction implies that <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.11.p2.19.m19.1"><semantics id="S3.11.p2.19.m19.1a"><mover accent="true" id="S3.11.p2.19.m19.1.1" xref="S3.11.p2.19.m19.1.1.cmml"><mi id="S3.11.p2.19.m19.1.1.2" xref="S3.11.p2.19.m19.1.1.2.cmml">L</mi><mo id="S3.11.p2.19.m19.1.1.1" xref="S3.11.p2.19.m19.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.11.p2.19.m19.1b"><apply id="S3.11.p2.19.m19.1.1.cmml" xref="S3.11.p2.19.m19.1.1"><ci id="S3.11.p2.19.m19.1.1.1.cmml" xref="S3.11.p2.19.m19.1.1.1">Β―</ci><ci id="S3.11.p2.19.m19.1.1.2.cmml" xref="S3.11.p2.19.m19.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.19.m19.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.19.m19.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> contains only finite decreasing chains. Hence <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.11.p2.20.m20.1"><semantics id="S3.11.p2.20.m20.1a"><mover accent="true" id="S3.11.p2.20.m20.1.1" xref="S3.11.p2.20.m20.1.1.cmml"><mi id="S3.11.p2.20.m20.1.1.2" xref="S3.11.p2.20.m20.1.1.2.cmml">L</mi><mo id="S3.11.p2.20.m20.1.1.1" xref="S3.11.p2.20.m20.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.11.p2.20.m20.1b"><apply id="S3.11.p2.20.m20.1.1.cmml" xref="S3.11.p2.20.m20.1.1"><ci id="S3.11.p2.20.m20.1.1.1.cmml" xref="S3.11.p2.20.m20.1.1.1">Β―</ci><ci id="S3.11.p2.20.m20.1.1.2.cmml" xref="S3.11.p2.20.m20.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.20.m20.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.20.m20.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> is isomorphic to <math alttext="(\theta,\min)" class="ltx_Math" display="inline" id="S3.11.p2.21.m21.2"><semantics id="S3.11.p2.21.m21.2a"><mrow id="S3.11.p2.21.m21.2.3.2" xref="S3.11.p2.21.m21.2.3.1.cmml"><mo id="S3.11.p2.21.m21.2.3.2.1" stretchy="false" xref="S3.11.p2.21.m21.2.3.1.cmml">(</mo><mi id="S3.11.p2.21.m21.1.1" xref="S3.11.p2.21.m21.1.1.cmml">ΞΈ</mi><mo id="S3.11.p2.21.m21.2.3.2.2" xref="S3.11.p2.21.m21.2.3.1.cmml">,</mo><mi id="S3.11.p2.21.m21.2.2" xref="S3.11.p2.21.m21.2.2.cmml">min</mi><mo id="S3.11.p2.21.m21.2.3.2.3" stretchy="false" xref="S3.11.p2.21.m21.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.21.m21.2b"><interval closure="open" id="S3.11.p2.21.m21.2.3.1.cmml" xref="S3.11.p2.21.m21.2.3.2"><ci id="S3.11.p2.21.m21.1.1.cmml" xref="S3.11.p2.21.m21.1.1">π</ci><min id="S3.11.p2.21.m21.2.2.cmml" xref="S3.11.p2.21.m21.2.2"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.21.m21.2c">(\theta,\min)</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.21.m21.2d">( italic_ΞΈ , roman_min )</annotation></semantics></math> for some ordinal <math alttext="\theta" class="ltx_Math" display="inline" id="S3.11.p2.22.m22.1"><semantics id="S3.11.p2.22.m22.1a"><mi id="S3.11.p2.22.m22.1.1" xref="S3.11.p2.22.m22.1.1.cmml">ΞΈ</mi><annotation-xml encoding="MathML-Content" id="S3.11.p2.22.m22.1b"><ci id="S3.11.p2.22.m22.1.1.cmml" xref="S3.11.p2.22.m22.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.22.m22.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.22.m22.1d">italic_ΞΈ</annotation></semantics></math>. Fix the order isomorphism <math alttext="\psi:\overline{L}\rightarrow\theta" class="ltx_Math" display="inline" id="S3.11.p2.23.m23.1"><semantics id="S3.11.p2.23.m23.1a"><mrow id="S3.11.p2.23.m23.1.1" xref="S3.11.p2.23.m23.1.1.cmml"><mi id="S3.11.p2.23.m23.1.1.2" xref="S3.11.p2.23.m23.1.1.2.cmml">Ο</mi><mo id="S3.11.p2.23.m23.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.11.p2.23.m23.1.1.1.cmml">:</mo><mrow id="S3.11.p2.23.m23.1.1.3" xref="S3.11.p2.23.m23.1.1.3.cmml"><mover accent="true" id="S3.11.p2.23.m23.1.1.3.2" xref="S3.11.p2.23.m23.1.1.3.2.cmml"><mi id="S3.11.p2.23.m23.1.1.3.2.2" xref="S3.11.p2.23.m23.1.1.3.2.2.cmml">L</mi><mo id="S3.11.p2.23.m23.1.1.3.2.1" xref="S3.11.p2.23.m23.1.1.3.2.1.cmml">Β―</mo></mover><mo id="S3.11.p2.23.m23.1.1.3.1" stretchy="false" xref="S3.11.p2.23.m23.1.1.3.1.cmml">β</mo><mi id="S3.11.p2.23.m23.1.1.3.3" xref="S3.11.p2.23.m23.1.1.3.3.cmml">ΞΈ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.23.m23.1b"><apply id="S3.11.p2.23.m23.1.1.cmml" xref="S3.11.p2.23.m23.1.1"><ci id="S3.11.p2.23.m23.1.1.1.cmml" xref="S3.11.p2.23.m23.1.1.1">:</ci><ci id="S3.11.p2.23.m23.1.1.2.cmml" xref="S3.11.p2.23.m23.1.1.2">π</ci><apply id="S3.11.p2.23.m23.1.1.3.cmml" xref="S3.11.p2.23.m23.1.1.3"><ci id="S3.11.p2.23.m23.1.1.3.1.cmml" xref="S3.11.p2.23.m23.1.1.3.1">β</ci><apply id="S3.11.p2.23.m23.1.1.3.2.cmml" xref="S3.11.p2.23.m23.1.1.3.2"><ci id="S3.11.p2.23.m23.1.1.3.2.1.cmml" xref="S3.11.p2.23.m23.1.1.3.2.1">Β―</ci><ci id="S3.11.p2.23.m23.1.1.3.2.2.cmml" xref="S3.11.p2.23.m23.1.1.3.2.2">πΏ</ci></apply><ci id="S3.11.p2.23.m23.1.1.3.3.cmml" xref="S3.11.p2.23.m23.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.23.m23.1c">\psi:\overline{L}\rightarrow\theta</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.23.m23.1d">italic_Ο : overΒ― start_ARG italic_L end_ARG β italic_ΞΈ</annotation></semantics></math>. It is easy to see that <math alttext="\psi" class="ltx_Math" display="inline" id="S3.11.p2.24.m24.1"><semantics id="S3.11.p2.24.m24.1a"><mi id="S3.11.p2.24.m24.1.1" xref="S3.11.p2.24.m24.1.1.cmml">Ο</mi><annotation-xml encoding="MathML-Content" id="S3.11.p2.24.m24.1b"><ci id="S3.11.p2.24.m24.1.1.cmml" xref="S3.11.p2.24.m24.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.24.m24.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.24.m24.1d">italic_Ο</annotation></semantics></math> can be defined recursively as follows: <math alttext="\psi(l_{0})=0" class="ltx_Math" display="inline" id="S3.11.p2.25.m25.1"><semantics id="S3.11.p2.25.m25.1a"><mrow id="S3.11.p2.25.m25.1.1" xref="S3.11.p2.25.m25.1.1.cmml"><mrow id="S3.11.p2.25.m25.1.1.1" xref="S3.11.p2.25.m25.1.1.1.cmml"><mi id="S3.11.p2.25.m25.1.1.1.3" xref="S3.11.p2.25.m25.1.1.1.3.cmml">Ο</mi><mo id="S3.11.p2.25.m25.1.1.1.2" xref="S3.11.p2.25.m25.1.1.1.2.cmml">β’</mo><mrow id="S3.11.p2.25.m25.1.1.1.1.1" xref="S3.11.p2.25.m25.1.1.1.1.1.1.cmml"><mo id="S3.11.p2.25.m25.1.1.1.1.1.2" stretchy="false" xref="S3.11.p2.25.m25.1.1.1.1.1.1.cmml">(</mo><msub id="S3.11.p2.25.m25.1.1.1.1.1.1" xref="S3.11.p2.25.m25.1.1.1.1.1.1.cmml"><mi id="S3.11.p2.25.m25.1.1.1.1.1.1.2" xref="S3.11.p2.25.m25.1.1.1.1.1.1.2.cmml">l</mi><mn id="S3.11.p2.25.m25.1.1.1.1.1.1.3" xref="S3.11.p2.25.m25.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S3.11.p2.25.m25.1.1.1.1.1.3" stretchy="false" xref="S3.11.p2.25.m25.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.11.p2.25.m25.1.1.2" xref="S3.11.p2.25.m25.1.1.2.cmml">=</mo><mn id="S3.11.p2.25.m25.1.1.3" xref="S3.11.p2.25.m25.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.25.m25.1b"><apply id="S3.11.p2.25.m25.1.1.cmml" xref="S3.11.p2.25.m25.1.1"><eq id="S3.11.p2.25.m25.1.1.2.cmml" xref="S3.11.p2.25.m25.1.1.2"></eq><apply id="S3.11.p2.25.m25.1.1.1.cmml" xref="S3.11.p2.25.m25.1.1.1"><times id="S3.11.p2.25.m25.1.1.1.2.cmml" xref="S3.11.p2.25.m25.1.1.1.2"></times><ci id="S3.11.p2.25.m25.1.1.1.3.cmml" xref="S3.11.p2.25.m25.1.1.1.3">π</ci><apply id="S3.11.p2.25.m25.1.1.1.1.1.1.cmml" xref="S3.11.p2.25.m25.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.11.p2.25.m25.1.1.1.1.1.1.1.cmml" xref="S3.11.p2.25.m25.1.1.1.1.1">subscript</csymbol><ci id="S3.11.p2.25.m25.1.1.1.1.1.1.2.cmml" xref="S3.11.p2.25.m25.1.1.1.1.1.1.2">π</ci><cn id="S3.11.p2.25.m25.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.11.p2.25.m25.1.1.1.1.1.1.3">0</cn></apply></apply><cn id="S3.11.p2.25.m25.1.1.3.cmml" type="integer" xref="S3.11.p2.25.m25.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.25.m25.1c">\psi(l_{0})=0</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.25.m25.1d">italic_Ο ( italic_l start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = 0</annotation></semantics></math> and for each <math alttext="y\in\overline{L}" class="ltx_Math" display="inline" id="S3.11.p2.26.m26.1"><semantics id="S3.11.p2.26.m26.1a"><mrow id="S3.11.p2.26.m26.1.1" xref="S3.11.p2.26.m26.1.1.cmml"><mi id="S3.11.p2.26.m26.1.1.2" xref="S3.11.p2.26.m26.1.1.2.cmml">y</mi><mo id="S3.11.p2.26.m26.1.1.1" xref="S3.11.p2.26.m26.1.1.1.cmml">β</mo><mover accent="true" id="S3.11.p2.26.m26.1.1.3" xref="S3.11.p2.26.m26.1.1.3.cmml"><mi id="S3.11.p2.26.m26.1.1.3.2" xref="S3.11.p2.26.m26.1.1.3.2.cmml">L</mi><mo id="S3.11.p2.26.m26.1.1.3.1" xref="S3.11.p2.26.m26.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.11.p2.26.m26.1b"><apply id="S3.11.p2.26.m26.1.1.cmml" xref="S3.11.p2.26.m26.1.1"><in id="S3.11.p2.26.m26.1.1.1.cmml" xref="S3.11.p2.26.m26.1.1.1"></in><ci id="S3.11.p2.26.m26.1.1.2.cmml" xref="S3.11.p2.26.m26.1.1.2">π¦</ci><apply id="S3.11.p2.26.m26.1.1.3.cmml" xref="S3.11.p2.26.m26.1.1.3"><ci id="S3.11.p2.26.m26.1.1.3.1.cmml" xref="S3.11.p2.26.m26.1.1.3.1">Β―</ci><ci id="S3.11.p2.26.m26.1.1.3.2.cmml" xref="S3.11.p2.26.m26.1.1.3.2">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.11.p2.26.m26.1c">y\in\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.11.p2.26.m26.1d">italic_y β overΒ― start_ARG italic_L end_ARG</annotation></semantics></math>,</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\psi(y)=\begin{cases}\psi(x)+1,&\hbox{if there exists }x<y\hbox{ such that }\{% z\in\overline{L}:x<z<y\}=\varnothing;\\ \sup\{\psi(x):x<y\},&\hbox{otherwise}.\end{cases}" class="ltx_Math" display="block" id="S3.Ex9.m1.5"><semantics id="S3.Ex9.m1.5a"><mrow id="S3.Ex9.m1.5.6" xref="S3.Ex9.m1.5.6.cmml"><mrow id="S3.Ex9.m1.5.6.2" xref="S3.Ex9.m1.5.6.2.cmml"><mi id="S3.Ex9.m1.5.6.2.2" xref="S3.Ex9.m1.5.6.2.2.cmml">Ο</mi><mo id="S3.Ex9.m1.5.6.2.1" xref="S3.Ex9.m1.5.6.2.1.cmml">β’</mo><mrow id="S3.Ex9.m1.5.6.2.3.2" xref="S3.Ex9.m1.5.6.2.cmml"><mo id="S3.Ex9.m1.5.6.2.3.2.1" stretchy="false" xref="S3.Ex9.m1.5.6.2.cmml">(</mo><mi id="S3.Ex9.m1.5.5" xref="S3.Ex9.m1.5.5.cmml">y</mi><mo id="S3.Ex9.m1.5.6.2.3.2.2" stretchy="false" xref="S3.Ex9.m1.5.6.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex9.m1.5.6.1" xref="S3.Ex9.m1.5.6.1.cmml">=</mo><mrow id="S3.Ex9.m1.4.4" xref="S3.Ex9.m1.5.6.3.1.cmml"><mo id="S3.Ex9.m1.4.4.5" xref="S3.Ex9.m1.5.6.3.1.1.cmml">{</mo><mtable columnspacing="5pt" displaystyle="true" id="S3.Ex9.m1.4.4.4" rowspacing="0pt" xref="S3.Ex9.m1.5.6.3.1.cmml"><mtr id="S3.Ex9.m1.4.4.4a" xref="S3.Ex9.m1.5.6.3.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S3.Ex9.m1.4.4.4b" xref="S3.Ex9.m1.5.6.3.1.cmml"><mrow id="S3.Ex9.m1.1.1.1.1.1.1.2" xref="S3.Ex9.m1.1.1.1.1.1.1.2.1.cmml"><mrow id="S3.Ex9.m1.1.1.1.1.1.1.2.1" xref="S3.Ex9.m1.1.1.1.1.1.1.2.1.cmml"><mrow id="S3.Ex9.m1.1.1.1.1.1.1.2.1.2" xref="S3.Ex9.m1.1.1.1.1.1.1.2.1.2.cmml"><mi id="S3.Ex9.m1.1.1.1.1.1.1.2.1.2.2" xref="S3.Ex9.m1.1.1.1.1.1.1.2.1.2.2.cmml">Ο</mi><mo id="S3.Ex9.m1.1.1.1.1.1.1.2.1.2.1" xref="S3.Ex9.m1.1.1.1.1.1.1.2.1.2.1.cmml">β’</mo><mrow id="S3.Ex9.m1.1.1.1.1.1.1.2.1.2.3.2" xref="S3.Ex9.m1.1.1.1.1.1.1.2.1.2.cmml"><mo id="S3.Ex9.m1.1.1.1.1.1.1.2.1.2.3.2.1" stretchy="false" xref="S3.Ex9.m1.1.1.1.1.1.1.2.1.2.cmml">(</mo><mi id="S3.Ex9.m1.1.1.1.1.1.1.1" xref="S3.Ex9.m1.1.1.1.1.1.1.1.cmml">x</mi><mo id="S3.Ex9.m1.1.1.1.1.1.1.2.1.2.3.2.2" stretchy="false" xref="S3.Ex9.m1.1.1.1.1.1.1.2.1.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex9.m1.1.1.1.1.1.1.2.1.1" xref="S3.Ex9.m1.1.1.1.1.1.1.2.1.1.cmml">+</mo><mn id="S3.Ex9.m1.1.1.1.1.1.1.2.1.3" xref="S3.Ex9.m1.1.1.1.1.1.1.2.1.3.cmml">1</mn></mrow><mo id="S3.Ex9.m1.1.1.1.1.1.1.2.2" xref="S3.Ex9.m1.1.1.1.1.1.1.2.1.cmml">,</mo></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S3.Ex9.m1.4.4.4c" xref="S3.Ex9.m1.5.6.3.1.cmml"><mrow id="S3.Ex9.m1.2.2.2.2.2.1.1" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.cmml"><mrow id="S3.Ex9.m1.2.2.2.2.2.1.1.1" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.cmml"><mrow id="S3.Ex9.m1.2.2.2.2.2.1.1.1.4" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.4.cmml"><mtext id="S3.Ex9.m1.2.2.2.2.2.1.1.1.4.2" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.4.2a.cmml">if there exists </mtext><mo id="S3.Ex9.m1.2.2.2.2.2.1.1.1.4.1" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.4.1.cmml">β’</mo><mi id="S3.Ex9.m1.2.2.2.2.2.1.1.1.4.3" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.4.3.cmml">x</mi></mrow><mo id="S3.Ex9.m1.2.2.2.2.2.1.1.1.5" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.5.cmml"><</mo><mrow id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.cmml"><mi id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.4" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.4.cmml">y</mi><mo id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.3" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.3.cmml">β’</mo><mtext id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.5" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.5a.cmml"> such that </mtext><mo id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.3a" 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id="S3.Ex9.m1.5.6.3.1.1.cmml" xref="S3.Ex9.m1.4.4.5">cases</csymbol><apply id="S3.Ex9.m1.1.1.1.1.1.1.2.1.cmml" xref="S3.Ex9.m1.1.1.1.1.1.1.2"><plus id="S3.Ex9.m1.1.1.1.1.1.1.2.1.1.cmml" xref="S3.Ex9.m1.1.1.1.1.1.1.2.1.1"></plus><apply id="S3.Ex9.m1.1.1.1.1.1.1.2.1.2.cmml" xref="S3.Ex9.m1.1.1.1.1.1.1.2.1.2"><times id="S3.Ex9.m1.1.1.1.1.1.1.2.1.2.1.cmml" xref="S3.Ex9.m1.1.1.1.1.1.1.2.1.2.1"></times><ci id="S3.Ex9.m1.1.1.1.1.1.1.2.1.2.2.cmml" xref="S3.Ex9.m1.1.1.1.1.1.1.2.1.2.2">π</ci><ci id="S3.Ex9.m1.1.1.1.1.1.1.1.cmml" xref="S3.Ex9.m1.1.1.1.1.1.1.1">π₯</ci></apply><cn id="S3.Ex9.m1.1.1.1.1.1.1.2.1.3.cmml" type="integer" xref="S3.Ex9.m1.1.1.1.1.1.1.2.1.3">1</cn></apply><apply id="S3.Ex9.m1.2.2.2.2.2.1.1.1.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1"><and id="S3.Ex9.m1.2.2.2.2.2.1.1.1a.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1"></and><apply id="S3.Ex9.m1.2.2.2.2.2.1.1.1b.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1"><lt id="S3.Ex9.m1.2.2.2.2.2.1.1.1.5.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.5"></lt><apply id="S3.Ex9.m1.2.2.2.2.2.1.1.1.4.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.4"><times id="S3.Ex9.m1.2.2.2.2.2.1.1.1.4.1.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.4.1"></times><ci id="S3.Ex9.m1.2.2.2.2.2.1.1.1.4.2a.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.4.2"><mtext id="S3.Ex9.m1.2.2.2.2.2.1.1.1.4.2.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.4.2">if there exists </mtext></ci><ci id="S3.Ex9.m1.2.2.2.2.2.1.1.1.4.3.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.4.3">π₯</ci></apply><apply id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2"><times id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.3.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.3"></times><ci id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.4.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.4">π¦</ci><ci id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.5a.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.5"><mtext id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.5.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.5"> such that </mtext></ci><apply id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.3.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2"><csymbol cd="latexml" id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.3.1.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.3">conditional-set</csymbol><apply id="S3.Ex9.m1.2.2.2.2.2.1.1.1.1.1.1.1.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.1.1.1.1"><in id="S3.Ex9.m1.2.2.2.2.2.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.1.1.1.1.1"></in><ci id="S3.Ex9.m1.2.2.2.2.2.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.1.1.1.1.2">π§</ci><apply id="S3.Ex9.m1.2.2.2.2.2.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.1.1.1.1.3"><ci id="S3.Ex9.m1.2.2.2.2.2.1.1.1.1.1.1.1.3.1.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.1.1.1.1.3.1">Β―</ci><ci id="S3.Ex9.m1.2.2.2.2.2.1.1.1.1.1.1.1.3.2.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.1.1.1.1.3.2">πΏ</ci></apply></apply><apply id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2"><and id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2a.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2"></and><apply id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2b.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2"><lt id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2.3.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2.3"></lt><ci id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2.2.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2.2">π₯</ci><ci id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2.4.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2.4">π§</ci></apply><apply id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2c.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2"><lt id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2.5.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2.5"></lt><share href="https://arxiv.org/html/2503.13666v1#S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2.4.cmml" id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2d.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2"></share><ci id="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2.6.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.2.2.2.2.6">π¦</ci></apply></apply></apply></apply></apply><apply id="S3.Ex9.m1.2.2.2.2.2.1.1.1c.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1"><eq id="S3.Ex9.m1.2.2.2.2.2.1.1.1.6.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.6"></eq><share href="https://arxiv.org/html/2503.13666v1#S3.Ex9.m1.2.2.2.2.2.1.1.1.2.cmml" id="S3.Ex9.m1.2.2.2.2.2.1.1.1d.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1"></share><emptyset id="S3.Ex9.m1.2.2.2.2.2.1.1.1.7.cmml" xref="S3.Ex9.m1.2.2.2.2.2.1.1.1.7"></emptyset></apply></apply><apply id="S3.Ex9.m1.3.3.3.3.1.1.2.1.cmml" xref="S3.Ex9.m1.3.3.3.3.1.1.2"><csymbol cd="latexml" id="S3.Ex9.m1.3.3.3.3.1.1.2.1.3.cmml" xref="S3.Ex9.m1.3.3.3.3.1.1.2.1.3">supremum</csymbol><apply id="S3.Ex9.m1.3.3.3.3.1.1.2.1.2.3.cmml" xref="S3.Ex9.m1.3.3.3.3.1.1.2.1.2.2"><csymbol cd="latexml" id="S3.Ex9.m1.3.3.3.3.1.1.2.1.2.3.1.cmml" xref="S3.Ex9.m1.3.3.3.3.1.1.2.1.2.2.3">conditional-set</csymbol><apply id="S3.Ex9.m1.3.3.3.3.1.1.2.1.1.1.1.cmml" xref="S3.Ex9.m1.3.3.3.3.1.1.2.1.1.1.1"><times id="S3.Ex9.m1.3.3.3.3.1.1.2.1.1.1.1.1.cmml" xref="S3.Ex9.m1.3.3.3.3.1.1.2.1.1.1.1.1"></times><ci id="S3.Ex9.m1.3.3.3.3.1.1.2.1.1.1.1.2.cmml" xref="S3.Ex9.m1.3.3.3.3.1.1.2.1.1.1.1.2">π</ci><ci id="S3.Ex9.m1.3.3.3.3.1.1.1.cmml" xref="S3.Ex9.m1.3.3.3.3.1.1.1">π₯</ci></apply><apply id="S3.Ex9.m1.3.3.3.3.1.1.2.1.2.2.2.cmml" xref="S3.Ex9.m1.3.3.3.3.1.1.2.1.2.2.2"><lt id="S3.Ex9.m1.3.3.3.3.1.1.2.1.2.2.2.1.cmml" xref="S3.Ex9.m1.3.3.3.3.1.1.2.1.2.2.2.1"></lt><ci id="S3.Ex9.m1.3.3.3.3.1.1.2.1.2.2.2.2.cmml" xref="S3.Ex9.m1.3.3.3.3.1.1.2.1.2.2.2.2">π₯</ci><ci id="S3.Ex9.m1.3.3.3.3.1.1.2.1.2.2.2.3.cmml" xref="S3.Ex9.m1.3.3.3.3.1.1.2.1.2.2.2.3">π¦</ci></apply></apply></apply><ci id="S3.Ex9.m1.4.4.4.4.2.1.1a.cmml" xref="S3.Ex9.m1.4.4.4.4.2.1.3"><mtext id="S3.Ex9.m1.4.4.4.4.2.1.1.cmml" xref="S3.Ex9.m1.4.4.4.4.2.1.1">otherwise</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex9.m1.5c">\psi(y)=\begin{cases}\psi(x)+1,&\hbox{if there exists }x<y\hbox{ such that }\{% z\in\overline{L}:x<z<y\}=\varnothing;\\ \sup\{\psi(x):x<y\},&\hbox{otherwise}.\end{cases}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex9.m1.5d">italic_Ο ( italic_y ) = { start_ROW start_CELL italic_Ο ( italic_x ) + 1 , end_CELL start_CELL if there exists italic_x < italic_y such that { italic_z β overΒ― start_ARG italic_L end_ARG : italic_x < italic_z < italic_y } = β ; end_CELL end_ROW start_ROW start_CELL roman_sup { italic_Ο ( italic_x ) : italic_x < italic_y } , end_CELL start_CELL otherwise . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.12.p3"> <p class="ltx_p" id="S3.12.p3.3">Since <math alttext="\psi(l_{\xi})\geq\xi" class="ltx_Math" display="inline" id="S3.12.p3.1.m1.1"><semantics id="S3.12.p3.1.m1.1a"><mrow id="S3.12.p3.1.m1.1.1" xref="S3.12.p3.1.m1.1.1.cmml"><mrow id="S3.12.p3.1.m1.1.1.1" xref="S3.12.p3.1.m1.1.1.1.cmml"><mi id="S3.12.p3.1.m1.1.1.1.3" xref="S3.12.p3.1.m1.1.1.1.3.cmml">Ο</mi><mo id="S3.12.p3.1.m1.1.1.1.2" xref="S3.12.p3.1.m1.1.1.1.2.cmml">β’</mo><mrow id="S3.12.p3.1.m1.1.1.1.1.1" xref="S3.12.p3.1.m1.1.1.1.1.1.1.cmml"><mo id="S3.12.p3.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.12.p3.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.12.p3.1.m1.1.1.1.1.1.1" xref="S3.12.p3.1.m1.1.1.1.1.1.1.cmml"><mi id="S3.12.p3.1.m1.1.1.1.1.1.1.2" xref="S3.12.p3.1.m1.1.1.1.1.1.1.2.cmml">l</mi><mi id="S3.12.p3.1.m1.1.1.1.1.1.1.3" xref="S3.12.p3.1.m1.1.1.1.1.1.1.3.cmml">ΞΎ</mi></msub><mo id="S3.12.p3.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.12.p3.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.12.p3.1.m1.1.1.2" xref="S3.12.p3.1.m1.1.1.2.cmml">β₯</mo><mi id="S3.12.p3.1.m1.1.1.3" xref="S3.12.p3.1.m1.1.1.3.cmml">ΞΎ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.12.p3.1.m1.1b"><apply id="S3.12.p3.1.m1.1.1.cmml" xref="S3.12.p3.1.m1.1.1"><geq id="S3.12.p3.1.m1.1.1.2.cmml" xref="S3.12.p3.1.m1.1.1.2"></geq><apply id="S3.12.p3.1.m1.1.1.1.cmml" xref="S3.12.p3.1.m1.1.1.1"><times id="S3.12.p3.1.m1.1.1.1.2.cmml" xref="S3.12.p3.1.m1.1.1.1.2"></times><ci id="S3.12.p3.1.m1.1.1.1.3.cmml" xref="S3.12.p3.1.m1.1.1.1.3">π</ci><apply id="S3.12.p3.1.m1.1.1.1.1.1.1.cmml" xref="S3.12.p3.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.12.p3.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.12.p3.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S3.12.p3.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.12.p3.1.m1.1.1.1.1.1.1.2">π</ci><ci id="S3.12.p3.1.m1.1.1.1.1.1.1.3.cmml" xref="S3.12.p3.1.m1.1.1.1.1.1.1.3">π</ci></apply></apply><ci id="S3.12.p3.1.m1.1.1.3.cmml" xref="S3.12.p3.1.m1.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p3.1.m1.1c">\psi(l_{\xi})\geq\xi</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.1.m1.1d">italic_Ο ( italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT ) β₯ italic_ΞΎ</annotation></semantics></math> for all <math alttext="\xi\in\alpha" class="ltx_Math" display="inline" id="S3.12.p3.2.m2.1"><semantics id="S3.12.p3.2.m2.1a"><mrow id="S3.12.p3.2.m2.1.1" xref="S3.12.p3.2.m2.1.1.cmml"><mi id="S3.12.p3.2.m2.1.1.2" xref="S3.12.p3.2.m2.1.1.2.cmml">ΞΎ</mi><mo id="S3.12.p3.2.m2.1.1.1" xref="S3.12.p3.2.m2.1.1.1.cmml">β</mo><mi id="S3.12.p3.2.m2.1.1.3" xref="S3.12.p3.2.m2.1.1.3.cmml">Ξ±</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.12.p3.2.m2.1b"><apply id="S3.12.p3.2.m2.1.1.cmml" xref="S3.12.p3.2.m2.1.1"><in id="S3.12.p3.2.m2.1.1.1.cmml" xref="S3.12.p3.2.m2.1.1.1"></in><ci id="S3.12.p3.2.m2.1.1.2.cmml" xref="S3.12.p3.2.m2.1.1.2">π</ci><ci id="S3.12.p3.2.m2.1.1.3.cmml" xref="S3.12.p3.2.m2.1.1.3">πΌ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p3.2.m2.1c">\xi\in\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.2.m2.1d">italic_ΞΎ β italic_Ξ±</annotation></semantics></math>, we get <math alttext="\alpha\leq\theta" class="ltx_Math" display="inline" id="S3.12.p3.3.m3.1"><semantics id="S3.12.p3.3.m3.1a"><mrow id="S3.12.p3.3.m3.1.1" xref="S3.12.p3.3.m3.1.1.cmml"><mi id="S3.12.p3.3.m3.1.1.2" xref="S3.12.p3.3.m3.1.1.2.cmml">Ξ±</mi><mo id="S3.12.p3.3.m3.1.1.1" xref="S3.12.p3.3.m3.1.1.1.cmml">β€</mo><mi id="S3.12.p3.3.m3.1.1.3" xref="S3.12.p3.3.m3.1.1.3.cmml">ΞΈ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.12.p3.3.m3.1b"><apply id="S3.12.p3.3.m3.1.1.cmml" xref="S3.12.p3.3.m3.1.1"><leq id="S3.12.p3.3.m3.1.1.1.cmml" xref="S3.12.p3.3.m3.1.1.1"></leq><ci id="S3.12.p3.3.m3.1.1.2.cmml" xref="S3.12.p3.3.m3.1.1.2">πΌ</ci><ci id="S3.12.p3.3.m3.1.1.3.cmml" xref="S3.12.p3.3.m3.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.12.p3.3.m3.1c">\alpha\leq\theta</annotation><annotation encoding="application/x-llamapun" id="S3.12.p3.3.m3.1d">italic_Ξ± β€ italic_ΞΈ</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_claim" id="S3.Thmtheorem10"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem10.1.1.1">Claim 3.10</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem10.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem10.p1"> <p class="ltx_p" id="S3.Thmtheorem10.p1.2"><math alttext="\psi(l_{\xi})\leq\xi+1" class="ltx_Math" display="inline" id="S3.Thmtheorem10.p1.1.m1.1"><semantics id="S3.Thmtheorem10.p1.1.m1.1a"><mrow id="S3.Thmtheorem10.p1.1.m1.1.1" xref="S3.Thmtheorem10.p1.1.m1.1.1.cmml"><mrow id="S3.Thmtheorem10.p1.1.m1.1.1.1" xref="S3.Thmtheorem10.p1.1.m1.1.1.1.cmml"><mi id="S3.Thmtheorem10.p1.1.m1.1.1.1.3" xref="S3.Thmtheorem10.p1.1.m1.1.1.1.3.cmml">Ο</mi><mo id="S3.Thmtheorem10.p1.1.m1.1.1.1.2" xref="S3.Thmtheorem10.p1.1.m1.1.1.1.2.cmml">β’</mo><mrow id="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1" xref="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1.1" xref="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1.1.2" xref="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1.1.2.cmml">l</mi><mi id="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1.1.3" xref="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1.1.3.cmml">ΞΎ</mi></msub><mo id="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem10.p1.1.m1.1.1.2" xref="S3.Thmtheorem10.p1.1.m1.1.1.2.cmml">β€</mo><mrow id="S3.Thmtheorem10.p1.1.m1.1.1.3" xref="S3.Thmtheorem10.p1.1.m1.1.1.3.cmml"><mi id="S3.Thmtheorem10.p1.1.m1.1.1.3.2" xref="S3.Thmtheorem10.p1.1.m1.1.1.3.2.cmml">ΞΎ</mi><mo id="S3.Thmtheorem10.p1.1.m1.1.1.3.1" xref="S3.Thmtheorem10.p1.1.m1.1.1.3.1.cmml">+</mo><mn id="S3.Thmtheorem10.p1.1.m1.1.1.3.3" xref="S3.Thmtheorem10.p1.1.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem10.p1.1.m1.1b"><apply id="S3.Thmtheorem10.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem10.p1.1.m1.1.1"><leq id="S3.Thmtheorem10.p1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem10.p1.1.m1.1.1.2"></leq><apply id="S3.Thmtheorem10.p1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem10.p1.1.m1.1.1.1"><times id="S3.Thmtheorem10.p1.1.m1.1.1.1.2.cmml" xref="S3.Thmtheorem10.p1.1.m1.1.1.1.2"></times><ci id="S3.Thmtheorem10.p1.1.m1.1.1.1.3.cmml" xref="S3.Thmtheorem10.p1.1.m1.1.1.1.3">π</ci><apply id="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1.1.2">π</ci><ci id="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem10.p1.1.m1.1.1.1.1.1.1.3">π</ci></apply></apply><apply id="S3.Thmtheorem10.p1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem10.p1.1.m1.1.1.3"><plus id="S3.Thmtheorem10.p1.1.m1.1.1.3.1.cmml" xref="S3.Thmtheorem10.p1.1.m1.1.1.3.1"></plus><ci id="S3.Thmtheorem10.p1.1.m1.1.1.3.2.cmml" xref="S3.Thmtheorem10.p1.1.m1.1.1.3.2">π</ci><cn id="S3.Thmtheorem10.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem10.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem10.p1.1.m1.1c">\psi(l_{\xi})\leq\xi+1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem10.p1.1.m1.1d">italic_Ο ( italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT ) β€ italic_ΞΎ + 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem10.p1.2.1"> for every <math alttext="\xi<\alpha" class="ltx_Math" display="inline" id="S3.Thmtheorem10.p1.2.1.m1.1"><semantics id="S3.Thmtheorem10.p1.2.1.m1.1a"><mrow id="S3.Thmtheorem10.p1.2.1.m1.1.1" xref="S3.Thmtheorem10.p1.2.1.m1.1.1.cmml"><mi id="S3.Thmtheorem10.p1.2.1.m1.1.1.2" xref="S3.Thmtheorem10.p1.2.1.m1.1.1.2.cmml">ΞΎ</mi><mo id="S3.Thmtheorem10.p1.2.1.m1.1.1.1" xref="S3.Thmtheorem10.p1.2.1.m1.1.1.1.cmml"><</mo><mi id="S3.Thmtheorem10.p1.2.1.m1.1.1.3" xref="S3.Thmtheorem10.p1.2.1.m1.1.1.3.cmml">Ξ±</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem10.p1.2.1.m1.1b"><apply id="S3.Thmtheorem10.p1.2.1.m1.1.1.cmml" xref="S3.Thmtheorem10.p1.2.1.m1.1.1"><lt id="S3.Thmtheorem10.p1.2.1.m1.1.1.1.cmml" xref="S3.Thmtheorem10.p1.2.1.m1.1.1.1"></lt><ci id="S3.Thmtheorem10.p1.2.1.m1.1.1.2.cmml" xref="S3.Thmtheorem10.p1.2.1.m1.1.1.2">π</ci><ci id="S3.Thmtheorem10.p1.2.1.m1.1.1.3.cmml" xref="S3.Thmtheorem10.p1.2.1.m1.1.1.3">πΌ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem10.p1.2.1.m1.1c">\xi<\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem10.p1.2.1.m1.1d">italic_ΞΎ < italic_Ξ±</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S3.15.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.13.1.p1"> <p class="ltx_p" id="S3.13.1.p1.2">Assume that <math alttext="\psi(l_{\delta})<\delta+1" class="ltx_Math" display="inline" id="S3.13.1.p1.1.m1.1"><semantics id="S3.13.1.p1.1.m1.1a"><mrow id="S3.13.1.p1.1.m1.1.1" xref="S3.13.1.p1.1.m1.1.1.cmml"><mrow id="S3.13.1.p1.1.m1.1.1.1" xref="S3.13.1.p1.1.m1.1.1.1.cmml"><mi id="S3.13.1.p1.1.m1.1.1.1.3" xref="S3.13.1.p1.1.m1.1.1.1.3.cmml">Ο</mi><mo id="S3.13.1.p1.1.m1.1.1.1.2" xref="S3.13.1.p1.1.m1.1.1.1.2.cmml">β’</mo><mrow id="S3.13.1.p1.1.m1.1.1.1.1.1" xref="S3.13.1.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S3.13.1.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.13.1.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.13.1.p1.1.m1.1.1.1.1.1.1" xref="S3.13.1.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="S3.13.1.p1.1.m1.1.1.1.1.1.1.2" xref="S3.13.1.p1.1.m1.1.1.1.1.1.1.2.cmml">l</mi><mi id="S3.13.1.p1.1.m1.1.1.1.1.1.1.3" xref="S3.13.1.p1.1.m1.1.1.1.1.1.1.3.cmml">Ξ΄</mi></msub><mo id="S3.13.1.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.13.1.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.13.1.p1.1.m1.1.1.2" xref="S3.13.1.p1.1.m1.1.1.2.cmml"><</mo><mrow id="S3.13.1.p1.1.m1.1.1.3" xref="S3.13.1.p1.1.m1.1.1.3.cmml"><mi id="S3.13.1.p1.1.m1.1.1.3.2" xref="S3.13.1.p1.1.m1.1.1.3.2.cmml">Ξ΄</mi><mo id="S3.13.1.p1.1.m1.1.1.3.1" xref="S3.13.1.p1.1.m1.1.1.3.1.cmml">+</mo><mn id="S3.13.1.p1.1.m1.1.1.3.3" xref="S3.13.1.p1.1.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.13.1.p1.1.m1.1b"><apply id="S3.13.1.p1.1.m1.1.1.cmml" xref="S3.13.1.p1.1.m1.1.1"><lt id="S3.13.1.p1.1.m1.1.1.2.cmml" xref="S3.13.1.p1.1.m1.1.1.2"></lt><apply id="S3.13.1.p1.1.m1.1.1.1.cmml" xref="S3.13.1.p1.1.m1.1.1.1"><times id="S3.13.1.p1.1.m1.1.1.1.2.cmml" xref="S3.13.1.p1.1.m1.1.1.1.2"></times><ci id="S3.13.1.p1.1.m1.1.1.1.3.cmml" xref="S3.13.1.p1.1.m1.1.1.1.3">π</ci><apply id="S3.13.1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S3.13.1.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.13.1.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.13.1.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S3.13.1.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.13.1.p1.1.m1.1.1.1.1.1.1.2">π</ci><ci id="S3.13.1.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="S3.13.1.p1.1.m1.1.1.1.1.1.1.3">πΏ</ci></apply></apply><apply id="S3.13.1.p1.1.m1.1.1.3.cmml" xref="S3.13.1.p1.1.m1.1.1.3"><plus id="S3.13.1.p1.1.m1.1.1.3.1.cmml" xref="S3.13.1.p1.1.m1.1.1.3.1"></plus><ci id="S3.13.1.p1.1.m1.1.1.3.2.cmml" xref="S3.13.1.p1.1.m1.1.1.3.2">πΏ</ci><cn id="S3.13.1.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S3.13.1.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.13.1.p1.1.m1.1c">\psi(l_{\delta})<\delta+1</annotation><annotation encoding="application/x-llamapun" id="S3.13.1.p1.1.m1.1d">italic_Ο ( italic_l start_POSTSUBSCRIPT italic_Ξ΄ end_POSTSUBSCRIPT ) < italic_Ξ΄ + 1</annotation></semantics></math> for all <math alttext="\delta<\xi" class="ltx_Math" display="inline" id="S3.13.1.p1.2.m2.1"><semantics id="S3.13.1.p1.2.m2.1a"><mrow id="S3.13.1.p1.2.m2.1.1" xref="S3.13.1.p1.2.m2.1.1.cmml"><mi id="S3.13.1.p1.2.m2.1.1.2" xref="S3.13.1.p1.2.m2.1.1.2.cmml">Ξ΄</mi><mo id="S3.13.1.p1.2.m2.1.1.1" xref="S3.13.1.p1.2.m2.1.1.1.cmml"><</mo><mi id="S3.13.1.p1.2.m2.1.1.3" xref="S3.13.1.p1.2.m2.1.1.3.cmml">ΞΎ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.13.1.p1.2.m2.1b"><apply id="S3.13.1.p1.2.m2.1.1.cmml" xref="S3.13.1.p1.2.m2.1.1"><lt id="S3.13.1.p1.2.m2.1.1.1.cmml" xref="S3.13.1.p1.2.m2.1.1.1"></lt><ci id="S3.13.1.p1.2.m2.1.1.2.cmml" xref="S3.13.1.p1.2.m2.1.1.2">πΏ</ci><ci id="S3.13.1.p1.2.m2.1.1.3.cmml" xref="S3.13.1.p1.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.13.1.p1.2.m2.1c">\delta<\xi</annotation><annotation encoding="application/x-llamapun" id="S3.13.1.p1.2.m2.1d">italic_Ξ΄ < italic_ΞΎ</annotation></semantics></math>. There are two cases to consider:</p> <ul class="ltx_itemize" id="S3.I4"> <li class="ltx_item" id="S3.I4.ix1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S3.I4.ix1.p1"> <p class="ltx_p" id="S3.I4.ix1.p1.2"><math alttext="\xi=\mu+1" class="ltx_Math" display="inline" id="S3.I4.ix1.p1.1.m1.1"><semantics id="S3.I4.ix1.p1.1.m1.1a"><mrow id="S3.I4.ix1.p1.1.m1.1.1" xref="S3.I4.ix1.p1.1.m1.1.1.cmml"><mi id="S3.I4.ix1.p1.1.m1.1.1.2" xref="S3.I4.ix1.p1.1.m1.1.1.2.cmml">ΞΎ</mi><mo id="S3.I4.ix1.p1.1.m1.1.1.1" xref="S3.I4.ix1.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S3.I4.ix1.p1.1.m1.1.1.3" xref="S3.I4.ix1.p1.1.m1.1.1.3.cmml"><mi id="S3.I4.ix1.p1.1.m1.1.1.3.2" xref="S3.I4.ix1.p1.1.m1.1.1.3.2.cmml">ΞΌ</mi><mo id="S3.I4.ix1.p1.1.m1.1.1.3.1" xref="S3.I4.ix1.p1.1.m1.1.1.3.1.cmml">+</mo><mn id="S3.I4.ix1.p1.1.m1.1.1.3.3" xref="S3.I4.ix1.p1.1.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.ix1.p1.1.m1.1b"><apply id="S3.I4.ix1.p1.1.m1.1.1.cmml" xref="S3.I4.ix1.p1.1.m1.1.1"><eq id="S3.I4.ix1.p1.1.m1.1.1.1.cmml" xref="S3.I4.ix1.p1.1.m1.1.1.1"></eq><ci id="S3.I4.ix1.p1.1.m1.1.1.2.cmml" xref="S3.I4.ix1.p1.1.m1.1.1.2">π</ci><apply id="S3.I4.ix1.p1.1.m1.1.1.3.cmml" xref="S3.I4.ix1.p1.1.m1.1.1.3"><plus id="S3.I4.ix1.p1.1.m1.1.1.3.1.cmml" xref="S3.I4.ix1.p1.1.m1.1.1.3.1"></plus><ci id="S3.I4.ix1.p1.1.m1.1.1.3.2.cmml" xref="S3.I4.ix1.p1.1.m1.1.1.3.2">π</ci><cn id="S3.I4.ix1.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S3.I4.ix1.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.ix1.p1.1.m1.1c">\xi=\mu+1</annotation><annotation encoding="application/x-llamapun" id="S3.I4.ix1.p1.1.m1.1d">italic_ΞΎ = italic_ΞΌ + 1</annotation></semantics></math> for some ordinal <math alttext="\mu<\alpha" class="ltx_Math" display="inline" id="S3.I4.ix1.p1.2.m2.1"><semantics id="S3.I4.ix1.p1.2.m2.1a"><mrow id="S3.I4.ix1.p1.2.m2.1.1" xref="S3.I4.ix1.p1.2.m2.1.1.cmml"><mi id="S3.I4.ix1.p1.2.m2.1.1.2" xref="S3.I4.ix1.p1.2.m2.1.1.2.cmml">ΞΌ</mi><mo id="S3.I4.ix1.p1.2.m2.1.1.1" xref="S3.I4.ix1.p1.2.m2.1.1.1.cmml"><</mo><mi id="S3.I4.ix1.p1.2.m2.1.1.3" xref="S3.I4.ix1.p1.2.m2.1.1.3.cmml">Ξ±</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.ix1.p1.2.m2.1b"><apply id="S3.I4.ix1.p1.2.m2.1.1.cmml" xref="S3.I4.ix1.p1.2.m2.1.1"><lt id="S3.I4.ix1.p1.2.m2.1.1.1.cmml" xref="S3.I4.ix1.p1.2.m2.1.1.1"></lt><ci id="S3.I4.ix1.p1.2.m2.1.1.2.cmml" xref="S3.I4.ix1.p1.2.m2.1.1.2">π</ci><ci id="S3.I4.ix1.p1.2.m2.1.1.3.cmml" xref="S3.I4.ix1.p1.2.m2.1.1.3">πΌ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.ix1.p1.2.m2.1c">\mu<\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.I4.ix1.p1.2.m2.1d">italic_ΞΌ < italic_Ξ±</annotation></semantics></math>;</p> </div> </li> <li class="ltx_item" id="S3.I4.ix2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S3.I4.ix2.p1"> <p class="ltx_p" id="S3.I4.ix2.p1.1"><math alttext="\xi" class="ltx_Math" display="inline" id="S3.I4.ix2.p1.1.m1.1"><semantics id="S3.I4.ix2.p1.1.m1.1a"><mi id="S3.I4.ix2.p1.1.m1.1.1" xref="S3.I4.ix2.p1.1.m1.1.1.cmml">ΞΎ</mi><annotation-xml encoding="MathML-Content" id="S3.I4.ix2.p1.1.m1.1b"><ci id="S3.I4.ix2.p1.1.m1.1.1.cmml" xref="S3.I4.ix2.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.ix2.p1.1.m1.1c">\xi</annotation><annotation encoding="application/x-llamapun" id="S3.I4.ix2.p1.1.m1.1d">italic_ΞΎ</annotation></semantics></math> is a limit ordinal.</p> </div> </li> </ul> </div> <div class="ltx_para" id="S3.14.2.p2"> <p class="ltx_p" id="S3.14.2.p2.9">In case (i) we have <math alttext="\psi(l_{\mu})\leq\mu+1=\xi" class="ltx_Math" display="inline" id="S3.14.2.p2.1.m1.1"><semantics id="S3.14.2.p2.1.m1.1a"><mrow id="S3.14.2.p2.1.m1.1.1" xref="S3.14.2.p2.1.m1.1.1.cmml"><mrow id="S3.14.2.p2.1.m1.1.1.1" xref="S3.14.2.p2.1.m1.1.1.1.cmml"><mi id="S3.14.2.p2.1.m1.1.1.1.3" xref="S3.14.2.p2.1.m1.1.1.1.3.cmml">Ο</mi><mo id="S3.14.2.p2.1.m1.1.1.1.2" xref="S3.14.2.p2.1.m1.1.1.1.2.cmml">β’</mo><mrow id="S3.14.2.p2.1.m1.1.1.1.1.1" xref="S3.14.2.p2.1.m1.1.1.1.1.1.1.cmml"><mo id="S3.14.2.p2.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.14.2.p2.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.14.2.p2.1.m1.1.1.1.1.1.1" xref="S3.14.2.p2.1.m1.1.1.1.1.1.1.cmml"><mi id="S3.14.2.p2.1.m1.1.1.1.1.1.1.2" xref="S3.14.2.p2.1.m1.1.1.1.1.1.1.2.cmml">l</mi><mi id="S3.14.2.p2.1.m1.1.1.1.1.1.1.3" xref="S3.14.2.p2.1.m1.1.1.1.1.1.1.3.cmml">ΞΌ</mi></msub><mo id="S3.14.2.p2.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.14.2.p2.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.14.2.p2.1.m1.1.1.3" xref="S3.14.2.p2.1.m1.1.1.3.cmml">β€</mo><mrow id="S3.14.2.p2.1.m1.1.1.4" xref="S3.14.2.p2.1.m1.1.1.4.cmml"><mi id="S3.14.2.p2.1.m1.1.1.4.2" xref="S3.14.2.p2.1.m1.1.1.4.2.cmml">ΞΌ</mi><mo id="S3.14.2.p2.1.m1.1.1.4.1" xref="S3.14.2.p2.1.m1.1.1.4.1.cmml">+</mo><mn id="S3.14.2.p2.1.m1.1.1.4.3" xref="S3.14.2.p2.1.m1.1.1.4.3.cmml">1</mn></mrow><mo id="S3.14.2.p2.1.m1.1.1.5" xref="S3.14.2.p2.1.m1.1.1.5.cmml">=</mo><mi id="S3.14.2.p2.1.m1.1.1.6" xref="S3.14.2.p2.1.m1.1.1.6.cmml">ΞΎ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.14.2.p2.1.m1.1b"><apply id="S3.14.2.p2.1.m1.1.1.cmml" xref="S3.14.2.p2.1.m1.1.1"><and id="S3.14.2.p2.1.m1.1.1a.cmml" xref="S3.14.2.p2.1.m1.1.1"></and><apply id="S3.14.2.p2.1.m1.1.1b.cmml" xref="S3.14.2.p2.1.m1.1.1"><leq id="S3.14.2.p2.1.m1.1.1.3.cmml" xref="S3.14.2.p2.1.m1.1.1.3"></leq><apply id="S3.14.2.p2.1.m1.1.1.1.cmml" xref="S3.14.2.p2.1.m1.1.1.1"><times id="S3.14.2.p2.1.m1.1.1.1.2.cmml" xref="S3.14.2.p2.1.m1.1.1.1.2"></times><ci id="S3.14.2.p2.1.m1.1.1.1.3.cmml" xref="S3.14.2.p2.1.m1.1.1.1.3">π</ci><apply id="S3.14.2.p2.1.m1.1.1.1.1.1.1.cmml" xref="S3.14.2.p2.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.14.2.p2.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.14.2.p2.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S3.14.2.p2.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.14.2.p2.1.m1.1.1.1.1.1.1.2">π</ci><ci id="S3.14.2.p2.1.m1.1.1.1.1.1.1.3.cmml" xref="S3.14.2.p2.1.m1.1.1.1.1.1.1.3">π</ci></apply></apply><apply id="S3.14.2.p2.1.m1.1.1.4.cmml" xref="S3.14.2.p2.1.m1.1.1.4"><plus id="S3.14.2.p2.1.m1.1.1.4.1.cmml" xref="S3.14.2.p2.1.m1.1.1.4.1"></plus><ci id="S3.14.2.p2.1.m1.1.1.4.2.cmml" xref="S3.14.2.p2.1.m1.1.1.4.2">π</ci><cn id="S3.14.2.p2.1.m1.1.1.4.3.cmml" type="integer" xref="S3.14.2.p2.1.m1.1.1.4.3">1</cn></apply></apply><apply id="S3.14.2.p2.1.m1.1.1c.cmml" xref="S3.14.2.p2.1.m1.1.1"><eq id="S3.14.2.p2.1.m1.1.1.5.cmml" xref="S3.14.2.p2.1.m1.1.1.5"></eq><share href="https://arxiv.org/html/2503.13666v1#S3.14.2.p2.1.m1.1.1.4.cmml" id="S3.14.2.p2.1.m1.1.1d.cmml" xref="S3.14.2.p2.1.m1.1.1"></share><ci id="S3.14.2.p2.1.m1.1.1.6.cmml" xref="S3.14.2.p2.1.m1.1.1.6">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.14.2.p2.1.m1.1c">\psi(l_{\mu})\leq\mu+1=\xi</annotation><annotation encoding="application/x-llamapun" id="S3.14.2.p2.1.m1.1d">italic_Ο ( italic_l start_POSTSUBSCRIPT italic_ΞΌ end_POSTSUBSCRIPT ) β€ italic_ΞΌ + 1 = italic_ΞΎ</annotation></semantics></math>. Observe that the set <math alttext="A=\overline{L}\setminus({\uparrow}l_{\xi}\cup{\downarrow}l_{\mu})" class="ltx_math_unparsed" display="inline" id="S3.14.2.p2.2.m2.1"><semantics id="S3.14.2.p2.2.m2.1a"><mrow id="S3.14.2.p2.2.m2.1b"><mi id="S3.14.2.p2.2.m2.1.1">A</mi><mo id="S3.14.2.p2.2.m2.1.2">=</mo><mover accent="true" id="S3.14.2.p2.2.m2.1.3"><mi id="S3.14.2.p2.2.m2.1.3.2">L</mi><mo id="S3.14.2.p2.2.m2.1.3.1">Β―</mo></mover><mo id="S3.14.2.p2.2.m2.1.4">β</mo><mrow id="S3.14.2.p2.2.m2.1.5"><mo id="S3.14.2.p2.2.m2.1.5.1" stretchy="false">(</mo><mo id="S3.14.2.p2.2.m2.1.5.2" lspace="0em" stretchy="false">β</mo><msub id="S3.14.2.p2.2.m2.1.5.3"><mi id="S3.14.2.p2.2.m2.1.5.3.2">l</mi><mi id="S3.14.2.p2.2.m2.1.5.3.3">ΞΎ</mi></msub><mo id="S3.14.2.p2.2.m2.1.5.4" rspace="0em">βͺ</mo><mo id="S3.14.2.p2.2.m2.1.5.5" lspace="0em" stretchy="false">β</mo><msub id="S3.14.2.p2.2.m2.1.5.6"><mi id="S3.14.2.p2.2.m2.1.5.6.2">l</mi><mi id="S3.14.2.p2.2.m2.1.5.6.3">ΞΌ</mi></msub><mo id="S3.14.2.p2.2.m2.1.5.7" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.14.2.p2.2.m2.1c">A=\overline{L}\setminus({\uparrow}l_{\xi}\cup{\downarrow}l_{\mu})</annotation><annotation encoding="application/x-llamapun" id="S3.14.2.p2.2.m2.1d">italic_A = overΒ― start_ARG italic_L end_ARG β ( β italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT βͺ β italic_l start_POSTSUBSCRIPT italic_ΞΌ end_POSTSUBSCRIPT )</annotation></semantics></math> is open in <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.14.2.p2.3.m3.1"><semantics id="S3.14.2.p2.3.m3.1a"><mover accent="true" id="S3.14.2.p2.3.m3.1.1" xref="S3.14.2.p2.3.m3.1.1.cmml"><mi id="S3.14.2.p2.3.m3.1.1.2" xref="S3.14.2.p2.3.m3.1.1.2.cmml">L</mi><mo id="S3.14.2.p2.3.m3.1.1.1" xref="S3.14.2.p2.3.m3.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.14.2.p2.3.m3.1b"><apply id="S3.14.2.p2.3.m3.1.1.cmml" xref="S3.14.2.p2.3.m3.1.1"><ci id="S3.14.2.p2.3.m3.1.1.1.cmml" xref="S3.14.2.p2.3.m3.1.1.1">Β―</ci><ci id="S3.14.2.p2.3.m3.1.1.2.cmml" xref="S3.14.2.p2.3.m3.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.14.2.p2.3.m3.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.14.2.p2.3.m3.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> and disjoint with <math alttext="L" class="ltx_Math" display="inline" id="S3.14.2.p2.4.m4.1"><semantics id="S3.14.2.p2.4.m4.1a"><mi id="S3.14.2.p2.4.m4.1.1" xref="S3.14.2.p2.4.m4.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.14.2.p2.4.m4.1b"><ci id="S3.14.2.p2.4.m4.1.1.cmml" xref="S3.14.2.p2.4.m4.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.14.2.p2.4.m4.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.14.2.p2.4.m4.1d">italic_L</annotation></semantics></math>. Therefore <math alttext="A=\varnothing" class="ltx_Math" display="inline" id="S3.14.2.p2.5.m5.1"><semantics id="S3.14.2.p2.5.m5.1a"><mrow id="S3.14.2.p2.5.m5.1.1" xref="S3.14.2.p2.5.m5.1.1.cmml"><mi id="S3.14.2.p2.5.m5.1.1.2" xref="S3.14.2.p2.5.m5.1.1.2.cmml">A</mi><mo id="S3.14.2.p2.5.m5.1.1.1" xref="S3.14.2.p2.5.m5.1.1.1.cmml">=</mo><mi id="S3.14.2.p2.5.m5.1.1.3" mathvariant="normal" xref="S3.14.2.p2.5.m5.1.1.3.cmml">β </mi></mrow><annotation-xml encoding="MathML-Content" id="S3.14.2.p2.5.m5.1b"><apply id="S3.14.2.p2.5.m5.1.1.cmml" xref="S3.14.2.p2.5.m5.1.1"><eq id="S3.14.2.p2.5.m5.1.1.1.cmml" xref="S3.14.2.p2.5.m5.1.1.1"></eq><ci id="S3.14.2.p2.5.m5.1.1.2.cmml" xref="S3.14.2.p2.5.m5.1.1.2">π΄</ci><emptyset id="S3.14.2.p2.5.m5.1.1.3.cmml" xref="S3.14.2.p2.5.m5.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.14.2.p2.5.m5.1c">A=\varnothing</annotation><annotation encoding="application/x-llamapun" id="S3.14.2.p2.5.m5.1d">italic_A = β </annotation></semantics></math>. Hence there exists no <math alttext="z\in\overline{L}" class="ltx_Math" display="inline" id="S3.14.2.p2.6.m6.1"><semantics id="S3.14.2.p2.6.m6.1a"><mrow id="S3.14.2.p2.6.m6.1.1" xref="S3.14.2.p2.6.m6.1.1.cmml"><mi id="S3.14.2.p2.6.m6.1.1.2" xref="S3.14.2.p2.6.m6.1.1.2.cmml">z</mi><mo id="S3.14.2.p2.6.m6.1.1.1" xref="S3.14.2.p2.6.m6.1.1.1.cmml">β</mo><mover accent="true" id="S3.14.2.p2.6.m6.1.1.3" xref="S3.14.2.p2.6.m6.1.1.3.cmml"><mi id="S3.14.2.p2.6.m6.1.1.3.2" xref="S3.14.2.p2.6.m6.1.1.3.2.cmml">L</mi><mo id="S3.14.2.p2.6.m6.1.1.3.1" xref="S3.14.2.p2.6.m6.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.14.2.p2.6.m6.1b"><apply id="S3.14.2.p2.6.m6.1.1.cmml" xref="S3.14.2.p2.6.m6.1.1"><in id="S3.14.2.p2.6.m6.1.1.1.cmml" xref="S3.14.2.p2.6.m6.1.1.1"></in><ci id="S3.14.2.p2.6.m6.1.1.2.cmml" xref="S3.14.2.p2.6.m6.1.1.2">π§</ci><apply id="S3.14.2.p2.6.m6.1.1.3.cmml" xref="S3.14.2.p2.6.m6.1.1.3"><ci id="S3.14.2.p2.6.m6.1.1.3.1.cmml" xref="S3.14.2.p2.6.m6.1.1.3.1">Β―</ci><ci id="S3.14.2.p2.6.m6.1.1.3.2.cmml" xref="S3.14.2.p2.6.m6.1.1.3.2">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.14.2.p2.6.m6.1c">z\in\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.14.2.p2.6.m6.1d">italic_z β overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> such that <math alttext="l_{\mu}<z<l_{\xi}" class="ltx_Math" display="inline" id="S3.14.2.p2.7.m7.1"><semantics id="S3.14.2.p2.7.m7.1a"><mrow id="S3.14.2.p2.7.m7.1.1" xref="S3.14.2.p2.7.m7.1.1.cmml"><msub id="S3.14.2.p2.7.m7.1.1.2" xref="S3.14.2.p2.7.m7.1.1.2.cmml"><mi id="S3.14.2.p2.7.m7.1.1.2.2" xref="S3.14.2.p2.7.m7.1.1.2.2.cmml">l</mi><mi id="S3.14.2.p2.7.m7.1.1.2.3" xref="S3.14.2.p2.7.m7.1.1.2.3.cmml">ΞΌ</mi></msub><mo id="S3.14.2.p2.7.m7.1.1.3" xref="S3.14.2.p2.7.m7.1.1.3.cmml"><</mo><mi id="S3.14.2.p2.7.m7.1.1.4" xref="S3.14.2.p2.7.m7.1.1.4.cmml">z</mi><mo id="S3.14.2.p2.7.m7.1.1.5" xref="S3.14.2.p2.7.m7.1.1.5.cmml"><</mo><msub id="S3.14.2.p2.7.m7.1.1.6" xref="S3.14.2.p2.7.m7.1.1.6.cmml"><mi id="S3.14.2.p2.7.m7.1.1.6.2" xref="S3.14.2.p2.7.m7.1.1.6.2.cmml">l</mi><mi id="S3.14.2.p2.7.m7.1.1.6.3" xref="S3.14.2.p2.7.m7.1.1.6.3.cmml">ΞΎ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.14.2.p2.7.m7.1b"><apply id="S3.14.2.p2.7.m7.1.1.cmml" xref="S3.14.2.p2.7.m7.1.1"><and id="S3.14.2.p2.7.m7.1.1a.cmml" xref="S3.14.2.p2.7.m7.1.1"></and><apply id="S3.14.2.p2.7.m7.1.1b.cmml" xref="S3.14.2.p2.7.m7.1.1"><lt id="S3.14.2.p2.7.m7.1.1.3.cmml" xref="S3.14.2.p2.7.m7.1.1.3"></lt><apply id="S3.14.2.p2.7.m7.1.1.2.cmml" xref="S3.14.2.p2.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.14.2.p2.7.m7.1.1.2.1.cmml" xref="S3.14.2.p2.7.m7.1.1.2">subscript</csymbol><ci id="S3.14.2.p2.7.m7.1.1.2.2.cmml" xref="S3.14.2.p2.7.m7.1.1.2.2">π</ci><ci id="S3.14.2.p2.7.m7.1.1.2.3.cmml" xref="S3.14.2.p2.7.m7.1.1.2.3">π</ci></apply><ci id="S3.14.2.p2.7.m7.1.1.4.cmml" xref="S3.14.2.p2.7.m7.1.1.4">π§</ci></apply><apply id="S3.14.2.p2.7.m7.1.1c.cmml" xref="S3.14.2.p2.7.m7.1.1"><lt id="S3.14.2.p2.7.m7.1.1.5.cmml" xref="S3.14.2.p2.7.m7.1.1.5"></lt><share href="https://arxiv.org/html/2503.13666v1#S3.14.2.p2.7.m7.1.1.4.cmml" id="S3.14.2.p2.7.m7.1.1d.cmml" xref="S3.14.2.p2.7.m7.1.1"></share><apply id="S3.14.2.p2.7.m7.1.1.6.cmml" xref="S3.14.2.p2.7.m7.1.1.6"><csymbol cd="ambiguous" id="S3.14.2.p2.7.m7.1.1.6.1.cmml" xref="S3.14.2.p2.7.m7.1.1.6">subscript</csymbol><ci id="S3.14.2.p2.7.m7.1.1.6.2.cmml" xref="S3.14.2.p2.7.m7.1.1.6.2">π</ci><ci id="S3.14.2.p2.7.m7.1.1.6.3.cmml" xref="S3.14.2.p2.7.m7.1.1.6.3">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.14.2.p2.7.m7.1c">l_{\mu}<z<l_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S3.14.2.p2.7.m7.1d">italic_l start_POSTSUBSCRIPT italic_ΞΌ end_POSTSUBSCRIPT < italic_z < italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT</annotation></semantics></math>. The definition of <math alttext="\psi" class="ltx_Math" display="inline" id="S3.14.2.p2.8.m8.1"><semantics id="S3.14.2.p2.8.m8.1a"><mi id="S3.14.2.p2.8.m8.1.1" xref="S3.14.2.p2.8.m8.1.1.cmml">Ο</mi><annotation-xml encoding="MathML-Content" id="S3.14.2.p2.8.m8.1b"><ci id="S3.14.2.p2.8.m8.1.1.cmml" xref="S3.14.2.p2.8.m8.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.14.2.p2.8.m8.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S3.14.2.p2.8.m8.1d">italic_Ο</annotation></semantics></math> implies that <math alttext="\psi(l_{\xi})=\psi(l_{\mu})+1\leq\xi+1" class="ltx_Math" display="inline" id="S3.14.2.p2.9.m9.2"><semantics id="S3.14.2.p2.9.m9.2a"><mrow id="S3.14.2.p2.9.m9.2.2" xref="S3.14.2.p2.9.m9.2.2.cmml"><mrow id="S3.14.2.p2.9.m9.1.1.1" xref="S3.14.2.p2.9.m9.1.1.1.cmml"><mi id="S3.14.2.p2.9.m9.1.1.1.3" xref="S3.14.2.p2.9.m9.1.1.1.3.cmml">Ο</mi><mo id="S3.14.2.p2.9.m9.1.1.1.2" xref="S3.14.2.p2.9.m9.1.1.1.2.cmml">β’</mo><mrow id="S3.14.2.p2.9.m9.1.1.1.1.1" xref="S3.14.2.p2.9.m9.1.1.1.1.1.1.cmml"><mo id="S3.14.2.p2.9.m9.1.1.1.1.1.2" stretchy="false" xref="S3.14.2.p2.9.m9.1.1.1.1.1.1.cmml">(</mo><msub id="S3.14.2.p2.9.m9.1.1.1.1.1.1" xref="S3.14.2.p2.9.m9.1.1.1.1.1.1.cmml"><mi id="S3.14.2.p2.9.m9.1.1.1.1.1.1.2" xref="S3.14.2.p2.9.m9.1.1.1.1.1.1.2.cmml">l</mi><mi id="S3.14.2.p2.9.m9.1.1.1.1.1.1.3" xref="S3.14.2.p2.9.m9.1.1.1.1.1.1.3.cmml">ΞΎ</mi></msub><mo id="S3.14.2.p2.9.m9.1.1.1.1.1.3" stretchy="false" xref="S3.14.2.p2.9.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.14.2.p2.9.m9.2.2.4" xref="S3.14.2.p2.9.m9.2.2.4.cmml">=</mo><mrow id="S3.14.2.p2.9.m9.2.2.2" xref="S3.14.2.p2.9.m9.2.2.2.cmml"><mrow id="S3.14.2.p2.9.m9.2.2.2.1" xref="S3.14.2.p2.9.m9.2.2.2.1.cmml"><mi id="S3.14.2.p2.9.m9.2.2.2.1.3" xref="S3.14.2.p2.9.m9.2.2.2.1.3.cmml">Ο</mi><mo id="S3.14.2.p2.9.m9.2.2.2.1.2" xref="S3.14.2.p2.9.m9.2.2.2.1.2.cmml">β’</mo><mrow id="S3.14.2.p2.9.m9.2.2.2.1.1.1" xref="S3.14.2.p2.9.m9.2.2.2.1.1.1.1.cmml"><mo id="S3.14.2.p2.9.m9.2.2.2.1.1.1.2" stretchy="false" xref="S3.14.2.p2.9.m9.2.2.2.1.1.1.1.cmml">(</mo><msub id="S3.14.2.p2.9.m9.2.2.2.1.1.1.1" xref="S3.14.2.p2.9.m9.2.2.2.1.1.1.1.cmml"><mi id="S3.14.2.p2.9.m9.2.2.2.1.1.1.1.2" xref="S3.14.2.p2.9.m9.2.2.2.1.1.1.1.2.cmml">l</mi><mi id="S3.14.2.p2.9.m9.2.2.2.1.1.1.1.3" xref="S3.14.2.p2.9.m9.2.2.2.1.1.1.1.3.cmml">ΞΌ</mi></msub><mo id="S3.14.2.p2.9.m9.2.2.2.1.1.1.3" stretchy="false" xref="S3.14.2.p2.9.m9.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.14.2.p2.9.m9.2.2.2.2" xref="S3.14.2.p2.9.m9.2.2.2.2.cmml">+</mo><mn id="S3.14.2.p2.9.m9.2.2.2.3" xref="S3.14.2.p2.9.m9.2.2.2.3.cmml">1</mn></mrow><mo id="S3.14.2.p2.9.m9.2.2.5" xref="S3.14.2.p2.9.m9.2.2.5.cmml">β€</mo><mrow id="S3.14.2.p2.9.m9.2.2.6" xref="S3.14.2.p2.9.m9.2.2.6.cmml"><mi id="S3.14.2.p2.9.m9.2.2.6.2" xref="S3.14.2.p2.9.m9.2.2.6.2.cmml">ΞΎ</mi><mo id="S3.14.2.p2.9.m9.2.2.6.1" xref="S3.14.2.p2.9.m9.2.2.6.1.cmml">+</mo><mn id="S3.14.2.p2.9.m9.2.2.6.3" xref="S3.14.2.p2.9.m9.2.2.6.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.14.2.p2.9.m9.2b"><apply id="S3.14.2.p2.9.m9.2.2.cmml" xref="S3.14.2.p2.9.m9.2.2"><and id="S3.14.2.p2.9.m9.2.2a.cmml" xref="S3.14.2.p2.9.m9.2.2"></and><apply id="S3.14.2.p2.9.m9.2.2b.cmml" xref="S3.14.2.p2.9.m9.2.2"><eq id="S3.14.2.p2.9.m9.2.2.4.cmml" xref="S3.14.2.p2.9.m9.2.2.4"></eq><apply id="S3.14.2.p2.9.m9.1.1.1.cmml" xref="S3.14.2.p2.9.m9.1.1.1"><times id="S3.14.2.p2.9.m9.1.1.1.2.cmml" xref="S3.14.2.p2.9.m9.1.1.1.2"></times><ci id="S3.14.2.p2.9.m9.1.1.1.3.cmml" xref="S3.14.2.p2.9.m9.1.1.1.3">π</ci><apply id="S3.14.2.p2.9.m9.1.1.1.1.1.1.cmml" xref="S3.14.2.p2.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.14.2.p2.9.m9.1.1.1.1.1.1.1.cmml" xref="S3.14.2.p2.9.m9.1.1.1.1.1">subscript</csymbol><ci id="S3.14.2.p2.9.m9.1.1.1.1.1.1.2.cmml" xref="S3.14.2.p2.9.m9.1.1.1.1.1.1.2">π</ci><ci id="S3.14.2.p2.9.m9.1.1.1.1.1.1.3.cmml" xref="S3.14.2.p2.9.m9.1.1.1.1.1.1.3">π</ci></apply></apply><apply id="S3.14.2.p2.9.m9.2.2.2.cmml" xref="S3.14.2.p2.9.m9.2.2.2"><plus id="S3.14.2.p2.9.m9.2.2.2.2.cmml" xref="S3.14.2.p2.9.m9.2.2.2.2"></plus><apply id="S3.14.2.p2.9.m9.2.2.2.1.cmml" xref="S3.14.2.p2.9.m9.2.2.2.1"><times id="S3.14.2.p2.9.m9.2.2.2.1.2.cmml" xref="S3.14.2.p2.9.m9.2.2.2.1.2"></times><ci id="S3.14.2.p2.9.m9.2.2.2.1.3.cmml" xref="S3.14.2.p2.9.m9.2.2.2.1.3">π</ci><apply id="S3.14.2.p2.9.m9.2.2.2.1.1.1.1.cmml" xref="S3.14.2.p2.9.m9.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.14.2.p2.9.m9.2.2.2.1.1.1.1.1.cmml" xref="S3.14.2.p2.9.m9.2.2.2.1.1.1">subscript</csymbol><ci id="S3.14.2.p2.9.m9.2.2.2.1.1.1.1.2.cmml" xref="S3.14.2.p2.9.m9.2.2.2.1.1.1.1.2">π</ci><ci id="S3.14.2.p2.9.m9.2.2.2.1.1.1.1.3.cmml" xref="S3.14.2.p2.9.m9.2.2.2.1.1.1.1.3">π</ci></apply></apply><cn id="S3.14.2.p2.9.m9.2.2.2.3.cmml" type="integer" xref="S3.14.2.p2.9.m9.2.2.2.3">1</cn></apply></apply><apply id="S3.14.2.p2.9.m9.2.2c.cmml" xref="S3.14.2.p2.9.m9.2.2"><leq id="S3.14.2.p2.9.m9.2.2.5.cmml" xref="S3.14.2.p2.9.m9.2.2.5"></leq><share href="https://arxiv.org/html/2503.13666v1#S3.14.2.p2.9.m9.2.2.2.cmml" id="S3.14.2.p2.9.m9.2.2d.cmml" xref="S3.14.2.p2.9.m9.2.2"></share><apply id="S3.14.2.p2.9.m9.2.2.6.cmml" xref="S3.14.2.p2.9.m9.2.2.6"><plus id="S3.14.2.p2.9.m9.2.2.6.1.cmml" xref="S3.14.2.p2.9.m9.2.2.6.1"></plus><ci id="S3.14.2.p2.9.m9.2.2.6.2.cmml" xref="S3.14.2.p2.9.m9.2.2.6.2">π</ci><cn id="S3.14.2.p2.9.m9.2.2.6.3.cmml" type="integer" xref="S3.14.2.p2.9.m9.2.2.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.14.2.p2.9.m9.2c">\psi(l_{\xi})=\psi(l_{\mu})+1\leq\xi+1</annotation><annotation encoding="application/x-llamapun" id="S3.14.2.p2.9.m9.2d">italic_Ο ( italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT ) = italic_Ο ( italic_l start_POSTSUBSCRIPT italic_ΞΌ end_POSTSUBSCRIPT ) + 1 β€ italic_ΞΎ + 1</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.15.3.p3"> <p class="ltx_p" id="S3.15.3.p3.7">Consider case (ii). Let <math alttext="\pi" class="ltx_Math" display="inline" id="S3.15.3.p3.1.m1.1"><semantics id="S3.15.3.p3.1.m1.1a"><mi id="S3.15.3.p3.1.m1.1.1" xref="S3.15.3.p3.1.m1.1.1.cmml">Ο</mi><annotation-xml encoding="MathML-Content" id="S3.15.3.p3.1.m1.1b"><ci id="S3.15.3.p3.1.m1.1.1.cmml" xref="S3.15.3.p3.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.15.3.p3.1.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S3.15.3.p3.1.m1.1d">italic_Ο</annotation></semantics></math> be the supremum of the set <math alttext="\{l_{\delta}:\delta<\xi\}" class="ltx_Math" display="inline" id="S3.15.3.p3.2.m2.2"><semantics id="S3.15.3.p3.2.m2.2a"><mrow id="S3.15.3.p3.2.m2.2.2.2" xref="S3.15.3.p3.2.m2.2.2.3.cmml"><mo id="S3.15.3.p3.2.m2.2.2.2.3" stretchy="false" xref="S3.15.3.p3.2.m2.2.2.3.1.cmml">{</mo><msub id="S3.15.3.p3.2.m2.1.1.1.1" xref="S3.15.3.p3.2.m2.1.1.1.1.cmml"><mi id="S3.15.3.p3.2.m2.1.1.1.1.2" xref="S3.15.3.p3.2.m2.1.1.1.1.2.cmml">l</mi><mi id="S3.15.3.p3.2.m2.1.1.1.1.3" xref="S3.15.3.p3.2.m2.1.1.1.1.3.cmml">Ξ΄</mi></msub><mo id="S3.15.3.p3.2.m2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.15.3.p3.2.m2.2.2.3.1.cmml">:</mo><mrow id="S3.15.3.p3.2.m2.2.2.2.2" xref="S3.15.3.p3.2.m2.2.2.2.2.cmml"><mi id="S3.15.3.p3.2.m2.2.2.2.2.2" xref="S3.15.3.p3.2.m2.2.2.2.2.2.cmml">Ξ΄</mi><mo id="S3.15.3.p3.2.m2.2.2.2.2.1" xref="S3.15.3.p3.2.m2.2.2.2.2.1.cmml"><</mo><mi id="S3.15.3.p3.2.m2.2.2.2.2.3" xref="S3.15.3.p3.2.m2.2.2.2.2.3.cmml">ΞΎ</mi></mrow><mo id="S3.15.3.p3.2.m2.2.2.2.5" stretchy="false" xref="S3.15.3.p3.2.m2.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.15.3.p3.2.m2.2b"><apply id="S3.15.3.p3.2.m2.2.2.3.cmml" xref="S3.15.3.p3.2.m2.2.2.2"><csymbol cd="latexml" id="S3.15.3.p3.2.m2.2.2.3.1.cmml" xref="S3.15.3.p3.2.m2.2.2.2.3">conditional-set</csymbol><apply id="S3.15.3.p3.2.m2.1.1.1.1.cmml" xref="S3.15.3.p3.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S3.15.3.p3.2.m2.1.1.1.1.1.cmml" xref="S3.15.3.p3.2.m2.1.1.1.1">subscript</csymbol><ci id="S3.15.3.p3.2.m2.1.1.1.1.2.cmml" xref="S3.15.3.p3.2.m2.1.1.1.1.2">π</ci><ci id="S3.15.3.p3.2.m2.1.1.1.1.3.cmml" xref="S3.15.3.p3.2.m2.1.1.1.1.3">πΏ</ci></apply><apply id="S3.15.3.p3.2.m2.2.2.2.2.cmml" xref="S3.15.3.p3.2.m2.2.2.2.2"><lt id="S3.15.3.p3.2.m2.2.2.2.2.1.cmml" xref="S3.15.3.p3.2.m2.2.2.2.2.1"></lt><ci id="S3.15.3.p3.2.m2.2.2.2.2.2.cmml" xref="S3.15.3.p3.2.m2.2.2.2.2.2">πΏ</ci><ci id="S3.15.3.p3.2.m2.2.2.2.2.3.cmml" xref="S3.15.3.p3.2.m2.2.2.2.2.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.15.3.p3.2.m2.2c">\{l_{\delta}:\delta<\xi\}</annotation><annotation encoding="application/x-llamapun" id="S3.15.3.p3.2.m2.2d">{ italic_l start_POSTSUBSCRIPT italic_Ξ΄ end_POSTSUBSCRIPT : italic_Ξ΄ < italic_ΞΎ }</annotation></semantics></math> in <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.15.3.p3.3.m3.1"><semantics id="S3.15.3.p3.3.m3.1a"><mover accent="true" id="S3.15.3.p3.3.m3.1.1" xref="S3.15.3.p3.3.m3.1.1.cmml"><mi id="S3.15.3.p3.3.m3.1.1.2" xref="S3.15.3.p3.3.m3.1.1.2.cmml">L</mi><mo id="S3.15.3.p3.3.m3.1.1.1" xref="S3.15.3.p3.3.m3.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.15.3.p3.3.m3.1b"><apply id="S3.15.3.p3.3.m3.1.1.cmml" xref="S3.15.3.p3.3.m3.1.1"><ci id="S3.15.3.p3.3.m3.1.1.1.cmml" xref="S3.15.3.p3.3.m3.1.1.1">Β―</ci><ci id="S3.15.3.p3.3.m3.1.1.2.cmml" xref="S3.15.3.p3.3.m3.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.15.3.p3.3.m3.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.15.3.p3.3.m3.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math>. It is clear that <math alttext="\pi\leq l_{\xi}" class="ltx_Math" display="inline" id="S3.15.3.p3.4.m4.1"><semantics id="S3.15.3.p3.4.m4.1a"><mrow id="S3.15.3.p3.4.m4.1.1" xref="S3.15.3.p3.4.m4.1.1.cmml"><mi id="S3.15.3.p3.4.m4.1.1.2" xref="S3.15.3.p3.4.m4.1.1.2.cmml">Ο</mi><mo id="S3.15.3.p3.4.m4.1.1.1" xref="S3.15.3.p3.4.m4.1.1.1.cmml">β€</mo><msub id="S3.15.3.p3.4.m4.1.1.3" xref="S3.15.3.p3.4.m4.1.1.3.cmml"><mi id="S3.15.3.p3.4.m4.1.1.3.2" xref="S3.15.3.p3.4.m4.1.1.3.2.cmml">l</mi><mi id="S3.15.3.p3.4.m4.1.1.3.3" xref="S3.15.3.p3.4.m4.1.1.3.3.cmml">ΞΎ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.15.3.p3.4.m4.1b"><apply id="S3.15.3.p3.4.m4.1.1.cmml" xref="S3.15.3.p3.4.m4.1.1"><leq id="S3.15.3.p3.4.m4.1.1.1.cmml" xref="S3.15.3.p3.4.m4.1.1.1"></leq><ci id="S3.15.3.p3.4.m4.1.1.2.cmml" xref="S3.15.3.p3.4.m4.1.1.2">π</ci><apply id="S3.15.3.p3.4.m4.1.1.3.cmml" xref="S3.15.3.p3.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.15.3.p3.4.m4.1.1.3.1.cmml" xref="S3.15.3.p3.4.m4.1.1.3">subscript</csymbol><ci id="S3.15.3.p3.4.m4.1.1.3.2.cmml" xref="S3.15.3.p3.4.m4.1.1.3.2">π</ci><ci id="S3.15.3.p3.4.m4.1.1.3.3.cmml" xref="S3.15.3.p3.4.m4.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.15.3.p3.4.m4.1c">\pi\leq l_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S3.15.3.p3.4.m4.1d">italic_Ο β€ italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT</annotation></semantics></math>. Since the set <math alttext="\{l_{\delta}:\delta<\xi\}" class="ltx_Math" display="inline" id="S3.15.3.p3.5.m5.2"><semantics id="S3.15.3.p3.5.m5.2a"><mrow id="S3.15.3.p3.5.m5.2.2.2" xref="S3.15.3.p3.5.m5.2.2.3.cmml"><mo id="S3.15.3.p3.5.m5.2.2.2.3" stretchy="false" xref="S3.15.3.p3.5.m5.2.2.3.1.cmml">{</mo><msub id="S3.15.3.p3.5.m5.1.1.1.1" xref="S3.15.3.p3.5.m5.1.1.1.1.cmml"><mi id="S3.15.3.p3.5.m5.1.1.1.1.2" xref="S3.15.3.p3.5.m5.1.1.1.1.2.cmml">l</mi><mi id="S3.15.3.p3.5.m5.1.1.1.1.3" xref="S3.15.3.p3.5.m5.1.1.1.1.3.cmml">Ξ΄</mi></msub><mo id="S3.15.3.p3.5.m5.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.15.3.p3.5.m5.2.2.3.1.cmml">:</mo><mrow id="S3.15.3.p3.5.m5.2.2.2.2" xref="S3.15.3.p3.5.m5.2.2.2.2.cmml"><mi id="S3.15.3.p3.5.m5.2.2.2.2.2" xref="S3.15.3.p3.5.m5.2.2.2.2.2.cmml">Ξ΄</mi><mo id="S3.15.3.p3.5.m5.2.2.2.2.1" xref="S3.15.3.p3.5.m5.2.2.2.2.1.cmml"><</mo><mi id="S3.15.3.p3.5.m5.2.2.2.2.3" xref="S3.15.3.p3.5.m5.2.2.2.2.3.cmml">ΞΎ</mi></mrow><mo id="S3.15.3.p3.5.m5.2.2.2.5" stretchy="false" xref="S3.15.3.p3.5.m5.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.15.3.p3.5.m5.2b"><apply id="S3.15.3.p3.5.m5.2.2.3.cmml" xref="S3.15.3.p3.5.m5.2.2.2"><csymbol cd="latexml" id="S3.15.3.p3.5.m5.2.2.3.1.cmml" xref="S3.15.3.p3.5.m5.2.2.2.3">conditional-set</csymbol><apply id="S3.15.3.p3.5.m5.1.1.1.1.cmml" xref="S3.15.3.p3.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S3.15.3.p3.5.m5.1.1.1.1.1.cmml" xref="S3.15.3.p3.5.m5.1.1.1.1">subscript</csymbol><ci id="S3.15.3.p3.5.m5.1.1.1.1.2.cmml" xref="S3.15.3.p3.5.m5.1.1.1.1.2">π</ci><ci id="S3.15.3.p3.5.m5.1.1.1.1.3.cmml" xref="S3.15.3.p3.5.m5.1.1.1.1.3">πΏ</ci></apply><apply id="S3.15.3.p3.5.m5.2.2.2.2.cmml" xref="S3.15.3.p3.5.m5.2.2.2.2"><lt id="S3.15.3.p3.5.m5.2.2.2.2.1.cmml" xref="S3.15.3.p3.5.m5.2.2.2.2.1"></lt><ci id="S3.15.3.p3.5.m5.2.2.2.2.2.cmml" xref="S3.15.3.p3.5.m5.2.2.2.2.2">πΏ</ci><ci id="S3.15.3.p3.5.m5.2.2.2.2.3.cmml" xref="S3.15.3.p3.5.m5.2.2.2.2.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.15.3.p3.5.m5.2c">\{l_{\delta}:\delta<\xi\}</annotation><annotation encoding="application/x-llamapun" id="S3.15.3.p3.5.m5.2d">{ italic_l start_POSTSUBSCRIPT italic_Ξ΄ end_POSTSUBSCRIPT : italic_Ξ΄ < italic_ΞΎ }</annotation></semantics></math> is cofinal in <math alttext="\{x\in\overline{L}:x<\pi\}" class="ltx_Math" display="inline" id="S3.15.3.p3.6.m6.2"><semantics id="S3.15.3.p3.6.m6.2a"><mrow id="S3.15.3.p3.6.m6.2.2.2" xref="S3.15.3.p3.6.m6.2.2.3.cmml"><mo id="S3.15.3.p3.6.m6.2.2.2.3" stretchy="false" xref="S3.15.3.p3.6.m6.2.2.3.1.cmml">{</mo><mrow id="S3.15.3.p3.6.m6.1.1.1.1" xref="S3.15.3.p3.6.m6.1.1.1.1.cmml"><mi id="S3.15.3.p3.6.m6.1.1.1.1.2" xref="S3.15.3.p3.6.m6.1.1.1.1.2.cmml">x</mi><mo id="S3.15.3.p3.6.m6.1.1.1.1.1" xref="S3.15.3.p3.6.m6.1.1.1.1.1.cmml">β</mo><mover accent="true" id="S3.15.3.p3.6.m6.1.1.1.1.3" xref="S3.15.3.p3.6.m6.1.1.1.1.3.cmml"><mi id="S3.15.3.p3.6.m6.1.1.1.1.3.2" xref="S3.15.3.p3.6.m6.1.1.1.1.3.2.cmml">L</mi><mo id="S3.15.3.p3.6.m6.1.1.1.1.3.1" xref="S3.15.3.p3.6.m6.1.1.1.1.3.1.cmml">Β―</mo></mover></mrow><mo id="S3.15.3.p3.6.m6.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.15.3.p3.6.m6.2.2.3.1.cmml">:</mo><mrow id="S3.15.3.p3.6.m6.2.2.2.2" xref="S3.15.3.p3.6.m6.2.2.2.2.cmml"><mi id="S3.15.3.p3.6.m6.2.2.2.2.2" xref="S3.15.3.p3.6.m6.2.2.2.2.2.cmml">x</mi><mo id="S3.15.3.p3.6.m6.2.2.2.2.1" xref="S3.15.3.p3.6.m6.2.2.2.2.1.cmml"><</mo><mi id="S3.15.3.p3.6.m6.2.2.2.2.3" xref="S3.15.3.p3.6.m6.2.2.2.2.3.cmml">Ο</mi></mrow><mo id="S3.15.3.p3.6.m6.2.2.2.5" stretchy="false" xref="S3.15.3.p3.6.m6.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.15.3.p3.6.m6.2b"><apply id="S3.15.3.p3.6.m6.2.2.3.cmml" xref="S3.15.3.p3.6.m6.2.2.2"><csymbol cd="latexml" id="S3.15.3.p3.6.m6.2.2.3.1.cmml" xref="S3.15.3.p3.6.m6.2.2.2.3">conditional-set</csymbol><apply id="S3.15.3.p3.6.m6.1.1.1.1.cmml" xref="S3.15.3.p3.6.m6.1.1.1.1"><in id="S3.15.3.p3.6.m6.1.1.1.1.1.cmml" xref="S3.15.3.p3.6.m6.1.1.1.1.1"></in><ci id="S3.15.3.p3.6.m6.1.1.1.1.2.cmml" xref="S3.15.3.p3.6.m6.1.1.1.1.2">π₯</ci><apply id="S3.15.3.p3.6.m6.1.1.1.1.3.cmml" xref="S3.15.3.p3.6.m6.1.1.1.1.3"><ci id="S3.15.3.p3.6.m6.1.1.1.1.3.1.cmml" xref="S3.15.3.p3.6.m6.1.1.1.1.3.1">Β―</ci><ci id="S3.15.3.p3.6.m6.1.1.1.1.3.2.cmml" xref="S3.15.3.p3.6.m6.1.1.1.1.3.2">πΏ</ci></apply></apply><apply id="S3.15.3.p3.6.m6.2.2.2.2.cmml" xref="S3.15.3.p3.6.m6.2.2.2.2"><lt id="S3.15.3.p3.6.m6.2.2.2.2.1.cmml" xref="S3.15.3.p3.6.m6.2.2.2.2.1"></lt><ci id="S3.15.3.p3.6.m6.2.2.2.2.2.cmml" xref="S3.15.3.p3.6.m6.2.2.2.2.2">π₯</ci><ci id="S3.15.3.p3.6.m6.2.2.2.2.3.cmml" xref="S3.15.3.p3.6.m6.2.2.2.2.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.15.3.p3.6.m6.2c">\{x\in\overline{L}:x<\pi\}</annotation><annotation encoding="application/x-llamapun" id="S3.15.3.p3.6.m6.2d">{ italic_x β overΒ― start_ARG italic_L end_ARG : italic_x < italic_Ο }</annotation></semantics></math> and <math alttext="\psi" class="ltx_Math" display="inline" id="S3.15.3.p3.7.m7.1"><semantics id="S3.15.3.p3.7.m7.1a"><mi id="S3.15.3.p3.7.m7.1.1" xref="S3.15.3.p3.7.m7.1.1.cmml">Ο</mi><annotation-xml encoding="MathML-Content" id="S3.15.3.p3.7.m7.1b"><ci id="S3.15.3.p3.7.m7.1.1.cmml" xref="S3.15.3.p3.7.m7.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.15.3.p3.7.m7.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S3.15.3.p3.7.m7.1d">italic_Ο</annotation></semantics></math> is an order isomorphism, we get that</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\psi(\pi)=\sup\{\psi(l_{\delta}):\delta<\xi\}\leq\sup\{\delta+1:\delta<\xi\}=\xi," class="ltx_Math" display="block" id="S3.Ex10.m1.2"><semantics id="S3.Ex10.m1.2a"><mrow id="S3.Ex10.m1.2.2.1" xref="S3.Ex10.m1.2.2.1.1.cmml"><mrow id="S3.Ex10.m1.2.2.1.1" xref="S3.Ex10.m1.2.2.1.1.cmml"><mrow id="S3.Ex10.m1.2.2.1.1.6" xref="S3.Ex10.m1.2.2.1.1.6.cmml"><mi id="S3.Ex10.m1.2.2.1.1.6.2" xref="S3.Ex10.m1.2.2.1.1.6.2.cmml">Ο</mi><mo id="S3.Ex10.m1.2.2.1.1.6.1" xref="S3.Ex10.m1.2.2.1.1.6.1.cmml">β’</mo><mrow id="S3.Ex10.m1.2.2.1.1.6.3.2" xref="S3.Ex10.m1.2.2.1.1.6.cmml"><mo id="S3.Ex10.m1.2.2.1.1.6.3.2.1" stretchy="false" xref="S3.Ex10.m1.2.2.1.1.6.cmml">(</mo><mi id="S3.Ex10.m1.1.1" xref="S3.Ex10.m1.1.1.cmml">Ο</mi><mo id="S3.Ex10.m1.2.2.1.1.6.3.2.2" stretchy="false" xref="S3.Ex10.m1.2.2.1.1.6.cmml">)</mo></mrow></mrow><mo id="S3.Ex10.m1.2.2.1.1.7" rspace="0.1389em" xref="S3.Ex10.m1.2.2.1.1.7.cmml">=</mo><mrow id="S3.Ex10.m1.2.2.1.1.2" xref="S3.Ex10.m1.2.2.1.1.2.cmml"><mo id="S3.Ex10.m1.2.2.1.1.2.3" lspace="0.1389em" movablelimits="false" rspace="0em" xref="S3.Ex10.m1.2.2.1.1.2.3.cmml">sup</mo><mrow id="S3.Ex10.m1.2.2.1.1.2.2.2" xref="S3.Ex10.m1.2.2.1.1.2.2.3.cmml"><mo id="S3.Ex10.m1.2.2.1.1.2.2.2.3" stretchy="false" xref="S3.Ex10.m1.2.2.1.1.2.2.3.1.cmml">{</mo><mrow id="S3.Ex10.m1.2.2.1.1.1.1.1.1" xref="S3.Ex10.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S3.Ex10.m1.2.2.1.1.1.1.1.1.3" xref="S3.Ex10.m1.2.2.1.1.1.1.1.1.3.cmml">Ο</mi><mo id="S3.Ex10.m1.2.2.1.1.1.1.1.1.2" xref="S3.Ex10.m1.2.2.1.1.1.1.1.1.2.cmml">β’</mo><mrow id="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1" xref="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1.1" xref="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1.1.2" xref="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1.1.2.cmml">l</mi><mi id="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1.1.3" xref="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1.1.3.cmml">Ξ΄</mi></msub><mo id="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1.3" rspace="0.278em" stretchy="false" xref="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex10.m1.2.2.1.1.2.2.2.4" rspace="0.278em" xref="S3.Ex10.m1.2.2.1.1.2.2.3.1.cmml">:</mo><mrow id="S3.Ex10.m1.2.2.1.1.2.2.2.2" xref="S3.Ex10.m1.2.2.1.1.2.2.2.2.cmml"><mi id="S3.Ex10.m1.2.2.1.1.2.2.2.2.2" xref="S3.Ex10.m1.2.2.1.1.2.2.2.2.2.cmml">Ξ΄</mi><mo id="S3.Ex10.m1.2.2.1.1.2.2.2.2.1" xref="S3.Ex10.m1.2.2.1.1.2.2.2.2.1.cmml"><</mo><mi id="S3.Ex10.m1.2.2.1.1.2.2.2.2.3" xref="S3.Ex10.m1.2.2.1.1.2.2.2.2.3.cmml">ΞΎ</mi></mrow><mo id="S3.Ex10.m1.2.2.1.1.2.2.2.5" stretchy="false" xref="S3.Ex10.m1.2.2.1.1.2.2.3.1.cmml">}</mo></mrow></mrow><mo id="S3.Ex10.m1.2.2.1.1.8" rspace="0.1389em" xref="S3.Ex10.m1.2.2.1.1.8.cmml">β€</mo><mrow id="S3.Ex10.m1.2.2.1.1.4" xref="S3.Ex10.m1.2.2.1.1.4.cmml"><mo id="S3.Ex10.m1.2.2.1.1.4.3" lspace="0.1389em" movablelimits="false" rspace="0em" xref="S3.Ex10.m1.2.2.1.1.4.3.cmml">sup</mo><mrow id="S3.Ex10.m1.2.2.1.1.4.2.2" xref="S3.Ex10.m1.2.2.1.1.4.2.3.cmml"><mo 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xref="S3.Ex10.m1.2.2.1.1.2.3">supremum</csymbol><apply id="S3.Ex10.m1.2.2.1.1.2.2.3.cmml" xref="S3.Ex10.m1.2.2.1.1.2.2.2"><csymbol cd="latexml" id="S3.Ex10.m1.2.2.1.1.2.2.3.1.cmml" xref="S3.Ex10.m1.2.2.1.1.2.2.2.3">conditional-set</csymbol><apply id="S3.Ex10.m1.2.2.1.1.1.1.1.1.cmml" xref="S3.Ex10.m1.2.2.1.1.1.1.1.1"><times id="S3.Ex10.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S3.Ex10.m1.2.2.1.1.1.1.1.1.2"></times><ci id="S3.Ex10.m1.2.2.1.1.1.1.1.1.3.cmml" xref="S3.Ex10.m1.2.2.1.1.1.1.1.1.3">π</ci><apply id="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1.1.2">π</ci><ci id="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex10.m1.2.2.1.1.1.1.1.1.1.1.1.3">πΏ</ci></apply></apply><apply id="S3.Ex10.m1.2.2.1.1.2.2.2.2.cmml" xref="S3.Ex10.m1.2.2.1.1.2.2.2.2"><lt id="S3.Ex10.m1.2.2.1.1.2.2.2.2.1.cmml" xref="S3.Ex10.m1.2.2.1.1.2.2.2.2.1"></lt><ci id="S3.Ex10.m1.2.2.1.1.2.2.2.2.2.cmml" xref="S3.Ex10.m1.2.2.1.1.2.2.2.2.2">πΏ</ci><ci id="S3.Ex10.m1.2.2.1.1.2.2.2.2.3.cmml" xref="S3.Ex10.m1.2.2.1.1.2.2.2.2.3">π</ci></apply></apply></apply></apply><apply id="S3.Ex10.m1.2.2.1.1c.cmml" xref="S3.Ex10.m1.2.2.1"><leq id="S3.Ex10.m1.2.2.1.1.8.cmml" xref="S3.Ex10.m1.2.2.1.1.8"></leq><share href="https://arxiv.org/html/2503.13666v1#S3.Ex10.m1.2.2.1.1.2.cmml" id="S3.Ex10.m1.2.2.1.1d.cmml" xref="S3.Ex10.m1.2.2.1"></share><apply id="S3.Ex10.m1.2.2.1.1.4.cmml" xref="S3.Ex10.m1.2.2.1.1.4"><csymbol cd="latexml" id="S3.Ex10.m1.2.2.1.1.4.3.cmml" xref="S3.Ex10.m1.2.2.1.1.4.3">supremum</csymbol><apply id="S3.Ex10.m1.2.2.1.1.4.2.3.cmml" xref="S3.Ex10.m1.2.2.1.1.4.2.2"><csymbol cd="latexml" id="S3.Ex10.m1.2.2.1.1.4.2.3.1.cmml" xref="S3.Ex10.m1.2.2.1.1.4.2.2.3">conditional-set</csymbol><apply id="S3.Ex10.m1.2.2.1.1.3.1.1.1.cmml" xref="S3.Ex10.m1.2.2.1.1.3.1.1.1"><plus id="S3.Ex10.m1.2.2.1.1.3.1.1.1.1.cmml" xref="S3.Ex10.m1.2.2.1.1.3.1.1.1.1"></plus><ci id="S3.Ex10.m1.2.2.1.1.3.1.1.1.2.cmml" xref="S3.Ex10.m1.2.2.1.1.3.1.1.1.2">πΏ</ci><cn id="S3.Ex10.m1.2.2.1.1.3.1.1.1.3.cmml" type="integer" xref="S3.Ex10.m1.2.2.1.1.3.1.1.1.3">1</cn></apply><apply id="S3.Ex10.m1.2.2.1.1.4.2.2.2.cmml" xref="S3.Ex10.m1.2.2.1.1.4.2.2.2"><lt id="S3.Ex10.m1.2.2.1.1.4.2.2.2.1.cmml" xref="S3.Ex10.m1.2.2.1.1.4.2.2.2.1"></lt><ci id="S3.Ex10.m1.2.2.1.1.4.2.2.2.2.cmml" xref="S3.Ex10.m1.2.2.1.1.4.2.2.2.2">πΏ</ci><ci id="S3.Ex10.m1.2.2.1.1.4.2.2.2.3.cmml" xref="S3.Ex10.m1.2.2.1.1.4.2.2.2.3">π</ci></apply></apply></apply></apply><apply id="S3.Ex10.m1.2.2.1.1e.cmml" xref="S3.Ex10.m1.2.2.1"><eq id="S3.Ex10.m1.2.2.1.1.9.cmml" xref="S3.Ex10.m1.2.2.1.1.9"></eq><share href="https://arxiv.org/html/2503.13666v1#S3.Ex10.m1.2.2.1.1.4.cmml" id="S3.Ex10.m1.2.2.1.1f.cmml" xref="S3.Ex10.m1.2.2.1"></share><ci id="S3.Ex10.m1.2.2.1.1.10.cmml" xref="S3.Ex10.m1.2.2.1.1.10">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex10.m1.2c">\psi(\pi)=\sup\{\psi(l_{\delta}):\delta<\xi\}\leq\sup\{\delta+1:\delta<\xi\}=\xi,</annotation><annotation encoding="application/x-llamapun" id="S3.Ex10.m1.2d">italic_Ο ( italic_Ο ) = roman_sup { italic_Ο ( italic_l start_POSTSUBSCRIPT italic_Ξ΄ end_POSTSUBSCRIPT ) : italic_Ξ΄ < italic_ΞΎ } β€ roman_sup { italic_Ξ΄ + 1 : italic_Ξ΄ < italic_ΞΎ } = italic_ΞΎ ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.15.3.p3.15">where the inequality above follows from the inductive assumption. So, if <math alttext="\pi=l_{\xi}" class="ltx_Math" display="inline" id="S3.15.3.p3.8.m1.1"><semantics id="S3.15.3.p3.8.m1.1a"><mrow id="S3.15.3.p3.8.m1.1.1" xref="S3.15.3.p3.8.m1.1.1.cmml"><mi id="S3.15.3.p3.8.m1.1.1.2" xref="S3.15.3.p3.8.m1.1.1.2.cmml">Ο</mi><mo id="S3.15.3.p3.8.m1.1.1.1" xref="S3.15.3.p3.8.m1.1.1.1.cmml">=</mo><msub id="S3.15.3.p3.8.m1.1.1.3" xref="S3.15.3.p3.8.m1.1.1.3.cmml"><mi id="S3.15.3.p3.8.m1.1.1.3.2" xref="S3.15.3.p3.8.m1.1.1.3.2.cmml">l</mi><mi id="S3.15.3.p3.8.m1.1.1.3.3" xref="S3.15.3.p3.8.m1.1.1.3.3.cmml">ΞΎ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.15.3.p3.8.m1.1b"><apply id="S3.15.3.p3.8.m1.1.1.cmml" xref="S3.15.3.p3.8.m1.1.1"><eq id="S3.15.3.p3.8.m1.1.1.1.cmml" xref="S3.15.3.p3.8.m1.1.1.1"></eq><ci id="S3.15.3.p3.8.m1.1.1.2.cmml" xref="S3.15.3.p3.8.m1.1.1.2">π</ci><apply id="S3.15.3.p3.8.m1.1.1.3.cmml" xref="S3.15.3.p3.8.m1.1.1.3"><csymbol cd="ambiguous" id="S3.15.3.p3.8.m1.1.1.3.1.cmml" xref="S3.15.3.p3.8.m1.1.1.3">subscript</csymbol><ci id="S3.15.3.p3.8.m1.1.1.3.2.cmml" xref="S3.15.3.p3.8.m1.1.1.3.2">π</ci><ci id="S3.15.3.p3.8.m1.1.1.3.3.cmml" xref="S3.15.3.p3.8.m1.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.15.3.p3.8.m1.1c">\pi=l_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S3.15.3.p3.8.m1.1d">italic_Ο = italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT</annotation></semantics></math>, then we are done. Assume that <math alttext="\pi<l_{\xi}" class="ltx_Math" display="inline" id="S3.15.3.p3.9.m2.1"><semantics id="S3.15.3.p3.9.m2.1a"><mrow id="S3.15.3.p3.9.m2.1.1" xref="S3.15.3.p3.9.m2.1.1.cmml"><mi id="S3.15.3.p3.9.m2.1.1.2" xref="S3.15.3.p3.9.m2.1.1.2.cmml">Ο</mi><mo id="S3.15.3.p3.9.m2.1.1.1" xref="S3.15.3.p3.9.m2.1.1.1.cmml"><</mo><msub id="S3.15.3.p3.9.m2.1.1.3" xref="S3.15.3.p3.9.m2.1.1.3.cmml"><mi id="S3.15.3.p3.9.m2.1.1.3.2" xref="S3.15.3.p3.9.m2.1.1.3.2.cmml">l</mi><mi id="S3.15.3.p3.9.m2.1.1.3.3" xref="S3.15.3.p3.9.m2.1.1.3.3.cmml">ΞΎ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.15.3.p3.9.m2.1b"><apply id="S3.15.3.p3.9.m2.1.1.cmml" xref="S3.15.3.p3.9.m2.1.1"><lt id="S3.15.3.p3.9.m2.1.1.1.cmml" xref="S3.15.3.p3.9.m2.1.1.1"></lt><ci id="S3.15.3.p3.9.m2.1.1.2.cmml" xref="S3.15.3.p3.9.m2.1.1.2">π</ci><apply id="S3.15.3.p3.9.m2.1.1.3.cmml" xref="S3.15.3.p3.9.m2.1.1.3"><csymbol cd="ambiguous" id="S3.15.3.p3.9.m2.1.1.3.1.cmml" xref="S3.15.3.p3.9.m2.1.1.3">subscript</csymbol><ci id="S3.15.3.p3.9.m2.1.1.3.2.cmml" xref="S3.15.3.p3.9.m2.1.1.3.2">π</ci><ci id="S3.15.3.p3.9.m2.1.1.3.3.cmml" xref="S3.15.3.p3.9.m2.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.15.3.p3.9.m2.1c">\pi<l_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S3.15.3.p3.9.m2.1d">italic_Ο < italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT</annotation></semantics></math>. Observe that the set <math alttext="\overline{L}\setminus({\downarrow}\pi\cup{\uparrow}l_{\xi})" class="ltx_math_unparsed" display="inline" id="S3.15.3.p3.10.m3.1"><semantics id="S3.15.3.p3.10.m3.1a"><mrow id="S3.15.3.p3.10.m3.1b"><mover accent="true" id="S3.15.3.p3.10.m3.1.1"><mi id="S3.15.3.p3.10.m3.1.1.2">L</mi><mo id="S3.15.3.p3.10.m3.1.1.1">Β―</mo></mover><mo id="S3.15.3.p3.10.m3.1.2">β</mo><mrow id="S3.15.3.p3.10.m3.1.3"><mo id="S3.15.3.p3.10.m3.1.3.1" stretchy="false">(</mo><mo id="S3.15.3.p3.10.m3.1.3.2" lspace="0em" stretchy="false">β</mo><mi id="S3.15.3.p3.10.m3.1.3.3">Ο</mi><mo id="S3.15.3.p3.10.m3.1.3.4" rspace="0em">βͺ</mo><mo id="S3.15.3.p3.10.m3.1.3.5" lspace="0em" stretchy="false">β</mo><msub id="S3.15.3.p3.10.m3.1.3.6"><mi id="S3.15.3.p3.10.m3.1.3.6.2">l</mi><mi id="S3.15.3.p3.10.m3.1.3.6.3">ΞΎ</mi></msub><mo id="S3.15.3.p3.10.m3.1.3.7" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.15.3.p3.10.m3.1c">\overline{L}\setminus({\downarrow}\pi\cup{\uparrow}l_{\xi})</annotation><annotation encoding="application/x-llamapun" id="S3.15.3.p3.10.m3.1d">overΒ― start_ARG italic_L end_ARG β ( β italic_Ο βͺ β italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT )</annotation></semantics></math> is empty, as it is open in <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.15.3.p3.11.m4.1"><semantics id="S3.15.3.p3.11.m4.1a"><mover accent="true" id="S3.15.3.p3.11.m4.1.1" xref="S3.15.3.p3.11.m4.1.1.cmml"><mi id="S3.15.3.p3.11.m4.1.1.2" xref="S3.15.3.p3.11.m4.1.1.2.cmml">L</mi><mo id="S3.15.3.p3.11.m4.1.1.1" xref="S3.15.3.p3.11.m4.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.15.3.p3.11.m4.1b"><apply id="S3.15.3.p3.11.m4.1.1.cmml" xref="S3.15.3.p3.11.m4.1.1"><ci id="S3.15.3.p3.11.m4.1.1.1.cmml" xref="S3.15.3.p3.11.m4.1.1.1">Β―</ci><ci id="S3.15.3.p3.11.m4.1.1.2.cmml" xref="S3.15.3.p3.11.m4.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.15.3.p3.11.m4.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.15.3.p3.11.m4.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> and disjoint with <math alttext="L" class="ltx_Math" display="inline" id="S3.15.3.p3.12.m5.1"><semantics id="S3.15.3.p3.12.m5.1a"><mi id="S3.15.3.p3.12.m5.1.1" xref="S3.15.3.p3.12.m5.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.15.3.p3.12.m5.1b"><ci id="S3.15.3.p3.12.m5.1.1.cmml" xref="S3.15.3.p3.12.m5.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.15.3.p3.12.m5.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.15.3.p3.12.m5.1d">italic_L</annotation></semantics></math>. Hence there exists no <math alttext="z\in\overline{L}" class="ltx_Math" display="inline" id="S3.15.3.p3.13.m6.1"><semantics id="S3.15.3.p3.13.m6.1a"><mrow id="S3.15.3.p3.13.m6.1.1" xref="S3.15.3.p3.13.m6.1.1.cmml"><mi id="S3.15.3.p3.13.m6.1.1.2" xref="S3.15.3.p3.13.m6.1.1.2.cmml">z</mi><mo id="S3.15.3.p3.13.m6.1.1.1" xref="S3.15.3.p3.13.m6.1.1.1.cmml">β</mo><mover accent="true" id="S3.15.3.p3.13.m6.1.1.3" xref="S3.15.3.p3.13.m6.1.1.3.cmml"><mi id="S3.15.3.p3.13.m6.1.1.3.2" xref="S3.15.3.p3.13.m6.1.1.3.2.cmml">L</mi><mo id="S3.15.3.p3.13.m6.1.1.3.1" xref="S3.15.3.p3.13.m6.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.15.3.p3.13.m6.1b"><apply id="S3.15.3.p3.13.m6.1.1.cmml" xref="S3.15.3.p3.13.m6.1.1"><in id="S3.15.3.p3.13.m6.1.1.1.cmml" xref="S3.15.3.p3.13.m6.1.1.1"></in><ci id="S3.15.3.p3.13.m6.1.1.2.cmml" xref="S3.15.3.p3.13.m6.1.1.2">π§</ci><apply id="S3.15.3.p3.13.m6.1.1.3.cmml" xref="S3.15.3.p3.13.m6.1.1.3"><ci id="S3.15.3.p3.13.m6.1.1.3.1.cmml" xref="S3.15.3.p3.13.m6.1.1.3.1">Β―</ci><ci id="S3.15.3.p3.13.m6.1.1.3.2.cmml" xref="S3.15.3.p3.13.m6.1.1.3.2">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.15.3.p3.13.m6.1c">z\in\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.15.3.p3.13.m6.1d">italic_z β overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> such that <math alttext="\pi<z<l_{\xi}" class="ltx_Math" display="inline" id="S3.15.3.p3.14.m7.1"><semantics id="S3.15.3.p3.14.m7.1a"><mrow id="S3.15.3.p3.14.m7.1.1" xref="S3.15.3.p3.14.m7.1.1.cmml"><mi id="S3.15.3.p3.14.m7.1.1.2" xref="S3.15.3.p3.14.m7.1.1.2.cmml">Ο</mi><mo id="S3.15.3.p3.14.m7.1.1.3" xref="S3.15.3.p3.14.m7.1.1.3.cmml"><</mo><mi id="S3.15.3.p3.14.m7.1.1.4" xref="S3.15.3.p3.14.m7.1.1.4.cmml">z</mi><mo id="S3.15.3.p3.14.m7.1.1.5" xref="S3.15.3.p3.14.m7.1.1.5.cmml"><</mo><msub id="S3.15.3.p3.14.m7.1.1.6" xref="S3.15.3.p3.14.m7.1.1.6.cmml"><mi id="S3.15.3.p3.14.m7.1.1.6.2" xref="S3.15.3.p3.14.m7.1.1.6.2.cmml">l</mi><mi id="S3.15.3.p3.14.m7.1.1.6.3" xref="S3.15.3.p3.14.m7.1.1.6.3.cmml">ΞΎ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.15.3.p3.14.m7.1b"><apply id="S3.15.3.p3.14.m7.1.1.cmml" xref="S3.15.3.p3.14.m7.1.1"><and id="S3.15.3.p3.14.m7.1.1a.cmml" xref="S3.15.3.p3.14.m7.1.1"></and><apply id="S3.15.3.p3.14.m7.1.1b.cmml" xref="S3.15.3.p3.14.m7.1.1"><lt id="S3.15.3.p3.14.m7.1.1.3.cmml" xref="S3.15.3.p3.14.m7.1.1.3"></lt><ci id="S3.15.3.p3.14.m7.1.1.2.cmml" xref="S3.15.3.p3.14.m7.1.1.2">π</ci><ci id="S3.15.3.p3.14.m7.1.1.4.cmml" xref="S3.15.3.p3.14.m7.1.1.4">π§</ci></apply><apply id="S3.15.3.p3.14.m7.1.1c.cmml" xref="S3.15.3.p3.14.m7.1.1"><lt id="S3.15.3.p3.14.m7.1.1.5.cmml" xref="S3.15.3.p3.14.m7.1.1.5"></lt><share href="https://arxiv.org/html/2503.13666v1#S3.15.3.p3.14.m7.1.1.4.cmml" id="S3.15.3.p3.14.m7.1.1d.cmml" xref="S3.15.3.p3.14.m7.1.1"></share><apply id="S3.15.3.p3.14.m7.1.1.6.cmml" xref="S3.15.3.p3.14.m7.1.1.6"><csymbol cd="ambiguous" id="S3.15.3.p3.14.m7.1.1.6.1.cmml" xref="S3.15.3.p3.14.m7.1.1.6">subscript</csymbol><ci id="S3.15.3.p3.14.m7.1.1.6.2.cmml" xref="S3.15.3.p3.14.m7.1.1.6.2">π</ci><ci id="S3.15.3.p3.14.m7.1.1.6.3.cmml" xref="S3.15.3.p3.14.m7.1.1.6.3">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.15.3.p3.14.m7.1c">\pi<z<l_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S3.15.3.p3.14.m7.1d">italic_Ο < italic_z < italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT</annotation></semantics></math>. Then <math alttext="\psi(l_{\xi})=\psi(\pi)+1\leq\xi+1" class="ltx_Math" display="inline" id="S3.15.3.p3.15.m8.2"><semantics id="S3.15.3.p3.15.m8.2a"><mrow id="S3.15.3.p3.15.m8.2.2" xref="S3.15.3.p3.15.m8.2.2.cmml"><mrow id="S3.15.3.p3.15.m8.2.2.1" xref="S3.15.3.p3.15.m8.2.2.1.cmml"><mi id="S3.15.3.p3.15.m8.2.2.1.3" xref="S3.15.3.p3.15.m8.2.2.1.3.cmml">Ο</mi><mo id="S3.15.3.p3.15.m8.2.2.1.2" xref="S3.15.3.p3.15.m8.2.2.1.2.cmml">β’</mo><mrow id="S3.15.3.p3.15.m8.2.2.1.1.1" xref="S3.15.3.p3.15.m8.2.2.1.1.1.1.cmml"><mo id="S3.15.3.p3.15.m8.2.2.1.1.1.2" stretchy="false" xref="S3.15.3.p3.15.m8.2.2.1.1.1.1.cmml">(</mo><msub id="S3.15.3.p3.15.m8.2.2.1.1.1.1" xref="S3.15.3.p3.15.m8.2.2.1.1.1.1.cmml"><mi id="S3.15.3.p3.15.m8.2.2.1.1.1.1.2" xref="S3.15.3.p3.15.m8.2.2.1.1.1.1.2.cmml">l</mi><mi id="S3.15.3.p3.15.m8.2.2.1.1.1.1.3" xref="S3.15.3.p3.15.m8.2.2.1.1.1.1.3.cmml">ΞΎ</mi></msub><mo id="S3.15.3.p3.15.m8.2.2.1.1.1.3" stretchy="false" xref="S3.15.3.p3.15.m8.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.15.3.p3.15.m8.2.2.3" xref="S3.15.3.p3.15.m8.2.2.3.cmml">=</mo><mrow id="S3.15.3.p3.15.m8.2.2.4" xref="S3.15.3.p3.15.m8.2.2.4.cmml"><mrow id="S3.15.3.p3.15.m8.2.2.4.2" xref="S3.15.3.p3.15.m8.2.2.4.2.cmml"><mi id="S3.15.3.p3.15.m8.2.2.4.2.2" xref="S3.15.3.p3.15.m8.2.2.4.2.2.cmml">Ο</mi><mo id="S3.15.3.p3.15.m8.2.2.4.2.1" xref="S3.15.3.p3.15.m8.2.2.4.2.1.cmml">β’</mo><mrow id="S3.15.3.p3.15.m8.2.2.4.2.3.2" xref="S3.15.3.p3.15.m8.2.2.4.2.cmml"><mo id="S3.15.3.p3.15.m8.2.2.4.2.3.2.1" stretchy="false" xref="S3.15.3.p3.15.m8.2.2.4.2.cmml">(</mo><mi id="S3.15.3.p3.15.m8.1.1" xref="S3.15.3.p3.15.m8.1.1.cmml">Ο</mi><mo id="S3.15.3.p3.15.m8.2.2.4.2.3.2.2" stretchy="false" xref="S3.15.3.p3.15.m8.2.2.4.2.cmml">)</mo></mrow></mrow><mo id="S3.15.3.p3.15.m8.2.2.4.1" xref="S3.15.3.p3.15.m8.2.2.4.1.cmml">+</mo><mn id="S3.15.3.p3.15.m8.2.2.4.3" xref="S3.15.3.p3.15.m8.2.2.4.3.cmml">1</mn></mrow><mo id="S3.15.3.p3.15.m8.2.2.5" xref="S3.15.3.p3.15.m8.2.2.5.cmml">β€</mo><mrow id="S3.15.3.p3.15.m8.2.2.6" xref="S3.15.3.p3.15.m8.2.2.6.cmml"><mi id="S3.15.3.p3.15.m8.2.2.6.2" xref="S3.15.3.p3.15.m8.2.2.6.2.cmml">ΞΎ</mi><mo id="S3.15.3.p3.15.m8.2.2.6.1" xref="S3.15.3.p3.15.m8.2.2.6.1.cmml">+</mo><mn id="S3.15.3.p3.15.m8.2.2.6.3" xref="S3.15.3.p3.15.m8.2.2.6.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.15.3.p3.15.m8.2b"><apply id="S3.15.3.p3.15.m8.2.2.cmml" xref="S3.15.3.p3.15.m8.2.2"><and id="S3.15.3.p3.15.m8.2.2a.cmml" xref="S3.15.3.p3.15.m8.2.2"></and><apply id="S3.15.3.p3.15.m8.2.2b.cmml" xref="S3.15.3.p3.15.m8.2.2"><eq id="S3.15.3.p3.15.m8.2.2.3.cmml" xref="S3.15.3.p3.15.m8.2.2.3"></eq><apply id="S3.15.3.p3.15.m8.2.2.1.cmml" xref="S3.15.3.p3.15.m8.2.2.1"><times id="S3.15.3.p3.15.m8.2.2.1.2.cmml" xref="S3.15.3.p3.15.m8.2.2.1.2"></times><ci id="S3.15.3.p3.15.m8.2.2.1.3.cmml" xref="S3.15.3.p3.15.m8.2.2.1.3">π</ci><apply id="S3.15.3.p3.15.m8.2.2.1.1.1.1.cmml" xref="S3.15.3.p3.15.m8.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.15.3.p3.15.m8.2.2.1.1.1.1.1.cmml" xref="S3.15.3.p3.15.m8.2.2.1.1.1">subscript</csymbol><ci id="S3.15.3.p3.15.m8.2.2.1.1.1.1.2.cmml" xref="S3.15.3.p3.15.m8.2.2.1.1.1.1.2">π</ci><ci id="S3.15.3.p3.15.m8.2.2.1.1.1.1.3.cmml" xref="S3.15.3.p3.15.m8.2.2.1.1.1.1.3">π</ci></apply></apply><apply id="S3.15.3.p3.15.m8.2.2.4.cmml" xref="S3.15.3.p3.15.m8.2.2.4"><plus id="S3.15.3.p3.15.m8.2.2.4.1.cmml" xref="S3.15.3.p3.15.m8.2.2.4.1"></plus><apply id="S3.15.3.p3.15.m8.2.2.4.2.cmml" xref="S3.15.3.p3.15.m8.2.2.4.2"><times id="S3.15.3.p3.15.m8.2.2.4.2.1.cmml" xref="S3.15.3.p3.15.m8.2.2.4.2.1"></times><ci id="S3.15.3.p3.15.m8.2.2.4.2.2.cmml" xref="S3.15.3.p3.15.m8.2.2.4.2.2">π</ci><ci id="S3.15.3.p3.15.m8.1.1.cmml" xref="S3.15.3.p3.15.m8.1.1">π</ci></apply><cn id="S3.15.3.p3.15.m8.2.2.4.3.cmml" type="integer" xref="S3.15.3.p3.15.m8.2.2.4.3">1</cn></apply></apply><apply id="S3.15.3.p3.15.m8.2.2c.cmml" xref="S3.15.3.p3.15.m8.2.2"><leq id="S3.15.3.p3.15.m8.2.2.5.cmml" xref="S3.15.3.p3.15.m8.2.2.5"></leq><share href="https://arxiv.org/html/2503.13666v1#S3.15.3.p3.15.m8.2.2.4.cmml" id="S3.15.3.p3.15.m8.2.2d.cmml" xref="S3.15.3.p3.15.m8.2.2"></share><apply id="S3.15.3.p3.15.m8.2.2.6.cmml" xref="S3.15.3.p3.15.m8.2.2.6"><plus id="S3.15.3.p3.15.m8.2.2.6.1.cmml" xref="S3.15.3.p3.15.m8.2.2.6.1"></plus><ci id="S3.15.3.p3.15.m8.2.2.6.2.cmml" xref="S3.15.3.p3.15.m8.2.2.6.2">π</ci><cn id="S3.15.3.p3.15.m8.2.2.6.3.cmml" type="integer" xref="S3.15.3.p3.15.m8.2.2.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.15.3.p3.15.m8.2c">\psi(l_{\xi})=\psi(\pi)+1\leq\xi+1</annotation><annotation encoding="application/x-llamapun" id="S3.15.3.p3.15.m8.2d">italic_Ο ( italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT ) = italic_Ο ( italic_Ο ) + 1 β€ italic_ΞΎ + 1</annotation></semantics></math>. β</p> </div> </div> <div class="ltx_para" id="S3.16.p4"> <p class="ltx_p" id="S3.16.p4.7">Since the ordinal <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.16.p4.1.m1.1"><semantics id="S3.16.p4.1.m1.1a"><mi id="S3.16.p4.1.m1.1.1" xref="S3.16.p4.1.m1.1.1.cmml">Ξ±</mi><annotation-xml encoding="MathML-Content" id="S3.16.p4.1.m1.1b"><ci id="S3.16.p4.1.m1.1.1.cmml" xref="S3.16.p4.1.m1.1.1">πΌ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.16.p4.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.16.p4.1.m1.1d">italic_Ξ±</annotation></semantics></math> is limit, <math alttext="\alpha=\sup\{\xi+1:\xi<\alpha\}" class="ltx_Math" display="inline" id="S3.16.p4.2.m2.2"><semantics id="S3.16.p4.2.m2.2a"><mrow id="S3.16.p4.2.m2.2.2" xref="S3.16.p4.2.m2.2.2.cmml"><mi id="S3.16.p4.2.m2.2.2.4" xref="S3.16.p4.2.m2.2.2.4.cmml">Ξ±</mi><mo id="S3.16.p4.2.m2.2.2.3" rspace="0.1389em" xref="S3.16.p4.2.m2.2.2.3.cmml">=</mo><mrow id="S3.16.p4.2.m2.2.2.2" xref="S3.16.p4.2.m2.2.2.2.cmml"><mo id="S3.16.p4.2.m2.2.2.2.3" lspace="0.1389em" rspace="0em" xref="S3.16.p4.2.m2.2.2.2.3.cmml">sup</mo><mrow id="S3.16.p4.2.m2.2.2.2.2.2" xref="S3.16.p4.2.m2.2.2.2.2.3.cmml"><mo id="S3.16.p4.2.m2.2.2.2.2.2.3" stretchy="false" xref="S3.16.p4.2.m2.2.2.2.2.3.1.cmml">{</mo><mrow id="S3.16.p4.2.m2.1.1.1.1.1.1" xref="S3.16.p4.2.m2.1.1.1.1.1.1.cmml"><mi id="S3.16.p4.2.m2.1.1.1.1.1.1.2" xref="S3.16.p4.2.m2.1.1.1.1.1.1.2.cmml">ΞΎ</mi><mo id="S3.16.p4.2.m2.1.1.1.1.1.1.1" xref="S3.16.p4.2.m2.1.1.1.1.1.1.1.cmml">+</mo><mn id="S3.16.p4.2.m2.1.1.1.1.1.1.3" xref="S3.16.p4.2.m2.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.16.p4.2.m2.2.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.16.p4.2.m2.2.2.2.2.3.1.cmml">:</mo><mrow id="S3.16.p4.2.m2.2.2.2.2.2.2" xref="S3.16.p4.2.m2.2.2.2.2.2.2.cmml"><mi id="S3.16.p4.2.m2.2.2.2.2.2.2.2" xref="S3.16.p4.2.m2.2.2.2.2.2.2.2.cmml">ΞΎ</mi><mo id="S3.16.p4.2.m2.2.2.2.2.2.2.1" xref="S3.16.p4.2.m2.2.2.2.2.2.2.1.cmml"><</mo><mi id="S3.16.p4.2.m2.2.2.2.2.2.2.3" xref="S3.16.p4.2.m2.2.2.2.2.2.2.3.cmml">Ξ±</mi></mrow><mo id="S3.16.p4.2.m2.2.2.2.2.2.5" stretchy="false" xref="S3.16.p4.2.m2.2.2.2.2.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.16.p4.2.m2.2b"><apply id="S3.16.p4.2.m2.2.2.cmml" xref="S3.16.p4.2.m2.2.2"><eq id="S3.16.p4.2.m2.2.2.3.cmml" xref="S3.16.p4.2.m2.2.2.3"></eq><ci id="S3.16.p4.2.m2.2.2.4.cmml" xref="S3.16.p4.2.m2.2.2.4">πΌ</ci><apply id="S3.16.p4.2.m2.2.2.2.cmml" xref="S3.16.p4.2.m2.2.2.2"><csymbol cd="latexml" id="S3.16.p4.2.m2.2.2.2.3.cmml" xref="S3.16.p4.2.m2.2.2.2.3">supremum</csymbol><apply id="S3.16.p4.2.m2.2.2.2.2.3.cmml" xref="S3.16.p4.2.m2.2.2.2.2.2"><csymbol cd="latexml" id="S3.16.p4.2.m2.2.2.2.2.3.1.cmml" xref="S3.16.p4.2.m2.2.2.2.2.2.3">conditional-set</csymbol><apply id="S3.16.p4.2.m2.1.1.1.1.1.1.cmml" xref="S3.16.p4.2.m2.1.1.1.1.1.1"><plus id="S3.16.p4.2.m2.1.1.1.1.1.1.1.cmml" xref="S3.16.p4.2.m2.1.1.1.1.1.1.1"></plus><ci id="S3.16.p4.2.m2.1.1.1.1.1.1.2.cmml" xref="S3.16.p4.2.m2.1.1.1.1.1.1.2">π</ci><cn id="S3.16.p4.2.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.16.p4.2.m2.1.1.1.1.1.1.3">1</cn></apply><apply id="S3.16.p4.2.m2.2.2.2.2.2.2.cmml" xref="S3.16.p4.2.m2.2.2.2.2.2.2"><lt id="S3.16.p4.2.m2.2.2.2.2.2.2.1.cmml" xref="S3.16.p4.2.m2.2.2.2.2.2.2.1"></lt><ci id="S3.16.p4.2.m2.2.2.2.2.2.2.2.cmml" xref="S3.16.p4.2.m2.2.2.2.2.2.2.2">π</ci><ci id="S3.16.p4.2.m2.2.2.2.2.2.2.3.cmml" xref="S3.16.p4.2.m2.2.2.2.2.2.2.3">πΌ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.16.p4.2.m2.2c">\alpha=\sup\{\xi+1:\xi<\alpha\}</annotation><annotation encoding="application/x-llamapun" id="S3.16.p4.2.m2.2d">italic_Ξ± = roman_sup { italic_ΞΎ + 1 : italic_ΞΎ < italic_Ξ± }</annotation></semantics></math>. Observe that <math alttext="\psi(L)" class="ltx_Math" display="inline" id="S3.16.p4.3.m3.1"><semantics id="S3.16.p4.3.m3.1a"><mrow id="S3.16.p4.3.m3.1.2" xref="S3.16.p4.3.m3.1.2.cmml"><mi id="S3.16.p4.3.m3.1.2.2" xref="S3.16.p4.3.m3.1.2.2.cmml">Ο</mi><mo id="S3.16.p4.3.m3.1.2.1" xref="S3.16.p4.3.m3.1.2.1.cmml">β’</mo><mrow id="S3.16.p4.3.m3.1.2.3.2" xref="S3.16.p4.3.m3.1.2.cmml"><mo id="S3.16.p4.3.m3.1.2.3.2.1" stretchy="false" xref="S3.16.p4.3.m3.1.2.cmml">(</mo><mi id="S3.16.p4.3.m3.1.1" xref="S3.16.p4.3.m3.1.1.cmml">L</mi><mo id="S3.16.p4.3.m3.1.2.3.2.2" stretchy="false" xref="S3.16.p4.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.16.p4.3.m3.1b"><apply id="S3.16.p4.3.m3.1.2.cmml" xref="S3.16.p4.3.m3.1.2"><times id="S3.16.p4.3.m3.1.2.1.cmml" xref="S3.16.p4.3.m3.1.2.1"></times><ci id="S3.16.p4.3.m3.1.2.2.cmml" xref="S3.16.p4.3.m3.1.2.2">π</ci><ci id="S3.16.p4.3.m3.1.1.cmml" xref="S3.16.p4.3.m3.1.1">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.16.p4.3.m3.1c">\psi(L)</annotation><annotation encoding="application/x-llamapun" id="S3.16.p4.3.m3.1d">italic_Ο ( italic_L )</annotation></semantics></math> is cofinal in <math alttext="\theta" class="ltx_Math" display="inline" id="S3.16.p4.4.m4.1"><semantics id="S3.16.p4.4.m4.1a"><mi id="S3.16.p4.4.m4.1.1" xref="S3.16.p4.4.m4.1.1.cmml">ΞΈ</mi><annotation-xml encoding="MathML-Content" id="S3.16.p4.4.m4.1b"><ci id="S3.16.p4.4.m4.1.1.cmml" xref="S3.16.p4.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.16.p4.4.m4.1c">\theta</annotation><annotation encoding="application/x-llamapun" id="S3.16.p4.4.m4.1d">italic_ΞΈ</annotation></semantics></math>, as <math alttext="\psi" class="ltx_Math" display="inline" id="S3.16.p4.5.m5.1"><semantics id="S3.16.p4.5.m5.1a"><mi id="S3.16.p4.5.m5.1.1" xref="S3.16.p4.5.m5.1.1.cmml">Ο</mi><annotation-xml encoding="MathML-Content" id="S3.16.p4.5.m5.1b"><ci id="S3.16.p4.5.m5.1.1.cmml" xref="S3.16.p4.5.m5.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.16.p4.5.m5.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S3.16.p4.5.m5.1d">italic_Ο</annotation></semantics></math> is an order isomoprhism. Then Claim <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem10" title="Claim 3.10. β£ Proof. β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.10</span></a> yields <math alttext="\theta\leq\sup\{\xi+1:\xi<\alpha\}=\alpha" class="ltx_Math" display="inline" id="S3.16.p4.6.m6.2"><semantics id="S3.16.p4.6.m6.2a"><mrow id="S3.16.p4.6.m6.2.2" xref="S3.16.p4.6.m6.2.2.cmml"><mi id="S3.16.p4.6.m6.2.2.4" xref="S3.16.p4.6.m6.2.2.4.cmml">ΞΈ</mi><mo id="S3.16.p4.6.m6.2.2.5" rspace="0.1389em" xref="S3.16.p4.6.m6.2.2.5.cmml">β€</mo><mrow id="S3.16.p4.6.m6.2.2.2" xref="S3.16.p4.6.m6.2.2.2.cmml"><mo id="S3.16.p4.6.m6.2.2.2.3" lspace="0.1389em" rspace="0em" xref="S3.16.p4.6.m6.2.2.2.3.cmml">sup</mo><mrow id="S3.16.p4.6.m6.2.2.2.2.2" xref="S3.16.p4.6.m6.2.2.2.2.3.cmml"><mo id="S3.16.p4.6.m6.2.2.2.2.2.3" stretchy="false" xref="S3.16.p4.6.m6.2.2.2.2.3.1.cmml">{</mo><mrow id="S3.16.p4.6.m6.1.1.1.1.1.1" xref="S3.16.p4.6.m6.1.1.1.1.1.1.cmml"><mi id="S3.16.p4.6.m6.1.1.1.1.1.1.2" xref="S3.16.p4.6.m6.1.1.1.1.1.1.2.cmml">ΞΎ</mi><mo id="S3.16.p4.6.m6.1.1.1.1.1.1.1" xref="S3.16.p4.6.m6.1.1.1.1.1.1.1.cmml">+</mo><mn id="S3.16.p4.6.m6.1.1.1.1.1.1.3" xref="S3.16.p4.6.m6.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.16.p4.6.m6.2.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.16.p4.6.m6.2.2.2.2.3.1.cmml">:</mo><mrow id="S3.16.p4.6.m6.2.2.2.2.2.2" xref="S3.16.p4.6.m6.2.2.2.2.2.2.cmml"><mi id="S3.16.p4.6.m6.2.2.2.2.2.2.2" xref="S3.16.p4.6.m6.2.2.2.2.2.2.2.cmml">ΞΎ</mi><mo id="S3.16.p4.6.m6.2.2.2.2.2.2.1" xref="S3.16.p4.6.m6.2.2.2.2.2.2.1.cmml"><</mo><mi id="S3.16.p4.6.m6.2.2.2.2.2.2.3" xref="S3.16.p4.6.m6.2.2.2.2.2.2.3.cmml">Ξ±</mi></mrow><mo id="S3.16.p4.6.m6.2.2.2.2.2.5" stretchy="false" xref="S3.16.p4.6.m6.2.2.2.2.3.1.cmml">}</mo></mrow></mrow><mo id="S3.16.p4.6.m6.2.2.6" xref="S3.16.p4.6.m6.2.2.6.cmml">=</mo><mi id="S3.16.p4.6.m6.2.2.7" xref="S3.16.p4.6.m6.2.2.7.cmml">Ξ±</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.16.p4.6.m6.2b"><apply id="S3.16.p4.6.m6.2.2.cmml" xref="S3.16.p4.6.m6.2.2"><and id="S3.16.p4.6.m6.2.2a.cmml" xref="S3.16.p4.6.m6.2.2"></and><apply id="S3.16.p4.6.m6.2.2b.cmml" xref="S3.16.p4.6.m6.2.2"><leq id="S3.16.p4.6.m6.2.2.5.cmml" xref="S3.16.p4.6.m6.2.2.5"></leq><ci id="S3.16.p4.6.m6.2.2.4.cmml" xref="S3.16.p4.6.m6.2.2.4">π</ci><apply id="S3.16.p4.6.m6.2.2.2.cmml" xref="S3.16.p4.6.m6.2.2.2"><csymbol cd="latexml" id="S3.16.p4.6.m6.2.2.2.3.cmml" xref="S3.16.p4.6.m6.2.2.2.3">supremum</csymbol><apply id="S3.16.p4.6.m6.2.2.2.2.3.cmml" xref="S3.16.p4.6.m6.2.2.2.2.2"><csymbol cd="latexml" id="S3.16.p4.6.m6.2.2.2.2.3.1.cmml" xref="S3.16.p4.6.m6.2.2.2.2.2.3">conditional-set</csymbol><apply id="S3.16.p4.6.m6.1.1.1.1.1.1.cmml" xref="S3.16.p4.6.m6.1.1.1.1.1.1"><plus id="S3.16.p4.6.m6.1.1.1.1.1.1.1.cmml" xref="S3.16.p4.6.m6.1.1.1.1.1.1.1"></plus><ci id="S3.16.p4.6.m6.1.1.1.1.1.1.2.cmml" xref="S3.16.p4.6.m6.1.1.1.1.1.1.2">π</ci><cn id="S3.16.p4.6.m6.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.16.p4.6.m6.1.1.1.1.1.1.3">1</cn></apply><apply id="S3.16.p4.6.m6.2.2.2.2.2.2.cmml" xref="S3.16.p4.6.m6.2.2.2.2.2.2"><lt id="S3.16.p4.6.m6.2.2.2.2.2.2.1.cmml" xref="S3.16.p4.6.m6.2.2.2.2.2.2.1"></lt><ci id="S3.16.p4.6.m6.2.2.2.2.2.2.2.cmml" xref="S3.16.p4.6.m6.2.2.2.2.2.2.2">π</ci><ci id="S3.16.p4.6.m6.2.2.2.2.2.2.3.cmml" xref="S3.16.p4.6.m6.2.2.2.2.2.2.3">πΌ</ci></apply></apply></apply></apply><apply id="S3.16.p4.6.m6.2.2c.cmml" xref="S3.16.p4.6.m6.2.2"><eq id="S3.16.p4.6.m6.2.2.6.cmml" xref="S3.16.p4.6.m6.2.2.6"></eq><share href="https://arxiv.org/html/2503.13666v1#S3.16.p4.6.m6.2.2.2.cmml" id="S3.16.p4.6.m6.2.2d.cmml" xref="S3.16.p4.6.m6.2.2"></share><ci id="S3.16.p4.6.m6.2.2.7.cmml" xref="S3.16.p4.6.m6.2.2.7">πΌ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.16.p4.6.m6.2c">\theta\leq\sup\{\xi+1:\xi<\alpha\}=\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.16.p4.6.m6.2d">italic_ΞΈ β€ roman_sup { italic_ΞΎ + 1 : italic_ΞΎ < italic_Ξ± } = italic_Ξ±</annotation></semantics></math>. Thus <math alttext="\theta=\alpha" class="ltx_Math" display="inline" id="S3.16.p4.7.m7.1"><semantics id="S3.16.p4.7.m7.1a"><mrow id="S3.16.p4.7.m7.1.1" xref="S3.16.p4.7.m7.1.1.cmml"><mi id="S3.16.p4.7.m7.1.1.2" xref="S3.16.p4.7.m7.1.1.2.cmml">ΞΈ</mi><mo id="S3.16.p4.7.m7.1.1.1" xref="S3.16.p4.7.m7.1.1.1.cmml">=</mo><mi id="S3.16.p4.7.m7.1.1.3" xref="S3.16.p4.7.m7.1.1.3.cmml">Ξ±</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.16.p4.7.m7.1b"><apply id="S3.16.p4.7.m7.1.1.cmml" xref="S3.16.p4.7.m7.1.1"><eq id="S3.16.p4.7.m7.1.1.1.cmml" xref="S3.16.p4.7.m7.1.1.1"></eq><ci id="S3.16.p4.7.m7.1.1.2.cmml" xref="S3.16.p4.7.m7.1.1.2">π</ci><ci id="S3.16.p4.7.m7.1.1.3.cmml" xref="S3.16.p4.7.m7.1.1.3">πΌ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.16.p4.7.m7.1c">\theta=\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.16.p4.7.m7.1d">italic_ΞΈ = italic_Ξ±</annotation></semantics></math>, as required. β</p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem11"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem11.1.1.1">Lemma 3.11</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem11.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem11.p1"> <p class="ltx_p" id="S3.Thmtheorem11.p1.7"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem11.p1.7.7">Let <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.1.1.m1.1"><semantics id="S3.Thmtheorem11.p1.1.1.m1.1a"><mi id="S3.Thmtheorem11.p1.1.1.m1.1.1" xref="S3.Thmtheorem11.p1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.1.1.m1.1b"><ci id="S3.Thmtheorem11.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem11.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.1.1.m1.1d">italic_X</annotation></semantics></math> be a countably compact countably tight semitopological semilattice and <math alttext="L\subset X" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.2.2.m2.1"><semantics id="S3.Thmtheorem11.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem11.p1.2.2.m2.1.1" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem11.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.2.cmml">L</mi><mo id="S3.Thmtheorem11.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.1.cmml">β</mo><mi id="S3.Thmtheorem11.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.2.2.m2.1b"><apply id="S3.Thmtheorem11.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem11.p1.2.2.m2.1.1"><subset id="S3.Thmtheorem11.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.1"></subset><ci id="S3.Thmtheorem11.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.2">πΏ</ci><ci id="S3.Thmtheorem11.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem11.p1.2.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.2.2.m2.1c">L\subset X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.2.2.m2.1d">italic_L β italic_X</annotation></semantics></math> be a chain isomorphic to <math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.3.3.m3.2"><semantics id="S3.Thmtheorem11.p1.3.3.m3.2a"><mrow id="S3.Thmtheorem11.p1.3.3.m3.2.2.1" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.2.cmml"><mo id="S3.Thmtheorem11.p1.3.3.m3.2.2.1.2" stretchy="false" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.2.cmml">(</mo><msub id="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.cmml"><mi id="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.2" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.2.cmml">Ο</mi><mn id="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.3" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.Thmtheorem11.p1.3.3.m3.2.2.1.3" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.2.cmml">,</mo><mi id="S3.Thmtheorem11.p1.3.3.m3.1.1" xref="S3.Thmtheorem11.p1.3.3.m3.1.1.cmml">min</mi><mo id="S3.Thmtheorem11.p1.3.3.m3.2.2.1.4" stretchy="false" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.3.3.m3.2b"><interval closure="open" id="S3.Thmtheorem11.p1.3.3.m3.2.2.2.cmml" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.1"><apply id="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.cmml" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.1.cmml" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1">subscript</csymbol><ci id="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.2.cmml" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.2">π</ci><cn id="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.3.cmml" type="integer" xref="S3.Thmtheorem11.p1.3.3.m3.2.2.1.1.3">1</cn></apply><min id="S3.Thmtheorem11.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem11.p1.3.3.m3.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.3.3.m3.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.3.3.m3.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math>. Then <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.4.4.m4.1"><semantics id="S3.Thmtheorem11.p1.4.4.m4.1a"><mover accent="true" id="S3.Thmtheorem11.p1.4.4.m4.1.1" xref="S3.Thmtheorem11.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem11.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem11.p1.4.4.m4.1.1.2.cmml">L</mi><mo id="S3.Thmtheorem11.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem11.p1.4.4.m4.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.4.4.m4.1b"><apply id="S3.Thmtheorem11.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem11.p1.4.4.m4.1.1"><ci id="S3.Thmtheorem11.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem11.p1.4.4.m4.1.1.1">Β―</ci><ci id="S3.Thmtheorem11.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem11.p1.4.4.m4.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.4.4.m4.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.4.4.m4.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> is topologically isomorphic to <math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.5.5.m5.2"><semantics id="S3.Thmtheorem11.p1.5.5.m5.2a"><mrow id="S3.Thmtheorem11.p1.5.5.m5.2.2.1" xref="S3.Thmtheorem11.p1.5.5.m5.2.2.2.cmml"><mo id="S3.Thmtheorem11.p1.5.5.m5.2.2.1.2" stretchy="false" xref="S3.Thmtheorem11.p1.5.5.m5.2.2.2.cmml">(</mo><msub id="S3.Thmtheorem11.p1.5.5.m5.2.2.1.1" xref="S3.Thmtheorem11.p1.5.5.m5.2.2.1.1.cmml"><mi id="S3.Thmtheorem11.p1.5.5.m5.2.2.1.1.2" xref="S3.Thmtheorem11.p1.5.5.m5.2.2.1.1.2.cmml">Ο</mi><mn id="S3.Thmtheorem11.p1.5.5.m5.2.2.1.1.3" xref="S3.Thmtheorem11.p1.5.5.m5.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.Thmtheorem11.p1.5.5.m5.2.2.1.3" xref="S3.Thmtheorem11.p1.5.5.m5.2.2.2.cmml">,</mo><mi id="S3.Thmtheorem11.p1.5.5.m5.1.1" xref="S3.Thmtheorem11.p1.5.5.m5.1.1.cmml">min</mi><mo id="S3.Thmtheorem11.p1.5.5.m5.2.2.1.4" stretchy="false" xref="S3.Thmtheorem11.p1.5.5.m5.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.5.5.m5.2b"><interval closure="open" id="S3.Thmtheorem11.p1.5.5.m5.2.2.2.cmml" xref="S3.Thmtheorem11.p1.5.5.m5.2.2.1"><apply id="S3.Thmtheorem11.p1.5.5.m5.2.2.1.1.cmml" xref="S3.Thmtheorem11.p1.5.5.m5.2.2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem11.p1.5.5.m5.2.2.1.1.1.cmml" xref="S3.Thmtheorem11.p1.5.5.m5.2.2.1.1">subscript</csymbol><ci id="S3.Thmtheorem11.p1.5.5.m5.2.2.1.1.2.cmml" xref="S3.Thmtheorem11.p1.5.5.m5.2.2.1.1.2">π</ci><cn id="S3.Thmtheorem11.p1.5.5.m5.2.2.1.1.3.cmml" type="integer" xref="S3.Thmtheorem11.p1.5.5.m5.2.2.1.1.3">1</cn></apply><min id="S3.Thmtheorem11.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem11.p1.5.5.m5.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.5.5.m5.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.5.5.m5.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math> endowed with the order topology, and <math alttext="L" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.6.6.m6.1"><semantics id="S3.Thmtheorem11.p1.6.6.m6.1a"><mi id="S3.Thmtheorem11.p1.6.6.m6.1.1" xref="S3.Thmtheorem11.p1.6.6.m6.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.6.6.m6.1b"><ci id="S3.Thmtheorem11.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem11.p1.6.6.m6.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.6.6.m6.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.6.6.m6.1d">italic_L</annotation></semantics></math> is cofinal in <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.Thmtheorem11.p1.7.7.m7.1"><semantics id="S3.Thmtheorem11.p1.7.7.m7.1a"><mover accent="true" id="S3.Thmtheorem11.p1.7.7.m7.1.1" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.cmml"><mi id="S3.Thmtheorem11.p1.7.7.m7.1.1.2" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.2.cmml">L</mi><mo id="S3.Thmtheorem11.p1.7.7.m7.1.1.1" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem11.p1.7.7.m7.1b"><apply id="S3.Thmtheorem11.p1.7.7.m7.1.1.cmml" xref="S3.Thmtheorem11.p1.7.7.m7.1.1"><ci id="S3.Thmtheorem11.p1.7.7.m7.1.1.1.cmml" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.1">Β―</ci><ci id="S3.Thmtheorem11.p1.7.7.m7.1.1.2.cmml" xref="S3.Thmtheorem11.p1.7.7.m7.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem11.p1.7.7.m7.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem11.p1.7.7.m7.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S3.17"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.17.p1"> <p class="ltx_p" id="S3.17.p1.23">Fix any <math alttext="x\in\overline{L}\setminus L" class="ltx_Math" display="inline" id="S3.17.p1.1.m1.1"><semantics id="S3.17.p1.1.m1.1a"><mrow id="S3.17.p1.1.m1.1.1" xref="S3.17.p1.1.m1.1.1.cmml"><mi id="S3.17.p1.1.m1.1.1.2" xref="S3.17.p1.1.m1.1.1.2.cmml">x</mi><mo id="S3.17.p1.1.m1.1.1.1" xref="S3.17.p1.1.m1.1.1.1.cmml">β</mo><mrow id="S3.17.p1.1.m1.1.1.3" xref="S3.17.p1.1.m1.1.1.3.cmml"><mover accent="true" id="S3.17.p1.1.m1.1.1.3.2" xref="S3.17.p1.1.m1.1.1.3.2.cmml"><mi id="S3.17.p1.1.m1.1.1.3.2.2" xref="S3.17.p1.1.m1.1.1.3.2.2.cmml">L</mi><mo id="S3.17.p1.1.m1.1.1.3.2.1" xref="S3.17.p1.1.m1.1.1.3.2.1.cmml">Β―</mo></mover><mo id="S3.17.p1.1.m1.1.1.3.1" xref="S3.17.p1.1.m1.1.1.3.1.cmml">β</mo><mi id="S3.17.p1.1.m1.1.1.3.3" xref="S3.17.p1.1.m1.1.1.3.3.cmml">L</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.1.m1.1b"><apply id="S3.17.p1.1.m1.1.1.cmml" xref="S3.17.p1.1.m1.1.1"><in id="S3.17.p1.1.m1.1.1.1.cmml" xref="S3.17.p1.1.m1.1.1.1"></in><ci id="S3.17.p1.1.m1.1.1.2.cmml" xref="S3.17.p1.1.m1.1.1.2">π₯</ci><apply id="S3.17.p1.1.m1.1.1.3.cmml" xref="S3.17.p1.1.m1.1.1.3"><setdiff id="S3.17.p1.1.m1.1.1.3.1.cmml" xref="S3.17.p1.1.m1.1.1.3.1"></setdiff><apply id="S3.17.p1.1.m1.1.1.3.2.cmml" xref="S3.17.p1.1.m1.1.1.3.2"><ci id="S3.17.p1.1.m1.1.1.3.2.1.cmml" xref="S3.17.p1.1.m1.1.1.3.2.1">Β―</ci><ci id="S3.17.p1.1.m1.1.1.3.2.2.cmml" xref="S3.17.p1.1.m1.1.1.3.2.2">πΏ</ci></apply><ci id="S3.17.p1.1.m1.1.1.3.3.cmml" xref="S3.17.p1.1.m1.1.1.3.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.1.m1.1c">x\in\overline{L}\setminus L</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.1.m1.1d">italic_x β overΒ― start_ARG italic_L end_ARG β italic_L</annotation></semantics></math>. Since <math alttext="X" class="ltx_Math" display="inline" id="S3.17.p1.2.m2.1"><semantics id="S3.17.p1.2.m2.1a"><mi id="S3.17.p1.2.m2.1.1" xref="S3.17.p1.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.17.p1.2.m2.1b"><ci id="S3.17.p1.2.m2.1.1.cmml" xref="S3.17.p1.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.2.m2.1d">italic_X</annotation></semantics></math> is countably tight, there exists a countable subset <math alttext="A" class="ltx_Math" display="inline" id="S3.17.p1.3.m3.1"><semantics id="S3.17.p1.3.m3.1a"><mi id="S3.17.p1.3.m3.1.1" xref="S3.17.p1.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.17.p1.3.m3.1b"><ci id="S3.17.p1.3.m3.1.1.cmml" xref="S3.17.p1.3.m3.1.1">π΄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.3.m3.1d">italic_A</annotation></semantics></math> of <math alttext="L" class="ltx_Math" display="inline" id="S3.17.p1.4.m4.1"><semantics id="S3.17.p1.4.m4.1a"><mi id="S3.17.p1.4.m4.1.1" xref="S3.17.p1.4.m4.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.17.p1.4.m4.1b"><ci id="S3.17.p1.4.m4.1.1.cmml" xref="S3.17.p1.4.m4.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.4.m4.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.4.m4.1d">italic_L</annotation></semantics></math> such that <math alttext="x\in\overline{L}" class="ltx_Math" display="inline" id="S3.17.p1.5.m5.1"><semantics id="S3.17.p1.5.m5.1a"><mrow id="S3.17.p1.5.m5.1.1" xref="S3.17.p1.5.m5.1.1.cmml"><mi id="S3.17.p1.5.m5.1.1.2" xref="S3.17.p1.5.m5.1.1.2.cmml">x</mi><mo id="S3.17.p1.5.m5.1.1.1" xref="S3.17.p1.5.m5.1.1.1.cmml">β</mo><mover accent="true" id="S3.17.p1.5.m5.1.1.3" xref="S3.17.p1.5.m5.1.1.3.cmml"><mi id="S3.17.p1.5.m5.1.1.3.2" xref="S3.17.p1.5.m5.1.1.3.2.cmml">L</mi><mo id="S3.17.p1.5.m5.1.1.3.1" xref="S3.17.p1.5.m5.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.5.m5.1b"><apply id="S3.17.p1.5.m5.1.1.cmml" xref="S3.17.p1.5.m5.1.1"><in id="S3.17.p1.5.m5.1.1.1.cmml" xref="S3.17.p1.5.m5.1.1.1"></in><ci id="S3.17.p1.5.m5.1.1.2.cmml" xref="S3.17.p1.5.m5.1.1.2">π₯</ci><apply id="S3.17.p1.5.m5.1.1.3.cmml" xref="S3.17.p1.5.m5.1.1.3"><ci id="S3.17.p1.5.m5.1.1.3.1.cmml" xref="S3.17.p1.5.m5.1.1.3.1">Β―</ci><ci id="S3.17.p1.5.m5.1.1.3.2.cmml" xref="S3.17.p1.5.m5.1.1.3.2">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.5.m5.1c">x\in\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.5.m5.1d">italic_x β overΒ― start_ARG italic_L end_ARG</annotation></semantics></math>. It is clear that there exists <math alttext="l\in L" class="ltx_Math" display="inline" id="S3.17.p1.6.m6.1"><semantics id="S3.17.p1.6.m6.1a"><mrow id="S3.17.p1.6.m6.1.1" xref="S3.17.p1.6.m6.1.1.cmml"><mi id="S3.17.p1.6.m6.1.1.2" xref="S3.17.p1.6.m6.1.1.2.cmml">l</mi><mo id="S3.17.p1.6.m6.1.1.1" xref="S3.17.p1.6.m6.1.1.1.cmml">β</mo><mi id="S3.17.p1.6.m6.1.1.3" xref="S3.17.p1.6.m6.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.6.m6.1b"><apply id="S3.17.p1.6.m6.1.1.cmml" xref="S3.17.p1.6.m6.1.1"><in id="S3.17.p1.6.m6.1.1.1.cmml" xref="S3.17.p1.6.m6.1.1.1"></in><ci id="S3.17.p1.6.m6.1.1.2.cmml" xref="S3.17.p1.6.m6.1.1.2">π</ci><ci id="S3.17.p1.6.m6.1.1.3.cmml" xref="S3.17.p1.6.m6.1.1.3">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.6.m6.1c">l\in L</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.6.m6.1d">italic_l β italic_L</annotation></semantics></math> such that <math alttext="A\subseteq{\downarrow}l" class="ltx_math_unparsed" display="inline" id="S3.17.p1.7.m7.1"><semantics id="S3.17.p1.7.m7.1a"><mrow id="S3.17.p1.7.m7.1b"><mi id="S3.17.p1.7.m7.1.1">A</mi><mo id="S3.17.p1.7.m7.1.2" rspace="0em">β</mo><mo id="S3.17.p1.7.m7.1.3" lspace="0em" stretchy="false">β</mo><mi id="S3.17.p1.7.m7.1.4">l</mi></mrow><annotation encoding="application/x-tex" id="S3.17.p1.7.m7.1c">A\subseteq{\downarrow}l</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.7.m7.1d">italic_A β β italic_l</annotation></semantics></math>. Since <math alttext="X" class="ltx_Math" display="inline" id="S3.17.p1.8.m8.1"><semantics id="S3.17.p1.8.m8.1a"><mi id="S3.17.p1.8.m8.1.1" xref="S3.17.p1.8.m8.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.17.p1.8.m8.1b"><ci id="S3.17.p1.8.m8.1.1.cmml" xref="S3.17.p1.8.m8.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.8.m8.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.8.m8.1d">italic_X</annotation></semantics></math> is a semitopological semillatice, the set <math alttext="{\downarrow}l" class="ltx_Math" display="inline" id="S3.17.p1.9.m9.1"><semantics id="S3.17.p1.9.m9.1a"><mrow id="S3.17.p1.9.m9.1.1" xref="S3.17.p1.9.m9.1.1.cmml"><mi id="S3.17.p1.9.m9.1.1.2" xref="S3.17.p1.9.m9.1.1.2.cmml"></mi><mo id="S3.17.p1.9.m9.1.1.1" stretchy="false" xref="S3.17.p1.9.m9.1.1.1.cmml">β</mo><mi id="S3.17.p1.9.m9.1.1.3" xref="S3.17.p1.9.m9.1.1.3.cmml">l</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.9.m9.1b"><apply id="S3.17.p1.9.m9.1.1.cmml" xref="S3.17.p1.9.m9.1.1"><ci id="S3.17.p1.9.m9.1.1.1.cmml" xref="S3.17.p1.9.m9.1.1.1">β</ci><csymbol cd="latexml" id="S3.17.p1.9.m9.1.1.2.cmml" xref="S3.17.p1.9.m9.1.1.2">absent</csymbol><ci id="S3.17.p1.9.m9.1.1.3.cmml" xref="S3.17.p1.9.m9.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.9.m9.1c">{\downarrow}l</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.9.m9.1d">β italic_l</annotation></semantics></math> is closed and, as such, <math alttext="x\in\overline{A}\subseteq{\downarrow}l" class="ltx_math_unparsed" display="inline" id="S3.17.p1.10.m10.1"><semantics id="S3.17.p1.10.m10.1a"><mrow id="S3.17.p1.10.m10.1b"><mi id="S3.17.p1.10.m10.1.1">x</mi><mo id="S3.17.p1.10.m10.1.2">β</mo><mover accent="true" id="S3.17.p1.10.m10.1.3"><mi id="S3.17.p1.10.m10.1.3.2">A</mi><mo id="S3.17.p1.10.m10.1.3.1">Β―</mo></mover><mo id="S3.17.p1.10.m10.1.4" rspace="0em">β</mo><mo id="S3.17.p1.10.m10.1.5" lspace="0em" stretchy="false">β</mo><mi id="S3.17.p1.10.m10.1.6">l</mi></mrow><annotation encoding="application/x-tex" id="S3.17.p1.10.m10.1c">x\in\overline{A}\subseteq{\downarrow}l</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.10.m10.1d">italic_x β overΒ― start_ARG italic_A end_ARG β β italic_l</annotation></semantics></math>. It follows that for each <math alttext="x\in\overline{L}" class="ltx_Math" display="inline" id="S3.17.p1.11.m11.1"><semantics id="S3.17.p1.11.m11.1a"><mrow id="S3.17.p1.11.m11.1.1" xref="S3.17.p1.11.m11.1.1.cmml"><mi id="S3.17.p1.11.m11.1.1.2" xref="S3.17.p1.11.m11.1.1.2.cmml">x</mi><mo id="S3.17.p1.11.m11.1.1.1" xref="S3.17.p1.11.m11.1.1.1.cmml">β</mo><mover accent="true" id="S3.17.p1.11.m11.1.1.3" xref="S3.17.p1.11.m11.1.1.3.cmml"><mi id="S3.17.p1.11.m11.1.1.3.2" xref="S3.17.p1.11.m11.1.1.3.2.cmml">L</mi><mo id="S3.17.p1.11.m11.1.1.3.1" xref="S3.17.p1.11.m11.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.11.m11.1b"><apply id="S3.17.p1.11.m11.1.1.cmml" xref="S3.17.p1.11.m11.1.1"><in id="S3.17.p1.11.m11.1.1.1.cmml" xref="S3.17.p1.11.m11.1.1.1"></in><ci id="S3.17.p1.11.m11.1.1.2.cmml" xref="S3.17.p1.11.m11.1.1.2">π₯</ci><apply id="S3.17.p1.11.m11.1.1.3.cmml" xref="S3.17.p1.11.m11.1.1.3"><ci id="S3.17.p1.11.m11.1.1.3.1.cmml" xref="S3.17.p1.11.m11.1.1.3.1">Β―</ci><ci id="S3.17.p1.11.m11.1.1.3.2.cmml" xref="S3.17.p1.11.m11.1.1.3.2">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.11.m11.1c">x\in\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.11.m11.1d">italic_x β overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> there exists <math alttext="l\in L" class="ltx_Math" display="inline" id="S3.17.p1.12.m12.1"><semantics id="S3.17.p1.12.m12.1a"><mrow id="S3.17.p1.12.m12.1.1" xref="S3.17.p1.12.m12.1.1.cmml"><mi id="S3.17.p1.12.m12.1.1.2" xref="S3.17.p1.12.m12.1.1.2.cmml">l</mi><mo id="S3.17.p1.12.m12.1.1.1" xref="S3.17.p1.12.m12.1.1.1.cmml">β</mo><mi id="S3.17.p1.12.m12.1.1.3" xref="S3.17.p1.12.m12.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.12.m12.1b"><apply id="S3.17.p1.12.m12.1.1.cmml" xref="S3.17.p1.12.m12.1.1"><in id="S3.17.p1.12.m12.1.1.1.cmml" xref="S3.17.p1.12.m12.1.1.1"></in><ci id="S3.17.p1.12.m12.1.1.2.cmml" xref="S3.17.p1.12.m12.1.1.2">π</ci><ci id="S3.17.p1.12.m12.1.1.3.cmml" xref="S3.17.p1.12.m12.1.1.3">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.12.m12.1c">l\in L</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.12.m12.1d">italic_l β italic_L</annotation></semantics></math> with <math alttext="x\leq l" class="ltx_Math" display="inline" id="S3.17.p1.13.m13.1"><semantics id="S3.17.p1.13.m13.1a"><mrow id="S3.17.p1.13.m13.1.1" xref="S3.17.p1.13.m13.1.1.cmml"><mi id="S3.17.p1.13.m13.1.1.2" xref="S3.17.p1.13.m13.1.1.2.cmml">x</mi><mo id="S3.17.p1.13.m13.1.1.1" xref="S3.17.p1.13.m13.1.1.1.cmml">β€</mo><mi id="S3.17.p1.13.m13.1.1.3" xref="S3.17.p1.13.m13.1.1.3.cmml">l</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.13.m13.1b"><apply id="S3.17.p1.13.m13.1.1.cmml" xref="S3.17.p1.13.m13.1.1"><leq id="S3.17.p1.13.m13.1.1.1.cmml" xref="S3.17.p1.13.m13.1.1.1"></leq><ci id="S3.17.p1.13.m13.1.1.2.cmml" xref="S3.17.p1.13.m13.1.1.2">π₯</ci><ci id="S3.17.p1.13.m13.1.1.3.cmml" xref="S3.17.p1.13.m13.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.13.m13.1c">x\leq l</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.13.m13.1d">italic_x β€ italic_l</annotation></semantics></math>. Thus, <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.17.p1.14.m14.1"><semantics id="S3.17.p1.14.m14.1a"><mover accent="true" id="S3.17.p1.14.m14.1.1" xref="S3.17.p1.14.m14.1.1.cmml"><mi id="S3.17.p1.14.m14.1.1.2" xref="S3.17.p1.14.m14.1.1.2.cmml">L</mi><mo id="S3.17.p1.14.m14.1.1.1" xref="S3.17.p1.14.m14.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.17.p1.14.m14.1b"><apply id="S3.17.p1.14.m14.1.1.cmml" xref="S3.17.p1.14.m14.1.1"><ci id="S3.17.p1.14.m14.1.1.1.cmml" xref="S3.17.p1.14.m14.1.1.1">Β―</ci><ci id="S3.17.p1.14.m14.1.1.2.cmml" xref="S3.17.p1.14.m14.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.14.m14.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.14.m14.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> does not contain maximum. Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem9" title="Lemma 3.9. β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.9</span></a> implies that <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.17.p1.15.m15.1"><semantics id="S3.17.p1.15.m15.1a"><mover accent="true" id="S3.17.p1.15.m15.1.1" xref="S3.17.p1.15.m15.1.1.cmml"><mi id="S3.17.p1.15.m15.1.1.2" xref="S3.17.p1.15.m15.1.1.2.cmml">L</mi><mo id="S3.17.p1.15.m15.1.1.1" xref="S3.17.p1.15.m15.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.17.p1.15.m15.1b"><apply id="S3.17.p1.15.m15.1.1.cmml" xref="S3.17.p1.15.m15.1.1"><ci id="S3.17.p1.15.m15.1.1.1.cmml" xref="S3.17.p1.15.m15.1.1.1">Β―</ci><ci id="S3.17.p1.15.m15.1.1.2.cmml" xref="S3.17.p1.15.m15.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.15.m15.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.15.m15.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> is isomorphic to <math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S3.17.p1.16.m16.2"><semantics id="S3.17.p1.16.m16.2a"><mrow id="S3.17.p1.16.m16.2.2.1" xref="S3.17.p1.16.m16.2.2.2.cmml"><mo id="S3.17.p1.16.m16.2.2.1.2" stretchy="false" xref="S3.17.p1.16.m16.2.2.2.cmml">(</mo><msub id="S3.17.p1.16.m16.2.2.1.1" xref="S3.17.p1.16.m16.2.2.1.1.cmml"><mi id="S3.17.p1.16.m16.2.2.1.1.2" xref="S3.17.p1.16.m16.2.2.1.1.2.cmml">Ο</mi><mn id="S3.17.p1.16.m16.2.2.1.1.3" xref="S3.17.p1.16.m16.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.17.p1.16.m16.2.2.1.3" xref="S3.17.p1.16.m16.2.2.2.cmml">,</mo><mi id="S3.17.p1.16.m16.1.1" xref="S3.17.p1.16.m16.1.1.cmml">min</mi><mo id="S3.17.p1.16.m16.2.2.1.4" stretchy="false" xref="S3.17.p1.16.m16.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.16.m16.2b"><interval closure="open" id="S3.17.p1.16.m16.2.2.2.cmml" xref="S3.17.p1.16.m16.2.2.1"><apply id="S3.17.p1.16.m16.2.2.1.1.cmml" xref="S3.17.p1.16.m16.2.2.1.1"><csymbol cd="ambiguous" id="S3.17.p1.16.m16.2.2.1.1.1.cmml" xref="S3.17.p1.16.m16.2.2.1.1">subscript</csymbol><ci id="S3.17.p1.16.m16.2.2.1.1.2.cmml" xref="S3.17.p1.16.m16.2.2.1.1.2">π</ci><cn id="S3.17.p1.16.m16.2.2.1.1.3.cmml" type="integer" xref="S3.17.p1.16.m16.2.2.1.1.3">1</cn></apply><min id="S3.17.p1.16.m16.1.1.cmml" xref="S3.17.p1.16.m16.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.16.m16.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.16.m16.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math> and <math alttext="L" class="ltx_Math" display="inline" id="S3.17.p1.17.m17.1"><semantics id="S3.17.p1.17.m17.1a"><mi id="S3.17.p1.17.m17.1.1" xref="S3.17.p1.17.m17.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.17.p1.17.m17.1b"><ci id="S3.17.p1.17.m17.1.1.cmml" xref="S3.17.p1.17.m17.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.17.m17.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.17.m17.1d">italic_L</annotation></semantics></math> is cofinal in <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.17.p1.18.m18.1"><semantics id="S3.17.p1.18.m18.1a"><mover accent="true" id="S3.17.p1.18.m18.1.1" xref="S3.17.p1.18.m18.1.1.cmml"><mi id="S3.17.p1.18.m18.1.1.2" xref="S3.17.p1.18.m18.1.1.2.cmml">L</mi><mo id="S3.17.p1.18.m18.1.1.1" xref="S3.17.p1.18.m18.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.17.p1.18.m18.1b"><apply id="S3.17.p1.18.m18.1.1.cmml" xref="S3.17.p1.18.m18.1.1"><ci id="S3.17.p1.18.m18.1.1.1.cmml" xref="S3.17.p1.18.m18.1.1.1">Β―</ci><ci id="S3.17.p1.18.m18.1.1.2.cmml" xref="S3.17.p1.18.m18.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.18.m18.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.18.m18.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math>. Let <math alttext="\overline{L}=\{y_{\xi}:\xi<\omega_{1}\}" class="ltx_Math" display="inline" id="S3.17.p1.19.m19.2"><semantics id="S3.17.p1.19.m19.2a"><mrow id="S3.17.p1.19.m19.2.2" xref="S3.17.p1.19.m19.2.2.cmml"><mover accent="true" id="S3.17.p1.19.m19.2.2.4" xref="S3.17.p1.19.m19.2.2.4.cmml"><mi id="S3.17.p1.19.m19.2.2.4.2" xref="S3.17.p1.19.m19.2.2.4.2.cmml">L</mi><mo id="S3.17.p1.19.m19.2.2.4.1" xref="S3.17.p1.19.m19.2.2.4.1.cmml">Β―</mo></mover><mo id="S3.17.p1.19.m19.2.2.3" xref="S3.17.p1.19.m19.2.2.3.cmml">=</mo><mrow id="S3.17.p1.19.m19.2.2.2.2" xref="S3.17.p1.19.m19.2.2.2.3.cmml"><mo id="S3.17.p1.19.m19.2.2.2.2.3" stretchy="false" xref="S3.17.p1.19.m19.2.2.2.3.1.cmml">{</mo><msub id="S3.17.p1.19.m19.1.1.1.1.1" xref="S3.17.p1.19.m19.1.1.1.1.1.cmml"><mi id="S3.17.p1.19.m19.1.1.1.1.1.2" xref="S3.17.p1.19.m19.1.1.1.1.1.2.cmml">y</mi><mi id="S3.17.p1.19.m19.1.1.1.1.1.3" xref="S3.17.p1.19.m19.1.1.1.1.1.3.cmml">ΞΎ</mi></msub><mo id="S3.17.p1.19.m19.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.17.p1.19.m19.2.2.2.3.1.cmml">:</mo><mrow id="S3.17.p1.19.m19.2.2.2.2.2" xref="S3.17.p1.19.m19.2.2.2.2.2.cmml"><mi id="S3.17.p1.19.m19.2.2.2.2.2.2" xref="S3.17.p1.19.m19.2.2.2.2.2.2.cmml">ΞΎ</mi><mo id="S3.17.p1.19.m19.2.2.2.2.2.1" xref="S3.17.p1.19.m19.2.2.2.2.2.1.cmml"><</mo><msub id="S3.17.p1.19.m19.2.2.2.2.2.3" xref="S3.17.p1.19.m19.2.2.2.2.2.3.cmml"><mi id="S3.17.p1.19.m19.2.2.2.2.2.3.2" xref="S3.17.p1.19.m19.2.2.2.2.2.3.2.cmml">Ο</mi><mn id="S3.17.p1.19.m19.2.2.2.2.2.3.3" xref="S3.17.p1.19.m19.2.2.2.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S3.17.p1.19.m19.2.2.2.2.5" stretchy="false" xref="S3.17.p1.19.m19.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.19.m19.2b"><apply id="S3.17.p1.19.m19.2.2.cmml" xref="S3.17.p1.19.m19.2.2"><eq id="S3.17.p1.19.m19.2.2.3.cmml" xref="S3.17.p1.19.m19.2.2.3"></eq><apply id="S3.17.p1.19.m19.2.2.4.cmml" xref="S3.17.p1.19.m19.2.2.4"><ci id="S3.17.p1.19.m19.2.2.4.1.cmml" xref="S3.17.p1.19.m19.2.2.4.1">Β―</ci><ci id="S3.17.p1.19.m19.2.2.4.2.cmml" xref="S3.17.p1.19.m19.2.2.4.2">πΏ</ci></apply><apply id="S3.17.p1.19.m19.2.2.2.3.cmml" xref="S3.17.p1.19.m19.2.2.2.2"><csymbol cd="latexml" id="S3.17.p1.19.m19.2.2.2.3.1.cmml" xref="S3.17.p1.19.m19.2.2.2.2.3">conditional-set</csymbol><apply id="S3.17.p1.19.m19.1.1.1.1.1.cmml" xref="S3.17.p1.19.m19.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.17.p1.19.m19.1.1.1.1.1.1.cmml" xref="S3.17.p1.19.m19.1.1.1.1.1">subscript</csymbol><ci id="S3.17.p1.19.m19.1.1.1.1.1.2.cmml" xref="S3.17.p1.19.m19.1.1.1.1.1.2">π¦</ci><ci id="S3.17.p1.19.m19.1.1.1.1.1.3.cmml" xref="S3.17.p1.19.m19.1.1.1.1.1.3">π</ci></apply><apply id="S3.17.p1.19.m19.2.2.2.2.2.cmml" xref="S3.17.p1.19.m19.2.2.2.2.2"><lt id="S3.17.p1.19.m19.2.2.2.2.2.1.cmml" xref="S3.17.p1.19.m19.2.2.2.2.2.1"></lt><ci id="S3.17.p1.19.m19.2.2.2.2.2.2.cmml" xref="S3.17.p1.19.m19.2.2.2.2.2.2">π</ci><apply id="S3.17.p1.19.m19.2.2.2.2.2.3.cmml" xref="S3.17.p1.19.m19.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S3.17.p1.19.m19.2.2.2.2.2.3.1.cmml" xref="S3.17.p1.19.m19.2.2.2.2.2.3">subscript</csymbol><ci id="S3.17.p1.19.m19.2.2.2.2.2.3.2.cmml" xref="S3.17.p1.19.m19.2.2.2.2.2.3.2">π</ci><cn id="S3.17.p1.19.m19.2.2.2.2.2.3.3.cmml" type="integer" xref="S3.17.p1.19.m19.2.2.2.2.2.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.19.m19.2c">\overline{L}=\{y_{\xi}:\xi<\omega_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.19.m19.2d">overΒ― start_ARG italic_L end_ARG = { italic_y start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT : italic_ΞΎ < italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math>, where <math alttext="y_{\xi}\leq y_{\delta}" class="ltx_Math" display="inline" id="S3.17.p1.20.m20.1"><semantics id="S3.17.p1.20.m20.1a"><mrow id="S3.17.p1.20.m20.1.1" xref="S3.17.p1.20.m20.1.1.cmml"><msub id="S3.17.p1.20.m20.1.1.2" xref="S3.17.p1.20.m20.1.1.2.cmml"><mi id="S3.17.p1.20.m20.1.1.2.2" xref="S3.17.p1.20.m20.1.1.2.2.cmml">y</mi><mi id="S3.17.p1.20.m20.1.1.2.3" xref="S3.17.p1.20.m20.1.1.2.3.cmml">ΞΎ</mi></msub><mo id="S3.17.p1.20.m20.1.1.1" xref="S3.17.p1.20.m20.1.1.1.cmml">β€</mo><msub id="S3.17.p1.20.m20.1.1.3" xref="S3.17.p1.20.m20.1.1.3.cmml"><mi id="S3.17.p1.20.m20.1.1.3.2" xref="S3.17.p1.20.m20.1.1.3.2.cmml">y</mi><mi id="S3.17.p1.20.m20.1.1.3.3" xref="S3.17.p1.20.m20.1.1.3.3.cmml">Ξ΄</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.20.m20.1b"><apply id="S3.17.p1.20.m20.1.1.cmml" xref="S3.17.p1.20.m20.1.1"><leq id="S3.17.p1.20.m20.1.1.1.cmml" xref="S3.17.p1.20.m20.1.1.1"></leq><apply id="S3.17.p1.20.m20.1.1.2.cmml" xref="S3.17.p1.20.m20.1.1.2"><csymbol cd="ambiguous" id="S3.17.p1.20.m20.1.1.2.1.cmml" xref="S3.17.p1.20.m20.1.1.2">subscript</csymbol><ci id="S3.17.p1.20.m20.1.1.2.2.cmml" xref="S3.17.p1.20.m20.1.1.2.2">π¦</ci><ci id="S3.17.p1.20.m20.1.1.2.3.cmml" xref="S3.17.p1.20.m20.1.1.2.3">π</ci></apply><apply id="S3.17.p1.20.m20.1.1.3.cmml" xref="S3.17.p1.20.m20.1.1.3"><csymbol cd="ambiguous" id="S3.17.p1.20.m20.1.1.3.1.cmml" xref="S3.17.p1.20.m20.1.1.3">subscript</csymbol><ci id="S3.17.p1.20.m20.1.1.3.2.cmml" xref="S3.17.p1.20.m20.1.1.3.2">π¦</ci><ci id="S3.17.p1.20.m20.1.1.3.3.cmml" xref="S3.17.p1.20.m20.1.1.3.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.20.m20.1c">y_{\xi}\leq y_{\delta}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.20.m20.1d">italic_y start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT β€ italic_y start_POSTSUBSCRIPT italic_Ξ΄ end_POSTSUBSCRIPT</annotation></semantics></math> if and only if <math alttext="\xi\leq\delta" class="ltx_Math" display="inline" id="S3.17.p1.21.m21.1"><semantics id="S3.17.p1.21.m21.1a"><mrow id="S3.17.p1.21.m21.1.1" xref="S3.17.p1.21.m21.1.1.cmml"><mi id="S3.17.p1.21.m21.1.1.2" xref="S3.17.p1.21.m21.1.1.2.cmml">ΞΎ</mi><mo id="S3.17.p1.21.m21.1.1.1" xref="S3.17.p1.21.m21.1.1.1.cmml">β€</mo><mi id="S3.17.p1.21.m21.1.1.3" xref="S3.17.p1.21.m21.1.1.3.cmml">Ξ΄</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.21.m21.1b"><apply id="S3.17.p1.21.m21.1.1.cmml" xref="S3.17.p1.21.m21.1.1"><leq id="S3.17.p1.21.m21.1.1.1.cmml" xref="S3.17.p1.21.m21.1.1.1"></leq><ci id="S3.17.p1.21.m21.1.1.2.cmml" xref="S3.17.p1.21.m21.1.1.2">π</ci><ci id="S3.17.p1.21.m21.1.1.3.cmml" xref="S3.17.p1.21.m21.1.1.3">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.21.m21.1c">\xi\leq\delta</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.21.m21.1d">italic_ΞΎ β€ italic_Ξ΄</annotation></semantics></math>. Fix any <math alttext="\xi<\omega_{1}" class="ltx_Math" display="inline" id="S3.17.p1.22.m22.1"><semantics id="S3.17.p1.22.m22.1a"><mrow id="S3.17.p1.22.m22.1.1" xref="S3.17.p1.22.m22.1.1.cmml"><mi id="S3.17.p1.22.m22.1.1.2" xref="S3.17.p1.22.m22.1.1.2.cmml">ΞΎ</mi><mo id="S3.17.p1.22.m22.1.1.1" xref="S3.17.p1.22.m22.1.1.1.cmml"><</mo><msub id="S3.17.p1.22.m22.1.1.3" xref="S3.17.p1.22.m22.1.1.3.cmml"><mi id="S3.17.p1.22.m22.1.1.3.2" xref="S3.17.p1.22.m22.1.1.3.2.cmml">Ο</mi><mn id="S3.17.p1.22.m22.1.1.3.3" xref="S3.17.p1.22.m22.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.22.m22.1b"><apply id="S3.17.p1.22.m22.1.1.cmml" xref="S3.17.p1.22.m22.1.1"><lt id="S3.17.p1.22.m22.1.1.1.cmml" xref="S3.17.p1.22.m22.1.1.1"></lt><ci id="S3.17.p1.22.m22.1.1.2.cmml" xref="S3.17.p1.22.m22.1.1.2">π</ci><apply id="S3.17.p1.22.m22.1.1.3.cmml" xref="S3.17.p1.22.m22.1.1.3"><csymbol cd="ambiguous" id="S3.17.p1.22.m22.1.1.3.1.cmml" xref="S3.17.p1.22.m22.1.1.3">subscript</csymbol><ci id="S3.17.p1.22.m22.1.1.3.2.cmml" xref="S3.17.p1.22.m22.1.1.3.2">π</ci><cn id="S3.17.p1.22.m22.1.1.3.3.cmml" type="integer" xref="S3.17.p1.22.m22.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.22.m22.1c">\xi<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.22.m22.1d">italic_ΞΎ < italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Clearly, for each <math alttext="\delta<\xi" class="ltx_Math" display="inline" id="S3.17.p1.23.m23.1"><semantics id="S3.17.p1.23.m23.1a"><mrow id="S3.17.p1.23.m23.1.1" xref="S3.17.p1.23.m23.1.1.cmml"><mi id="S3.17.p1.23.m23.1.1.2" xref="S3.17.p1.23.m23.1.1.2.cmml">Ξ΄</mi><mo id="S3.17.p1.23.m23.1.1.1" xref="S3.17.p1.23.m23.1.1.1.cmml"><</mo><mi id="S3.17.p1.23.m23.1.1.3" xref="S3.17.p1.23.m23.1.1.3.cmml">ΞΎ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.23.m23.1b"><apply id="S3.17.p1.23.m23.1.1.cmml" xref="S3.17.p1.23.m23.1.1"><lt id="S3.17.p1.23.m23.1.1.1.cmml" xref="S3.17.p1.23.m23.1.1.1"></lt><ci id="S3.17.p1.23.m23.1.1.2.cmml" xref="S3.17.p1.23.m23.1.1.2">πΏ</ci><ci id="S3.17.p1.23.m23.1.1.3.cmml" xref="S3.17.p1.23.m23.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.23.m23.1c">\delta<\xi</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.23.m23.1d">italic_Ξ΄ < italic_ΞΎ</annotation></semantics></math> the set</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\{y_{\alpha}:\delta<\alpha<{\xi+1}\}=\overline{L}\setminus({\downarrow}y_{% \delta}\cup{\uparrow}y_{\xi+1})" class="ltx_math_unparsed" display="block" id="S3.Ex11.m1.1"><semantics id="S3.Ex11.m1.1a"><mrow id="S3.Ex11.m1.1b"><mrow id="S3.Ex11.m1.1.1"><mo id="S3.Ex11.m1.1.1.1" stretchy="false">{</mo><msub id="S3.Ex11.m1.1.1.2"><mi id="S3.Ex11.m1.1.1.2.2">y</mi><mi id="S3.Ex11.m1.1.1.2.3">Ξ±</mi></msub><mo id="S3.Ex11.m1.1.1.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S3.Ex11.m1.1.1.4">Ξ΄</mi><mo id="S3.Ex11.m1.1.1.5"><</mo><mi id="S3.Ex11.m1.1.1.6">Ξ±</mi><mo id="S3.Ex11.m1.1.1.7"><</mo><mi id="S3.Ex11.m1.1.1.8">ΞΎ</mi><mo id="S3.Ex11.m1.1.1.9">+</mo><mn id="S3.Ex11.m1.1.1.10">1</mn><mo id="S3.Ex11.m1.1.1.11" stretchy="false">}</mo></mrow><mo id="S3.Ex11.m1.1.2">=</mo><mover accent="true" id="S3.Ex11.m1.1.3"><mi id="S3.Ex11.m1.1.3.2">L</mi><mo id="S3.Ex11.m1.1.3.1">Β―</mo></mover><mo id="S3.Ex11.m1.1.4">β</mo><mrow id="S3.Ex11.m1.1.5"><mo id="S3.Ex11.m1.1.5.1" stretchy="false">(</mo><mo id="S3.Ex11.m1.1.5.2" lspace="0em" stretchy="false">β</mo><msub id="S3.Ex11.m1.1.5.3"><mi id="S3.Ex11.m1.1.5.3.2">y</mi><mi id="S3.Ex11.m1.1.5.3.3">Ξ΄</mi></msub><mo id="S3.Ex11.m1.1.5.4" rspace="0em">βͺ</mo><mo id="S3.Ex11.m1.1.5.5" lspace="0em" stretchy="false">β</mo><msub id="S3.Ex11.m1.1.5.6"><mi id="S3.Ex11.m1.1.5.6.2">y</mi><mrow id="S3.Ex11.m1.1.5.6.3"><mi id="S3.Ex11.m1.1.5.6.3.2">ΞΎ</mi><mo id="S3.Ex11.m1.1.5.6.3.1">+</mo><mn id="S3.Ex11.m1.1.5.6.3.3">1</mn></mrow></msub><mo id="S3.Ex11.m1.1.5.7" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.Ex11.m1.1c">\{y_{\alpha}:\delta<\alpha<{\xi+1}\}=\overline{L}\setminus({\downarrow}y_{% \delta}\cup{\uparrow}y_{\xi+1})</annotation><annotation encoding="application/x-llamapun" id="S3.Ex11.m1.1d">{ italic_y start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT : italic_Ξ΄ < italic_Ξ± < italic_ΞΎ + 1 } = overΒ― start_ARG italic_L end_ARG β ( β italic_y start_POSTSUBSCRIPT italic_Ξ΄ end_POSTSUBSCRIPT βͺ β italic_y start_POSTSUBSCRIPT italic_ΞΎ + 1 end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.17.p1.38">is open in <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.17.p1.24.m1.1"><semantics id="S3.17.p1.24.m1.1a"><mover accent="true" id="S3.17.p1.24.m1.1.1" xref="S3.17.p1.24.m1.1.1.cmml"><mi id="S3.17.p1.24.m1.1.1.2" xref="S3.17.p1.24.m1.1.1.2.cmml">L</mi><mo id="S3.17.p1.24.m1.1.1.1" xref="S3.17.p1.24.m1.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.17.p1.24.m1.1b"><apply id="S3.17.p1.24.m1.1.1.cmml" xref="S3.17.p1.24.m1.1.1"><ci id="S3.17.p1.24.m1.1.1.1.cmml" xref="S3.17.p1.24.m1.1.1.1">Β―</ci><ci id="S3.17.p1.24.m1.1.1.2.cmml" xref="S3.17.p1.24.m1.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.24.m1.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.24.m1.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math>. Hence the order topology on <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.17.p1.25.m2.1"><semantics id="S3.17.p1.25.m2.1a"><mover accent="true" id="S3.17.p1.25.m2.1.1" xref="S3.17.p1.25.m2.1.1.cmml"><mi id="S3.17.p1.25.m2.1.1.2" xref="S3.17.p1.25.m2.1.1.2.cmml">L</mi><mo id="S3.17.p1.25.m2.1.1.1" xref="S3.17.p1.25.m2.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.17.p1.25.m2.1b"><apply id="S3.17.p1.25.m2.1.1.cmml" xref="S3.17.p1.25.m2.1.1"><ci id="S3.17.p1.25.m2.1.1.1.cmml" xref="S3.17.p1.25.m2.1.1.1">Β―</ci><ci id="S3.17.p1.25.m2.1.1.2.cmml" xref="S3.17.p1.25.m2.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.25.m2.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.25.m2.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> is contained in the original one. To derive a contradiction, assume that the order topology on <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S3.17.p1.26.m3.1"><semantics id="S3.17.p1.26.m3.1a"><mover accent="true" id="S3.17.p1.26.m3.1.1" xref="S3.17.p1.26.m3.1.1.cmml"><mi id="S3.17.p1.26.m3.1.1.2" xref="S3.17.p1.26.m3.1.1.2.cmml">L</mi><mo id="S3.17.p1.26.m3.1.1.1" xref="S3.17.p1.26.m3.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S3.17.p1.26.m3.1b"><apply id="S3.17.p1.26.m3.1.1.cmml" xref="S3.17.p1.26.m3.1.1"><ci id="S3.17.p1.26.m3.1.1.1.cmml" xref="S3.17.p1.26.m3.1.1.1">Β―</ci><ci id="S3.17.p1.26.m3.1.1.2.cmml" xref="S3.17.p1.26.m3.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.26.m3.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.26.m3.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> is strictly coarser than the original one. Then, taking into account that for each successor ordinal <math alttext="\theta<\omega_{1}" class="ltx_Math" display="inline" id="S3.17.p1.27.m4.1"><semantics id="S3.17.p1.27.m4.1a"><mrow id="S3.17.p1.27.m4.1.1" xref="S3.17.p1.27.m4.1.1.cmml"><mi id="S3.17.p1.27.m4.1.1.2" xref="S3.17.p1.27.m4.1.1.2.cmml">ΞΈ</mi><mo id="S3.17.p1.27.m4.1.1.1" xref="S3.17.p1.27.m4.1.1.1.cmml"><</mo><msub id="S3.17.p1.27.m4.1.1.3" xref="S3.17.p1.27.m4.1.1.3.cmml"><mi id="S3.17.p1.27.m4.1.1.3.2" xref="S3.17.p1.27.m4.1.1.3.2.cmml">Ο</mi><mn id="S3.17.p1.27.m4.1.1.3.3" xref="S3.17.p1.27.m4.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.27.m4.1b"><apply id="S3.17.p1.27.m4.1.1.cmml" xref="S3.17.p1.27.m4.1.1"><lt id="S3.17.p1.27.m4.1.1.1.cmml" xref="S3.17.p1.27.m4.1.1.1"></lt><ci id="S3.17.p1.27.m4.1.1.2.cmml" xref="S3.17.p1.27.m4.1.1.2">π</ci><apply id="S3.17.p1.27.m4.1.1.3.cmml" xref="S3.17.p1.27.m4.1.1.3"><csymbol cd="ambiguous" id="S3.17.p1.27.m4.1.1.3.1.cmml" xref="S3.17.p1.27.m4.1.1.3">subscript</csymbol><ci id="S3.17.p1.27.m4.1.1.3.2.cmml" xref="S3.17.p1.27.m4.1.1.3.2">π</ci><cn id="S3.17.p1.27.m4.1.1.3.3.cmml" type="integer" xref="S3.17.p1.27.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.27.m4.1c">\theta<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.27.m4.1d">italic_ΞΈ < italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> the point <math alttext="y_{\theta}" class="ltx_Math" display="inline" id="S3.17.p1.28.m5.1"><semantics id="S3.17.p1.28.m5.1a"><msub id="S3.17.p1.28.m5.1.1" xref="S3.17.p1.28.m5.1.1.cmml"><mi id="S3.17.p1.28.m5.1.1.2" xref="S3.17.p1.28.m5.1.1.2.cmml">y</mi><mi id="S3.17.p1.28.m5.1.1.3" xref="S3.17.p1.28.m5.1.1.3.cmml">ΞΈ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.17.p1.28.m5.1b"><apply id="S3.17.p1.28.m5.1.1.cmml" xref="S3.17.p1.28.m5.1.1"><csymbol cd="ambiguous" id="S3.17.p1.28.m5.1.1.1.cmml" xref="S3.17.p1.28.m5.1.1">subscript</csymbol><ci id="S3.17.p1.28.m5.1.1.2.cmml" xref="S3.17.p1.28.m5.1.1.2">π¦</ci><ci id="S3.17.p1.28.m5.1.1.3.cmml" xref="S3.17.p1.28.m5.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.28.m5.1c">y_{\theta}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.28.m5.1d">italic_y start_POSTSUBSCRIPT italic_ΞΈ end_POSTSUBSCRIPT</annotation></semantics></math> is isolated in both topologies, there exists a limit ordinal <math alttext="\xi<\omega_{1}" class="ltx_Math" display="inline" id="S3.17.p1.29.m6.1"><semantics id="S3.17.p1.29.m6.1a"><mrow id="S3.17.p1.29.m6.1.1" xref="S3.17.p1.29.m6.1.1.cmml"><mi id="S3.17.p1.29.m6.1.1.2" xref="S3.17.p1.29.m6.1.1.2.cmml">ΞΎ</mi><mo id="S3.17.p1.29.m6.1.1.1" xref="S3.17.p1.29.m6.1.1.1.cmml"><</mo><msub id="S3.17.p1.29.m6.1.1.3" xref="S3.17.p1.29.m6.1.1.3.cmml"><mi id="S3.17.p1.29.m6.1.1.3.2" xref="S3.17.p1.29.m6.1.1.3.2.cmml">Ο</mi><mn id="S3.17.p1.29.m6.1.1.3.3" xref="S3.17.p1.29.m6.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.29.m6.1b"><apply id="S3.17.p1.29.m6.1.1.cmml" xref="S3.17.p1.29.m6.1.1"><lt id="S3.17.p1.29.m6.1.1.1.cmml" xref="S3.17.p1.29.m6.1.1.1"></lt><ci id="S3.17.p1.29.m6.1.1.2.cmml" xref="S3.17.p1.29.m6.1.1.2">π</ci><apply id="S3.17.p1.29.m6.1.1.3.cmml" xref="S3.17.p1.29.m6.1.1.3"><csymbol cd="ambiguous" id="S3.17.p1.29.m6.1.1.3.1.cmml" xref="S3.17.p1.29.m6.1.1.3">subscript</csymbol><ci id="S3.17.p1.29.m6.1.1.3.2.cmml" xref="S3.17.p1.29.m6.1.1.3.2">π</ci><cn id="S3.17.p1.29.m6.1.1.3.3.cmml" type="integer" xref="S3.17.p1.29.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.29.m6.1c">\xi<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.29.m6.1d">italic_ΞΎ < italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and an open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S3.17.p1.30.m7.1"><semantics id="S3.17.p1.30.m7.1a"><mi id="S3.17.p1.30.m7.1.1" xref="S3.17.p1.30.m7.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S3.17.p1.30.m7.1b"><ci id="S3.17.p1.30.m7.1.1.cmml" xref="S3.17.p1.30.m7.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.30.m7.1c">U</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.30.m7.1d">italic_U</annotation></semantics></math> of <math alttext="y_{\xi}" class="ltx_Math" display="inline" id="S3.17.p1.31.m8.1"><semantics id="S3.17.p1.31.m8.1a"><msub id="S3.17.p1.31.m8.1.1" xref="S3.17.p1.31.m8.1.1.cmml"><mi id="S3.17.p1.31.m8.1.1.2" xref="S3.17.p1.31.m8.1.1.2.cmml">y</mi><mi id="S3.17.p1.31.m8.1.1.3" xref="S3.17.p1.31.m8.1.1.3.cmml">ΞΎ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.17.p1.31.m8.1b"><apply id="S3.17.p1.31.m8.1.1.cmml" xref="S3.17.p1.31.m8.1.1"><csymbol cd="ambiguous" id="S3.17.p1.31.m8.1.1.1.cmml" xref="S3.17.p1.31.m8.1.1">subscript</csymbol><ci id="S3.17.p1.31.m8.1.1.2.cmml" xref="S3.17.p1.31.m8.1.1.2">π¦</ci><ci id="S3.17.p1.31.m8.1.1.3.cmml" xref="S3.17.p1.31.m8.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.31.m8.1c">y_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.31.m8.1d">italic_y start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT</annotation></semantics></math> such that for any <math alttext="\delta<\xi" class="ltx_Math" display="inline" id="S3.17.p1.32.m9.1"><semantics id="S3.17.p1.32.m9.1a"><mrow id="S3.17.p1.32.m9.1.1" xref="S3.17.p1.32.m9.1.1.cmml"><mi id="S3.17.p1.32.m9.1.1.2" xref="S3.17.p1.32.m9.1.1.2.cmml">Ξ΄</mi><mo id="S3.17.p1.32.m9.1.1.1" xref="S3.17.p1.32.m9.1.1.1.cmml"><</mo><mi id="S3.17.p1.32.m9.1.1.3" xref="S3.17.p1.32.m9.1.1.3.cmml">ΞΎ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.32.m9.1b"><apply id="S3.17.p1.32.m9.1.1.cmml" xref="S3.17.p1.32.m9.1.1"><lt id="S3.17.p1.32.m9.1.1.1.cmml" xref="S3.17.p1.32.m9.1.1.1"></lt><ci id="S3.17.p1.32.m9.1.1.2.cmml" xref="S3.17.p1.32.m9.1.1.2">πΏ</ci><ci id="S3.17.p1.32.m9.1.1.3.cmml" xref="S3.17.p1.32.m9.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.32.m9.1c">\delta<\xi</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.32.m9.1d">italic_Ξ΄ < italic_ΞΎ</annotation></semantics></math> the set <math alttext="\{y_{\alpha}:\delta<\alpha<\xi+1\}\setminus U" class="ltx_Math" display="inline" id="S3.17.p1.33.m10.2"><semantics id="S3.17.p1.33.m10.2a"><mrow id="S3.17.p1.33.m10.2.2" xref="S3.17.p1.33.m10.2.2.cmml"><mrow id="S3.17.p1.33.m10.2.2.2.2" xref="S3.17.p1.33.m10.2.2.2.3.cmml"><mo id="S3.17.p1.33.m10.2.2.2.2.3" stretchy="false" xref="S3.17.p1.33.m10.2.2.2.3.1.cmml">{</mo><msub id="S3.17.p1.33.m10.1.1.1.1.1" xref="S3.17.p1.33.m10.1.1.1.1.1.cmml"><mi id="S3.17.p1.33.m10.1.1.1.1.1.2" xref="S3.17.p1.33.m10.1.1.1.1.1.2.cmml">y</mi><mi id="S3.17.p1.33.m10.1.1.1.1.1.3" xref="S3.17.p1.33.m10.1.1.1.1.1.3.cmml">Ξ±</mi></msub><mo id="S3.17.p1.33.m10.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.17.p1.33.m10.2.2.2.3.1.cmml">:</mo><mrow id="S3.17.p1.33.m10.2.2.2.2.2" xref="S3.17.p1.33.m10.2.2.2.2.2.cmml"><mi id="S3.17.p1.33.m10.2.2.2.2.2.2" xref="S3.17.p1.33.m10.2.2.2.2.2.2.cmml">Ξ΄</mi><mo id="S3.17.p1.33.m10.2.2.2.2.2.3" xref="S3.17.p1.33.m10.2.2.2.2.2.3.cmml"><</mo><mi id="S3.17.p1.33.m10.2.2.2.2.2.4" xref="S3.17.p1.33.m10.2.2.2.2.2.4.cmml">Ξ±</mi><mo id="S3.17.p1.33.m10.2.2.2.2.2.5" xref="S3.17.p1.33.m10.2.2.2.2.2.5.cmml"><</mo><mrow id="S3.17.p1.33.m10.2.2.2.2.2.6" xref="S3.17.p1.33.m10.2.2.2.2.2.6.cmml"><mi id="S3.17.p1.33.m10.2.2.2.2.2.6.2" xref="S3.17.p1.33.m10.2.2.2.2.2.6.2.cmml">ΞΎ</mi><mo id="S3.17.p1.33.m10.2.2.2.2.2.6.1" xref="S3.17.p1.33.m10.2.2.2.2.2.6.1.cmml">+</mo><mn id="S3.17.p1.33.m10.2.2.2.2.2.6.3" xref="S3.17.p1.33.m10.2.2.2.2.2.6.3.cmml">1</mn></mrow></mrow><mo id="S3.17.p1.33.m10.2.2.2.2.5" stretchy="false" xref="S3.17.p1.33.m10.2.2.2.3.1.cmml">}</mo></mrow><mo id="S3.17.p1.33.m10.2.2.3" xref="S3.17.p1.33.m10.2.2.3.cmml">β</mo><mi id="S3.17.p1.33.m10.2.2.4" xref="S3.17.p1.33.m10.2.2.4.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.33.m10.2b"><apply id="S3.17.p1.33.m10.2.2.cmml" xref="S3.17.p1.33.m10.2.2"><setdiff id="S3.17.p1.33.m10.2.2.3.cmml" xref="S3.17.p1.33.m10.2.2.3"></setdiff><apply id="S3.17.p1.33.m10.2.2.2.3.cmml" xref="S3.17.p1.33.m10.2.2.2.2"><csymbol cd="latexml" id="S3.17.p1.33.m10.2.2.2.3.1.cmml" xref="S3.17.p1.33.m10.2.2.2.2.3">conditional-set</csymbol><apply id="S3.17.p1.33.m10.1.1.1.1.1.cmml" xref="S3.17.p1.33.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.17.p1.33.m10.1.1.1.1.1.1.cmml" xref="S3.17.p1.33.m10.1.1.1.1.1">subscript</csymbol><ci id="S3.17.p1.33.m10.1.1.1.1.1.2.cmml" xref="S3.17.p1.33.m10.1.1.1.1.1.2">π¦</ci><ci id="S3.17.p1.33.m10.1.1.1.1.1.3.cmml" xref="S3.17.p1.33.m10.1.1.1.1.1.3">πΌ</ci></apply><apply id="S3.17.p1.33.m10.2.2.2.2.2.cmml" xref="S3.17.p1.33.m10.2.2.2.2.2"><and id="S3.17.p1.33.m10.2.2.2.2.2a.cmml" xref="S3.17.p1.33.m10.2.2.2.2.2"></and><apply id="S3.17.p1.33.m10.2.2.2.2.2b.cmml" xref="S3.17.p1.33.m10.2.2.2.2.2"><lt id="S3.17.p1.33.m10.2.2.2.2.2.3.cmml" xref="S3.17.p1.33.m10.2.2.2.2.2.3"></lt><ci id="S3.17.p1.33.m10.2.2.2.2.2.2.cmml" xref="S3.17.p1.33.m10.2.2.2.2.2.2">πΏ</ci><ci id="S3.17.p1.33.m10.2.2.2.2.2.4.cmml" xref="S3.17.p1.33.m10.2.2.2.2.2.4">πΌ</ci></apply><apply id="S3.17.p1.33.m10.2.2.2.2.2c.cmml" xref="S3.17.p1.33.m10.2.2.2.2.2"><lt id="S3.17.p1.33.m10.2.2.2.2.2.5.cmml" xref="S3.17.p1.33.m10.2.2.2.2.2.5"></lt><share href="https://arxiv.org/html/2503.13666v1#S3.17.p1.33.m10.2.2.2.2.2.4.cmml" id="S3.17.p1.33.m10.2.2.2.2.2d.cmml" xref="S3.17.p1.33.m10.2.2.2.2.2"></share><apply id="S3.17.p1.33.m10.2.2.2.2.2.6.cmml" xref="S3.17.p1.33.m10.2.2.2.2.2.6"><plus id="S3.17.p1.33.m10.2.2.2.2.2.6.1.cmml" xref="S3.17.p1.33.m10.2.2.2.2.2.6.1"></plus><ci id="S3.17.p1.33.m10.2.2.2.2.2.6.2.cmml" xref="S3.17.p1.33.m10.2.2.2.2.2.6.2">π</ci><cn id="S3.17.p1.33.m10.2.2.2.2.2.6.3.cmml" type="integer" xref="S3.17.p1.33.m10.2.2.2.2.2.6.3">1</cn></apply></apply></apply></apply><ci id="S3.17.p1.33.m10.2.2.4.cmml" xref="S3.17.p1.33.m10.2.2.4">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.33.m10.2c">\{y_{\alpha}:\delta<\alpha<\xi+1\}\setminus U</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.33.m10.2d">{ italic_y start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT : italic_Ξ΄ < italic_Ξ± < italic_ΞΎ + 1 } β italic_U</annotation></semantics></math> is not empty. Since <math alttext="\xi" class="ltx_Math" display="inline" id="S3.17.p1.34.m11.1"><semantics id="S3.17.p1.34.m11.1a"><mi id="S3.17.p1.34.m11.1.1" xref="S3.17.p1.34.m11.1.1.cmml">ΞΎ</mi><annotation-xml encoding="MathML-Content" id="S3.17.p1.34.m11.1b"><ci id="S3.17.p1.34.m11.1.1.cmml" xref="S3.17.p1.34.m11.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.34.m11.1c">\xi</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.34.m11.1d">italic_ΞΎ</annotation></semantics></math> is countable, there exists a cofinal in <math alttext="{\downarrow}y_{\xi}\setminus\{y_{\xi}\}" class="ltx_Math" display="inline" id="S3.17.p1.35.m12.1"><semantics id="S3.17.p1.35.m12.1a"><mrow id="S3.17.p1.35.m12.1.1" xref="S3.17.p1.35.m12.1.1.cmml"><mi id="S3.17.p1.35.m12.1.1.3" xref="S3.17.p1.35.m12.1.1.3.cmml"></mi><mo id="S3.17.p1.35.m12.1.1.2" stretchy="false" xref="S3.17.p1.35.m12.1.1.2.cmml">β</mo><mrow id="S3.17.p1.35.m12.1.1.1" xref="S3.17.p1.35.m12.1.1.1.cmml"><msub id="S3.17.p1.35.m12.1.1.1.3" xref="S3.17.p1.35.m12.1.1.1.3.cmml"><mi id="S3.17.p1.35.m12.1.1.1.3.2" xref="S3.17.p1.35.m12.1.1.1.3.2.cmml">y</mi><mi id="S3.17.p1.35.m12.1.1.1.3.3" xref="S3.17.p1.35.m12.1.1.1.3.3.cmml">ΞΎ</mi></msub><mo id="S3.17.p1.35.m12.1.1.1.2" xref="S3.17.p1.35.m12.1.1.1.2.cmml">β</mo><mrow id="S3.17.p1.35.m12.1.1.1.1.1" xref="S3.17.p1.35.m12.1.1.1.1.2.cmml"><mo id="S3.17.p1.35.m12.1.1.1.1.1.2" stretchy="false" xref="S3.17.p1.35.m12.1.1.1.1.2.cmml">{</mo><msub id="S3.17.p1.35.m12.1.1.1.1.1.1" xref="S3.17.p1.35.m12.1.1.1.1.1.1.cmml"><mi id="S3.17.p1.35.m12.1.1.1.1.1.1.2" xref="S3.17.p1.35.m12.1.1.1.1.1.1.2.cmml">y</mi><mi id="S3.17.p1.35.m12.1.1.1.1.1.1.3" xref="S3.17.p1.35.m12.1.1.1.1.1.1.3.cmml">ΞΎ</mi></msub><mo id="S3.17.p1.35.m12.1.1.1.1.1.3" stretchy="false" xref="S3.17.p1.35.m12.1.1.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.35.m12.1b"><apply id="S3.17.p1.35.m12.1.1.cmml" xref="S3.17.p1.35.m12.1.1"><ci id="S3.17.p1.35.m12.1.1.2.cmml" xref="S3.17.p1.35.m12.1.1.2">β</ci><csymbol cd="latexml" id="S3.17.p1.35.m12.1.1.3.cmml" xref="S3.17.p1.35.m12.1.1.3">absent</csymbol><apply id="S3.17.p1.35.m12.1.1.1.cmml" xref="S3.17.p1.35.m12.1.1.1"><setdiff id="S3.17.p1.35.m12.1.1.1.2.cmml" xref="S3.17.p1.35.m12.1.1.1.2"></setdiff><apply id="S3.17.p1.35.m12.1.1.1.3.cmml" xref="S3.17.p1.35.m12.1.1.1.3"><csymbol cd="ambiguous" id="S3.17.p1.35.m12.1.1.1.3.1.cmml" xref="S3.17.p1.35.m12.1.1.1.3">subscript</csymbol><ci id="S3.17.p1.35.m12.1.1.1.3.2.cmml" xref="S3.17.p1.35.m12.1.1.1.3.2">π¦</ci><ci id="S3.17.p1.35.m12.1.1.1.3.3.cmml" xref="S3.17.p1.35.m12.1.1.1.3.3">π</ci></apply><set id="S3.17.p1.35.m12.1.1.1.1.2.cmml" xref="S3.17.p1.35.m12.1.1.1.1.1"><apply id="S3.17.p1.35.m12.1.1.1.1.1.1.cmml" xref="S3.17.p1.35.m12.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.17.p1.35.m12.1.1.1.1.1.1.1.cmml" xref="S3.17.p1.35.m12.1.1.1.1.1.1">subscript</csymbol><ci id="S3.17.p1.35.m12.1.1.1.1.1.1.2.cmml" xref="S3.17.p1.35.m12.1.1.1.1.1.1.2">π¦</ci><ci id="S3.17.p1.35.m12.1.1.1.1.1.1.3.cmml" xref="S3.17.p1.35.m12.1.1.1.1.1.1.3">π</ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.35.m12.1c">{\downarrow}y_{\xi}\setminus\{y_{\xi}\}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.35.m12.1d">β italic_y start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT β { italic_y start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT }</annotation></semantics></math> subset <math alttext="\{\pi_{n}:n\in\omega\}\subset\overline{L}\setminus U" class="ltx_Math" display="inline" id="S3.17.p1.36.m13.2"><semantics id="S3.17.p1.36.m13.2a"><mrow id="S3.17.p1.36.m13.2.2" xref="S3.17.p1.36.m13.2.2.cmml"><mrow id="S3.17.p1.36.m13.2.2.2.2" xref="S3.17.p1.36.m13.2.2.2.3.cmml"><mo id="S3.17.p1.36.m13.2.2.2.2.3" stretchy="false" xref="S3.17.p1.36.m13.2.2.2.3.1.cmml">{</mo><msub id="S3.17.p1.36.m13.1.1.1.1.1" xref="S3.17.p1.36.m13.1.1.1.1.1.cmml"><mi id="S3.17.p1.36.m13.1.1.1.1.1.2" xref="S3.17.p1.36.m13.1.1.1.1.1.2.cmml">Ο</mi><mi id="S3.17.p1.36.m13.1.1.1.1.1.3" xref="S3.17.p1.36.m13.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S3.17.p1.36.m13.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.17.p1.36.m13.2.2.2.3.1.cmml">:</mo><mrow id="S3.17.p1.36.m13.2.2.2.2.2" xref="S3.17.p1.36.m13.2.2.2.2.2.cmml"><mi id="S3.17.p1.36.m13.2.2.2.2.2.2" xref="S3.17.p1.36.m13.2.2.2.2.2.2.cmml">n</mi><mo id="S3.17.p1.36.m13.2.2.2.2.2.1" xref="S3.17.p1.36.m13.2.2.2.2.2.1.cmml">β</mo><mi id="S3.17.p1.36.m13.2.2.2.2.2.3" xref="S3.17.p1.36.m13.2.2.2.2.2.3.cmml">Ο</mi></mrow><mo id="S3.17.p1.36.m13.2.2.2.2.5" stretchy="false" xref="S3.17.p1.36.m13.2.2.2.3.1.cmml">}</mo></mrow><mo id="S3.17.p1.36.m13.2.2.3" xref="S3.17.p1.36.m13.2.2.3.cmml">β</mo><mrow id="S3.17.p1.36.m13.2.2.4" xref="S3.17.p1.36.m13.2.2.4.cmml"><mover accent="true" id="S3.17.p1.36.m13.2.2.4.2" xref="S3.17.p1.36.m13.2.2.4.2.cmml"><mi id="S3.17.p1.36.m13.2.2.4.2.2" xref="S3.17.p1.36.m13.2.2.4.2.2.cmml">L</mi><mo id="S3.17.p1.36.m13.2.2.4.2.1" xref="S3.17.p1.36.m13.2.2.4.2.1.cmml">Β―</mo></mover><mo id="S3.17.p1.36.m13.2.2.4.1" xref="S3.17.p1.36.m13.2.2.4.1.cmml">β</mo><mi id="S3.17.p1.36.m13.2.2.4.3" xref="S3.17.p1.36.m13.2.2.4.3.cmml">U</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.36.m13.2b"><apply id="S3.17.p1.36.m13.2.2.cmml" xref="S3.17.p1.36.m13.2.2"><subset id="S3.17.p1.36.m13.2.2.3.cmml" xref="S3.17.p1.36.m13.2.2.3"></subset><apply id="S3.17.p1.36.m13.2.2.2.3.cmml" xref="S3.17.p1.36.m13.2.2.2.2"><csymbol cd="latexml" id="S3.17.p1.36.m13.2.2.2.3.1.cmml" xref="S3.17.p1.36.m13.2.2.2.2.3">conditional-set</csymbol><apply id="S3.17.p1.36.m13.1.1.1.1.1.cmml" xref="S3.17.p1.36.m13.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.17.p1.36.m13.1.1.1.1.1.1.cmml" xref="S3.17.p1.36.m13.1.1.1.1.1">subscript</csymbol><ci id="S3.17.p1.36.m13.1.1.1.1.1.2.cmml" xref="S3.17.p1.36.m13.1.1.1.1.1.2">π</ci><ci id="S3.17.p1.36.m13.1.1.1.1.1.3.cmml" xref="S3.17.p1.36.m13.1.1.1.1.1.3">π</ci></apply><apply id="S3.17.p1.36.m13.2.2.2.2.2.cmml" xref="S3.17.p1.36.m13.2.2.2.2.2"><in id="S3.17.p1.36.m13.2.2.2.2.2.1.cmml" xref="S3.17.p1.36.m13.2.2.2.2.2.1"></in><ci id="S3.17.p1.36.m13.2.2.2.2.2.2.cmml" xref="S3.17.p1.36.m13.2.2.2.2.2.2">π</ci><ci id="S3.17.p1.36.m13.2.2.2.2.2.3.cmml" xref="S3.17.p1.36.m13.2.2.2.2.2.3">π</ci></apply></apply><apply id="S3.17.p1.36.m13.2.2.4.cmml" xref="S3.17.p1.36.m13.2.2.4"><setdiff id="S3.17.p1.36.m13.2.2.4.1.cmml" xref="S3.17.p1.36.m13.2.2.4.1"></setdiff><apply id="S3.17.p1.36.m13.2.2.4.2.cmml" xref="S3.17.p1.36.m13.2.2.4.2"><ci id="S3.17.p1.36.m13.2.2.4.2.1.cmml" xref="S3.17.p1.36.m13.2.2.4.2.1">Β―</ci><ci id="S3.17.p1.36.m13.2.2.4.2.2.cmml" xref="S3.17.p1.36.m13.2.2.4.2.2">πΏ</ci></apply><ci id="S3.17.p1.36.m13.2.2.4.3.cmml" xref="S3.17.p1.36.m13.2.2.4.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.36.m13.2c">\{\pi_{n}:n\in\omega\}\subset\overline{L}\setminus U</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.36.m13.2d">{ italic_Ο start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n β italic_Ο } β overΒ― start_ARG italic_L end_ARG β italic_U</annotation></semantics></math>. But this contradicts Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem8" title="Lemma 3.8. β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.8</span></a>, as <math alttext="y_{\xi}=\sup\{\pi_{n}:n\in\omega\}" class="ltx_Math" display="inline" id="S3.17.p1.37.m14.2"><semantics id="S3.17.p1.37.m14.2a"><mrow id="S3.17.p1.37.m14.2.2" xref="S3.17.p1.37.m14.2.2.cmml"><msub id="S3.17.p1.37.m14.2.2.4" xref="S3.17.p1.37.m14.2.2.4.cmml"><mi id="S3.17.p1.37.m14.2.2.4.2" xref="S3.17.p1.37.m14.2.2.4.2.cmml">y</mi><mi id="S3.17.p1.37.m14.2.2.4.3" xref="S3.17.p1.37.m14.2.2.4.3.cmml">ΞΎ</mi></msub><mo id="S3.17.p1.37.m14.2.2.3" rspace="0.1389em" xref="S3.17.p1.37.m14.2.2.3.cmml">=</mo><mrow id="S3.17.p1.37.m14.2.2.2" xref="S3.17.p1.37.m14.2.2.2.cmml"><mo id="S3.17.p1.37.m14.2.2.2.3" lspace="0.1389em" rspace="0em" xref="S3.17.p1.37.m14.2.2.2.3.cmml">sup</mo><mrow id="S3.17.p1.37.m14.2.2.2.2.2" xref="S3.17.p1.37.m14.2.2.2.2.3.cmml"><mo id="S3.17.p1.37.m14.2.2.2.2.2.3" stretchy="false" xref="S3.17.p1.37.m14.2.2.2.2.3.1.cmml">{</mo><msub id="S3.17.p1.37.m14.1.1.1.1.1.1" xref="S3.17.p1.37.m14.1.1.1.1.1.1.cmml"><mi id="S3.17.p1.37.m14.1.1.1.1.1.1.2" xref="S3.17.p1.37.m14.1.1.1.1.1.1.2.cmml">Ο</mi><mi id="S3.17.p1.37.m14.1.1.1.1.1.1.3" xref="S3.17.p1.37.m14.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S3.17.p1.37.m14.2.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.17.p1.37.m14.2.2.2.2.3.1.cmml">:</mo><mrow id="S3.17.p1.37.m14.2.2.2.2.2.2" xref="S3.17.p1.37.m14.2.2.2.2.2.2.cmml"><mi id="S3.17.p1.37.m14.2.2.2.2.2.2.2" xref="S3.17.p1.37.m14.2.2.2.2.2.2.2.cmml">n</mi><mo id="S3.17.p1.37.m14.2.2.2.2.2.2.1" xref="S3.17.p1.37.m14.2.2.2.2.2.2.1.cmml">β</mo><mi id="S3.17.p1.37.m14.2.2.2.2.2.2.3" xref="S3.17.p1.37.m14.2.2.2.2.2.2.3.cmml">Ο</mi></mrow><mo id="S3.17.p1.37.m14.2.2.2.2.2.5" stretchy="false" xref="S3.17.p1.37.m14.2.2.2.2.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.37.m14.2b"><apply id="S3.17.p1.37.m14.2.2.cmml" xref="S3.17.p1.37.m14.2.2"><eq id="S3.17.p1.37.m14.2.2.3.cmml" xref="S3.17.p1.37.m14.2.2.3"></eq><apply id="S3.17.p1.37.m14.2.2.4.cmml" xref="S3.17.p1.37.m14.2.2.4"><csymbol cd="ambiguous" id="S3.17.p1.37.m14.2.2.4.1.cmml" xref="S3.17.p1.37.m14.2.2.4">subscript</csymbol><ci id="S3.17.p1.37.m14.2.2.4.2.cmml" xref="S3.17.p1.37.m14.2.2.4.2">π¦</ci><ci id="S3.17.p1.37.m14.2.2.4.3.cmml" xref="S3.17.p1.37.m14.2.2.4.3">π</ci></apply><apply id="S3.17.p1.37.m14.2.2.2.cmml" xref="S3.17.p1.37.m14.2.2.2"><csymbol cd="latexml" id="S3.17.p1.37.m14.2.2.2.3.cmml" xref="S3.17.p1.37.m14.2.2.2.3">supremum</csymbol><apply id="S3.17.p1.37.m14.2.2.2.2.3.cmml" xref="S3.17.p1.37.m14.2.2.2.2.2"><csymbol cd="latexml" id="S3.17.p1.37.m14.2.2.2.2.3.1.cmml" xref="S3.17.p1.37.m14.2.2.2.2.2.3">conditional-set</csymbol><apply id="S3.17.p1.37.m14.1.1.1.1.1.1.cmml" xref="S3.17.p1.37.m14.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.17.p1.37.m14.1.1.1.1.1.1.1.cmml" xref="S3.17.p1.37.m14.1.1.1.1.1.1">subscript</csymbol><ci id="S3.17.p1.37.m14.1.1.1.1.1.1.2.cmml" xref="S3.17.p1.37.m14.1.1.1.1.1.1.2">π</ci><ci id="S3.17.p1.37.m14.1.1.1.1.1.1.3.cmml" xref="S3.17.p1.37.m14.1.1.1.1.1.1.3">π</ci></apply><apply id="S3.17.p1.37.m14.2.2.2.2.2.2.cmml" xref="S3.17.p1.37.m14.2.2.2.2.2.2"><in id="S3.17.p1.37.m14.2.2.2.2.2.2.1.cmml" xref="S3.17.p1.37.m14.2.2.2.2.2.2.1"></in><ci id="S3.17.p1.37.m14.2.2.2.2.2.2.2.cmml" xref="S3.17.p1.37.m14.2.2.2.2.2.2.2">π</ci><ci id="S3.17.p1.37.m14.2.2.2.2.2.2.3.cmml" xref="S3.17.p1.37.m14.2.2.2.2.2.2.3">π</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.37.m14.2c">y_{\xi}=\sup\{\pi_{n}:n\in\omega\}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.37.m14.2d">italic_y start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT = roman_sup { italic_Ο start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n β italic_Ο }</annotation></semantics></math>, but <math alttext="y_{\xi}\notin\overline{\{\pi_{n}:n\in\omega\}}" class="ltx_Math" display="inline" id="S3.17.p1.38.m15.2"><semantics id="S3.17.p1.38.m15.2a"><mrow id="S3.17.p1.38.m15.2.3" xref="S3.17.p1.38.m15.2.3.cmml"><msub id="S3.17.p1.38.m15.2.3.2" xref="S3.17.p1.38.m15.2.3.2.cmml"><mi id="S3.17.p1.38.m15.2.3.2.2" xref="S3.17.p1.38.m15.2.3.2.2.cmml">y</mi><mi id="S3.17.p1.38.m15.2.3.2.3" xref="S3.17.p1.38.m15.2.3.2.3.cmml">ΞΎ</mi></msub><mo id="S3.17.p1.38.m15.2.3.1" xref="S3.17.p1.38.m15.2.3.1.cmml">β</mo><mover accent="true" id="S3.17.p1.38.m15.2.2" xref="S3.17.p1.38.m15.2.2.cmml"><mrow id="S3.17.p1.38.m15.2.2.2.2" xref="S3.17.p1.38.m15.2.2.2.3.cmml"><mo id="S3.17.p1.38.m15.2.2.2.2.3" stretchy="false" xref="S3.17.p1.38.m15.2.2.2.3.1.cmml">{</mo><msub id="S3.17.p1.38.m15.1.1.1.1.1" xref="S3.17.p1.38.m15.1.1.1.1.1.cmml"><mi id="S3.17.p1.38.m15.1.1.1.1.1.2" xref="S3.17.p1.38.m15.1.1.1.1.1.2.cmml">Ο</mi><mi id="S3.17.p1.38.m15.1.1.1.1.1.3" xref="S3.17.p1.38.m15.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S3.17.p1.38.m15.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.17.p1.38.m15.2.2.2.3.1.cmml">:</mo><mrow id="S3.17.p1.38.m15.2.2.2.2.2" xref="S3.17.p1.38.m15.2.2.2.2.2.cmml"><mi id="S3.17.p1.38.m15.2.2.2.2.2.2" xref="S3.17.p1.38.m15.2.2.2.2.2.2.cmml">n</mi><mo id="S3.17.p1.38.m15.2.2.2.2.2.1" xref="S3.17.p1.38.m15.2.2.2.2.2.1.cmml">β</mo><mi id="S3.17.p1.38.m15.2.2.2.2.2.3" xref="S3.17.p1.38.m15.2.2.2.2.2.3.cmml">Ο</mi></mrow><mo id="S3.17.p1.38.m15.2.2.2.2.5" stretchy="false" xref="S3.17.p1.38.m15.2.2.2.3.1.cmml">}</mo></mrow><mo id="S3.17.p1.38.m15.2.2.3" xref="S3.17.p1.38.m15.2.2.3.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.17.p1.38.m15.2b"><apply id="S3.17.p1.38.m15.2.3.cmml" xref="S3.17.p1.38.m15.2.3"><notin id="S3.17.p1.38.m15.2.3.1.cmml" xref="S3.17.p1.38.m15.2.3.1"></notin><apply id="S3.17.p1.38.m15.2.3.2.cmml" xref="S3.17.p1.38.m15.2.3.2"><csymbol cd="ambiguous" id="S3.17.p1.38.m15.2.3.2.1.cmml" xref="S3.17.p1.38.m15.2.3.2">subscript</csymbol><ci id="S3.17.p1.38.m15.2.3.2.2.cmml" xref="S3.17.p1.38.m15.2.3.2.2">π¦</ci><ci id="S3.17.p1.38.m15.2.3.2.3.cmml" xref="S3.17.p1.38.m15.2.3.2.3">π</ci></apply><apply id="S3.17.p1.38.m15.2.2.cmml" xref="S3.17.p1.38.m15.2.2"><ci id="S3.17.p1.38.m15.2.2.3.cmml" xref="S3.17.p1.38.m15.2.2.3">Β―</ci><apply id="S3.17.p1.38.m15.2.2.2.3.cmml" xref="S3.17.p1.38.m15.2.2.2.2"><csymbol cd="latexml" id="S3.17.p1.38.m15.2.2.2.3.1.cmml" xref="S3.17.p1.38.m15.2.2.2.2.3">conditional-set</csymbol><apply id="S3.17.p1.38.m15.1.1.1.1.1.cmml" xref="S3.17.p1.38.m15.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.17.p1.38.m15.1.1.1.1.1.1.cmml" xref="S3.17.p1.38.m15.1.1.1.1.1">subscript</csymbol><ci id="S3.17.p1.38.m15.1.1.1.1.1.2.cmml" xref="S3.17.p1.38.m15.1.1.1.1.1.2">π</ci><ci id="S3.17.p1.38.m15.1.1.1.1.1.3.cmml" xref="S3.17.p1.38.m15.1.1.1.1.1.3">π</ci></apply><apply id="S3.17.p1.38.m15.2.2.2.2.2.cmml" xref="S3.17.p1.38.m15.2.2.2.2.2"><in id="S3.17.p1.38.m15.2.2.2.2.2.1.cmml" xref="S3.17.p1.38.m15.2.2.2.2.2.1"></in><ci id="S3.17.p1.38.m15.2.2.2.2.2.2.cmml" xref="S3.17.p1.38.m15.2.2.2.2.2.2">π</ci><ci id="S3.17.p1.38.m15.2.2.2.2.2.3.cmml" xref="S3.17.p1.38.m15.2.2.2.2.2.3">π</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.17.p1.38.m15.2c">y_{\xi}\notin\overline{\{\pi_{n}:n\in\omega\}}</annotation><annotation encoding="application/x-llamapun" id="S3.17.p1.38.m15.2d">italic_y start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT β overΒ― start_ARG { italic_Ο start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n β italic_Ο } end_ARG</annotation></semantics></math>. β</p> </div> </div> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4. </span>Nyikos semilattices</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">The aim of this section is to show that each locally compact Nyikos topological semilattice is compact (see Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S4.Thmtheorem12" title="Theorem 4.12. β£ 4. Nyikos semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">4.12</span></a>).</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem1.1.1.1">Lemma 4.1</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem1.p1"> <p class="ltx_p" id="S4.Thmtheorem1.p1.1"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem1.p1.1.1">Each separable countably compact semitopological semilattice possesses the minimum.</span></p> </div> </div> <div class="ltx_proof" id="S4.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.1.p1"> <p class="ltx_p" id="S4.1.p1.11">Let <math alttext="D" class="ltx_Math" display="inline" id="S4.1.p1.1.m1.1"><semantics id="S4.1.p1.1.m1.1a"><mi id="S4.1.p1.1.m1.1.1" xref="S4.1.p1.1.m1.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S4.1.p1.1.m1.1b"><ci id="S4.1.p1.1.m1.1.1.cmml" xref="S4.1.p1.1.m1.1.1">π·</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.1.m1.1c">D</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.1.m1.1d">italic_D</annotation></semantics></math> be a countable dense subset of a countably compact semitopological semilattice <math alttext="X" class="ltx_Math" display="inline" id="S4.1.p1.2.m2.1"><semantics id="S4.1.p1.2.m2.1a"><mi id="S4.1.p1.2.m2.1.1" xref="S4.1.p1.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.1.p1.2.m2.1b"><ci id="S4.1.p1.2.m2.1.1.cmml" xref="S4.1.p1.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.2.m2.1d">italic_X</annotation></semantics></math>. Recall that for each <math alttext="x\in X" class="ltx_Math" display="inline" id="S4.1.p1.3.m3.1"><semantics id="S4.1.p1.3.m3.1a"><mrow id="S4.1.p1.3.m3.1.1" xref="S4.1.p1.3.m3.1.1.cmml"><mi id="S4.1.p1.3.m3.1.1.2" xref="S4.1.p1.3.m3.1.1.2.cmml">x</mi><mo id="S4.1.p1.3.m3.1.1.1" xref="S4.1.p1.3.m3.1.1.1.cmml">β</mo><mi id="S4.1.p1.3.m3.1.1.3" xref="S4.1.p1.3.m3.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.3.m3.1b"><apply id="S4.1.p1.3.m3.1.1.cmml" xref="S4.1.p1.3.m3.1.1"><in id="S4.1.p1.3.m3.1.1.1.cmml" xref="S4.1.p1.3.m3.1.1.1"></in><ci id="S4.1.p1.3.m3.1.1.2.cmml" xref="S4.1.p1.3.m3.1.1.2">π₯</ci><ci id="S4.1.p1.3.m3.1.1.3.cmml" xref="S4.1.p1.3.m3.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.3.m3.1c">x\in X</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.3.m3.1d">italic_x β italic_X</annotation></semantics></math> the set <math alttext="{\downarrow}x" class="ltx_Math" display="inline" id="S4.1.p1.4.m4.1"><semantics id="S4.1.p1.4.m4.1a"><mrow id="S4.1.p1.4.m4.1.1" xref="S4.1.p1.4.m4.1.1.cmml"><mi id="S4.1.p1.4.m4.1.1.2" xref="S4.1.p1.4.m4.1.1.2.cmml"></mi><mo id="S4.1.p1.4.m4.1.1.1" stretchy="false" xref="S4.1.p1.4.m4.1.1.1.cmml">β</mo><mi id="S4.1.p1.4.m4.1.1.3" xref="S4.1.p1.4.m4.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.4.m4.1b"><apply id="S4.1.p1.4.m4.1.1.cmml" xref="S4.1.p1.4.m4.1.1"><ci id="S4.1.p1.4.m4.1.1.1.cmml" xref="S4.1.p1.4.m4.1.1.1">β</ci><csymbol cd="latexml" id="S4.1.p1.4.m4.1.1.2.cmml" xref="S4.1.p1.4.m4.1.1.2">absent</csymbol><ci id="S4.1.p1.4.m4.1.1.3.cmml" xref="S4.1.p1.4.m4.1.1.3">π₯</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.4.m4.1c">{\downarrow}x</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.4.m4.1d">β italic_x</annotation></semantics></math> is closed. Since <math alttext="X" class="ltx_Math" display="inline" id="S4.1.p1.5.m5.1"><semantics id="S4.1.p1.5.m5.1a"><mi id="S4.1.p1.5.m5.1.1" xref="S4.1.p1.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.1.p1.5.m5.1b"><ci id="S4.1.p1.5.m5.1.1.cmml" xref="S4.1.p1.5.m5.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.5.m5.1d">italic_X</annotation></semantics></math> is countably compact and the family <math alttext="\{{\downarrow}x:x\in X\}" class="ltx_math_unparsed" display="inline" id="S4.1.p1.6.m6.1"><semantics id="S4.1.p1.6.m6.1a"><mrow id="S4.1.p1.6.m6.1b"><mo id="S4.1.p1.6.m6.1.1" stretchy="false">{</mo><mo id="S4.1.p1.6.m6.1.2" lspace="0em" stretchy="false">β</mo><mi id="S4.1.p1.6.m6.1.3">x</mi><mo id="S4.1.p1.6.m6.1.4" lspace="0.278em" rspace="0.278em">:</mo><mi id="S4.1.p1.6.m6.1.5">x</mi><mo id="S4.1.p1.6.m6.1.6">β</mo><mi id="S4.1.p1.6.m6.1.7">X</mi><mo id="S4.1.p1.6.m6.1.8" stretchy="false">}</mo></mrow><annotation encoding="application/x-tex" id="S4.1.p1.6.m6.1c">\{{\downarrow}x:x\in X\}</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.6.m6.1d">{ β italic_x : italic_x β italic_X }</annotation></semantics></math> is centered, <math alttext="Z=\bigcap_{d\in D}{\downarrow}d\neq\varnothing" class="ltx_Math" display="inline" id="S4.1.p1.7.m7.1"><semantics id="S4.1.p1.7.m7.1a"><mrow id="S4.1.p1.7.m7.1.1" xref="S4.1.p1.7.m7.1.1.cmml"><mi id="S4.1.p1.7.m7.1.1.2" xref="S4.1.p1.7.m7.1.1.2.cmml">Z</mi><mo id="S4.1.p1.7.m7.1.1.3" rspace="0.111em" xref="S4.1.p1.7.m7.1.1.3.cmml">=</mo><msub id="S4.1.p1.7.m7.1.1.4" xref="S4.1.p1.7.m7.1.1.4.cmml"><mo id="S4.1.p1.7.m7.1.1.4.2" xref="S4.1.p1.7.m7.1.1.4.2.cmml">β</mo><mrow id="S4.1.p1.7.m7.1.1.4.3" xref="S4.1.p1.7.m7.1.1.4.3.cmml"><mi id="S4.1.p1.7.m7.1.1.4.3.2" xref="S4.1.p1.7.m7.1.1.4.3.2.cmml">d</mi><mo id="S4.1.p1.7.m7.1.1.4.3.1" xref="S4.1.p1.7.m7.1.1.4.3.1.cmml">β</mo><mi id="S4.1.p1.7.m7.1.1.4.3.3" xref="S4.1.p1.7.m7.1.1.4.3.3.cmml">D</mi></mrow></msub><mo id="S4.1.p1.7.m7.1.1.5" lspace="0.111em" stretchy="false" xref="S4.1.p1.7.m7.1.1.5.cmml">β</mo><mi id="S4.1.p1.7.m7.1.1.6" xref="S4.1.p1.7.m7.1.1.6.cmml">d</mi><mo id="S4.1.p1.7.m7.1.1.7" xref="S4.1.p1.7.m7.1.1.7.cmml">β </mo><mi id="S4.1.p1.7.m7.1.1.8" mathvariant="normal" xref="S4.1.p1.7.m7.1.1.8.cmml">β </mi></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.7.m7.1b"><apply id="S4.1.p1.7.m7.1.1.cmml" xref="S4.1.p1.7.m7.1.1"><and id="S4.1.p1.7.m7.1.1a.cmml" xref="S4.1.p1.7.m7.1.1"></and><apply id="S4.1.p1.7.m7.1.1b.cmml" xref="S4.1.p1.7.m7.1.1"><eq id="S4.1.p1.7.m7.1.1.3.cmml" xref="S4.1.p1.7.m7.1.1.3"></eq><ci id="S4.1.p1.7.m7.1.1.2.cmml" xref="S4.1.p1.7.m7.1.1.2">π</ci><apply id="S4.1.p1.7.m7.1.1.4.cmml" xref="S4.1.p1.7.m7.1.1.4"><csymbol cd="ambiguous" id="S4.1.p1.7.m7.1.1.4.1.cmml" xref="S4.1.p1.7.m7.1.1.4">subscript</csymbol><intersect id="S4.1.p1.7.m7.1.1.4.2.cmml" xref="S4.1.p1.7.m7.1.1.4.2"></intersect><apply id="S4.1.p1.7.m7.1.1.4.3.cmml" xref="S4.1.p1.7.m7.1.1.4.3"><in id="S4.1.p1.7.m7.1.1.4.3.1.cmml" xref="S4.1.p1.7.m7.1.1.4.3.1"></in><ci id="S4.1.p1.7.m7.1.1.4.3.2.cmml" xref="S4.1.p1.7.m7.1.1.4.3.2">π</ci><ci id="S4.1.p1.7.m7.1.1.4.3.3.cmml" xref="S4.1.p1.7.m7.1.1.4.3.3">π·</ci></apply></apply></apply><apply id="S4.1.p1.7.m7.1.1c.cmml" xref="S4.1.p1.7.m7.1.1"><ci id="S4.1.p1.7.m7.1.1.5.cmml" xref="S4.1.p1.7.m7.1.1.5">β</ci><share href="https://arxiv.org/html/2503.13666v1#S4.1.p1.7.m7.1.1.4.cmml" id="S4.1.p1.7.m7.1.1d.cmml" xref="S4.1.p1.7.m7.1.1"></share><ci id="S4.1.p1.7.m7.1.1.6.cmml" xref="S4.1.p1.7.m7.1.1.6">π</ci></apply><apply id="S4.1.p1.7.m7.1.1e.cmml" xref="S4.1.p1.7.m7.1.1"><neq id="S4.1.p1.7.m7.1.1.7.cmml" xref="S4.1.p1.7.m7.1.1.7"></neq><share href="https://arxiv.org/html/2503.13666v1#S4.1.p1.7.m7.1.1.6.cmml" id="S4.1.p1.7.m7.1.1f.cmml" xref="S4.1.p1.7.m7.1.1"></share><emptyset id="S4.1.p1.7.m7.1.1.8.cmml" xref="S4.1.p1.7.m7.1.1.8"></emptyset></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.7.m7.1c">Z=\bigcap_{d\in D}{\downarrow}d\neq\varnothing</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.7.m7.1d">italic_Z = β start_POSTSUBSCRIPT italic_d β italic_D end_POSTSUBSCRIPT β italic_d β β </annotation></semantics></math>. Pick any <math alttext="z\in Z" class="ltx_Math" display="inline" id="S4.1.p1.8.m8.1"><semantics id="S4.1.p1.8.m8.1a"><mrow id="S4.1.p1.8.m8.1.1" xref="S4.1.p1.8.m8.1.1.cmml"><mi id="S4.1.p1.8.m8.1.1.2" xref="S4.1.p1.8.m8.1.1.2.cmml">z</mi><mo id="S4.1.p1.8.m8.1.1.1" xref="S4.1.p1.8.m8.1.1.1.cmml">β</mo><mi id="S4.1.p1.8.m8.1.1.3" xref="S4.1.p1.8.m8.1.1.3.cmml">Z</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.8.m8.1b"><apply id="S4.1.p1.8.m8.1.1.cmml" xref="S4.1.p1.8.m8.1.1"><in id="S4.1.p1.8.m8.1.1.1.cmml" xref="S4.1.p1.8.m8.1.1.1"></in><ci id="S4.1.p1.8.m8.1.1.2.cmml" xref="S4.1.p1.8.m8.1.1.2">π§</ci><ci id="S4.1.p1.8.m8.1.1.3.cmml" xref="S4.1.p1.8.m8.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.8.m8.1c">z\in Z</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.8.m8.1d">italic_z β italic_Z</annotation></semantics></math> and note that <math alttext="zX=z\overline{D}\subset\overline{zD}=\overline{\{z\}}=\{z\}" class="ltx_Math" display="inline" id="S4.1.p1.9.m9.2"><semantics id="S4.1.p1.9.m9.2a"><mrow id="S4.1.p1.9.m9.2.3" xref="S4.1.p1.9.m9.2.3.cmml"><mrow id="S4.1.p1.9.m9.2.3.2" xref="S4.1.p1.9.m9.2.3.2.cmml"><mi id="S4.1.p1.9.m9.2.3.2.2" xref="S4.1.p1.9.m9.2.3.2.2.cmml">z</mi><mo id="S4.1.p1.9.m9.2.3.2.1" xref="S4.1.p1.9.m9.2.3.2.1.cmml">β’</mo><mi id="S4.1.p1.9.m9.2.3.2.3" xref="S4.1.p1.9.m9.2.3.2.3.cmml">X</mi></mrow><mo id="S4.1.p1.9.m9.2.3.3" xref="S4.1.p1.9.m9.2.3.3.cmml">=</mo><mrow id="S4.1.p1.9.m9.2.3.4" xref="S4.1.p1.9.m9.2.3.4.cmml"><mi id="S4.1.p1.9.m9.2.3.4.2" xref="S4.1.p1.9.m9.2.3.4.2.cmml">z</mi><mo id="S4.1.p1.9.m9.2.3.4.1" xref="S4.1.p1.9.m9.2.3.4.1.cmml">β’</mo><mover accent="true" id="S4.1.p1.9.m9.2.3.4.3" xref="S4.1.p1.9.m9.2.3.4.3.cmml"><mi id="S4.1.p1.9.m9.2.3.4.3.2" xref="S4.1.p1.9.m9.2.3.4.3.2.cmml">D</mi><mo id="S4.1.p1.9.m9.2.3.4.3.1" xref="S4.1.p1.9.m9.2.3.4.3.1.cmml">Β―</mo></mover></mrow><mo id="S4.1.p1.9.m9.2.3.5" xref="S4.1.p1.9.m9.2.3.5.cmml">β</mo><mover accent="true" id="S4.1.p1.9.m9.2.3.6" xref="S4.1.p1.9.m9.2.3.6.cmml"><mrow id="S4.1.p1.9.m9.2.3.6.2" xref="S4.1.p1.9.m9.2.3.6.2.cmml"><mi id="S4.1.p1.9.m9.2.3.6.2.2" xref="S4.1.p1.9.m9.2.3.6.2.2.cmml">z</mi><mo id="S4.1.p1.9.m9.2.3.6.2.1" xref="S4.1.p1.9.m9.2.3.6.2.1.cmml">β’</mo><mi id="S4.1.p1.9.m9.2.3.6.2.3" xref="S4.1.p1.9.m9.2.3.6.2.3.cmml">D</mi></mrow><mo id="S4.1.p1.9.m9.2.3.6.1" xref="S4.1.p1.9.m9.2.3.6.1.cmml">Β―</mo></mover><mo id="S4.1.p1.9.m9.2.3.7" xref="S4.1.p1.9.m9.2.3.7.cmml">=</mo><mover accent="true" id="S4.1.p1.9.m9.1.1" xref="S4.1.p1.9.m9.1.1.cmml"><mrow id="S4.1.p1.9.m9.1.1.1.3" xref="S4.1.p1.9.m9.1.1.1.2.cmml"><mo id="S4.1.p1.9.m9.1.1.1.3.1" stretchy="false" xref="S4.1.p1.9.m9.1.1.1.2.cmml">{</mo><mi id="S4.1.p1.9.m9.1.1.1.1" xref="S4.1.p1.9.m9.1.1.1.1.cmml">z</mi><mo id="S4.1.p1.9.m9.1.1.1.3.2" stretchy="false" xref="S4.1.p1.9.m9.1.1.1.2.cmml">}</mo></mrow><mo id="S4.1.p1.9.m9.1.1.2" xref="S4.1.p1.9.m9.1.1.2.cmml">Β―</mo></mover><mo id="S4.1.p1.9.m9.2.3.8" xref="S4.1.p1.9.m9.2.3.8.cmml">=</mo><mrow id="S4.1.p1.9.m9.2.3.9.2" xref="S4.1.p1.9.m9.2.3.9.1.cmml"><mo id="S4.1.p1.9.m9.2.3.9.2.1" stretchy="false" xref="S4.1.p1.9.m9.2.3.9.1.cmml">{</mo><mi id="S4.1.p1.9.m9.2.2" xref="S4.1.p1.9.m9.2.2.cmml">z</mi><mo id="S4.1.p1.9.m9.2.3.9.2.2" stretchy="false" xref="S4.1.p1.9.m9.2.3.9.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.1.p1.9.m9.2b"><apply id="S4.1.p1.9.m9.2.3.cmml" xref="S4.1.p1.9.m9.2.3"><and id="S4.1.p1.9.m9.2.3a.cmml" xref="S4.1.p1.9.m9.2.3"></and><apply id="S4.1.p1.9.m9.2.3b.cmml" xref="S4.1.p1.9.m9.2.3"><eq id="S4.1.p1.9.m9.2.3.3.cmml" xref="S4.1.p1.9.m9.2.3.3"></eq><apply id="S4.1.p1.9.m9.2.3.2.cmml" xref="S4.1.p1.9.m9.2.3.2"><times id="S4.1.p1.9.m9.2.3.2.1.cmml" xref="S4.1.p1.9.m9.2.3.2.1"></times><ci id="S4.1.p1.9.m9.2.3.2.2.cmml" xref="S4.1.p1.9.m9.2.3.2.2">π§</ci><ci id="S4.1.p1.9.m9.2.3.2.3.cmml" xref="S4.1.p1.9.m9.2.3.2.3">π</ci></apply><apply id="S4.1.p1.9.m9.2.3.4.cmml" xref="S4.1.p1.9.m9.2.3.4"><times id="S4.1.p1.9.m9.2.3.4.1.cmml" xref="S4.1.p1.9.m9.2.3.4.1"></times><ci id="S4.1.p1.9.m9.2.3.4.2.cmml" xref="S4.1.p1.9.m9.2.3.4.2">π§</ci><apply id="S4.1.p1.9.m9.2.3.4.3.cmml" xref="S4.1.p1.9.m9.2.3.4.3"><ci id="S4.1.p1.9.m9.2.3.4.3.1.cmml" xref="S4.1.p1.9.m9.2.3.4.3.1">Β―</ci><ci id="S4.1.p1.9.m9.2.3.4.3.2.cmml" xref="S4.1.p1.9.m9.2.3.4.3.2">π·</ci></apply></apply></apply><apply id="S4.1.p1.9.m9.2.3c.cmml" xref="S4.1.p1.9.m9.2.3"><subset id="S4.1.p1.9.m9.2.3.5.cmml" xref="S4.1.p1.9.m9.2.3.5"></subset><share href="https://arxiv.org/html/2503.13666v1#S4.1.p1.9.m9.2.3.4.cmml" id="S4.1.p1.9.m9.2.3d.cmml" xref="S4.1.p1.9.m9.2.3"></share><apply id="S4.1.p1.9.m9.2.3.6.cmml" xref="S4.1.p1.9.m9.2.3.6"><ci id="S4.1.p1.9.m9.2.3.6.1.cmml" xref="S4.1.p1.9.m9.2.3.6.1">Β―</ci><apply id="S4.1.p1.9.m9.2.3.6.2.cmml" xref="S4.1.p1.9.m9.2.3.6.2"><times id="S4.1.p1.9.m9.2.3.6.2.1.cmml" xref="S4.1.p1.9.m9.2.3.6.2.1"></times><ci id="S4.1.p1.9.m9.2.3.6.2.2.cmml" xref="S4.1.p1.9.m9.2.3.6.2.2">π§</ci><ci id="S4.1.p1.9.m9.2.3.6.2.3.cmml" xref="S4.1.p1.9.m9.2.3.6.2.3">π·</ci></apply></apply></apply><apply id="S4.1.p1.9.m9.2.3e.cmml" xref="S4.1.p1.9.m9.2.3"><eq id="S4.1.p1.9.m9.2.3.7.cmml" xref="S4.1.p1.9.m9.2.3.7"></eq><share href="https://arxiv.org/html/2503.13666v1#S4.1.p1.9.m9.2.3.6.cmml" id="S4.1.p1.9.m9.2.3f.cmml" xref="S4.1.p1.9.m9.2.3"></share><apply id="S4.1.p1.9.m9.1.1.cmml" xref="S4.1.p1.9.m9.1.1"><ci id="S4.1.p1.9.m9.1.1.2.cmml" xref="S4.1.p1.9.m9.1.1.2">Β―</ci><set id="S4.1.p1.9.m9.1.1.1.2.cmml" xref="S4.1.p1.9.m9.1.1.1.3"><ci id="S4.1.p1.9.m9.1.1.1.1.cmml" xref="S4.1.p1.9.m9.1.1.1.1">π§</ci></set></apply></apply><apply id="S4.1.p1.9.m9.2.3g.cmml" xref="S4.1.p1.9.m9.2.3"><eq id="S4.1.p1.9.m9.2.3.8.cmml" xref="S4.1.p1.9.m9.2.3.8"></eq><share href="https://arxiv.org/html/2503.13666v1#S4.1.p1.9.m9.1.1.cmml" id="S4.1.p1.9.m9.2.3h.cmml" xref="S4.1.p1.9.m9.2.3"></share><set id="S4.1.p1.9.m9.2.3.9.1.cmml" xref="S4.1.p1.9.m9.2.3.9.2"><ci id="S4.1.p1.9.m9.2.2.cmml" xref="S4.1.p1.9.m9.2.2">π§</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.9.m9.2c">zX=z\overline{D}\subset\overline{zD}=\overline{\{z\}}=\{z\}</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.9.m9.2d">italic_z italic_X = italic_z overΒ― start_ARG italic_D end_ARG β overΒ― start_ARG italic_z italic_D end_ARG = overΒ― start_ARG { italic_z } end_ARG = { italic_z }</annotation></semantics></math>. Hence <math alttext="z" class="ltx_Math" display="inline" id="S4.1.p1.10.m10.1"><semantics id="S4.1.p1.10.m10.1a"><mi id="S4.1.p1.10.m10.1.1" xref="S4.1.p1.10.m10.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.1.p1.10.m10.1b"><ci id="S4.1.p1.10.m10.1.1.cmml" xref="S4.1.p1.10.m10.1.1">π§</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.10.m10.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.10.m10.1d">italic_z</annotation></semantics></math> is the minimum of <math alttext="X" class="ltx_Math" display="inline" id="S4.1.p1.11.m11.1"><semantics id="S4.1.p1.11.m11.1a"><mi id="S4.1.p1.11.m11.1.1" xref="S4.1.p1.11.m11.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.1.p1.11.m11.1b"><ci id="S4.1.p1.11.m11.1.1.cmml" xref="S4.1.p1.11.m11.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.1.p1.11.m11.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.1.p1.11.m11.1d">italic_X</annotation></semantics></math>. β</p> </div> </div> <div class="ltx_theorem ltx_theorem_definition" id="S4.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem2.1.1.1">Definition 4.2</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem2.p1"> <p class="ltx_p" id="S4.Thmtheorem2.p1.3">Let <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.1.m1.1"><semantics id="S4.Thmtheorem2.p1.1.m1.1a"><mi id="S4.Thmtheorem2.p1.1.m1.1.1" xref="S4.Thmtheorem2.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.1.m1.1b"><ci id="S4.Thmtheorem2.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.1.m1.1d">italic_X</annotation></semantics></math> be a semilattice and <math alttext="L\subseteq X" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.2.m2.1"><semantics id="S4.Thmtheorem2.p1.2.m2.1a"><mrow id="S4.Thmtheorem2.p1.2.m2.1.1" xref="S4.Thmtheorem2.p1.2.m2.1.1.cmml"><mi id="S4.Thmtheorem2.p1.2.m2.1.1.2" xref="S4.Thmtheorem2.p1.2.m2.1.1.2.cmml">L</mi><mo id="S4.Thmtheorem2.p1.2.m2.1.1.1" xref="S4.Thmtheorem2.p1.2.m2.1.1.1.cmml">β</mo><mi id="S4.Thmtheorem2.p1.2.m2.1.1.3" xref="S4.Thmtheorem2.p1.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.2.m2.1b"><apply id="S4.Thmtheorem2.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1"><subset id="S4.Thmtheorem2.p1.2.m2.1.1.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.1"></subset><ci id="S4.Thmtheorem2.p1.2.m2.1.1.2.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.2">πΏ</ci><ci id="S4.Thmtheorem2.p1.2.m2.1.1.3.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.2.m2.1c">L\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.2.m2.1d">italic_L β italic_X</annotation></semantics></math> be a chain. Denote <math alttext="I_{L}=\{x\in X:|xL|<|L|\}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.3.m3.3"><semantics id="S4.Thmtheorem2.p1.3.m3.3a"><mrow id="S4.Thmtheorem2.p1.3.m3.3.3" xref="S4.Thmtheorem2.p1.3.m3.3.3.cmml"><msub id="S4.Thmtheorem2.p1.3.m3.3.3.4" xref="S4.Thmtheorem2.p1.3.m3.3.3.4.cmml"><mi id="S4.Thmtheorem2.p1.3.m3.3.3.4.2" xref="S4.Thmtheorem2.p1.3.m3.3.3.4.2.cmml">I</mi><mi id="S4.Thmtheorem2.p1.3.m3.3.3.4.3" xref="S4.Thmtheorem2.p1.3.m3.3.3.4.3.cmml">L</mi></msub><mo id="S4.Thmtheorem2.p1.3.m3.3.3.3" xref="S4.Thmtheorem2.p1.3.m3.3.3.3.cmml">=</mo><mrow id="S4.Thmtheorem2.p1.3.m3.3.3.2.2" xref="S4.Thmtheorem2.p1.3.m3.3.3.2.3.cmml"><mo id="S4.Thmtheorem2.p1.3.m3.3.3.2.2.3" stretchy="false" xref="S4.Thmtheorem2.p1.3.m3.3.3.2.3.1.cmml">{</mo><mrow id="S4.Thmtheorem2.p1.3.m3.2.2.1.1.1" xref="S4.Thmtheorem2.p1.3.m3.2.2.1.1.1.cmml"><mi id="S4.Thmtheorem2.p1.3.m3.2.2.1.1.1.2" xref="S4.Thmtheorem2.p1.3.m3.2.2.1.1.1.2.cmml">x</mi><mo 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id="S4.Thmtheorem2.p1.3.m3.3.3.cmml" xref="S4.Thmtheorem2.p1.3.m3.3.3"><eq id="S4.Thmtheorem2.p1.3.m3.3.3.3.cmml" xref="S4.Thmtheorem2.p1.3.m3.3.3.3"></eq><apply id="S4.Thmtheorem2.p1.3.m3.3.3.4.cmml" xref="S4.Thmtheorem2.p1.3.m3.3.3.4"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.3.m3.3.3.4.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.3.3.4">subscript</csymbol><ci id="S4.Thmtheorem2.p1.3.m3.3.3.4.2.cmml" xref="S4.Thmtheorem2.p1.3.m3.3.3.4.2">πΌ</ci><ci id="S4.Thmtheorem2.p1.3.m3.3.3.4.3.cmml" xref="S4.Thmtheorem2.p1.3.m3.3.3.4.3">πΏ</ci></apply><apply id="S4.Thmtheorem2.p1.3.m3.3.3.2.3.cmml" xref="S4.Thmtheorem2.p1.3.m3.3.3.2.2"><csymbol cd="latexml" id="S4.Thmtheorem2.p1.3.m3.3.3.2.3.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.3.3.2.2.3">conditional-set</csymbol><apply id="S4.Thmtheorem2.p1.3.m3.2.2.1.1.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.2.2.1.1.1"><in id="S4.Thmtheorem2.p1.3.m3.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.2.2.1.1.1.1"></in><ci id="S4.Thmtheorem2.p1.3.m3.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.3.m3.2.2.1.1.1.2">π₯</ci><ci id="S4.Thmtheorem2.p1.3.m3.2.2.1.1.1.3.cmml" xref="S4.Thmtheorem2.p1.3.m3.2.2.1.1.1.3">π</ci></apply><apply id="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.cmml" xref="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2"><lt id="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.2.cmml" xref="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.2"></lt><apply id="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.1.2.cmml" xref="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.1.1"><abs id="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.1.2.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.1.1.2"></abs><apply id="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.1.1.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.1.1.1"><times id="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.1.1.1.1"></times><ci id="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.1.1.1.2">π₯</ci><ci id="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.1.1.1.3.cmml" xref="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.1.1.1.3">πΏ</ci></apply></apply><apply id="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.3.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.3.2"><abs id="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.3.1.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.3.3.2.2.2.3.2.1"></abs><ci id="S4.Thmtheorem2.p1.3.m3.1.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1">πΏ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.3.m3.3c">I_{L}=\{x\in X:|xL|<|L|\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.3.m3.3d">italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT = { italic_x β italic_X : | italic_x italic_L | < | italic_L | }</annotation></semantics></math>.</p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem3.1.1.1">Lemma 4.3</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem3.p1"> <p class="ltx_p" id="S4.Thmtheorem3.p1.3"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem3.p1.3.3">For every semilattice <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.1.1.m1.1"><semantics id="S4.Thmtheorem3.p1.1.1.m1.1a"><mi id="S4.Thmtheorem3.p1.1.1.m1.1.1" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.1.1.m1.1b"><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.1.1.m1.1d">italic_X</annotation></semantics></math> and chain <math alttext="L\subseteq X" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.2.2.m2.1"><semantics id="S4.Thmtheorem3.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem3.p1.2.2.m2.1.1" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem3.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.2.cmml">L</mi><mo id="S4.Thmtheorem3.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.1.cmml">β</mo><mi id="S4.Thmtheorem3.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.2.2.m2.1b"><apply id="S4.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.1.1"><subset id="S4.Thmtheorem3.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.1"></subset><ci id="S4.Thmtheorem3.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.2">πΏ</ci><ci id="S4.Thmtheorem3.p1.2.2.m2.1.1.3.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.2.2.m2.1c">L\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.2.2.m2.1d">italic_L β italic_X</annotation></semantics></math> the set <math alttext="I_{L}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.3.3.m3.1"><semantics id="S4.Thmtheorem3.p1.3.3.m3.1a"><msub id="S4.Thmtheorem3.p1.3.3.m3.1.1" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem3.p1.3.3.m3.1.1.2" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.2.cmml">I</mi><mi id="S4.Thmtheorem3.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.3.3.m3.1b"><apply id="S4.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.1.1">subscript</csymbol><ci id="S4.Thmtheorem3.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.2">πΌ</ci><ci id="S4.Thmtheorem3.p1.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem3.p1.3.3.m3.1.1.3">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.3.3.m3.1c">I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.3.3.m3.1d">italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> is an ideal.</span></p> </div> </div> <div class="ltx_proof" id="S4.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.2.p1"> <p class="ltx_p" id="S4.2.p1.7">If <math alttext="I_{L}=\varnothing" class="ltx_Math" display="inline" id="S4.2.p1.1.m1.1"><semantics id="S4.2.p1.1.m1.1a"><mrow id="S4.2.p1.1.m1.1.1" xref="S4.2.p1.1.m1.1.1.cmml"><msub id="S4.2.p1.1.m1.1.1.2" xref="S4.2.p1.1.m1.1.1.2.cmml"><mi id="S4.2.p1.1.m1.1.1.2.2" xref="S4.2.p1.1.m1.1.1.2.2.cmml">I</mi><mi id="S4.2.p1.1.m1.1.1.2.3" xref="S4.2.p1.1.m1.1.1.2.3.cmml">L</mi></msub><mo id="S4.2.p1.1.m1.1.1.1" xref="S4.2.p1.1.m1.1.1.1.cmml">=</mo><mi id="S4.2.p1.1.m1.1.1.3" mathvariant="normal" xref="S4.2.p1.1.m1.1.1.3.cmml">β </mi></mrow><annotation-xml encoding="MathML-Content" id="S4.2.p1.1.m1.1b"><apply id="S4.2.p1.1.m1.1.1.cmml" xref="S4.2.p1.1.m1.1.1"><eq id="S4.2.p1.1.m1.1.1.1.cmml" xref="S4.2.p1.1.m1.1.1.1"></eq><apply id="S4.2.p1.1.m1.1.1.2.cmml" xref="S4.2.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.2.p1.1.m1.1.1.2.1.cmml" xref="S4.2.p1.1.m1.1.1.2">subscript</csymbol><ci id="S4.2.p1.1.m1.1.1.2.2.cmml" xref="S4.2.p1.1.m1.1.1.2.2">πΌ</ci><ci id="S4.2.p1.1.m1.1.1.2.3.cmml" xref="S4.2.p1.1.m1.1.1.2.3">πΏ</ci></apply><emptyset id="S4.2.p1.1.m1.1.1.3.cmml" xref="S4.2.p1.1.m1.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p1.1.m1.1c">I_{L}=\varnothing</annotation><annotation encoding="application/x-llamapun" id="S4.2.p1.1.m1.1d">italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT = β </annotation></semantics></math>, then there is nothing to prove. Fix any <math alttext="x\in X" class="ltx_Math" display="inline" id="S4.2.p1.2.m2.1"><semantics id="S4.2.p1.2.m2.1a"><mrow id="S4.2.p1.2.m2.1.1" xref="S4.2.p1.2.m2.1.1.cmml"><mi id="S4.2.p1.2.m2.1.1.2" xref="S4.2.p1.2.m2.1.1.2.cmml">x</mi><mo id="S4.2.p1.2.m2.1.1.1" xref="S4.2.p1.2.m2.1.1.1.cmml">β</mo><mi id="S4.2.p1.2.m2.1.1.3" xref="S4.2.p1.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.2.p1.2.m2.1b"><apply id="S4.2.p1.2.m2.1.1.cmml" xref="S4.2.p1.2.m2.1.1"><in id="S4.2.p1.2.m2.1.1.1.cmml" xref="S4.2.p1.2.m2.1.1.1"></in><ci id="S4.2.p1.2.m2.1.1.2.cmml" xref="S4.2.p1.2.m2.1.1.2">π₯</ci><ci id="S4.2.p1.2.m2.1.1.3.cmml" xref="S4.2.p1.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p1.2.m2.1c">x\in X</annotation><annotation encoding="application/x-llamapun" id="S4.2.p1.2.m2.1d">italic_x β italic_X</annotation></semantics></math> and <math alttext="y\in I_{L}" class="ltx_Math" display="inline" id="S4.2.p1.3.m3.1"><semantics id="S4.2.p1.3.m3.1a"><mrow id="S4.2.p1.3.m3.1.1" xref="S4.2.p1.3.m3.1.1.cmml"><mi id="S4.2.p1.3.m3.1.1.2" xref="S4.2.p1.3.m3.1.1.2.cmml">y</mi><mo id="S4.2.p1.3.m3.1.1.1" xref="S4.2.p1.3.m3.1.1.1.cmml">β</mo><msub id="S4.2.p1.3.m3.1.1.3" xref="S4.2.p1.3.m3.1.1.3.cmml"><mi id="S4.2.p1.3.m3.1.1.3.2" xref="S4.2.p1.3.m3.1.1.3.2.cmml">I</mi><mi id="S4.2.p1.3.m3.1.1.3.3" xref="S4.2.p1.3.m3.1.1.3.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.2.p1.3.m3.1b"><apply id="S4.2.p1.3.m3.1.1.cmml" xref="S4.2.p1.3.m3.1.1"><in id="S4.2.p1.3.m3.1.1.1.cmml" xref="S4.2.p1.3.m3.1.1.1"></in><ci id="S4.2.p1.3.m3.1.1.2.cmml" xref="S4.2.p1.3.m3.1.1.2">π¦</ci><apply id="S4.2.p1.3.m3.1.1.3.cmml" xref="S4.2.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.2.p1.3.m3.1.1.3.1.cmml" xref="S4.2.p1.3.m3.1.1.3">subscript</csymbol><ci id="S4.2.p1.3.m3.1.1.3.2.cmml" xref="S4.2.p1.3.m3.1.1.3.2">πΌ</ci><ci id="S4.2.p1.3.m3.1.1.3.3.cmml" xref="S4.2.p1.3.m3.1.1.3.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p1.3.m3.1c">y\in I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.2.p1.3.m3.1d">italic_y β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="|yL|<|L|" class="ltx_Math" display="inline" id="S4.2.p1.4.m4.2"><semantics id="S4.2.p1.4.m4.2a"><mrow id="S4.2.p1.4.m4.2.2" xref="S4.2.p1.4.m4.2.2.cmml"><mrow id="S4.2.p1.4.m4.2.2.1.1" xref="S4.2.p1.4.m4.2.2.1.2.cmml"><mo id="S4.2.p1.4.m4.2.2.1.1.2" stretchy="false" xref="S4.2.p1.4.m4.2.2.1.2.1.cmml">|</mo><mrow id="S4.2.p1.4.m4.2.2.1.1.1" xref="S4.2.p1.4.m4.2.2.1.1.1.cmml"><mi id="S4.2.p1.4.m4.2.2.1.1.1.2" xref="S4.2.p1.4.m4.2.2.1.1.1.2.cmml">y</mi><mo id="S4.2.p1.4.m4.2.2.1.1.1.1" xref="S4.2.p1.4.m4.2.2.1.1.1.1.cmml">β’</mo><mi id="S4.2.p1.4.m4.2.2.1.1.1.3" xref="S4.2.p1.4.m4.2.2.1.1.1.3.cmml">L</mi></mrow><mo id="S4.2.p1.4.m4.2.2.1.1.3" stretchy="false" xref="S4.2.p1.4.m4.2.2.1.2.1.cmml">|</mo></mrow><mo id="S4.2.p1.4.m4.2.2.2" xref="S4.2.p1.4.m4.2.2.2.cmml"><</mo><mrow id="S4.2.p1.4.m4.2.2.3.2" xref="S4.2.p1.4.m4.2.2.3.1.cmml"><mo id="S4.2.p1.4.m4.2.2.3.2.1" stretchy="false" xref="S4.2.p1.4.m4.2.2.3.1.1.cmml">|</mo><mi id="S4.2.p1.4.m4.1.1" xref="S4.2.p1.4.m4.1.1.cmml">L</mi><mo id="S4.2.p1.4.m4.2.2.3.2.2" stretchy="false" xref="S4.2.p1.4.m4.2.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.2.p1.4.m4.2b"><apply id="S4.2.p1.4.m4.2.2.cmml" xref="S4.2.p1.4.m4.2.2"><lt id="S4.2.p1.4.m4.2.2.2.cmml" xref="S4.2.p1.4.m4.2.2.2"></lt><apply id="S4.2.p1.4.m4.2.2.1.2.cmml" xref="S4.2.p1.4.m4.2.2.1.1"><abs id="S4.2.p1.4.m4.2.2.1.2.1.cmml" xref="S4.2.p1.4.m4.2.2.1.1.2"></abs><apply id="S4.2.p1.4.m4.2.2.1.1.1.cmml" xref="S4.2.p1.4.m4.2.2.1.1.1"><times id="S4.2.p1.4.m4.2.2.1.1.1.1.cmml" xref="S4.2.p1.4.m4.2.2.1.1.1.1"></times><ci id="S4.2.p1.4.m4.2.2.1.1.1.2.cmml" xref="S4.2.p1.4.m4.2.2.1.1.1.2">π¦</ci><ci id="S4.2.p1.4.m4.2.2.1.1.1.3.cmml" xref="S4.2.p1.4.m4.2.2.1.1.1.3">πΏ</ci></apply></apply><apply id="S4.2.p1.4.m4.2.2.3.1.cmml" xref="S4.2.p1.4.m4.2.2.3.2"><abs id="S4.2.p1.4.m4.2.2.3.1.1.cmml" xref="S4.2.p1.4.m4.2.2.3.2.1"></abs><ci id="S4.2.p1.4.m4.1.1.cmml" xref="S4.2.p1.4.m4.1.1">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p1.4.m4.2c">|yL|<|L|</annotation><annotation encoding="application/x-llamapun" id="S4.2.p1.4.m4.2d">| italic_y italic_L | < | italic_L |</annotation></semantics></math>, we get <math alttext="|xyL|=|x(yL)|\leq|yL|<|L|" class="ltx_Math" display="inline" id="S4.2.p1.5.m5.4"><semantics id="S4.2.p1.5.m5.4a"><mrow id="S4.2.p1.5.m5.4.4" xref="S4.2.p1.5.m5.4.4.cmml"><mrow id="S4.2.p1.5.m5.2.2.1.1" xref="S4.2.p1.5.m5.2.2.1.2.cmml"><mo id="S4.2.p1.5.m5.2.2.1.1.2" stretchy="false" xref="S4.2.p1.5.m5.2.2.1.2.1.cmml">|</mo><mrow id="S4.2.p1.5.m5.2.2.1.1.1" xref="S4.2.p1.5.m5.2.2.1.1.1.cmml"><mi id="S4.2.p1.5.m5.2.2.1.1.1.2" xref="S4.2.p1.5.m5.2.2.1.1.1.2.cmml">x</mi><mo id="S4.2.p1.5.m5.2.2.1.1.1.1" xref="S4.2.p1.5.m5.2.2.1.1.1.1.cmml">β’</mo><mi id="S4.2.p1.5.m5.2.2.1.1.1.3" xref="S4.2.p1.5.m5.2.2.1.1.1.3.cmml">y</mi><mo id="S4.2.p1.5.m5.2.2.1.1.1.1a" xref="S4.2.p1.5.m5.2.2.1.1.1.1.cmml">β’</mo><mi id="S4.2.p1.5.m5.2.2.1.1.1.4" xref="S4.2.p1.5.m5.2.2.1.1.1.4.cmml">L</mi></mrow><mo id="S4.2.p1.5.m5.2.2.1.1.3" stretchy="false" xref="S4.2.p1.5.m5.2.2.1.2.1.cmml">|</mo></mrow><mo id="S4.2.p1.5.m5.4.4.5" xref="S4.2.p1.5.m5.4.4.5.cmml">=</mo><mrow id="S4.2.p1.5.m5.3.3.2.1" xref="S4.2.p1.5.m5.3.3.2.2.cmml"><mo id="S4.2.p1.5.m5.3.3.2.1.2" stretchy="false" xref="S4.2.p1.5.m5.3.3.2.2.1.cmml">|</mo><mrow id="S4.2.p1.5.m5.3.3.2.1.1" xref="S4.2.p1.5.m5.3.3.2.1.1.cmml"><mi id="S4.2.p1.5.m5.3.3.2.1.1.3" xref="S4.2.p1.5.m5.3.3.2.1.1.3.cmml">x</mi><mo id="S4.2.p1.5.m5.3.3.2.1.1.2" xref="S4.2.p1.5.m5.3.3.2.1.1.2.cmml">β’</mo><mrow id="S4.2.p1.5.m5.3.3.2.1.1.1.1" xref="S4.2.p1.5.m5.3.3.2.1.1.1.1.1.cmml"><mo id="S4.2.p1.5.m5.3.3.2.1.1.1.1.2" stretchy="false" xref="S4.2.p1.5.m5.3.3.2.1.1.1.1.1.cmml">(</mo><mrow id="S4.2.p1.5.m5.3.3.2.1.1.1.1.1" xref="S4.2.p1.5.m5.3.3.2.1.1.1.1.1.cmml"><mi id="S4.2.p1.5.m5.3.3.2.1.1.1.1.1.2" xref="S4.2.p1.5.m5.3.3.2.1.1.1.1.1.2.cmml">y</mi><mo id="S4.2.p1.5.m5.3.3.2.1.1.1.1.1.1" xref="S4.2.p1.5.m5.3.3.2.1.1.1.1.1.1.cmml">β’</mo><mi id="S4.2.p1.5.m5.3.3.2.1.1.1.1.1.3" xref="S4.2.p1.5.m5.3.3.2.1.1.1.1.1.3.cmml">L</mi></mrow><mo id="S4.2.p1.5.m5.3.3.2.1.1.1.1.3" stretchy="false" xref="S4.2.p1.5.m5.3.3.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.2.p1.5.m5.3.3.2.1.3" stretchy="false" xref="S4.2.p1.5.m5.3.3.2.2.1.cmml">|</mo></mrow><mo id="S4.2.p1.5.m5.4.4.6" xref="S4.2.p1.5.m5.4.4.6.cmml">β€</mo><mrow id="S4.2.p1.5.m5.4.4.3.1" xref="S4.2.p1.5.m5.4.4.3.2.cmml"><mo id="S4.2.p1.5.m5.4.4.3.1.2" stretchy="false" xref="S4.2.p1.5.m5.4.4.3.2.1.cmml">|</mo><mrow id="S4.2.p1.5.m5.4.4.3.1.1" xref="S4.2.p1.5.m5.4.4.3.1.1.cmml"><mi id="S4.2.p1.5.m5.4.4.3.1.1.2" xref="S4.2.p1.5.m5.4.4.3.1.1.2.cmml">y</mi><mo id="S4.2.p1.5.m5.4.4.3.1.1.1" xref="S4.2.p1.5.m5.4.4.3.1.1.1.cmml">β’</mo><mi id="S4.2.p1.5.m5.4.4.3.1.1.3" xref="S4.2.p1.5.m5.4.4.3.1.1.3.cmml">L</mi></mrow><mo id="S4.2.p1.5.m5.4.4.3.1.3" stretchy="false" xref="S4.2.p1.5.m5.4.4.3.2.1.cmml">|</mo></mrow><mo id="S4.2.p1.5.m5.4.4.7" xref="S4.2.p1.5.m5.4.4.7.cmml"><</mo><mrow id="S4.2.p1.5.m5.4.4.8.2" xref="S4.2.p1.5.m5.4.4.8.1.cmml"><mo id="S4.2.p1.5.m5.4.4.8.2.1" stretchy="false" xref="S4.2.p1.5.m5.4.4.8.1.1.cmml">|</mo><mi id="S4.2.p1.5.m5.1.1" xref="S4.2.p1.5.m5.1.1.cmml">L</mi><mo id="S4.2.p1.5.m5.4.4.8.2.2" stretchy="false" xref="S4.2.p1.5.m5.4.4.8.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.2.p1.5.m5.4b"><apply id="S4.2.p1.5.m5.4.4.cmml" xref="S4.2.p1.5.m5.4.4"><and id="S4.2.p1.5.m5.4.4a.cmml" xref="S4.2.p1.5.m5.4.4"></and><apply id="S4.2.p1.5.m5.4.4b.cmml" xref="S4.2.p1.5.m5.4.4"><eq id="S4.2.p1.5.m5.4.4.5.cmml" xref="S4.2.p1.5.m5.4.4.5"></eq><apply id="S4.2.p1.5.m5.2.2.1.2.cmml" xref="S4.2.p1.5.m5.2.2.1.1"><abs id="S4.2.p1.5.m5.2.2.1.2.1.cmml" xref="S4.2.p1.5.m5.2.2.1.1.2"></abs><apply id="S4.2.p1.5.m5.2.2.1.1.1.cmml" xref="S4.2.p1.5.m5.2.2.1.1.1"><times id="S4.2.p1.5.m5.2.2.1.1.1.1.cmml" xref="S4.2.p1.5.m5.2.2.1.1.1.1"></times><ci id="S4.2.p1.5.m5.2.2.1.1.1.2.cmml" xref="S4.2.p1.5.m5.2.2.1.1.1.2">π₯</ci><ci id="S4.2.p1.5.m5.2.2.1.1.1.3.cmml" xref="S4.2.p1.5.m5.2.2.1.1.1.3">π¦</ci><ci id="S4.2.p1.5.m5.2.2.1.1.1.4.cmml" xref="S4.2.p1.5.m5.2.2.1.1.1.4">πΏ</ci></apply></apply><apply id="S4.2.p1.5.m5.3.3.2.2.cmml" xref="S4.2.p1.5.m5.3.3.2.1"><abs id="S4.2.p1.5.m5.3.3.2.2.1.cmml" xref="S4.2.p1.5.m5.3.3.2.1.2"></abs><apply id="S4.2.p1.5.m5.3.3.2.1.1.cmml" xref="S4.2.p1.5.m5.3.3.2.1.1"><times id="S4.2.p1.5.m5.3.3.2.1.1.2.cmml" xref="S4.2.p1.5.m5.3.3.2.1.1.2"></times><ci id="S4.2.p1.5.m5.3.3.2.1.1.3.cmml" xref="S4.2.p1.5.m5.3.3.2.1.1.3">π₯</ci><apply id="S4.2.p1.5.m5.3.3.2.1.1.1.1.1.cmml" xref="S4.2.p1.5.m5.3.3.2.1.1.1.1"><times id="S4.2.p1.5.m5.3.3.2.1.1.1.1.1.1.cmml" xref="S4.2.p1.5.m5.3.3.2.1.1.1.1.1.1"></times><ci id="S4.2.p1.5.m5.3.3.2.1.1.1.1.1.2.cmml" xref="S4.2.p1.5.m5.3.3.2.1.1.1.1.1.2">π¦</ci><ci id="S4.2.p1.5.m5.3.3.2.1.1.1.1.1.3.cmml" xref="S4.2.p1.5.m5.3.3.2.1.1.1.1.1.3">πΏ</ci></apply></apply></apply></apply><apply id="S4.2.p1.5.m5.4.4c.cmml" xref="S4.2.p1.5.m5.4.4"><leq id="S4.2.p1.5.m5.4.4.6.cmml" xref="S4.2.p1.5.m5.4.4.6"></leq><share href="https://arxiv.org/html/2503.13666v1#S4.2.p1.5.m5.3.3.2.cmml" id="S4.2.p1.5.m5.4.4d.cmml" xref="S4.2.p1.5.m5.4.4"></share><apply id="S4.2.p1.5.m5.4.4.3.2.cmml" xref="S4.2.p1.5.m5.4.4.3.1"><abs id="S4.2.p1.5.m5.4.4.3.2.1.cmml" xref="S4.2.p1.5.m5.4.4.3.1.2"></abs><apply id="S4.2.p1.5.m5.4.4.3.1.1.cmml" xref="S4.2.p1.5.m5.4.4.3.1.1"><times id="S4.2.p1.5.m5.4.4.3.1.1.1.cmml" xref="S4.2.p1.5.m5.4.4.3.1.1.1"></times><ci id="S4.2.p1.5.m5.4.4.3.1.1.2.cmml" xref="S4.2.p1.5.m5.4.4.3.1.1.2">π¦</ci><ci id="S4.2.p1.5.m5.4.4.3.1.1.3.cmml" xref="S4.2.p1.5.m5.4.4.3.1.1.3">πΏ</ci></apply></apply></apply><apply id="S4.2.p1.5.m5.4.4e.cmml" xref="S4.2.p1.5.m5.4.4"><lt id="S4.2.p1.5.m5.4.4.7.cmml" xref="S4.2.p1.5.m5.4.4.7"></lt><share href="https://arxiv.org/html/2503.13666v1#S4.2.p1.5.m5.4.4.3.cmml" id="S4.2.p1.5.m5.4.4f.cmml" xref="S4.2.p1.5.m5.4.4"></share><apply id="S4.2.p1.5.m5.4.4.8.1.cmml" xref="S4.2.p1.5.m5.4.4.8.2"><abs id="S4.2.p1.5.m5.4.4.8.1.1.cmml" xref="S4.2.p1.5.m5.4.4.8.2.1"></abs><ci id="S4.2.p1.5.m5.1.1.cmml" xref="S4.2.p1.5.m5.1.1">πΏ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p1.5.m5.4c">|xyL|=|x(yL)|\leq|yL|<|L|</annotation><annotation encoding="application/x-llamapun" id="S4.2.p1.5.m5.4d">| italic_x italic_y italic_L | = | italic_x ( italic_y italic_L ) | β€ | italic_y italic_L | < | italic_L |</annotation></semantics></math>. Hence <math alttext="xy\in I_{L}" class="ltx_Math" display="inline" id="S4.2.p1.6.m6.1"><semantics id="S4.2.p1.6.m6.1a"><mrow id="S4.2.p1.6.m6.1.1" xref="S4.2.p1.6.m6.1.1.cmml"><mrow id="S4.2.p1.6.m6.1.1.2" xref="S4.2.p1.6.m6.1.1.2.cmml"><mi id="S4.2.p1.6.m6.1.1.2.2" xref="S4.2.p1.6.m6.1.1.2.2.cmml">x</mi><mo id="S4.2.p1.6.m6.1.1.2.1" xref="S4.2.p1.6.m6.1.1.2.1.cmml">β’</mo><mi id="S4.2.p1.6.m6.1.1.2.3" xref="S4.2.p1.6.m6.1.1.2.3.cmml">y</mi></mrow><mo id="S4.2.p1.6.m6.1.1.1" xref="S4.2.p1.6.m6.1.1.1.cmml">β</mo><msub id="S4.2.p1.6.m6.1.1.3" xref="S4.2.p1.6.m6.1.1.3.cmml"><mi id="S4.2.p1.6.m6.1.1.3.2" xref="S4.2.p1.6.m6.1.1.3.2.cmml">I</mi><mi id="S4.2.p1.6.m6.1.1.3.3" xref="S4.2.p1.6.m6.1.1.3.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.2.p1.6.m6.1b"><apply id="S4.2.p1.6.m6.1.1.cmml" xref="S4.2.p1.6.m6.1.1"><in id="S4.2.p1.6.m6.1.1.1.cmml" xref="S4.2.p1.6.m6.1.1.1"></in><apply id="S4.2.p1.6.m6.1.1.2.cmml" xref="S4.2.p1.6.m6.1.1.2"><times id="S4.2.p1.6.m6.1.1.2.1.cmml" xref="S4.2.p1.6.m6.1.1.2.1"></times><ci id="S4.2.p1.6.m6.1.1.2.2.cmml" xref="S4.2.p1.6.m6.1.1.2.2">π₯</ci><ci id="S4.2.p1.6.m6.1.1.2.3.cmml" xref="S4.2.p1.6.m6.1.1.2.3">π¦</ci></apply><apply id="S4.2.p1.6.m6.1.1.3.cmml" xref="S4.2.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.2.p1.6.m6.1.1.3.1.cmml" xref="S4.2.p1.6.m6.1.1.3">subscript</csymbol><ci id="S4.2.p1.6.m6.1.1.3.2.cmml" xref="S4.2.p1.6.m6.1.1.3.2">πΌ</ci><ci id="S4.2.p1.6.m6.1.1.3.3.cmml" xref="S4.2.p1.6.m6.1.1.3.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p1.6.m6.1c">xy\in I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.2.p1.6.m6.1d">italic_x italic_y β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>, witnessing that <math alttext="I_{L}" class="ltx_Math" display="inline" id="S4.2.p1.7.m7.1"><semantics id="S4.2.p1.7.m7.1a"><msub id="S4.2.p1.7.m7.1.1" xref="S4.2.p1.7.m7.1.1.cmml"><mi id="S4.2.p1.7.m7.1.1.2" xref="S4.2.p1.7.m7.1.1.2.cmml">I</mi><mi id="S4.2.p1.7.m7.1.1.3" xref="S4.2.p1.7.m7.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S4.2.p1.7.m7.1b"><apply id="S4.2.p1.7.m7.1.1.cmml" xref="S4.2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.2.p1.7.m7.1.1.1.cmml" xref="S4.2.p1.7.m7.1.1">subscript</csymbol><ci id="S4.2.p1.7.m7.1.1.2.cmml" xref="S4.2.p1.7.m7.1.1.2">πΌ</ci><ci id="S4.2.p1.7.m7.1.1.3.cmml" xref="S4.2.p1.7.m7.1.1.3">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.2.p1.7.m7.1c">I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.2.p1.7.m7.1d">italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> is an ideal. β</p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem4.1.1.1">Lemma 4.4</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem4.p1"> <p class="ltx_p" id="S4.Thmtheorem4.p1.4"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem4.p1.4.4">Let <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.1.1.m1.1"><semantics id="S4.Thmtheorem4.p1.1.1.m1.1a"><mi id="S4.Thmtheorem4.p1.1.1.m1.1.1" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.1.1.m1.1b"><ci id="S4.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.1.1.m1.1d">italic_X</annotation></semantics></math> be a countably tight semitopological semilattice and <math alttext="L\subseteq X" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.2.2.m2.1"><semantics id="S4.Thmtheorem4.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem4.p1.2.2.m2.1.1" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem4.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.2.cmml">L</mi><mo id="S4.Thmtheorem4.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.1.cmml">β</mo><mi id="S4.Thmtheorem4.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.2.2.m2.1b"><apply id="S4.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1"><subset id="S4.Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.1"></subset><ci id="S4.Thmtheorem4.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.2">πΏ</ci><ci id="S4.Thmtheorem4.p1.2.2.m2.1.1.3.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.2.2.m2.1c">L\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.2.2.m2.1d">italic_L β italic_X</annotation></semantics></math> be a chain isomorphic to <math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.3.3.m3.2"><semantics id="S4.Thmtheorem4.p1.3.3.m3.2a"><mrow id="S4.Thmtheorem4.p1.3.3.m3.2.2.1" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.2.cmml"><mo id="S4.Thmtheorem4.p1.3.3.m3.2.2.1.2" stretchy="false" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.2.cmml">(</mo><msub id="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.cmml"><mi id="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.2" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.2.cmml">Ο</mi><mn id="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.3" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem4.p1.3.3.m3.2.2.1.3" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.2.cmml">,</mo><mi id="S4.Thmtheorem4.p1.3.3.m3.1.1" xref="S4.Thmtheorem4.p1.3.3.m3.1.1.cmml">min</mi><mo id="S4.Thmtheorem4.p1.3.3.m3.2.2.1.4" stretchy="false" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.3.3.m3.2b"><interval closure="open" id="S4.Thmtheorem4.p1.3.3.m3.2.2.2.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.1"><apply id="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1">subscript</csymbol><ci id="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.2.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.2">π</ci><cn id="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.3.3.m3.2.2.1.1.3">1</cn></apply><min id="S4.Thmtheorem4.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem4.p1.3.3.m3.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.3.3.m3.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.3.3.m3.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math>. Then <math alttext="I_{L}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.4.4.m4.1"><semantics id="S4.Thmtheorem4.p1.4.4.m4.1a"><msub id="S4.Thmtheorem4.p1.4.4.m4.1.1" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.cmml"><mi id="S4.Thmtheorem4.p1.4.4.m4.1.1.2" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.2.cmml">I</mi><mi id="S4.Thmtheorem4.p1.4.4.m4.1.1.3" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.4.4.m4.1b"><apply id="S4.Thmtheorem4.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.1.1">subscript</csymbol><ci id="S4.Thmtheorem4.p1.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.2">πΌ</ci><ci id="S4.Thmtheorem4.p1.4.4.m4.1.1.3.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.3">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.4.4.m4.1c">I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.4.4.m4.1d">italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> is a closed ideal.</span></p> </div> </div> <div class="ltx_proof" id="S4.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.3.p1"> <p class="ltx_p" id="S4.3.p1.27">By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S4.Thmtheorem3" title="Lemma 4.3. β£ 4. Nyikos semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">4.3</span></a>, <math alttext="I_{L}" class="ltx_Math" display="inline" id="S4.3.p1.1.m1.1"><semantics id="S4.3.p1.1.m1.1a"><msub id="S4.3.p1.1.m1.1.1" xref="S4.3.p1.1.m1.1.1.cmml"><mi id="S4.3.p1.1.m1.1.1.2" xref="S4.3.p1.1.m1.1.1.2.cmml">I</mi><mi id="S4.3.p1.1.m1.1.1.3" xref="S4.3.p1.1.m1.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S4.3.p1.1.m1.1b"><apply id="S4.3.p1.1.m1.1.1.cmml" xref="S4.3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.3.p1.1.m1.1.1.1.cmml" xref="S4.3.p1.1.m1.1.1">subscript</csymbol><ci id="S4.3.p1.1.m1.1.1.2.cmml" xref="S4.3.p1.1.m1.1.1.2">πΌ</ci><ci id="S4.3.p1.1.m1.1.1.3.cmml" xref="S4.3.p1.1.m1.1.1.3">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.1.m1.1c">I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.1.m1.1d">italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> is an ideal. Let <math alttext="L=\{l_{\xi}:\xi<\omega_{1}\}" class="ltx_Math" display="inline" id="S4.3.p1.2.m2.2"><semantics id="S4.3.p1.2.m2.2a"><mrow id="S4.3.p1.2.m2.2.2" xref="S4.3.p1.2.m2.2.2.cmml"><mi id="S4.3.p1.2.m2.2.2.4" xref="S4.3.p1.2.m2.2.2.4.cmml">L</mi><mo id="S4.3.p1.2.m2.2.2.3" xref="S4.3.p1.2.m2.2.2.3.cmml">=</mo><mrow id="S4.3.p1.2.m2.2.2.2.2" xref="S4.3.p1.2.m2.2.2.2.3.cmml"><mo id="S4.3.p1.2.m2.2.2.2.2.3" stretchy="false" xref="S4.3.p1.2.m2.2.2.2.3.1.cmml">{</mo><msub id="S4.3.p1.2.m2.1.1.1.1.1" xref="S4.3.p1.2.m2.1.1.1.1.1.cmml"><mi id="S4.3.p1.2.m2.1.1.1.1.1.2" xref="S4.3.p1.2.m2.1.1.1.1.1.2.cmml">l</mi><mi id="S4.3.p1.2.m2.1.1.1.1.1.3" xref="S4.3.p1.2.m2.1.1.1.1.1.3.cmml">ΞΎ</mi></msub><mo id="S4.3.p1.2.m2.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.3.p1.2.m2.2.2.2.3.1.cmml">:</mo><mrow id="S4.3.p1.2.m2.2.2.2.2.2" xref="S4.3.p1.2.m2.2.2.2.2.2.cmml"><mi id="S4.3.p1.2.m2.2.2.2.2.2.2" xref="S4.3.p1.2.m2.2.2.2.2.2.2.cmml">ΞΎ</mi><mo id="S4.3.p1.2.m2.2.2.2.2.2.1" xref="S4.3.p1.2.m2.2.2.2.2.2.1.cmml"><</mo><msub id="S4.3.p1.2.m2.2.2.2.2.2.3" xref="S4.3.p1.2.m2.2.2.2.2.2.3.cmml"><mi id="S4.3.p1.2.m2.2.2.2.2.2.3.2" xref="S4.3.p1.2.m2.2.2.2.2.2.3.2.cmml">Ο</mi><mn id="S4.3.p1.2.m2.2.2.2.2.2.3.3" xref="S4.3.p1.2.m2.2.2.2.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S4.3.p1.2.m2.2.2.2.2.5" stretchy="false" xref="S4.3.p1.2.m2.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.2.m2.2b"><apply id="S4.3.p1.2.m2.2.2.cmml" xref="S4.3.p1.2.m2.2.2"><eq id="S4.3.p1.2.m2.2.2.3.cmml" xref="S4.3.p1.2.m2.2.2.3"></eq><ci id="S4.3.p1.2.m2.2.2.4.cmml" xref="S4.3.p1.2.m2.2.2.4">πΏ</ci><apply id="S4.3.p1.2.m2.2.2.2.3.cmml" xref="S4.3.p1.2.m2.2.2.2.2"><csymbol cd="latexml" id="S4.3.p1.2.m2.2.2.2.3.1.cmml" xref="S4.3.p1.2.m2.2.2.2.2.3">conditional-set</csymbol><apply id="S4.3.p1.2.m2.1.1.1.1.1.cmml" xref="S4.3.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.3.p1.2.m2.1.1.1.1.1.1.cmml" xref="S4.3.p1.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S4.3.p1.2.m2.1.1.1.1.1.2.cmml" xref="S4.3.p1.2.m2.1.1.1.1.1.2">π</ci><ci id="S4.3.p1.2.m2.1.1.1.1.1.3.cmml" xref="S4.3.p1.2.m2.1.1.1.1.1.3">π</ci></apply><apply id="S4.3.p1.2.m2.2.2.2.2.2.cmml" xref="S4.3.p1.2.m2.2.2.2.2.2"><lt id="S4.3.p1.2.m2.2.2.2.2.2.1.cmml" xref="S4.3.p1.2.m2.2.2.2.2.2.1"></lt><ci id="S4.3.p1.2.m2.2.2.2.2.2.2.cmml" xref="S4.3.p1.2.m2.2.2.2.2.2.2">π</ci><apply id="S4.3.p1.2.m2.2.2.2.2.2.3.cmml" xref="S4.3.p1.2.m2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.3.p1.2.m2.2.2.2.2.2.3.1.cmml" xref="S4.3.p1.2.m2.2.2.2.2.2.3">subscript</csymbol><ci id="S4.3.p1.2.m2.2.2.2.2.2.3.2.cmml" xref="S4.3.p1.2.m2.2.2.2.2.2.3.2">π</ci><cn id="S4.3.p1.2.m2.2.2.2.2.2.3.3.cmml" type="integer" xref="S4.3.p1.2.m2.2.2.2.2.2.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.2.m2.2c">L=\{l_{\xi}:\xi<\omega_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.2.m2.2d">italic_L = { italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT : italic_ΞΎ < italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math>, where <math alttext="l_{\xi}\leq l_{\beta}" class="ltx_Math" display="inline" id="S4.3.p1.3.m3.1"><semantics id="S4.3.p1.3.m3.1a"><mrow id="S4.3.p1.3.m3.1.1" xref="S4.3.p1.3.m3.1.1.cmml"><msub id="S4.3.p1.3.m3.1.1.2" xref="S4.3.p1.3.m3.1.1.2.cmml"><mi id="S4.3.p1.3.m3.1.1.2.2" xref="S4.3.p1.3.m3.1.1.2.2.cmml">l</mi><mi id="S4.3.p1.3.m3.1.1.2.3" xref="S4.3.p1.3.m3.1.1.2.3.cmml">ΞΎ</mi></msub><mo id="S4.3.p1.3.m3.1.1.1" xref="S4.3.p1.3.m3.1.1.1.cmml">β€</mo><msub id="S4.3.p1.3.m3.1.1.3" xref="S4.3.p1.3.m3.1.1.3.cmml"><mi id="S4.3.p1.3.m3.1.1.3.2" xref="S4.3.p1.3.m3.1.1.3.2.cmml">l</mi><mi id="S4.3.p1.3.m3.1.1.3.3" xref="S4.3.p1.3.m3.1.1.3.3.cmml">Ξ²</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.3.m3.1b"><apply id="S4.3.p1.3.m3.1.1.cmml" xref="S4.3.p1.3.m3.1.1"><leq id="S4.3.p1.3.m3.1.1.1.cmml" xref="S4.3.p1.3.m3.1.1.1"></leq><apply id="S4.3.p1.3.m3.1.1.2.cmml" xref="S4.3.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.3.p1.3.m3.1.1.2.1.cmml" xref="S4.3.p1.3.m3.1.1.2">subscript</csymbol><ci id="S4.3.p1.3.m3.1.1.2.2.cmml" xref="S4.3.p1.3.m3.1.1.2.2">π</ci><ci id="S4.3.p1.3.m3.1.1.2.3.cmml" xref="S4.3.p1.3.m3.1.1.2.3">π</ci></apply><apply id="S4.3.p1.3.m3.1.1.3.cmml" xref="S4.3.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.3.p1.3.m3.1.1.3.1.cmml" xref="S4.3.p1.3.m3.1.1.3">subscript</csymbol><ci id="S4.3.p1.3.m3.1.1.3.2.cmml" xref="S4.3.p1.3.m3.1.1.3.2">π</ci><ci id="S4.3.p1.3.m3.1.1.3.3.cmml" xref="S4.3.p1.3.m3.1.1.3.3">π½</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.3.m3.1c">l_{\xi}\leq l_{\beta}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.3.m3.1d">italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT β€ italic_l start_POSTSUBSCRIPT italic_Ξ² end_POSTSUBSCRIPT</annotation></semantics></math> if and only if <math alttext="\xi\leq\beta" class="ltx_Math" display="inline" id="S4.3.p1.4.m4.1"><semantics id="S4.3.p1.4.m4.1a"><mrow id="S4.3.p1.4.m4.1.1" xref="S4.3.p1.4.m4.1.1.cmml"><mi id="S4.3.p1.4.m4.1.1.2" xref="S4.3.p1.4.m4.1.1.2.cmml">ΞΎ</mi><mo id="S4.3.p1.4.m4.1.1.1" xref="S4.3.p1.4.m4.1.1.1.cmml">β€</mo><mi id="S4.3.p1.4.m4.1.1.3" xref="S4.3.p1.4.m4.1.1.3.cmml">Ξ²</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.4.m4.1b"><apply id="S4.3.p1.4.m4.1.1.cmml" xref="S4.3.p1.4.m4.1.1"><leq id="S4.3.p1.4.m4.1.1.1.cmml" xref="S4.3.p1.4.m4.1.1.1"></leq><ci id="S4.3.p1.4.m4.1.1.2.cmml" xref="S4.3.p1.4.m4.1.1.2">π</ci><ci id="S4.3.p1.4.m4.1.1.3.cmml" xref="S4.3.p1.4.m4.1.1.3">π½</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.4.m4.1c">\xi\leq\beta</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.4.m4.1d">italic_ΞΎ β€ italic_Ξ²</annotation></semantics></math>. Fix any accumulation point <math alttext="z" class="ltx_Math" display="inline" id="S4.3.p1.5.m5.1"><semantics id="S4.3.p1.5.m5.1a"><mi id="S4.3.p1.5.m5.1.1" xref="S4.3.p1.5.m5.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.3.p1.5.m5.1b"><ci id="S4.3.p1.5.m5.1.1.cmml" xref="S4.3.p1.5.m5.1.1">π§</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.5.m5.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.5.m5.1d">italic_z</annotation></semantics></math> of <math alttext="I_{L}" class="ltx_Math" display="inline" id="S4.3.p1.6.m6.1"><semantics id="S4.3.p1.6.m6.1a"><msub id="S4.3.p1.6.m6.1.1" xref="S4.3.p1.6.m6.1.1.cmml"><mi id="S4.3.p1.6.m6.1.1.2" xref="S4.3.p1.6.m6.1.1.2.cmml">I</mi><mi id="S4.3.p1.6.m6.1.1.3" xref="S4.3.p1.6.m6.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S4.3.p1.6.m6.1b"><apply id="S4.3.p1.6.m6.1.1.cmml" xref="S4.3.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.3.p1.6.m6.1.1.1.cmml" xref="S4.3.p1.6.m6.1.1">subscript</csymbol><ci id="S4.3.p1.6.m6.1.1.2.cmml" xref="S4.3.p1.6.m6.1.1.2">πΌ</ci><ci id="S4.3.p1.6.m6.1.1.3.cmml" xref="S4.3.p1.6.m6.1.1.3">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.6.m6.1c">I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.6.m6.1d">italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="X" class="ltx_Math" display="inline" id="S4.3.p1.7.m7.1"><semantics id="S4.3.p1.7.m7.1a"><mi id="S4.3.p1.7.m7.1.1" xref="S4.3.p1.7.m7.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.3.p1.7.m7.1b"><ci id="S4.3.p1.7.m7.1.1.cmml" xref="S4.3.p1.7.m7.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.7.m7.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.7.m7.1d">italic_X</annotation></semantics></math> is countably tight, there exists a countable subset <math alttext="B\subseteq I_{L}" class="ltx_Math" display="inline" id="S4.3.p1.8.m8.1"><semantics id="S4.3.p1.8.m8.1a"><mrow id="S4.3.p1.8.m8.1.1" xref="S4.3.p1.8.m8.1.1.cmml"><mi id="S4.3.p1.8.m8.1.1.2" xref="S4.3.p1.8.m8.1.1.2.cmml">B</mi><mo id="S4.3.p1.8.m8.1.1.1" xref="S4.3.p1.8.m8.1.1.1.cmml">β</mo><msub id="S4.3.p1.8.m8.1.1.3" xref="S4.3.p1.8.m8.1.1.3.cmml"><mi id="S4.3.p1.8.m8.1.1.3.2" xref="S4.3.p1.8.m8.1.1.3.2.cmml">I</mi><mi id="S4.3.p1.8.m8.1.1.3.3" xref="S4.3.p1.8.m8.1.1.3.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.8.m8.1b"><apply id="S4.3.p1.8.m8.1.1.cmml" xref="S4.3.p1.8.m8.1.1"><subset id="S4.3.p1.8.m8.1.1.1.cmml" xref="S4.3.p1.8.m8.1.1.1"></subset><ci id="S4.3.p1.8.m8.1.1.2.cmml" xref="S4.3.p1.8.m8.1.1.2">π΅</ci><apply id="S4.3.p1.8.m8.1.1.3.cmml" xref="S4.3.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.3.p1.8.m8.1.1.3.1.cmml" xref="S4.3.p1.8.m8.1.1.3">subscript</csymbol><ci id="S4.3.p1.8.m8.1.1.3.2.cmml" xref="S4.3.p1.8.m8.1.1.3.2">πΌ</ci><ci id="S4.3.p1.8.m8.1.1.3.3.cmml" xref="S4.3.p1.8.m8.1.1.3.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.8.m8.1c">B\subseteq I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.8.m8.1d">italic_B β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="z\in\overline{B}" class="ltx_Math" display="inline" id="S4.3.p1.9.m9.1"><semantics id="S4.3.p1.9.m9.1a"><mrow id="S4.3.p1.9.m9.1.1" xref="S4.3.p1.9.m9.1.1.cmml"><mi id="S4.3.p1.9.m9.1.1.2" xref="S4.3.p1.9.m9.1.1.2.cmml">z</mi><mo id="S4.3.p1.9.m9.1.1.1" xref="S4.3.p1.9.m9.1.1.1.cmml">β</mo><mover accent="true" id="S4.3.p1.9.m9.1.1.3" xref="S4.3.p1.9.m9.1.1.3.cmml"><mi id="S4.3.p1.9.m9.1.1.3.2" xref="S4.3.p1.9.m9.1.1.3.2.cmml">B</mi><mo id="S4.3.p1.9.m9.1.1.3.1" xref="S4.3.p1.9.m9.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.9.m9.1b"><apply id="S4.3.p1.9.m9.1.1.cmml" xref="S4.3.p1.9.m9.1.1"><in id="S4.3.p1.9.m9.1.1.1.cmml" xref="S4.3.p1.9.m9.1.1.1"></in><ci id="S4.3.p1.9.m9.1.1.2.cmml" xref="S4.3.p1.9.m9.1.1.2">π§</ci><apply id="S4.3.p1.9.m9.1.1.3.cmml" xref="S4.3.p1.9.m9.1.1.3"><ci id="S4.3.p1.9.m9.1.1.3.1.cmml" xref="S4.3.p1.9.m9.1.1.3.1">Β―</ci><ci id="S4.3.p1.9.m9.1.1.3.2.cmml" xref="S4.3.p1.9.m9.1.1.3.2">π΅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.9.m9.1c">z\in\overline{B}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.9.m9.1d">italic_z β overΒ― start_ARG italic_B end_ARG</annotation></semantics></math>. For each <math alttext="b\in B" class="ltx_Math" display="inline" id="S4.3.p1.10.m10.1"><semantics id="S4.3.p1.10.m10.1a"><mrow id="S4.3.p1.10.m10.1.1" xref="S4.3.p1.10.m10.1.1.cmml"><mi id="S4.3.p1.10.m10.1.1.2" xref="S4.3.p1.10.m10.1.1.2.cmml">b</mi><mo id="S4.3.p1.10.m10.1.1.1" xref="S4.3.p1.10.m10.1.1.1.cmml">β</mo><mi id="S4.3.p1.10.m10.1.1.3" xref="S4.3.p1.10.m10.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.10.m10.1b"><apply id="S4.3.p1.10.m10.1.1.cmml" xref="S4.3.p1.10.m10.1.1"><in id="S4.3.p1.10.m10.1.1.1.cmml" xref="S4.3.p1.10.m10.1.1.1"></in><ci id="S4.3.p1.10.m10.1.1.2.cmml" xref="S4.3.p1.10.m10.1.1.2">π</ci><ci id="S4.3.p1.10.m10.1.1.3.cmml" xref="S4.3.p1.10.m10.1.1.3">π΅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.10.m10.1c">b\in B</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.10.m10.1d">italic_b β italic_B</annotation></semantics></math> there exists an ordinal <math alttext="\alpha_{b}<\omega_{1}" class="ltx_Math" display="inline" id="S4.3.p1.11.m11.1"><semantics id="S4.3.p1.11.m11.1a"><mrow id="S4.3.p1.11.m11.1.1" xref="S4.3.p1.11.m11.1.1.cmml"><msub id="S4.3.p1.11.m11.1.1.2" xref="S4.3.p1.11.m11.1.1.2.cmml"><mi id="S4.3.p1.11.m11.1.1.2.2" xref="S4.3.p1.11.m11.1.1.2.2.cmml">Ξ±</mi><mi id="S4.3.p1.11.m11.1.1.2.3" xref="S4.3.p1.11.m11.1.1.2.3.cmml">b</mi></msub><mo id="S4.3.p1.11.m11.1.1.1" xref="S4.3.p1.11.m11.1.1.1.cmml"><</mo><msub id="S4.3.p1.11.m11.1.1.3" xref="S4.3.p1.11.m11.1.1.3.cmml"><mi id="S4.3.p1.11.m11.1.1.3.2" xref="S4.3.p1.11.m11.1.1.3.2.cmml">Ο</mi><mn id="S4.3.p1.11.m11.1.1.3.3" xref="S4.3.p1.11.m11.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.11.m11.1b"><apply id="S4.3.p1.11.m11.1.1.cmml" xref="S4.3.p1.11.m11.1.1"><lt id="S4.3.p1.11.m11.1.1.1.cmml" xref="S4.3.p1.11.m11.1.1.1"></lt><apply id="S4.3.p1.11.m11.1.1.2.cmml" xref="S4.3.p1.11.m11.1.1.2"><csymbol cd="ambiguous" id="S4.3.p1.11.m11.1.1.2.1.cmml" xref="S4.3.p1.11.m11.1.1.2">subscript</csymbol><ci id="S4.3.p1.11.m11.1.1.2.2.cmml" xref="S4.3.p1.11.m11.1.1.2.2">πΌ</ci><ci id="S4.3.p1.11.m11.1.1.2.3.cmml" xref="S4.3.p1.11.m11.1.1.2.3">π</ci></apply><apply id="S4.3.p1.11.m11.1.1.3.cmml" xref="S4.3.p1.11.m11.1.1.3"><csymbol cd="ambiguous" id="S4.3.p1.11.m11.1.1.3.1.cmml" xref="S4.3.p1.11.m11.1.1.3">subscript</csymbol><ci id="S4.3.p1.11.m11.1.1.3.2.cmml" xref="S4.3.p1.11.m11.1.1.3.2">π</ci><cn id="S4.3.p1.11.m11.1.1.3.3.cmml" type="integer" xref="S4.3.p1.11.m11.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.11.m11.1c">\alpha_{b}<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.11.m11.1d">italic_Ξ± start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT < italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="bl_{\xi}=bl_{\alpha_{b}}" class="ltx_Math" display="inline" id="S4.3.p1.12.m12.1"><semantics id="S4.3.p1.12.m12.1a"><mrow id="S4.3.p1.12.m12.1.1" xref="S4.3.p1.12.m12.1.1.cmml"><mrow id="S4.3.p1.12.m12.1.1.2" xref="S4.3.p1.12.m12.1.1.2.cmml"><mi id="S4.3.p1.12.m12.1.1.2.2" xref="S4.3.p1.12.m12.1.1.2.2.cmml">b</mi><mo id="S4.3.p1.12.m12.1.1.2.1" xref="S4.3.p1.12.m12.1.1.2.1.cmml">β’</mo><msub id="S4.3.p1.12.m12.1.1.2.3" xref="S4.3.p1.12.m12.1.1.2.3.cmml"><mi id="S4.3.p1.12.m12.1.1.2.3.2" xref="S4.3.p1.12.m12.1.1.2.3.2.cmml">l</mi><mi id="S4.3.p1.12.m12.1.1.2.3.3" xref="S4.3.p1.12.m12.1.1.2.3.3.cmml">ΞΎ</mi></msub></mrow><mo id="S4.3.p1.12.m12.1.1.1" xref="S4.3.p1.12.m12.1.1.1.cmml">=</mo><mrow id="S4.3.p1.12.m12.1.1.3" xref="S4.3.p1.12.m12.1.1.3.cmml"><mi id="S4.3.p1.12.m12.1.1.3.2" xref="S4.3.p1.12.m12.1.1.3.2.cmml">b</mi><mo id="S4.3.p1.12.m12.1.1.3.1" xref="S4.3.p1.12.m12.1.1.3.1.cmml">β’</mo><msub id="S4.3.p1.12.m12.1.1.3.3" xref="S4.3.p1.12.m12.1.1.3.3.cmml"><mi id="S4.3.p1.12.m12.1.1.3.3.2" xref="S4.3.p1.12.m12.1.1.3.3.2.cmml">l</mi><msub id="S4.3.p1.12.m12.1.1.3.3.3" xref="S4.3.p1.12.m12.1.1.3.3.3.cmml"><mi id="S4.3.p1.12.m12.1.1.3.3.3.2" xref="S4.3.p1.12.m12.1.1.3.3.3.2.cmml">Ξ±</mi><mi id="S4.3.p1.12.m12.1.1.3.3.3.3" xref="S4.3.p1.12.m12.1.1.3.3.3.3.cmml">b</mi></msub></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.12.m12.1b"><apply id="S4.3.p1.12.m12.1.1.cmml" xref="S4.3.p1.12.m12.1.1"><eq id="S4.3.p1.12.m12.1.1.1.cmml" xref="S4.3.p1.12.m12.1.1.1"></eq><apply id="S4.3.p1.12.m12.1.1.2.cmml" xref="S4.3.p1.12.m12.1.1.2"><times id="S4.3.p1.12.m12.1.1.2.1.cmml" xref="S4.3.p1.12.m12.1.1.2.1"></times><ci id="S4.3.p1.12.m12.1.1.2.2.cmml" xref="S4.3.p1.12.m12.1.1.2.2">π</ci><apply id="S4.3.p1.12.m12.1.1.2.3.cmml" xref="S4.3.p1.12.m12.1.1.2.3"><csymbol cd="ambiguous" id="S4.3.p1.12.m12.1.1.2.3.1.cmml" xref="S4.3.p1.12.m12.1.1.2.3">subscript</csymbol><ci id="S4.3.p1.12.m12.1.1.2.3.2.cmml" xref="S4.3.p1.12.m12.1.1.2.3.2">π</ci><ci id="S4.3.p1.12.m12.1.1.2.3.3.cmml" xref="S4.3.p1.12.m12.1.1.2.3.3">π</ci></apply></apply><apply id="S4.3.p1.12.m12.1.1.3.cmml" xref="S4.3.p1.12.m12.1.1.3"><times id="S4.3.p1.12.m12.1.1.3.1.cmml" xref="S4.3.p1.12.m12.1.1.3.1"></times><ci id="S4.3.p1.12.m12.1.1.3.2.cmml" xref="S4.3.p1.12.m12.1.1.3.2">π</ci><apply id="S4.3.p1.12.m12.1.1.3.3.cmml" xref="S4.3.p1.12.m12.1.1.3.3"><csymbol cd="ambiguous" id="S4.3.p1.12.m12.1.1.3.3.1.cmml" xref="S4.3.p1.12.m12.1.1.3.3">subscript</csymbol><ci id="S4.3.p1.12.m12.1.1.3.3.2.cmml" xref="S4.3.p1.12.m12.1.1.3.3.2">π</ci><apply id="S4.3.p1.12.m12.1.1.3.3.3.cmml" xref="S4.3.p1.12.m12.1.1.3.3.3"><csymbol cd="ambiguous" id="S4.3.p1.12.m12.1.1.3.3.3.1.cmml" xref="S4.3.p1.12.m12.1.1.3.3.3">subscript</csymbol><ci id="S4.3.p1.12.m12.1.1.3.3.3.2.cmml" xref="S4.3.p1.12.m12.1.1.3.3.3.2">πΌ</ci><ci id="S4.3.p1.12.m12.1.1.3.3.3.3.cmml" xref="S4.3.p1.12.m12.1.1.3.3.3.3">π</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.12.m12.1c">bl_{\xi}=bl_{\alpha_{b}}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.12.m12.1d">italic_b italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT = italic_b italic_l start_POSTSUBSCRIPT italic_Ξ± start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="\xi\geq\alpha_{b}" class="ltx_Math" display="inline" id="S4.3.p1.13.m13.1"><semantics id="S4.3.p1.13.m13.1a"><mrow id="S4.3.p1.13.m13.1.1" xref="S4.3.p1.13.m13.1.1.cmml"><mi id="S4.3.p1.13.m13.1.1.2" xref="S4.3.p1.13.m13.1.1.2.cmml">ΞΎ</mi><mo id="S4.3.p1.13.m13.1.1.1" xref="S4.3.p1.13.m13.1.1.1.cmml">β₯</mo><msub id="S4.3.p1.13.m13.1.1.3" xref="S4.3.p1.13.m13.1.1.3.cmml"><mi id="S4.3.p1.13.m13.1.1.3.2" xref="S4.3.p1.13.m13.1.1.3.2.cmml">Ξ±</mi><mi id="S4.3.p1.13.m13.1.1.3.3" xref="S4.3.p1.13.m13.1.1.3.3.cmml">b</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.13.m13.1b"><apply id="S4.3.p1.13.m13.1.1.cmml" xref="S4.3.p1.13.m13.1.1"><geq id="S4.3.p1.13.m13.1.1.1.cmml" xref="S4.3.p1.13.m13.1.1.1"></geq><ci id="S4.3.p1.13.m13.1.1.2.cmml" xref="S4.3.p1.13.m13.1.1.2">π</ci><apply id="S4.3.p1.13.m13.1.1.3.cmml" xref="S4.3.p1.13.m13.1.1.3"><csymbol cd="ambiguous" id="S4.3.p1.13.m13.1.1.3.1.cmml" xref="S4.3.p1.13.m13.1.1.3">subscript</csymbol><ci id="S4.3.p1.13.m13.1.1.3.2.cmml" xref="S4.3.p1.13.m13.1.1.3.2">πΌ</ci><ci id="S4.3.p1.13.m13.1.1.3.3.cmml" xref="S4.3.p1.13.m13.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.13.m13.1c">\xi\geq\alpha_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.13.m13.1d">italic_ΞΎ β₯ italic_Ξ± start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="\alpha=\sup\{\alpha_{b}:b\in B\}" class="ltx_Math" display="inline" id="S4.3.p1.14.m14.2"><semantics id="S4.3.p1.14.m14.2a"><mrow id="S4.3.p1.14.m14.2.2" xref="S4.3.p1.14.m14.2.2.cmml"><mi id="S4.3.p1.14.m14.2.2.4" xref="S4.3.p1.14.m14.2.2.4.cmml">Ξ±</mi><mo id="S4.3.p1.14.m14.2.2.3" rspace="0.1389em" xref="S4.3.p1.14.m14.2.2.3.cmml">=</mo><mrow id="S4.3.p1.14.m14.2.2.2" xref="S4.3.p1.14.m14.2.2.2.cmml"><mo id="S4.3.p1.14.m14.2.2.2.3" lspace="0.1389em" rspace="0em" xref="S4.3.p1.14.m14.2.2.2.3.cmml">sup</mo><mrow id="S4.3.p1.14.m14.2.2.2.2.2" xref="S4.3.p1.14.m14.2.2.2.2.3.cmml"><mo id="S4.3.p1.14.m14.2.2.2.2.2.3" stretchy="false" xref="S4.3.p1.14.m14.2.2.2.2.3.1.cmml">{</mo><msub id="S4.3.p1.14.m14.1.1.1.1.1.1" xref="S4.3.p1.14.m14.1.1.1.1.1.1.cmml"><mi id="S4.3.p1.14.m14.1.1.1.1.1.1.2" xref="S4.3.p1.14.m14.1.1.1.1.1.1.2.cmml">Ξ±</mi><mi id="S4.3.p1.14.m14.1.1.1.1.1.1.3" xref="S4.3.p1.14.m14.1.1.1.1.1.1.3.cmml">b</mi></msub><mo id="S4.3.p1.14.m14.2.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.3.p1.14.m14.2.2.2.2.3.1.cmml">:</mo><mrow id="S4.3.p1.14.m14.2.2.2.2.2.2" xref="S4.3.p1.14.m14.2.2.2.2.2.2.cmml"><mi id="S4.3.p1.14.m14.2.2.2.2.2.2.2" xref="S4.3.p1.14.m14.2.2.2.2.2.2.2.cmml">b</mi><mo id="S4.3.p1.14.m14.2.2.2.2.2.2.1" xref="S4.3.p1.14.m14.2.2.2.2.2.2.1.cmml">β</mo><mi id="S4.3.p1.14.m14.2.2.2.2.2.2.3" xref="S4.3.p1.14.m14.2.2.2.2.2.2.3.cmml">B</mi></mrow><mo id="S4.3.p1.14.m14.2.2.2.2.2.5" stretchy="false" xref="S4.3.p1.14.m14.2.2.2.2.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.14.m14.2b"><apply id="S4.3.p1.14.m14.2.2.cmml" xref="S4.3.p1.14.m14.2.2"><eq id="S4.3.p1.14.m14.2.2.3.cmml" xref="S4.3.p1.14.m14.2.2.3"></eq><ci id="S4.3.p1.14.m14.2.2.4.cmml" xref="S4.3.p1.14.m14.2.2.4">πΌ</ci><apply id="S4.3.p1.14.m14.2.2.2.cmml" xref="S4.3.p1.14.m14.2.2.2"><csymbol cd="latexml" id="S4.3.p1.14.m14.2.2.2.3.cmml" xref="S4.3.p1.14.m14.2.2.2.3">supremum</csymbol><apply id="S4.3.p1.14.m14.2.2.2.2.3.cmml" xref="S4.3.p1.14.m14.2.2.2.2.2"><csymbol cd="latexml" id="S4.3.p1.14.m14.2.2.2.2.3.1.cmml" xref="S4.3.p1.14.m14.2.2.2.2.2.3">conditional-set</csymbol><apply id="S4.3.p1.14.m14.1.1.1.1.1.1.cmml" xref="S4.3.p1.14.m14.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.3.p1.14.m14.1.1.1.1.1.1.1.cmml" xref="S4.3.p1.14.m14.1.1.1.1.1.1">subscript</csymbol><ci id="S4.3.p1.14.m14.1.1.1.1.1.1.2.cmml" xref="S4.3.p1.14.m14.1.1.1.1.1.1.2">πΌ</ci><ci id="S4.3.p1.14.m14.1.1.1.1.1.1.3.cmml" xref="S4.3.p1.14.m14.1.1.1.1.1.1.3">π</ci></apply><apply id="S4.3.p1.14.m14.2.2.2.2.2.2.cmml" xref="S4.3.p1.14.m14.2.2.2.2.2.2"><in id="S4.3.p1.14.m14.2.2.2.2.2.2.1.cmml" xref="S4.3.p1.14.m14.2.2.2.2.2.2.1"></in><ci id="S4.3.p1.14.m14.2.2.2.2.2.2.2.cmml" xref="S4.3.p1.14.m14.2.2.2.2.2.2.2">π</ci><ci id="S4.3.p1.14.m14.2.2.2.2.2.2.3.cmml" xref="S4.3.p1.14.m14.2.2.2.2.2.2.3">π΅</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.14.m14.2c">\alpha=\sup\{\alpha_{b}:b\in B\}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.14.m14.2d">italic_Ξ± = roman_sup { italic_Ξ± start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT : italic_b β italic_B }</annotation></semantics></math>. Then for each <math alttext="\xi\geq\alpha" class="ltx_Math" display="inline" id="S4.3.p1.15.m15.1"><semantics id="S4.3.p1.15.m15.1a"><mrow id="S4.3.p1.15.m15.1.1" xref="S4.3.p1.15.m15.1.1.cmml"><mi id="S4.3.p1.15.m15.1.1.2" xref="S4.3.p1.15.m15.1.1.2.cmml">ΞΎ</mi><mo id="S4.3.p1.15.m15.1.1.1" xref="S4.3.p1.15.m15.1.1.1.cmml">β₯</mo><mi id="S4.3.p1.15.m15.1.1.3" xref="S4.3.p1.15.m15.1.1.3.cmml">Ξ±</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.15.m15.1b"><apply id="S4.3.p1.15.m15.1.1.cmml" xref="S4.3.p1.15.m15.1.1"><geq id="S4.3.p1.15.m15.1.1.1.cmml" xref="S4.3.p1.15.m15.1.1.1"></geq><ci id="S4.3.p1.15.m15.1.1.2.cmml" xref="S4.3.p1.15.m15.1.1.2">π</ci><ci id="S4.3.p1.15.m15.1.1.3.cmml" xref="S4.3.p1.15.m15.1.1.3">πΌ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.15.m15.1c">\xi\geq\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.15.m15.1d">italic_ΞΎ β₯ italic_Ξ±</annotation></semantics></math> and <math alttext="b\in B" class="ltx_Math" display="inline" id="S4.3.p1.16.m16.1"><semantics id="S4.3.p1.16.m16.1a"><mrow id="S4.3.p1.16.m16.1.1" xref="S4.3.p1.16.m16.1.1.cmml"><mi id="S4.3.p1.16.m16.1.1.2" xref="S4.3.p1.16.m16.1.1.2.cmml">b</mi><mo id="S4.3.p1.16.m16.1.1.1" xref="S4.3.p1.16.m16.1.1.1.cmml">β</mo><mi id="S4.3.p1.16.m16.1.1.3" xref="S4.3.p1.16.m16.1.1.3.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.16.m16.1b"><apply id="S4.3.p1.16.m16.1.1.cmml" xref="S4.3.p1.16.m16.1.1"><in id="S4.3.p1.16.m16.1.1.1.cmml" xref="S4.3.p1.16.m16.1.1.1"></in><ci id="S4.3.p1.16.m16.1.1.2.cmml" xref="S4.3.p1.16.m16.1.1.2">π</ci><ci id="S4.3.p1.16.m16.1.1.3.cmml" xref="S4.3.p1.16.m16.1.1.3">π΅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.16.m16.1c">b\in B</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.16.m16.1d">italic_b β italic_B</annotation></semantics></math> we have <math alttext="bl_{\xi}=bl_{\alpha}" class="ltx_Math" display="inline" id="S4.3.p1.17.m17.1"><semantics id="S4.3.p1.17.m17.1a"><mrow id="S4.3.p1.17.m17.1.1" xref="S4.3.p1.17.m17.1.1.cmml"><mrow id="S4.3.p1.17.m17.1.1.2" xref="S4.3.p1.17.m17.1.1.2.cmml"><mi id="S4.3.p1.17.m17.1.1.2.2" xref="S4.3.p1.17.m17.1.1.2.2.cmml">b</mi><mo id="S4.3.p1.17.m17.1.1.2.1" xref="S4.3.p1.17.m17.1.1.2.1.cmml">β’</mo><msub id="S4.3.p1.17.m17.1.1.2.3" xref="S4.3.p1.17.m17.1.1.2.3.cmml"><mi id="S4.3.p1.17.m17.1.1.2.3.2" xref="S4.3.p1.17.m17.1.1.2.3.2.cmml">l</mi><mi id="S4.3.p1.17.m17.1.1.2.3.3" xref="S4.3.p1.17.m17.1.1.2.3.3.cmml">ΞΎ</mi></msub></mrow><mo id="S4.3.p1.17.m17.1.1.1" xref="S4.3.p1.17.m17.1.1.1.cmml">=</mo><mrow id="S4.3.p1.17.m17.1.1.3" xref="S4.3.p1.17.m17.1.1.3.cmml"><mi id="S4.3.p1.17.m17.1.1.3.2" xref="S4.3.p1.17.m17.1.1.3.2.cmml">b</mi><mo id="S4.3.p1.17.m17.1.1.3.1" xref="S4.3.p1.17.m17.1.1.3.1.cmml">β’</mo><msub id="S4.3.p1.17.m17.1.1.3.3" xref="S4.3.p1.17.m17.1.1.3.3.cmml"><mi id="S4.3.p1.17.m17.1.1.3.3.2" xref="S4.3.p1.17.m17.1.1.3.3.2.cmml">l</mi><mi id="S4.3.p1.17.m17.1.1.3.3.3" xref="S4.3.p1.17.m17.1.1.3.3.3.cmml">Ξ±</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.17.m17.1b"><apply id="S4.3.p1.17.m17.1.1.cmml" xref="S4.3.p1.17.m17.1.1"><eq id="S4.3.p1.17.m17.1.1.1.cmml" xref="S4.3.p1.17.m17.1.1.1"></eq><apply id="S4.3.p1.17.m17.1.1.2.cmml" xref="S4.3.p1.17.m17.1.1.2"><times id="S4.3.p1.17.m17.1.1.2.1.cmml" xref="S4.3.p1.17.m17.1.1.2.1"></times><ci id="S4.3.p1.17.m17.1.1.2.2.cmml" xref="S4.3.p1.17.m17.1.1.2.2">π</ci><apply id="S4.3.p1.17.m17.1.1.2.3.cmml" xref="S4.3.p1.17.m17.1.1.2.3"><csymbol cd="ambiguous" id="S4.3.p1.17.m17.1.1.2.3.1.cmml" xref="S4.3.p1.17.m17.1.1.2.3">subscript</csymbol><ci id="S4.3.p1.17.m17.1.1.2.3.2.cmml" xref="S4.3.p1.17.m17.1.1.2.3.2">π</ci><ci id="S4.3.p1.17.m17.1.1.2.3.3.cmml" xref="S4.3.p1.17.m17.1.1.2.3.3">π</ci></apply></apply><apply id="S4.3.p1.17.m17.1.1.3.cmml" xref="S4.3.p1.17.m17.1.1.3"><times id="S4.3.p1.17.m17.1.1.3.1.cmml" xref="S4.3.p1.17.m17.1.1.3.1"></times><ci id="S4.3.p1.17.m17.1.1.3.2.cmml" xref="S4.3.p1.17.m17.1.1.3.2">π</ci><apply id="S4.3.p1.17.m17.1.1.3.3.cmml" xref="S4.3.p1.17.m17.1.1.3.3"><csymbol cd="ambiguous" id="S4.3.p1.17.m17.1.1.3.3.1.cmml" xref="S4.3.p1.17.m17.1.1.3.3">subscript</csymbol><ci id="S4.3.p1.17.m17.1.1.3.3.2.cmml" xref="S4.3.p1.17.m17.1.1.3.3.2">π</ci><ci id="S4.3.p1.17.m17.1.1.3.3.3.cmml" xref="S4.3.p1.17.m17.1.1.3.3.3">πΌ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.17.m17.1c">bl_{\xi}=bl_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.17.m17.1d">italic_b italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT = italic_b italic_l start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="z\in\overline{B}" class="ltx_Math" display="inline" id="S4.3.p1.18.m18.1"><semantics id="S4.3.p1.18.m18.1a"><mrow id="S4.3.p1.18.m18.1.1" xref="S4.3.p1.18.m18.1.1.cmml"><mi id="S4.3.p1.18.m18.1.1.2" xref="S4.3.p1.18.m18.1.1.2.cmml">z</mi><mo id="S4.3.p1.18.m18.1.1.1" xref="S4.3.p1.18.m18.1.1.1.cmml">β</mo><mover accent="true" id="S4.3.p1.18.m18.1.1.3" xref="S4.3.p1.18.m18.1.1.3.cmml"><mi id="S4.3.p1.18.m18.1.1.3.2" xref="S4.3.p1.18.m18.1.1.3.2.cmml">B</mi><mo id="S4.3.p1.18.m18.1.1.3.1" xref="S4.3.p1.18.m18.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.18.m18.1b"><apply id="S4.3.p1.18.m18.1.1.cmml" xref="S4.3.p1.18.m18.1.1"><in id="S4.3.p1.18.m18.1.1.1.cmml" xref="S4.3.p1.18.m18.1.1.1"></in><ci id="S4.3.p1.18.m18.1.1.2.cmml" xref="S4.3.p1.18.m18.1.1.2">π§</ci><apply id="S4.3.p1.18.m18.1.1.3.cmml" xref="S4.3.p1.18.m18.1.1.3"><ci id="S4.3.p1.18.m18.1.1.3.1.cmml" xref="S4.3.p1.18.m18.1.1.3.1">Β―</ci><ci id="S4.3.p1.18.m18.1.1.3.2.cmml" xref="S4.3.p1.18.m18.1.1.3.2">π΅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.18.m18.1c">z\in\overline{B}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.18.m18.1d">italic_z β overΒ― start_ARG italic_B end_ARG</annotation></semantics></math>, for each open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S4.3.p1.19.m19.1"><semantics id="S4.3.p1.19.m19.1a"><mi id="S4.3.p1.19.m19.1.1" xref="S4.3.p1.19.m19.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S4.3.p1.19.m19.1b"><ci id="S4.3.p1.19.m19.1.1.cmml" xref="S4.3.p1.19.m19.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.19.m19.1c">U</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.19.m19.1d">italic_U</annotation></semantics></math> of <math alttext="z" class="ltx_Math" display="inline" id="S4.3.p1.20.m20.1"><semantics id="S4.3.p1.20.m20.1a"><mi id="S4.3.p1.20.m20.1.1" xref="S4.3.p1.20.m20.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.3.p1.20.m20.1b"><ci id="S4.3.p1.20.m20.1.1.cmml" xref="S4.3.p1.20.m20.1.1">π§</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.20.m20.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.20.m20.1d">italic_z</annotation></semantics></math> and <math alttext="\xi\geq\alpha" class="ltx_Math" display="inline" id="S4.3.p1.21.m21.1"><semantics id="S4.3.p1.21.m21.1a"><mrow id="S4.3.p1.21.m21.1.1" xref="S4.3.p1.21.m21.1.1.cmml"><mi id="S4.3.p1.21.m21.1.1.2" xref="S4.3.p1.21.m21.1.1.2.cmml">ΞΎ</mi><mo id="S4.3.p1.21.m21.1.1.1" xref="S4.3.p1.21.m21.1.1.1.cmml">β₯</mo><mi id="S4.3.p1.21.m21.1.1.3" xref="S4.3.p1.21.m21.1.1.3.cmml">Ξ±</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.21.m21.1b"><apply id="S4.3.p1.21.m21.1.1.cmml" xref="S4.3.p1.21.m21.1.1"><geq id="S4.3.p1.21.m21.1.1.1.cmml" xref="S4.3.p1.21.m21.1.1.1"></geq><ci id="S4.3.p1.21.m21.1.1.2.cmml" xref="S4.3.p1.21.m21.1.1.2">π</ci><ci id="S4.3.p1.21.m21.1.1.3.cmml" xref="S4.3.p1.21.m21.1.1.3">πΌ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.21.m21.1c">\xi\geq\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.21.m21.1d">italic_ΞΎ β₯ italic_Ξ±</annotation></semantics></math> we have <math alttext="Ul_{\alpha}\cap Ul_{\xi}\neq\varnothing" class="ltx_Math" display="inline" id="S4.3.p1.22.m22.1"><semantics id="S4.3.p1.22.m22.1a"><mrow id="S4.3.p1.22.m22.1.1" xref="S4.3.p1.22.m22.1.1.cmml"><mrow id="S4.3.p1.22.m22.1.1.2" xref="S4.3.p1.22.m22.1.1.2.cmml"><mrow id="S4.3.p1.22.m22.1.1.2.2" xref="S4.3.p1.22.m22.1.1.2.2.cmml"><mi id="S4.3.p1.22.m22.1.1.2.2.2" xref="S4.3.p1.22.m22.1.1.2.2.2.cmml">U</mi><mo id="S4.3.p1.22.m22.1.1.2.2.1" xref="S4.3.p1.22.m22.1.1.2.2.1.cmml">β’</mo><msub id="S4.3.p1.22.m22.1.1.2.2.3" xref="S4.3.p1.22.m22.1.1.2.2.3.cmml"><mi id="S4.3.p1.22.m22.1.1.2.2.3.2" xref="S4.3.p1.22.m22.1.1.2.2.3.2.cmml">l</mi><mi id="S4.3.p1.22.m22.1.1.2.2.3.3" xref="S4.3.p1.22.m22.1.1.2.2.3.3.cmml">Ξ±</mi></msub></mrow><mo id="S4.3.p1.22.m22.1.1.2.1" xref="S4.3.p1.22.m22.1.1.2.1.cmml">β©</mo><mrow id="S4.3.p1.22.m22.1.1.2.3" xref="S4.3.p1.22.m22.1.1.2.3.cmml"><mi id="S4.3.p1.22.m22.1.1.2.3.2" xref="S4.3.p1.22.m22.1.1.2.3.2.cmml">U</mi><mo id="S4.3.p1.22.m22.1.1.2.3.1" xref="S4.3.p1.22.m22.1.1.2.3.1.cmml">β’</mo><msub id="S4.3.p1.22.m22.1.1.2.3.3" xref="S4.3.p1.22.m22.1.1.2.3.3.cmml"><mi id="S4.3.p1.22.m22.1.1.2.3.3.2" xref="S4.3.p1.22.m22.1.1.2.3.3.2.cmml">l</mi><mi id="S4.3.p1.22.m22.1.1.2.3.3.3" xref="S4.3.p1.22.m22.1.1.2.3.3.3.cmml">ΞΎ</mi></msub></mrow></mrow><mo id="S4.3.p1.22.m22.1.1.1" xref="S4.3.p1.22.m22.1.1.1.cmml">β </mo><mi id="S4.3.p1.22.m22.1.1.3" mathvariant="normal" xref="S4.3.p1.22.m22.1.1.3.cmml">β </mi></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.22.m22.1b"><apply id="S4.3.p1.22.m22.1.1.cmml" xref="S4.3.p1.22.m22.1.1"><neq id="S4.3.p1.22.m22.1.1.1.cmml" xref="S4.3.p1.22.m22.1.1.1"></neq><apply id="S4.3.p1.22.m22.1.1.2.cmml" xref="S4.3.p1.22.m22.1.1.2"><intersect id="S4.3.p1.22.m22.1.1.2.1.cmml" xref="S4.3.p1.22.m22.1.1.2.1"></intersect><apply id="S4.3.p1.22.m22.1.1.2.2.cmml" xref="S4.3.p1.22.m22.1.1.2.2"><times id="S4.3.p1.22.m22.1.1.2.2.1.cmml" xref="S4.3.p1.22.m22.1.1.2.2.1"></times><ci id="S4.3.p1.22.m22.1.1.2.2.2.cmml" xref="S4.3.p1.22.m22.1.1.2.2.2">π</ci><apply id="S4.3.p1.22.m22.1.1.2.2.3.cmml" xref="S4.3.p1.22.m22.1.1.2.2.3"><csymbol cd="ambiguous" id="S4.3.p1.22.m22.1.1.2.2.3.1.cmml" xref="S4.3.p1.22.m22.1.1.2.2.3">subscript</csymbol><ci id="S4.3.p1.22.m22.1.1.2.2.3.2.cmml" xref="S4.3.p1.22.m22.1.1.2.2.3.2">π</ci><ci id="S4.3.p1.22.m22.1.1.2.2.3.3.cmml" xref="S4.3.p1.22.m22.1.1.2.2.3.3">πΌ</ci></apply></apply><apply id="S4.3.p1.22.m22.1.1.2.3.cmml" xref="S4.3.p1.22.m22.1.1.2.3"><times id="S4.3.p1.22.m22.1.1.2.3.1.cmml" xref="S4.3.p1.22.m22.1.1.2.3.1"></times><ci id="S4.3.p1.22.m22.1.1.2.3.2.cmml" xref="S4.3.p1.22.m22.1.1.2.3.2">π</ci><apply id="S4.3.p1.22.m22.1.1.2.3.3.cmml" xref="S4.3.p1.22.m22.1.1.2.3.3"><csymbol cd="ambiguous" id="S4.3.p1.22.m22.1.1.2.3.3.1.cmml" xref="S4.3.p1.22.m22.1.1.2.3.3">subscript</csymbol><ci id="S4.3.p1.22.m22.1.1.2.3.3.2.cmml" xref="S4.3.p1.22.m22.1.1.2.3.3.2">π</ci><ci id="S4.3.p1.22.m22.1.1.2.3.3.3.cmml" xref="S4.3.p1.22.m22.1.1.2.3.3.3">π</ci></apply></apply></apply><emptyset id="S4.3.p1.22.m22.1.1.3.cmml" xref="S4.3.p1.22.m22.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.22.m22.1c">Ul_{\alpha}\cap Ul_{\xi}\neq\varnothing</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.22.m22.1d">italic_U italic_l start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT β© italic_U italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT β β </annotation></semantics></math>. Taking into account that <math alttext="X" class="ltx_Math" display="inline" id="S4.3.p1.23.m23.1"><semantics id="S4.3.p1.23.m23.1a"><mi id="S4.3.p1.23.m23.1.1" xref="S4.3.p1.23.m23.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.3.p1.23.m23.1b"><ci id="S4.3.p1.23.m23.1.1.cmml" xref="S4.3.p1.23.m23.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.23.m23.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.23.m23.1d">italic_X</annotation></semantics></math> is a Hausdorff semitopological semilattice, we conclude that <math alttext="zl_{\alpha}=zl_{\xi}" class="ltx_Math" display="inline" id="S4.3.p1.24.m24.1"><semantics id="S4.3.p1.24.m24.1a"><mrow id="S4.3.p1.24.m24.1.1" xref="S4.3.p1.24.m24.1.1.cmml"><mrow id="S4.3.p1.24.m24.1.1.2" xref="S4.3.p1.24.m24.1.1.2.cmml"><mi id="S4.3.p1.24.m24.1.1.2.2" xref="S4.3.p1.24.m24.1.1.2.2.cmml">z</mi><mo id="S4.3.p1.24.m24.1.1.2.1" xref="S4.3.p1.24.m24.1.1.2.1.cmml">β’</mo><msub id="S4.3.p1.24.m24.1.1.2.3" xref="S4.3.p1.24.m24.1.1.2.3.cmml"><mi id="S4.3.p1.24.m24.1.1.2.3.2" xref="S4.3.p1.24.m24.1.1.2.3.2.cmml">l</mi><mi id="S4.3.p1.24.m24.1.1.2.3.3" xref="S4.3.p1.24.m24.1.1.2.3.3.cmml">Ξ±</mi></msub></mrow><mo id="S4.3.p1.24.m24.1.1.1" xref="S4.3.p1.24.m24.1.1.1.cmml">=</mo><mrow id="S4.3.p1.24.m24.1.1.3" xref="S4.3.p1.24.m24.1.1.3.cmml"><mi id="S4.3.p1.24.m24.1.1.3.2" xref="S4.3.p1.24.m24.1.1.3.2.cmml">z</mi><mo id="S4.3.p1.24.m24.1.1.3.1" xref="S4.3.p1.24.m24.1.1.3.1.cmml">β’</mo><msub id="S4.3.p1.24.m24.1.1.3.3" xref="S4.3.p1.24.m24.1.1.3.3.cmml"><mi id="S4.3.p1.24.m24.1.1.3.3.2" xref="S4.3.p1.24.m24.1.1.3.3.2.cmml">l</mi><mi id="S4.3.p1.24.m24.1.1.3.3.3" xref="S4.3.p1.24.m24.1.1.3.3.3.cmml">ΞΎ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.24.m24.1b"><apply id="S4.3.p1.24.m24.1.1.cmml" xref="S4.3.p1.24.m24.1.1"><eq id="S4.3.p1.24.m24.1.1.1.cmml" xref="S4.3.p1.24.m24.1.1.1"></eq><apply id="S4.3.p1.24.m24.1.1.2.cmml" xref="S4.3.p1.24.m24.1.1.2"><times id="S4.3.p1.24.m24.1.1.2.1.cmml" xref="S4.3.p1.24.m24.1.1.2.1"></times><ci id="S4.3.p1.24.m24.1.1.2.2.cmml" xref="S4.3.p1.24.m24.1.1.2.2">π§</ci><apply id="S4.3.p1.24.m24.1.1.2.3.cmml" xref="S4.3.p1.24.m24.1.1.2.3"><csymbol cd="ambiguous" id="S4.3.p1.24.m24.1.1.2.3.1.cmml" xref="S4.3.p1.24.m24.1.1.2.3">subscript</csymbol><ci id="S4.3.p1.24.m24.1.1.2.3.2.cmml" xref="S4.3.p1.24.m24.1.1.2.3.2">π</ci><ci id="S4.3.p1.24.m24.1.1.2.3.3.cmml" xref="S4.3.p1.24.m24.1.1.2.3.3">πΌ</ci></apply></apply><apply id="S4.3.p1.24.m24.1.1.3.cmml" xref="S4.3.p1.24.m24.1.1.3"><times id="S4.3.p1.24.m24.1.1.3.1.cmml" xref="S4.3.p1.24.m24.1.1.3.1"></times><ci id="S4.3.p1.24.m24.1.1.3.2.cmml" xref="S4.3.p1.24.m24.1.1.3.2">π§</ci><apply id="S4.3.p1.24.m24.1.1.3.3.cmml" xref="S4.3.p1.24.m24.1.1.3.3"><csymbol cd="ambiguous" id="S4.3.p1.24.m24.1.1.3.3.1.cmml" xref="S4.3.p1.24.m24.1.1.3.3">subscript</csymbol><ci id="S4.3.p1.24.m24.1.1.3.3.2.cmml" xref="S4.3.p1.24.m24.1.1.3.3.2">π</ci><ci id="S4.3.p1.24.m24.1.1.3.3.3.cmml" xref="S4.3.p1.24.m24.1.1.3.3.3">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.24.m24.1c">zl_{\alpha}=zl_{\xi}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.24.m24.1d">italic_z italic_l start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT = italic_z italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="\xi\geq\alpha" class="ltx_Math" display="inline" id="S4.3.p1.25.m25.1"><semantics id="S4.3.p1.25.m25.1a"><mrow id="S4.3.p1.25.m25.1.1" xref="S4.3.p1.25.m25.1.1.cmml"><mi id="S4.3.p1.25.m25.1.1.2" xref="S4.3.p1.25.m25.1.1.2.cmml">ΞΎ</mi><mo id="S4.3.p1.25.m25.1.1.1" xref="S4.3.p1.25.m25.1.1.1.cmml">β₯</mo><mi id="S4.3.p1.25.m25.1.1.3" xref="S4.3.p1.25.m25.1.1.3.cmml">Ξ±</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.25.m25.1b"><apply id="S4.3.p1.25.m25.1.1.cmml" xref="S4.3.p1.25.m25.1.1"><geq id="S4.3.p1.25.m25.1.1.1.cmml" xref="S4.3.p1.25.m25.1.1.1"></geq><ci id="S4.3.p1.25.m25.1.1.2.cmml" xref="S4.3.p1.25.m25.1.1.2">π</ci><ci id="S4.3.p1.25.m25.1.1.3.cmml" xref="S4.3.p1.25.m25.1.1.3">πΌ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.25.m25.1c">\xi\geq\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.25.m25.1d">italic_ΞΎ β₯ italic_Ξ±</annotation></semantics></math>. It follows that <math alttext="|zL|<\omega_{1}" class="ltx_Math" display="inline" id="S4.3.p1.26.m26.1"><semantics id="S4.3.p1.26.m26.1a"><mrow id="S4.3.p1.26.m26.1.1" xref="S4.3.p1.26.m26.1.1.cmml"><mrow id="S4.3.p1.26.m26.1.1.1.1" xref="S4.3.p1.26.m26.1.1.1.2.cmml"><mo id="S4.3.p1.26.m26.1.1.1.1.2" stretchy="false" xref="S4.3.p1.26.m26.1.1.1.2.1.cmml">|</mo><mrow id="S4.3.p1.26.m26.1.1.1.1.1" xref="S4.3.p1.26.m26.1.1.1.1.1.cmml"><mi id="S4.3.p1.26.m26.1.1.1.1.1.2" xref="S4.3.p1.26.m26.1.1.1.1.1.2.cmml">z</mi><mo id="S4.3.p1.26.m26.1.1.1.1.1.1" xref="S4.3.p1.26.m26.1.1.1.1.1.1.cmml">β’</mo><mi id="S4.3.p1.26.m26.1.1.1.1.1.3" xref="S4.3.p1.26.m26.1.1.1.1.1.3.cmml">L</mi></mrow><mo id="S4.3.p1.26.m26.1.1.1.1.3" stretchy="false" xref="S4.3.p1.26.m26.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.3.p1.26.m26.1.1.2" xref="S4.3.p1.26.m26.1.1.2.cmml"><</mo><msub id="S4.3.p1.26.m26.1.1.3" xref="S4.3.p1.26.m26.1.1.3.cmml"><mi id="S4.3.p1.26.m26.1.1.3.2" xref="S4.3.p1.26.m26.1.1.3.2.cmml">Ο</mi><mn id="S4.3.p1.26.m26.1.1.3.3" xref="S4.3.p1.26.m26.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.26.m26.1b"><apply id="S4.3.p1.26.m26.1.1.cmml" xref="S4.3.p1.26.m26.1.1"><lt id="S4.3.p1.26.m26.1.1.2.cmml" xref="S4.3.p1.26.m26.1.1.2"></lt><apply id="S4.3.p1.26.m26.1.1.1.2.cmml" xref="S4.3.p1.26.m26.1.1.1.1"><abs id="S4.3.p1.26.m26.1.1.1.2.1.cmml" xref="S4.3.p1.26.m26.1.1.1.1.2"></abs><apply id="S4.3.p1.26.m26.1.1.1.1.1.cmml" xref="S4.3.p1.26.m26.1.1.1.1.1"><times id="S4.3.p1.26.m26.1.1.1.1.1.1.cmml" xref="S4.3.p1.26.m26.1.1.1.1.1.1"></times><ci id="S4.3.p1.26.m26.1.1.1.1.1.2.cmml" xref="S4.3.p1.26.m26.1.1.1.1.1.2">π§</ci><ci id="S4.3.p1.26.m26.1.1.1.1.1.3.cmml" xref="S4.3.p1.26.m26.1.1.1.1.1.3">πΏ</ci></apply></apply><apply id="S4.3.p1.26.m26.1.1.3.cmml" xref="S4.3.p1.26.m26.1.1.3"><csymbol cd="ambiguous" id="S4.3.p1.26.m26.1.1.3.1.cmml" xref="S4.3.p1.26.m26.1.1.3">subscript</csymbol><ci id="S4.3.p1.26.m26.1.1.3.2.cmml" xref="S4.3.p1.26.m26.1.1.3.2">π</ci><cn id="S4.3.p1.26.m26.1.1.3.3.cmml" type="integer" xref="S4.3.p1.26.m26.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.26.m26.1c">|zL|<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.26.m26.1d">| italic_z italic_L | < italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and, as such, <math alttext="z\in I_{L}" class="ltx_Math" display="inline" id="S4.3.p1.27.m27.1"><semantics id="S4.3.p1.27.m27.1a"><mrow id="S4.3.p1.27.m27.1.1" xref="S4.3.p1.27.m27.1.1.cmml"><mi id="S4.3.p1.27.m27.1.1.2" xref="S4.3.p1.27.m27.1.1.2.cmml">z</mi><mo id="S4.3.p1.27.m27.1.1.1" xref="S4.3.p1.27.m27.1.1.1.cmml">β</mo><msub id="S4.3.p1.27.m27.1.1.3" xref="S4.3.p1.27.m27.1.1.3.cmml"><mi id="S4.3.p1.27.m27.1.1.3.2" xref="S4.3.p1.27.m27.1.1.3.2.cmml">I</mi><mi id="S4.3.p1.27.m27.1.1.3.3" xref="S4.3.p1.27.m27.1.1.3.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.3.p1.27.m27.1b"><apply id="S4.3.p1.27.m27.1.1.cmml" xref="S4.3.p1.27.m27.1.1"><in id="S4.3.p1.27.m27.1.1.1.cmml" xref="S4.3.p1.27.m27.1.1.1"></in><ci id="S4.3.p1.27.m27.1.1.2.cmml" xref="S4.3.p1.27.m27.1.1.2">π§</ci><apply id="S4.3.p1.27.m27.1.1.3.cmml" xref="S4.3.p1.27.m27.1.1.3"><csymbol cd="ambiguous" id="S4.3.p1.27.m27.1.1.3.1.cmml" xref="S4.3.p1.27.m27.1.1.3">subscript</csymbol><ci id="S4.3.p1.27.m27.1.1.3.2.cmml" xref="S4.3.p1.27.m27.1.1.3.2">πΌ</ci><ci id="S4.3.p1.27.m27.1.1.3.3.cmml" xref="S4.3.p1.27.m27.1.1.3.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.3.p1.27.m27.1c">z\in I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.3.p1.27.m27.1d">italic_z β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>. β</p> </div> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.1">Similarly one can show the following.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem5.1.1.1">Lemma 4.5</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem5.p1"> <p class="ltx_p" id="S4.Thmtheorem5.p1.4"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem5.p1.4.4">Let <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.1.1.m1.1"><semantics id="S4.Thmtheorem5.p1.1.1.m1.1a"><mi id="S4.Thmtheorem5.p1.1.1.m1.1.1" xref="S4.Thmtheorem5.p1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.1.1.m1.1b"><ci id="S4.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.1.1.m1.1d">italic_X</annotation></semantics></math> be a countably tight semitopological semilattice and <math alttext="L\subseteq X" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.2.2.m2.1"><semantics id="S4.Thmtheorem5.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem5.p1.2.2.m2.1.1" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem5.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.2.cmml">L</mi><mo id="S4.Thmtheorem5.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.1.cmml">β</mo><mi id="S4.Thmtheorem5.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.2.2.m2.1b"><apply id="S4.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1"><subset id="S4.Thmtheorem5.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.1"></subset><ci id="S4.Thmtheorem5.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.2">πΏ</ci><ci id="S4.Thmtheorem5.p1.2.2.m2.1.1.3.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.2.2.m2.1c">L\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.2.2.m2.1d">italic_L β italic_X</annotation></semantics></math> be a chain isomorphic to <math alttext="(\omega_{1},\max)" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.3.3.m3.2"><semantics id="S4.Thmtheorem5.p1.3.3.m3.2a"><mrow id="S4.Thmtheorem5.p1.3.3.m3.2.2.1" xref="S4.Thmtheorem5.p1.3.3.m3.2.2.2.cmml"><mo id="S4.Thmtheorem5.p1.3.3.m3.2.2.1.2" stretchy="false" xref="S4.Thmtheorem5.p1.3.3.m3.2.2.2.cmml">(</mo><msub id="S4.Thmtheorem5.p1.3.3.m3.2.2.1.1" xref="S4.Thmtheorem5.p1.3.3.m3.2.2.1.1.cmml"><mi id="S4.Thmtheorem5.p1.3.3.m3.2.2.1.1.2" xref="S4.Thmtheorem5.p1.3.3.m3.2.2.1.1.2.cmml">Ο</mi><mn id="S4.Thmtheorem5.p1.3.3.m3.2.2.1.1.3" xref="S4.Thmtheorem5.p1.3.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem5.p1.3.3.m3.2.2.1.3" xref="S4.Thmtheorem5.p1.3.3.m3.2.2.2.cmml">,</mo><mi id="S4.Thmtheorem5.p1.3.3.m3.1.1" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.cmml">max</mi><mo id="S4.Thmtheorem5.p1.3.3.m3.2.2.1.4" stretchy="false" xref="S4.Thmtheorem5.p1.3.3.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.3.3.m3.2b"><interval closure="open" id="S4.Thmtheorem5.p1.3.3.m3.2.2.2.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.2.2.1"><apply id="S4.Thmtheorem5.p1.3.3.m3.2.2.1.1.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.3.3.m3.2.2.1.1.1.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.2.2.1.1">subscript</csymbol><ci id="S4.Thmtheorem5.p1.3.3.m3.2.2.1.1.2.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.2.2.1.1.2">π</ci><cn id="S4.Thmtheorem5.p1.3.3.m3.2.2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem5.p1.3.3.m3.2.2.1.1.3">1</cn></apply><max id="S4.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1"></max></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.3.3.m3.2c">(\omega_{1},\max)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.3.3.m3.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_max )</annotation></semantics></math>. Then <math alttext="I_{L}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.4.4.m4.1"><semantics id="S4.Thmtheorem5.p1.4.4.m4.1a"><msub id="S4.Thmtheorem5.p1.4.4.m4.1.1" xref="S4.Thmtheorem5.p1.4.4.m4.1.1.cmml"><mi id="S4.Thmtheorem5.p1.4.4.m4.1.1.2" xref="S4.Thmtheorem5.p1.4.4.m4.1.1.2.cmml">I</mi><mi id="S4.Thmtheorem5.p1.4.4.m4.1.1.3" xref="S4.Thmtheorem5.p1.4.4.m4.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.4.4.m4.1b"><apply id="S4.Thmtheorem5.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem5.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem5.p1.4.4.m4.1.1">subscript</csymbol><ci id="S4.Thmtheorem5.p1.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem5.p1.4.4.m4.1.1.2">πΌ</ci><ci id="S4.Thmtheorem5.p1.4.4.m4.1.1.3.cmml" xref="S4.Thmtheorem5.p1.4.4.m4.1.1.3">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.4.4.m4.1c">I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.4.4.m4.1d">italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> is a closed ideal.</span></p> </div> </div> <div class="ltx_para" id="S4.p3"> <p class="ltx_p" id="S4.p3.2">Note that the order topologies on semilattices <math alttext="(\omega_{1},\max)" class="ltx_Math" display="inline" id="S4.p3.1.m1.2"><semantics id="S4.p3.1.m1.2a"><mrow id="S4.p3.1.m1.2.2.1" xref="S4.p3.1.m1.2.2.2.cmml"><mo id="S4.p3.1.m1.2.2.1.2" stretchy="false" xref="S4.p3.1.m1.2.2.2.cmml">(</mo><msub id="S4.p3.1.m1.2.2.1.1" xref="S4.p3.1.m1.2.2.1.1.cmml"><mi id="S4.p3.1.m1.2.2.1.1.2" xref="S4.p3.1.m1.2.2.1.1.2.cmml">Ο</mi><mn id="S4.p3.1.m1.2.2.1.1.3" xref="S4.p3.1.m1.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.p3.1.m1.2.2.1.3" xref="S4.p3.1.m1.2.2.2.cmml">,</mo><mi id="S4.p3.1.m1.1.1" xref="S4.p3.1.m1.1.1.cmml">max</mi><mo id="S4.p3.1.m1.2.2.1.4" stretchy="false" xref="S4.p3.1.m1.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.1.m1.2b"><interval closure="open" id="S4.p3.1.m1.2.2.2.cmml" xref="S4.p3.1.m1.2.2.1"><apply id="S4.p3.1.m1.2.2.1.1.cmml" xref="S4.p3.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S4.p3.1.m1.2.2.1.1.1.cmml" xref="S4.p3.1.m1.2.2.1.1">subscript</csymbol><ci id="S4.p3.1.m1.2.2.1.1.2.cmml" xref="S4.p3.1.m1.2.2.1.1.2">π</ci><cn id="S4.p3.1.m1.2.2.1.1.3.cmml" type="integer" xref="S4.p3.1.m1.2.2.1.1.3">1</cn></apply><max id="S4.p3.1.m1.1.1.cmml" xref="S4.p3.1.m1.1.1"></max></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.1.m1.2c">(\omega_{1},\max)</annotation><annotation encoding="application/x-llamapun" id="S4.p3.1.m1.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_max )</annotation></semantics></math> and <math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S4.p3.2.m2.2"><semantics id="S4.p3.2.m2.2a"><mrow id="S4.p3.2.m2.2.2.1" xref="S4.p3.2.m2.2.2.2.cmml"><mo id="S4.p3.2.m2.2.2.1.2" stretchy="false" xref="S4.p3.2.m2.2.2.2.cmml">(</mo><msub id="S4.p3.2.m2.2.2.1.1" xref="S4.p3.2.m2.2.2.1.1.cmml"><mi id="S4.p3.2.m2.2.2.1.1.2" xref="S4.p3.2.m2.2.2.1.1.2.cmml">Ο</mi><mn id="S4.p3.2.m2.2.2.1.1.3" xref="S4.p3.2.m2.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.p3.2.m2.2.2.1.3" xref="S4.p3.2.m2.2.2.2.cmml">,</mo><mi id="S4.p3.2.m2.1.1" xref="S4.p3.2.m2.1.1.cmml">min</mi><mo id="S4.p3.2.m2.2.2.1.4" stretchy="false" xref="S4.p3.2.m2.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.2.m2.2b"><interval closure="open" id="S4.p3.2.m2.2.2.2.cmml" xref="S4.p3.2.m2.2.2.1"><apply id="S4.p3.2.m2.2.2.1.1.cmml" xref="S4.p3.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S4.p3.2.m2.2.2.1.1.1.cmml" xref="S4.p3.2.m2.2.2.1.1">subscript</csymbol><ci id="S4.p3.2.m2.2.2.1.1.2.cmml" xref="S4.p3.2.m2.2.2.1.1.2">π</ci><cn id="S4.p3.2.m2.2.2.1.1.3.cmml" type="integer" xref="S4.p3.2.m2.2.2.1.1.3">1</cn></apply><min id="S4.p3.2.m2.1.1.cmml" xref="S4.p3.2.m2.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.2.m2.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S4.p3.2.m2.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math> coincide. The following result follows from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib2" title="">2</a>, Theorem 1]</cite></p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S4.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem6.1.1.1">Theorem 4.6</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem6.2.2"> </span>(Banakh, Bonnet, KubiΕ)<span class="ltx_text ltx_font_bold" id="S4.Thmtheorem6.3.3">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem6.p1"> <p class="ltx_p" id="S4.Thmtheorem6.p1.4"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem6.p1.4.4">Let <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.1.1.m1.1"><semantics id="S4.Thmtheorem6.p1.1.1.m1.1a"><mi id="S4.Thmtheorem6.p1.1.1.m1.1.1" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.1.1.m1.1b"><ci id="S4.Thmtheorem6.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.1.1.m1.1d">italic_X</annotation></semantics></math> be a separable topological semilattice that contains a subspace <math alttext="L" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.2.2.m2.1"><semantics id="S4.Thmtheorem6.p1.2.2.m2.1a"><mi id="S4.Thmtheorem6.p1.2.2.m2.1.1" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.2.2.m2.1b"><ci id="S4.Thmtheorem6.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.2.2.m2.1c">L</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.2.2.m2.1d">italic_L</annotation></semantics></math> homeomorphic to <math alttext="\omega_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.3.3.m3.1"><semantics id="S4.Thmtheorem6.p1.3.3.m3.1a"><msub id="S4.Thmtheorem6.p1.3.3.m3.1.1" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem6.p1.3.3.m3.1.1.2" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.2.cmml">Ο</mi><mn id="S4.Thmtheorem6.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.3.3.m3.1b"><apply id="S4.Thmtheorem6.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1">subscript</csymbol><ci id="S4.Thmtheorem6.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.2">π</ci><cn id="S4.Thmtheorem6.p1.3.3.m3.1.1.3.cmml" type="integer" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.3.3.m3.1c">\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.3.3.m3.1d">italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> endowed with the order topology. Then <math alttext="|X\setminus L|>\omega" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.4.4.m4.1"><semantics id="S4.Thmtheorem6.p1.4.4.m4.1a"><mrow id="S4.Thmtheorem6.p1.4.4.m4.1.1" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.cmml"><mrow id="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.1.2.cmml"><mo id="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.1.2.1.cmml">|</mo><mrow id="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.1" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.1.2" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.1.2.cmml">X</mi><mo id="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.1.1" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.1.1.cmml">β</mo><mi id="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.1.3" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.1.3.cmml">L</mi></mrow><mo id="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.Thmtheorem6.p1.4.4.m4.1.1.2" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.2.cmml">></mo><mi id="S4.Thmtheorem6.p1.4.4.m4.1.1.3" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.3.cmml">Ο</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.4.4.m4.1b"><apply id="S4.Thmtheorem6.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.1.1"><gt id="S4.Thmtheorem6.p1.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.2"></gt><apply id="S4.Thmtheorem6.p1.4.4.m4.1.1.1.2.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1"><abs id="S4.Thmtheorem6.p1.4.4.m4.1.1.1.2.1.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.2"></abs><apply id="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.1.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.1"><setdiff id="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.1.1"></setdiff><ci id="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.1.2">π</ci><ci id="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.1.1.1.3">πΏ</ci></apply></apply><ci id="S4.Thmtheorem6.p1.4.4.m4.1.1.3.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.4.4.m4.1c">|X\setminus L|>\omega</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.4.4.m4.1d">| italic_X β italic_L | > italic_Ο</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem7.1.1.1">Lemma 4.7</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem7.p1"> <p class="ltx_p" id="S4.Thmtheorem7.p1.5"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem7.p1.5.5">Let <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.1.1.m1.1"><semantics id="S4.Thmtheorem7.p1.1.1.m1.1a"><mi id="S4.Thmtheorem7.p1.1.1.m1.1.1" xref="S4.Thmtheorem7.p1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.1.1.m1.1b"><ci id="S4.Thmtheorem7.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.1.1.m1.1d">italic_X</annotation></semantics></math> be a Nyikos topological semilattice and <math alttext="L\subseteq X" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.2.2.m2.1"><semantics id="S4.Thmtheorem7.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem7.p1.2.2.m2.1.1" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem7.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.2.cmml">L</mi><mo id="S4.Thmtheorem7.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.1.cmml">β</mo><mi id="S4.Thmtheorem7.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.2.2.m2.1b"><apply id="S4.Thmtheorem7.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem7.p1.2.2.m2.1.1"><subset id="S4.Thmtheorem7.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.1"></subset><ci id="S4.Thmtheorem7.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.2">πΏ</ci><ci id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.cmml" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.2.2.m2.1c">L\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.2.2.m2.1d">italic_L β italic_X</annotation></semantics></math> be a chain isomorphic to <math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.3.3.m3.2"><semantics id="S4.Thmtheorem7.p1.3.3.m3.2a"><mrow id="S4.Thmtheorem7.p1.3.3.m3.2.2.1" xref="S4.Thmtheorem7.p1.3.3.m3.2.2.2.cmml"><mo id="S4.Thmtheorem7.p1.3.3.m3.2.2.1.2" stretchy="false" xref="S4.Thmtheorem7.p1.3.3.m3.2.2.2.cmml">(</mo><msub id="S4.Thmtheorem7.p1.3.3.m3.2.2.1.1" xref="S4.Thmtheorem7.p1.3.3.m3.2.2.1.1.cmml"><mi id="S4.Thmtheorem7.p1.3.3.m3.2.2.1.1.2" xref="S4.Thmtheorem7.p1.3.3.m3.2.2.1.1.2.cmml">Ο</mi><mn id="S4.Thmtheorem7.p1.3.3.m3.2.2.1.1.3" xref="S4.Thmtheorem7.p1.3.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem7.p1.3.3.m3.2.2.1.3" xref="S4.Thmtheorem7.p1.3.3.m3.2.2.2.cmml">,</mo><mi id="S4.Thmtheorem7.p1.3.3.m3.1.1" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.cmml">min</mi><mo id="S4.Thmtheorem7.p1.3.3.m3.2.2.1.4" stretchy="false" xref="S4.Thmtheorem7.p1.3.3.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.3.3.m3.2b"><interval closure="open" id="S4.Thmtheorem7.p1.3.3.m3.2.2.2.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.2.2.1"><apply id="S4.Thmtheorem7.p1.3.3.m3.2.2.1.1.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.2.2.1.1">subscript</csymbol><ci id="S4.Thmtheorem7.p1.3.3.m3.2.2.1.1.2.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.2.2.1.1.2">π</ci><cn id="S4.Thmtheorem7.p1.3.3.m3.2.2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem7.p1.3.3.m3.2.2.1.1.3">1</cn></apply><min id="S4.Thmtheorem7.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.3.3.m3.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.3.3.m3.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math>. Then there exists <math alttext="x\in L" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.4.4.m4.1"><semantics id="S4.Thmtheorem7.p1.4.4.m4.1a"><mrow id="S4.Thmtheorem7.p1.4.4.m4.1.1" xref="S4.Thmtheorem7.p1.4.4.m4.1.1.cmml"><mi id="S4.Thmtheorem7.p1.4.4.m4.1.1.2" xref="S4.Thmtheorem7.p1.4.4.m4.1.1.2.cmml">x</mi><mo id="S4.Thmtheorem7.p1.4.4.m4.1.1.1" xref="S4.Thmtheorem7.p1.4.4.m4.1.1.1.cmml">β</mo><mi id="S4.Thmtheorem7.p1.4.4.m4.1.1.3" xref="S4.Thmtheorem7.p1.4.4.m4.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.4.4.m4.1b"><apply id="S4.Thmtheorem7.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.1.1"><in id="S4.Thmtheorem7.p1.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.1.1.1"></in><ci id="S4.Thmtheorem7.p1.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.1.1.2">π₯</ci><ci id="S4.Thmtheorem7.p1.4.4.m4.1.1.3.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.1.1.3">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.4.4.m4.1c">x\in L</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.4.4.m4.1d">italic_x β italic_L</annotation></semantics></math> such that <math alttext="x\in\overline{X\setminus I_{L}}" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.5.5.m5.1"><semantics id="S4.Thmtheorem7.p1.5.5.m5.1a"><mrow id="S4.Thmtheorem7.p1.5.5.m5.1.1" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.cmml"><mi id="S4.Thmtheorem7.p1.5.5.m5.1.1.2" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.2.cmml">x</mi><mo id="S4.Thmtheorem7.p1.5.5.m5.1.1.1" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.1.cmml">β</mo><mover accent="true" id="S4.Thmtheorem7.p1.5.5.m5.1.1.3" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.3.cmml"><mrow id="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.cmml"><mi id="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.2" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.2.cmml">X</mi><mo id="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.1" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.1.cmml">β</mo><msub id="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.3" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.3.cmml"><mi id="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.3.2" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.3.2.cmml">I</mi><mi id="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.3.3" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.3.3.cmml">L</mi></msub></mrow><mo id="S4.Thmtheorem7.p1.5.5.m5.1.1.3.1" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.5.5.m5.1b"><apply id="S4.Thmtheorem7.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem7.p1.5.5.m5.1.1"><in id="S4.Thmtheorem7.p1.5.5.m5.1.1.1.cmml" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.1"></in><ci id="S4.Thmtheorem7.p1.5.5.m5.1.1.2.cmml" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.2">π₯</ci><apply id="S4.Thmtheorem7.p1.5.5.m5.1.1.3.cmml" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.3"><ci id="S4.Thmtheorem7.p1.5.5.m5.1.1.3.1.cmml" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.3.1">Β―</ci><apply id="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.cmml" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2"><setdiff id="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.1.cmml" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.1"></setdiff><ci id="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.2.cmml" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.2">π</ci><apply id="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.3.cmml" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.3.1.cmml" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.3">subscript</csymbol><ci id="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.3.2.cmml" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.3.2">πΌ</ci><ci id="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.3.3.cmml" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.3.2.3.3">πΏ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.5.5.m5.1c">x\in\overline{X\setminus I_{L}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.5.5.m5.1d">italic_x β overΒ― start_ARG italic_X β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_ARG</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S4.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.4.p1"> <p class="ltx_p" id="S4.4.p1.24">Let <math alttext="L=\{l_{\alpha}:\alpha<\omega_{1}\}\subseteq X" class="ltx_Math" display="inline" id="S4.4.p1.1.m1.2"><semantics id="S4.4.p1.1.m1.2a"><mrow id="S4.4.p1.1.m1.2.2" xref="S4.4.p1.1.m1.2.2.cmml"><mi id="S4.4.p1.1.m1.2.2.4" xref="S4.4.p1.1.m1.2.2.4.cmml">L</mi><mo id="S4.4.p1.1.m1.2.2.5" xref="S4.4.p1.1.m1.2.2.5.cmml">=</mo><mrow id="S4.4.p1.1.m1.2.2.2.2" xref="S4.4.p1.1.m1.2.2.2.3.cmml"><mo id="S4.4.p1.1.m1.2.2.2.2.3" stretchy="false" xref="S4.4.p1.1.m1.2.2.2.3.1.cmml">{</mo><msub id="S4.4.p1.1.m1.1.1.1.1.1" xref="S4.4.p1.1.m1.1.1.1.1.1.cmml"><mi id="S4.4.p1.1.m1.1.1.1.1.1.2" xref="S4.4.p1.1.m1.1.1.1.1.1.2.cmml">l</mi><mi id="S4.4.p1.1.m1.1.1.1.1.1.3" xref="S4.4.p1.1.m1.1.1.1.1.1.3.cmml">Ξ±</mi></msub><mo id="S4.4.p1.1.m1.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.4.p1.1.m1.2.2.2.3.1.cmml">:</mo><mrow id="S4.4.p1.1.m1.2.2.2.2.2" xref="S4.4.p1.1.m1.2.2.2.2.2.cmml"><mi id="S4.4.p1.1.m1.2.2.2.2.2.2" xref="S4.4.p1.1.m1.2.2.2.2.2.2.cmml">Ξ±</mi><mo id="S4.4.p1.1.m1.2.2.2.2.2.1" xref="S4.4.p1.1.m1.2.2.2.2.2.1.cmml"><</mo><msub id="S4.4.p1.1.m1.2.2.2.2.2.3" xref="S4.4.p1.1.m1.2.2.2.2.2.3.cmml"><mi id="S4.4.p1.1.m1.2.2.2.2.2.3.2" xref="S4.4.p1.1.m1.2.2.2.2.2.3.2.cmml">Ο</mi><mn id="S4.4.p1.1.m1.2.2.2.2.2.3.3" xref="S4.4.p1.1.m1.2.2.2.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S4.4.p1.1.m1.2.2.2.2.5" stretchy="false" xref="S4.4.p1.1.m1.2.2.2.3.1.cmml">}</mo></mrow><mo id="S4.4.p1.1.m1.2.2.6" xref="S4.4.p1.1.m1.2.2.6.cmml">β</mo><mi id="S4.4.p1.1.m1.2.2.7" xref="S4.4.p1.1.m1.2.2.7.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.1.m1.2b"><apply id="S4.4.p1.1.m1.2.2.cmml" xref="S4.4.p1.1.m1.2.2"><and id="S4.4.p1.1.m1.2.2a.cmml" xref="S4.4.p1.1.m1.2.2"></and><apply id="S4.4.p1.1.m1.2.2b.cmml" xref="S4.4.p1.1.m1.2.2"><eq id="S4.4.p1.1.m1.2.2.5.cmml" xref="S4.4.p1.1.m1.2.2.5"></eq><ci id="S4.4.p1.1.m1.2.2.4.cmml" xref="S4.4.p1.1.m1.2.2.4">πΏ</ci><apply id="S4.4.p1.1.m1.2.2.2.3.cmml" xref="S4.4.p1.1.m1.2.2.2.2"><csymbol cd="latexml" id="S4.4.p1.1.m1.2.2.2.3.1.cmml" xref="S4.4.p1.1.m1.2.2.2.2.3">conditional-set</csymbol><apply id="S4.4.p1.1.m1.1.1.1.1.1.cmml" xref="S4.4.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.4.p1.1.m1.1.1.1.1.1.1.cmml" xref="S4.4.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S4.4.p1.1.m1.1.1.1.1.1.2.cmml" xref="S4.4.p1.1.m1.1.1.1.1.1.2">π</ci><ci id="S4.4.p1.1.m1.1.1.1.1.1.3.cmml" xref="S4.4.p1.1.m1.1.1.1.1.1.3">πΌ</ci></apply><apply id="S4.4.p1.1.m1.2.2.2.2.2.cmml" xref="S4.4.p1.1.m1.2.2.2.2.2"><lt id="S4.4.p1.1.m1.2.2.2.2.2.1.cmml" xref="S4.4.p1.1.m1.2.2.2.2.2.1"></lt><ci id="S4.4.p1.1.m1.2.2.2.2.2.2.cmml" xref="S4.4.p1.1.m1.2.2.2.2.2.2">πΌ</ci><apply id="S4.4.p1.1.m1.2.2.2.2.2.3.cmml" xref="S4.4.p1.1.m1.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.4.p1.1.m1.2.2.2.2.2.3.1.cmml" xref="S4.4.p1.1.m1.2.2.2.2.2.3">subscript</csymbol><ci id="S4.4.p1.1.m1.2.2.2.2.2.3.2.cmml" xref="S4.4.p1.1.m1.2.2.2.2.2.3.2">π</ci><cn id="S4.4.p1.1.m1.2.2.2.2.2.3.3.cmml" type="integer" xref="S4.4.p1.1.m1.2.2.2.2.2.3.3">1</cn></apply></apply></apply></apply><apply id="S4.4.p1.1.m1.2.2c.cmml" xref="S4.4.p1.1.m1.2.2"><subset id="S4.4.p1.1.m1.2.2.6.cmml" xref="S4.4.p1.1.m1.2.2.6"></subset><share href="https://arxiv.org/html/2503.13666v1#S4.4.p1.1.m1.2.2.2.cmml" id="S4.4.p1.1.m1.2.2d.cmml" xref="S4.4.p1.1.m1.2.2"></share><ci id="S4.4.p1.1.m1.2.2.7.cmml" xref="S4.4.p1.1.m1.2.2.7">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.1.m1.2c">L=\{l_{\alpha}:\alpha<\omega_{1}\}\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.1.m1.2d">italic_L = { italic_l start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT : italic_Ξ± < italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT } β italic_X</annotation></semantics></math>, where <math alttext="l_{\alpha}\leq l_{\beta}" class="ltx_Math" display="inline" id="S4.4.p1.2.m2.1"><semantics id="S4.4.p1.2.m2.1a"><mrow id="S4.4.p1.2.m2.1.1" xref="S4.4.p1.2.m2.1.1.cmml"><msub id="S4.4.p1.2.m2.1.1.2" xref="S4.4.p1.2.m2.1.1.2.cmml"><mi id="S4.4.p1.2.m2.1.1.2.2" xref="S4.4.p1.2.m2.1.1.2.2.cmml">l</mi><mi id="S4.4.p1.2.m2.1.1.2.3" xref="S4.4.p1.2.m2.1.1.2.3.cmml">Ξ±</mi></msub><mo id="S4.4.p1.2.m2.1.1.1" xref="S4.4.p1.2.m2.1.1.1.cmml">β€</mo><msub id="S4.4.p1.2.m2.1.1.3" xref="S4.4.p1.2.m2.1.1.3.cmml"><mi id="S4.4.p1.2.m2.1.1.3.2" xref="S4.4.p1.2.m2.1.1.3.2.cmml">l</mi><mi id="S4.4.p1.2.m2.1.1.3.3" xref="S4.4.p1.2.m2.1.1.3.3.cmml">Ξ²</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.2.m2.1b"><apply id="S4.4.p1.2.m2.1.1.cmml" xref="S4.4.p1.2.m2.1.1"><leq id="S4.4.p1.2.m2.1.1.1.cmml" xref="S4.4.p1.2.m2.1.1.1"></leq><apply id="S4.4.p1.2.m2.1.1.2.cmml" xref="S4.4.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.4.p1.2.m2.1.1.2.1.cmml" xref="S4.4.p1.2.m2.1.1.2">subscript</csymbol><ci id="S4.4.p1.2.m2.1.1.2.2.cmml" xref="S4.4.p1.2.m2.1.1.2.2">π</ci><ci id="S4.4.p1.2.m2.1.1.2.3.cmml" xref="S4.4.p1.2.m2.1.1.2.3">πΌ</ci></apply><apply id="S4.4.p1.2.m2.1.1.3.cmml" xref="S4.4.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.4.p1.2.m2.1.1.3.1.cmml" xref="S4.4.p1.2.m2.1.1.3">subscript</csymbol><ci id="S4.4.p1.2.m2.1.1.3.2.cmml" xref="S4.4.p1.2.m2.1.1.3.2">π</ci><ci id="S4.4.p1.2.m2.1.1.3.3.cmml" xref="S4.4.p1.2.m2.1.1.3.3">π½</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.2.m2.1c">l_{\alpha}\leq l_{\beta}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.2.m2.1d">italic_l start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT β€ italic_l start_POSTSUBSCRIPT italic_Ξ² end_POSTSUBSCRIPT</annotation></semantics></math> if and only if <math alttext="\alpha\leq\beta" class="ltx_Math" display="inline" id="S4.4.p1.3.m3.1"><semantics id="S4.4.p1.3.m3.1a"><mrow id="S4.4.p1.3.m3.1.1" xref="S4.4.p1.3.m3.1.1.cmml"><mi id="S4.4.p1.3.m3.1.1.2" xref="S4.4.p1.3.m3.1.1.2.cmml">Ξ±</mi><mo id="S4.4.p1.3.m3.1.1.1" xref="S4.4.p1.3.m3.1.1.1.cmml">β€</mo><mi id="S4.4.p1.3.m3.1.1.3" xref="S4.4.p1.3.m3.1.1.3.cmml">Ξ²</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.3.m3.1b"><apply id="S4.4.p1.3.m3.1.1.cmml" xref="S4.4.p1.3.m3.1.1"><leq id="S4.4.p1.3.m3.1.1.1.cmml" xref="S4.4.p1.3.m3.1.1.1"></leq><ci id="S4.4.p1.3.m3.1.1.2.cmml" xref="S4.4.p1.3.m3.1.1.2">πΌ</ci><ci id="S4.4.p1.3.m3.1.1.3.cmml" xref="S4.4.p1.3.m3.1.1.3">π½</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.3.m3.1c">\alpha\leq\beta</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.3.m3.1d">italic_Ξ± β€ italic_Ξ²</annotation></semantics></math>. It is clear that <math alttext="L\subset I_{L}" class="ltx_Math" display="inline" id="S4.4.p1.4.m4.1"><semantics id="S4.4.p1.4.m4.1a"><mrow id="S4.4.p1.4.m4.1.1" xref="S4.4.p1.4.m4.1.1.cmml"><mi id="S4.4.p1.4.m4.1.1.2" xref="S4.4.p1.4.m4.1.1.2.cmml">L</mi><mo id="S4.4.p1.4.m4.1.1.1" xref="S4.4.p1.4.m4.1.1.1.cmml">β</mo><msub id="S4.4.p1.4.m4.1.1.3" xref="S4.4.p1.4.m4.1.1.3.cmml"><mi id="S4.4.p1.4.m4.1.1.3.2" xref="S4.4.p1.4.m4.1.1.3.2.cmml">I</mi><mi id="S4.4.p1.4.m4.1.1.3.3" xref="S4.4.p1.4.m4.1.1.3.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.4.m4.1b"><apply id="S4.4.p1.4.m4.1.1.cmml" xref="S4.4.p1.4.m4.1.1"><subset id="S4.4.p1.4.m4.1.1.1.cmml" xref="S4.4.p1.4.m4.1.1.1"></subset><ci id="S4.4.p1.4.m4.1.1.2.cmml" xref="S4.4.p1.4.m4.1.1.2">πΏ</ci><apply id="S4.4.p1.4.m4.1.1.3.cmml" xref="S4.4.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.4.p1.4.m4.1.1.3.1.cmml" xref="S4.4.p1.4.m4.1.1.3">subscript</csymbol><ci id="S4.4.p1.4.m4.1.1.3.2.cmml" xref="S4.4.p1.4.m4.1.1.3.2">πΌ</ci><ci id="S4.4.p1.4.m4.1.1.3.3.cmml" xref="S4.4.p1.4.m4.1.1.3.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.4.m4.1c">L\subset I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.4.m4.1d">italic_L β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>. Fix any <math alttext="x\in I_{L}" class="ltx_Math" display="inline" id="S4.4.p1.5.m5.1"><semantics id="S4.4.p1.5.m5.1a"><mrow id="S4.4.p1.5.m5.1.1" xref="S4.4.p1.5.m5.1.1.cmml"><mi id="S4.4.p1.5.m5.1.1.2" xref="S4.4.p1.5.m5.1.1.2.cmml">x</mi><mo id="S4.4.p1.5.m5.1.1.1" xref="S4.4.p1.5.m5.1.1.1.cmml">β</mo><msub id="S4.4.p1.5.m5.1.1.3" xref="S4.4.p1.5.m5.1.1.3.cmml"><mi id="S4.4.p1.5.m5.1.1.3.2" xref="S4.4.p1.5.m5.1.1.3.2.cmml">I</mi><mi id="S4.4.p1.5.m5.1.1.3.3" xref="S4.4.p1.5.m5.1.1.3.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.5.m5.1b"><apply id="S4.4.p1.5.m5.1.1.cmml" xref="S4.4.p1.5.m5.1.1"><in id="S4.4.p1.5.m5.1.1.1.cmml" xref="S4.4.p1.5.m5.1.1.1"></in><ci id="S4.4.p1.5.m5.1.1.2.cmml" xref="S4.4.p1.5.m5.1.1.2">π₯</ci><apply id="S4.4.p1.5.m5.1.1.3.cmml" xref="S4.4.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.4.p1.5.m5.1.1.3.1.cmml" xref="S4.4.p1.5.m5.1.1.3">subscript</csymbol><ci id="S4.4.p1.5.m5.1.1.3.2.cmml" xref="S4.4.p1.5.m5.1.1.3.2">πΌ</ci><ci id="S4.4.p1.5.m5.1.1.3.3.cmml" xref="S4.4.p1.5.m5.1.1.3.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.5.m5.1c">x\in I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.5.m5.1d">italic_x β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="xL" class="ltx_Math" display="inline" id="S4.4.p1.6.m6.1"><semantics id="S4.4.p1.6.m6.1a"><mrow id="S4.4.p1.6.m6.1.1" xref="S4.4.p1.6.m6.1.1.cmml"><mi id="S4.4.p1.6.m6.1.1.2" xref="S4.4.p1.6.m6.1.1.2.cmml">x</mi><mo id="S4.4.p1.6.m6.1.1.1" xref="S4.4.p1.6.m6.1.1.1.cmml">β’</mo><mi id="S4.4.p1.6.m6.1.1.3" xref="S4.4.p1.6.m6.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.6.m6.1b"><apply id="S4.4.p1.6.m6.1.1.cmml" xref="S4.4.p1.6.m6.1.1"><times id="S4.4.p1.6.m6.1.1.1.cmml" xref="S4.4.p1.6.m6.1.1.1"></times><ci id="S4.4.p1.6.m6.1.1.2.cmml" xref="S4.4.p1.6.m6.1.1.2">π₯</ci><ci id="S4.4.p1.6.m6.1.1.3.cmml" xref="S4.4.p1.6.m6.1.1.3">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.6.m6.1c">xL</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.6.m6.1d">italic_x italic_L</annotation></semantics></math> is countable, there exists <math alttext="\delta<\omega_{1}" class="ltx_Math" display="inline" id="S4.4.p1.7.m7.1"><semantics id="S4.4.p1.7.m7.1a"><mrow id="S4.4.p1.7.m7.1.1" xref="S4.4.p1.7.m7.1.1.cmml"><mi id="S4.4.p1.7.m7.1.1.2" xref="S4.4.p1.7.m7.1.1.2.cmml">Ξ΄</mi><mo id="S4.4.p1.7.m7.1.1.1" xref="S4.4.p1.7.m7.1.1.1.cmml"><</mo><msub id="S4.4.p1.7.m7.1.1.3" xref="S4.4.p1.7.m7.1.1.3.cmml"><mi id="S4.4.p1.7.m7.1.1.3.2" xref="S4.4.p1.7.m7.1.1.3.2.cmml">Ο</mi><mn id="S4.4.p1.7.m7.1.1.3.3" xref="S4.4.p1.7.m7.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.7.m7.1b"><apply id="S4.4.p1.7.m7.1.1.cmml" xref="S4.4.p1.7.m7.1.1"><lt id="S4.4.p1.7.m7.1.1.1.cmml" xref="S4.4.p1.7.m7.1.1.1"></lt><ci id="S4.4.p1.7.m7.1.1.2.cmml" xref="S4.4.p1.7.m7.1.1.2">πΏ</ci><apply id="S4.4.p1.7.m7.1.1.3.cmml" xref="S4.4.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.4.p1.7.m7.1.1.3.1.cmml" xref="S4.4.p1.7.m7.1.1.3">subscript</csymbol><ci id="S4.4.p1.7.m7.1.1.3.2.cmml" xref="S4.4.p1.7.m7.1.1.3.2">π</ci><cn id="S4.4.p1.7.m7.1.1.3.3.cmml" type="integer" xref="S4.4.p1.7.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.7.m7.1c">\delta<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.7.m7.1d">italic_Ξ΄ < italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="xl_{\xi}=xl_{\delta}" class="ltx_Math" display="inline" id="S4.4.p1.8.m8.1"><semantics id="S4.4.p1.8.m8.1a"><mrow id="S4.4.p1.8.m8.1.1" xref="S4.4.p1.8.m8.1.1.cmml"><mrow id="S4.4.p1.8.m8.1.1.2" xref="S4.4.p1.8.m8.1.1.2.cmml"><mi id="S4.4.p1.8.m8.1.1.2.2" xref="S4.4.p1.8.m8.1.1.2.2.cmml">x</mi><mo id="S4.4.p1.8.m8.1.1.2.1" xref="S4.4.p1.8.m8.1.1.2.1.cmml">β’</mo><msub id="S4.4.p1.8.m8.1.1.2.3" xref="S4.4.p1.8.m8.1.1.2.3.cmml"><mi id="S4.4.p1.8.m8.1.1.2.3.2" xref="S4.4.p1.8.m8.1.1.2.3.2.cmml">l</mi><mi id="S4.4.p1.8.m8.1.1.2.3.3" xref="S4.4.p1.8.m8.1.1.2.3.3.cmml">ΞΎ</mi></msub></mrow><mo id="S4.4.p1.8.m8.1.1.1" xref="S4.4.p1.8.m8.1.1.1.cmml">=</mo><mrow id="S4.4.p1.8.m8.1.1.3" xref="S4.4.p1.8.m8.1.1.3.cmml"><mi id="S4.4.p1.8.m8.1.1.3.2" xref="S4.4.p1.8.m8.1.1.3.2.cmml">x</mi><mo id="S4.4.p1.8.m8.1.1.3.1" xref="S4.4.p1.8.m8.1.1.3.1.cmml">β’</mo><msub id="S4.4.p1.8.m8.1.1.3.3" xref="S4.4.p1.8.m8.1.1.3.3.cmml"><mi id="S4.4.p1.8.m8.1.1.3.3.2" xref="S4.4.p1.8.m8.1.1.3.3.2.cmml">l</mi><mi id="S4.4.p1.8.m8.1.1.3.3.3" xref="S4.4.p1.8.m8.1.1.3.3.3.cmml">Ξ΄</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.8.m8.1b"><apply id="S4.4.p1.8.m8.1.1.cmml" xref="S4.4.p1.8.m8.1.1"><eq id="S4.4.p1.8.m8.1.1.1.cmml" xref="S4.4.p1.8.m8.1.1.1"></eq><apply id="S4.4.p1.8.m8.1.1.2.cmml" xref="S4.4.p1.8.m8.1.1.2"><times id="S4.4.p1.8.m8.1.1.2.1.cmml" xref="S4.4.p1.8.m8.1.1.2.1"></times><ci id="S4.4.p1.8.m8.1.1.2.2.cmml" xref="S4.4.p1.8.m8.1.1.2.2">π₯</ci><apply id="S4.4.p1.8.m8.1.1.2.3.cmml" xref="S4.4.p1.8.m8.1.1.2.3"><csymbol cd="ambiguous" id="S4.4.p1.8.m8.1.1.2.3.1.cmml" xref="S4.4.p1.8.m8.1.1.2.3">subscript</csymbol><ci id="S4.4.p1.8.m8.1.1.2.3.2.cmml" xref="S4.4.p1.8.m8.1.1.2.3.2">π</ci><ci id="S4.4.p1.8.m8.1.1.2.3.3.cmml" xref="S4.4.p1.8.m8.1.1.2.3.3">π</ci></apply></apply><apply id="S4.4.p1.8.m8.1.1.3.cmml" xref="S4.4.p1.8.m8.1.1.3"><times id="S4.4.p1.8.m8.1.1.3.1.cmml" xref="S4.4.p1.8.m8.1.1.3.1"></times><ci id="S4.4.p1.8.m8.1.1.3.2.cmml" xref="S4.4.p1.8.m8.1.1.3.2">π₯</ci><apply id="S4.4.p1.8.m8.1.1.3.3.cmml" xref="S4.4.p1.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="S4.4.p1.8.m8.1.1.3.3.1.cmml" xref="S4.4.p1.8.m8.1.1.3.3">subscript</csymbol><ci id="S4.4.p1.8.m8.1.1.3.3.2.cmml" xref="S4.4.p1.8.m8.1.1.3.3.2">π</ci><ci id="S4.4.p1.8.m8.1.1.3.3.3.cmml" xref="S4.4.p1.8.m8.1.1.3.3.3">πΏ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.8.m8.1c">xl_{\xi}=xl_{\delta}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.8.m8.1d">italic_x italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT = italic_x italic_l start_POSTSUBSCRIPT italic_Ξ΄ end_POSTSUBSCRIPT</annotation></semantics></math> for each <math alttext="\xi\geq\delta" class="ltx_Math" display="inline" id="S4.4.p1.9.m9.1"><semantics id="S4.4.p1.9.m9.1a"><mrow id="S4.4.p1.9.m9.1.1" xref="S4.4.p1.9.m9.1.1.cmml"><mi id="S4.4.p1.9.m9.1.1.2" xref="S4.4.p1.9.m9.1.1.2.cmml">ΞΎ</mi><mo id="S4.4.p1.9.m9.1.1.1" xref="S4.4.p1.9.m9.1.1.1.cmml">β₯</mo><mi id="S4.4.p1.9.m9.1.1.3" xref="S4.4.p1.9.m9.1.1.3.cmml">Ξ΄</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.9.m9.1b"><apply id="S4.4.p1.9.m9.1.1.cmml" xref="S4.4.p1.9.m9.1.1"><geq id="S4.4.p1.9.m9.1.1.1.cmml" xref="S4.4.p1.9.m9.1.1.1"></geq><ci id="S4.4.p1.9.m9.1.1.2.cmml" xref="S4.4.p1.9.m9.1.1.2">π</ci><ci id="S4.4.p1.9.m9.1.1.3.cmml" xref="S4.4.p1.9.m9.1.1.3">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.9.m9.1c">\xi\geq\delta</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.9.m9.1d">italic_ΞΎ β₯ italic_Ξ΄</annotation></semantics></math>. By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem11" title="Lemma 3.11. β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.11</span></a>, the chain <math alttext="\overline{L}" class="ltx_Math" display="inline" id="S4.4.p1.10.m10.1"><semantics id="S4.4.p1.10.m10.1a"><mover accent="true" id="S4.4.p1.10.m10.1.1" xref="S4.4.p1.10.m10.1.1.cmml"><mi id="S4.4.p1.10.m10.1.1.2" xref="S4.4.p1.10.m10.1.1.2.cmml">L</mi><mo id="S4.4.p1.10.m10.1.1.1" xref="S4.4.p1.10.m10.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S4.4.p1.10.m10.1b"><apply id="S4.4.p1.10.m10.1.1.cmml" xref="S4.4.p1.10.m10.1.1"><ci id="S4.4.p1.10.m10.1.1.1.cmml" xref="S4.4.p1.10.m10.1.1.1">Β―</ci><ci id="S4.4.p1.10.m10.1.1.2.cmml" xref="S4.4.p1.10.m10.1.1.2">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.10.m10.1c">\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.10.m10.1d">overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> is topologically isomorphic to <math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S4.4.p1.11.m11.2"><semantics id="S4.4.p1.11.m11.2a"><mrow id="S4.4.p1.11.m11.2.2.1" xref="S4.4.p1.11.m11.2.2.2.cmml"><mo id="S4.4.p1.11.m11.2.2.1.2" stretchy="false" xref="S4.4.p1.11.m11.2.2.2.cmml">(</mo><msub id="S4.4.p1.11.m11.2.2.1.1" xref="S4.4.p1.11.m11.2.2.1.1.cmml"><mi id="S4.4.p1.11.m11.2.2.1.1.2" xref="S4.4.p1.11.m11.2.2.1.1.2.cmml">Ο</mi><mn id="S4.4.p1.11.m11.2.2.1.1.3" xref="S4.4.p1.11.m11.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.4.p1.11.m11.2.2.1.3" xref="S4.4.p1.11.m11.2.2.2.cmml">,</mo><mi id="S4.4.p1.11.m11.1.1" xref="S4.4.p1.11.m11.1.1.cmml">min</mi><mo id="S4.4.p1.11.m11.2.2.1.4" stretchy="false" xref="S4.4.p1.11.m11.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.11.m11.2b"><interval closure="open" id="S4.4.p1.11.m11.2.2.2.cmml" xref="S4.4.p1.11.m11.2.2.1"><apply id="S4.4.p1.11.m11.2.2.1.1.cmml" xref="S4.4.p1.11.m11.2.2.1.1"><csymbol cd="ambiguous" id="S4.4.p1.11.m11.2.2.1.1.1.cmml" xref="S4.4.p1.11.m11.2.2.1.1">subscript</csymbol><ci id="S4.4.p1.11.m11.2.2.1.1.2.cmml" xref="S4.4.p1.11.m11.2.2.1.1.2">π</ci><cn id="S4.4.p1.11.m11.2.2.1.1.3.cmml" type="integer" xref="S4.4.p1.11.m11.2.2.1.1.3">1</cn></apply><min id="S4.4.p1.11.m11.1.1.cmml" xref="S4.4.p1.11.m11.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.11.m11.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.11.m11.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math> equipped with the order topology, and contains <math alttext="L" class="ltx_Math" display="inline" id="S4.4.p1.12.m12.1"><semantics id="S4.4.p1.12.m12.1a"><mi id="S4.4.p1.12.m12.1.1" xref="S4.4.p1.12.m12.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S4.4.p1.12.m12.1b"><ci id="S4.4.p1.12.m12.1.1.cmml" xref="S4.4.p1.12.m12.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.12.m12.1c">L</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.12.m12.1d">italic_L</annotation></semantics></math> as a cofinal subset. Hence <math alttext="xl_{\delta}=xa" class="ltx_Math" display="inline" id="S4.4.p1.13.m13.1"><semantics id="S4.4.p1.13.m13.1a"><mrow id="S4.4.p1.13.m13.1.1" xref="S4.4.p1.13.m13.1.1.cmml"><mrow id="S4.4.p1.13.m13.1.1.2" xref="S4.4.p1.13.m13.1.1.2.cmml"><mi id="S4.4.p1.13.m13.1.1.2.2" xref="S4.4.p1.13.m13.1.1.2.2.cmml">x</mi><mo id="S4.4.p1.13.m13.1.1.2.1" xref="S4.4.p1.13.m13.1.1.2.1.cmml">β’</mo><msub id="S4.4.p1.13.m13.1.1.2.3" xref="S4.4.p1.13.m13.1.1.2.3.cmml"><mi id="S4.4.p1.13.m13.1.1.2.3.2" xref="S4.4.p1.13.m13.1.1.2.3.2.cmml">l</mi><mi id="S4.4.p1.13.m13.1.1.2.3.3" xref="S4.4.p1.13.m13.1.1.2.3.3.cmml">Ξ΄</mi></msub></mrow><mo id="S4.4.p1.13.m13.1.1.1" xref="S4.4.p1.13.m13.1.1.1.cmml">=</mo><mrow id="S4.4.p1.13.m13.1.1.3" xref="S4.4.p1.13.m13.1.1.3.cmml"><mi id="S4.4.p1.13.m13.1.1.3.2" xref="S4.4.p1.13.m13.1.1.3.2.cmml">x</mi><mo id="S4.4.p1.13.m13.1.1.3.1" xref="S4.4.p1.13.m13.1.1.3.1.cmml">β’</mo><mi id="S4.4.p1.13.m13.1.1.3.3" xref="S4.4.p1.13.m13.1.1.3.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.13.m13.1b"><apply id="S4.4.p1.13.m13.1.1.cmml" xref="S4.4.p1.13.m13.1.1"><eq id="S4.4.p1.13.m13.1.1.1.cmml" xref="S4.4.p1.13.m13.1.1.1"></eq><apply id="S4.4.p1.13.m13.1.1.2.cmml" xref="S4.4.p1.13.m13.1.1.2"><times id="S4.4.p1.13.m13.1.1.2.1.cmml" xref="S4.4.p1.13.m13.1.1.2.1"></times><ci id="S4.4.p1.13.m13.1.1.2.2.cmml" xref="S4.4.p1.13.m13.1.1.2.2">π₯</ci><apply id="S4.4.p1.13.m13.1.1.2.3.cmml" xref="S4.4.p1.13.m13.1.1.2.3"><csymbol cd="ambiguous" id="S4.4.p1.13.m13.1.1.2.3.1.cmml" xref="S4.4.p1.13.m13.1.1.2.3">subscript</csymbol><ci id="S4.4.p1.13.m13.1.1.2.3.2.cmml" xref="S4.4.p1.13.m13.1.1.2.3.2">π</ci><ci id="S4.4.p1.13.m13.1.1.2.3.3.cmml" xref="S4.4.p1.13.m13.1.1.2.3.3">πΏ</ci></apply></apply><apply id="S4.4.p1.13.m13.1.1.3.cmml" xref="S4.4.p1.13.m13.1.1.3"><times id="S4.4.p1.13.m13.1.1.3.1.cmml" xref="S4.4.p1.13.m13.1.1.3.1"></times><ci id="S4.4.p1.13.m13.1.1.3.2.cmml" xref="S4.4.p1.13.m13.1.1.3.2">π₯</ci><ci id="S4.4.p1.13.m13.1.1.3.3.cmml" xref="S4.4.p1.13.m13.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.13.m13.1c">xl_{\delta}=xa</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.13.m13.1d">italic_x italic_l start_POSTSUBSCRIPT italic_Ξ΄ end_POSTSUBSCRIPT = italic_x italic_a</annotation></semantics></math> for each <math alttext="a\in\overline{L}\cap{\uparrow}l_{\delta}" class="ltx_Math" display="inline" id="S4.4.p1.14.m14.1"><semantics id="S4.4.p1.14.m14.1a"><mrow id="S4.4.p1.14.m14.1.1" xref="S4.4.p1.14.m14.1.1.cmml"><mi id="S4.4.p1.14.m14.1.1.2" xref="S4.4.p1.14.m14.1.1.2.cmml">a</mi><mo id="S4.4.p1.14.m14.1.1.3" xref="S4.4.p1.14.m14.1.1.3.cmml">β</mo><mrow id="S4.4.p1.14.m14.1.1.4" xref="S4.4.p1.14.m14.1.1.4.cmml"><mover accent="true" id="S4.4.p1.14.m14.1.1.4.2" xref="S4.4.p1.14.m14.1.1.4.2.cmml"><mi id="S4.4.p1.14.m14.1.1.4.2.2" xref="S4.4.p1.14.m14.1.1.4.2.2.cmml">L</mi><mo id="S4.4.p1.14.m14.1.1.4.2.1" xref="S4.4.p1.14.m14.1.1.4.2.1.cmml">Β―</mo></mover><mo id="S4.4.p1.14.m14.1.1.4.3" rspace="0em" xref="S4.4.p1.14.m14.1.1.4.3.cmml">β©</mo></mrow><mo id="S4.4.p1.14.m14.1.1.5" lspace="0em" stretchy="false" xref="S4.4.p1.14.m14.1.1.5.cmml">β</mo><msub id="S4.4.p1.14.m14.1.1.6" xref="S4.4.p1.14.m14.1.1.6.cmml"><mi id="S4.4.p1.14.m14.1.1.6.2" xref="S4.4.p1.14.m14.1.1.6.2.cmml">l</mi><mi id="S4.4.p1.14.m14.1.1.6.3" xref="S4.4.p1.14.m14.1.1.6.3.cmml">Ξ΄</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.14.m14.1b"><apply id="S4.4.p1.14.m14.1.1.cmml" xref="S4.4.p1.14.m14.1.1"><and id="S4.4.p1.14.m14.1.1a.cmml" xref="S4.4.p1.14.m14.1.1"></and><apply id="S4.4.p1.14.m14.1.1b.cmml" xref="S4.4.p1.14.m14.1.1"><in id="S4.4.p1.14.m14.1.1.3.cmml" xref="S4.4.p1.14.m14.1.1.3"></in><ci id="S4.4.p1.14.m14.1.1.2.cmml" xref="S4.4.p1.14.m14.1.1.2">π</ci><apply id="S4.4.p1.14.m14.1.1.4.cmml" xref="S4.4.p1.14.m14.1.1.4"><csymbol cd="latexml" id="S4.4.p1.14.m14.1.1.4.1.cmml" xref="S4.4.p1.14.m14.1.1.4">limit-from</csymbol><apply id="S4.4.p1.14.m14.1.1.4.2.cmml" xref="S4.4.p1.14.m14.1.1.4.2"><ci id="S4.4.p1.14.m14.1.1.4.2.1.cmml" xref="S4.4.p1.14.m14.1.1.4.2.1">Β―</ci><ci id="S4.4.p1.14.m14.1.1.4.2.2.cmml" xref="S4.4.p1.14.m14.1.1.4.2.2">πΏ</ci></apply><intersect id="S4.4.p1.14.m14.1.1.4.3.cmml" xref="S4.4.p1.14.m14.1.1.4.3"></intersect></apply></apply><apply id="S4.4.p1.14.m14.1.1c.cmml" xref="S4.4.p1.14.m14.1.1"><ci id="S4.4.p1.14.m14.1.1.5.cmml" xref="S4.4.p1.14.m14.1.1.5">β</ci><share href="https://arxiv.org/html/2503.13666v1#S4.4.p1.14.m14.1.1.4.cmml" id="S4.4.p1.14.m14.1.1d.cmml" xref="S4.4.p1.14.m14.1.1"></share><apply id="S4.4.p1.14.m14.1.1.6.cmml" xref="S4.4.p1.14.m14.1.1.6"><csymbol cd="ambiguous" id="S4.4.p1.14.m14.1.1.6.1.cmml" xref="S4.4.p1.14.m14.1.1.6">subscript</csymbol><ci id="S4.4.p1.14.m14.1.1.6.2.cmml" xref="S4.4.p1.14.m14.1.1.6.2">π</ci><ci id="S4.4.p1.14.m14.1.1.6.3.cmml" xref="S4.4.p1.14.m14.1.1.6.3">πΏ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.14.m14.1c">a\in\overline{L}\cap{\uparrow}l_{\delta}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.14.m14.1d">italic_a β overΒ― start_ARG italic_L end_ARG β© β italic_l start_POSTSUBSCRIPT italic_Ξ΄ end_POSTSUBSCRIPT</annotation></semantics></math>, implying that the set <math alttext="x\overline{L}" class="ltx_Math" display="inline" id="S4.4.p1.15.m15.1"><semantics id="S4.4.p1.15.m15.1a"><mrow id="S4.4.p1.15.m15.1.1" xref="S4.4.p1.15.m15.1.1.cmml"><mi id="S4.4.p1.15.m15.1.1.2" xref="S4.4.p1.15.m15.1.1.2.cmml">x</mi><mo id="S4.4.p1.15.m15.1.1.1" xref="S4.4.p1.15.m15.1.1.1.cmml">β’</mo><mover accent="true" id="S4.4.p1.15.m15.1.1.3" xref="S4.4.p1.15.m15.1.1.3.cmml"><mi id="S4.4.p1.15.m15.1.1.3.2" xref="S4.4.p1.15.m15.1.1.3.2.cmml">L</mi><mo id="S4.4.p1.15.m15.1.1.3.1" xref="S4.4.p1.15.m15.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.15.m15.1b"><apply id="S4.4.p1.15.m15.1.1.cmml" xref="S4.4.p1.15.m15.1.1"><times id="S4.4.p1.15.m15.1.1.1.cmml" xref="S4.4.p1.15.m15.1.1.1"></times><ci id="S4.4.p1.15.m15.1.1.2.cmml" xref="S4.4.p1.15.m15.1.1.2">π₯</ci><apply id="S4.4.p1.15.m15.1.1.3.cmml" xref="S4.4.p1.15.m15.1.1.3"><ci id="S4.4.p1.15.m15.1.1.3.1.cmml" xref="S4.4.p1.15.m15.1.1.3.1">Β―</ci><ci id="S4.4.p1.15.m15.1.1.3.2.cmml" xref="S4.4.p1.15.m15.1.1.3.2">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.15.m15.1c">x\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.15.m15.1d">italic_x overΒ― start_ARG italic_L end_ARG</annotation></semantics></math> is countable. It follows that <math alttext="I_{L}=I_{\overline{L}}" class="ltx_Math" display="inline" id="S4.4.p1.16.m16.1"><semantics id="S4.4.p1.16.m16.1a"><mrow id="S4.4.p1.16.m16.1.1" xref="S4.4.p1.16.m16.1.1.cmml"><msub id="S4.4.p1.16.m16.1.1.2" xref="S4.4.p1.16.m16.1.1.2.cmml"><mi id="S4.4.p1.16.m16.1.1.2.2" xref="S4.4.p1.16.m16.1.1.2.2.cmml">I</mi><mi id="S4.4.p1.16.m16.1.1.2.3" xref="S4.4.p1.16.m16.1.1.2.3.cmml">L</mi></msub><mo id="S4.4.p1.16.m16.1.1.1" xref="S4.4.p1.16.m16.1.1.1.cmml">=</mo><msub id="S4.4.p1.16.m16.1.1.3" xref="S4.4.p1.16.m16.1.1.3.cmml"><mi id="S4.4.p1.16.m16.1.1.3.2" xref="S4.4.p1.16.m16.1.1.3.2.cmml">I</mi><mover accent="true" id="S4.4.p1.16.m16.1.1.3.3" xref="S4.4.p1.16.m16.1.1.3.3.cmml"><mi id="S4.4.p1.16.m16.1.1.3.3.2" xref="S4.4.p1.16.m16.1.1.3.3.2.cmml">L</mi><mo id="S4.4.p1.16.m16.1.1.3.3.1" xref="S4.4.p1.16.m16.1.1.3.3.1.cmml">Β―</mo></mover></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.16.m16.1b"><apply id="S4.4.p1.16.m16.1.1.cmml" xref="S4.4.p1.16.m16.1.1"><eq id="S4.4.p1.16.m16.1.1.1.cmml" xref="S4.4.p1.16.m16.1.1.1"></eq><apply id="S4.4.p1.16.m16.1.1.2.cmml" xref="S4.4.p1.16.m16.1.1.2"><csymbol cd="ambiguous" id="S4.4.p1.16.m16.1.1.2.1.cmml" xref="S4.4.p1.16.m16.1.1.2">subscript</csymbol><ci id="S4.4.p1.16.m16.1.1.2.2.cmml" xref="S4.4.p1.16.m16.1.1.2.2">πΌ</ci><ci id="S4.4.p1.16.m16.1.1.2.3.cmml" xref="S4.4.p1.16.m16.1.1.2.3">πΏ</ci></apply><apply id="S4.4.p1.16.m16.1.1.3.cmml" xref="S4.4.p1.16.m16.1.1.3"><csymbol cd="ambiguous" id="S4.4.p1.16.m16.1.1.3.1.cmml" xref="S4.4.p1.16.m16.1.1.3">subscript</csymbol><ci id="S4.4.p1.16.m16.1.1.3.2.cmml" xref="S4.4.p1.16.m16.1.1.3.2">πΌ</ci><apply id="S4.4.p1.16.m16.1.1.3.3.cmml" xref="S4.4.p1.16.m16.1.1.3.3"><ci id="S4.4.p1.16.m16.1.1.3.3.1.cmml" xref="S4.4.p1.16.m16.1.1.3.3.1">Β―</ci><ci id="S4.4.p1.16.m16.1.1.3.3.2.cmml" xref="S4.4.p1.16.m16.1.1.3.3.2">πΏ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.16.m16.1c">I_{L}=I_{\overline{L}}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.16.m16.1d">italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT = italic_I start_POSTSUBSCRIPT overΒ― start_ARG italic_L end_ARG end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="X" class="ltx_Math" display="inline" id="S4.4.p1.17.m17.1"><semantics id="S4.4.p1.17.m17.1a"><mi id="S4.4.p1.17.m17.1.1" xref="S4.4.p1.17.m17.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.4.p1.17.m17.1b"><ci id="S4.4.p1.17.m17.1.1.cmml" xref="S4.4.p1.17.m17.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.17.m17.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.17.m17.1d">italic_X</annotation></semantics></math> is separable, it contains a countable dense subsemilattice <math alttext="D" class="ltx_Math" display="inline" id="S4.4.p1.18.m18.1"><semantics id="S4.4.p1.18.m18.1a"><mi id="S4.4.p1.18.m18.1.1" xref="S4.4.p1.18.m18.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S4.4.p1.18.m18.1b"><ci id="S4.4.p1.18.m18.1.1.cmml" xref="S4.4.p1.18.m18.1.1">π·</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.18.m18.1c">D</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.18.m18.1d">italic_D</annotation></semantics></math>. Seeking a contradiction, assume that there exists an open set <math alttext="U\supseteq L" class="ltx_Math" display="inline" id="S4.4.p1.19.m19.1"><semantics id="S4.4.p1.19.m19.1a"><mrow id="S4.4.p1.19.m19.1.1" xref="S4.4.p1.19.m19.1.1.cmml"><mi id="S4.4.p1.19.m19.1.1.2" xref="S4.4.p1.19.m19.1.1.2.cmml">U</mi><mo id="S4.4.p1.19.m19.1.1.1" xref="S4.4.p1.19.m19.1.1.cmml">β</mo><mi id="S4.4.p1.19.m19.1.1.3" xref="S4.4.p1.19.m19.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.19.m19.1b"><apply id="S4.4.p1.19.m19.1.1.cmml" xref="S4.4.p1.19.m19.1.1"><subset id="S4.4.p1.19.m19.1.1a.cmml" xref="S4.4.p1.19.m19.1.1"></subset><ci id="S4.4.p1.19.m19.1.1.3.cmml" xref="S4.4.p1.19.m19.1.1.3">πΏ</ci><ci id="S4.4.p1.19.m19.1.1.2.cmml" xref="S4.4.p1.19.m19.1.1.2">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.19.m19.1c">U\supseteq L</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.19.m19.1d">italic_U β italic_L</annotation></semantics></math> such that <math alttext="U\subset I_{L}" class="ltx_Math" display="inline" id="S4.4.p1.20.m20.1"><semantics id="S4.4.p1.20.m20.1a"><mrow id="S4.4.p1.20.m20.1.1" xref="S4.4.p1.20.m20.1.1.cmml"><mi id="S4.4.p1.20.m20.1.1.2" xref="S4.4.p1.20.m20.1.1.2.cmml">U</mi><mo id="S4.4.p1.20.m20.1.1.1" xref="S4.4.p1.20.m20.1.1.1.cmml">β</mo><msub id="S4.4.p1.20.m20.1.1.3" xref="S4.4.p1.20.m20.1.1.3.cmml"><mi id="S4.4.p1.20.m20.1.1.3.2" xref="S4.4.p1.20.m20.1.1.3.2.cmml">I</mi><mi id="S4.4.p1.20.m20.1.1.3.3" xref="S4.4.p1.20.m20.1.1.3.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.20.m20.1b"><apply id="S4.4.p1.20.m20.1.1.cmml" xref="S4.4.p1.20.m20.1.1"><subset id="S4.4.p1.20.m20.1.1.1.cmml" xref="S4.4.p1.20.m20.1.1.1"></subset><ci id="S4.4.p1.20.m20.1.1.2.cmml" xref="S4.4.p1.20.m20.1.1.2">π</ci><apply id="S4.4.p1.20.m20.1.1.3.cmml" xref="S4.4.p1.20.m20.1.1.3"><csymbol cd="ambiguous" id="S4.4.p1.20.m20.1.1.3.1.cmml" xref="S4.4.p1.20.m20.1.1.3">subscript</csymbol><ci id="S4.4.p1.20.m20.1.1.3.2.cmml" xref="S4.4.p1.20.m20.1.1.3.2">πΌ</ci><ci id="S4.4.p1.20.m20.1.1.3.3.cmml" xref="S4.4.p1.20.m20.1.1.3.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.20.m20.1c">U\subset I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.20.m20.1d">italic_U β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>. Consider the subsemilattice <math alttext="Y" class="ltx_Math" display="inline" id="S4.4.p1.21.m21.1"><semantics id="S4.4.p1.21.m21.1a"><mi id="S4.4.p1.21.m21.1.1" xref="S4.4.p1.21.m21.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S4.4.p1.21.m21.1b"><ci id="S4.4.p1.21.m21.1.1.cmml" xref="S4.4.p1.21.m21.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.21.m21.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.21.m21.1d">italic_Y</annotation></semantics></math> of <math alttext="X" class="ltx_Math" display="inline" id="S4.4.p1.22.m22.1"><semantics id="S4.4.p1.22.m22.1a"><mi id="S4.4.p1.22.m22.1.1" xref="S4.4.p1.22.m22.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.4.p1.22.m22.1b"><ci id="S4.4.p1.22.m22.1.1.cmml" xref="S4.4.p1.22.m22.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.22.m22.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.22.m22.1d">italic_X</annotation></semantics></math> generated by <math alttext="\overline{L}\cup(D\cap U)" class="ltx_Math" display="inline" id="S4.4.p1.23.m23.1"><semantics id="S4.4.p1.23.m23.1a"><mrow id="S4.4.p1.23.m23.1.1" xref="S4.4.p1.23.m23.1.1.cmml"><mover accent="true" id="S4.4.p1.23.m23.1.1.3" xref="S4.4.p1.23.m23.1.1.3.cmml"><mi id="S4.4.p1.23.m23.1.1.3.2" xref="S4.4.p1.23.m23.1.1.3.2.cmml">L</mi><mo id="S4.4.p1.23.m23.1.1.3.1" xref="S4.4.p1.23.m23.1.1.3.1.cmml">Β―</mo></mover><mo id="S4.4.p1.23.m23.1.1.2" xref="S4.4.p1.23.m23.1.1.2.cmml">βͺ</mo><mrow id="S4.4.p1.23.m23.1.1.1.1" xref="S4.4.p1.23.m23.1.1.1.1.1.cmml"><mo id="S4.4.p1.23.m23.1.1.1.1.2" stretchy="false" xref="S4.4.p1.23.m23.1.1.1.1.1.cmml">(</mo><mrow id="S4.4.p1.23.m23.1.1.1.1.1" xref="S4.4.p1.23.m23.1.1.1.1.1.cmml"><mi id="S4.4.p1.23.m23.1.1.1.1.1.2" xref="S4.4.p1.23.m23.1.1.1.1.1.2.cmml">D</mi><mo id="S4.4.p1.23.m23.1.1.1.1.1.1" xref="S4.4.p1.23.m23.1.1.1.1.1.1.cmml">β©</mo><mi id="S4.4.p1.23.m23.1.1.1.1.1.3" xref="S4.4.p1.23.m23.1.1.1.1.1.3.cmml">U</mi></mrow><mo id="S4.4.p1.23.m23.1.1.1.1.3" stretchy="false" xref="S4.4.p1.23.m23.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.23.m23.1b"><apply id="S4.4.p1.23.m23.1.1.cmml" xref="S4.4.p1.23.m23.1.1"><union id="S4.4.p1.23.m23.1.1.2.cmml" xref="S4.4.p1.23.m23.1.1.2"></union><apply id="S4.4.p1.23.m23.1.1.3.cmml" xref="S4.4.p1.23.m23.1.1.3"><ci id="S4.4.p1.23.m23.1.1.3.1.cmml" xref="S4.4.p1.23.m23.1.1.3.1">Β―</ci><ci id="S4.4.p1.23.m23.1.1.3.2.cmml" xref="S4.4.p1.23.m23.1.1.3.2">πΏ</ci></apply><apply id="S4.4.p1.23.m23.1.1.1.1.1.cmml" xref="S4.4.p1.23.m23.1.1.1.1"><intersect id="S4.4.p1.23.m23.1.1.1.1.1.1.cmml" xref="S4.4.p1.23.m23.1.1.1.1.1.1"></intersect><ci id="S4.4.p1.23.m23.1.1.1.1.1.2.cmml" xref="S4.4.p1.23.m23.1.1.1.1.1.2">π·</ci><ci id="S4.4.p1.23.m23.1.1.1.1.1.3.cmml" xref="S4.4.p1.23.m23.1.1.1.1.1.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.23.m23.1c">\overline{L}\cup(D\cap U)</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.23.m23.1d">overΒ― start_ARG italic_L end_ARG βͺ ( italic_D β© italic_U )</annotation></semantics></math>. Since <math alttext="U\subseteq I_{L}=I_{\overline{L}}" class="ltx_Math" display="inline" id="S4.4.p1.24.m24.1"><semantics id="S4.4.p1.24.m24.1a"><mrow id="S4.4.p1.24.m24.1.1" xref="S4.4.p1.24.m24.1.1.cmml"><mi id="S4.4.p1.24.m24.1.1.2" xref="S4.4.p1.24.m24.1.1.2.cmml">U</mi><mo id="S4.4.p1.24.m24.1.1.3" xref="S4.4.p1.24.m24.1.1.3.cmml">β</mo><msub id="S4.4.p1.24.m24.1.1.4" xref="S4.4.p1.24.m24.1.1.4.cmml"><mi id="S4.4.p1.24.m24.1.1.4.2" xref="S4.4.p1.24.m24.1.1.4.2.cmml">I</mi><mi id="S4.4.p1.24.m24.1.1.4.3" xref="S4.4.p1.24.m24.1.1.4.3.cmml">L</mi></msub><mo id="S4.4.p1.24.m24.1.1.5" xref="S4.4.p1.24.m24.1.1.5.cmml">=</mo><msub id="S4.4.p1.24.m24.1.1.6" xref="S4.4.p1.24.m24.1.1.6.cmml"><mi id="S4.4.p1.24.m24.1.1.6.2" xref="S4.4.p1.24.m24.1.1.6.2.cmml">I</mi><mover accent="true" id="S4.4.p1.24.m24.1.1.6.3" xref="S4.4.p1.24.m24.1.1.6.3.cmml"><mi id="S4.4.p1.24.m24.1.1.6.3.2" xref="S4.4.p1.24.m24.1.1.6.3.2.cmml">L</mi><mo id="S4.4.p1.24.m24.1.1.6.3.1" xref="S4.4.p1.24.m24.1.1.6.3.1.cmml">Β―</mo></mover></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.24.m24.1b"><apply id="S4.4.p1.24.m24.1.1.cmml" xref="S4.4.p1.24.m24.1.1"><and id="S4.4.p1.24.m24.1.1a.cmml" xref="S4.4.p1.24.m24.1.1"></and><apply id="S4.4.p1.24.m24.1.1b.cmml" xref="S4.4.p1.24.m24.1.1"><subset id="S4.4.p1.24.m24.1.1.3.cmml" xref="S4.4.p1.24.m24.1.1.3"></subset><ci id="S4.4.p1.24.m24.1.1.2.cmml" xref="S4.4.p1.24.m24.1.1.2">π</ci><apply id="S4.4.p1.24.m24.1.1.4.cmml" xref="S4.4.p1.24.m24.1.1.4"><csymbol cd="ambiguous" id="S4.4.p1.24.m24.1.1.4.1.cmml" xref="S4.4.p1.24.m24.1.1.4">subscript</csymbol><ci id="S4.4.p1.24.m24.1.1.4.2.cmml" xref="S4.4.p1.24.m24.1.1.4.2">πΌ</ci><ci id="S4.4.p1.24.m24.1.1.4.3.cmml" xref="S4.4.p1.24.m24.1.1.4.3">πΏ</ci></apply></apply><apply id="S4.4.p1.24.m24.1.1c.cmml" xref="S4.4.p1.24.m24.1.1"><eq id="S4.4.p1.24.m24.1.1.5.cmml" xref="S4.4.p1.24.m24.1.1.5"></eq><share href="https://arxiv.org/html/2503.13666v1#S4.4.p1.24.m24.1.1.4.cmml" id="S4.4.p1.24.m24.1.1d.cmml" xref="S4.4.p1.24.m24.1.1"></share><apply id="S4.4.p1.24.m24.1.1.6.cmml" xref="S4.4.p1.24.m24.1.1.6"><csymbol cd="ambiguous" id="S4.4.p1.24.m24.1.1.6.1.cmml" xref="S4.4.p1.24.m24.1.1.6">subscript</csymbol><ci id="S4.4.p1.24.m24.1.1.6.2.cmml" xref="S4.4.p1.24.m24.1.1.6.2">πΌ</ci><apply id="S4.4.p1.24.m24.1.1.6.3.cmml" xref="S4.4.p1.24.m24.1.1.6.3"><ci id="S4.4.p1.24.m24.1.1.6.3.1.cmml" xref="S4.4.p1.24.m24.1.1.6.3.1">Β―</ci><ci id="S4.4.p1.24.m24.1.1.6.3.2.cmml" xref="S4.4.p1.24.m24.1.1.6.3.2">πΏ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.24.m24.1c">U\subseteq I_{L}=I_{\overline{L}}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.24.m24.1d">italic_U β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT = italic_I start_POSTSUBSCRIPT overΒ― start_ARG italic_L end_ARG end_POSTSUBSCRIPT</annotation></semantics></math> we obtain that the set</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="D^{\prime}=Y\setminus\overline{L}\subseteq(D\cap I_{L})\cup\bigcup_{x\in D\cap I% _{L}}x\overline{L}" class="ltx_Math" display="block" id="S4.Ex12.m1.1"><semantics id="S4.Ex12.m1.1a"><mrow id="S4.Ex12.m1.1.1" xref="S4.Ex12.m1.1.1.cmml"><msup id="S4.Ex12.m1.1.1.3" xref="S4.Ex12.m1.1.1.3.cmml"><mi id="S4.Ex12.m1.1.1.3.2" xref="S4.Ex12.m1.1.1.3.2.cmml">D</mi><mo id="S4.Ex12.m1.1.1.3.3" xref="S4.Ex12.m1.1.1.3.3.cmml">β²</mo></msup><mo id="S4.Ex12.m1.1.1.4" xref="S4.Ex12.m1.1.1.4.cmml">=</mo><mrow id="S4.Ex12.m1.1.1.5" xref="S4.Ex12.m1.1.1.5.cmml"><mi id="S4.Ex12.m1.1.1.5.2" xref="S4.Ex12.m1.1.1.5.2.cmml">Y</mi><mo id="S4.Ex12.m1.1.1.5.1" xref="S4.Ex12.m1.1.1.5.1.cmml">β</mo><mover accent="true" id="S4.Ex12.m1.1.1.5.3" xref="S4.Ex12.m1.1.1.5.3.cmml"><mi id="S4.Ex12.m1.1.1.5.3.2" xref="S4.Ex12.m1.1.1.5.3.2.cmml">L</mi><mo id="S4.Ex12.m1.1.1.5.3.1" xref="S4.Ex12.m1.1.1.5.3.1.cmml">Β―</mo></mover></mrow><mo id="S4.Ex12.m1.1.1.6" xref="S4.Ex12.m1.1.1.6.cmml">β</mo><mrow id="S4.Ex12.m1.1.1.1" xref="S4.Ex12.m1.1.1.1.cmml"><mrow id="S4.Ex12.m1.1.1.1.1.1" xref="S4.Ex12.m1.1.1.1.1.1.1.cmml"><mo id="S4.Ex12.m1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex12.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex12.m1.1.1.1.1.1.1" xref="S4.Ex12.m1.1.1.1.1.1.1.cmml"><mi id="S4.Ex12.m1.1.1.1.1.1.1.2" xref="S4.Ex12.m1.1.1.1.1.1.1.2.cmml">D</mi><mo id="S4.Ex12.m1.1.1.1.1.1.1.1" xref="S4.Ex12.m1.1.1.1.1.1.1.1.cmml">β©</mo><msub id="S4.Ex12.m1.1.1.1.1.1.1.3" xref="S4.Ex12.m1.1.1.1.1.1.1.3.cmml"><mi id="S4.Ex12.m1.1.1.1.1.1.1.3.2" xref="S4.Ex12.m1.1.1.1.1.1.1.3.2.cmml">I</mi><mi id="S4.Ex12.m1.1.1.1.1.1.1.3.3" xref="S4.Ex12.m1.1.1.1.1.1.1.3.3.cmml">L</mi></msub></mrow><mo id="S4.Ex12.m1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex12.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Ex12.m1.1.1.1.2" rspace="0.055em" xref="S4.Ex12.m1.1.1.1.2.cmml">βͺ</mo><mrow id="S4.Ex12.m1.1.1.1.3" xref="S4.Ex12.m1.1.1.1.3.cmml"><munder id="S4.Ex12.m1.1.1.1.3.1" xref="S4.Ex12.m1.1.1.1.3.1.cmml"><mo id="S4.Ex12.m1.1.1.1.3.1.2" movablelimits="false" xref="S4.Ex12.m1.1.1.1.3.1.2.cmml">β</mo><mrow id="S4.Ex12.m1.1.1.1.3.1.3" xref="S4.Ex12.m1.1.1.1.3.1.3.cmml"><mi id="S4.Ex12.m1.1.1.1.3.1.3.2" xref="S4.Ex12.m1.1.1.1.3.1.3.2.cmml">x</mi><mo id="S4.Ex12.m1.1.1.1.3.1.3.1" xref="S4.Ex12.m1.1.1.1.3.1.3.1.cmml">β</mo><mrow id="S4.Ex12.m1.1.1.1.3.1.3.3" xref="S4.Ex12.m1.1.1.1.3.1.3.3.cmml"><mi id="S4.Ex12.m1.1.1.1.3.1.3.3.2" xref="S4.Ex12.m1.1.1.1.3.1.3.3.2.cmml">D</mi><mo id="S4.Ex12.m1.1.1.1.3.1.3.3.1" xref="S4.Ex12.m1.1.1.1.3.1.3.3.1.cmml">β©</mo><msub id="S4.Ex12.m1.1.1.1.3.1.3.3.3" xref="S4.Ex12.m1.1.1.1.3.1.3.3.3.cmml"><mi id="S4.Ex12.m1.1.1.1.3.1.3.3.3.2" xref="S4.Ex12.m1.1.1.1.3.1.3.3.3.2.cmml">I</mi><mi id="S4.Ex12.m1.1.1.1.3.1.3.3.3.3" xref="S4.Ex12.m1.1.1.1.3.1.3.3.3.3.cmml">L</mi></msub></mrow></mrow></munder><mrow id="S4.Ex12.m1.1.1.1.3.2" xref="S4.Ex12.m1.1.1.1.3.2.cmml"><mi id="S4.Ex12.m1.1.1.1.3.2.2" xref="S4.Ex12.m1.1.1.1.3.2.2.cmml">x</mi><mo id="S4.Ex12.m1.1.1.1.3.2.1" xref="S4.Ex12.m1.1.1.1.3.2.1.cmml">β’</mo><mover accent="true" id="S4.Ex12.m1.1.1.1.3.2.3" xref="S4.Ex12.m1.1.1.1.3.2.3.cmml"><mi id="S4.Ex12.m1.1.1.1.3.2.3.2" xref="S4.Ex12.m1.1.1.1.3.2.3.2.cmml">L</mi><mo id="S4.Ex12.m1.1.1.1.3.2.3.1" xref="S4.Ex12.m1.1.1.1.3.2.3.1.cmml">Β―</mo></mover></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex12.m1.1b"><apply id="S4.Ex12.m1.1.1.cmml" xref="S4.Ex12.m1.1.1"><and id="S4.Ex12.m1.1.1a.cmml" xref="S4.Ex12.m1.1.1"></and><apply id="S4.Ex12.m1.1.1b.cmml" xref="S4.Ex12.m1.1.1"><eq id="S4.Ex12.m1.1.1.4.cmml" xref="S4.Ex12.m1.1.1.4"></eq><apply id="S4.Ex12.m1.1.1.3.cmml" xref="S4.Ex12.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Ex12.m1.1.1.3.1.cmml" xref="S4.Ex12.m1.1.1.3">superscript</csymbol><ci id="S4.Ex12.m1.1.1.3.2.cmml" xref="S4.Ex12.m1.1.1.3.2">π·</ci><ci id="S4.Ex12.m1.1.1.3.3.cmml" xref="S4.Ex12.m1.1.1.3.3">β²</ci></apply><apply id="S4.Ex12.m1.1.1.5.cmml" xref="S4.Ex12.m1.1.1.5"><setdiff id="S4.Ex12.m1.1.1.5.1.cmml" xref="S4.Ex12.m1.1.1.5.1"></setdiff><ci id="S4.Ex12.m1.1.1.5.2.cmml" xref="S4.Ex12.m1.1.1.5.2">π</ci><apply id="S4.Ex12.m1.1.1.5.3.cmml" xref="S4.Ex12.m1.1.1.5.3"><ci id="S4.Ex12.m1.1.1.5.3.1.cmml" xref="S4.Ex12.m1.1.1.5.3.1">Β―</ci><ci id="S4.Ex12.m1.1.1.5.3.2.cmml" xref="S4.Ex12.m1.1.1.5.3.2">πΏ</ci></apply></apply></apply><apply id="S4.Ex12.m1.1.1c.cmml" xref="S4.Ex12.m1.1.1"><subset id="S4.Ex12.m1.1.1.6.cmml" xref="S4.Ex12.m1.1.1.6"></subset><share href="https://arxiv.org/html/2503.13666v1#S4.Ex12.m1.1.1.5.cmml" id="S4.Ex12.m1.1.1d.cmml" xref="S4.Ex12.m1.1.1"></share><apply id="S4.Ex12.m1.1.1.1.cmml" xref="S4.Ex12.m1.1.1.1"><union id="S4.Ex12.m1.1.1.1.2.cmml" xref="S4.Ex12.m1.1.1.1.2"></union><apply id="S4.Ex12.m1.1.1.1.1.1.1.cmml" xref="S4.Ex12.m1.1.1.1.1.1"><intersect id="S4.Ex12.m1.1.1.1.1.1.1.1.cmml" xref="S4.Ex12.m1.1.1.1.1.1.1.1"></intersect><ci id="S4.Ex12.m1.1.1.1.1.1.1.2.cmml" xref="S4.Ex12.m1.1.1.1.1.1.1.2">π·</ci><apply id="S4.Ex12.m1.1.1.1.1.1.1.3.cmml" xref="S4.Ex12.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.Ex12.m1.1.1.1.1.1.1.3.1.cmml" xref="S4.Ex12.m1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S4.Ex12.m1.1.1.1.1.1.1.3.2.cmml" xref="S4.Ex12.m1.1.1.1.1.1.1.3.2">πΌ</ci><ci id="S4.Ex12.m1.1.1.1.1.1.1.3.3.cmml" xref="S4.Ex12.m1.1.1.1.1.1.1.3.3">πΏ</ci></apply></apply><apply id="S4.Ex12.m1.1.1.1.3.cmml" xref="S4.Ex12.m1.1.1.1.3"><apply id="S4.Ex12.m1.1.1.1.3.1.cmml" xref="S4.Ex12.m1.1.1.1.3.1"><csymbol cd="ambiguous" id="S4.Ex12.m1.1.1.1.3.1.1.cmml" xref="S4.Ex12.m1.1.1.1.3.1">subscript</csymbol><union id="S4.Ex12.m1.1.1.1.3.1.2.cmml" xref="S4.Ex12.m1.1.1.1.3.1.2"></union><apply id="S4.Ex12.m1.1.1.1.3.1.3.cmml" xref="S4.Ex12.m1.1.1.1.3.1.3"><in id="S4.Ex12.m1.1.1.1.3.1.3.1.cmml" xref="S4.Ex12.m1.1.1.1.3.1.3.1"></in><ci id="S4.Ex12.m1.1.1.1.3.1.3.2.cmml" xref="S4.Ex12.m1.1.1.1.3.1.3.2">π₯</ci><apply id="S4.Ex12.m1.1.1.1.3.1.3.3.cmml" xref="S4.Ex12.m1.1.1.1.3.1.3.3"><intersect id="S4.Ex12.m1.1.1.1.3.1.3.3.1.cmml" xref="S4.Ex12.m1.1.1.1.3.1.3.3.1"></intersect><ci id="S4.Ex12.m1.1.1.1.3.1.3.3.2.cmml" xref="S4.Ex12.m1.1.1.1.3.1.3.3.2">π·</ci><apply id="S4.Ex12.m1.1.1.1.3.1.3.3.3.cmml" xref="S4.Ex12.m1.1.1.1.3.1.3.3.3"><csymbol cd="ambiguous" id="S4.Ex12.m1.1.1.1.3.1.3.3.3.1.cmml" xref="S4.Ex12.m1.1.1.1.3.1.3.3.3">subscript</csymbol><ci id="S4.Ex12.m1.1.1.1.3.1.3.3.3.2.cmml" xref="S4.Ex12.m1.1.1.1.3.1.3.3.3.2">πΌ</ci><ci id="S4.Ex12.m1.1.1.1.3.1.3.3.3.3.cmml" xref="S4.Ex12.m1.1.1.1.3.1.3.3.3.3">πΏ</ci></apply></apply></apply></apply><apply id="S4.Ex12.m1.1.1.1.3.2.cmml" xref="S4.Ex12.m1.1.1.1.3.2"><times id="S4.Ex12.m1.1.1.1.3.2.1.cmml" xref="S4.Ex12.m1.1.1.1.3.2.1"></times><ci id="S4.Ex12.m1.1.1.1.3.2.2.cmml" xref="S4.Ex12.m1.1.1.1.3.2.2">π₯</ci><apply id="S4.Ex12.m1.1.1.1.3.2.3.cmml" xref="S4.Ex12.m1.1.1.1.3.2.3"><ci id="S4.Ex12.m1.1.1.1.3.2.3.1.cmml" xref="S4.Ex12.m1.1.1.1.3.2.3.1">Β―</ci><ci id="S4.Ex12.m1.1.1.1.3.2.3.2.cmml" xref="S4.Ex12.m1.1.1.1.3.2.3.2">πΏ</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex12.m1.1c">D^{\prime}=Y\setminus\overline{L}\subseteq(D\cap I_{L})\cup\bigcup_{x\in D\cap I% _{L}}x\overline{L}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex12.m1.1d">italic_D start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT = italic_Y β overΒ― start_ARG italic_L end_ARG β ( italic_D β© italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) βͺ β start_POSTSUBSCRIPT italic_x β italic_D β© italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_x overΒ― start_ARG italic_L end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.4.p1.31">is countable. Since <math alttext="D" class="ltx_Math" display="inline" id="S4.4.p1.25.m1.1"><semantics id="S4.4.p1.25.m1.1a"><mi id="S4.4.p1.25.m1.1.1" xref="S4.4.p1.25.m1.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S4.4.p1.25.m1.1b"><ci id="S4.4.p1.25.m1.1.1.cmml" xref="S4.4.p1.25.m1.1.1">π·</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.25.m1.1c">D</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.25.m1.1d">italic_D</annotation></semantics></math> is dense in <math alttext="X" class="ltx_Math" display="inline" id="S4.4.p1.26.m2.1"><semantics id="S4.4.p1.26.m2.1a"><mi id="S4.4.p1.26.m2.1.1" xref="S4.4.p1.26.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.4.p1.26.m2.1b"><ci id="S4.4.p1.26.m2.1.1.cmml" xref="S4.4.p1.26.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.26.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.26.m2.1d">italic_X</annotation></semantics></math> and <math alttext="U\supseteq L" class="ltx_Math" display="inline" id="S4.4.p1.27.m3.1"><semantics id="S4.4.p1.27.m3.1a"><mrow id="S4.4.p1.27.m3.1.1" xref="S4.4.p1.27.m3.1.1.cmml"><mi id="S4.4.p1.27.m3.1.1.2" xref="S4.4.p1.27.m3.1.1.2.cmml">U</mi><mo id="S4.4.p1.27.m3.1.1.1" xref="S4.4.p1.27.m3.1.1.cmml">β</mo><mi id="S4.4.p1.27.m3.1.1.3" xref="S4.4.p1.27.m3.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.27.m3.1b"><apply id="S4.4.p1.27.m3.1.1.cmml" xref="S4.4.p1.27.m3.1.1"><subset id="S4.4.p1.27.m3.1.1a.cmml" xref="S4.4.p1.27.m3.1.1"></subset><ci id="S4.4.p1.27.m3.1.1.3.cmml" xref="S4.4.p1.27.m3.1.1.3">πΏ</ci><ci id="S4.4.p1.27.m3.1.1.2.cmml" xref="S4.4.p1.27.m3.1.1.2">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.27.m3.1c">U\supseteq L</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.27.m3.1d">italic_U β italic_L</annotation></semantics></math> is open, we get that <math alttext="L\subset\overline{U\cap D}" class="ltx_Math" display="inline" id="S4.4.p1.28.m4.1"><semantics id="S4.4.p1.28.m4.1a"><mrow id="S4.4.p1.28.m4.1.1" xref="S4.4.p1.28.m4.1.1.cmml"><mi id="S4.4.p1.28.m4.1.1.2" xref="S4.4.p1.28.m4.1.1.2.cmml">L</mi><mo id="S4.4.p1.28.m4.1.1.1" xref="S4.4.p1.28.m4.1.1.1.cmml">β</mo><mover accent="true" id="S4.4.p1.28.m4.1.1.3" xref="S4.4.p1.28.m4.1.1.3.cmml"><mrow id="S4.4.p1.28.m4.1.1.3.2" xref="S4.4.p1.28.m4.1.1.3.2.cmml"><mi id="S4.4.p1.28.m4.1.1.3.2.2" xref="S4.4.p1.28.m4.1.1.3.2.2.cmml">U</mi><mo id="S4.4.p1.28.m4.1.1.3.2.1" xref="S4.4.p1.28.m4.1.1.3.2.1.cmml">β©</mo><mi id="S4.4.p1.28.m4.1.1.3.2.3" xref="S4.4.p1.28.m4.1.1.3.2.3.cmml">D</mi></mrow><mo id="S4.4.p1.28.m4.1.1.3.1" xref="S4.4.p1.28.m4.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.28.m4.1b"><apply id="S4.4.p1.28.m4.1.1.cmml" xref="S4.4.p1.28.m4.1.1"><subset id="S4.4.p1.28.m4.1.1.1.cmml" xref="S4.4.p1.28.m4.1.1.1"></subset><ci id="S4.4.p1.28.m4.1.1.2.cmml" xref="S4.4.p1.28.m4.1.1.2">πΏ</ci><apply id="S4.4.p1.28.m4.1.1.3.cmml" xref="S4.4.p1.28.m4.1.1.3"><ci id="S4.4.p1.28.m4.1.1.3.1.cmml" xref="S4.4.p1.28.m4.1.1.3.1">Β―</ci><apply id="S4.4.p1.28.m4.1.1.3.2.cmml" xref="S4.4.p1.28.m4.1.1.3.2"><intersect id="S4.4.p1.28.m4.1.1.3.2.1.cmml" xref="S4.4.p1.28.m4.1.1.3.2.1"></intersect><ci id="S4.4.p1.28.m4.1.1.3.2.2.cmml" xref="S4.4.p1.28.m4.1.1.3.2.2">π</ci><ci id="S4.4.p1.28.m4.1.1.3.2.3.cmml" xref="S4.4.p1.28.m4.1.1.3.2.3">π·</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.28.m4.1c">L\subset\overline{U\cap D}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.28.m4.1d">italic_L β overΒ― start_ARG italic_U β© italic_D end_ARG</annotation></semantics></math>. It follows that <math alttext="\overline{L}\subset\overline{U\cap D}" class="ltx_Math" display="inline" id="S4.4.p1.29.m5.1"><semantics id="S4.4.p1.29.m5.1a"><mrow id="S4.4.p1.29.m5.1.1" xref="S4.4.p1.29.m5.1.1.cmml"><mover accent="true" id="S4.4.p1.29.m5.1.1.2" xref="S4.4.p1.29.m5.1.1.2.cmml"><mi id="S4.4.p1.29.m5.1.1.2.2" xref="S4.4.p1.29.m5.1.1.2.2.cmml">L</mi><mo id="S4.4.p1.29.m5.1.1.2.1" xref="S4.4.p1.29.m5.1.1.2.1.cmml">Β―</mo></mover><mo id="S4.4.p1.29.m5.1.1.1" xref="S4.4.p1.29.m5.1.1.1.cmml">β</mo><mover accent="true" id="S4.4.p1.29.m5.1.1.3" xref="S4.4.p1.29.m5.1.1.3.cmml"><mrow id="S4.4.p1.29.m5.1.1.3.2" xref="S4.4.p1.29.m5.1.1.3.2.cmml"><mi id="S4.4.p1.29.m5.1.1.3.2.2" xref="S4.4.p1.29.m5.1.1.3.2.2.cmml">U</mi><mo id="S4.4.p1.29.m5.1.1.3.2.1" xref="S4.4.p1.29.m5.1.1.3.2.1.cmml">β©</mo><mi id="S4.4.p1.29.m5.1.1.3.2.3" xref="S4.4.p1.29.m5.1.1.3.2.3.cmml">D</mi></mrow><mo id="S4.4.p1.29.m5.1.1.3.1" xref="S4.4.p1.29.m5.1.1.3.1.cmml">Β―</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S4.4.p1.29.m5.1b"><apply id="S4.4.p1.29.m5.1.1.cmml" xref="S4.4.p1.29.m5.1.1"><subset id="S4.4.p1.29.m5.1.1.1.cmml" xref="S4.4.p1.29.m5.1.1.1"></subset><apply id="S4.4.p1.29.m5.1.1.2.cmml" xref="S4.4.p1.29.m5.1.1.2"><ci id="S4.4.p1.29.m5.1.1.2.1.cmml" xref="S4.4.p1.29.m5.1.1.2.1">Β―</ci><ci id="S4.4.p1.29.m5.1.1.2.2.cmml" xref="S4.4.p1.29.m5.1.1.2.2">πΏ</ci></apply><apply id="S4.4.p1.29.m5.1.1.3.cmml" xref="S4.4.p1.29.m5.1.1.3"><ci id="S4.4.p1.29.m5.1.1.3.1.cmml" xref="S4.4.p1.29.m5.1.1.3.1">Β―</ci><apply id="S4.4.p1.29.m5.1.1.3.2.cmml" xref="S4.4.p1.29.m5.1.1.3.2"><intersect id="S4.4.p1.29.m5.1.1.3.2.1.cmml" xref="S4.4.p1.29.m5.1.1.3.2.1"></intersect><ci id="S4.4.p1.29.m5.1.1.3.2.2.cmml" xref="S4.4.p1.29.m5.1.1.3.2.2">π</ci><ci id="S4.4.p1.29.m5.1.1.3.2.3.cmml" xref="S4.4.p1.29.m5.1.1.3.2.3">π·</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.29.m5.1c">\overline{L}\subset\overline{U\cap D}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.29.m5.1d">overΒ― start_ARG italic_L end_ARG β overΒ― start_ARG italic_U β© italic_D end_ARG</annotation></semantics></math>. Thus the countable set <math alttext="D^{\prime}" class="ltx_Math" display="inline" id="S4.4.p1.30.m6.1"><semantics id="S4.4.p1.30.m6.1a"><msup id="S4.4.p1.30.m6.1.1" xref="S4.4.p1.30.m6.1.1.cmml"><mi id="S4.4.p1.30.m6.1.1.2" xref="S4.4.p1.30.m6.1.1.2.cmml">D</mi><mo id="S4.4.p1.30.m6.1.1.3" xref="S4.4.p1.30.m6.1.1.3.cmml">β²</mo></msup><annotation-xml encoding="MathML-Content" id="S4.4.p1.30.m6.1b"><apply id="S4.4.p1.30.m6.1.1.cmml" xref="S4.4.p1.30.m6.1.1"><csymbol cd="ambiguous" id="S4.4.p1.30.m6.1.1.1.cmml" xref="S4.4.p1.30.m6.1.1">superscript</csymbol><ci id="S4.4.p1.30.m6.1.1.2.cmml" xref="S4.4.p1.30.m6.1.1.2">π·</ci><ci id="S4.4.p1.30.m6.1.1.3.cmml" xref="S4.4.p1.30.m6.1.1.3">β²</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.30.m6.1c">D^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.30.m6.1d">italic_D start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT</annotation></semantics></math> is dense in the topological semilattice <math alttext="Y" class="ltx_Math" display="inline" id="S4.4.p1.31.m7.1"><semantics id="S4.4.p1.31.m7.1a"><mi id="S4.4.p1.31.m7.1.1" xref="S4.4.p1.31.m7.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S4.4.p1.31.m7.1b"><ci id="S4.4.p1.31.m7.1.1.cmml" xref="S4.4.p1.31.m7.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.4.p1.31.m7.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S4.4.p1.31.m7.1d">italic_Y</annotation></semantics></math>, which contradicts Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S4.Thmtheorem6" title="Theorem 4.6 (Banakh, Bonnet, KubiΕ). β£ 4. Nyikos semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">4.6</span></a>. β</p> </div> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S4.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem8.1.1.1">Proposition 4.8</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem8.p1"> <p class="ltx_p" id="S4.Thmtheorem8.p1.3"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem8.p1.3.3">Let <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.1.1.m1.1"><semantics id="S4.Thmtheorem8.p1.1.1.m1.1a"><mi id="S4.Thmtheorem8.p1.1.1.m1.1.1" xref="S4.Thmtheorem8.p1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.1.1.m1.1b"><ci id="S4.Thmtheorem8.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem8.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.1.1.m1.1d">italic_X</annotation></semantics></math> be a locally compact Nyikos topological semilattice. Then <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.2.2.m2.1"><semantics id="S4.Thmtheorem8.p1.2.2.m2.1a"><mi id="S4.Thmtheorem8.p1.2.2.m2.1.1" xref="S4.Thmtheorem8.p1.2.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.2.2.m2.1b"><ci id="S4.Thmtheorem8.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem8.p1.2.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.2.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.2.2.m2.1d">italic_X</annotation></semantics></math> contains no chain isomorphic to <math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.3.3.m3.2"><semantics id="S4.Thmtheorem8.p1.3.3.m3.2a"><mrow id="S4.Thmtheorem8.p1.3.3.m3.2.2.1" xref="S4.Thmtheorem8.p1.3.3.m3.2.2.2.cmml"><mo id="S4.Thmtheorem8.p1.3.3.m3.2.2.1.2" stretchy="false" xref="S4.Thmtheorem8.p1.3.3.m3.2.2.2.cmml">(</mo><msub id="S4.Thmtheorem8.p1.3.3.m3.2.2.1.1" xref="S4.Thmtheorem8.p1.3.3.m3.2.2.1.1.cmml"><mi id="S4.Thmtheorem8.p1.3.3.m3.2.2.1.1.2" xref="S4.Thmtheorem8.p1.3.3.m3.2.2.1.1.2.cmml">Ο</mi><mn id="S4.Thmtheorem8.p1.3.3.m3.2.2.1.1.3" xref="S4.Thmtheorem8.p1.3.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem8.p1.3.3.m3.2.2.1.3" xref="S4.Thmtheorem8.p1.3.3.m3.2.2.2.cmml">,</mo><mi id="S4.Thmtheorem8.p1.3.3.m3.1.1" xref="S4.Thmtheorem8.p1.3.3.m3.1.1.cmml">min</mi><mo id="S4.Thmtheorem8.p1.3.3.m3.2.2.1.4" stretchy="false" xref="S4.Thmtheorem8.p1.3.3.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.3.3.m3.2b"><interval closure="open" id="S4.Thmtheorem8.p1.3.3.m3.2.2.2.cmml" xref="S4.Thmtheorem8.p1.3.3.m3.2.2.1"><apply id="S4.Thmtheorem8.p1.3.3.m3.2.2.1.1.cmml" xref="S4.Thmtheorem8.p1.3.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem8.p1.3.3.m3.2.2.1.1.1.cmml" xref="S4.Thmtheorem8.p1.3.3.m3.2.2.1.1">subscript</csymbol><ci id="S4.Thmtheorem8.p1.3.3.m3.2.2.1.1.2.cmml" xref="S4.Thmtheorem8.p1.3.3.m3.2.2.1.1.2">π</ci><cn id="S4.Thmtheorem8.p1.3.3.m3.2.2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p1.3.3.m3.2.2.1.1.3">1</cn></apply><min id="S4.Thmtheorem8.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem8.p1.3.3.m3.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.3.3.m3.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.3.3.m3.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S4.5"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.5.p1"> <p class="ltx_p" id="S4.5.p1.43">To derive a contradiction assume that <math alttext="X" class="ltx_Math" display="inline" id="S4.5.p1.1.m1.1"><semantics id="S4.5.p1.1.m1.1a"><mi id="S4.5.p1.1.m1.1.1" xref="S4.5.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.5.p1.1.m1.1b"><ci id="S4.5.p1.1.m1.1.1.cmml" xref="S4.5.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.1.m1.1d">italic_X</annotation></semantics></math> contains a chain <math alttext="L=\{l_{\alpha}:\alpha<\omega_{1}\}" class="ltx_Math" display="inline" id="S4.5.p1.2.m2.2"><semantics id="S4.5.p1.2.m2.2a"><mrow id="S4.5.p1.2.m2.2.2" xref="S4.5.p1.2.m2.2.2.cmml"><mi id="S4.5.p1.2.m2.2.2.4" xref="S4.5.p1.2.m2.2.2.4.cmml">L</mi><mo id="S4.5.p1.2.m2.2.2.3" xref="S4.5.p1.2.m2.2.2.3.cmml">=</mo><mrow id="S4.5.p1.2.m2.2.2.2.2" xref="S4.5.p1.2.m2.2.2.2.3.cmml"><mo id="S4.5.p1.2.m2.2.2.2.2.3" stretchy="false" xref="S4.5.p1.2.m2.2.2.2.3.1.cmml">{</mo><msub id="S4.5.p1.2.m2.1.1.1.1.1" xref="S4.5.p1.2.m2.1.1.1.1.1.cmml"><mi id="S4.5.p1.2.m2.1.1.1.1.1.2" xref="S4.5.p1.2.m2.1.1.1.1.1.2.cmml">l</mi><mi id="S4.5.p1.2.m2.1.1.1.1.1.3" xref="S4.5.p1.2.m2.1.1.1.1.1.3.cmml">Ξ±</mi></msub><mo id="S4.5.p1.2.m2.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.5.p1.2.m2.2.2.2.3.1.cmml">:</mo><mrow id="S4.5.p1.2.m2.2.2.2.2.2" xref="S4.5.p1.2.m2.2.2.2.2.2.cmml"><mi id="S4.5.p1.2.m2.2.2.2.2.2.2" xref="S4.5.p1.2.m2.2.2.2.2.2.2.cmml">Ξ±</mi><mo id="S4.5.p1.2.m2.2.2.2.2.2.1" xref="S4.5.p1.2.m2.2.2.2.2.2.1.cmml"><</mo><msub id="S4.5.p1.2.m2.2.2.2.2.2.3" xref="S4.5.p1.2.m2.2.2.2.2.2.3.cmml"><mi id="S4.5.p1.2.m2.2.2.2.2.2.3.2" xref="S4.5.p1.2.m2.2.2.2.2.2.3.2.cmml">Ο</mi><mn id="S4.5.p1.2.m2.2.2.2.2.2.3.3" xref="S4.5.p1.2.m2.2.2.2.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S4.5.p1.2.m2.2.2.2.2.5" stretchy="false" xref="S4.5.p1.2.m2.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.2.m2.2b"><apply id="S4.5.p1.2.m2.2.2.cmml" xref="S4.5.p1.2.m2.2.2"><eq id="S4.5.p1.2.m2.2.2.3.cmml" xref="S4.5.p1.2.m2.2.2.3"></eq><ci id="S4.5.p1.2.m2.2.2.4.cmml" xref="S4.5.p1.2.m2.2.2.4">πΏ</ci><apply id="S4.5.p1.2.m2.2.2.2.3.cmml" xref="S4.5.p1.2.m2.2.2.2.2"><csymbol cd="latexml" id="S4.5.p1.2.m2.2.2.2.3.1.cmml" xref="S4.5.p1.2.m2.2.2.2.2.3">conditional-set</csymbol><apply id="S4.5.p1.2.m2.1.1.1.1.1.cmml" xref="S4.5.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.5.p1.2.m2.1.1.1.1.1.1.cmml" xref="S4.5.p1.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S4.5.p1.2.m2.1.1.1.1.1.2.cmml" xref="S4.5.p1.2.m2.1.1.1.1.1.2">π</ci><ci id="S4.5.p1.2.m2.1.1.1.1.1.3.cmml" xref="S4.5.p1.2.m2.1.1.1.1.1.3">πΌ</ci></apply><apply id="S4.5.p1.2.m2.2.2.2.2.2.cmml" xref="S4.5.p1.2.m2.2.2.2.2.2"><lt id="S4.5.p1.2.m2.2.2.2.2.2.1.cmml" xref="S4.5.p1.2.m2.2.2.2.2.2.1"></lt><ci id="S4.5.p1.2.m2.2.2.2.2.2.2.cmml" xref="S4.5.p1.2.m2.2.2.2.2.2.2">πΌ</ci><apply id="S4.5.p1.2.m2.2.2.2.2.2.3.cmml" xref="S4.5.p1.2.m2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.5.p1.2.m2.2.2.2.2.2.3.1.cmml" xref="S4.5.p1.2.m2.2.2.2.2.2.3">subscript</csymbol><ci id="S4.5.p1.2.m2.2.2.2.2.2.3.2.cmml" xref="S4.5.p1.2.m2.2.2.2.2.2.3.2">π</ci><cn id="S4.5.p1.2.m2.2.2.2.2.2.3.3.cmml" type="integer" xref="S4.5.p1.2.m2.2.2.2.2.2.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.2.m2.2c">L=\{l_{\alpha}:\alpha<\omega_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.2.m2.2d">italic_L = { italic_l start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT : italic_Ξ± < italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math> such that <math alttext="l_{\alpha}\leq l_{\beta}" class="ltx_Math" display="inline" id="S4.5.p1.3.m3.1"><semantics id="S4.5.p1.3.m3.1a"><mrow id="S4.5.p1.3.m3.1.1" xref="S4.5.p1.3.m3.1.1.cmml"><msub id="S4.5.p1.3.m3.1.1.2" xref="S4.5.p1.3.m3.1.1.2.cmml"><mi id="S4.5.p1.3.m3.1.1.2.2" xref="S4.5.p1.3.m3.1.1.2.2.cmml">l</mi><mi id="S4.5.p1.3.m3.1.1.2.3" xref="S4.5.p1.3.m3.1.1.2.3.cmml">Ξ±</mi></msub><mo id="S4.5.p1.3.m3.1.1.1" xref="S4.5.p1.3.m3.1.1.1.cmml">β€</mo><msub id="S4.5.p1.3.m3.1.1.3" xref="S4.5.p1.3.m3.1.1.3.cmml"><mi id="S4.5.p1.3.m3.1.1.3.2" xref="S4.5.p1.3.m3.1.1.3.2.cmml">l</mi><mi id="S4.5.p1.3.m3.1.1.3.3" xref="S4.5.p1.3.m3.1.1.3.3.cmml">Ξ²</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.3.m3.1b"><apply id="S4.5.p1.3.m3.1.1.cmml" xref="S4.5.p1.3.m3.1.1"><leq id="S4.5.p1.3.m3.1.1.1.cmml" xref="S4.5.p1.3.m3.1.1.1"></leq><apply id="S4.5.p1.3.m3.1.1.2.cmml" xref="S4.5.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.5.p1.3.m3.1.1.2.1.cmml" xref="S4.5.p1.3.m3.1.1.2">subscript</csymbol><ci id="S4.5.p1.3.m3.1.1.2.2.cmml" xref="S4.5.p1.3.m3.1.1.2.2">π</ci><ci id="S4.5.p1.3.m3.1.1.2.3.cmml" xref="S4.5.p1.3.m3.1.1.2.3">πΌ</ci></apply><apply id="S4.5.p1.3.m3.1.1.3.cmml" xref="S4.5.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.5.p1.3.m3.1.1.3.1.cmml" xref="S4.5.p1.3.m3.1.1.3">subscript</csymbol><ci id="S4.5.p1.3.m3.1.1.3.2.cmml" xref="S4.5.p1.3.m3.1.1.3.2">π</ci><ci id="S4.5.p1.3.m3.1.1.3.3.cmml" xref="S4.5.p1.3.m3.1.1.3.3">π½</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.3.m3.1c">l_{\alpha}\leq l_{\beta}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.3.m3.1d">italic_l start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT β€ italic_l start_POSTSUBSCRIPT italic_Ξ² end_POSTSUBSCRIPT</annotation></semantics></math> if and only if <math alttext="\alpha\leq\beta" class="ltx_Math" display="inline" id="S4.5.p1.4.m4.1"><semantics id="S4.5.p1.4.m4.1a"><mrow id="S4.5.p1.4.m4.1.1" xref="S4.5.p1.4.m4.1.1.cmml"><mi id="S4.5.p1.4.m4.1.1.2" xref="S4.5.p1.4.m4.1.1.2.cmml">Ξ±</mi><mo id="S4.5.p1.4.m4.1.1.1" xref="S4.5.p1.4.m4.1.1.1.cmml">β€</mo><mi id="S4.5.p1.4.m4.1.1.3" xref="S4.5.p1.4.m4.1.1.3.cmml">Ξ²</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.4.m4.1b"><apply id="S4.5.p1.4.m4.1.1.cmml" xref="S4.5.p1.4.m4.1.1"><leq id="S4.5.p1.4.m4.1.1.1.cmml" xref="S4.5.p1.4.m4.1.1.1"></leq><ci id="S4.5.p1.4.m4.1.1.2.cmml" xref="S4.5.p1.4.m4.1.1.2">πΌ</ci><ci id="S4.5.p1.4.m4.1.1.3.cmml" xref="S4.5.p1.4.m4.1.1.3">π½</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.4.m4.1c">\alpha\leq\beta</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.4.m4.1d">italic_Ξ± β€ italic_Ξ²</annotation></semantics></math>. By <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib22" title="">22</a>, Corollary 3.10.15]</cite>, <math alttext="X{\times}X" class="ltx_Math" display="inline" id="S4.5.p1.5.m5.1"><semantics id="S4.5.p1.5.m5.1a"><mrow id="S4.5.p1.5.m5.1.1" xref="S4.5.p1.5.m5.1.1.cmml"><mi id="S4.5.p1.5.m5.1.1.2" xref="S4.5.p1.5.m5.1.1.2.cmml">X</mi><mo id="S4.5.p1.5.m5.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.5.p1.5.m5.1.1.1.cmml">Γ</mo><mi id="S4.5.p1.5.m5.1.1.3" xref="S4.5.p1.5.m5.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.5.m5.1b"><apply id="S4.5.p1.5.m5.1.1.cmml" xref="S4.5.p1.5.m5.1.1"><times id="S4.5.p1.5.m5.1.1.1.cmml" xref="S4.5.p1.5.m5.1.1.1"></times><ci id="S4.5.p1.5.m5.1.1.2.cmml" xref="S4.5.p1.5.m5.1.1.2">π</ci><ci id="S4.5.p1.5.m5.1.1.3.cmml" xref="S4.5.p1.5.m5.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.5.m5.1c">X{\times}X</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.5.m5.1d">italic_X Γ italic_X</annotation></semantics></math> is countable compact and thus pseudocompact. Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S2.Thmtheorem9" title="Theorem 2.9 (Banakh, Dimitrova, Gutik). β£ 2. Compact-like semigroups with dense inverse subsemigroups and the automatic continuity of inversion β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">2.9</span></a> implies that <math alttext="\beta X" class="ltx_Math" display="inline" id="S4.5.p1.6.m6.1"><semantics id="S4.5.p1.6.m6.1a"><mrow id="S4.5.p1.6.m6.1.1" xref="S4.5.p1.6.m6.1.1.cmml"><mi id="S4.5.p1.6.m6.1.1.2" xref="S4.5.p1.6.m6.1.1.2.cmml">Ξ²</mi><mo id="S4.5.p1.6.m6.1.1.1" xref="S4.5.p1.6.m6.1.1.1.cmml">β’</mo><mi id="S4.5.p1.6.m6.1.1.3" xref="S4.5.p1.6.m6.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.6.m6.1b"><apply id="S4.5.p1.6.m6.1.1.cmml" xref="S4.5.p1.6.m6.1.1"><times id="S4.5.p1.6.m6.1.1.1.cmml" xref="S4.5.p1.6.m6.1.1.1"></times><ci id="S4.5.p1.6.m6.1.1.2.cmml" xref="S4.5.p1.6.m6.1.1.2">π½</ci><ci id="S4.5.p1.6.m6.1.1.3.cmml" xref="S4.5.p1.6.m6.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.6.m6.1c">\beta X</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.6.m6.1d">italic_Ξ² italic_X</annotation></semantics></math> is a compact topological semigroup. Since <math alttext="X" class="ltx_Math" display="inline" id="S4.5.p1.7.m7.1"><semantics id="S4.5.p1.7.m7.1a"><mi id="S4.5.p1.7.m7.1.1" xref="S4.5.p1.7.m7.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.5.p1.7.m7.1b"><ci id="S4.5.p1.7.m7.1.1.cmml" xref="S4.5.p1.7.m7.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.7.m7.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.7.m7.1d">italic_X</annotation></semantics></math> is dense in <math alttext="\beta X" class="ltx_Math" display="inline" id="S4.5.p1.8.m8.1"><semantics id="S4.5.p1.8.m8.1a"><mrow id="S4.5.p1.8.m8.1.1" xref="S4.5.p1.8.m8.1.1.cmml"><mi id="S4.5.p1.8.m8.1.1.2" xref="S4.5.p1.8.m8.1.1.2.cmml">Ξ²</mi><mo id="S4.5.p1.8.m8.1.1.1" xref="S4.5.p1.8.m8.1.1.1.cmml">β’</mo><mi id="S4.5.p1.8.m8.1.1.3" xref="S4.5.p1.8.m8.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.8.m8.1b"><apply id="S4.5.p1.8.m8.1.1.cmml" xref="S4.5.p1.8.m8.1.1"><times id="S4.5.p1.8.m8.1.1.1.cmml" xref="S4.5.p1.8.m8.1.1.1"></times><ci id="S4.5.p1.8.m8.1.1.2.cmml" xref="S4.5.p1.8.m8.1.1.2">π½</ci><ci id="S4.5.p1.8.m8.1.1.3.cmml" xref="S4.5.p1.8.m8.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.8.m8.1c">\beta X</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.8.m8.1d">italic_Ξ² italic_X</annotation></semantics></math>, we obtain that <math alttext="\beta X" class="ltx_Math" display="inline" id="S4.5.p1.9.m9.1"><semantics id="S4.5.p1.9.m9.1a"><mrow id="S4.5.p1.9.m9.1.1" xref="S4.5.p1.9.m9.1.1.cmml"><mi id="S4.5.p1.9.m9.1.1.2" xref="S4.5.p1.9.m9.1.1.2.cmml">Ξ²</mi><mo id="S4.5.p1.9.m9.1.1.1" xref="S4.5.p1.9.m9.1.1.1.cmml">β’</mo><mi id="S4.5.p1.9.m9.1.1.3" xref="S4.5.p1.9.m9.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.9.m9.1b"><apply id="S4.5.p1.9.m9.1.1.cmml" xref="S4.5.p1.9.m9.1.1"><times id="S4.5.p1.9.m9.1.1.1.cmml" xref="S4.5.p1.9.m9.1.1.1"></times><ci id="S4.5.p1.9.m9.1.1.2.cmml" xref="S4.5.p1.9.m9.1.1.2">π½</ci><ci id="S4.5.p1.9.m9.1.1.3.cmml" xref="S4.5.p1.9.m9.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.9.m9.1c">\beta X</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.9.m9.1d">italic_Ξ² italic_X</annotation></semantics></math> is a semilattice. By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem4" title="Lemma 3.4. β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.4</span></a>, there exists <math alttext="z=\sup L\in\beta X" class="ltx_Math" display="inline" id="S4.5.p1.10.m10.1"><semantics id="S4.5.p1.10.m10.1a"><mrow id="S4.5.p1.10.m10.1.1" xref="S4.5.p1.10.m10.1.1.cmml"><mi id="S4.5.p1.10.m10.1.1.2" xref="S4.5.p1.10.m10.1.1.2.cmml">z</mi><mo id="S4.5.p1.10.m10.1.1.3" rspace="0.1389em" xref="S4.5.p1.10.m10.1.1.3.cmml">=</mo><mrow id="S4.5.p1.10.m10.1.1.4" xref="S4.5.p1.10.m10.1.1.4.cmml"><mo id="S4.5.p1.10.m10.1.1.4.1" lspace="0.1389em" rspace="0.167em" xref="S4.5.p1.10.m10.1.1.4.1.cmml">sup</mo><mi id="S4.5.p1.10.m10.1.1.4.2" xref="S4.5.p1.10.m10.1.1.4.2.cmml">L</mi></mrow><mo id="S4.5.p1.10.m10.1.1.5" xref="S4.5.p1.10.m10.1.1.5.cmml">β</mo><mrow id="S4.5.p1.10.m10.1.1.6" xref="S4.5.p1.10.m10.1.1.6.cmml"><mi id="S4.5.p1.10.m10.1.1.6.2" xref="S4.5.p1.10.m10.1.1.6.2.cmml">Ξ²</mi><mo id="S4.5.p1.10.m10.1.1.6.1" xref="S4.5.p1.10.m10.1.1.6.1.cmml">β’</mo><mi id="S4.5.p1.10.m10.1.1.6.3" xref="S4.5.p1.10.m10.1.1.6.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.10.m10.1b"><apply id="S4.5.p1.10.m10.1.1.cmml" xref="S4.5.p1.10.m10.1.1"><and id="S4.5.p1.10.m10.1.1a.cmml" xref="S4.5.p1.10.m10.1.1"></and><apply id="S4.5.p1.10.m10.1.1b.cmml" xref="S4.5.p1.10.m10.1.1"><eq id="S4.5.p1.10.m10.1.1.3.cmml" xref="S4.5.p1.10.m10.1.1.3"></eq><ci id="S4.5.p1.10.m10.1.1.2.cmml" xref="S4.5.p1.10.m10.1.1.2">π§</ci><apply id="S4.5.p1.10.m10.1.1.4.cmml" xref="S4.5.p1.10.m10.1.1.4"><csymbol cd="latexml" id="S4.5.p1.10.m10.1.1.4.1.cmml" xref="S4.5.p1.10.m10.1.1.4.1">supremum</csymbol><ci id="S4.5.p1.10.m10.1.1.4.2.cmml" xref="S4.5.p1.10.m10.1.1.4.2">πΏ</ci></apply></apply><apply id="S4.5.p1.10.m10.1.1c.cmml" xref="S4.5.p1.10.m10.1.1"><in id="S4.5.p1.10.m10.1.1.5.cmml" xref="S4.5.p1.10.m10.1.1.5"></in><share href="https://arxiv.org/html/2503.13666v1#S4.5.p1.10.m10.1.1.4.cmml" id="S4.5.p1.10.m10.1.1d.cmml" xref="S4.5.p1.10.m10.1.1"></share><apply id="S4.5.p1.10.m10.1.1.6.cmml" xref="S4.5.p1.10.m10.1.1.6"><times id="S4.5.p1.10.m10.1.1.6.1.cmml" xref="S4.5.p1.10.m10.1.1.6.1"></times><ci id="S4.5.p1.10.m10.1.1.6.2.cmml" xref="S4.5.p1.10.m10.1.1.6.2">π½</ci><ci id="S4.5.p1.10.m10.1.1.6.3.cmml" xref="S4.5.p1.10.m10.1.1.6.3">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.10.m10.1c">z=\sup L\in\beta X</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.10.m10.1d">italic_z = roman_sup italic_L β italic_Ξ² italic_X</annotation></semantics></math>. Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S4.Thmtheorem7" title="Lemma 4.7. β£ 4. Nyikos semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">4.7</span></a> implies that there exists <math alttext="l_{\alpha}\in L" class="ltx_Math" display="inline" id="S4.5.p1.11.m11.1"><semantics id="S4.5.p1.11.m11.1a"><mrow id="S4.5.p1.11.m11.1.1" xref="S4.5.p1.11.m11.1.1.cmml"><msub id="S4.5.p1.11.m11.1.1.2" xref="S4.5.p1.11.m11.1.1.2.cmml"><mi id="S4.5.p1.11.m11.1.1.2.2" xref="S4.5.p1.11.m11.1.1.2.2.cmml">l</mi><mi id="S4.5.p1.11.m11.1.1.2.3" xref="S4.5.p1.11.m11.1.1.2.3.cmml">Ξ±</mi></msub><mo id="S4.5.p1.11.m11.1.1.1" xref="S4.5.p1.11.m11.1.1.1.cmml">β</mo><mi id="S4.5.p1.11.m11.1.1.3" xref="S4.5.p1.11.m11.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.11.m11.1b"><apply id="S4.5.p1.11.m11.1.1.cmml" xref="S4.5.p1.11.m11.1.1"><in id="S4.5.p1.11.m11.1.1.1.cmml" xref="S4.5.p1.11.m11.1.1.1"></in><apply id="S4.5.p1.11.m11.1.1.2.cmml" xref="S4.5.p1.11.m11.1.1.2"><csymbol cd="ambiguous" id="S4.5.p1.11.m11.1.1.2.1.cmml" xref="S4.5.p1.11.m11.1.1.2">subscript</csymbol><ci id="S4.5.p1.11.m11.1.1.2.2.cmml" xref="S4.5.p1.11.m11.1.1.2.2">π</ci><ci id="S4.5.p1.11.m11.1.1.2.3.cmml" xref="S4.5.p1.11.m11.1.1.2.3">πΌ</ci></apply><ci id="S4.5.p1.11.m11.1.1.3.cmml" xref="S4.5.p1.11.m11.1.1.3">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.11.m11.1c">l_{\alpha}\in L</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.11.m11.1d">italic_l start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT β italic_L</annotation></semantics></math> such that for each open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S4.5.p1.12.m12.1"><semantics id="S4.5.p1.12.m12.1a"><mi id="S4.5.p1.12.m12.1.1" xref="S4.5.p1.12.m12.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S4.5.p1.12.m12.1b"><ci id="S4.5.p1.12.m12.1.1.cmml" xref="S4.5.p1.12.m12.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.12.m12.1c">U</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.12.m12.1d">italic_U</annotation></semantics></math> of <math alttext="l_{\alpha}" class="ltx_Math" display="inline" id="S4.5.p1.13.m13.1"><semantics id="S4.5.p1.13.m13.1a"><msub id="S4.5.p1.13.m13.1.1" xref="S4.5.p1.13.m13.1.1.cmml"><mi id="S4.5.p1.13.m13.1.1.2" xref="S4.5.p1.13.m13.1.1.2.cmml">l</mi><mi id="S4.5.p1.13.m13.1.1.3" xref="S4.5.p1.13.m13.1.1.3.cmml">Ξ±</mi></msub><annotation-xml encoding="MathML-Content" id="S4.5.p1.13.m13.1b"><apply id="S4.5.p1.13.m13.1.1.cmml" xref="S4.5.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S4.5.p1.13.m13.1.1.1.cmml" xref="S4.5.p1.13.m13.1.1">subscript</csymbol><ci id="S4.5.p1.13.m13.1.1.2.cmml" xref="S4.5.p1.13.m13.1.1.2">π</ci><ci id="S4.5.p1.13.m13.1.1.3.cmml" xref="S4.5.p1.13.m13.1.1.3">πΌ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.13.m13.1c">l_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.13.m13.1d">italic_l start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT</annotation></semantics></math> the set <math alttext="U\setminus I_{L}" class="ltx_Math" display="inline" id="S4.5.p1.14.m14.1"><semantics id="S4.5.p1.14.m14.1a"><mrow id="S4.5.p1.14.m14.1.1" xref="S4.5.p1.14.m14.1.1.cmml"><mi id="S4.5.p1.14.m14.1.1.2" xref="S4.5.p1.14.m14.1.1.2.cmml">U</mi><mo id="S4.5.p1.14.m14.1.1.1" xref="S4.5.p1.14.m14.1.1.1.cmml">β</mo><msub id="S4.5.p1.14.m14.1.1.3" xref="S4.5.p1.14.m14.1.1.3.cmml"><mi id="S4.5.p1.14.m14.1.1.3.2" xref="S4.5.p1.14.m14.1.1.3.2.cmml">I</mi><mi id="S4.5.p1.14.m14.1.1.3.3" xref="S4.5.p1.14.m14.1.1.3.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.14.m14.1b"><apply id="S4.5.p1.14.m14.1.1.cmml" xref="S4.5.p1.14.m14.1.1"><setdiff id="S4.5.p1.14.m14.1.1.1.cmml" xref="S4.5.p1.14.m14.1.1.1"></setdiff><ci id="S4.5.p1.14.m14.1.1.2.cmml" xref="S4.5.p1.14.m14.1.1.2">π</ci><apply id="S4.5.p1.14.m14.1.1.3.cmml" xref="S4.5.p1.14.m14.1.1.3"><csymbol cd="ambiguous" id="S4.5.p1.14.m14.1.1.3.1.cmml" xref="S4.5.p1.14.m14.1.1.3">subscript</csymbol><ci id="S4.5.p1.14.m14.1.1.3.2.cmml" xref="S4.5.p1.14.m14.1.1.3.2">πΌ</ci><ci id="S4.5.p1.14.m14.1.1.3.3.cmml" xref="S4.5.p1.14.m14.1.1.3.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.14.m14.1c">U\setminus I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.14.m14.1d">italic_U β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> contains an element <math alttext="d_{U}" class="ltx_Math" display="inline" id="S4.5.p1.15.m15.1"><semantics id="S4.5.p1.15.m15.1a"><msub id="S4.5.p1.15.m15.1.1" xref="S4.5.p1.15.m15.1.1.cmml"><mi id="S4.5.p1.15.m15.1.1.2" xref="S4.5.p1.15.m15.1.1.2.cmml">d</mi><mi id="S4.5.p1.15.m15.1.1.3" xref="S4.5.p1.15.m15.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S4.5.p1.15.m15.1b"><apply id="S4.5.p1.15.m15.1.1.cmml" xref="S4.5.p1.15.m15.1.1"><csymbol cd="ambiguous" id="S4.5.p1.15.m15.1.1.1.cmml" xref="S4.5.p1.15.m15.1.1">subscript</csymbol><ci id="S4.5.p1.15.m15.1.1.2.cmml" xref="S4.5.p1.15.m15.1.1.2">π</ci><ci id="S4.5.p1.15.m15.1.1.3.cmml" xref="S4.5.p1.15.m15.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.15.m15.1c">d_{U}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.15.m15.1d">italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="X" class="ltx_Math" display="inline" id="S4.5.p1.16.m16.1"><semantics id="S4.5.p1.16.m16.1a"><mi id="S4.5.p1.16.m16.1.1" xref="S4.5.p1.16.m16.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.5.p1.16.m16.1b"><ci id="S4.5.p1.16.m16.1.1.cmml" xref="S4.5.p1.16.m16.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.16.m16.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.16.m16.1d">italic_X</annotation></semantics></math> is locally compact, there exists an open neighborhood <math alttext="W" class="ltx_Math" display="inline" id="S4.5.p1.17.m17.1"><semantics id="S4.5.p1.17.m17.1a"><mi id="S4.5.p1.17.m17.1.1" xref="S4.5.p1.17.m17.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S4.5.p1.17.m17.1b"><ci id="S4.5.p1.17.m17.1.1.cmml" xref="S4.5.p1.17.m17.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.17.m17.1c">W</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.17.m17.1d">italic_W</annotation></semantics></math> of <math alttext="l_{\alpha}" class="ltx_Math" display="inline" id="S4.5.p1.18.m18.1"><semantics id="S4.5.p1.18.m18.1a"><msub id="S4.5.p1.18.m18.1.1" xref="S4.5.p1.18.m18.1.1.cmml"><mi id="S4.5.p1.18.m18.1.1.2" xref="S4.5.p1.18.m18.1.1.2.cmml">l</mi><mi id="S4.5.p1.18.m18.1.1.3" xref="S4.5.p1.18.m18.1.1.3.cmml">Ξ±</mi></msub><annotation-xml encoding="MathML-Content" id="S4.5.p1.18.m18.1b"><apply id="S4.5.p1.18.m18.1.1.cmml" xref="S4.5.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S4.5.p1.18.m18.1.1.1.cmml" xref="S4.5.p1.18.m18.1.1">subscript</csymbol><ci id="S4.5.p1.18.m18.1.1.2.cmml" xref="S4.5.p1.18.m18.1.1.2">π</ci><ci id="S4.5.p1.18.m18.1.1.3.cmml" xref="S4.5.p1.18.m18.1.1.3">πΌ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.18.m18.1c">l_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.18.m18.1d">italic_l start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="\operatorname{cl}_{X}(W)" class="ltx_Math" display="inline" id="S4.5.p1.19.m19.2"><semantics id="S4.5.p1.19.m19.2a"><mrow id="S4.5.p1.19.m19.2.2.1" xref="S4.5.p1.19.m19.2.2.2.cmml"><msub id="S4.5.p1.19.m19.2.2.1.1" xref="S4.5.p1.19.m19.2.2.1.1.cmml"><mi id="S4.5.p1.19.m19.2.2.1.1.2" xref="S4.5.p1.19.m19.2.2.1.1.2.cmml">cl</mi><mi id="S4.5.p1.19.m19.2.2.1.1.3" xref="S4.5.p1.19.m19.2.2.1.1.3.cmml">X</mi></msub><mo id="S4.5.p1.19.m19.2.2.1a" xref="S4.5.p1.19.m19.2.2.2.cmml">β‘</mo><mrow id="S4.5.p1.19.m19.2.2.1.2" xref="S4.5.p1.19.m19.2.2.2.cmml"><mo id="S4.5.p1.19.m19.2.2.1.2.1" stretchy="false" xref="S4.5.p1.19.m19.2.2.2.cmml">(</mo><mi id="S4.5.p1.19.m19.1.1" xref="S4.5.p1.19.m19.1.1.cmml">W</mi><mo id="S4.5.p1.19.m19.2.2.1.2.2" stretchy="false" xref="S4.5.p1.19.m19.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.19.m19.2b"><apply id="S4.5.p1.19.m19.2.2.2.cmml" xref="S4.5.p1.19.m19.2.2.1"><apply id="S4.5.p1.19.m19.2.2.1.1.cmml" xref="S4.5.p1.19.m19.2.2.1.1"><csymbol cd="ambiguous" id="S4.5.p1.19.m19.2.2.1.1.1.cmml" xref="S4.5.p1.19.m19.2.2.1.1">subscript</csymbol><ci id="S4.5.p1.19.m19.2.2.1.1.2.cmml" xref="S4.5.p1.19.m19.2.2.1.1.2">cl</ci><ci id="S4.5.p1.19.m19.2.2.1.1.3.cmml" xref="S4.5.p1.19.m19.2.2.1.1.3">π</ci></apply><ci id="S4.5.p1.19.m19.1.1.cmml" xref="S4.5.p1.19.m19.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.19.m19.2c">\operatorname{cl}_{X}(W)</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.19.m19.2d">roman_cl start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_W )</annotation></semantics></math> is compact. The continuity of the semilattice operation in <math alttext="\beta X" class="ltx_Math" display="inline" id="S4.5.p1.20.m20.1"><semantics id="S4.5.p1.20.m20.1a"><mrow id="S4.5.p1.20.m20.1.1" xref="S4.5.p1.20.m20.1.1.cmml"><mi id="S4.5.p1.20.m20.1.1.2" xref="S4.5.p1.20.m20.1.1.2.cmml">Ξ²</mi><mo id="S4.5.p1.20.m20.1.1.1" xref="S4.5.p1.20.m20.1.1.1.cmml">β’</mo><mi id="S4.5.p1.20.m20.1.1.3" xref="S4.5.p1.20.m20.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.20.m20.1b"><apply id="S4.5.p1.20.m20.1.1.cmml" xref="S4.5.p1.20.m20.1.1"><times id="S4.5.p1.20.m20.1.1.1.cmml" xref="S4.5.p1.20.m20.1.1.1"></times><ci id="S4.5.p1.20.m20.1.1.2.cmml" xref="S4.5.p1.20.m20.1.1.2">π½</ci><ci id="S4.5.p1.20.m20.1.1.3.cmml" xref="S4.5.p1.20.m20.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.20.m20.1c">\beta X</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.20.m20.1d">italic_Ξ² italic_X</annotation></semantics></math> yields open neighborhoods <math alttext="U" class="ltx_Math" display="inline" id="S4.5.p1.21.m21.1"><semantics id="S4.5.p1.21.m21.1a"><mi id="S4.5.p1.21.m21.1.1" xref="S4.5.p1.21.m21.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S4.5.p1.21.m21.1b"><ci id="S4.5.p1.21.m21.1.1.cmml" xref="S4.5.p1.21.m21.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.21.m21.1c">U</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.21.m21.1d">italic_U</annotation></semantics></math> of <math alttext="l_{\alpha}" class="ltx_Math" display="inline" id="S4.5.p1.22.m22.1"><semantics id="S4.5.p1.22.m22.1a"><msub id="S4.5.p1.22.m22.1.1" xref="S4.5.p1.22.m22.1.1.cmml"><mi id="S4.5.p1.22.m22.1.1.2" xref="S4.5.p1.22.m22.1.1.2.cmml">l</mi><mi id="S4.5.p1.22.m22.1.1.3" xref="S4.5.p1.22.m22.1.1.3.cmml">Ξ±</mi></msub><annotation-xml encoding="MathML-Content" id="S4.5.p1.22.m22.1b"><apply id="S4.5.p1.22.m22.1.1.cmml" xref="S4.5.p1.22.m22.1.1"><csymbol cd="ambiguous" id="S4.5.p1.22.m22.1.1.1.cmml" xref="S4.5.p1.22.m22.1.1">subscript</csymbol><ci id="S4.5.p1.22.m22.1.1.2.cmml" xref="S4.5.p1.22.m22.1.1.2">π</ci><ci id="S4.5.p1.22.m22.1.1.3.cmml" xref="S4.5.p1.22.m22.1.1.3">πΌ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.22.m22.1c">l_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.22.m22.1d">italic_l start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="V" class="ltx_Math" display="inline" id="S4.5.p1.23.m23.1"><semantics id="S4.5.p1.23.m23.1a"><mi id="S4.5.p1.23.m23.1.1" xref="S4.5.p1.23.m23.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S4.5.p1.23.m23.1b"><ci id="S4.5.p1.23.m23.1.1.cmml" xref="S4.5.p1.23.m23.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.23.m23.1c">V</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.23.m23.1d">italic_V</annotation></semantics></math> of <math alttext="z" class="ltx_Math" display="inline" id="S4.5.p1.24.m24.1"><semantics id="S4.5.p1.24.m24.1a"><mi id="S4.5.p1.24.m24.1.1" xref="S4.5.p1.24.m24.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.5.p1.24.m24.1b"><ci id="S4.5.p1.24.m24.1.1.cmml" xref="S4.5.p1.24.m24.1.1">π§</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.24.m24.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.24.m24.1d">italic_z</annotation></semantics></math> such that <math alttext="UV\subseteq W" class="ltx_Math" display="inline" id="S4.5.p1.25.m25.1"><semantics id="S4.5.p1.25.m25.1a"><mrow id="S4.5.p1.25.m25.1.1" xref="S4.5.p1.25.m25.1.1.cmml"><mrow id="S4.5.p1.25.m25.1.1.2" xref="S4.5.p1.25.m25.1.1.2.cmml"><mi id="S4.5.p1.25.m25.1.1.2.2" xref="S4.5.p1.25.m25.1.1.2.2.cmml">U</mi><mo id="S4.5.p1.25.m25.1.1.2.1" xref="S4.5.p1.25.m25.1.1.2.1.cmml">β’</mo><mi id="S4.5.p1.25.m25.1.1.2.3" xref="S4.5.p1.25.m25.1.1.2.3.cmml">V</mi></mrow><mo id="S4.5.p1.25.m25.1.1.1" xref="S4.5.p1.25.m25.1.1.1.cmml">β</mo><mi id="S4.5.p1.25.m25.1.1.3" xref="S4.5.p1.25.m25.1.1.3.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.25.m25.1b"><apply id="S4.5.p1.25.m25.1.1.cmml" xref="S4.5.p1.25.m25.1.1"><subset id="S4.5.p1.25.m25.1.1.1.cmml" xref="S4.5.p1.25.m25.1.1.1"></subset><apply id="S4.5.p1.25.m25.1.1.2.cmml" xref="S4.5.p1.25.m25.1.1.2"><times id="S4.5.p1.25.m25.1.1.2.1.cmml" xref="S4.5.p1.25.m25.1.1.2.1"></times><ci id="S4.5.p1.25.m25.1.1.2.2.cmml" xref="S4.5.p1.25.m25.1.1.2.2">π</ci><ci id="S4.5.p1.25.m25.1.1.2.3.cmml" xref="S4.5.p1.25.m25.1.1.2.3">π</ci></apply><ci id="S4.5.p1.25.m25.1.1.3.cmml" xref="S4.5.p1.25.m25.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.25.m25.1c">UV\subseteq W</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.25.m25.1d">italic_U italic_V β italic_W</annotation></semantics></math>. By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem4" title="Lemma 3.4. β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.4</span></a>, there exists an ordinal <math alttext="\delta<\omega_{1}" class="ltx_Math" display="inline" id="S4.5.p1.26.m26.1"><semantics id="S4.5.p1.26.m26.1a"><mrow id="S4.5.p1.26.m26.1.1" xref="S4.5.p1.26.m26.1.1.cmml"><mi id="S4.5.p1.26.m26.1.1.2" xref="S4.5.p1.26.m26.1.1.2.cmml">Ξ΄</mi><mo id="S4.5.p1.26.m26.1.1.1" xref="S4.5.p1.26.m26.1.1.1.cmml"><</mo><msub id="S4.5.p1.26.m26.1.1.3" xref="S4.5.p1.26.m26.1.1.3.cmml"><mi id="S4.5.p1.26.m26.1.1.3.2" xref="S4.5.p1.26.m26.1.1.3.2.cmml">Ο</mi><mn id="S4.5.p1.26.m26.1.1.3.3" xref="S4.5.p1.26.m26.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.26.m26.1b"><apply id="S4.5.p1.26.m26.1.1.cmml" xref="S4.5.p1.26.m26.1.1"><lt id="S4.5.p1.26.m26.1.1.1.cmml" xref="S4.5.p1.26.m26.1.1.1"></lt><ci id="S4.5.p1.26.m26.1.1.2.cmml" xref="S4.5.p1.26.m26.1.1.2">πΏ</ci><apply id="S4.5.p1.26.m26.1.1.3.cmml" xref="S4.5.p1.26.m26.1.1.3"><csymbol cd="ambiguous" id="S4.5.p1.26.m26.1.1.3.1.cmml" xref="S4.5.p1.26.m26.1.1.3">subscript</csymbol><ci id="S4.5.p1.26.m26.1.1.3.2.cmml" xref="S4.5.p1.26.m26.1.1.3.2">π</ci><cn id="S4.5.p1.26.m26.1.1.3.3.cmml" type="integer" xref="S4.5.p1.26.m26.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.26.m26.1c">\delta<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.26.m26.1d">italic_Ξ΄ < italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="L^{\prime}=\{l_{\xi}:\delta<\xi<\omega_{1}\}\subset V" class="ltx_Math" display="inline" id="S4.5.p1.27.m27.2"><semantics id="S4.5.p1.27.m27.2a"><mrow id="S4.5.p1.27.m27.2.2" xref="S4.5.p1.27.m27.2.2.cmml"><msup id="S4.5.p1.27.m27.2.2.4" xref="S4.5.p1.27.m27.2.2.4.cmml"><mi id="S4.5.p1.27.m27.2.2.4.2" xref="S4.5.p1.27.m27.2.2.4.2.cmml">L</mi><mo id="S4.5.p1.27.m27.2.2.4.3" xref="S4.5.p1.27.m27.2.2.4.3.cmml">β²</mo></msup><mo id="S4.5.p1.27.m27.2.2.5" xref="S4.5.p1.27.m27.2.2.5.cmml">=</mo><mrow id="S4.5.p1.27.m27.2.2.2.2" xref="S4.5.p1.27.m27.2.2.2.3.cmml"><mo id="S4.5.p1.27.m27.2.2.2.2.3" stretchy="false" xref="S4.5.p1.27.m27.2.2.2.3.1.cmml">{</mo><msub id="S4.5.p1.27.m27.1.1.1.1.1" xref="S4.5.p1.27.m27.1.1.1.1.1.cmml"><mi id="S4.5.p1.27.m27.1.1.1.1.1.2" xref="S4.5.p1.27.m27.1.1.1.1.1.2.cmml">l</mi><mi id="S4.5.p1.27.m27.1.1.1.1.1.3" xref="S4.5.p1.27.m27.1.1.1.1.1.3.cmml">ΞΎ</mi></msub><mo id="S4.5.p1.27.m27.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.5.p1.27.m27.2.2.2.3.1.cmml">:</mo><mrow id="S4.5.p1.27.m27.2.2.2.2.2" xref="S4.5.p1.27.m27.2.2.2.2.2.cmml"><mi id="S4.5.p1.27.m27.2.2.2.2.2.2" xref="S4.5.p1.27.m27.2.2.2.2.2.2.cmml">Ξ΄</mi><mo id="S4.5.p1.27.m27.2.2.2.2.2.3" xref="S4.5.p1.27.m27.2.2.2.2.2.3.cmml"><</mo><mi id="S4.5.p1.27.m27.2.2.2.2.2.4" xref="S4.5.p1.27.m27.2.2.2.2.2.4.cmml">ΞΎ</mi><mo id="S4.5.p1.27.m27.2.2.2.2.2.5" xref="S4.5.p1.27.m27.2.2.2.2.2.5.cmml"><</mo><msub id="S4.5.p1.27.m27.2.2.2.2.2.6" xref="S4.5.p1.27.m27.2.2.2.2.2.6.cmml"><mi id="S4.5.p1.27.m27.2.2.2.2.2.6.2" xref="S4.5.p1.27.m27.2.2.2.2.2.6.2.cmml">Ο</mi><mn id="S4.5.p1.27.m27.2.2.2.2.2.6.3" xref="S4.5.p1.27.m27.2.2.2.2.2.6.3.cmml">1</mn></msub></mrow><mo id="S4.5.p1.27.m27.2.2.2.2.5" stretchy="false" xref="S4.5.p1.27.m27.2.2.2.3.1.cmml">}</mo></mrow><mo id="S4.5.p1.27.m27.2.2.6" xref="S4.5.p1.27.m27.2.2.6.cmml">β</mo><mi id="S4.5.p1.27.m27.2.2.7" xref="S4.5.p1.27.m27.2.2.7.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.27.m27.2b"><apply id="S4.5.p1.27.m27.2.2.cmml" xref="S4.5.p1.27.m27.2.2"><and id="S4.5.p1.27.m27.2.2a.cmml" xref="S4.5.p1.27.m27.2.2"></and><apply id="S4.5.p1.27.m27.2.2b.cmml" xref="S4.5.p1.27.m27.2.2"><eq id="S4.5.p1.27.m27.2.2.5.cmml" xref="S4.5.p1.27.m27.2.2.5"></eq><apply id="S4.5.p1.27.m27.2.2.4.cmml" xref="S4.5.p1.27.m27.2.2.4"><csymbol cd="ambiguous" id="S4.5.p1.27.m27.2.2.4.1.cmml" xref="S4.5.p1.27.m27.2.2.4">superscript</csymbol><ci id="S4.5.p1.27.m27.2.2.4.2.cmml" xref="S4.5.p1.27.m27.2.2.4.2">πΏ</ci><ci id="S4.5.p1.27.m27.2.2.4.3.cmml" xref="S4.5.p1.27.m27.2.2.4.3">β²</ci></apply><apply id="S4.5.p1.27.m27.2.2.2.3.cmml" xref="S4.5.p1.27.m27.2.2.2.2"><csymbol cd="latexml" id="S4.5.p1.27.m27.2.2.2.3.1.cmml" xref="S4.5.p1.27.m27.2.2.2.2.3">conditional-set</csymbol><apply id="S4.5.p1.27.m27.1.1.1.1.1.cmml" xref="S4.5.p1.27.m27.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.5.p1.27.m27.1.1.1.1.1.1.cmml" xref="S4.5.p1.27.m27.1.1.1.1.1">subscript</csymbol><ci id="S4.5.p1.27.m27.1.1.1.1.1.2.cmml" xref="S4.5.p1.27.m27.1.1.1.1.1.2">π</ci><ci id="S4.5.p1.27.m27.1.1.1.1.1.3.cmml" xref="S4.5.p1.27.m27.1.1.1.1.1.3">π</ci></apply><apply id="S4.5.p1.27.m27.2.2.2.2.2.cmml" xref="S4.5.p1.27.m27.2.2.2.2.2"><and id="S4.5.p1.27.m27.2.2.2.2.2a.cmml" xref="S4.5.p1.27.m27.2.2.2.2.2"></and><apply id="S4.5.p1.27.m27.2.2.2.2.2b.cmml" xref="S4.5.p1.27.m27.2.2.2.2.2"><lt id="S4.5.p1.27.m27.2.2.2.2.2.3.cmml" xref="S4.5.p1.27.m27.2.2.2.2.2.3"></lt><ci id="S4.5.p1.27.m27.2.2.2.2.2.2.cmml" xref="S4.5.p1.27.m27.2.2.2.2.2.2">πΏ</ci><ci id="S4.5.p1.27.m27.2.2.2.2.2.4.cmml" xref="S4.5.p1.27.m27.2.2.2.2.2.4">π</ci></apply><apply id="S4.5.p1.27.m27.2.2.2.2.2c.cmml" xref="S4.5.p1.27.m27.2.2.2.2.2"><lt id="S4.5.p1.27.m27.2.2.2.2.2.5.cmml" xref="S4.5.p1.27.m27.2.2.2.2.2.5"></lt><share href="https://arxiv.org/html/2503.13666v1#S4.5.p1.27.m27.2.2.2.2.2.4.cmml" id="S4.5.p1.27.m27.2.2.2.2.2d.cmml" xref="S4.5.p1.27.m27.2.2.2.2.2"></share><apply id="S4.5.p1.27.m27.2.2.2.2.2.6.cmml" xref="S4.5.p1.27.m27.2.2.2.2.2.6"><csymbol cd="ambiguous" id="S4.5.p1.27.m27.2.2.2.2.2.6.1.cmml" xref="S4.5.p1.27.m27.2.2.2.2.2.6">subscript</csymbol><ci id="S4.5.p1.27.m27.2.2.2.2.2.6.2.cmml" xref="S4.5.p1.27.m27.2.2.2.2.2.6.2">π</ci><cn id="S4.5.p1.27.m27.2.2.2.2.2.6.3.cmml" type="integer" xref="S4.5.p1.27.m27.2.2.2.2.2.6.3">1</cn></apply></apply></apply></apply></apply><apply id="S4.5.p1.27.m27.2.2c.cmml" xref="S4.5.p1.27.m27.2.2"><subset id="S4.5.p1.27.m27.2.2.6.cmml" xref="S4.5.p1.27.m27.2.2.6"></subset><share href="https://arxiv.org/html/2503.13666v1#S4.5.p1.27.m27.2.2.2.cmml" id="S4.5.p1.27.m27.2.2d.cmml" xref="S4.5.p1.27.m27.2.2"></share><ci id="S4.5.p1.27.m27.2.2.7.cmml" xref="S4.5.p1.27.m27.2.2.7">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.27.m27.2c">L^{\prime}=\{l_{\xi}:\delta<\xi<\omega_{1}\}\subset V</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.27.m27.2d">italic_L start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT = { italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT : italic_Ξ΄ < italic_ΞΎ < italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT } β italic_V</annotation></semantics></math>. Note that <math alttext="d_{U}L^{\prime}\subseteq W" class="ltx_Math" display="inline" id="S4.5.p1.28.m28.1"><semantics id="S4.5.p1.28.m28.1a"><mrow id="S4.5.p1.28.m28.1.1" xref="S4.5.p1.28.m28.1.1.cmml"><mrow id="S4.5.p1.28.m28.1.1.2" xref="S4.5.p1.28.m28.1.1.2.cmml"><msub id="S4.5.p1.28.m28.1.1.2.2" xref="S4.5.p1.28.m28.1.1.2.2.cmml"><mi id="S4.5.p1.28.m28.1.1.2.2.2" xref="S4.5.p1.28.m28.1.1.2.2.2.cmml">d</mi><mi id="S4.5.p1.28.m28.1.1.2.2.3" xref="S4.5.p1.28.m28.1.1.2.2.3.cmml">U</mi></msub><mo id="S4.5.p1.28.m28.1.1.2.1" xref="S4.5.p1.28.m28.1.1.2.1.cmml">β’</mo><msup id="S4.5.p1.28.m28.1.1.2.3" xref="S4.5.p1.28.m28.1.1.2.3.cmml"><mi id="S4.5.p1.28.m28.1.1.2.3.2" xref="S4.5.p1.28.m28.1.1.2.3.2.cmml">L</mi><mo id="S4.5.p1.28.m28.1.1.2.3.3" xref="S4.5.p1.28.m28.1.1.2.3.3.cmml">β²</mo></msup></mrow><mo id="S4.5.p1.28.m28.1.1.1" xref="S4.5.p1.28.m28.1.1.1.cmml">β</mo><mi id="S4.5.p1.28.m28.1.1.3" xref="S4.5.p1.28.m28.1.1.3.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.28.m28.1b"><apply id="S4.5.p1.28.m28.1.1.cmml" xref="S4.5.p1.28.m28.1.1"><subset id="S4.5.p1.28.m28.1.1.1.cmml" xref="S4.5.p1.28.m28.1.1.1"></subset><apply id="S4.5.p1.28.m28.1.1.2.cmml" xref="S4.5.p1.28.m28.1.1.2"><times id="S4.5.p1.28.m28.1.1.2.1.cmml" xref="S4.5.p1.28.m28.1.1.2.1"></times><apply id="S4.5.p1.28.m28.1.1.2.2.cmml" xref="S4.5.p1.28.m28.1.1.2.2"><csymbol cd="ambiguous" id="S4.5.p1.28.m28.1.1.2.2.1.cmml" xref="S4.5.p1.28.m28.1.1.2.2">subscript</csymbol><ci id="S4.5.p1.28.m28.1.1.2.2.2.cmml" xref="S4.5.p1.28.m28.1.1.2.2.2">π</ci><ci id="S4.5.p1.28.m28.1.1.2.2.3.cmml" xref="S4.5.p1.28.m28.1.1.2.2.3">π</ci></apply><apply id="S4.5.p1.28.m28.1.1.2.3.cmml" xref="S4.5.p1.28.m28.1.1.2.3"><csymbol cd="ambiguous" id="S4.5.p1.28.m28.1.1.2.3.1.cmml" xref="S4.5.p1.28.m28.1.1.2.3">superscript</csymbol><ci id="S4.5.p1.28.m28.1.1.2.3.2.cmml" xref="S4.5.p1.28.m28.1.1.2.3.2">πΏ</ci><ci id="S4.5.p1.28.m28.1.1.2.3.3.cmml" xref="S4.5.p1.28.m28.1.1.2.3.3">β²</ci></apply></apply><ci id="S4.5.p1.28.m28.1.1.3.cmml" xref="S4.5.p1.28.m28.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.28.m28.1c">d_{U}L^{\prime}\subseteq W</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.28.m28.1d">italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT β italic_W</annotation></semantics></math>, <math alttext="L^{\prime}" class="ltx_Math" display="inline" id="S4.5.p1.29.m29.1"><semantics id="S4.5.p1.29.m29.1a"><msup id="S4.5.p1.29.m29.1.1" xref="S4.5.p1.29.m29.1.1.cmml"><mi id="S4.5.p1.29.m29.1.1.2" xref="S4.5.p1.29.m29.1.1.2.cmml">L</mi><mo id="S4.5.p1.29.m29.1.1.3" xref="S4.5.p1.29.m29.1.1.3.cmml">β²</mo></msup><annotation-xml encoding="MathML-Content" id="S4.5.p1.29.m29.1b"><apply id="S4.5.p1.29.m29.1.1.cmml" xref="S4.5.p1.29.m29.1.1"><csymbol cd="ambiguous" id="S4.5.p1.29.m29.1.1.1.cmml" xref="S4.5.p1.29.m29.1.1">superscript</csymbol><ci id="S4.5.p1.29.m29.1.1.2.cmml" xref="S4.5.p1.29.m29.1.1.2">πΏ</ci><ci id="S4.5.p1.29.m29.1.1.3.cmml" xref="S4.5.p1.29.m29.1.1.3">β²</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.29.m29.1c">L^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.29.m29.1d">italic_L start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT</annotation></semantics></math> is isomorphic to <math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S4.5.p1.30.m30.2"><semantics id="S4.5.p1.30.m30.2a"><mrow id="S4.5.p1.30.m30.2.2.1" xref="S4.5.p1.30.m30.2.2.2.cmml"><mo id="S4.5.p1.30.m30.2.2.1.2" stretchy="false" xref="S4.5.p1.30.m30.2.2.2.cmml">(</mo><msub id="S4.5.p1.30.m30.2.2.1.1" xref="S4.5.p1.30.m30.2.2.1.1.cmml"><mi id="S4.5.p1.30.m30.2.2.1.1.2" xref="S4.5.p1.30.m30.2.2.1.1.2.cmml">Ο</mi><mn id="S4.5.p1.30.m30.2.2.1.1.3" xref="S4.5.p1.30.m30.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.5.p1.30.m30.2.2.1.3" xref="S4.5.p1.30.m30.2.2.2.cmml">,</mo><mi id="S4.5.p1.30.m30.1.1" xref="S4.5.p1.30.m30.1.1.cmml">min</mi><mo id="S4.5.p1.30.m30.2.2.1.4" stretchy="false" xref="S4.5.p1.30.m30.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.30.m30.2b"><interval closure="open" id="S4.5.p1.30.m30.2.2.2.cmml" xref="S4.5.p1.30.m30.2.2.1"><apply id="S4.5.p1.30.m30.2.2.1.1.cmml" xref="S4.5.p1.30.m30.2.2.1.1"><csymbol cd="ambiguous" id="S4.5.p1.30.m30.2.2.1.1.1.cmml" xref="S4.5.p1.30.m30.2.2.1.1">subscript</csymbol><ci id="S4.5.p1.30.m30.2.2.1.1.2.cmml" xref="S4.5.p1.30.m30.2.2.1.1.2">π</ci><cn id="S4.5.p1.30.m30.2.2.1.1.3.cmml" type="integer" xref="S4.5.p1.30.m30.2.2.1.1.3">1</cn></apply><min id="S4.5.p1.30.m30.1.1.cmml" xref="S4.5.p1.30.m30.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.30.m30.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.30.m30.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math> and <math alttext="|d_{U}L^{\prime}|=|d_{U}L|=\omega_{1}" class="ltx_Math" display="inline" id="S4.5.p1.31.m31.2"><semantics id="S4.5.p1.31.m31.2a"><mrow id="S4.5.p1.31.m31.2.2" xref="S4.5.p1.31.m31.2.2.cmml"><mrow id="S4.5.p1.31.m31.1.1.1.1" xref="S4.5.p1.31.m31.1.1.1.2.cmml"><mo id="S4.5.p1.31.m31.1.1.1.1.2" stretchy="false" xref="S4.5.p1.31.m31.1.1.1.2.1.cmml">|</mo><mrow id="S4.5.p1.31.m31.1.1.1.1.1" xref="S4.5.p1.31.m31.1.1.1.1.1.cmml"><msub id="S4.5.p1.31.m31.1.1.1.1.1.2" xref="S4.5.p1.31.m31.1.1.1.1.1.2.cmml"><mi id="S4.5.p1.31.m31.1.1.1.1.1.2.2" xref="S4.5.p1.31.m31.1.1.1.1.1.2.2.cmml">d</mi><mi id="S4.5.p1.31.m31.1.1.1.1.1.2.3" xref="S4.5.p1.31.m31.1.1.1.1.1.2.3.cmml">U</mi></msub><mo id="S4.5.p1.31.m31.1.1.1.1.1.1" xref="S4.5.p1.31.m31.1.1.1.1.1.1.cmml">β’</mo><msup id="S4.5.p1.31.m31.1.1.1.1.1.3" xref="S4.5.p1.31.m31.1.1.1.1.1.3.cmml"><mi id="S4.5.p1.31.m31.1.1.1.1.1.3.2" xref="S4.5.p1.31.m31.1.1.1.1.1.3.2.cmml">L</mi><mo id="S4.5.p1.31.m31.1.1.1.1.1.3.3" xref="S4.5.p1.31.m31.1.1.1.1.1.3.3.cmml">β²</mo></msup></mrow><mo id="S4.5.p1.31.m31.1.1.1.1.3" stretchy="false" xref="S4.5.p1.31.m31.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.5.p1.31.m31.2.2.4" xref="S4.5.p1.31.m31.2.2.4.cmml">=</mo><mrow id="S4.5.p1.31.m31.2.2.2.1" xref="S4.5.p1.31.m31.2.2.2.2.cmml"><mo id="S4.5.p1.31.m31.2.2.2.1.2" stretchy="false" xref="S4.5.p1.31.m31.2.2.2.2.1.cmml">|</mo><mrow id="S4.5.p1.31.m31.2.2.2.1.1" xref="S4.5.p1.31.m31.2.2.2.1.1.cmml"><msub id="S4.5.p1.31.m31.2.2.2.1.1.2" xref="S4.5.p1.31.m31.2.2.2.1.1.2.cmml"><mi id="S4.5.p1.31.m31.2.2.2.1.1.2.2" xref="S4.5.p1.31.m31.2.2.2.1.1.2.2.cmml">d</mi><mi id="S4.5.p1.31.m31.2.2.2.1.1.2.3" xref="S4.5.p1.31.m31.2.2.2.1.1.2.3.cmml">U</mi></msub><mo id="S4.5.p1.31.m31.2.2.2.1.1.1" xref="S4.5.p1.31.m31.2.2.2.1.1.1.cmml">β’</mo><mi id="S4.5.p1.31.m31.2.2.2.1.1.3" xref="S4.5.p1.31.m31.2.2.2.1.1.3.cmml">L</mi></mrow><mo id="S4.5.p1.31.m31.2.2.2.1.3" stretchy="false" xref="S4.5.p1.31.m31.2.2.2.2.1.cmml">|</mo></mrow><mo id="S4.5.p1.31.m31.2.2.5" xref="S4.5.p1.31.m31.2.2.5.cmml">=</mo><msub id="S4.5.p1.31.m31.2.2.6" xref="S4.5.p1.31.m31.2.2.6.cmml"><mi id="S4.5.p1.31.m31.2.2.6.2" xref="S4.5.p1.31.m31.2.2.6.2.cmml">Ο</mi><mn id="S4.5.p1.31.m31.2.2.6.3" xref="S4.5.p1.31.m31.2.2.6.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.31.m31.2b"><apply id="S4.5.p1.31.m31.2.2.cmml" xref="S4.5.p1.31.m31.2.2"><and id="S4.5.p1.31.m31.2.2a.cmml" xref="S4.5.p1.31.m31.2.2"></and><apply id="S4.5.p1.31.m31.2.2b.cmml" xref="S4.5.p1.31.m31.2.2"><eq id="S4.5.p1.31.m31.2.2.4.cmml" xref="S4.5.p1.31.m31.2.2.4"></eq><apply id="S4.5.p1.31.m31.1.1.1.2.cmml" xref="S4.5.p1.31.m31.1.1.1.1"><abs id="S4.5.p1.31.m31.1.1.1.2.1.cmml" xref="S4.5.p1.31.m31.1.1.1.1.2"></abs><apply id="S4.5.p1.31.m31.1.1.1.1.1.cmml" xref="S4.5.p1.31.m31.1.1.1.1.1"><times id="S4.5.p1.31.m31.1.1.1.1.1.1.cmml" xref="S4.5.p1.31.m31.1.1.1.1.1.1"></times><apply id="S4.5.p1.31.m31.1.1.1.1.1.2.cmml" xref="S4.5.p1.31.m31.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.5.p1.31.m31.1.1.1.1.1.2.1.cmml" xref="S4.5.p1.31.m31.1.1.1.1.1.2">subscript</csymbol><ci id="S4.5.p1.31.m31.1.1.1.1.1.2.2.cmml" xref="S4.5.p1.31.m31.1.1.1.1.1.2.2">π</ci><ci id="S4.5.p1.31.m31.1.1.1.1.1.2.3.cmml" xref="S4.5.p1.31.m31.1.1.1.1.1.2.3">π</ci></apply><apply id="S4.5.p1.31.m31.1.1.1.1.1.3.cmml" xref="S4.5.p1.31.m31.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.5.p1.31.m31.1.1.1.1.1.3.1.cmml" xref="S4.5.p1.31.m31.1.1.1.1.1.3">superscript</csymbol><ci id="S4.5.p1.31.m31.1.1.1.1.1.3.2.cmml" xref="S4.5.p1.31.m31.1.1.1.1.1.3.2">πΏ</ci><ci id="S4.5.p1.31.m31.1.1.1.1.1.3.3.cmml" xref="S4.5.p1.31.m31.1.1.1.1.1.3.3">β²</ci></apply></apply></apply><apply id="S4.5.p1.31.m31.2.2.2.2.cmml" xref="S4.5.p1.31.m31.2.2.2.1"><abs id="S4.5.p1.31.m31.2.2.2.2.1.cmml" xref="S4.5.p1.31.m31.2.2.2.1.2"></abs><apply id="S4.5.p1.31.m31.2.2.2.1.1.cmml" xref="S4.5.p1.31.m31.2.2.2.1.1"><times id="S4.5.p1.31.m31.2.2.2.1.1.1.cmml" xref="S4.5.p1.31.m31.2.2.2.1.1.1"></times><apply id="S4.5.p1.31.m31.2.2.2.1.1.2.cmml" xref="S4.5.p1.31.m31.2.2.2.1.1.2"><csymbol cd="ambiguous" id="S4.5.p1.31.m31.2.2.2.1.1.2.1.cmml" xref="S4.5.p1.31.m31.2.2.2.1.1.2">subscript</csymbol><ci id="S4.5.p1.31.m31.2.2.2.1.1.2.2.cmml" xref="S4.5.p1.31.m31.2.2.2.1.1.2.2">π</ci><ci id="S4.5.p1.31.m31.2.2.2.1.1.2.3.cmml" xref="S4.5.p1.31.m31.2.2.2.1.1.2.3">π</ci></apply><ci id="S4.5.p1.31.m31.2.2.2.1.1.3.cmml" xref="S4.5.p1.31.m31.2.2.2.1.1.3">πΏ</ci></apply></apply></apply><apply id="S4.5.p1.31.m31.2.2c.cmml" xref="S4.5.p1.31.m31.2.2"><eq id="S4.5.p1.31.m31.2.2.5.cmml" xref="S4.5.p1.31.m31.2.2.5"></eq><share href="https://arxiv.org/html/2503.13666v1#S4.5.p1.31.m31.2.2.2.cmml" id="S4.5.p1.31.m31.2.2d.cmml" xref="S4.5.p1.31.m31.2.2"></share><apply id="S4.5.p1.31.m31.2.2.6.cmml" xref="S4.5.p1.31.m31.2.2.6"><csymbol cd="ambiguous" id="S4.5.p1.31.m31.2.2.6.1.cmml" xref="S4.5.p1.31.m31.2.2.6">subscript</csymbol><ci id="S4.5.p1.31.m31.2.2.6.2.cmml" xref="S4.5.p1.31.m31.2.2.6.2">π</ci><cn id="S4.5.p1.31.m31.2.2.6.3.cmml" type="integer" xref="S4.5.p1.31.m31.2.2.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.31.m31.2c">|d_{U}L^{\prime}|=|d_{U}L|=\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.31.m31.2d">| italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT | = | italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT italic_L | = italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, by the choice of <math alttext="d_{U}" class="ltx_Math" display="inline" id="S4.5.p1.32.m32.1"><semantics id="S4.5.p1.32.m32.1a"><msub id="S4.5.p1.32.m32.1.1" xref="S4.5.p1.32.m32.1.1.cmml"><mi id="S4.5.p1.32.m32.1.1.2" xref="S4.5.p1.32.m32.1.1.2.cmml">d</mi><mi id="S4.5.p1.32.m32.1.1.3" xref="S4.5.p1.32.m32.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S4.5.p1.32.m32.1b"><apply id="S4.5.p1.32.m32.1.1.cmml" xref="S4.5.p1.32.m32.1.1"><csymbol cd="ambiguous" id="S4.5.p1.32.m32.1.1.1.cmml" xref="S4.5.p1.32.m32.1.1">subscript</csymbol><ci id="S4.5.p1.32.m32.1.1.2.cmml" xref="S4.5.p1.32.m32.1.1.2">π</ci><ci id="S4.5.p1.32.m32.1.1.3.cmml" xref="S4.5.p1.32.m32.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.32.m32.1c">d_{U}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.32.m32.1d">italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math>. Since for each <math alttext="a\in X" class="ltx_Math" display="inline" id="S4.5.p1.33.m33.1"><semantics id="S4.5.p1.33.m33.1a"><mrow id="S4.5.p1.33.m33.1.1" xref="S4.5.p1.33.m33.1.1.cmml"><mi id="S4.5.p1.33.m33.1.1.2" xref="S4.5.p1.33.m33.1.1.2.cmml">a</mi><mo id="S4.5.p1.33.m33.1.1.1" xref="S4.5.p1.33.m33.1.1.1.cmml">β</mo><mi id="S4.5.p1.33.m33.1.1.3" xref="S4.5.p1.33.m33.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.33.m33.1b"><apply id="S4.5.p1.33.m33.1.1.cmml" xref="S4.5.p1.33.m33.1.1"><in id="S4.5.p1.33.m33.1.1.1.cmml" xref="S4.5.p1.33.m33.1.1.1"></in><ci id="S4.5.p1.33.m33.1.1.2.cmml" xref="S4.5.p1.33.m33.1.1.2">π</ci><ci id="S4.5.p1.33.m33.1.1.3.cmml" xref="S4.5.p1.33.m33.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.33.m33.1c">a\in X</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.33.m33.1d">italic_a β italic_X</annotation></semantics></math> the map <math alttext="\phi_{a}:X\rightarrow X" class="ltx_Math" display="inline" id="S4.5.p1.34.m34.1"><semantics id="S4.5.p1.34.m34.1a"><mrow id="S4.5.p1.34.m34.1.1" xref="S4.5.p1.34.m34.1.1.cmml"><msub id="S4.5.p1.34.m34.1.1.2" xref="S4.5.p1.34.m34.1.1.2.cmml"><mi id="S4.5.p1.34.m34.1.1.2.2" xref="S4.5.p1.34.m34.1.1.2.2.cmml">Ο</mi><mi id="S4.5.p1.34.m34.1.1.2.3" xref="S4.5.p1.34.m34.1.1.2.3.cmml">a</mi></msub><mo id="S4.5.p1.34.m34.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.5.p1.34.m34.1.1.1.cmml">:</mo><mrow id="S4.5.p1.34.m34.1.1.3" xref="S4.5.p1.34.m34.1.1.3.cmml"><mi id="S4.5.p1.34.m34.1.1.3.2" xref="S4.5.p1.34.m34.1.1.3.2.cmml">X</mi><mo id="S4.5.p1.34.m34.1.1.3.1" stretchy="false" xref="S4.5.p1.34.m34.1.1.3.1.cmml">β</mo><mi id="S4.5.p1.34.m34.1.1.3.3" xref="S4.5.p1.34.m34.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.34.m34.1b"><apply id="S4.5.p1.34.m34.1.1.cmml" xref="S4.5.p1.34.m34.1.1"><ci id="S4.5.p1.34.m34.1.1.1.cmml" xref="S4.5.p1.34.m34.1.1.1">:</ci><apply id="S4.5.p1.34.m34.1.1.2.cmml" xref="S4.5.p1.34.m34.1.1.2"><csymbol cd="ambiguous" id="S4.5.p1.34.m34.1.1.2.1.cmml" xref="S4.5.p1.34.m34.1.1.2">subscript</csymbol><ci id="S4.5.p1.34.m34.1.1.2.2.cmml" xref="S4.5.p1.34.m34.1.1.2.2">italic-Ο</ci><ci id="S4.5.p1.34.m34.1.1.2.3.cmml" xref="S4.5.p1.34.m34.1.1.2.3">π</ci></apply><apply id="S4.5.p1.34.m34.1.1.3.cmml" xref="S4.5.p1.34.m34.1.1.3"><ci id="S4.5.p1.34.m34.1.1.3.1.cmml" xref="S4.5.p1.34.m34.1.1.3.1">β</ci><ci id="S4.5.p1.34.m34.1.1.3.2.cmml" xref="S4.5.p1.34.m34.1.1.3.2">π</ci><ci id="S4.5.p1.34.m34.1.1.3.3.cmml" xref="S4.5.p1.34.m34.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.34.m34.1c">\phi_{a}:X\rightarrow X</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.34.m34.1d">italic_Ο start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT : italic_X β italic_X</annotation></semantics></math>, <math alttext="\phi_{a}(x)=ax" class="ltx_Math" display="inline" id="S4.5.p1.35.m35.1"><semantics id="S4.5.p1.35.m35.1a"><mrow id="S4.5.p1.35.m35.1.2" xref="S4.5.p1.35.m35.1.2.cmml"><mrow id="S4.5.p1.35.m35.1.2.2" xref="S4.5.p1.35.m35.1.2.2.cmml"><msub id="S4.5.p1.35.m35.1.2.2.2" xref="S4.5.p1.35.m35.1.2.2.2.cmml"><mi id="S4.5.p1.35.m35.1.2.2.2.2" xref="S4.5.p1.35.m35.1.2.2.2.2.cmml">Ο</mi><mi id="S4.5.p1.35.m35.1.2.2.2.3" xref="S4.5.p1.35.m35.1.2.2.2.3.cmml">a</mi></msub><mo id="S4.5.p1.35.m35.1.2.2.1" xref="S4.5.p1.35.m35.1.2.2.1.cmml">β’</mo><mrow id="S4.5.p1.35.m35.1.2.2.3.2" xref="S4.5.p1.35.m35.1.2.2.cmml"><mo id="S4.5.p1.35.m35.1.2.2.3.2.1" stretchy="false" xref="S4.5.p1.35.m35.1.2.2.cmml">(</mo><mi id="S4.5.p1.35.m35.1.1" xref="S4.5.p1.35.m35.1.1.cmml">x</mi><mo id="S4.5.p1.35.m35.1.2.2.3.2.2" stretchy="false" xref="S4.5.p1.35.m35.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.5.p1.35.m35.1.2.1" xref="S4.5.p1.35.m35.1.2.1.cmml">=</mo><mrow id="S4.5.p1.35.m35.1.2.3" xref="S4.5.p1.35.m35.1.2.3.cmml"><mi id="S4.5.p1.35.m35.1.2.3.2" xref="S4.5.p1.35.m35.1.2.3.2.cmml">a</mi><mo id="S4.5.p1.35.m35.1.2.3.1" xref="S4.5.p1.35.m35.1.2.3.1.cmml">β’</mo><mi id="S4.5.p1.35.m35.1.2.3.3" xref="S4.5.p1.35.m35.1.2.3.3.cmml">x</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.35.m35.1b"><apply id="S4.5.p1.35.m35.1.2.cmml" xref="S4.5.p1.35.m35.1.2"><eq id="S4.5.p1.35.m35.1.2.1.cmml" xref="S4.5.p1.35.m35.1.2.1"></eq><apply id="S4.5.p1.35.m35.1.2.2.cmml" xref="S4.5.p1.35.m35.1.2.2"><times id="S4.5.p1.35.m35.1.2.2.1.cmml" xref="S4.5.p1.35.m35.1.2.2.1"></times><apply id="S4.5.p1.35.m35.1.2.2.2.cmml" xref="S4.5.p1.35.m35.1.2.2.2"><csymbol cd="ambiguous" id="S4.5.p1.35.m35.1.2.2.2.1.cmml" xref="S4.5.p1.35.m35.1.2.2.2">subscript</csymbol><ci id="S4.5.p1.35.m35.1.2.2.2.2.cmml" xref="S4.5.p1.35.m35.1.2.2.2.2">italic-Ο</ci><ci id="S4.5.p1.35.m35.1.2.2.2.3.cmml" xref="S4.5.p1.35.m35.1.2.2.2.3">π</ci></apply><ci id="S4.5.p1.35.m35.1.1.cmml" xref="S4.5.p1.35.m35.1.1">π₯</ci></apply><apply id="S4.5.p1.35.m35.1.2.3.cmml" xref="S4.5.p1.35.m35.1.2.3"><times id="S4.5.p1.35.m35.1.2.3.1.cmml" xref="S4.5.p1.35.m35.1.2.3.1"></times><ci id="S4.5.p1.35.m35.1.2.3.2.cmml" xref="S4.5.p1.35.m35.1.2.3.2">π</ci><ci id="S4.5.p1.35.m35.1.2.3.3.cmml" xref="S4.5.p1.35.m35.1.2.3.3">π₯</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.35.m35.1c">\phi_{a}(x)=ax</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.35.m35.1d">italic_Ο start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( italic_x ) = italic_a italic_x</annotation></semantics></math> is a homomorphism, Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem7" title="Lemma 3.7. β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.7</span></a> implies that the chain <math alttext="d_{U}L^{\prime}" class="ltx_Math" display="inline" id="S4.5.p1.36.m36.1"><semantics id="S4.5.p1.36.m36.1a"><mrow id="S4.5.p1.36.m36.1.1" xref="S4.5.p1.36.m36.1.1.cmml"><msub id="S4.5.p1.36.m36.1.1.2" xref="S4.5.p1.36.m36.1.1.2.cmml"><mi id="S4.5.p1.36.m36.1.1.2.2" xref="S4.5.p1.36.m36.1.1.2.2.cmml">d</mi><mi id="S4.5.p1.36.m36.1.1.2.3" xref="S4.5.p1.36.m36.1.1.2.3.cmml">U</mi></msub><mo id="S4.5.p1.36.m36.1.1.1" xref="S4.5.p1.36.m36.1.1.1.cmml">β’</mo><msup id="S4.5.p1.36.m36.1.1.3" xref="S4.5.p1.36.m36.1.1.3.cmml"><mi id="S4.5.p1.36.m36.1.1.3.2" xref="S4.5.p1.36.m36.1.1.3.2.cmml">L</mi><mo id="S4.5.p1.36.m36.1.1.3.3" xref="S4.5.p1.36.m36.1.1.3.3.cmml">β²</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.36.m36.1b"><apply id="S4.5.p1.36.m36.1.1.cmml" xref="S4.5.p1.36.m36.1.1"><times id="S4.5.p1.36.m36.1.1.1.cmml" xref="S4.5.p1.36.m36.1.1.1"></times><apply id="S4.5.p1.36.m36.1.1.2.cmml" xref="S4.5.p1.36.m36.1.1.2"><csymbol cd="ambiguous" id="S4.5.p1.36.m36.1.1.2.1.cmml" xref="S4.5.p1.36.m36.1.1.2">subscript</csymbol><ci id="S4.5.p1.36.m36.1.1.2.2.cmml" xref="S4.5.p1.36.m36.1.1.2.2">π</ci><ci id="S4.5.p1.36.m36.1.1.2.3.cmml" xref="S4.5.p1.36.m36.1.1.2.3">π</ci></apply><apply id="S4.5.p1.36.m36.1.1.3.cmml" xref="S4.5.p1.36.m36.1.1.3"><csymbol cd="ambiguous" id="S4.5.p1.36.m36.1.1.3.1.cmml" xref="S4.5.p1.36.m36.1.1.3">superscript</csymbol><ci id="S4.5.p1.36.m36.1.1.3.2.cmml" xref="S4.5.p1.36.m36.1.1.3.2">πΏ</ci><ci id="S4.5.p1.36.m36.1.1.3.3.cmml" xref="S4.5.p1.36.m36.1.1.3.3">β²</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.36.m36.1c">d_{U}L^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.36.m36.1d">italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT</annotation></semantics></math> is isomorphic to <math alttext="(\beta,\min)" class="ltx_Math" display="inline" id="S4.5.p1.37.m37.2"><semantics id="S4.5.p1.37.m37.2a"><mrow id="S4.5.p1.37.m37.2.3.2" xref="S4.5.p1.37.m37.2.3.1.cmml"><mo id="S4.5.p1.37.m37.2.3.2.1" stretchy="false" xref="S4.5.p1.37.m37.2.3.1.cmml">(</mo><mi id="S4.5.p1.37.m37.1.1" xref="S4.5.p1.37.m37.1.1.cmml">Ξ²</mi><mo id="S4.5.p1.37.m37.2.3.2.2" xref="S4.5.p1.37.m37.2.3.1.cmml">,</mo><mi id="S4.5.p1.37.m37.2.2" xref="S4.5.p1.37.m37.2.2.cmml">min</mi><mo id="S4.5.p1.37.m37.2.3.2.3" stretchy="false" xref="S4.5.p1.37.m37.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.37.m37.2b"><interval closure="open" id="S4.5.p1.37.m37.2.3.1.cmml" xref="S4.5.p1.37.m37.2.3.2"><ci id="S4.5.p1.37.m37.1.1.cmml" xref="S4.5.p1.37.m37.1.1">π½</ci><min id="S4.5.p1.37.m37.2.2.cmml" xref="S4.5.p1.37.m37.2.2"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.37.m37.2c">(\beta,\min)</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.37.m37.2d">( italic_Ξ² , roman_min )</annotation></semantics></math> for some ordinal <math alttext="\beta\leq\omega_{1}" class="ltx_Math" display="inline" id="S4.5.p1.38.m38.1"><semantics id="S4.5.p1.38.m38.1a"><mrow id="S4.5.p1.38.m38.1.1" xref="S4.5.p1.38.m38.1.1.cmml"><mi id="S4.5.p1.38.m38.1.1.2" xref="S4.5.p1.38.m38.1.1.2.cmml">Ξ²</mi><mo id="S4.5.p1.38.m38.1.1.1" xref="S4.5.p1.38.m38.1.1.1.cmml">β€</mo><msub id="S4.5.p1.38.m38.1.1.3" xref="S4.5.p1.38.m38.1.1.3.cmml"><mi id="S4.5.p1.38.m38.1.1.3.2" xref="S4.5.p1.38.m38.1.1.3.2.cmml">Ο</mi><mn id="S4.5.p1.38.m38.1.1.3.3" xref="S4.5.p1.38.m38.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.38.m38.1b"><apply id="S4.5.p1.38.m38.1.1.cmml" xref="S4.5.p1.38.m38.1.1"><leq id="S4.5.p1.38.m38.1.1.1.cmml" xref="S4.5.p1.38.m38.1.1.1"></leq><ci id="S4.5.p1.38.m38.1.1.2.cmml" xref="S4.5.p1.38.m38.1.1.2">π½</ci><apply id="S4.5.p1.38.m38.1.1.3.cmml" xref="S4.5.p1.38.m38.1.1.3"><csymbol cd="ambiguous" id="S4.5.p1.38.m38.1.1.3.1.cmml" xref="S4.5.p1.38.m38.1.1.3">subscript</csymbol><ci id="S4.5.p1.38.m38.1.1.3.2.cmml" xref="S4.5.p1.38.m38.1.1.3.2">π</ci><cn id="S4.5.p1.38.m38.1.1.3.3.cmml" type="integer" xref="S4.5.p1.38.m38.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.38.m38.1c">\beta\leq\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.38.m38.1d">italic_Ξ² β€ italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Taking into account that <math alttext="|d_{U}L^{\prime}|=\omega_{1}" class="ltx_Math" display="inline" id="S4.5.p1.39.m39.1"><semantics id="S4.5.p1.39.m39.1a"><mrow id="S4.5.p1.39.m39.1.1" xref="S4.5.p1.39.m39.1.1.cmml"><mrow id="S4.5.p1.39.m39.1.1.1.1" xref="S4.5.p1.39.m39.1.1.1.2.cmml"><mo id="S4.5.p1.39.m39.1.1.1.1.2" stretchy="false" xref="S4.5.p1.39.m39.1.1.1.2.1.cmml">|</mo><mrow id="S4.5.p1.39.m39.1.1.1.1.1" xref="S4.5.p1.39.m39.1.1.1.1.1.cmml"><msub id="S4.5.p1.39.m39.1.1.1.1.1.2" xref="S4.5.p1.39.m39.1.1.1.1.1.2.cmml"><mi id="S4.5.p1.39.m39.1.1.1.1.1.2.2" xref="S4.5.p1.39.m39.1.1.1.1.1.2.2.cmml">d</mi><mi id="S4.5.p1.39.m39.1.1.1.1.1.2.3" xref="S4.5.p1.39.m39.1.1.1.1.1.2.3.cmml">U</mi></msub><mo id="S4.5.p1.39.m39.1.1.1.1.1.1" xref="S4.5.p1.39.m39.1.1.1.1.1.1.cmml">β’</mo><msup id="S4.5.p1.39.m39.1.1.1.1.1.3" xref="S4.5.p1.39.m39.1.1.1.1.1.3.cmml"><mi id="S4.5.p1.39.m39.1.1.1.1.1.3.2" xref="S4.5.p1.39.m39.1.1.1.1.1.3.2.cmml">L</mi><mo id="S4.5.p1.39.m39.1.1.1.1.1.3.3" xref="S4.5.p1.39.m39.1.1.1.1.1.3.3.cmml">β²</mo></msup></mrow><mo id="S4.5.p1.39.m39.1.1.1.1.3" stretchy="false" xref="S4.5.p1.39.m39.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.5.p1.39.m39.1.1.2" xref="S4.5.p1.39.m39.1.1.2.cmml">=</mo><msub id="S4.5.p1.39.m39.1.1.3" xref="S4.5.p1.39.m39.1.1.3.cmml"><mi id="S4.5.p1.39.m39.1.1.3.2" xref="S4.5.p1.39.m39.1.1.3.2.cmml">Ο</mi><mn id="S4.5.p1.39.m39.1.1.3.3" xref="S4.5.p1.39.m39.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.39.m39.1b"><apply id="S4.5.p1.39.m39.1.1.cmml" xref="S4.5.p1.39.m39.1.1"><eq id="S4.5.p1.39.m39.1.1.2.cmml" xref="S4.5.p1.39.m39.1.1.2"></eq><apply id="S4.5.p1.39.m39.1.1.1.2.cmml" xref="S4.5.p1.39.m39.1.1.1.1"><abs id="S4.5.p1.39.m39.1.1.1.2.1.cmml" xref="S4.5.p1.39.m39.1.1.1.1.2"></abs><apply id="S4.5.p1.39.m39.1.1.1.1.1.cmml" xref="S4.5.p1.39.m39.1.1.1.1.1"><times id="S4.5.p1.39.m39.1.1.1.1.1.1.cmml" xref="S4.5.p1.39.m39.1.1.1.1.1.1"></times><apply id="S4.5.p1.39.m39.1.1.1.1.1.2.cmml" xref="S4.5.p1.39.m39.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.5.p1.39.m39.1.1.1.1.1.2.1.cmml" xref="S4.5.p1.39.m39.1.1.1.1.1.2">subscript</csymbol><ci id="S4.5.p1.39.m39.1.1.1.1.1.2.2.cmml" xref="S4.5.p1.39.m39.1.1.1.1.1.2.2">π</ci><ci id="S4.5.p1.39.m39.1.1.1.1.1.2.3.cmml" xref="S4.5.p1.39.m39.1.1.1.1.1.2.3">π</ci></apply><apply id="S4.5.p1.39.m39.1.1.1.1.1.3.cmml" xref="S4.5.p1.39.m39.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.5.p1.39.m39.1.1.1.1.1.3.1.cmml" xref="S4.5.p1.39.m39.1.1.1.1.1.3">superscript</csymbol><ci id="S4.5.p1.39.m39.1.1.1.1.1.3.2.cmml" xref="S4.5.p1.39.m39.1.1.1.1.1.3.2">πΏ</ci><ci id="S4.5.p1.39.m39.1.1.1.1.1.3.3.cmml" xref="S4.5.p1.39.m39.1.1.1.1.1.3.3">β²</ci></apply></apply></apply><apply id="S4.5.p1.39.m39.1.1.3.cmml" xref="S4.5.p1.39.m39.1.1.3"><csymbol cd="ambiguous" id="S4.5.p1.39.m39.1.1.3.1.cmml" xref="S4.5.p1.39.m39.1.1.3">subscript</csymbol><ci id="S4.5.p1.39.m39.1.1.3.2.cmml" xref="S4.5.p1.39.m39.1.1.3.2">π</ci><cn id="S4.5.p1.39.m39.1.1.3.3.cmml" type="integer" xref="S4.5.p1.39.m39.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.39.m39.1c">|d_{U}L^{\prime}|=\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.39.m39.1d">| italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT | = italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, we get that <math alttext="d_{U}L^{\prime}" class="ltx_Math" display="inline" id="S4.5.p1.40.m40.1"><semantics id="S4.5.p1.40.m40.1a"><mrow id="S4.5.p1.40.m40.1.1" xref="S4.5.p1.40.m40.1.1.cmml"><msub id="S4.5.p1.40.m40.1.1.2" xref="S4.5.p1.40.m40.1.1.2.cmml"><mi id="S4.5.p1.40.m40.1.1.2.2" xref="S4.5.p1.40.m40.1.1.2.2.cmml">d</mi><mi id="S4.5.p1.40.m40.1.1.2.3" xref="S4.5.p1.40.m40.1.1.2.3.cmml">U</mi></msub><mo id="S4.5.p1.40.m40.1.1.1" xref="S4.5.p1.40.m40.1.1.1.cmml">β’</mo><msup id="S4.5.p1.40.m40.1.1.3" xref="S4.5.p1.40.m40.1.1.3.cmml"><mi id="S4.5.p1.40.m40.1.1.3.2" xref="S4.5.p1.40.m40.1.1.3.2.cmml">L</mi><mo id="S4.5.p1.40.m40.1.1.3.3" xref="S4.5.p1.40.m40.1.1.3.3.cmml">β²</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.40.m40.1b"><apply id="S4.5.p1.40.m40.1.1.cmml" xref="S4.5.p1.40.m40.1.1"><times id="S4.5.p1.40.m40.1.1.1.cmml" xref="S4.5.p1.40.m40.1.1.1"></times><apply id="S4.5.p1.40.m40.1.1.2.cmml" xref="S4.5.p1.40.m40.1.1.2"><csymbol cd="ambiguous" id="S4.5.p1.40.m40.1.1.2.1.cmml" xref="S4.5.p1.40.m40.1.1.2">subscript</csymbol><ci id="S4.5.p1.40.m40.1.1.2.2.cmml" xref="S4.5.p1.40.m40.1.1.2.2">π</ci><ci id="S4.5.p1.40.m40.1.1.2.3.cmml" xref="S4.5.p1.40.m40.1.1.2.3">π</ci></apply><apply id="S4.5.p1.40.m40.1.1.3.cmml" xref="S4.5.p1.40.m40.1.1.3"><csymbol cd="ambiguous" id="S4.5.p1.40.m40.1.1.3.1.cmml" xref="S4.5.p1.40.m40.1.1.3">superscript</csymbol><ci id="S4.5.p1.40.m40.1.1.3.2.cmml" xref="S4.5.p1.40.m40.1.1.3.2">πΏ</ci><ci id="S4.5.p1.40.m40.1.1.3.3.cmml" xref="S4.5.p1.40.m40.1.1.3.3">β²</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.40.m40.1c">d_{U}L^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.40.m40.1d">italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT</annotation></semantics></math> is isomorphic to <math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S4.5.p1.41.m41.2"><semantics id="S4.5.p1.41.m41.2a"><mrow id="S4.5.p1.41.m41.2.2.1" xref="S4.5.p1.41.m41.2.2.2.cmml"><mo id="S4.5.p1.41.m41.2.2.1.2" stretchy="false" xref="S4.5.p1.41.m41.2.2.2.cmml">(</mo><msub id="S4.5.p1.41.m41.2.2.1.1" xref="S4.5.p1.41.m41.2.2.1.1.cmml"><mi id="S4.5.p1.41.m41.2.2.1.1.2" xref="S4.5.p1.41.m41.2.2.1.1.2.cmml">Ο</mi><mn id="S4.5.p1.41.m41.2.2.1.1.3" xref="S4.5.p1.41.m41.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.5.p1.41.m41.2.2.1.3" xref="S4.5.p1.41.m41.2.2.2.cmml">,</mo><mi id="S4.5.p1.41.m41.1.1" xref="S4.5.p1.41.m41.1.1.cmml">min</mi><mo id="S4.5.p1.41.m41.2.2.1.4" stretchy="false" xref="S4.5.p1.41.m41.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.41.m41.2b"><interval closure="open" id="S4.5.p1.41.m41.2.2.2.cmml" xref="S4.5.p1.41.m41.2.2.1"><apply id="S4.5.p1.41.m41.2.2.1.1.cmml" xref="S4.5.p1.41.m41.2.2.1.1"><csymbol cd="ambiguous" id="S4.5.p1.41.m41.2.2.1.1.1.cmml" xref="S4.5.p1.41.m41.2.2.1.1">subscript</csymbol><ci id="S4.5.p1.41.m41.2.2.1.1.2.cmml" xref="S4.5.p1.41.m41.2.2.1.1.2">π</ci><cn id="S4.5.p1.41.m41.2.2.1.1.3.cmml" type="integer" xref="S4.5.p1.41.m41.2.2.1.1.3">1</cn></apply><min id="S4.5.p1.41.m41.1.1.cmml" xref="S4.5.p1.41.m41.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.41.m41.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.41.m41.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math>. Since <math alttext="\operatorname{cl}_{X}(W)" class="ltx_Math" display="inline" id="S4.5.p1.42.m42.2"><semantics id="S4.5.p1.42.m42.2a"><mrow id="S4.5.p1.42.m42.2.2.1" xref="S4.5.p1.42.m42.2.2.2.cmml"><msub id="S4.5.p1.42.m42.2.2.1.1" xref="S4.5.p1.42.m42.2.2.1.1.cmml"><mi id="S4.5.p1.42.m42.2.2.1.1.2" xref="S4.5.p1.42.m42.2.2.1.1.2.cmml">cl</mi><mi id="S4.5.p1.42.m42.2.2.1.1.3" xref="S4.5.p1.42.m42.2.2.1.1.3.cmml">X</mi></msub><mo id="S4.5.p1.42.m42.2.2.1a" xref="S4.5.p1.42.m42.2.2.2.cmml">β‘</mo><mrow id="S4.5.p1.42.m42.2.2.1.2" xref="S4.5.p1.42.m42.2.2.2.cmml"><mo id="S4.5.p1.42.m42.2.2.1.2.1" stretchy="false" xref="S4.5.p1.42.m42.2.2.2.cmml">(</mo><mi id="S4.5.p1.42.m42.1.1" xref="S4.5.p1.42.m42.1.1.cmml">W</mi><mo id="S4.5.p1.42.m42.2.2.1.2.2" stretchy="false" xref="S4.5.p1.42.m42.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.42.m42.2b"><apply id="S4.5.p1.42.m42.2.2.2.cmml" xref="S4.5.p1.42.m42.2.2.1"><apply id="S4.5.p1.42.m42.2.2.1.1.cmml" xref="S4.5.p1.42.m42.2.2.1.1"><csymbol cd="ambiguous" id="S4.5.p1.42.m42.2.2.1.1.1.cmml" xref="S4.5.p1.42.m42.2.2.1.1">subscript</csymbol><ci id="S4.5.p1.42.m42.2.2.1.1.2.cmml" xref="S4.5.p1.42.m42.2.2.1.1.2">cl</ci><ci id="S4.5.p1.42.m42.2.2.1.1.3.cmml" xref="S4.5.p1.42.m42.2.2.1.1.3">π</ci></apply><ci id="S4.5.p1.42.m42.1.1.cmml" xref="S4.5.p1.42.m42.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.42.m42.2c">\operatorname{cl}_{X}(W)</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.42.m42.2d">roman_cl start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_W )</annotation></semantics></math> is compact, the chain <math alttext="\operatorname{cl}_{X}(d_{U}L^{\prime})\subseteq\operatorname{cl}_{X}(W)" class="ltx_Math" display="inline" id="S4.5.p1.43.m43.4"><semantics id="S4.5.p1.43.m43.4a"><mrow id="S4.5.p1.43.m43.4.4" xref="S4.5.p1.43.m43.4.4.cmml"><mrow id="S4.5.p1.43.m43.3.3.2.2" xref="S4.5.p1.43.m43.3.3.2.3.cmml"><msub id="S4.5.p1.43.m43.2.2.1.1.1" xref="S4.5.p1.43.m43.2.2.1.1.1.cmml"><mi id="S4.5.p1.43.m43.2.2.1.1.1.2" xref="S4.5.p1.43.m43.2.2.1.1.1.2.cmml">cl</mi><mi id="S4.5.p1.43.m43.2.2.1.1.1.3" xref="S4.5.p1.43.m43.2.2.1.1.1.3.cmml">X</mi></msub><mo id="S4.5.p1.43.m43.3.3.2.2a" xref="S4.5.p1.43.m43.3.3.2.3.cmml">β‘</mo><mrow id="S4.5.p1.43.m43.3.3.2.2.2" xref="S4.5.p1.43.m43.3.3.2.3.cmml"><mo id="S4.5.p1.43.m43.3.3.2.2.2.2" stretchy="false" xref="S4.5.p1.43.m43.3.3.2.3.cmml">(</mo><mrow id="S4.5.p1.43.m43.3.3.2.2.2.1" xref="S4.5.p1.43.m43.3.3.2.2.2.1.cmml"><msub id="S4.5.p1.43.m43.3.3.2.2.2.1.2" xref="S4.5.p1.43.m43.3.3.2.2.2.1.2.cmml"><mi id="S4.5.p1.43.m43.3.3.2.2.2.1.2.2" xref="S4.5.p1.43.m43.3.3.2.2.2.1.2.2.cmml">d</mi><mi id="S4.5.p1.43.m43.3.3.2.2.2.1.2.3" xref="S4.5.p1.43.m43.3.3.2.2.2.1.2.3.cmml">U</mi></msub><mo id="S4.5.p1.43.m43.3.3.2.2.2.1.1" xref="S4.5.p1.43.m43.3.3.2.2.2.1.1.cmml">β’</mo><msup id="S4.5.p1.43.m43.3.3.2.2.2.1.3" xref="S4.5.p1.43.m43.3.3.2.2.2.1.3.cmml"><mi id="S4.5.p1.43.m43.3.3.2.2.2.1.3.2" xref="S4.5.p1.43.m43.3.3.2.2.2.1.3.2.cmml">L</mi><mo id="S4.5.p1.43.m43.3.3.2.2.2.1.3.3" xref="S4.5.p1.43.m43.3.3.2.2.2.1.3.3.cmml">β²</mo></msup></mrow><mo id="S4.5.p1.43.m43.3.3.2.2.2.3" stretchy="false" xref="S4.5.p1.43.m43.3.3.2.3.cmml">)</mo></mrow></mrow><mo id="S4.5.p1.43.m43.4.4.4" xref="S4.5.p1.43.m43.4.4.4.cmml">β</mo><mrow id="S4.5.p1.43.m43.4.4.3.1" xref="S4.5.p1.43.m43.4.4.3.2.cmml"><msub id="S4.5.p1.43.m43.4.4.3.1.1" xref="S4.5.p1.43.m43.4.4.3.1.1.cmml"><mi id="S4.5.p1.43.m43.4.4.3.1.1.2" xref="S4.5.p1.43.m43.4.4.3.1.1.2.cmml">cl</mi><mi id="S4.5.p1.43.m43.4.4.3.1.1.3" xref="S4.5.p1.43.m43.4.4.3.1.1.3.cmml">X</mi></msub><mo id="S4.5.p1.43.m43.4.4.3.1a" xref="S4.5.p1.43.m43.4.4.3.2.cmml">β‘</mo><mrow id="S4.5.p1.43.m43.4.4.3.1.2" xref="S4.5.p1.43.m43.4.4.3.2.cmml"><mo id="S4.5.p1.43.m43.4.4.3.1.2.1" stretchy="false" xref="S4.5.p1.43.m43.4.4.3.2.cmml">(</mo><mi id="S4.5.p1.43.m43.1.1" xref="S4.5.p1.43.m43.1.1.cmml">W</mi><mo id="S4.5.p1.43.m43.4.4.3.1.2.2" stretchy="false" xref="S4.5.p1.43.m43.4.4.3.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.5.p1.43.m43.4b"><apply id="S4.5.p1.43.m43.4.4.cmml" xref="S4.5.p1.43.m43.4.4"><subset id="S4.5.p1.43.m43.4.4.4.cmml" xref="S4.5.p1.43.m43.4.4.4"></subset><apply id="S4.5.p1.43.m43.3.3.2.3.cmml" xref="S4.5.p1.43.m43.3.3.2.2"><apply id="S4.5.p1.43.m43.2.2.1.1.1.cmml" xref="S4.5.p1.43.m43.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.5.p1.43.m43.2.2.1.1.1.1.cmml" xref="S4.5.p1.43.m43.2.2.1.1.1">subscript</csymbol><ci id="S4.5.p1.43.m43.2.2.1.1.1.2.cmml" xref="S4.5.p1.43.m43.2.2.1.1.1.2">cl</ci><ci id="S4.5.p1.43.m43.2.2.1.1.1.3.cmml" xref="S4.5.p1.43.m43.2.2.1.1.1.3">π</ci></apply><apply id="S4.5.p1.43.m43.3.3.2.2.2.1.cmml" xref="S4.5.p1.43.m43.3.3.2.2.2.1"><times id="S4.5.p1.43.m43.3.3.2.2.2.1.1.cmml" xref="S4.5.p1.43.m43.3.3.2.2.2.1.1"></times><apply id="S4.5.p1.43.m43.3.3.2.2.2.1.2.cmml" xref="S4.5.p1.43.m43.3.3.2.2.2.1.2"><csymbol cd="ambiguous" id="S4.5.p1.43.m43.3.3.2.2.2.1.2.1.cmml" xref="S4.5.p1.43.m43.3.3.2.2.2.1.2">subscript</csymbol><ci id="S4.5.p1.43.m43.3.3.2.2.2.1.2.2.cmml" xref="S4.5.p1.43.m43.3.3.2.2.2.1.2.2">π</ci><ci id="S4.5.p1.43.m43.3.3.2.2.2.1.2.3.cmml" xref="S4.5.p1.43.m43.3.3.2.2.2.1.2.3">π</ci></apply><apply id="S4.5.p1.43.m43.3.3.2.2.2.1.3.cmml" xref="S4.5.p1.43.m43.3.3.2.2.2.1.3"><csymbol cd="ambiguous" id="S4.5.p1.43.m43.3.3.2.2.2.1.3.1.cmml" xref="S4.5.p1.43.m43.3.3.2.2.2.1.3">superscript</csymbol><ci id="S4.5.p1.43.m43.3.3.2.2.2.1.3.2.cmml" xref="S4.5.p1.43.m43.3.3.2.2.2.1.3.2">πΏ</ci><ci id="S4.5.p1.43.m43.3.3.2.2.2.1.3.3.cmml" xref="S4.5.p1.43.m43.3.3.2.2.2.1.3.3">β²</ci></apply></apply></apply><apply id="S4.5.p1.43.m43.4.4.3.2.cmml" xref="S4.5.p1.43.m43.4.4.3.1"><apply id="S4.5.p1.43.m43.4.4.3.1.1.cmml" xref="S4.5.p1.43.m43.4.4.3.1.1"><csymbol cd="ambiguous" id="S4.5.p1.43.m43.4.4.3.1.1.1.cmml" xref="S4.5.p1.43.m43.4.4.3.1.1">subscript</csymbol><ci id="S4.5.p1.43.m43.4.4.3.1.1.2.cmml" xref="S4.5.p1.43.m43.4.4.3.1.1.2">cl</ci><ci id="S4.5.p1.43.m43.4.4.3.1.1.3.cmml" xref="S4.5.p1.43.m43.4.4.3.1.1.3">π</ci></apply><ci id="S4.5.p1.43.m43.1.1.cmml" xref="S4.5.p1.43.m43.1.1">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.5.p1.43.m43.4c">\operatorname{cl}_{X}(d_{U}L^{\prime})\subseteq\operatorname{cl}_{X}(W)</annotation><annotation encoding="application/x-llamapun" id="S4.5.p1.43.m43.4d">roman_cl start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT ) β roman_cl start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_W )</annotation></semantics></math> is compact, which contradicts Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem6" title="Lemma 3.6. β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.6</span></a>. β</p> </div> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S4.Thmtheorem9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem9.1.1.1">Proposition 4.9</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem9.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem9.p1"> <p class="ltx_p" id="S4.Thmtheorem9.p1.3"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem9.p1.3.3">Let <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.1.1.m1.1"><semantics id="S4.Thmtheorem9.p1.1.1.m1.1a"><mi id="S4.Thmtheorem9.p1.1.1.m1.1.1" xref="S4.Thmtheorem9.p1.1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.1.1.m1.1b"><ci id="S4.Thmtheorem9.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem9.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.1.1.m1.1d">italic_X</annotation></semantics></math> be a locally compact Nyikos topological semilattice. Then <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.2.2.m2.1"><semantics id="S4.Thmtheorem9.p1.2.2.m2.1a"><mi id="S4.Thmtheorem9.p1.2.2.m2.1.1" xref="S4.Thmtheorem9.p1.2.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.2.2.m2.1b"><ci id="S4.Thmtheorem9.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.2.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.2.2.m2.1d">italic_X</annotation></semantics></math> contains no chain isomorphic to <math alttext="(\omega_{1},\max)" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.3.3.m3.2"><semantics id="S4.Thmtheorem9.p1.3.3.m3.2a"><mrow id="S4.Thmtheorem9.p1.3.3.m3.2.2.1" xref="S4.Thmtheorem9.p1.3.3.m3.2.2.2.cmml"><mo id="S4.Thmtheorem9.p1.3.3.m3.2.2.1.2" stretchy="false" xref="S4.Thmtheorem9.p1.3.3.m3.2.2.2.cmml">(</mo><msub id="S4.Thmtheorem9.p1.3.3.m3.2.2.1.1" xref="S4.Thmtheorem9.p1.3.3.m3.2.2.1.1.cmml"><mi id="S4.Thmtheorem9.p1.3.3.m3.2.2.1.1.2" xref="S4.Thmtheorem9.p1.3.3.m3.2.2.1.1.2.cmml">Ο</mi><mn id="S4.Thmtheorem9.p1.3.3.m3.2.2.1.1.3" xref="S4.Thmtheorem9.p1.3.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem9.p1.3.3.m3.2.2.1.3" xref="S4.Thmtheorem9.p1.3.3.m3.2.2.2.cmml">,</mo><mi id="S4.Thmtheorem9.p1.3.3.m3.1.1" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.cmml">max</mi><mo id="S4.Thmtheorem9.p1.3.3.m3.2.2.1.4" stretchy="false" xref="S4.Thmtheorem9.p1.3.3.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.3.3.m3.2b"><interval closure="open" id="S4.Thmtheorem9.p1.3.3.m3.2.2.2.cmml" xref="S4.Thmtheorem9.p1.3.3.m3.2.2.1"><apply id="S4.Thmtheorem9.p1.3.3.m3.2.2.1.1.cmml" xref="S4.Thmtheorem9.p1.3.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.3.3.m3.2.2.1.1.1.cmml" xref="S4.Thmtheorem9.p1.3.3.m3.2.2.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p1.3.3.m3.2.2.1.1.2.cmml" xref="S4.Thmtheorem9.p1.3.3.m3.2.2.1.1.2">π</ci><cn id="S4.Thmtheorem9.p1.3.3.m3.2.2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.3.3.m3.2.2.1.1.3">1</cn></apply><max id="S4.Thmtheorem9.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem9.p1.3.3.m3.1.1"></max></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.3.3.m3.2c">(\omega_{1},\max)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.3.3.m3.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_max )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S4.11"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.6.p1"> <p class="ltx_p" id="S4.6.p1.9">Seeking a contradiction, assume that a locally compact Nyikos semilattice <math alttext="X" class="ltx_Math" display="inline" id="S4.6.p1.1.m1.1"><semantics id="S4.6.p1.1.m1.1a"><mi id="S4.6.p1.1.m1.1.1" xref="S4.6.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.6.p1.1.m1.1b"><ci id="S4.6.p1.1.m1.1.1.cmml" xref="S4.6.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.1.m1.1d">italic_X</annotation></semantics></math> contains a chain <math alttext="L=\{l_{\alpha}:\alpha<\omega_{1}\}" class="ltx_Math" display="inline" id="S4.6.p1.2.m2.2"><semantics id="S4.6.p1.2.m2.2a"><mrow id="S4.6.p1.2.m2.2.2" xref="S4.6.p1.2.m2.2.2.cmml"><mi id="S4.6.p1.2.m2.2.2.4" xref="S4.6.p1.2.m2.2.2.4.cmml">L</mi><mo id="S4.6.p1.2.m2.2.2.3" xref="S4.6.p1.2.m2.2.2.3.cmml">=</mo><mrow id="S4.6.p1.2.m2.2.2.2.2" xref="S4.6.p1.2.m2.2.2.2.3.cmml"><mo id="S4.6.p1.2.m2.2.2.2.2.3" stretchy="false" xref="S4.6.p1.2.m2.2.2.2.3.1.cmml">{</mo><msub id="S4.6.p1.2.m2.1.1.1.1.1" xref="S4.6.p1.2.m2.1.1.1.1.1.cmml"><mi id="S4.6.p1.2.m2.1.1.1.1.1.2" xref="S4.6.p1.2.m2.1.1.1.1.1.2.cmml">l</mi><mi id="S4.6.p1.2.m2.1.1.1.1.1.3" xref="S4.6.p1.2.m2.1.1.1.1.1.3.cmml">Ξ±</mi></msub><mo id="S4.6.p1.2.m2.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.6.p1.2.m2.2.2.2.3.1.cmml">:</mo><mrow id="S4.6.p1.2.m2.2.2.2.2.2" xref="S4.6.p1.2.m2.2.2.2.2.2.cmml"><mi id="S4.6.p1.2.m2.2.2.2.2.2.2" xref="S4.6.p1.2.m2.2.2.2.2.2.2.cmml">Ξ±</mi><mo id="S4.6.p1.2.m2.2.2.2.2.2.1" xref="S4.6.p1.2.m2.2.2.2.2.2.1.cmml"><</mo><msub id="S4.6.p1.2.m2.2.2.2.2.2.3" xref="S4.6.p1.2.m2.2.2.2.2.2.3.cmml"><mi id="S4.6.p1.2.m2.2.2.2.2.2.3.2" xref="S4.6.p1.2.m2.2.2.2.2.2.3.2.cmml">Ο</mi><mn id="S4.6.p1.2.m2.2.2.2.2.2.3.3" xref="S4.6.p1.2.m2.2.2.2.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S4.6.p1.2.m2.2.2.2.2.5" stretchy="false" xref="S4.6.p1.2.m2.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p1.2.m2.2b"><apply id="S4.6.p1.2.m2.2.2.cmml" xref="S4.6.p1.2.m2.2.2"><eq id="S4.6.p1.2.m2.2.2.3.cmml" xref="S4.6.p1.2.m2.2.2.3"></eq><ci id="S4.6.p1.2.m2.2.2.4.cmml" xref="S4.6.p1.2.m2.2.2.4">πΏ</ci><apply id="S4.6.p1.2.m2.2.2.2.3.cmml" xref="S4.6.p1.2.m2.2.2.2.2"><csymbol cd="latexml" id="S4.6.p1.2.m2.2.2.2.3.1.cmml" xref="S4.6.p1.2.m2.2.2.2.2.3">conditional-set</csymbol><apply id="S4.6.p1.2.m2.1.1.1.1.1.cmml" xref="S4.6.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.6.p1.2.m2.1.1.1.1.1.1.cmml" xref="S4.6.p1.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S4.6.p1.2.m2.1.1.1.1.1.2.cmml" xref="S4.6.p1.2.m2.1.1.1.1.1.2">π</ci><ci id="S4.6.p1.2.m2.1.1.1.1.1.3.cmml" xref="S4.6.p1.2.m2.1.1.1.1.1.3">πΌ</ci></apply><apply id="S4.6.p1.2.m2.2.2.2.2.2.cmml" xref="S4.6.p1.2.m2.2.2.2.2.2"><lt id="S4.6.p1.2.m2.2.2.2.2.2.1.cmml" xref="S4.6.p1.2.m2.2.2.2.2.2.1"></lt><ci id="S4.6.p1.2.m2.2.2.2.2.2.2.cmml" xref="S4.6.p1.2.m2.2.2.2.2.2.2">πΌ</ci><apply id="S4.6.p1.2.m2.2.2.2.2.2.3.cmml" xref="S4.6.p1.2.m2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.6.p1.2.m2.2.2.2.2.2.3.1.cmml" xref="S4.6.p1.2.m2.2.2.2.2.2.3">subscript</csymbol><ci id="S4.6.p1.2.m2.2.2.2.2.2.3.2.cmml" xref="S4.6.p1.2.m2.2.2.2.2.2.3.2">π</ci><cn id="S4.6.p1.2.m2.2.2.2.2.2.3.3.cmml" type="integer" xref="S4.6.p1.2.m2.2.2.2.2.2.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.2.m2.2c">L=\{l_{\alpha}:\alpha<\omega_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.2.m2.2d">italic_L = { italic_l start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT : italic_Ξ± < italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math> such that <math alttext="l_{\alpha}\leq l_{\beta}" class="ltx_Math" display="inline" id="S4.6.p1.3.m3.1"><semantics id="S4.6.p1.3.m3.1a"><mrow id="S4.6.p1.3.m3.1.1" xref="S4.6.p1.3.m3.1.1.cmml"><msub id="S4.6.p1.3.m3.1.1.2" xref="S4.6.p1.3.m3.1.1.2.cmml"><mi id="S4.6.p1.3.m3.1.1.2.2" xref="S4.6.p1.3.m3.1.1.2.2.cmml">l</mi><mi id="S4.6.p1.3.m3.1.1.2.3" xref="S4.6.p1.3.m3.1.1.2.3.cmml">Ξ±</mi></msub><mo id="S4.6.p1.3.m3.1.1.1" xref="S4.6.p1.3.m3.1.1.1.cmml">β€</mo><msub id="S4.6.p1.3.m3.1.1.3" xref="S4.6.p1.3.m3.1.1.3.cmml"><mi id="S4.6.p1.3.m3.1.1.3.2" xref="S4.6.p1.3.m3.1.1.3.2.cmml">l</mi><mi id="S4.6.p1.3.m3.1.1.3.3" xref="S4.6.p1.3.m3.1.1.3.3.cmml">Ξ²</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p1.3.m3.1b"><apply id="S4.6.p1.3.m3.1.1.cmml" xref="S4.6.p1.3.m3.1.1"><leq id="S4.6.p1.3.m3.1.1.1.cmml" xref="S4.6.p1.3.m3.1.1.1"></leq><apply id="S4.6.p1.3.m3.1.1.2.cmml" xref="S4.6.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.6.p1.3.m3.1.1.2.1.cmml" xref="S4.6.p1.3.m3.1.1.2">subscript</csymbol><ci id="S4.6.p1.3.m3.1.1.2.2.cmml" xref="S4.6.p1.3.m3.1.1.2.2">π</ci><ci id="S4.6.p1.3.m3.1.1.2.3.cmml" xref="S4.6.p1.3.m3.1.1.2.3">πΌ</ci></apply><apply id="S4.6.p1.3.m3.1.1.3.cmml" xref="S4.6.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.6.p1.3.m3.1.1.3.1.cmml" xref="S4.6.p1.3.m3.1.1.3">subscript</csymbol><ci id="S4.6.p1.3.m3.1.1.3.2.cmml" xref="S4.6.p1.3.m3.1.1.3.2">π</ci><ci id="S4.6.p1.3.m3.1.1.3.3.cmml" xref="S4.6.p1.3.m3.1.1.3.3">π½</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.3.m3.1c">l_{\alpha}\leq l_{\beta}</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.3.m3.1d">italic_l start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT β€ italic_l start_POSTSUBSCRIPT italic_Ξ² end_POSTSUBSCRIPT</annotation></semantics></math> if and only if <math alttext="\alpha\geq\beta" class="ltx_Math" display="inline" id="S4.6.p1.4.m4.1"><semantics id="S4.6.p1.4.m4.1a"><mrow id="S4.6.p1.4.m4.1.1" xref="S4.6.p1.4.m4.1.1.cmml"><mi id="S4.6.p1.4.m4.1.1.2" xref="S4.6.p1.4.m4.1.1.2.cmml">Ξ±</mi><mo id="S4.6.p1.4.m4.1.1.1" xref="S4.6.p1.4.m4.1.1.1.cmml">β₯</mo><mi id="S4.6.p1.4.m4.1.1.3" xref="S4.6.p1.4.m4.1.1.3.cmml">Ξ²</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p1.4.m4.1b"><apply id="S4.6.p1.4.m4.1.1.cmml" xref="S4.6.p1.4.m4.1.1"><geq id="S4.6.p1.4.m4.1.1.1.cmml" xref="S4.6.p1.4.m4.1.1.1"></geq><ci id="S4.6.p1.4.m4.1.1.2.cmml" xref="S4.6.p1.4.m4.1.1.2">πΌ</ci><ci id="S4.6.p1.4.m4.1.1.3.cmml" xref="S4.6.p1.4.m4.1.1.3">π½</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.4.m4.1c">\alpha\geq\beta</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.4.m4.1d">italic_Ξ± β₯ italic_Ξ²</annotation></semantics></math>. By Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S2.Thmtheorem9" title="Theorem 2.9 (Banakh, Dimitrova, Gutik). β£ 2. Compact-like semigroups with dense inverse subsemigroups and the automatic continuity of inversion β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">2.9</span></a>, <math alttext="\beta X" class="ltx_Math" display="inline" id="S4.6.p1.5.m5.1"><semantics id="S4.6.p1.5.m5.1a"><mrow id="S4.6.p1.5.m5.1.1" xref="S4.6.p1.5.m5.1.1.cmml"><mi id="S4.6.p1.5.m5.1.1.2" xref="S4.6.p1.5.m5.1.1.2.cmml">Ξ²</mi><mo id="S4.6.p1.5.m5.1.1.1" xref="S4.6.p1.5.m5.1.1.1.cmml">β’</mo><mi id="S4.6.p1.5.m5.1.1.3" xref="S4.6.p1.5.m5.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p1.5.m5.1b"><apply id="S4.6.p1.5.m5.1.1.cmml" xref="S4.6.p1.5.m5.1.1"><times id="S4.6.p1.5.m5.1.1.1.cmml" xref="S4.6.p1.5.m5.1.1.1"></times><ci id="S4.6.p1.5.m5.1.1.2.cmml" xref="S4.6.p1.5.m5.1.1.2">π½</ci><ci id="S4.6.p1.5.m5.1.1.3.cmml" xref="S4.6.p1.5.m5.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.5.m5.1c">\beta X</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.5.m5.1d">italic_Ξ² italic_X</annotation></semantics></math> is a compact topological semigroup. Since <math alttext="X" class="ltx_Math" display="inline" id="S4.6.p1.6.m6.1"><semantics id="S4.6.p1.6.m6.1a"><mi id="S4.6.p1.6.m6.1.1" xref="S4.6.p1.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.6.p1.6.m6.1b"><ci id="S4.6.p1.6.m6.1.1.cmml" xref="S4.6.p1.6.m6.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.6.m6.1d">italic_X</annotation></semantics></math> is dense in <math alttext="\beta X" class="ltx_Math" display="inline" id="S4.6.p1.7.m7.1"><semantics id="S4.6.p1.7.m7.1a"><mrow id="S4.6.p1.7.m7.1.1" xref="S4.6.p1.7.m7.1.1.cmml"><mi id="S4.6.p1.7.m7.1.1.2" xref="S4.6.p1.7.m7.1.1.2.cmml">Ξ²</mi><mo id="S4.6.p1.7.m7.1.1.1" xref="S4.6.p1.7.m7.1.1.1.cmml">β’</mo><mi id="S4.6.p1.7.m7.1.1.3" xref="S4.6.p1.7.m7.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p1.7.m7.1b"><apply id="S4.6.p1.7.m7.1.1.cmml" xref="S4.6.p1.7.m7.1.1"><times id="S4.6.p1.7.m7.1.1.1.cmml" xref="S4.6.p1.7.m7.1.1.1"></times><ci id="S4.6.p1.7.m7.1.1.2.cmml" xref="S4.6.p1.7.m7.1.1.2">π½</ci><ci id="S4.6.p1.7.m7.1.1.3.cmml" xref="S4.6.p1.7.m7.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.7.m7.1c">\beta X</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.7.m7.1d">italic_Ξ² italic_X</annotation></semantics></math>, we obtain that <math alttext="\beta X" class="ltx_Math" display="inline" id="S4.6.p1.8.m8.1"><semantics id="S4.6.p1.8.m8.1a"><mrow id="S4.6.p1.8.m8.1.1" xref="S4.6.p1.8.m8.1.1.cmml"><mi id="S4.6.p1.8.m8.1.1.2" xref="S4.6.p1.8.m8.1.1.2.cmml">Ξ²</mi><mo id="S4.6.p1.8.m8.1.1.1" xref="S4.6.p1.8.m8.1.1.1.cmml">β’</mo><mi id="S4.6.p1.8.m8.1.1.3" xref="S4.6.p1.8.m8.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p1.8.m8.1b"><apply id="S4.6.p1.8.m8.1.1.cmml" xref="S4.6.p1.8.m8.1.1"><times id="S4.6.p1.8.m8.1.1.1.cmml" xref="S4.6.p1.8.m8.1.1.1"></times><ci id="S4.6.p1.8.m8.1.1.2.cmml" xref="S4.6.p1.8.m8.1.1.2">π½</ci><ci id="S4.6.p1.8.m8.1.1.3.cmml" xref="S4.6.p1.8.m8.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.8.m8.1c">\beta X</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.8.m8.1d">italic_Ξ² italic_X</annotation></semantics></math> is a semilattice. By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem5" title="Lemma 3.5. β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.5</span></a>, there exists <math alttext="z=\inf L\in\beta X\setminus X" class="ltx_Math" display="inline" id="S4.6.p1.9.m9.1"><semantics id="S4.6.p1.9.m9.1a"><mrow id="S4.6.p1.9.m9.1.1" xref="S4.6.p1.9.m9.1.1.cmml"><mi id="S4.6.p1.9.m9.1.1.2" xref="S4.6.p1.9.m9.1.1.2.cmml">z</mi><mo id="S4.6.p1.9.m9.1.1.3" rspace="0.1389em" xref="S4.6.p1.9.m9.1.1.3.cmml">=</mo><mrow id="S4.6.p1.9.m9.1.1.4" xref="S4.6.p1.9.m9.1.1.4.cmml"><mo id="S4.6.p1.9.m9.1.1.4.1" lspace="0.1389em" rspace="0.167em" xref="S4.6.p1.9.m9.1.1.4.1.cmml">inf</mo><mi id="S4.6.p1.9.m9.1.1.4.2" xref="S4.6.p1.9.m9.1.1.4.2.cmml">L</mi></mrow><mo id="S4.6.p1.9.m9.1.1.5" xref="S4.6.p1.9.m9.1.1.5.cmml">β</mo><mrow id="S4.6.p1.9.m9.1.1.6" xref="S4.6.p1.9.m9.1.1.6.cmml"><mrow id="S4.6.p1.9.m9.1.1.6.2" xref="S4.6.p1.9.m9.1.1.6.2.cmml"><mi id="S4.6.p1.9.m9.1.1.6.2.2" xref="S4.6.p1.9.m9.1.1.6.2.2.cmml">Ξ²</mi><mo id="S4.6.p1.9.m9.1.1.6.2.1" xref="S4.6.p1.9.m9.1.1.6.2.1.cmml">β’</mo><mi id="S4.6.p1.9.m9.1.1.6.2.3" xref="S4.6.p1.9.m9.1.1.6.2.3.cmml">X</mi></mrow><mo id="S4.6.p1.9.m9.1.1.6.1" xref="S4.6.p1.9.m9.1.1.6.1.cmml">β</mo><mi id="S4.6.p1.9.m9.1.1.6.3" xref="S4.6.p1.9.m9.1.1.6.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.6.p1.9.m9.1b"><apply id="S4.6.p1.9.m9.1.1.cmml" xref="S4.6.p1.9.m9.1.1"><and id="S4.6.p1.9.m9.1.1a.cmml" xref="S4.6.p1.9.m9.1.1"></and><apply id="S4.6.p1.9.m9.1.1b.cmml" xref="S4.6.p1.9.m9.1.1"><eq id="S4.6.p1.9.m9.1.1.3.cmml" xref="S4.6.p1.9.m9.1.1.3"></eq><ci id="S4.6.p1.9.m9.1.1.2.cmml" xref="S4.6.p1.9.m9.1.1.2">π§</ci><apply id="S4.6.p1.9.m9.1.1.4.cmml" xref="S4.6.p1.9.m9.1.1.4"><csymbol cd="latexml" id="S4.6.p1.9.m9.1.1.4.1.cmml" xref="S4.6.p1.9.m9.1.1.4.1">infimum</csymbol><ci id="S4.6.p1.9.m9.1.1.4.2.cmml" xref="S4.6.p1.9.m9.1.1.4.2">πΏ</ci></apply></apply><apply id="S4.6.p1.9.m9.1.1c.cmml" xref="S4.6.p1.9.m9.1.1"><in id="S4.6.p1.9.m9.1.1.5.cmml" xref="S4.6.p1.9.m9.1.1.5"></in><share href="https://arxiv.org/html/2503.13666v1#S4.6.p1.9.m9.1.1.4.cmml" id="S4.6.p1.9.m9.1.1d.cmml" xref="S4.6.p1.9.m9.1.1"></share><apply id="S4.6.p1.9.m9.1.1.6.cmml" xref="S4.6.p1.9.m9.1.1.6"><setdiff id="S4.6.p1.9.m9.1.1.6.1.cmml" xref="S4.6.p1.9.m9.1.1.6.1"></setdiff><apply id="S4.6.p1.9.m9.1.1.6.2.cmml" xref="S4.6.p1.9.m9.1.1.6.2"><times id="S4.6.p1.9.m9.1.1.6.2.1.cmml" xref="S4.6.p1.9.m9.1.1.6.2.1"></times><ci id="S4.6.p1.9.m9.1.1.6.2.2.cmml" xref="S4.6.p1.9.m9.1.1.6.2.2">π½</ci><ci id="S4.6.p1.9.m9.1.1.6.2.3.cmml" xref="S4.6.p1.9.m9.1.1.6.2.3">π</ci></apply><ci id="S4.6.p1.9.m9.1.1.6.3.cmml" xref="S4.6.p1.9.m9.1.1.6.3">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.6.p1.9.m9.1c">z=\inf L\in\beta X\setminus X</annotation><annotation encoding="application/x-llamapun" id="S4.6.p1.9.m9.1d">italic_z = roman_inf italic_L β italic_Ξ² italic_X β italic_X</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_claim" id="S4.Thmtheorem10"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem10.1.1.1">Claim 4.10</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem10.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem10.p1"> <p class="ltx_p" id="S4.Thmtheorem10.p1.2"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem10.p1.2.2">The set <math alttext="I_{L}" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p1.1.1.m1.1"><semantics id="S4.Thmtheorem10.p1.1.1.m1.1a"><msub id="S4.Thmtheorem10.p1.1.1.m1.1.1" xref="S4.Thmtheorem10.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem10.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem10.p1.1.1.m1.1.1.2.cmml">I</mi><mi id="S4.Thmtheorem10.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem10.p1.1.1.m1.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p1.1.1.m1.1b"><apply id="S4.Thmtheorem10.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem10.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem10.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem10.p1.1.1.m1.1.1">subscript</csymbol><ci id="S4.Thmtheorem10.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem10.p1.1.1.m1.1.1.2">πΌ</ci><ci id="S4.Thmtheorem10.p1.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem10.p1.1.1.m1.1.1.3">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p1.1.1.m1.1c">I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p1.1.1.m1.1d">italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> is clopen in <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p1.2.2.m2.1"><semantics id="S4.Thmtheorem10.p1.2.2.m2.1a"><mi id="S4.Thmtheorem10.p1.2.2.m2.1.1" xref="S4.Thmtheorem10.p1.2.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p1.2.2.m2.1b"><ci id="S4.Thmtheorem10.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem10.p1.2.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p1.2.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p1.2.2.m2.1d">italic_X</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S4.7.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.7.1.p1"> <p class="ltx_p" id="S4.7.1.p1.42">Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S4.Thmtheorem5" title="Lemma 4.5. β£ 4. Nyikos semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">4.5</span></a> implies that <math alttext="I_{L}" class="ltx_Math" display="inline" id="S4.7.1.p1.1.m1.1"><semantics id="S4.7.1.p1.1.m1.1a"><msub id="S4.7.1.p1.1.m1.1.1" xref="S4.7.1.p1.1.m1.1.1.cmml"><mi id="S4.7.1.p1.1.m1.1.1.2" xref="S4.7.1.p1.1.m1.1.1.2.cmml">I</mi><mi id="S4.7.1.p1.1.m1.1.1.3" xref="S4.7.1.p1.1.m1.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.1.m1.1b"><apply id="S4.7.1.p1.1.m1.1.1.cmml" xref="S4.7.1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.7.1.p1.1.m1.1.1.1.cmml" xref="S4.7.1.p1.1.m1.1.1">subscript</csymbol><ci id="S4.7.1.p1.1.m1.1.1.2.cmml" xref="S4.7.1.p1.1.m1.1.1.2">πΌ</ci><ci id="S4.7.1.p1.1.m1.1.1.3.cmml" xref="S4.7.1.p1.1.m1.1.1.3">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.1.m1.1c">I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.1.m1.1d">italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> is closed. To derive a contradiction, assume that <math alttext="I_{L}" class="ltx_Math" display="inline" id="S4.7.1.p1.2.m2.1"><semantics id="S4.7.1.p1.2.m2.1a"><msub id="S4.7.1.p1.2.m2.1.1" xref="S4.7.1.p1.2.m2.1.1.cmml"><mi id="S4.7.1.p1.2.m2.1.1.2" xref="S4.7.1.p1.2.m2.1.1.2.cmml">I</mi><mi id="S4.7.1.p1.2.m2.1.1.3" xref="S4.7.1.p1.2.m2.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.2.m2.1b"><apply id="S4.7.1.p1.2.m2.1.1.cmml" xref="S4.7.1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.7.1.p1.2.m2.1.1.1.cmml" xref="S4.7.1.p1.2.m2.1.1">subscript</csymbol><ci id="S4.7.1.p1.2.m2.1.1.2.cmml" xref="S4.7.1.p1.2.m2.1.1.2">πΌ</ci><ci id="S4.7.1.p1.2.m2.1.1.3.cmml" xref="S4.7.1.p1.2.m2.1.1.3">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.2.m2.1c">I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.2.m2.1d">italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> is not open. Then there exists <math alttext="y\in I_{L}" class="ltx_Math" display="inline" id="S4.7.1.p1.3.m3.1"><semantics id="S4.7.1.p1.3.m3.1a"><mrow id="S4.7.1.p1.3.m3.1.1" xref="S4.7.1.p1.3.m3.1.1.cmml"><mi id="S4.7.1.p1.3.m3.1.1.2" xref="S4.7.1.p1.3.m3.1.1.2.cmml">y</mi><mo id="S4.7.1.p1.3.m3.1.1.1" xref="S4.7.1.p1.3.m3.1.1.1.cmml">β</mo><msub id="S4.7.1.p1.3.m3.1.1.3" xref="S4.7.1.p1.3.m3.1.1.3.cmml"><mi id="S4.7.1.p1.3.m3.1.1.3.2" xref="S4.7.1.p1.3.m3.1.1.3.2.cmml">I</mi><mi id="S4.7.1.p1.3.m3.1.1.3.3" xref="S4.7.1.p1.3.m3.1.1.3.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.3.m3.1b"><apply id="S4.7.1.p1.3.m3.1.1.cmml" xref="S4.7.1.p1.3.m3.1.1"><in id="S4.7.1.p1.3.m3.1.1.1.cmml" xref="S4.7.1.p1.3.m3.1.1.1"></in><ci id="S4.7.1.p1.3.m3.1.1.2.cmml" xref="S4.7.1.p1.3.m3.1.1.2">π¦</ci><apply id="S4.7.1.p1.3.m3.1.1.3.cmml" xref="S4.7.1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.7.1.p1.3.m3.1.1.3.1.cmml" xref="S4.7.1.p1.3.m3.1.1.3">subscript</csymbol><ci id="S4.7.1.p1.3.m3.1.1.3.2.cmml" xref="S4.7.1.p1.3.m3.1.1.3.2">πΌ</ci><ci id="S4.7.1.p1.3.m3.1.1.3.3.cmml" xref="S4.7.1.p1.3.m3.1.1.3.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.3.m3.1c">y\in I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.3.m3.1d">italic_y β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> such that each open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S4.7.1.p1.4.m4.1"><semantics id="S4.7.1.p1.4.m4.1a"><mi id="S4.7.1.p1.4.m4.1.1" xref="S4.7.1.p1.4.m4.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.4.m4.1b"><ci id="S4.7.1.p1.4.m4.1.1.cmml" xref="S4.7.1.p1.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.4.m4.1c">U</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.4.m4.1d">italic_U</annotation></semantics></math> of <math alttext="y" class="ltx_Math" display="inline" id="S4.7.1.p1.5.m5.1"><semantics id="S4.7.1.p1.5.m5.1a"><mi id="S4.7.1.p1.5.m5.1.1" xref="S4.7.1.p1.5.m5.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.5.m5.1b"><ci id="S4.7.1.p1.5.m5.1.1.cmml" xref="S4.7.1.p1.5.m5.1.1">π¦</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.5.m5.1c">y</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.5.m5.1d">italic_y</annotation></semantics></math> contains an element <math alttext="d_{U}\in X\setminus I_{L}" class="ltx_Math" display="inline" id="S4.7.1.p1.6.m6.1"><semantics id="S4.7.1.p1.6.m6.1a"><mrow id="S4.7.1.p1.6.m6.1.1" xref="S4.7.1.p1.6.m6.1.1.cmml"><msub id="S4.7.1.p1.6.m6.1.1.2" xref="S4.7.1.p1.6.m6.1.1.2.cmml"><mi id="S4.7.1.p1.6.m6.1.1.2.2" xref="S4.7.1.p1.6.m6.1.1.2.2.cmml">d</mi><mi id="S4.7.1.p1.6.m6.1.1.2.3" xref="S4.7.1.p1.6.m6.1.1.2.3.cmml">U</mi></msub><mo id="S4.7.1.p1.6.m6.1.1.1" xref="S4.7.1.p1.6.m6.1.1.1.cmml">β</mo><mrow id="S4.7.1.p1.6.m6.1.1.3" xref="S4.7.1.p1.6.m6.1.1.3.cmml"><mi id="S4.7.1.p1.6.m6.1.1.3.2" xref="S4.7.1.p1.6.m6.1.1.3.2.cmml">X</mi><mo id="S4.7.1.p1.6.m6.1.1.3.1" xref="S4.7.1.p1.6.m6.1.1.3.1.cmml">β</mo><msub id="S4.7.1.p1.6.m6.1.1.3.3" xref="S4.7.1.p1.6.m6.1.1.3.3.cmml"><mi id="S4.7.1.p1.6.m6.1.1.3.3.2" xref="S4.7.1.p1.6.m6.1.1.3.3.2.cmml">I</mi><mi id="S4.7.1.p1.6.m6.1.1.3.3.3" xref="S4.7.1.p1.6.m6.1.1.3.3.3.cmml">L</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.6.m6.1b"><apply id="S4.7.1.p1.6.m6.1.1.cmml" xref="S4.7.1.p1.6.m6.1.1"><in id="S4.7.1.p1.6.m6.1.1.1.cmml" xref="S4.7.1.p1.6.m6.1.1.1"></in><apply id="S4.7.1.p1.6.m6.1.1.2.cmml" xref="S4.7.1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S4.7.1.p1.6.m6.1.1.2.1.cmml" xref="S4.7.1.p1.6.m6.1.1.2">subscript</csymbol><ci id="S4.7.1.p1.6.m6.1.1.2.2.cmml" xref="S4.7.1.p1.6.m6.1.1.2.2">π</ci><ci id="S4.7.1.p1.6.m6.1.1.2.3.cmml" xref="S4.7.1.p1.6.m6.1.1.2.3">π</ci></apply><apply id="S4.7.1.p1.6.m6.1.1.3.cmml" xref="S4.7.1.p1.6.m6.1.1.3"><setdiff id="S4.7.1.p1.6.m6.1.1.3.1.cmml" xref="S4.7.1.p1.6.m6.1.1.3.1"></setdiff><ci id="S4.7.1.p1.6.m6.1.1.3.2.cmml" xref="S4.7.1.p1.6.m6.1.1.3.2">π</ci><apply id="S4.7.1.p1.6.m6.1.1.3.3.cmml" xref="S4.7.1.p1.6.m6.1.1.3.3"><csymbol cd="ambiguous" id="S4.7.1.p1.6.m6.1.1.3.3.1.cmml" xref="S4.7.1.p1.6.m6.1.1.3.3">subscript</csymbol><ci id="S4.7.1.p1.6.m6.1.1.3.3.2.cmml" xref="S4.7.1.p1.6.m6.1.1.3.3.2">πΌ</ci><ci id="S4.7.1.p1.6.m6.1.1.3.3.3.cmml" xref="S4.7.1.p1.6.m6.1.1.3.3.3">πΏ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.6.m6.1c">d_{U}\in X\setminus I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.6.m6.1d">italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT β italic_X β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>. Let us show that <math alttext="yz\in X" class="ltx_Math" display="inline" id="S4.7.1.p1.7.m7.1"><semantics id="S4.7.1.p1.7.m7.1a"><mrow id="S4.7.1.p1.7.m7.1.1" xref="S4.7.1.p1.7.m7.1.1.cmml"><mrow id="S4.7.1.p1.7.m7.1.1.2" xref="S4.7.1.p1.7.m7.1.1.2.cmml"><mi id="S4.7.1.p1.7.m7.1.1.2.2" xref="S4.7.1.p1.7.m7.1.1.2.2.cmml">y</mi><mo id="S4.7.1.p1.7.m7.1.1.2.1" xref="S4.7.1.p1.7.m7.1.1.2.1.cmml">β’</mo><mi id="S4.7.1.p1.7.m7.1.1.2.3" xref="S4.7.1.p1.7.m7.1.1.2.3.cmml">z</mi></mrow><mo id="S4.7.1.p1.7.m7.1.1.1" xref="S4.7.1.p1.7.m7.1.1.1.cmml">β</mo><mi id="S4.7.1.p1.7.m7.1.1.3" xref="S4.7.1.p1.7.m7.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.7.m7.1b"><apply id="S4.7.1.p1.7.m7.1.1.cmml" xref="S4.7.1.p1.7.m7.1.1"><in id="S4.7.1.p1.7.m7.1.1.1.cmml" xref="S4.7.1.p1.7.m7.1.1.1"></in><apply id="S4.7.1.p1.7.m7.1.1.2.cmml" xref="S4.7.1.p1.7.m7.1.1.2"><times id="S4.7.1.p1.7.m7.1.1.2.1.cmml" xref="S4.7.1.p1.7.m7.1.1.2.1"></times><ci id="S4.7.1.p1.7.m7.1.1.2.2.cmml" xref="S4.7.1.p1.7.m7.1.1.2.2">π¦</ci><ci id="S4.7.1.p1.7.m7.1.1.2.3.cmml" xref="S4.7.1.p1.7.m7.1.1.2.3">π§</ci></apply><ci id="S4.7.1.p1.7.m7.1.1.3.cmml" xref="S4.7.1.p1.7.m7.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.7.m7.1c">yz\in X</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.7.m7.1d">italic_y italic_z β italic_X</annotation></semantics></math>. Since <math alttext="y\in I_{L}" class="ltx_Math" display="inline" id="S4.7.1.p1.8.m8.1"><semantics id="S4.7.1.p1.8.m8.1a"><mrow id="S4.7.1.p1.8.m8.1.1" xref="S4.7.1.p1.8.m8.1.1.cmml"><mi id="S4.7.1.p1.8.m8.1.1.2" xref="S4.7.1.p1.8.m8.1.1.2.cmml">y</mi><mo id="S4.7.1.p1.8.m8.1.1.1" xref="S4.7.1.p1.8.m8.1.1.1.cmml">β</mo><msub id="S4.7.1.p1.8.m8.1.1.3" xref="S4.7.1.p1.8.m8.1.1.3.cmml"><mi id="S4.7.1.p1.8.m8.1.1.3.2" xref="S4.7.1.p1.8.m8.1.1.3.2.cmml">I</mi><mi id="S4.7.1.p1.8.m8.1.1.3.3" xref="S4.7.1.p1.8.m8.1.1.3.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.8.m8.1b"><apply id="S4.7.1.p1.8.m8.1.1.cmml" xref="S4.7.1.p1.8.m8.1.1"><in id="S4.7.1.p1.8.m8.1.1.1.cmml" xref="S4.7.1.p1.8.m8.1.1.1"></in><ci id="S4.7.1.p1.8.m8.1.1.2.cmml" xref="S4.7.1.p1.8.m8.1.1.2">π¦</ci><apply id="S4.7.1.p1.8.m8.1.1.3.cmml" xref="S4.7.1.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.7.1.p1.8.m8.1.1.3.1.cmml" xref="S4.7.1.p1.8.m8.1.1.3">subscript</csymbol><ci id="S4.7.1.p1.8.m8.1.1.3.2.cmml" xref="S4.7.1.p1.8.m8.1.1.3.2">πΌ</ci><ci id="S4.7.1.p1.8.m8.1.1.3.3.cmml" xref="S4.7.1.p1.8.m8.1.1.3.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.8.m8.1c">y\in I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.8.m8.1d">italic_y β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>, there exists an ordinal <math alttext="\alpha<\omega_{1}" class="ltx_Math" display="inline" id="S4.7.1.p1.9.m9.1"><semantics id="S4.7.1.p1.9.m9.1a"><mrow id="S4.7.1.p1.9.m9.1.1" xref="S4.7.1.p1.9.m9.1.1.cmml"><mi id="S4.7.1.p1.9.m9.1.1.2" xref="S4.7.1.p1.9.m9.1.1.2.cmml">Ξ±</mi><mo id="S4.7.1.p1.9.m9.1.1.1" xref="S4.7.1.p1.9.m9.1.1.1.cmml"><</mo><msub id="S4.7.1.p1.9.m9.1.1.3" xref="S4.7.1.p1.9.m9.1.1.3.cmml"><mi id="S4.7.1.p1.9.m9.1.1.3.2" xref="S4.7.1.p1.9.m9.1.1.3.2.cmml">Ο</mi><mn id="S4.7.1.p1.9.m9.1.1.3.3" xref="S4.7.1.p1.9.m9.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.9.m9.1b"><apply id="S4.7.1.p1.9.m9.1.1.cmml" xref="S4.7.1.p1.9.m9.1.1"><lt id="S4.7.1.p1.9.m9.1.1.1.cmml" xref="S4.7.1.p1.9.m9.1.1.1"></lt><ci id="S4.7.1.p1.9.m9.1.1.2.cmml" xref="S4.7.1.p1.9.m9.1.1.2">πΌ</ci><apply id="S4.7.1.p1.9.m9.1.1.3.cmml" xref="S4.7.1.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="S4.7.1.p1.9.m9.1.1.3.1.cmml" xref="S4.7.1.p1.9.m9.1.1.3">subscript</csymbol><ci id="S4.7.1.p1.9.m9.1.1.3.2.cmml" xref="S4.7.1.p1.9.m9.1.1.3.2">π</ci><cn id="S4.7.1.p1.9.m9.1.1.3.3.cmml" type="integer" xref="S4.7.1.p1.9.m9.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.9.m9.1c">\alpha<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.9.m9.1d">italic_Ξ± < italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="yl_{\xi}=yl_{\alpha}" class="ltx_Math" display="inline" id="S4.7.1.p1.10.m10.1"><semantics id="S4.7.1.p1.10.m10.1a"><mrow id="S4.7.1.p1.10.m10.1.1" xref="S4.7.1.p1.10.m10.1.1.cmml"><mrow id="S4.7.1.p1.10.m10.1.1.2" xref="S4.7.1.p1.10.m10.1.1.2.cmml"><mi id="S4.7.1.p1.10.m10.1.1.2.2" xref="S4.7.1.p1.10.m10.1.1.2.2.cmml">y</mi><mo id="S4.7.1.p1.10.m10.1.1.2.1" xref="S4.7.1.p1.10.m10.1.1.2.1.cmml">β’</mo><msub id="S4.7.1.p1.10.m10.1.1.2.3" xref="S4.7.1.p1.10.m10.1.1.2.3.cmml"><mi id="S4.7.1.p1.10.m10.1.1.2.3.2" xref="S4.7.1.p1.10.m10.1.1.2.3.2.cmml">l</mi><mi id="S4.7.1.p1.10.m10.1.1.2.3.3" xref="S4.7.1.p1.10.m10.1.1.2.3.3.cmml">ΞΎ</mi></msub></mrow><mo id="S4.7.1.p1.10.m10.1.1.1" xref="S4.7.1.p1.10.m10.1.1.1.cmml">=</mo><mrow id="S4.7.1.p1.10.m10.1.1.3" xref="S4.7.1.p1.10.m10.1.1.3.cmml"><mi id="S4.7.1.p1.10.m10.1.1.3.2" xref="S4.7.1.p1.10.m10.1.1.3.2.cmml">y</mi><mo id="S4.7.1.p1.10.m10.1.1.3.1" xref="S4.7.1.p1.10.m10.1.1.3.1.cmml">β’</mo><msub id="S4.7.1.p1.10.m10.1.1.3.3" xref="S4.7.1.p1.10.m10.1.1.3.3.cmml"><mi id="S4.7.1.p1.10.m10.1.1.3.3.2" xref="S4.7.1.p1.10.m10.1.1.3.3.2.cmml">l</mi><mi id="S4.7.1.p1.10.m10.1.1.3.3.3" xref="S4.7.1.p1.10.m10.1.1.3.3.3.cmml">Ξ±</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.10.m10.1b"><apply id="S4.7.1.p1.10.m10.1.1.cmml" xref="S4.7.1.p1.10.m10.1.1"><eq id="S4.7.1.p1.10.m10.1.1.1.cmml" xref="S4.7.1.p1.10.m10.1.1.1"></eq><apply id="S4.7.1.p1.10.m10.1.1.2.cmml" xref="S4.7.1.p1.10.m10.1.1.2"><times id="S4.7.1.p1.10.m10.1.1.2.1.cmml" xref="S4.7.1.p1.10.m10.1.1.2.1"></times><ci id="S4.7.1.p1.10.m10.1.1.2.2.cmml" xref="S4.7.1.p1.10.m10.1.1.2.2">π¦</ci><apply id="S4.7.1.p1.10.m10.1.1.2.3.cmml" xref="S4.7.1.p1.10.m10.1.1.2.3"><csymbol cd="ambiguous" id="S4.7.1.p1.10.m10.1.1.2.3.1.cmml" xref="S4.7.1.p1.10.m10.1.1.2.3">subscript</csymbol><ci id="S4.7.1.p1.10.m10.1.1.2.3.2.cmml" xref="S4.7.1.p1.10.m10.1.1.2.3.2">π</ci><ci id="S4.7.1.p1.10.m10.1.1.2.3.3.cmml" xref="S4.7.1.p1.10.m10.1.1.2.3.3">π</ci></apply></apply><apply id="S4.7.1.p1.10.m10.1.1.3.cmml" xref="S4.7.1.p1.10.m10.1.1.3"><times id="S4.7.1.p1.10.m10.1.1.3.1.cmml" xref="S4.7.1.p1.10.m10.1.1.3.1"></times><ci id="S4.7.1.p1.10.m10.1.1.3.2.cmml" xref="S4.7.1.p1.10.m10.1.1.3.2">π¦</ci><apply id="S4.7.1.p1.10.m10.1.1.3.3.cmml" xref="S4.7.1.p1.10.m10.1.1.3.3"><csymbol cd="ambiguous" id="S4.7.1.p1.10.m10.1.1.3.3.1.cmml" xref="S4.7.1.p1.10.m10.1.1.3.3">subscript</csymbol><ci id="S4.7.1.p1.10.m10.1.1.3.3.2.cmml" xref="S4.7.1.p1.10.m10.1.1.3.3.2">π</ci><ci id="S4.7.1.p1.10.m10.1.1.3.3.3.cmml" xref="S4.7.1.p1.10.m10.1.1.3.3.3">πΌ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.10.m10.1c">yl_{\xi}=yl_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.10.m10.1d">italic_y italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT = italic_y italic_l start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT</annotation></semantics></math> for each <math alttext="\xi\geq\alpha" class="ltx_Math" display="inline" id="S4.7.1.p1.11.m11.1"><semantics id="S4.7.1.p1.11.m11.1a"><mrow id="S4.7.1.p1.11.m11.1.1" xref="S4.7.1.p1.11.m11.1.1.cmml"><mi id="S4.7.1.p1.11.m11.1.1.2" xref="S4.7.1.p1.11.m11.1.1.2.cmml">ΞΎ</mi><mo id="S4.7.1.p1.11.m11.1.1.1" xref="S4.7.1.p1.11.m11.1.1.1.cmml">β₯</mo><mi id="S4.7.1.p1.11.m11.1.1.3" xref="S4.7.1.p1.11.m11.1.1.3.cmml">Ξ±</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.11.m11.1b"><apply id="S4.7.1.p1.11.m11.1.1.cmml" xref="S4.7.1.p1.11.m11.1.1"><geq id="S4.7.1.p1.11.m11.1.1.1.cmml" xref="S4.7.1.p1.11.m11.1.1.1"></geq><ci id="S4.7.1.p1.11.m11.1.1.2.cmml" xref="S4.7.1.p1.11.m11.1.1.2">π</ci><ci id="S4.7.1.p1.11.m11.1.1.3.cmml" xref="S4.7.1.p1.11.m11.1.1.3">πΌ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.11.m11.1c">\xi\geq\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.11.m11.1d">italic_ΞΎ β₯ italic_Ξ±</annotation></semantics></math>. By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem5" title="Lemma 3.5. β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.5</span></a>, for each open neighborhood <math alttext="V" class="ltx_Math" display="inline" id="S4.7.1.p1.12.m12.1"><semantics id="S4.7.1.p1.12.m12.1a"><mi id="S4.7.1.p1.12.m12.1.1" xref="S4.7.1.p1.12.m12.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.12.m12.1b"><ci id="S4.7.1.p1.12.m12.1.1.cmml" xref="S4.7.1.p1.12.m12.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.12.m12.1c">V</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.12.m12.1d">italic_V</annotation></semantics></math> of <math alttext="z" class="ltx_Math" display="inline" id="S4.7.1.p1.13.m13.1"><semantics id="S4.7.1.p1.13.m13.1a"><mi id="S4.7.1.p1.13.m13.1.1" xref="S4.7.1.p1.13.m13.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.13.m13.1b"><ci id="S4.7.1.p1.13.m13.1.1.cmml" xref="S4.7.1.p1.13.m13.1.1">π§</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.13.m13.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.13.m13.1d">italic_z</annotation></semantics></math> there exists <math alttext="\delta_{V}<\omega_{1}" class="ltx_Math" display="inline" id="S4.7.1.p1.14.m14.1"><semantics id="S4.7.1.p1.14.m14.1a"><mrow id="S4.7.1.p1.14.m14.1.1" xref="S4.7.1.p1.14.m14.1.1.cmml"><msub id="S4.7.1.p1.14.m14.1.1.2" xref="S4.7.1.p1.14.m14.1.1.2.cmml"><mi id="S4.7.1.p1.14.m14.1.1.2.2" xref="S4.7.1.p1.14.m14.1.1.2.2.cmml">Ξ΄</mi><mi id="S4.7.1.p1.14.m14.1.1.2.3" xref="S4.7.1.p1.14.m14.1.1.2.3.cmml">V</mi></msub><mo id="S4.7.1.p1.14.m14.1.1.1" xref="S4.7.1.p1.14.m14.1.1.1.cmml"><</mo><msub id="S4.7.1.p1.14.m14.1.1.3" xref="S4.7.1.p1.14.m14.1.1.3.cmml"><mi id="S4.7.1.p1.14.m14.1.1.3.2" xref="S4.7.1.p1.14.m14.1.1.3.2.cmml">Ο</mi><mn id="S4.7.1.p1.14.m14.1.1.3.3" xref="S4.7.1.p1.14.m14.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.14.m14.1b"><apply id="S4.7.1.p1.14.m14.1.1.cmml" xref="S4.7.1.p1.14.m14.1.1"><lt id="S4.7.1.p1.14.m14.1.1.1.cmml" xref="S4.7.1.p1.14.m14.1.1.1"></lt><apply id="S4.7.1.p1.14.m14.1.1.2.cmml" xref="S4.7.1.p1.14.m14.1.1.2"><csymbol cd="ambiguous" id="S4.7.1.p1.14.m14.1.1.2.1.cmml" xref="S4.7.1.p1.14.m14.1.1.2">subscript</csymbol><ci id="S4.7.1.p1.14.m14.1.1.2.2.cmml" xref="S4.7.1.p1.14.m14.1.1.2.2">πΏ</ci><ci id="S4.7.1.p1.14.m14.1.1.2.3.cmml" xref="S4.7.1.p1.14.m14.1.1.2.3">π</ci></apply><apply id="S4.7.1.p1.14.m14.1.1.3.cmml" xref="S4.7.1.p1.14.m14.1.1.3"><csymbol cd="ambiguous" id="S4.7.1.p1.14.m14.1.1.3.1.cmml" xref="S4.7.1.p1.14.m14.1.1.3">subscript</csymbol><ci id="S4.7.1.p1.14.m14.1.1.3.2.cmml" xref="S4.7.1.p1.14.m14.1.1.3.2">π</ci><cn id="S4.7.1.p1.14.m14.1.1.3.3.cmml" type="integer" xref="S4.7.1.p1.14.m14.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.14.m14.1c">\delta_{V}<\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.14.m14.1d">italic_Ξ΄ start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT < italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="\{l_{\xi}:\xi\geq\delta_{V}\}\subset V" class="ltx_Math" display="inline" id="S4.7.1.p1.15.m15.2"><semantics id="S4.7.1.p1.15.m15.2a"><mrow id="S4.7.1.p1.15.m15.2.2" xref="S4.7.1.p1.15.m15.2.2.cmml"><mrow id="S4.7.1.p1.15.m15.2.2.2.2" xref="S4.7.1.p1.15.m15.2.2.2.3.cmml"><mo id="S4.7.1.p1.15.m15.2.2.2.2.3" stretchy="false" xref="S4.7.1.p1.15.m15.2.2.2.3.1.cmml">{</mo><msub id="S4.7.1.p1.15.m15.1.1.1.1.1" xref="S4.7.1.p1.15.m15.1.1.1.1.1.cmml"><mi id="S4.7.1.p1.15.m15.1.1.1.1.1.2" xref="S4.7.1.p1.15.m15.1.1.1.1.1.2.cmml">l</mi><mi id="S4.7.1.p1.15.m15.1.1.1.1.1.3" xref="S4.7.1.p1.15.m15.1.1.1.1.1.3.cmml">ΞΎ</mi></msub><mo id="S4.7.1.p1.15.m15.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.7.1.p1.15.m15.2.2.2.3.1.cmml">:</mo><mrow id="S4.7.1.p1.15.m15.2.2.2.2.2" xref="S4.7.1.p1.15.m15.2.2.2.2.2.cmml"><mi id="S4.7.1.p1.15.m15.2.2.2.2.2.2" xref="S4.7.1.p1.15.m15.2.2.2.2.2.2.cmml">ΞΎ</mi><mo id="S4.7.1.p1.15.m15.2.2.2.2.2.1" xref="S4.7.1.p1.15.m15.2.2.2.2.2.1.cmml">β₯</mo><msub id="S4.7.1.p1.15.m15.2.2.2.2.2.3" xref="S4.7.1.p1.15.m15.2.2.2.2.2.3.cmml"><mi id="S4.7.1.p1.15.m15.2.2.2.2.2.3.2" xref="S4.7.1.p1.15.m15.2.2.2.2.2.3.2.cmml">Ξ΄</mi><mi id="S4.7.1.p1.15.m15.2.2.2.2.2.3.3" xref="S4.7.1.p1.15.m15.2.2.2.2.2.3.3.cmml">V</mi></msub></mrow><mo id="S4.7.1.p1.15.m15.2.2.2.2.5" stretchy="false" xref="S4.7.1.p1.15.m15.2.2.2.3.1.cmml">}</mo></mrow><mo id="S4.7.1.p1.15.m15.2.2.3" xref="S4.7.1.p1.15.m15.2.2.3.cmml">β</mo><mi id="S4.7.1.p1.15.m15.2.2.4" xref="S4.7.1.p1.15.m15.2.2.4.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.15.m15.2b"><apply id="S4.7.1.p1.15.m15.2.2.cmml" xref="S4.7.1.p1.15.m15.2.2"><subset id="S4.7.1.p1.15.m15.2.2.3.cmml" xref="S4.7.1.p1.15.m15.2.2.3"></subset><apply id="S4.7.1.p1.15.m15.2.2.2.3.cmml" xref="S4.7.1.p1.15.m15.2.2.2.2"><csymbol cd="latexml" id="S4.7.1.p1.15.m15.2.2.2.3.1.cmml" xref="S4.7.1.p1.15.m15.2.2.2.2.3">conditional-set</csymbol><apply id="S4.7.1.p1.15.m15.1.1.1.1.1.cmml" xref="S4.7.1.p1.15.m15.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.7.1.p1.15.m15.1.1.1.1.1.1.cmml" xref="S4.7.1.p1.15.m15.1.1.1.1.1">subscript</csymbol><ci id="S4.7.1.p1.15.m15.1.1.1.1.1.2.cmml" xref="S4.7.1.p1.15.m15.1.1.1.1.1.2">π</ci><ci id="S4.7.1.p1.15.m15.1.1.1.1.1.3.cmml" xref="S4.7.1.p1.15.m15.1.1.1.1.1.3">π</ci></apply><apply id="S4.7.1.p1.15.m15.2.2.2.2.2.cmml" xref="S4.7.1.p1.15.m15.2.2.2.2.2"><geq id="S4.7.1.p1.15.m15.2.2.2.2.2.1.cmml" xref="S4.7.1.p1.15.m15.2.2.2.2.2.1"></geq><ci id="S4.7.1.p1.15.m15.2.2.2.2.2.2.cmml" xref="S4.7.1.p1.15.m15.2.2.2.2.2.2">π</ci><apply id="S4.7.1.p1.15.m15.2.2.2.2.2.3.cmml" xref="S4.7.1.p1.15.m15.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.7.1.p1.15.m15.2.2.2.2.2.3.1.cmml" xref="S4.7.1.p1.15.m15.2.2.2.2.2.3">subscript</csymbol><ci id="S4.7.1.p1.15.m15.2.2.2.2.2.3.2.cmml" xref="S4.7.1.p1.15.m15.2.2.2.2.2.3.2">πΏ</ci><ci id="S4.7.1.p1.15.m15.2.2.2.2.2.3.3.cmml" xref="S4.7.1.p1.15.m15.2.2.2.2.2.3.3">π</ci></apply></apply></apply><ci id="S4.7.1.p1.15.m15.2.2.4.cmml" xref="S4.7.1.p1.15.m15.2.2.4">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.15.m15.2c">\{l_{\xi}:\xi\geq\delta_{V}\}\subset V</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.15.m15.2d">{ italic_l start_POSTSUBSCRIPT italic_ΞΎ end_POSTSUBSCRIPT : italic_ΞΎ β₯ italic_Ξ΄ start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT } β italic_V</annotation></semantics></math>. Then <math alttext="yl_{\alpha}\in yV" class="ltx_Math" display="inline" id="S4.7.1.p1.16.m16.1"><semantics id="S4.7.1.p1.16.m16.1a"><mrow id="S4.7.1.p1.16.m16.1.1" xref="S4.7.1.p1.16.m16.1.1.cmml"><mrow id="S4.7.1.p1.16.m16.1.1.2" xref="S4.7.1.p1.16.m16.1.1.2.cmml"><mi id="S4.7.1.p1.16.m16.1.1.2.2" xref="S4.7.1.p1.16.m16.1.1.2.2.cmml">y</mi><mo id="S4.7.1.p1.16.m16.1.1.2.1" xref="S4.7.1.p1.16.m16.1.1.2.1.cmml">β’</mo><msub id="S4.7.1.p1.16.m16.1.1.2.3" xref="S4.7.1.p1.16.m16.1.1.2.3.cmml"><mi id="S4.7.1.p1.16.m16.1.1.2.3.2" xref="S4.7.1.p1.16.m16.1.1.2.3.2.cmml">l</mi><mi id="S4.7.1.p1.16.m16.1.1.2.3.3" xref="S4.7.1.p1.16.m16.1.1.2.3.3.cmml">Ξ±</mi></msub></mrow><mo id="S4.7.1.p1.16.m16.1.1.1" xref="S4.7.1.p1.16.m16.1.1.1.cmml">β</mo><mrow id="S4.7.1.p1.16.m16.1.1.3" xref="S4.7.1.p1.16.m16.1.1.3.cmml"><mi id="S4.7.1.p1.16.m16.1.1.3.2" xref="S4.7.1.p1.16.m16.1.1.3.2.cmml">y</mi><mo id="S4.7.1.p1.16.m16.1.1.3.1" xref="S4.7.1.p1.16.m16.1.1.3.1.cmml">β’</mo><mi id="S4.7.1.p1.16.m16.1.1.3.3" xref="S4.7.1.p1.16.m16.1.1.3.3.cmml">V</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.16.m16.1b"><apply id="S4.7.1.p1.16.m16.1.1.cmml" xref="S4.7.1.p1.16.m16.1.1"><in id="S4.7.1.p1.16.m16.1.1.1.cmml" xref="S4.7.1.p1.16.m16.1.1.1"></in><apply id="S4.7.1.p1.16.m16.1.1.2.cmml" xref="S4.7.1.p1.16.m16.1.1.2"><times id="S4.7.1.p1.16.m16.1.1.2.1.cmml" xref="S4.7.1.p1.16.m16.1.1.2.1"></times><ci id="S4.7.1.p1.16.m16.1.1.2.2.cmml" xref="S4.7.1.p1.16.m16.1.1.2.2">π¦</ci><apply id="S4.7.1.p1.16.m16.1.1.2.3.cmml" xref="S4.7.1.p1.16.m16.1.1.2.3"><csymbol cd="ambiguous" id="S4.7.1.p1.16.m16.1.1.2.3.1.cmml" xref="S4.7.1.p1.16.m16.1.1.2.3">subscript</csymbol><ci id="S4.7.1.p1.16.m16.1.1.2.3.2.cmml" xref="S4.7.1.p1.16.m16.1.1.2.3.2">π</ci><ci id="S4.7.1.p1.16.m16.1.1.2.3.3.cmml" xref="S4.7.1.p1.16.m16.1.1.2.3.3">πΌ</ci></apply></apply><apply id="S4.7.1.p1.16.m16.1.1.3.cmml" xref="S4.7.1.p1.16.m16.1.1.3"><times id="S4.7.1.p1.16.m16.1.1.3.1.cmml" xref="S4.7.1.p1.16.m16.1.1.3.1"></times><ci id="S4.7.1.p1.16.m16.1.1.3.2.cmml" xref="S4.7.1.p1.16.m16.1.1.3.2">π¦</ci><ci id="S4.7.1.p1.16.m16.1.1.3.3.cmml" xref="S4.7.1.p1.16.m16.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.16.m16.1c">yl_{\alpha}\in yV</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.16.m16.1d">italic_y italic_l start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT β italic_y italic_V</annotation></semantics></math> for each open neighborhood <math alttext="V\subset\beta X" class="ltx_Math" display="inline" id="S4.7.1.p1.17.m17.1"><semantics id="S4.7.1.p1.17.m17.1a"><mrow id="S4.7.1.p1.17.m17.1.1" xref="S4.7.1.p1.17.m17.1.1.cmml"><mi id="S4.7.1.p1.17.m17.1.1.2" xref="S4.7.1.p1.17.m17.1.1.2.cmml">V</mi><mo id="S4.7.1.p1.17.m17.1.1.1" xref="S4.7.1.p1.17.m17.1.1.1.cmml">β</mo><mrow id="S4.7.1.p1.17.m17.1.1.3" xref="S4.7.1.p1.17.m17.1.1.3.cmml"><mi id="S4.7.1.p1.17.m17.1.1.3.2" xref="S4.7.1.p1.17.m17.1.1.3.2.cmml">Ξ²</mi><mo id="S4.7.1.p1.17.m17.1.1.3.1" xref="S4.7.1.p1.17.m17.1.1.3.1.cmml">β’</mo><mi id="S4.7.1.p1.17.m17.1.1.3.3" xref="S4.7.1.p1.17.m17.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.17.m17.1b"><apply id="S4.7.1.p1.17.m17.1.1.cmml" xref="S4.7.1.p1.17.m17.1.1"><subset id="S4.7.1.p1.17.m17.1.1.1.cmml" xref="S4.7.1.p1.17.m17.1.1.1"></subset><ci id="S4.7.1.p1.17.m17.1.1.2.cmml" xref="S4.7.1.p1.17.m17.1.1.2">π</ci><apply id="S4.7.1.p1.17.m17.1.1.3.cmml" xref="S4.7.1.p1.17.m17.1.1.3"><times id="S4.7.1.p1.17.m17.1.1.3.1.cmml" xref="S4.7.1.p1.17.m17.1.1.3.1"></times><ci id="S4.7.1.p1.17.m17.1.1.3.2.cmml" xref="S4.7.1.p1.17.m17.1.1.3.2">π½</ci><ci id="S4.7.1.p1.17.m17.1.1.3.3.cmml" xref="S4.7.1.p1.17.m17.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.17.m17.1c">V\subset\beta X</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.17.m17.1d">italic_V β italic_Ξ² italic_X</annotation></semantics></math> of <math alttext="z" class="ltx_Math" display="inline" id="S4.7.1.p1.18.m18.1"><semantics id="S4.7.1.p1.18.m18.1a"><mi id="S4.7.1.p1.18.m18.1.1" xref="S4.7.1.p1.18.m18.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.18.m18.1b"><ci id="S4.7.1.p1.18.m18.1.1.cmml" xref="S4.7.1.p1.18.m18.1.1">π§</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.18.m18.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.18.m18.1d">italic_z</annotation></semantics></math>. Since <math alttext="\beta X" class="ltx_Math" display="inline" id="S4.7.1.p1.19.m19.1"><semantics id="S4.7.1.p1.19.m19.1a"><mrow id="S4.7.1.p1.19.m19.1.1" xref="S4.7.1.p1.19.m19.1.1.cmml"><mi id="S4.7.1.p1.19.m19.1.1.2" xref="S4.7.1.p1.19.m19.1.1.2.cmml">Ξ²</mi><mo id="S4.7.1.p1.19.m19.1.1.1" xref="S4.7.1.p1.19.m19.1.1.1.cmml">β’</mo><mi id="S4.7.1.p1.19.m19.1.1.3" xref="S4.7.1.p1.19.m19.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.19.m19.1b"><apply id="S4.7.1.p1.19.m19.1.1.cmml" xref="S4.7.1.p1.19.m19.1.1"><times id="S4.7.1.p1.19.m19.1.1.1.cmml" xref="S4.7.1.p1.19.m19.1.1.1"></times><ci id="S4.7.1.p1.19.m19.1.1.2.cmml" xref="S4.7.1.p1.19.m19.1.1.2">π½</ci><ci id="S4.7.1.p1.19.m19.1.1.3.cmml" xref="S4.7.1.p1.19.m19.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.19.m19.1c">\beta X</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.19.m19.1d">italic_Ξ² italic_X</annotation></semantics></math> is a Hausdorff topological semilattice, we get <math alttext="yz=yl_{\alpha}\in X" class="ltx_Math" display="inline" id="S4.7.1.p1.20.m20.1"><semantics id="S4.7.1.p1.20.m20.1a"><mrow id="S4.7.1.p1.20.m20.1.1" xref="S4.7.1.p1.20.m20.1.1.cmml"><mrow id="S4.7.1.p1.20.m20.1.1.2" xref="S4.7.1.p1.20.m20.1.1.2.cmml"><mi id="S4.7.1.p1.20.m20.1.1.2.2" xref="S4.7.1.p1.20.m20.1.1.2.2.cmml">y</mi><mo id="S4.7.1.p1.20.m20.1.1.2.1" xref="S4.7.1.p1.20.m20.1.1.2.1.cmml">β’</mo><mi id="S4.7.1.p1.20.m20.1.1.2.3" xref="S4.7.1.p1.20.m20.1.1.2.3.cmml">z</mi></mrow><mo id="S4.7.1.p1.20.m20.1.1.3" xref="S4.7.1.p1.20.m20.1.1.3.cmml">=</mo><mrow id="S4.7.1.p1.20.m20.1.1.4" xref="S4.7.1.p1.20.m20.1.1.4.cmml"><mi id="S4.7.1.p1.20.m20.1.1.4.2" xref="S4.7.1.p1.20.m20.1.1.4.2.cmml">y</mi><mo id="S4.7.1.p1.20.m20.1.1.4.1" xref="S4.7.1.p1.20.m20.1.1.4.1.cmml">β’</mo><msub id="S4.7.1.p1.20.m20.1.1.4.3" xref="S4.7.1.p1.20.m20.1.1.4.3.cmml"><mi id="S4.7.1.p1.20.m20.1.1.4.3.2" xref="S4.7.1.p1.20.m20.1.1.4.3.2.cmml">l</mi><mi id="S4.7.1.p1.20.m20.1.1.4.3.3" xref="S4.7.1.p1.20.m20.1.1.4.3.3.cmml">Ξ±</mi></msub></mrow><mo id="S4.7.1.p1.20.m20.1.1.5" xref="S4.7.1.p1.20.m20.1.1.5.cmml">β</mo><mi id="S4.7.1.p1.20.m20.1.1.6" xref="S4.7.1.p1.20.m20.1.1.6.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.20.m20.1b"><apply id="S4.7.1.p1.20.m20.1.1.cmml" xref="S4.7.1.p1.20.m20.1.1"><and id="S4.7.1.p1.20.m20.1.1a.cmml" xref="S4.7.1.p1.20.m20.1.1"></and><apply id="S4.7.1.p1.20.m20.1.1b.cmml" xref="S4.7.1.p1.20.m20.1.1"><eq id="S4.7.1.p1.20.m20.1.1.3.cmml" xref="S4.7.1.p1.20.m20.1.1.3"></eq><apply id="S4.7.1.p1.20.m20.1.1.2.cmml" xref="S4.7.1.p1.20.m20.1.1.2"><times id="S4.7.1.p1.20.m20.1.1.2.1.cmml" xref="S4.7.1.p1.20.m20.1.1.2.1"></times><ci id="S4.7.1.p1.20.m20.1.1.2.2.cmml" xref="S4.7.1.p1.20.m20.1.1.2.2">π¦</ci><ci id="S4.7.1.p1.20.m20.1.1.2.3.cmml" xref="S4.7.1.p1.20.m20.1.1.2.3">π§</ci></apply><apply id="S4.7.1.p1.20.m20.1.1.4.cmml" xref="S4.7.1.p1.20.m20.1.1.4"><times id="S4.7.1.p1.20.m20.1.1.4.1.cmml" xref="S4.7.1.p1.20.m20.1.1.4.1"></times><ci id="S4.7.1.p1.20.m20.1.1.4.2.cmml" xref="S4.7.1.p1.20.m20.1.1.4.2">π¦</ci><apply id="S4.7.1.p1.20.m20.1.1.4.3.cmml" xref="S4.7.1.p1.20.m20.1.1.4.3"><csymbol cd="ambiguous" id="S4.7.1.p1.20.m20.1.1.4.3.1.cmml" xref="S4.7.1.p1.20.m20.1.1.4.3">subscript</csymbol><ci id="S4.7.1.p1.20.m20.1.1.4.3.2.cmml" xref="S4.7.1.p1.20.m20.1.1.4.3.2">π</ci><ci id="S4.7.1.p1.20.m20.1.1.4.3.3.cmml" xref="S4.7.1.p1.20.m20.1.1.4.3.3">πΌ</ci></apply></apply></apply><apply id="S4.7.1.p1.20.m20.1.1c.cmml" xref="S4.7.1.p1.20.m20.1.1"><in id="S4.7.1.p1.20.m20.1.1.5.cmml" xref="S4.7.1.p1.20.m20.1.1.5"></in><share href="https://arxiv.org/html/2503.13666v1#S4.7.1.p1.20.m20.1.1.4.cmml" id="S4.7.1.p1.20.m20.1.1d.cmml" xref="S4.7.1.p1.20.m20.1.1"></share><ci id="S4.7.1.p1.20.m20.1.1.6.cmml" xref="S4.7.1.p1.20.m20.1.1.6">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.20.m20.1c">yz=yl_{\alpha}\in X</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.20.m20.1d">italic_y italic_z = italic_y italic_l start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT β italic_X</annotation></semantics></math>. Since <math alttext="X" class="ltx_Math" display="inline" id="S4.7.1.p1.21.m21.1"><semantics id="S4.7.1.p1.21.m21.1a"><mi id="S4.7.1.p1.21.m21.1.1" xref="S4.7.1.p1.21.m21.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.21.m21.1b"><ci id="S4.7.1.p1.21.m21.1.1.cmml" xref="S4.7.1.p1.21.m21.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.21.m21.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.21.m21.1d">italic_X</annotation></semantics></math> is locally compact, there is an open neighborhood <math alttext="W\subseteq X" class="ltx_Math" display="inline" id="S4.7.1.p1.22.m22.1"><semantics id="S4.7.1.p1.22.m22.1a"><mrow id="S4.7.1.p1.22.m22.1.1" xref="S4.7.1.p1.22.m22.1.1.cmml"><mi id="S4.7.1.p1.22.m22.1.1.2" xref="S4.7.1.p1.22.m22.1.1.2.cmml">W</mi><mo id="S4.7.1.p1.22.m22.1.1.1" xref="S4.7.1.p1.22.m22.1.1.1.cmml">β</mo><mi id="S4.7.1.p1.22.m22.1.1.3" xref="S4.7.1.p1.22.m22.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.22.m22.1b"><apply id="S4.7.1.p1.22.m22.1.1.cmml" xref="S4.7.1.p1.22.m22.1.1"><subset id="S4.7.1.p1.22.m22.1.1.1.cmml" xref="S4.7.1.p1.22.m22.1.1.1"></subset><ci id="S4.7.1.p1.22.m22.1.1.2.cmml" xref="S4.7.1.p1.22.m22.1.1.2">π</ci><ci id="S4.7.1.p1.22.m22.1.1.3.cmml" xref="S4.7.1.p1.22.m22.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.22.m22.1c">W\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.22.m22.1d">italic_W β italic_X</annotation></semantics></math> of <math alttext="yz" class="ltx_Math" display="inline" id="S4.7.1.p1.23.m23.1"><semantics id="S4.7.1.p1.23.m23.1a"><mrow id="S4.7.1.p1.23.m23.1.1" xref="S4.7.1.p1.23.m23.1.1.cmml"><mi id="S4.7.1.p1.23.m23.1.1.2" xref="S4.7.1.p1.23.m23.1.1.2.cmml">y</mi><mo id="S4.7.1.p1.23.m23.1.1.1" xref="S4.7.1.p1.23.m23.1.1.1.cmml">β’</mo><mi id="S4.7.1.p1.23.m23.1.1.3" xref="S4.7.1.p1.23.m23.1.1.3.cmml">z</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.23.m23.1b"><apply id="S4.7.1.p1.23.m23.1.1.cmml" xref="S4.7.1.p1.23.m23.1.1"><times id="S4.7.1.p1.23.m23.1.1.1.cmml" xref="S4.7.1.p1.23.m23.1.1.1"></times><ci id="S4.7.1.p1.23.m23.1.1.2.cmml" xref="S4.7.1.p1.23.m23.1.1.2">π¦</ci><ci id="S4.7.1.p1.23.m23.1.1.3.cmml" xref="S4.7.1.p1.23.m23.1.1.3">π§</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.23.m23.1c">yz</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.23.m23.1d">italic_y italic_z</annotation></semantics></math> such that <math alttext="\operatorname{cl}_{X}(W)" class="ltx_Math" display="inline" id="S4.7.1.p1.24.m24.2"><semantics id="S4.7.1.p1.24.m24.2a"><mrow id="S4.7.1.p1.24.m24.2.2.1" xref="S4.7.1.p1.24.m24.2.2.2.cmml"><msub id="S4.7.1.p1.24.m24.2.2.1.1" xref="S4.7.1.p1.24.m24.2.2.1.1.cmml"><mi id="S4.7.1.p1.24.m24.2.2.1.1.2" xref="S4.7.1.p1.24.m24.2.2.1.1.2.cmml">cl</mi><mi id="S4.7.1.p1.24.m24.2.2.1.1.3" xref="S4.7.1.p1.24.m24.2.2.1.1.3.cmml">X</mi></msub><mo id="S4.7.1.p1.24.m24.2.2.1a" xref="S4.7.1.p1.24.m24.2.2.2.cmml">β‘</mo><mrow id="S4.7.1.p1.24.m24.2.2.1.2" xref="S4.7.1.p1.24.m24.2.2.2.cmml"><mo id="S4.7.1.p1.24.m24.2.2.1.2.1" stretchy="false" xref="S4.7.1.p1.24.m24.2.2.2.cmml">(</mo><mi id="S4.7.1.p1.24.m24.1.1" xref="S4.7.1.p1.24.m24.1.1.cmml">W</mi><mo id="S4.7.1.p1.24.m24.2.2.1.2.2" stretchy="false" xref="S4.7.1.p1.24.m24.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.24.m24.2b"><apply id="S4.7.1.p1.24.m24.2.2.2.cmml" xref="S4.7.1.p1.24.m24.2.2.1"><apply id="S4.7.1.p1.24.m24.2.2.1.1.cmml" xref="S4.7.1.p1.24.m24.2.2.1.1"><csymbol cd="ambiguous" id="S4.7.1.p1.24.m24.2.2.1.1.1.cmml" xref="S4.7.1.p1.24.m24.2.2.1.1">subscript</csymbol><ci id="S4.7.1.p1.24.m24.2.2.1.1.2.cmml" xref="S4.7.1.p1.24.m24.2.2.1.1.2">cl</ci><ci id="S4.7.1.p1.24.m24.2.2.1.1.3.cmml" xref="S4.7.1.p1.24.m24.2.2.1.1.3">π</ci></apply><ci id="S4.7.1.p1.24.m24.1.1.cmml" xref="S4.7.1.p1.24.m24.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.24.m24.2c">\operatorname{cl}_{X}(W)</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.24.m24.2d">roman_cl start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_W )</annotation></semantics></math> is compact. The continuity of the semilattice operation in <math alttext="\beta X" class="ltx_Math" display="inline" id="S4.7.1.p1.25.m25.1"><semantics id="S4.7.1.p1.25.m25.1a"><mrow id="S4.7.1.p1.25.m25.1.1" xref="S4.7.1.p1.25.m25.1.1.cmml"><mi id="S4.7.1.p1.25.m25.1.1.2" xref="S4.7.1.p1.25.m25.1.1.2.cmml">Ξ²</mi><mo id="S4.7.1.p1.25.m25.1.1.1" xref="S4.7.1.p1.25.m25.1.1.1.cmml">β’</mo><mi id="S4.7.1.p1.25.m25.1.1.3" xref="S4.7.1.p1.25.m25.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.25.m25.1b"><apply id="S4.7.1.p1.25.m25.1.1.cmml" xref="S4.7.1.p1.25.m25.1.1"><times id="S4.7.1.p1.25.m25.1.1.1.cmml" xref="S4.7.1.p1.25.m25.1.1.1"></times><ci id="S4.7.1.p1.25.m25.1.1.2.cmml" xref="S4.7.1.p1.25.m25.1.1.2">π½</ci><ci id="S4.7.1.p1.25.m25.1.1.3.cmml" xref="S4.7.1.p1.25.m25.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.25.m25.1c">\beta X</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.25.m25.1d">italic_Ξ² italic_X</annotation></semantics></math> yields open neighborhoods <math alttext="U" class="ltx_Math" display="inline" id="S4.7.1.p1.26.m26.1"><semantics id="S4.7.1.p1.26.m26.1a"><mi id="S4.7.1.p1.26.m26.1.1" xref="S4.7.1.p1.26.m26.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.26.m26.1b"><ci id="S4.7.1.p1.26.m26.1.1.cmml" xref="S4.7.1.p1.26.m26.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.26.m26.1c">U</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.26.m26.1d">italic_U</annotation></semantics></math> of <math alttext="y" class="ltx_Math" display="inline" id="S4.7.1.p1.27.m27.1"><semantics id="S4.7.1.p1.27.m27.1a"><mi id="S4.7.1.p1.27.m27.1.1" xref="S4.7.1.p1.27.m27.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.27.m27.1b"><ci id="S4.7.1.p1.27.m27.1.1.cmml" xref="S4.7.1.p1.27.m27.1.1">π¦</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.27.m27.1c">y</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.27.m27.1d">italic_y</annotation></semantics></math> and <math alttext="V" class="ltx_Math" display="inline" id="S4.7.1.p1.28.m28.1"><semantics id="S4.7.1.p1.28.m28.1a"><mi id="S4.7.1.p1.28.m28.1.1" xref="S4.7.1.p1.28.m28.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.28.m28.1b"><ci id="S4.7.1.p1.28.m28.1.1.cmml" xref="S4.7.1.p1.28.m28.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.28.m28.1c">V</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.28.m28.1d">italic_V</annotation></semantics></math> of <math alttext="z" class="ltx_Math" display="inline" id="S4.7.1.p1.29.m29.1"><semantics id="S4.7.1.p1.29.m29.1a"><mi id="S4.7.1.p1.29.m29.1.1" xref="S4.7.1.p1.29.m29.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.29.m29.1b"><ci id="S4.7.1.p1.29.m29.1.1.cmml" xref="S4.7.1.p1.29.m29.1.1">π§</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.29.m29.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.29.m29.1d">italic_z</annotation></semantics></math> such that <math alttext="UV\subseteq W" class="ltx_Math" display="inline" id="S4.7.1.p1.30.m30.1"><semantics id="S4.7.1.p1.30.m30.1a"><mrow id="S4.7.1.p1.30.m30.1.1" xref="S4.7.1.p1.30.m30.1.1.cmml"><mrow id="S4.7.1.p1.30.m30.1.1.2" xref="S4.7.1.p1.30.m30.1.1.2.cmml"><mi id="S4.7.1.p1.30.m30.1.1.2.2" xref="S4.7.1.p1.30.m30.1.1.2.2.cmml">U</mi><mo id="S4.7.1.p1.30.m30.1.1.2.1" xref="S4.7.1.p1.30.m30.1.1.2.1.cmml">β’</mo><mi id="S4.7.1.p1.30.m30.1.1.2.3" xref="S4.7.1.p1.30.m30.1.1.2.3.cmml">V</mi></mrow><mo id="S4.7.1.p1.30.m30.1.1.1" xref="S4.7.1.p1.30.m30.1.1.1.cmml">β</mo><mi id="S4.7.1.p1.30.m30.1.1.3" xref="S4.7.1.p1.30.m30.1.1.3.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.30.m30.1b"><apply id="S4.7.1.p1.30.m30.1.1.cmml" xref="S4.7.1.p1.30.m30.1.1"><subset id="S4.7.1.p1.30.m30.1.1.1.cmml" xref="S4.7.1.p1.30.m30.1.1.1"></subset><apply id="S4.7.1.p1.30.m30.1.1.2.cmml" xref="S4.7.1.p1.30.m30.1.1.2"><times id="S4.7.1.p1.30.m30.1.1.2.1.cmml" xref="S4.7.1.p1.30.m30.1.1.2.1"></times><ci id="S4.7.1.p1.30.m30.1.1.2.2.cmml" xref="S4.7.1.p1.30.m30.1.1.2.2">π</ci><ci id="S4.7.1.p1.30.m30.1.1.2.3.cmml" xref="S4.7.1.p1.30.m30.1.1.2.3">π</ci></apply><ci id="S4.7.1.p1.30.m30.1.1.3.cmml" xref="S4.7.1.p1.30.m30.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.30.m30.1c">UV\subseteq W</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.30.m30.1d">italic_U italic_V β italic_W</annotation></semantics></math>. Let <math alttext="L^{\prime}=\{l_{\alpha}:\alpha\geq\delta_{V}\}\subset V" class="ltx_Math" display="inline" id="S4.7.1.p1.31.m31.2"><semantics id="S4.7.1.p1.31.m31.2a"><mrow id="S4.7.1.p1.31.m31.2.2" xref="S4.7.1.p1.31.m31.2.2.cmml"><msup id="S4.7.1.p1.31.m31.2.2.4" xref="S4.7.1.p1.31.m31.2.2.4.cmml"><mi id="S4.7.1.p1.31.m31.2.2.4.2" xref="S4.7.1.p1.31.m31.2.2.4.2.cmml">L</mi><mo id="S4.7.1.p1.31.m31.2.2.4.3" xref="S4.7.1.p1.31.m31.2.2.4.3.cmml">β²</mo></msup><mo id="S4.7.1.p1.31.m31.2.2.5" xref="S4.7.1.p1.31.m31.2.2.5.cmml">=</mo><mrow id="S4.7.1.p1.31.m31.2.2.2.2" xref="S4.7.1.p1.31.m31.2.2.2.3.cmml"><mo id="S4.7.1.p1.31.m31.2.2.2.2.3" stretchy="false" xref="S4.7.1.p1.31.m31.2.2.2.3.1.cmml">{</mo><msub id="S4.7.1.p1.31.m31.1.1.1.1.1" xref="S4.7.1.p1.31.m31.1.1.1.1.1.cmml"><mi id="S4.7.1.p1.31.m31.1.1.1.1.1.2" xref="S4.7.1.p1.31.m31.1.1.1.1.1.2.cmml">l</mi><mi id="S4.7.1.p1.31.m31.1.1.1.1.1.3" xref="S4.7.1.p1.31.m31.1.1.1.1.1.3.cmml">Ξ±</mi></msub><mo id="S4.7.1.p1.31.m31.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.7.1.p1.31.m31.2.2.2.3.1.cmml">:</mo><mrow id="S4.7.1.p1.31.m31.2.2.2.2.2" xref="S4.7.1.p1.31.m31.2.2.2.2.2.cmml"><mi id="S4.7.1.p1.31.m31.2.2.2.2.2.2" xref="S4.7.1.p1.31.m31.2.2.2.2.2.2.cmml">Ξ±</mi><mo id="S4.7.1.p1.31.m31.2.2.2.2.2.1" xref="S4.7.1.p1.31.m31.2.2.2.2.2.1.cmml">β₯</mo><msub id="S4.7.1.p1.31.m31.2.2.2.2.2.3" xref="S4.7.1.p1.31.m31.2.2.2.2.2.3.cmml"><mi id="S4.7.1.p1.31.m31.2.2.2.2.2.3.2" xref="S4.7.1.p1.31.m31.2.2.2.2.2.3.2.cmml">Ξ΄</mi><mi id="S4.7.1.p1.31.m31.2.2.2.2.2.3.3" xref="S4.7.1.p1.31.m31.2.2.2.2.2.3.3.cmml">V</mi></msub></mrow><mo id="S4.7.1.p1.31.m31.2.2.2.2.5" stretchy="false" xref="S4.7.1.p1.31.m31.2.2.2.3.1.cmml">}</mo></mrow><mo id="S4.7.1.p1.31.m31.2.2.6" xref="S4.7.1.p1.31.m31.2.2.6.cmml">β</mo><mi id="S4.7.1.p1.31.m31.2.2.7" xref="S4.7.1.p1.31.m31.2.2.7.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.31.m31.2b"><apply id="S4.7.1.p1.31.m31.2.2.cmml" xref="S4.7.1.p1.31.m31.2.2"><and id="S4.7.1.p1.31.m31.2.2a.cmml" xref="S4.7.1.p1.31.m31.2.2"></and><apply id="S4.7.1.p1.31.m31.2.2b.cmml" xref="S4.7.1.p1.31.m31.2.2"><eq id="S4.7.1.p1.31.m31.2.2.5.cmml" xref="S4.7.1.p1.31.m31.2.2.5"></eq><apply id="S4.7.1.p1.31.m31.2.2.4.cmml" xref="S4.7.1.p1.31.m31.2.2.4"><csymbol cd="ambiguous" id="S4.7.1.p1.31.m31.2.2.4.1.cmml" xref="S4.7.1.p1.31.m31.2.2.4">superscript</csymbol><ci id="S4.7.1.p1.31.m31.2.2.4.2.cmml" xref="S4.7.1.p1.31.m31.2.2.4.2">πΏ</ci><ci id="S4.7.1.p1.31.m31.2.2.4.3.cmml" xref="S4.7.1.p1.31.m31.2.2.4.3">β²</ci></apply><apply id="S4.7.1.p1.31.m31.2.2.2.3.cmml" xref="S4.7.1.p1.31.m31.2.2.2.2"><csymbol cd="latexml" id="S4.7.1.p1.31.m31.2.2.2.3.1.cmml" xref="S4.7.1.p1.31.m31.2.2.2.2.3">conditional-set</csymbol><apply id="S4.7.1.p1.31.m31.1.1.1.1.1.cmml" xref="S4.7.1.p1.31.m31.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.7.1.p1.31.m31.1.1.1.1.1.1.cmml" xref="S4.7.1.p1.31.m31.1.1.1.1.1">subscript</csymbol><ci id="S4.7.1.p1.31.m31.1.1.1.1.1.2.cmml" xref="S4.7.1.p1.31.m31.1.1.1.1.1.2">π</ci><ci id="S4.7.1.p1.31.m31.1.1.1.1.1.3.cmml" xref="S4.7.1.p1.31.m31.1.1.1.1.1.3">πΌ</ci></apply><apply id="S4.7.1.p1.31.m31.2.2.2.2.2.cmml" xref="S4.7.1.p1.31.m31.2.2.2.2.2"><geq id="S4.7.1.p1.31.m31.2.2.2.2.2.1.cmml" xref="S4.7.1.p1.31.m31.2.2.2.2.2.1"></geq><ci id="S4.7.1.p1.31.m31.2.2.2.2.2.2.cmml" xref="S4.7.1.p1.31.m31.2.2.2.2.2.2">πΌ</ci><apply id="S4.7.1.p1.31.m31.2.2.2.2.2.3.cmml" xref="S4.7.1.p1.31.m31.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.7.1.p1.31.m31.2.2.2.2.2.3.1.cmml" xref="S4.7.1.p1.31.m31.2.2.2.2.2.3">subscript</csymbol><ci id="S4.7.1.p1.31.m31.2.2.2.2.2.3.2.cmml" xref="S4.7.1.p1.31.m31.2.2.2.2.2.3.2">πΏ</ci><ci id="S4.7.1.p1.31.m31.2.2.2.2.2.3.3.cmml" xref="S4.7.1.p1.31.m31.2.2.2.2.2.3.3">π</ci></apply></apply></apply></apply><apply id="S4.7.1.p1.31.m31.2.2c.cmml" xref="S4.7.1.p1.31.m31.2.2"><subset id="S4.7.1.p1.31.m31.2.2.6.cmml" xref="S4.7.1.p1.31.m31.2.2.6"></subset><share href="https://arxiv.org/html/2503.13666v1#S4.7.1.p1.31.m31.2.2.2.cmml" id="S4.7.1.p1.31.m31.2.2d.cmml" xref="S4.7.1.p1.31.m31.2.2"></share><ci id="S4.7.1.p1.31.m31.2.2.7.cmml" xref="S4.7.1.p1.31.m31.2.2.7">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.31.m31.2c">L^{\prime}=\{l_{\alpha}:\alpha\geq\delta_{V}\}\subset V</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.31.m31.2d">italic_L start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT = { italic_l start_POSTSUBSCRIPT italic_Ξ± end_POSTSUBSCRIPT : italic_Ξ± β₯ italic_Ξ΄ start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT } β italic_V</annotation></semantics></math>. Note that <math alttext="d_{U}L^{\prime}\subseteq W" class="ltx_Math" display="inline" id="S4.7.1.p1.32.m32.1"><semantics id="S4.7.1.p1.32.m32.1a"><mrow id="S4.7.1.p1.32.m32.1.1" xref="S4.7.1.p1.32.m32.1.1.cmml"><mrow id="S4.7.1.p1.32.m32.1.1.2" xref="S4.7.1.p1.32.m32.1.1.2.cmml"><msub id="S4.7.1.p1.32.m32.1.1.2.2" xref="S4.7.1.p1.32.m32.1.1.2.2.cmml"><mi id="S4.7.1.p1.32.m32.1.1.2.2.2" xref="S4.7.1.p1.32.m32.1.1.2.2.2.cmml">d</mi><mi id="S4.7.1.p1.32.m32.1.1.2.2.3" xref="S4.7.1.p1.32.m32.1.1.2.2.3.cmml">U</mi></msub><mo id="S4.7.1.p1.32.m32.1.1.2.1" xref="S4.7.1.p1.32.m32.1.1.2.1.cmml">β’</mo><msup id="S4.7.1.p1.32.m32.1.1.2.3" xref="S4.7.1.p1.32.m32.1.1.2.3.cmml"><mi id="S4.7.1.p1.32.m32.1.1.2.3.2" xref="S4.7.1.p1.32.m32.1.1.2.3.2.cmml">L</mi><mo id="S4.7.1.p1.32.m32.1.1.2.3.3" xref="S4.7.1.p1.32.m32.1.1.2.3.3.cmml">β²</mo></msup></mrow><mo id="S4.7.1.p1.32.m32.1.1.1" xref="S4.7.1.p1.32.m32.1.1.1.cmml">β</mo><mi id="S4.7.1.p1.32.m32.1.1.3" xref="S4.7.1.p1.32.m32.1.1.3.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.32.m32.1b"><apply id="S4.7.1.p1.32.m32.1.1.cmml" xref="S4.7.1.p1.32.m32.1.1"><subset id="S4.7.1.p1.32.m32.1.1.1.cmml" xref="S4.7.1.p1.32.m32.1.1.1"></subset><apply id="S4.7.1.p1.32.m32.1.1.2.cmml" xref="S4.7.1.p1.32.m32.1.1.2"><times id="S4.7.1.p1.32.m32.1.1.2.1.cmml" xref="S4.7.1.p1.32.m32.1.1.2.1"></times><apply id="S4.7.1.p1.32.m32.1.1.2.2.cmml" xref="S4.7.1.p1.32.m32.1.1.2.2"><csymbol cd="ambiguous" id="S4.7.1.p1.32.m32.1.1.2.2.1.cmml" xref="S4.7.1.p1.32.m32.1.1.2.2">subscript</csymbol><ci id="S4.7.1.p1.32.m32.1.1.2.2.2.cmml" xref="S4.7.1.p1.32.m32.1.1.2.2.2">π</ci><ci id="S4.7.1.p1.32.m32.1.1.2.2.3.cmml" xref="S4.7.1.p1.32.m32.1.1.2.2.3">π</ci></apply><apply id="S4.7.1.p1.32.m32.1.1.2.3.cmml" xref="S4.7.1.p1.32.m32.1.1.2.3"><csymbol cd="ambiguous" id="S4.7.1.p1.32.m32.1.1.2.3.1.cmml" xref="S4.7.1.p1.32.m32.1.1.2.3">superscript</csymbol><ci id="S4.7.1.p1.32.m32.1.1.2.3.2.cmml" xref="S4.7.1.p1.32.m32.1.1.2.3.2">πΏ</ci><ci id="S4.7.1.p1.32.m32.1.1.2.3.3.cmml" xref="S4.7.1.p1.32.m32.1.1.2.3.3">β²</ci></apply></apply><ci id="S4.7.1.p1.32.m32.1.1.3.cmml" xref="S4.7.1.p1.32.m32.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.32.m32.1c">d_{U}L^{\prime}\subseteq W</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.32.m32.1d">italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT β italic_W</annotation></semantics></math>. By the choice of <math alttext="d_{U}" class="ltx_Math" display="inline" id="S4.7.1.p1.33.m33.1"><semantics id="S4.7.1.p1.33.m33.1a"><msub id="S4.7.1.p1.33.m33.1.1" xref="S4.7.1.p1.33.m33.1.1.cmml"><mi id="S4.7.1.p1.33.m33.1.1.2" xref="S4.7.1.p1.33.m33.1.1.2.cmml">d</mi><mi id="S4.7.1.p1.33.m33.1.1.3" xref="S4.7.1.p1.33.m33.1.1.3.cmml">U</mi></msub><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.33.m33.1b"><apply id="S4.7.1.p1.33.m33.1.1.cmml" xref="S4.7.1.p1.33.m33.1.1"><csymbol cd="ambiguous" id="S4.7.1.p1.33.m33.1.1.1.cmml" xref="S4.7.1.p1.33.m33.1.1">subscript</csymbol><ci id="S4.7.1.p1.33.m33.1.1.2.cmml" xref="S4.7.1.p1.33.m33.1.1.2">π</ci><ci id="S4.7.1.p1.33.m33.1.1.3.cmml" xref="S4.7.1.p1.33.m33.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.33.m33.1c">d_{U}</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.33.m33.1d">italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="|d_{U}L|=\omega_{1}" class="ltx_Math" display="inline" id="S4.7.1.p1.34.m34.1"><semantics id="S4.7.1.p1.34.m34.1a"><mrow id="S4.7.1.p1.34.m34.1.1" xref="S4.7.1.p1.34.m34.1.1.cmml"><mrow id="S4.7.1.p1.34.m34.1.1.1.1" xref="S4.7.1.p1.34.m34.1.1.1.2.cmml"><mo id="S4.7.1.p1.34.m34.1.1.1.1.2" stretchy="false" xref="S4.7.1.p1.34.m34.1.1.1.2.1.cmml">|</mo><mrow id="S4.7.1.p1.34.m34.1.1.1.1.1" xref="S4.7.1.p1.34.m34.1.1.1.1.1.cmml"><msub id="S4.7.1.p1.34.m34.1.1.1.1.1.2" xref="S4.7.1.p1.34.m34.1.1.1.1.1.2.cmml"><mi id="S4.7.1.p1.34.m34.1.1.1.1.1.2.2" xref="S4.7.1.p1.34.m34.1.1.1.1.1.2.2.cmml">d</mi><mi id="S4.7.1.p1.34.m34.1.1.1.1.1.2.3" xref="S4.7.1.p1.34.m34.1.1.1.1.1.2.3.cmml">U</mi></msub><mo id="S4.7.1.p1.34.m34.1.1.1.1.1.1" xref="S4.7.1.p1.34.m34.1.1.1.1.1.1.cmml">β’</mo><mi id="S4.7.1.p1.34.m34.1.1.1.1.1.3" xref="S4.7.1.p1.34.m34.1.1.1.1.1.3.cmml">L</mi></mrow><mo id="S4.7.1.p1.34.m34.1.1.1.1.3" stretchy="false" xref="S4.7.1.p1.34.m34.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.7.1.p1.34.m34.1.1.2" xref="S4.7.1.p1.34.m34.1.1.2.cmml">=</mo><msub id="S4.7.1.p1.34.m34.1.1.3" xref="S4.7.1.p1.34.m34.1.1.3.cmml"><mi id="S4.7.1.p1.34.m34.1.1.3.2" xref="S4.7.1.p1.34.m34.1.1.3.2.cmml">Ο</mi><mn id="S4.7.1.p1.34.m34.1.1.3.3" xref="S4.7.1.p1.34.m34.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.34.m34.1b"><apply id="S4.7.1.p1.34.m34.1.1.cmml" xref="S4.7.1.p1.34.m34.1.1"><eq id="S4.7.1.p1.34.m34.1.1.2.cmml" xref="S4.7.1.p1.34.m34.1.1.2"></eq><apply id="S4.7.1.p1.34.m34.1.1.1.2.cmml" xref="S4.7.1.p1.34.m34.1.1.1.1"><abs id="S4.7.1.p1.34.m34.1.1.1.2.1.cmml" xref="S4.7.1.p1.34.m34.1.1.1.1.2"></abs><apply id="S4.7.1.p1.34.m34.1.1.1.1.1.cmml" xref="S4.7.1.p1.34.m34.1.1.1.1.1"><times id="S4.7.1.p1.34.m34.1.1.1.1.1.1.cmml" xref="S4.7.1.p1.34.m34.1.1.1.1.1.1"></times><apply id="S4.7.1.p1.34.m34.1.1.1.1.1.2.cmml" xref="S4.7.1.p1.34.m34.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.7.1.p1.34.m34.1.1.1.1.1.2.1.cmml" xref="S4.7.1.p1.34.m34.1.1.1.1.1.2">subscript</csymbol><ci id="S4.7.1.p1.34.m34.1.1.1.1.1.2.2.cmml" xref="S4.7.1.p1.34.m34.1.1.1.1.1.2.2">π</ci><ci id="S4.7.1.p1.34.m34.1.1.1.1.1.2.3.cmml" xref="S4.7.1.p1.34.m34.1.1.1.1.1.2.3">π</ci></apply><ci id="S4.7.1.p1.34.m34.1.1.1.1.1.3.cmml" xref="S4.7.1.p1.34.m34.1.1.1.1.1.3">πΏ</ci></apply></apply><apply id="S4.7.1.p1.34.m34.1.1.3.cmml" xref="S4.7.1.p1.34.m34.1.1.3"><csymbol cd="ambiguous" id="S4.7.1.p1.34.m34.1.1.3.1.cmml" xref="S4.7.1.p1.34.m34.1.1.3">subscript</csymbol><ci id="S4.7.1.p1.34.m34.1.1.3.2.cmml" xref="S4.7.1.p1.34.m34.1.1.3.2">π</ci><cn id="S4.7.1.p1.34.m34.1.1.3.3.cmml" type="integer" xref="S4.7.1.p1.34.m34.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.34.m34.1c">|d_{U}L|=\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.34.m34.1d">| italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT italic_L | = italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, which implies <math alttext="|d_{U}L^{\prime}|=\omega_{1}" class="ltx_Math" display="inline" id="S4.7.1.p1.35.m35.1"><semantics id="S4.7.1.p1.35.m35.1a"><mrow id="S4.7.1.p1.35.m35.1.1" xref="S4.7.1.p1.35.m35.1.1.cmml"><mrow id="S4.7.1.p1.35.m35.1.1.1.1" xref="S4.7.1.p1.35.m35.1.1.1.2.cmml"><mo id="S4.7.1.p1.35.m35.1.1.1.1.2" stretchy="false" xref="S4.7.1.p1.35.m35.1.1.1.2.1.cmml">|</mo><mrow id="S4.7.1.p1.35.m35.1.1.1.1.1" xref="S4.7.1.p1.35.m35.1.1.1.1.1.cmml"><msub id="S4.7.1.p1.35.m35.1.1.1.1.1.2" xref="S4.7.1.p1.35.m35.1.1.1.1.1.2.cmml"><mi id="S4.7.1.p1.35.m35.1.1.1.1.1.2.2" xref="S4.7.1.p1.35.m35.1.1.1.1.1.2.2.cmml">d</mi><mi id="S4.7.1.p1.35.m35.1.1.1.1.1.2.3" xref="S4.7.1.p1.35.m35.1.1.1.1.1.2.3.cmml">U</mi></msub><mo id="S4.7.1.p1.35.m35.1.1.1.1.1.1" xref="S4.7.1.p1.35.m35.1.1.1.1.1.1.cmml">β’</mo><msup id="S4.7.1.p1.35.m35.1.1.1.1.1.3" xref="S4.7.1.p1.35.m35.1.1.1.1.1.3.cmml"><mi id="S4.7.1.p1.35.m35.1.1.1.1.1.3.2" xref="S4.7.1.p1.35.m35.1.1.1.1.1.3.2.cmml">L</mi><mo id="S4.7.1.p1.35.m35.1.1.1.1.1.3.3" xref="S4.7.1.p1.35.m35.1.1.1.1.1.3.3.cmml">β²</mo></msup></mrow><mo id="S4.7.1.p1.35.m35.1.1.1.1.3" stretchy="false" xref="S4.7.1.p1.35.m35.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.7.1.p1.35.m35.1.1.2" xref="S4.7.1.p1.35.m35.1.1.2.cmml">=</mo><msub id="S4.7.1.p1.35.m35.1.1.3" xref="S4.7.1.p1.35.m35.1.1.3.cmml"><mi id="S4.7.1.p1.35.m35.1.1.3.2" xref="S4.7.1.p1.35.m35.1.1.3.2.cmml">Ο</mi><mn id="S4.7.1.p1.35.m35.1.1.3.3" xref="S4.7.1.p1.35.m35.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.35.m35.1b"><apply id="S4.7.1.p1.35.m35.1.1.cmml" xref="S4.7.1.p1.35.m35.1.1"><eq id="S4.7.1.p1.35.m35.1.1.2.cmml" xref="S4.7.1.p1.35.m35.1.1.2"></eq><apply id="S4.7.1.p1.35.m35.1.1.1.2.cmml" xref="S4.7.1.p1.35.m35.1.1.1.1"><abs id="S4.7.1.p1.35.m35.1.1.1.2.1.cmml" xref="S4.7.1.p1.35.m35.1.1.1.1.2"></abs><apply id="S4.7.1.p1.35.m35.1.1.1.1.1.cmml" xref="S4.7.1.p1.35.m35.1.1.1.1.1"><times id="S4.7.1.p1.35.m35.1.1.1.1.1.1.cmml" xref="S4.7.1.p1.35.m35.1.1.1.1.1.1"></times><apply id="S4.7.1.p1.35.m35.1.1.1.1.1.2.cmml" xref="S4.7.1.p1.35.m35.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.7.1.p1.35.m35.1.1.1.1.1.2.1.cmml" xref="S4.7.1.p1.35.m35.1.1.1.1.1.2">subscript</csymbol><ci id="S4.7.1.p1.35.m35.1.1.1.1.1.2.2.cmml" xref="S4.7.1.p1.35.m35.1.1.1.1.1.2.2">π</ci><ci id="S4.7.1.p1.35.m35.1.1.1.1.1.2.3.cmml" xref="S4.7.1.p1.35.m35.1.1.1.1.1.2.3">π</ci></apply><apply id="S4.7.1.p1.35.m35.1.1.1.1.1.3.cmml" xref="S4.7.1.p1.35.m35.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.7.1.p1.35.m35.1.1.1.1.1.3.1.cmml" xref="S4.7.1.p1.35.m35.1.1.1.1.1.3">superscript</csymbol><ci id="S4.7.1.p1.35.m35.1.1.1.1.1.3.2.cmml" xref="S4.7.1.p1.35.m35.1.1.1.1.1.3.2">πΏ</ci><ci id="S4.7.1.p1.35.m35.1.1.1.1.1.3.3.cmml" xref="S4.7.1.p1.35.m35.1.1.1.1.1.3.3">β²</ci></apply></apply></apply><apply id="S4.7.1.p1.35.m35.1.1.3.cmml" xref="S4.7.1.p1.35.m35.1.1.3"><csymbol cd="ambiguous" id="S4.7.1.p1.35.m35.1.1.3.1.cmml" xref="S4.7.1.p1.35.m35.1.1.3">subscript</csymbol><ci id="S4.7.1.p1.35.m35.1.1.3.2.cmml" xref="S4.7.1.p1.35.m35.1.1.3.2">π</ci><cn id="S4.7.1.p1.35.m35.1.1.3.3.cmml" type="integer" xref="S4.7.1.p1.35.m35.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.35.m35.1c">|d_{U}L^{\prime}|=\omega_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.35.m35.1d">| italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT | = italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. As left shifts in <math alttext="\beta X" class="ltx_Math" display="inline" id="S4.7.1.p1.36.m36.1"><semantics id="S4.7.1.p1.36.m36.1a"><mrow id="S4.7.1.p1.36.m36.1.1" xref="S4.7.1.p1.36.m36.1.1.cmml"><mi id="S4.7.1.p1.36.m36.1.1.2" xref="S4.7.1.p1.36.m36.1.1.2.cmml">Ξ²</mi><mo id="S4.7.1.p1.36.m36.1.1.1" xref="S4.7.1.p1.36.m36.1.1.1.cmml">β’</mo><mi id="S4.7.1.p1.36.m36.1.1.3" xref="S4.7.1.p1.36.m36.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.36.m36.1b"><apply id="S4.7.1.p1.36.m36.1.1.cmml" xref="S4.7.1.p1.36.m36.1.1"><times id="S4.7.1.p1.36.m36.1.1.1.cmml" xref="S4.7.1.p1.36.m36.1.1.1"></times><ci id="S4.7.1.p1.36.m36.1.1.2.cmml" xref="S4.7.1.p1.36.m36.1.1.2">π½</ci><ci id="S4.7.1.p1.36.m36.1.1.3.cmml" xref="S4.7.1.p1.36.m36.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.36.m36.1c">\beta X</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.36.m36.1d">italic_Ξ² italic_X</annotation></semantics></math> are homomorphisms and <math alttext="L^{\prime}" class="ltx_Math" display="inline" id="S4.7.1.p1.37.m37.1"><semantics id="S4.7.1.p1.37.m37.1a"><msup id="S4.7.1.p1.37.m37.1.1" xref="S4.7.1.p1.37.m37.1.1.cmml"><mi id="S4.7.1.p1.37.m37.1.1.2" xref="S4.7.1.p1.37.m37.1.1.2.cmml">L</mi><mo id="S4.7.1.p1.37.m37.1.1.3" xref="S4.7.1.p1.37.m37.1.1.3.cmml">β²</mo></msup><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.37.m37.1b"><apply id="S4.7.1.p1.37.m37.1.1.cmml" xref="S4.7.1.p1.37.m37.1.1"><csymbol cd="ambiguous" id="S4.7.1.p1.37.m37.1.1.1.cmml" xref="S4.7.1.p1.37.m37.1.1">superscript</csymbol><ci id="S4.7.1.p1.37.m37.1.1.2.cmml" xref="S4.7.1.p1.37.m37.1.1.2">πΏ</ci><ci id="S4.7.1.p1.37.m37.1.1.3.cmml" xref="S4.7.1.p1.37.m37.1.1.3">β²</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.37.m37.1c">L^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.37.m37.1d">italic_L start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT</annotation></semantics></math> is order isomorphic to <math alttext="(\omega_{1},\max)" class="ltx_Math" display="inline" id="S4.7.1.p1.38.m38.2"><semantics id="S4.7.1.p1.38.m38.2a"><mrow id="S4.7.1.p1.38.m38.2.2.1" xref="S4.7.1.p1.38.m38.2.2.2.cmml"><mo id="S4.7.1.p1.38.m38.2.2.1.2" stretchy="false" xref="S4.7.1.p1.38.m38.2.2.2.cmml">(</mo><msub id="S4.7.1.p1.38.m38.2.2.1.1" xref="S4.7.1.p1.38.m38.2.2.1.1.cmml"><mi id="S4.7.1.p1.38.m38.2.2.1.1.2" xref="S4.7.1.p1.38.m38.2.2.1.1.2.cmml">Ο</mi><mn id="S4.7.1.p1.38.m38.2.2.1.1.3" xref="S4.7.1.p1.38.m38.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.7.1.p1.38.m38.2.2.1.3" xref="S4.7.1.p1.38.m38.2.2.2.cmml">,</mo><mi id="S4.7.1.p1.38.m38.1.1" xref="S4.7.1.p1.38.m38.1.1.cmml">max</mi><mo id="S4.7.1.p1.38.m38.2.2.1.4" stretchy="false" xref="S4.7.1.p1.38.m38.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.38.m38.2b"><interval closure="open" id="S4.7.1.p1.38.m38.2.2.2.cmml" xref="S4.7.1.p1.38.m38.2.2.1"><apply id="S4.7.1.p1.38.m38.2.2.1.1.cmml" xref="S4.7.1.p1.38.m38.2.2.1.1"><csymbol cd="ambiguous" id="S4.7.1.p1.38.m38.2.2.1.1.1.cmml" xref="S4.7.1.p1.38.m38.2.2.1.1">subscript</csymbol><ci id="S4.7.1.p1.38.m38.2.2.1.1.2.cmml" xref="S4.7.1.p1.38.m38.2.2.1.1.2">π</ci><cn id="S4.7.1.p1.38.m38.2.2.1.1.3.cmml" type="integer" xref="S4.7.1.p1.38.m38.2.2.1.1.3">1</cn></apply><max id="S4.7.1.p1.38.m38.1.1.cmml" xref="S4.7.1.p1.38.m38.1.1"></max></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.38.m38.2c">(\omega_{1},\max)</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.38.m38.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_max )</annotation></semantics></math>, Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem7" title="Lemma 3.7. β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.7</span></a>(2) implies that <math alttext="d_{U}L^{\prime}" class="ltx_Math" display="inline" id="S4.7.1.p1.39.m39.1"><semantics id="S4.7.1.p1.39.m39.1a"><mrow id="S4.7.1.p1.39.m39.1.1" xref="S4.7.1.p1.39.m39.1.1.cmml"><msub id="S4.7.1.p1.39.m39.1.1.2" xref="S4.7.1.p1.39.m39.1.1.2.cmml"><mi id="S4.7.1.p1.39.m39.1.1.2.2" xref="S4.7.1.p1.39.m39.1.1.2.2.cmml">d</mi><mi id="S4.7.1.p1.39.m39.1.1.2.3" xref="S4.7.1.p1.39.m39.1.1.2.3.cmml">U</mi></msub><mo id="S4.7.1.p1.39.m39.1.1.1" xref="S4.7.1.p1.39.m39.1.1.1.cmml">β’</mo><msup id="S4.7.1.p1.39.m39.1.1.3" xref="S4.7.1.p1.39.m39.1.1.3.cmml"><mi id="S4.7.1.p1.39.m39.1.1.3.2" xref="S4.7.1.p1.39.m39.1.1.3.2.cmml">L</mi><mo id="S4.7.1.p1.39.m39.1.1.3.3" xref="S4.7.1.p1.39.m39.1.1.3.3.cmml">β²</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.39.m39.1b"><apply id="S4.7.1.p1.39.m39.1.1.cmml" xref="S4.7.1.p1.39.m39.1.1"><times id="S4.7.1.p1.39.m39.1.1.1.cmml" xref="S4.7.1.p1.39.m39.1.1.1"></times><apply id="S4.7.1.p1.39.m39.1.1.2.cmml" xref="S4.7.1.p1.39.m39.1.1.2"><csymbol cd="ambiguous" id="S4.7.1.p1.39.m39.1.1.2.1.cmml" xref="S4.7.1.p1.39.m39.1.1.2">subscript</csymbol><ci id="S4.7.1.p1.39.m39.1.1.2.2.cmml" xref="S4.7.1.p1.39.m39.1.1.2.2">π</ci><ci id="S4.7.1.p1.39.m39.1.1.2.3.cmml" xref="S4.7.1.p1.39.m39.1.1.2.3">π</ci></apply><apply id="S4.7.1.p1.39.m39.1.1.3.cmml" xref="S4.7.1.p1.39.m39.1.1.3"><csymbol cd="ambiguous" id="S4.7.1.p1.39.m39.1.1.3.1.cmml" xref="S4.7.1.p1.39.m39.1.1.3">superscript</csymbol><ci id="S4.7.1.p1.39.m39.1.1.3.2.cmml" xref="S4.7.1.p1.39.m39.1.1.3.2">πΏ</ci><ci id="S4.7.1.p1.39.m39.1.1.3.3.cmml" xref="S4.7.1.p1.39.m39.1.1.3.3">β²</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.39.m39.1c">d_{U}L^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.39.m39.1d">italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT</annotation></semantics></math> is also order isomorphic to <math alttext="(\omega_{1},\max)" class="ltx_Math" display="inline" id="S4.7.1.p1.40.m40.2"><semantics id="S4.7.1.p1.40.m40.2a"><mrow id="S4.7.1.p1.40.m40.2.2.1" xref="S4.7.1.p1.40.m40.2.2.2.cmml"><mo id="S4.7.1.p1.40.m40.2.2.1.2" stretchy="false" xref="S4.7.1.p1.40.m40.2.2.2.cmml">(</mo><msub id="S4.7.1.p1.40.m40.2.2.1.1" xref="S4.7.1.p1.40.m40.2.2.1.1.cmml"><mi id="S4.7.1.p1.40.m40.2.2.1.1.2" xref="S4.7.1.p1.40.m40.2.2.1.1.2.cmml">Ο</mi><mn id="S4.7.1.p1.40.m40.2.2.1.1.3" xref="S4.7.1.p1.40.m40.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.7.1.p1.40.m40.2.2.1.3" xref="S4.7.1.p1.40.m40.2.2.2.cmml">,</mo><mi id="S4.7.1.p1.40.m40.1.1" xref="S4.7.1.p1.40.m40.1.1.cmml">max</mi><mo id="S4.7.1.p1.40.m40.2.2.1.4" stretchy="false" xref="S4.7.1.p1.40.m40.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.40.m40.2b"><interval closure="open" id="S4.7.1.p1.40.m40.2.2.2.cmml" xref="S4.7.1.p1.40.m40.2.2.1"><apply id="S4.7.1.p1.40.m40.2.2.1.1.cmml" xref="S4.7.1.p1.40.m40.2.2.1.1"><csymbol cd="ambiguous" id="S4.7.1.p1.40.m40.2.2.1.1.1.cmml" xref="S4.7.1.p1.40.m40.2.2.1.1">subscript</csymbol><ci id="S4.7.1.p1.40.m40.2.2.1.1.2.cmml" xref="S4.7.1.p1.40.m40.2.2.1.1.2">π</ci><cn id="S4.7.1.p1.40.m40.2.2.1.1.3.cmml" type="integer" xref="S4.7.1.p1.40.m40.2.2.1.1.3">1</cn></apply><max id="S4.7.1.p1.40.m40.1.1.cmml" xref="S4.7.1.p1.40.m40.1.1"></max></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.40.m40.2c">(\omega_{1},\max)</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.40.m40.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_max )</annotation></semantics></math>. Since <math alttext="\operatorname{cl}_{X}(W)" class="ltx_Math" display="inline" id="S4.7.1.p1.41.m41.2"><semantics id="S4.7.1.p1.41.m41.2a"><mrow id="S4.7.1.p1.41.m41.2.2.1" xref="S4.7.1.p1.41.m41.2.2.2.cmml"><msub id="S4.7.1.p1.41.m41.2.2.1.1" xref="S4.7.1.p1.41.m41.2.2.1.1.cmml"><mi id="S4.7.1.p1.41.m41.2.2.1.1.2" xref="S4.7.1.p1.41.m41.2.2.1.1.2.cmml">cl</mi><mi id="S4.7.1.p1.41.m41.2.2.1.1.3" xref="S4.7.1.p1.41.m41.2.2.1.1.3.cmml">X</mi></msub><mo id="S4.7.1.p1.41.m41.2.2.1a" xref="S4.7.1.p1.41.m41.2.2.2.cmml">β‘</mo><mrow id="S4.7.1.p1.41.m41.2.2.1.2" xref="S4.7.1.p1.41.m41.2.2.2.cmml"><mo id="S4.7.1.p1.41.m41.2.2.1.2.1" stretchy="false" xref="S4.7.1.p1.41.m41.2.2.2.cmml">(</mo><mi id="S4.7.1.p1.41.m41.1.1" xref="S4.7.1.p1.41.m41.1.1.cmml">W</mi><mo id="S4.7.1.p1.41.m41.2.2.1.2.2" stretchy="false" xref="S4.7.1.p1.41.m41.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.7.1.p1.41.m41.2b"><apply id="S4.7.1.p1.41.m41.2.2.2.cmml" xref="S4.7.1.p1.41.m41.2.2.1"><apply id="S4.7.1.p1.41.m41.2.2.1.1.cmml" xref="S4.7.1.p1.41.m41.2.2.1.1"><csymbol cd="ambiguous" id="S4.7.1.p1.41.m41.2.2.1.1.1.cmml" xref="S4.7.1.p1.41.m41.2.2.1.1">subscript</csymbol><ci id="S4.7.1.p1.41.m41.2.2.1.1.2.cmml" xref="S4.7.1.p1.41.m41.2.2.1.1.2">cl</ci><ci id="S4.7.1.p1.41.m41.2.2.1.1.3.cmml" xref="S4.7.1.p1.41.m41.2.2.1.1.3">π</ci></apply><ci id="S4.7.1.p1.41.m41.1.1.cmml" xref="S4.7.1.p1.41.m41.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.41.m41.2c">\operatorname{cl}_{X}(W)</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.41.m41.2d">roman_cl start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_W )</annotation></semantics></math> is compact, the chain <math alttext="\operatorname{cl}_{X}(d_{U}L^{\prime})\subset\operatorname{cl}_{X}(W)" class="ltx_Math" display="inline" id="S4.7.1.p1.42.m42.4"><semantics id="S4.7.1.p1.42.m42.4a"><mrow id="S4.7.1.p1.42.m42.4.4" xref="S4.7.1.p1.42.m42.4.4.cmml"><mrow id="S4.7.1.p1.42.m42.3.3.2.2" xref="S4.7.1.p1.42.m42.3.3.2.3.cmml"><msub id="S4.7.1.p1.42.m42.2.2.1.1.1" xref="S4.7.1.p1.42.m42.2.2.1.1.1.cmml"><mi id="S4.7.1.p1.42.m42.2.2.1.1.1.2" xref="S4.7.1.p1.42.m42.2.2.1.1.1.2.cmml">cl</mi><mi id="S4.7.1.p1.42.m42.2.2.1.1.1.3" xref="S4.7.1.p1.42.m42.2.2.1.1.1.3.cmml">X</mi></msub><mo id="S4.7.1.p1.42.m42.3.3.2.2a" xref="S4.7.1.p1.42.m42.3.3.2.3.cmml">β‘</mo><mrow id="S4.7.1.p1.42.m42.3.3.2.2.2" xref="S4.7.1.p1.42.m42.3.3.2.3.cmml"><mo id="S4.7.1.p1.42.m42.3.3.2.2.2.2" stretchy="false" xref="S4.7.1.p1.42.m42.3.3.2.3.cmml">(</mo><mrow id="S4.7.1.p1.42.m42.3.3.2.2.2.1" xref="S4.7.1.p1.42.m42.3.3.2.2.2.1.cmml"><msub id="S4.7.1.p1.42.m42.3.3.2.2.2.1.2" xref="S4.7.1.p1.42.m42.3.3.2.2.2.1.2.cmml"><mi id="S4.7.1.p1.42.m42.3.3.2.2.2.1.2.2" xref="S4.7.1.p1.42.m42.3.3.2.2.2.1.2.2.cmml">d</mi><mi id="S4.7.1.p1.42.m42.3.3.2.2.2.1.2.3" xref="S4.7.1.p1.42.m42.3.3.2.2.2.1.2.3.cmml">U</mi></msub><mo 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xref="S4.7.1.p1.42.m42.2.2.1.1.1.3">π</ci></apply><apply id="S4.7.1.p1.42.m42.3.3.2.2.2.1.cmml" xref="S4.7.1.p1.42.m42.3.3.2.2.2.1"><times id="S4.7.1.p1.42.m42.3.3.2.2.2.1.1.cmml" xref="S4.7.1.p1.42.m42.3.3.2.2.2.1.1"></times><apply id="S4.7.1.p1.42.m42.3.3.2.2.2.1.2.cmml" xref="S4.7.1.p1.42.m42.3.3.2.2.2.1.2"><csymbol cd="ambiguous" id="S4.7.1.p1.42.m42.3.3.2.2.2.1.2.1.cmml" xref="S4.7.1.p1.42.m42.3.3.2.2.2.1.2">subscript</csymbol><ci id="S4.7.1.p1.42.m42.3.3.2.2.2.1.2.2.cmml" xref="S4.7.1.p1.42.m42.3.3.2.2.2.1.2.2">π</ci><ci id="S4.7.1.p1.42.m42.3.3.2.2.2.1.2.3.cmml" xref="S4.7.1.p1.42.m42.3.3.2.2.2.1.2.3">π</ci></apply><apply id="S4.7.1.p1.42.m42.3.3.2.2.2.1.3.cmml" xref="S4.7.1.p1.42.m42.3.3.2.2.2.1.3"><csymbol cd="ambiguous" id="S4.7.1.p1.42.m42.3.3.2.2.2.1.3.1.cmml" xref="S4.7.1.p1.42.m42.3.3.2.2.2.1.3">superscript</csymbol><ci id="S4.7.1.p1.42.m42.3.3.2.2.2.1.3.2.cmml" xref="S4.7.1.p1.42.m42.3.3.2.2.2.1.3.2">πΏ</ci><ci id="S4.7.1.p1.42.m42.3.3.2.2.2.1.3.3.cmml" xref="S4.7.1.p1.42.m42.3.3.2.2.2.1.3.3">β²</ci></apply></apply></apply><apply id="S4.7.1.p1.42.m42.4.4.3.2.cmml" xref="S4.7.1.p1.42.m42.4.4.3.1"><apply id="S4.7.1.p1.42.m42.4.4.3.1.1.cmml" xref="S4.7.1.p1.42.m42.4.4.3.1.1"><csymbol cd="ambiguous" id="S4.7.1.p1.42.m42.4.4.3.1.1.1.cmml" xref="S4.7.1.p1.42.m42.4.4.3.1.1">subscript</csymbol><ci id="S4.7.1.p1.42.m42.4.4.3.1.1.2.cmml" xref="S4.7.1.p1.42.m42.4.4.3.1.1.2">cl</ci><ci id="S4.7.1.p1.42.m42.4.4.3.1.1.3.cmml" xref="S4.7.1.p1.42.m42.4.4.3.1.1.3">π</ci></apply><ci id="S4.7.1.p1.42.m42.1.1.cmml" xref="S4.7.1.p1.42.m42.1.1">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.7.1.p1.42.m42.4c">\operatorname{cl}_{X}(d_{U}L^{\prime})\subset\operatorname{cl}_{X}(W)</annotation><annotation encoding="application/x-llamapun" id="S4.7.1.p1.42.m42.4d">roman_cl start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT β² end_POSTSUPERSCRIPT ) β roman_cl start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_W )</annotation></semantics></math> is compact, which contradicts Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem6" title="Lemma 3.6. β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.6</span></a>. β</p> </div> </div> <div class="ltx_para" id="S4.8.p2"> <p class="ltx_p" id="S4.8.p2.1">By <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib22" title="">22</a>, Corollary 3.6.5]</cite>, the set <math alttext="\operatorname{cl}_{\beta X}(I_{L})" class="ltx_Math" display="inline" id="S4.8.p2.1.m1.2"><semantics id="S4.8.p2.1.m1.2a"><mrow id="S4.8.p2.1.m1.2.2.2" xref="S4.8.p2.1.m1.2.2.3.cmml"><msub id="S4.8.p2.1.m1.1.1.1.1" xref="S4.8.p2.1.m1.1.1.1.1.cmml"><mi id="S4.8.p2.1.m1.1.1.1.1.2" xref="S4.8.p2.1.m1.1.1.1.1.2.cmml">cl</mi><mrow id="S4.8.p2.1.m1.1.1.1.1.3" xref="S4.8.p2.1.m1.1.1.1.1.3.cmml"><mi id="S4.8.p2.1.m1.1.1.1.1.3.2" xref="S4.8.p2.1.m1.1.1.1.1.3.2.cmml">Ξ²</mi><mo id="S4.8.p2.1.m1.1.1.1.1.3.1" xref="S4.8.p2.1.m1.1.1.1.1.3.1.cmml">β’</mo><mi id="S4.8.p2.1.m1.1.1.1.1.3.3" xref="S4.8.p2.1.m1.1.1.1.1.3.3.cmml">X</mi></mrow></msub><mo id="S4.8.p2.1.m1.2.2.2a" xref="S4.8.p2.1.m1.2.2.3.cmml">β‘</mo><mrow id="S4.8.p2.1.m1.2.2.2.2" xref="S4.8.p2.1.m1.2.2.3.cmml"><mo id="S4.8.p2.1.m1.2.2.2.2.2" stretchy="false" xref="S4.8.p2.1.m1.2.2.3.cmml">(</mo><msub id="S4.8.p2.1.m1.2.2.2.2.1" xref="S4.8.p2.1.m1.2.2.2.2.1.cmml"><mi id="S4.8.p2.1.m1.2.2.2.2.1.2" xref="S4.8.p2.1.m1.2.2.2.2.1.2.cmml">I</mi><mi id="S4.8.p2.1.m1.2.2.2.2.1.3" xref="S4.8.p2.1.m1.2.2.2.2.1.3.cmml">L</mi></msub><mo id="S4.8.p2.1.m1.2.2.2.2.3" stretchy="false" xref="S4.8.p2.1.m1.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.8.p2.1.m1.2b"><apply id="S4.8.p2.1.m1.2.2.3.cmml" xref="S4.8.p2.1.m1.2.2.2"><apply id="S4.8.p2.1.m1.1.1.1.1.cmml" xref="S4.8.p2.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.8.p2.1.m1.1.1.1.1.1.cmml" xref="S4.8.p2.1.m1.1.1.1.1">subscript</csymbol><ci id="S4.8.p2.1.m1.1.1.1.1.2.cmml" xref="S4.8.p2.1.m1.1.1.1.1.2">cl</ci><apply id="S4.8.p2.1.m1.1.1.1.1.3.cmml" xref="S4.8.p2.1.m1.1.1.1.1.3"><times id="S4.8.p2.1.m1.1.1.1.1.3.1.cmml" xref="S4.8.p2.1.m1.1.1.1.1.3.1"></times><ci id="S4.8.p2.1.m1.1.1.1.1.3.2.cmml" xref="S4.8.p2.1.m1.1.1.1.1.3.2">π½</ci><ci id="S4.8.p2.1.m1.1.1.1.1.3.3.cmml" xref="S4.8.p2.1.m1.1.1.1.1.3.3">π</ci></apply></apply><apply id="S4.8.p2.1.m1.2.2.2.2.1.cmml" xref="S4.8.p2.1.m1.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.8.p2.1.m1.2.2.2.2.1.1.cmml" xref="S4.8.p2.1.m1.2.2.2.2.1">subscript</csymbol><ci id="S4.8.p2.1.m1.2.2.2.2.1.2.cmml" xref="S4.8.p2.1.m1.2.2.2.2.1.2">πΌ</ci><ci id="S4.8.p2.1.m1.2.2.2.2.1.3.cmml" xref="S4.8.p2.1.m1.2.2.2.2.1.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.8.p2.1.m1.2c">\operatorname{cl}_{\beta X}(I_{L})</annotation><annotation encoding="application/x-llamapun" id="S4.8.p2.1.m1.2d">roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_X end_POSTSUBSCRIPT ( italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT )</annotation></semantics></math> is clopen.</p> </div> <div class="ltx_theorem ltx_theorem_claim" id="S4.Thmtheorem11"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem11.1.1.1">Claim 4.11</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem11.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem11.p1"> <p class="ltx_p" id="S4.Thmtheorem11.p1.2"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem11.p1.2.2">There exists a finite set <math alttext="F\subset X\setminus I_{L}" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.1.1.m1.1"><semantics id="S4.Thmtheorem11.p1.1.1.m1.1a"><mrow id="S4.Thmtheorem11.p1.1.1.m1.1.1" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem11.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.2.cmml">F</mi><mo id="S4.Thmtheorem11.p1.1.1.m1.1.1.1" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.1.cmml">β</mo><mrow id="S4.Thmtheorem11.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3.cmml"><mi id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.2" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3.2.cmml">X</mi><mo id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.1" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3.1.cmml">β</mo><msub id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.3" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3.3.cmml"><mi id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.3.2" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3.3.2.cmml">I</mi><mi id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.3.3" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3.3.3.cmml">L</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.1.1.m1.1b"><apply id="S4.Thmtheorem11.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem11.p1.1.1.m1.1.1"><subset id="S4.Thmtheorem11.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.1"></subset><ci id="S4.Thmtheorem11.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.2">πΉ</ci><apply id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3"><setdiff id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3.1"></setdiff><ci id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3.2">π</ci><apply id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.3.cmml" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.3.1.cmml" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3.3">subscript</csymbol><ci id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.3.2.cmml" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3.3.2">πΌ</ci><ci id="S4.Thmtheorem11.p1.1.1.m1.1.1.3.3.3.cmml" xref="S4.Thmtheorem11.p1.1.1.m1.1.1.3.3.3">πΏ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.1.1.m1.1c">F\subset X\setminus I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.1.1.m1.1d">italic_F β italic_X β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="\beta X\setminus\operatorname{cl}_{\beta X}(I_{L})=\bigcup_{x\in F}{\uparrow}x" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.2.2.m2.2"><semantics id="S4.Thmtheorem11.p1.2.2.m2.2a"><mrow id="S4.Thmtheorem11.p1.2.2.m2.2.2" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.cmml"><mrow id="S4.Thmtheorem11.p1.2.2.m2.2.2.2" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.cmml"><mrow id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.4" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.4.cmml"><mi id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.4.2" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.4.2.cmml">Ξ²</mi><mo id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.4.1" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.4.1.cmml">β’</mo><mi id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.4.3" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.4.3.cmml">X</mi></mrow><mo id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.3" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.3.cmml">β</mo><mrow id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.3.cmml"><msub id="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1" xref="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.2" xref="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.2.cmml">cl</mi><mrow id="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.3" xref="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.3.2" xref="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.3.2.cmml">Ξ²</mi><mo id="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.3.1" xref="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.3.1.cmml">β’</mo><mi id="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.3.3" xref="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.3.3.cmml">X</mi></mrow></msub><mo id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2a" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.3.cmml">β‘</mo><mrow id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2.2" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.3.cmml"><mo id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2.2.2" stretchy="false" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.3.cmml">(</mo><msub id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2.2.1" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2.2.1.cmml"><mi id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2.2.1.2" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2.2.1.2.cmml">I</mi><mi id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2.2.1.3" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2.2.1.3.cmml">L</mi></msub><mo id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2.2.3" stretchy="false" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Thmtheorem11.p1.2.2.m2.2.2.4" rspace="0.111em" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.4.cmml">=</mo><msub id="S4.Thmtheorem11.p1.2.2.m2.2.2.5" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.5.cmml"><mo id="S4.Thmtheorem11.p1.2.2.m2.2.2.5.2" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.5.2.cmml">β</mo><mrow id="S4.Thmtheorem11.p1.2.2.m2.2.2.5.3" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.5.3.cmml"><mi id="S4.Thmtheorem11.p1.2.2.m2.2.2.5.3.2" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.5.3.2.cmml">x</mi><mo id="S4.Thmtheorem11.p1.2.2.m2.2.2.5.3.1" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.5.3.1.cmml">β</mo><mi id="S4.Thmtheorem11.p1.2.2.m2.2.2.5.3.3" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.5.3.3.cmml">F</mi></mrow></msub><mo id="S4.Thmtheorem11.p1.2.2.m2.2.2.6" lspace="0.111em" stretchy="false" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.6.cmml">β</mo><mi id="S4.Thmtheorem11.p1.2.2.m2.2.2.7" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.7.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.2.2.m2.2b"><apply id="S4.Thmtheorem11.p1.2.2.m2.2.2.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2"><and id="S4.Thmtheorem11.p1.2.2.m2.2.2a.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2"></and><apply id="S4.Thmtheorem11.p1.2.2.m2.2.2b.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2"><eq id="S4.Thmtheorem11.p1.2.2.m2.2.2.4.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.4"></eq><apply id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2"><setdiff id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.3.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.3"></setdiff><apply id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.4.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.4"><times id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.4.1.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.4.1"></times><ci id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.4.2.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.4.2">π½</ci><ci id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.4.3.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.4.3">π</ci></apply><apply id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.3.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2"><apply id="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.2">cl</ci><apply id="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.3"><times id="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.3.1.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.3.1"></times><ci id="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.3.2">π½</ci><ci id="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.3.3.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.1.1.1.1.1.1.3.3">π</ci></apply></apply><apply id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2.2.1.1.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2.2.1">subscript</csymbol><ci id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2.2.1.2.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2.2.1.2">πΌ</ci><ci id="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2.2.1.3.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.2.2.2.2.1.3">πΏ</ci></apply></apply></apply><apply id="S4.Thmtheorem11.p1.2.2.m2.2.2.5.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.5"><csymbol cd="ambiguous" id="S4.Thmtheorem11.p1.2.2.m2.2.2.5.1.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.5">subscript</csymbol><union id="S4.Thmtheorem11.p1.2.2.m2.2.2.5.2.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.5.2"></union><apply id="S4.Thmtheorem11.p1.2.2.m2.2.2.5.3.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.5.3"><in id="S4.Thmtheorem11.p1.2.2.m2.2.2.5.3.1.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.5.3.1"></in><ci id="S4.Thmtheorem11.p1.2.2.m2.2.2.5.3.2.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.5.3.2">π₯</ci><ci id="S4.Thmtheorem11.p1.2.2.m2.2.2.5.3.3.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.5.3.3">πΉ</ci></apply></apply></apply><apply id="S4.Thmtheorem11.p1.2.2.m2.2.2c.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2"><ci id="S4.Thmtheorem11.p1.2.2.m2.2.2.6.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.6">β</ci><share href="https://arxiv.org/html/2503.13666v1#S4.Thmtheorem11.p1.2.2.m2.2.2.5.cmml" id="S4.Thmtheorem11.p1.2.2.m2.2.2d.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2"></share><ci id="S4.Thmtheorem11.p1.2.2.m2.2.2.7.cmml" xref="S4.Thmtheorem11.p1.2.2.m2.2.2.7">π₯</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.2.2.m2.2c">\beta X\setminus\operatorname{cl}_{\beta X}(I_{L})=\bigcup_{x\in F}{\uparrow}x</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.2.2.m2.2d">italic_Ξ² italic_X β roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_X end_POSTSUBSCRIPT ( italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) = β start_POSTSUBSCRIPT italic_x β italic_F end_POSTSUBSCRIPT β italic_x</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S4.10.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.9.2.p1"> <p class="ltx_p" id="S4.9.2.p1.32">Let us first check that <math alttext="\operatorname{cl}_{\beta X}(I_{L})" class="ltx_Math" display="inline" id="S4.9.2.p1.1.m1.2"><semantics id="S4.9.2.p1.1.m1.2a"><mrow id="S4.9.2.p1.1.m1.2.2.2" xref="S4.9.2.p1.1.m1.2.2.3.cmml"><msub id="S4.9.2.p1.1.m1.1.1.1.1" xref="S4.9.2.p1.1.m1.1.1.1.1.cmml"><mi id="S4.9.2.p1.1.m1.1.1.1.1.2" xref="S4.9.2.p1.1.m1.1.1.1.1.2.cmml">cl</mi><mrow id="S4.9.2.p1.1.m1.1.1.1.1.3" xref="S4.9.2.p1.1.m1.1.1.1.1.3.cmml"><mi id="S4.9.2.p1.1.m1.1.1.1.1.3.2" xref="S4.9.2.p1.1.m1.1.1.1.1.3.2.cmml">Ξ²</mi><mo id="S4.9.2.p1.1.m1.1.1.1.1.3.1" xref="S4.9.2.p1.1.m1.1.1.1.1.3.1.cmml">β’</mo><mi id="S4.9.2.p1.1.m1.1.1.1.1.3.3" xref="S4.9.2.p1.1.m1.1.1.1.1.3.3.cmml">X</mi></mrow></msub><mo id="S4.9.2.p1.1.m1.2.2.2a" xref="S4.9.2.p1.1.m1.2.2.3.cmml">β‘</mo><mrow id="S4.9.2.p1.1.m1.2.2.2.2" xref="S4.9.2.p1.1.m1.2.2.3.cmml"><mo id="S4.9.2.p1.1.m1.2.2.2.2.2" stretchy="false" xref="S4.9.2.p1.1.m1.2.2.3.cmml">(</mo><msub id="S4.9.2.p1.1.m1.2.2.2.2.1" xref="S4.9.2.p1.1.m1.2.2.2.2.1.cmml"><mi id="S4.9.2.p1.1.m1.2.2.2.2.1.2" xref="S4.9.2.p1.1.m1.2.2.2.2.1.2.cmml">I</mi><mi id="S4.9.2.p1.1.m1.2.2.2.2.1.3" xref="S4.9.2.p1.1.m1.2.2.2.2.1.3.cmml">L</mi></msub><mo id="S4.9.2.p1.1.m1.2.2.2.2.3" stretchy="false" xref="S4.9.2.p1.1.m1.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.1.m1.2b"><apply id="S4.9.2.p1.1.m1.2.2.3.cmml" xref="S4.9.2.p1.1.m1.2.2.2"><apply id="S4.9.2.p1.1.m1.1.1.1.1.cmml" xref="S4.9.2.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.9.2.p1.1.m1.1.1.1.1.1.cmml" xref="S4.9.2.p1.1.m1.1.1.1.1">subscript</csymbol><ci id="S4.9.2.p1.1.m1.1.1.1.1.2.cmml" xref="S4.9.2.p1.1.m1.1.1.1.1.2">cl</ci><apply id="S4.9.2.p1.1.m1.1.1.1.1.3.cmml" xref="S4.9.2.p1.1.m1.1.1.1.1.3"><times id="S4.9.2.p1.1.m1.1.1.1.1.3.1.cmml" xref="S4.9.2.p1.1.m1.1.1.1.1.3.1"></times><ci id="S4.9.2.p1.1.m1.1.1.1.1.3.2.cmml" xref="S4.9.2.p1.1.m1.1.1.1.1.3.2">π½</ci><ci id="S4.9.2.p1.1.m1.1.1.1.1.3.3.cmml" xref="S4.9.2.p1.1.m1.1.1.1.1.3.3">π</ci></apply></apply><apply id="S4.9.2.p1.1.m1.2.2.2.2.1.cmml" xref="S4.9.2.p1.1.m1.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.9.2.p1.1.m1.2.2.2.2.1.1.cmml" xref="S4.9.2.p1.1.m1.2.2.2.2.1">subscript</csymbol><ci id="S4.9.2.p1.1.m1.2.2.2.2.1.2.cmml" xref="S4.9.2.p1.1.m1.2.2.2.2.1.2">πΌ</ci><ci id="S4.9.2.p1.1.m1.2.2.2.2.1.3.cmml" xref="S4.9.2.p1.1.m1.2.2.2.2.1.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.1.m1.2c">\operatorname{cl}_{\beta X}(I_{L})</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.1.m1.2d">roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_X end_POSTSUBSCRIPT ( italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT )</annotation></semantics></math> is an ideal in <math alttext="\beta X" class="ltx_Math" display="inline" id="S4.9.2.p1.2.m2.1"><semantics id="S4.9.2.p1.2.m2.1a"><mrow id="S4.9.2.p1.2.m2.1.1" xref="S4.9.2.p1.2.m2.1.1.cmml"><mi id="S4.9.2.p1.2.m2.1.1.2" xref="S4.9.2.p1.2.m2.1.1.2.cmml">Ξ²</mi><mo id="S4.9.2.p1.2.m2.1.1.1" xref="S4.9.2.p1.2.m2.1.1.1.cmml">β’</mo><mi id="S4.9.2.p1.2.m2.1.1.3" xref="S4.9.2.p1.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.2.m2.1b"><apply id="S4.9.2.p1.2.m2.1.1.cmml" xref="S4.9.2.p1.2.m2.1.1"><times id="S4.9.2.p1.2.m2.1.1.1.cmml" xref="S4.9.2.p1.2.m2.1.1.1"></times><ci id="S4.9.2.p1.2.m2.1.1.2.cmml" xref="S4.9.2.p1.2.m2.1.1.2">π½</ci><ci id="S4.9.2.p1.2.m2.1.1.3.cmml" xref="S4.9.2.p1.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.2.m2.1c">\beta X</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.2.m2.1d">italic_Ξ² italic_X</annotation></semantics></math>. It suffices to show that <math alttext="ab\in\operatorname{cl}_{\beta X}(I_{L})" class="ltx_Math" display="inline" id="S4.9.2.p1.3.m3.2"><semantics id="S4.9.2.p1.3.m3.2a"><mrow id="S4.9.2.p1.3.m3.2.2" xref="S4.9.2.p1.3.m3.2.2.cmml"><mrow id="S4.9.2.p1.3.m3.2.2.4" xref="S4.9.2.p1.3.m3.2.2.4.cmml"><mi id="S4.9.2.p1.3.m3.2.2.4.2" xref="S4.9.2.p1.3.m3.2.2.4.2.cmml">a</mi><mo id="S4.9.2.p1.3.m3.2.2.4.1" xref="S4.9.2.p1.3.m3.2.2.4.1.cmml">β’</mo><mi id="S4.9.2.p1.3.m3.2.2.4.3" xref="S4.9.2.p1.3.m3.2.2.4.3.cmml">b</mi></mrow><mo id="S4.9.2.p1.3.m3.2.2.3" xref="S4.9.2.p1.3.m3.2.2.3.cmml">β</mo><mrow id="S4.9.2.p1.3.m3.2.2.2.2" xref="S4.9.2.p1.3.m3.2.2.2.3.cmml"><msub id="S4.9.2.p1.3.m3.1.1.1.1.1" xref="S4.9.2.p1.3.m3.1.1.1.1.1.cmml"><mi id="S4.9.2.p1.3.m3.1.1.1.1.1.2" xref="S4.9.2.p1.3.m3.1.1.1.1.1.2.cmml">cl</mi><mrow id="S4.9.2.p1.3.m3.1.1.1.1.1.3" xref="S4.9.2.p1.3.m3.1.1.1.1.1.3.cmml"><mi id="S4.9.2.p1.3.m3.1.1.1.1.1.3.2" xref="S4.9.2.p1.3.m3.1.1.1.1.1.3.2.cmml">Ξ²</mi><mo id="S4.9.2.p1.3.m3.1.1.1.1.1.3.1" xref="S4.9.2.p1.3.m3.1.1.1.1.1.3.1.cmml">β’</mo><mi id="S4.9.2.p1.3.m3.1.1.1.1.1.3.3" xref="S4.9.2.p1.3.m3.1.1.1.1.1.3.3.cmml">X</mi></mrow></msub><mo id="S4.9.2.p1.3.m3.2.2.2.2a" xref="S4.9.2.p1.3.m3.2.2.2.3.cmml">β‘</mo><mrow id="S4.9.2.p1.3.m3.2.2.2.2.2" xref="S4.9.2.p1.3.m3.2.2.2.3.cmml"><mo id="S4.9.2.p1.3.m3.2.2.2.2.2.2" stretchy="false" xref="S4.9.2.p1.3.m3.2.2.2.3.cmml">(</mo><msub id="S4.9.2.p1.3.m3.2.2.2.2.2.1" xref="S4.9.2.p1.3.m3.2.2.2.2.2.1.cmml"><mi id="S4.9.2.p1.3.m3.2.2.2.2.2.1.2" xref="S4.9.2.p1.3.m3.2.2.2.2.2.1.2.cmml">I</mi><mi id="S4.9.2.p1.3.m3.2.2.2.2.2.1.3" xref="S4.9.2.p1.3.m3.2.2.2.2.2.1.3.cmml">L</mi></msub><mo id="S4.9.2.p1.3.m3.2.2.2.2.2.3" stretchy="false" xref="S4.9.2.p1.3.m3.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.3.m3.2b"><apply id="S4.9.2.p1.3.m3.2.2.cmml" xref="S4.9.2.p1.3.m3.2.2"><in id="S4.9.2.p1.3.m3.2.2.3.cmml" xref="S4.9.2.p1.3.m3.2.2.3"></in><apply id="S4.9.2.p1.3.m3.2.2.4.cmml" xref="S4.9.2.p1.3.m3.2.2.4"><times id="S4.9.2.p1.3.m3.2.2.4.1.cmml" xref="S4.9.2.p1.3.m3.2.2.4.1"></times><ci id="S4.9.2.p1.3.m3.2.2.4.2.cmml" xref="S4.9.2.p1.3.m3.2.2.4.2">π</ci><ci id="S4.9.2.p1.3.m3.2.2.4.3.cmml" xref="S4.9.2.p1.3.m3.2.2.4.3">π</ci></apply><apply id="S4.9.2.p1.3.m3.2.2.2.3.cmml" xref="S4.9.2.p1.3.m3.2.2.2.2"><apply id="S4.9.2.p1.3.m3.1.1.1.1.1.cmml" xref="S4.9.2.p1.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.9.2.p1.3.m3.1.1.1.1.1.1.cmml" xref="S4.9.2.p1.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S4.9.2.p1.3.m3.1.1.1.1.1.2.cmml" xref="S4.9.2.p1.3.m3.1.1.1.1.1.2">cl</ci><apply id="S4.9.2.p1.3.m3.1.1.1.1.1.3.cmml" xref="S4.9.2.p1.3.m3.1.1.1.1.1.3"><times id="S4.9.2.p1.3.m3.1.1.1.1.1.3.1.cmml" xref="S4.9.2.p1.3.m3.1.1.1.1.1.3.1"></times><ci id="S4.9.2.p1.3.m3.1.1.1.1.1.3.2.cmml" xref="S4.9.2.p1.3.m3.1.1.1.1.1.3.2">π½</ci><ci id="S4.9.2.p1.3.m3.1.1.1.1.1.3.3.cmml" xref="S4.9.2.p1.3.m3.1.1.1.1.1.3.3">π</ci></apply></apply><apply id="S4.9.2.p1.3.m3.2.2.2.2.2.1.cmml" xref="S4.9.2.p1.3.m3.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.9.2.p1.3.m3.2.2.2.2.2.1.1.cmml" xref="S4.9.2.p1.3.m3.2.2.2.2.2.1">subscript</csymbol><ci id="S4.9.2.p1.3.m3.2.2.2.2.2.1.2.cmml" xref="S4.9.2.p1.3.m3.2.2.2.2.2.1.2">πΌ</ci><ci id="S4.9.2.p1.3.m3.2.2.2.2.2.1.3.cmml" xref="S4.9.2.p1.3.m3.2.2.2.2.2.1.3">πΏ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.3.m3.2c">ab\in\operatorname{cl}_{\beta X}(I_{L})</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.3.m3.2d">italic_a italic_b β roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_X end_POSTSUBSCRIPT ( italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT )</annotation></semantics></math> for any <math alttext="a\in\operatorname{cl}_{\beta X}(I_{L})" class="ltx_Math" display="inline" id="S4.9.2.p1.4.m4.2"><semantics id="S4.9.2.p1.4.m4.2a"><mrow id="S4.9.2.p1.4.m4.2.2" xref="S4.9.2.p1.4.m4.2.2.cmml"><mi id="S4.9.2.p1.4.m4.2.2.4" xref="S4.9.2.p1.4.m4.2.2.4.cmml">a</mi><mo id="S4.9.2.p1.4.m4.2.2.3" xref="S4.9.2.p1.4.m4.2.2.3.cmml">β</mo><mrow id="S4.9.2.p1.4.m4.2.2.2.2" xref="S4.9.2.p1.4.m4.2.2.2.3.cmml"><msub id="S4.9.2.p1.4.m4.1.1.1.1.1" xref="S4.9.2.p1.4.m4.1.1.1.1.1.cmml"><mi id="S4.9.2.p1.4.m4.1.1.1.1.1.2" xref="S4.9.2.p1.4.m4.1.1.1.1.1.2.cmml">cl</mi><mrow id="S4.9.2.p1.4.m4.1.1.1.1.1.3" xref="S4.9.2.p1.4.m4.1.1.1.1.1.3.cmml"><mi id="S4.9.2.p1.4.m4.1.1.1.1.1.3.2" xref="S4.9.2.p1.4.m4.1.1.1.1.1.3.2.cmml">Ξ²</mi><mo id="S4.9.2.p1.4.m4.1.1.1.1.1.3.1" xref="S4.9.2.p1.4.m4.1.1.1.1.1.3.1.cmml">β’</mo><mi id="S4.9.2.p1.4.m4.1.1.1.1.1.3.3" xref="S4.9.2.p1.4.m4.1.1.1.1.1.3.3.cmml">X</mi></mrow></msub><mo id="S4.9.2.p1.4.m4.2.2.2.2a" xref="S4.9.2.p1.4.m4.2.2.2.3.cmml">β‘</mo><mrow id="S4.9.2.p1.4.m4.2.2.2.2.2" xref="S4.9.2.p1.4.m4.2.2.2.3.cmml"><mo id="S4.9.2.p1.4.m4.2.2.2.2.2.2" stretchy="false" xref="S4.9.2.p1.4.m4.2.2.2.3.cmml">(</mo><msub id="S4.9.2.p1.4.m4.2.2.2.2.2.1" xref="S4.9.2.p1.4.m4.2.2.2.2.2.1.cmml"><mi id="S4.9.2.p1.4.m4.2.2.2.2.2.1.2" xref="S4.9.2.p1.4.m4.2.2.2.2.2.1.2.cmml">I</mi><mi id="S4.9.2.p1.4.m4.2.2.2.2.2.1.3" xref="S4.9.2.p1.4.m4.2.2.2.2.2.1.3.cmml">L</mi></msub><mo id="S4.9.2.p1.4.m4.2.2.2.2.2.3" stretchy="false" xref="S4.9.2.p1.4.m4.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.4.m4.2b"><apply id="S4.9.2.p1.4.m4.2.2.cmml" xref="S4.9.2.p1.4.m4.2.2"><in id="S4.9.2.p1.4.m4.2.2.3.cmml" xref="S4.9.2.p1.4.m4.2.2.3"></in><ci id="S4.9.2.p1.4.m4.2.2.4.cmml" xref="S4.9.2.p1.4.m4.2.2.4">π</ci><apply id="S4.9.2.p1.4.m4.2.2.2.3.cmml" xref="S4.9.2.p1.4.m4.2.2.2.2"><apply id="S4.9.2.p1.4.m4.1.1.1.1.1.cmml" xref="S4.9.2.p1.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.9.2.p1.4.m4.1.1.1.1.1.1.cmml" xref="S4.9.2.p1.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S4.9.2.p1.4.m4.1.1.1.1.1.2.cmml" xref="S4.9.2.p1.4.m4.1.1.1.1.1.2">cl</ci><apply id="S4.9.2.p1.4.m4.1.1.1.1.1.3.cmml" xref="S4.9.2.p1.4.m4.1.1.1.1.1.3"><times id="S4.9.2.p1.4.m4.1.1.1.1.1.3.1.cmml" xref="S4.9.2.p1.4.m4.1.1.1.1.1.3.1"></times><ci id="S4.9.2.p1.4.m4.1.1.1.1.1.3.2.cmml" xref="S4.9.2.p1.4.m4.1.1.1.1.1.3.2">π½</ci><ci id="S4.9.2.p1.4.m4.1.1.1.1.1.3.3.cmml" xref="S4.9.2.p1.4.m4.1.1.1.1.1.3.3">π</ci></apply></apply><apply id="S4.9.2.p1.4.m4.2.2.2.2.2.1.cmml" xref="S4.9.2.p1.4.m4.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.9.2.p1.4.m4.2.2.2.2.2.1.1.cmml" xref="S4.9.2.p1.4.m4.2.2.2.2.2.1">subscript</csymbol><ci id="S4.9.2.p1.4.m4.2.2.2.2.2.1.2.cmml" xref="S4.9.2.p1.4.m4.2.2.2.2.2.1.2">πΌ</ci><ci id="S4.9.2.p1.4.m4.2.2.2.2.2.1.3.cmml" xref="S4.9.2.p1.4.m4.2.2.2.2.2.1.3">πΏ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.4.m4.2c">a\in\operatorname{cl}_{\beta X}(I_{L})</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.4.m4.2d">italic_a β roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_X end_POSTSUBSCRIPT ( italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="b\in\beta X" class="ltx_Math" display="inline" id="S4.9.2.p1.5.m5.1"><semantics id="S4.9.2.p1.5.m5.1a"><mrow id="S4.9.2.p1.5.m5.1.1" xref="S4.9.2.p1.5.m5.1.1.cmml"><mi id="S4.9.2.p1.5.m5.1.1.2" xref="S4.9.2.p1.5.m5.1.1.2.cmml">b</mi><mo id="S4.9.2.p1.5.m5.1.1.1" xref="S4.9.2.p1.5.m5.1.1.1.cmml">β</mo><mrow id="S4.9.2.p1.5.m5.1.1.3" xref="S4.9.2.p1.5.m5.1.1.3.cmml"><mi id="S4.9.2.p1.5.m5.1.1.3.2" xref="S4.9.2.p1.5.m5.1.1.3.2.cmml">Ξ²</mi><mo id="S4.9.2.p1.5.m5.1.1.3.1" xref="S4.9.2.p1.5.m5.1.1.3.1.cmml">β’</mo><mi id="S4.9.2.p1.5.m5.1.1.3.3" xref="S4.9.2.p1.5.m5.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.5.m5.1b"><apply id="S4.9.2.p1.5.m5.1.1.cmml" xref="S4.9.2.p1.5.m5.1.1"><in id="S4.9.2.p1.5.m5.1.1.1.cmml" xref="S4.9.2.p1.5.m5.1.1.1"></in><ci id="S4.9.2.p1.5.m5.1.1.2.cmml" xref="S4.9.2.p1.5.m5.1.1.2">π</ci><apply id="S4.9.2.p1.5.m5.1.1.3.cmml" xref="S4.9.2.p1.5.m5.1.1.3"><times id="S4.9.2.p1.5.m5.1.1.3.1.cmml" xref="S4.9.2.p1.5.m5.1.1.3.1"></times><ci id="S4.9.2.p1.5.m5.1.1.3.2.cmml" xref="S4.9.2.p1.5.m5.1.1.3.2">π½</ci><ci id="S4.9.2.p1.5.m5.1.1.3.3.cmml" xref="S4.9.2.p1.5.m5.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.5.m5.1c">b\in\beta X</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.5.m5.1d">italic_b β italic_Ξ² italic_X</annotation></semantics></math>. Fix any open neighborhood <math alttext="W" class="ltx_Math" display="inline" id="S4.9.2.p1.6.m6.1"><semantics id="S4.9.2.p1.6.m6.1a"><mi id="S4.9.2.p1.6.m6.1.1" xref="S4.9.2.p1.6.m6.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.6.m6.1b"><ci id="S4.9.2.p1.6.m6.1.1.cmml" xref="S4.9.2.p1.6.m6.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.6.m6.1c">W</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.6.m6.1d">italic_W</annotation></semantics></math> of <math alttext="ab" class="ltx_Math" display="inline" id="S4.9.2.p1.7.m7.1"><semantics id="S4.9.2.p1.7.m7.1a"><mrow id="S4.9.2.p1.7.m7.1.1" xref="S4.9.2.p1.7.m7.1.1.cmml"><mi id="S4.9.2.p1.7.m7.1.1.2" xref="S4.9.2.p1.7.m7.1.1.2.cmml">a</mi><mo id="S4.9.2.p1.7.m7.1.1.1" xref="S4.9.2.p1.7.m7.1.1.1.cmml">β’</mo><mi id="S4.9.2.p1.7.m7.1.1.3" xref="S4.9.2.p1.7.m7.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.7.m7.1b"><apply id="S4.9.2.p1.7.m7.1.1.cmml" xref="S4.9.2.p1.7.m7.1.1"><times id="S4.9.2.p1.7.m7.1.1.1.cmml" xref="S4.9.2.p1.7.m7.1.1.1"></times><ci id="S4.9.2.p1.7.m7.1.1.2.cmml" xref="S4.9.2.p1.7.m7.1.1.2">π</ci><ci id="S4.9.2.p1.7.m7.1.1.3.cmml" xref="S4.9.2.p1.7.m7.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.7.m7.1c">ab</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.7.m7.1d">italic_a italic_b</annotation></semantics></math>. By the continuity of the semigroup operation in <math alttext="\beta X" class="ltx_Math" display="inline" id="S4.9.2.p1.8.m8.1"><semantics id="S4.9.2.p1.8.m8.1a"><mrow id="S4.9.2.p1.8.m8.1.1" xref="S4.9.2.p1.8.m8.1.1.cmml"><mi id="S4.9.2.p1.8.m8.1.1.2" xref="S4.9.2.p1.8.m8.1.1.2.cmml">Ξ²</mi><mo id="S4.9.2.p1.8.m8.1.1.1" xref="S4.9.2.p1.8.m8.1.1.1.cmml">β’</mo><mi id="S4.9.2.p1.8.m8.1.1.3" xref="S4.9.2.p1.8.m8.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.8.m8.1b"><apply id="S4.9.2.p1.8.m8.1.1.cmml" xref="S4.9.2.p1.8.m8.1.1"><times id="S4.9.2.p1.8.m8.1.1.1.cmml" xref="S4.9.2.p1.8.m8.1.1.1"></times><ci id="S4.9.2.p1.8.m8.1.1.2.cmml" xref="S4.9.2.p1.8.m8.1.1.2">π½</ci><ci id="S4.9.2.p1.8.m8.1.1.3.cmml" xref="S4.9.2.p1.8.m8.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.8.m8.1c">\beta X</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.8.m8.1d">italic_Ξ² italic_X</annotation></semantics></math>, there exist open neighborhoods <math alttext="U" class="ltx_Math" display="inline" id="S4.9.2.p1.9.m9.1"><semantics id="S4.9.2.p1.9.m9.1a"><mi id="S4.9.2.p1.9.m9.1.1" xref="S4.9.2.p1.9.m9.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.9.m9.1b"><ci id="S4.9.2.p1.9.m9.1.1.cmml" xref="S4.9.2.p1.9.m9.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.9.m9.1c">U</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.9.m9.1d">italic_U</annotation></semantics></math> of <math alttext="a" class="ltx_Math" display="inline" id="S4.9.2.p1.10.m10.1"><semantics id="S4.9.2.p1.10.m10.1a"><mi id="S4.9.2.p1.10.m10.1.1" xref="S4.9.2.p1.10.m10.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.10.m10.1b"><ci id="S4.9.2.p1.10.m10.1.1.cmml" xref="S4.9.2.p1.10.m10.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.10.m10.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.10.m10.1d">italic_a</annotation></semantics></math> and <math alttext="V" class="ltx_Math" display="inline" id="S4.9.2.p1.11.m11.1"><semantics id="S4.9.2.p1.11.m11.1a"><mi id="S4.9.2.p1.11.m11.1.1" xref="S4.9.2.p1.11.m11.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.11.m11.1b"><ci id="S4.9.2.p1.11.m11.1.1.cmml" xref="S4.9.2.p1.11.m11.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.11.m11.1c">V</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.11.m11.1d">italic_V</annotation></semantics></math> of <math alttext="b" class="ltx_Math" display="inline" id="S4.9.2.p1.12.m12.1"><semantics id="S4.9.2.p1.12.m12.1a"><mi id="S4.9.2.p1.12.m12.1.1" xref="S4.9.2.p1.12.m12.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.12.m12.1b"><ci id="S4.9.2.p1.12.m12.1.1.cmml" xref="S4.9.2.p1.12.m12.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.12.m12.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.12.m12.1d">italic_b</annotation></semantics></math> such that <math alttext="UV\subseteq W" class="ltx_Math" display="inline" id="S4.9.2.p1.13.m13.1"><semantics id="S4.9.2.p1.13.m13.1a"><mrow id="S4.9.2.p1.13.m13.1.1" xref="S4.9.2.p1.13.m13.1.1.cmml"><mrow id="S4.9.2.p1.13.m13.1.1.2" xref="S4.9.2.p1.13.m13.1.1.2.cmml"><mi id="S4.9.2.p1.13.m13.1.1.2.2" xref="S4.9.2.p1.13.m13.1.1.2.2.cmml">U</mi><mo id="S4.9.2.p1.13.m13.1.1.2.1" xref="S4.9.2.p1.13.m13.1.1.2.1.cmml">β’</mo><mi id="S4.9.2.p1.13.m13.1.1.2.3" xref="S4.9.2.p1.13.m13.1.1.2.3.cmml">V</mi></mrow><mo id="S4.9.2.p1.13.m13.1.1.1" xref="S4.9.2.p1.13.m13.1.1.1.cmml">β</mo><mi id="S4.9.2.p1.13.m13.1.1.3" xref="S4.9.2.p1.13.m13.1.1.3.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.13.m13.1b"><apply id="S4.9.2.p1.13.m13.1.1.cmml" xref="S4.9.2.p1.13.m13.1.1"><subset id="S4.9.2.p1.13.m13.1.1.1.cmml" xref="S4.9.2.p1.13.m13.1.1.1"></subset><apply id="S4.9.2.p1.13.m13.1.1.2.cmml" xref="S4.9.2.p1.13.m13.1.1.2"><times id="S4.9.2.p1.13.m13.1.1.2.1.cmml" xref="S4.9.2.p1.13.m13.1.1.2.1"></times><ci id="S4.9.2.p1.13.m13.1.1.2.2.cmml" xref="S4.9.2.p1.13.m13.1.1.2.2">π</ci><ci id="S4.9.2.p1.13.m13.1.1.2.3.cmml" xref="S4.9.2.p1.13.m13.1.1.2.3">π</ci></apply><ci id="S4.9.2.p1.13.m13.1.1.3.cmml" xref="S4.9.2.p1.13.m13.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.13.m13.1c">UV\subseteq W</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.13.m13.1d">italic_U italic_V β italic_W</annotation></semantics></math>. Then there exist <math alttext="c\in U\cap I_{L}" class="ltx_Math" display="inline" id="S4.9.2.p1.14.m14.1"><semantics id="S4.9.2.p1.14.m14.1a"><mrow id="S4.9.2.p1.14.m14.1.1" xref="S4.9.2.p1.14.m14.1.1.cmml"><mi id="S4.9.2.p1.14.m14.1.1.2" xref="S4.9.2.p1.14.m14.1.1.2.cmml">c</mi><mo id="S4.9.2.p1.14.m14.1.1.1" xref="S4.9.2.p1.14.m14.1.1.1.cmml">β</mo><mrow id="S4.9.2.p1.14.m14.1.1.3" xref="S4.9.2.p1.14.m14.1.1.3.cmml"><mi id="S4.9.2.p1.14.m14.1.1.3.2" xref="S4.9.2.p1.14.m14.1.1.3.2.cmml">U</mi><mo id="S4.9.2.p1.14.m14.1.1.3.1" xref="S4.9.2.p1.14.m14.1.1.3.1.cmml">β©</mo><msub id="S4.9.2.p1.14.m14.1.1.3.3" xref="S4.9.2.p1.14.m14.1.1.3.3.cmml"><mi id="S4.9.2.p1.14.m14.1.1.3.3.2" xref="S4.9.2.p1.14.m14.1.1.3.3.2.cmml">I</mi><mi id="S4.9.2.p1.14.m14.1.1.3.3.3" xref="S4.9.2.p1.14.m14.1.1.3.3.3.cmml">L</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.14.m14.1b"><apply id="S4.9.2.p1.14.m14.1.1.cmml" xref="S4.9.2.p1.14.m14.1.1"><in id="S4.9.2.p1.14.m14.1.1.1.cmml" xref="S4.9.2.p1.14.m14.1.1.1"></in><ci id="S4.9.2.p1.14.m14.1.1.2.cmml" xref="S4.9.2.p1.14.m14.1.1.2">π</ci><apply id="S4.9.2.p1.14.m14.1.1.3.cmml" xref="S4.9.2.p1.14.m14.1.1.3"><intersect id="S4.9.2.p1.14.m14.1.1.3.1.cmml" xref="S4.9.2.p1.14.m14.1.1.3.1"></intersect><ci id="S4.9.2.p1.14.m14.1.1.3.2.cmml" xref="S4.9.2.p1.14.m14.1.1.3.2">π</ci><apply id="S4.9.2.p1.14.m14.1.1.3.3.cmml" xref="S4.9.2.p1.14.m14.1.1.3.3"><csymbol cd="ambiguous" id="S4.9.2.p1.14.m14.1.1.3.3.1.cmml" xref="S4.9.2.p1.14.m14.1.1.3.3">subscript</csymbol><ci id="S4.9.2.p1.14.m14.1.1.3.3.2.cmml" xref="S4.9.2.p1.14.m14.1.1.3.3.2">πΌ</ci><ci id="S4.9.2.p1.14.m14.1.1.3.3.3.cmml" xref="S4.9.2.p1.14.m14.1.1.3.3.3">πΏ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.14.m14.1c">c\in U\cap I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.14.m14.1d">italic_c β italic_U β© italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="d\in X\cap V" class="ltx_Math" display="inline" id="S4.9.2.p1.15.m15.1"><semantics id="S4.9.2.p1.15.m15.1a"><mrow id="S4.9.2.p1.15.m15.1.1" xref="S4.9.2.p1.15.m15.1.1.cmml"><mi id="S4.9.2.p1.15.m15.1.1.2" xref="S4.9.2.p1.15.m15.1.1.2.cmml">d</mi><mo id="S4.9.2.p1.15.m15.1.1.1" xref="S4.9.2.p1.15.m15.1.1.1.cmml">β</mo><mrow id="S4.9.2.p1.15.m15.1.1.3" xref="S4.9.2.p1.15.m15.1.1.3.cmml"><mi id="S4.9.2.p1.15.m15.1.1.3.2" xref="S4.9.2.p1.15.m15.1.1.3.2.cmml">X</mi><mo id="S4.9.2.p1.15.m15.1.1.3.1" xref="S4.9.2.p1.15.m15.1.1.3.1.cmml">β©</mo><mi id="S4.9.2.p1.15.m15.1.1.3.3" xref="S4.9.2.p1.15.m15.1.1.3.3.cmml">V</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.15.m15.1b"><apply id="S4.9.2.p1.15.m15.1.1.cmml" xref="S4.9.2.p1.15.m15.1.1"><in id="S4.9.2.p1.15.m15.1.1.1.cmml" xref="S4.9.2.p1.15.m15.1.1.1"></in><ci id="S4.9.2.p1.15.m15.1.1.2.cmml" xref="S4.9.2.p1.15.m15.1.1.2">π</ci><apply id="S4.9.2.p1.15.m15.1.1.3.cmml" xref="S4.9.2.p1.15.m15.1.1.3"><intersect id="S4.9.2.p1.15.m15.1.1.3.1.cmml" xref="S4.9.2.p1.15.m15.1.1.3.1"></intersect><ci id="S4.9.2.p1.15.m15.1.1.3.2.cmml" xref="S4.9.2.p1.15.m15.1.1.3.2">π</ci><ci id="S4.9.2.p1.15.m15.1.1.3.3.cmml" xref="S4.9.2.p1.15.m15.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.15.m15.1c">d\in X\cap V</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.15.m15.1d">italic_d β italic_X β© italic_V</annotation></semantics></math> such that <math alttext="cd\in W\cap I_{L}" class="ltx_Math" display="inline" id="S4.9.2.p1.16.m16.1"><semantics id="S4.9.2.p1.16.m16.1a"><mrow id="S4.9.2.p1.16.m16.1.1" xref="S4.9.2.p1.16.m16.1.1.cmml"><mrow id="S4.9.2.p1.16.m16.1.1.2" xref="S4.9.2.p1.16.m16.1.1.2.cmml"><mi id="S4.9.2.p1.16.m16.1.1.2.2" xref="S4.9.2.p1.16.m16.1.1.2.2.cmml">c</mi><mo id="S4.9.2.p1.16.m16.1.1.2.1" xref="S4.9.2.p1.16.m16.1.1.2.1.cmml">β’</mo><mi id="S4.9.2.p1.16.m16.1.1.2.3" xref="S4.9.2.p1.16.m16.1.1.2.3.cmml">d</mi></mrow><mo id="S4.9.2.p1.16.m16.1.1.1" xref="S4.9.2.p1.16.m16.1.1.1.cmml">β</mo><mrow id="S4.9.2.p1.16.m16.1.1.3" xref="S4.9.2.p1.16.m16.1.1.3.cmml"><mi id="S4.9.2.p1.16.m16.1.1.3.2" xref="S4.9.2.p1.16.m16.1.1.3.2.cmml">W</mi><mo id="S4.9.2.p1.16.m16.1.1.3.1" xref="S4.9.2.p1.16.m16.1.1.3.1.cmml">β©</mo><msub id="S4.9.2.p1.16.m16.1.1.3.3" xref="S4.9.2.p1.16.m16.1.1.3.3.cmml"><mi id="S4.9.2.p1.16.m16.1.1.3.3.2" xref="S4.9.2.p1.16.m16.1.1.3.3.2.cmml">I</mi><mi id="S4.9.2.p1.16.m16.1.1.3.3.3" xref="S4.9.2.p1.16.m16.1.1.3.3.3.cmml">L</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.16.m16.1b"><apply id="S4.9.2.p1.16.m16.1.1.cmml" xref="S4.9.2.p1.16.m16.1.1"><in id="S4.9.2.p1.16.m16.1.1.1.cmml" xref="S4.9.2.p1.16.m16.1.1.1"></in><apply id="S4.9.2.p1.16.m16.1.1.2.cmml" xref="S4.9.2.p1.16.m16.1.1.2"><times id="S4.9.2.p1.16.m16.1.1.2.1.cmml" xref="S4.9.2.p1.16.m16.1.1.2.1"></times><ci id="S4.9.2.p1.16.m16.1.1.2.2.cmml" xref="S4.9.2.p1.16.m16.1.1.2.2">π</ci><ci id="S4.9.2.p1.16.m16.1.1.2.3.cmml" xref="S4.9.2.p1.16.m16.1.1.2.3">π</ci></apply><apply id="S4.9.2.p1.16.m16.1.1.3.cmml" xref="S4.9.2.p1.16.m16.1.1.3"><intersect id="S4.9.2.p1.16.m16.1.1.3.1.cmml" xref="S4.9.2.p1.16.m16.1.1.3.1"></intersect><ci id="S4.9.2.p1.16.m16.1.1.3.2.cmml" xref="S4.9.2.p1.16.m16.1.1.3.2">π</ci><apply id="S4.9.2.p1.16.m16.1.1.3.3.cmml" xref="S4.9.2.p1.16.m16.1.1.3.3"><csymbol cd="ambiguous" id="S4.9.2.p1.16.m16.1.1.3.3.1.cmml" xref="S4.9.2.p1.16.m16.1.1.3.3">subscript</csymbol><ci id="S4.9.2.p1.16.m16.1.1.3.3.2.cmml" xref="S4.9.2.p1.16.m16.1.1.3.3.2">πΌ</ci><ci id="S4.9.2.p1.16.m16.1.1.3.3.3.cmml" xref="S4.9.2.p1.16.m16.1.1.3.3.3">πΏ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.16.m16.1c">cd\in W\cap I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.16.m16.1d">italic_c italic_d β italic_W β© italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>. Thus <math alttext="ab\in\operatorname{cl}_{\beta X}(I_{L})" class="ltx_Math" display="inline" id="S4.9.2.p1.17.m17.2"><semantics id="S4.9.2.p1.17.m17.2a"><mrow id="S4.9.2.p1.17.m17.2.2" xref="S4.9.2.p1.17.m17.2.2.cmml"><mrow id="S4.9.2.p1.17.m17.2.2.4" xref="S4.9.2.p1.17.m17.2.2.4.cmml"><mi id="S4.9.2.p1.17.m17.2.2.4.2" xref="S4.9.2.p1.17.m17.2.2.4.2.cmml">a</mi><mo id="S4.9.2.p1.17.m17.2.2.4.1" xref="S4.9.2.p1.17.m17.2.2.4.1.cmml">β’</mo><mi id="S4.9.2.p1.17.m17.2.2.4.3" xref="S4.9.2.p1.17.m17.2.2.4.3.cmml">b</mi></mrow><mo id="S4.9.2.p1.17.m17.2.2.3" xref="S4.9.2.p1.17.m17.2.2.3.cmml">β</mo><mrow id="S4.9.2.p1.17.m17.2.2.2.2" xref="S4.9.2.p1.17.m17.2.2.2.3.cmml"><msub id="S4.9.2.p1.17.m17.1.1.1.1.1" xref="S4.9.2.p1.17.m17.1.1.1.1.1.cmml"><mi id="S4.9.2.p1.17.m17.1.1.1.1.1.2" xref="S4.9.2.p1.17.m17.1.1.1.1.1.2.cmml">cl</mi><mrow id="S4.9.2.p1.17.m17.1.1.1.1.1.3" xref="S4.9.2.p1.17.m17.1.1.1.1.1.3.cmml"><mi id="S4.9.2.p1.17.m17.1.1.1.1.1.3.2" xref="S4.9.2.p1.17.m17.1.1.1.1.1.3.2.cmml">Ξ²</mi><mo id="S4.9.2.p1.17.m17.1.1.1.1.1.3.1" xref="S4.9.2.p1.17.m17.1.1.1.1.1.3.1.cmml">β’</mo><mi id="S4.9.2.p1.17.m17.1.1.1.1.1.3.3" xref="S4.9.2.p1.17.m17.1.1.1.1.1.3.3.cmml">X</mi></mrow></msub><mo id="S4.9.2.p1.17.m17.2.2.2.2a" xref="S4.9.2.p1.17.m17.2.2.2.3.cmml">β‘</mo><mrow id="S4.9.2.p1.17.m17.2.2.2.2.2" xref="S4.9.2.p1.17.m17.2.2.2.3.cmml"><mo id="S4.9.2.p1.17.m17.2.2.2.2.2.2" stretchy="false" xref="S4.9.2.p1.17.m17.2.2.2.3.cmml">(</mo><msub id="S4.9.2.p1.17.m17.2.2.2.2.2.1" xref="S4.9.2.p1.17.m17.2.2.2.2.2.1.cmml"><mi id="S4.9.2.p1.17.m17.2.2.2.2.2.1.2" xref="S4.9.2.p1.17.m17.2.2.2.2.2.1.2.cmml">I</mi><mi id="S4.9.2.p1.17.m17.2.2.2.2.2.1.3" xref="S4.9.2.p1.17.m17.2.2.2.2.2.1.3.cmml">L</mi></msub><mo id="S4.9.2.p1.17.m17.2.2.2.2.2.3" stretchy="false" xref="S4.9.2.p1.17.m17.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.17.m17.2b"><apply id="S4.9.2.p1.17.m17.2.2.cmml" xref="S4.9.2.p1.17.m17.2.2"><in id="S4.9.2.p1.17.m17.2.2.3.cmml" xref="S4.9.2.p1.17.m17.2.2.3"></in><apply id="S4.9.2.p1.17.m17.2.2.4.cmml" xref="S4.9.2.p1.17.m17.2.2.4"><times id="S4.9.2.p1.17.m17.2.2.4.1.cmml" xref="S4.9.2.p1.17.m17.2.2.4.1"></times><ci id="S4.9.2.p1.17.m17.2.2.4.2.cmml" xref="S4.9.2.p1.17.m17.2.2.4.2">π</ci><ci id="S4.9.2.p1.17.m17.2.2.4.3.cmml" xref="S4.9.2.p1.17.m17.2.2.4.3">π</ci></apply><apply id="S4.9.2.p1.17.m17.2.2.2.3.cmml" xref="S4.9.2.p1.17.m17.2.2.2.2"><apply id="S4.9.2.p1.17.m17.1.1.1.1.1.cmml" xref="S4.9.2.p1.17.m17.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.9.2.p1.17.m17.1.1.1.1.1.1.cmml" xref="S4.9.2.p1.17.m17.1.1.1.1.1">subscript</csymbol><ci id="S4.9.2.p1.17.m17.1.1.1.1.1.2.cmml" xref="S4.9.2.p1.17.m17.1.1.1.1.1.2">cl</ci><apply id="S4.9.2.p1.17.m17.1.1.1.1.1.3.cmml" xref="S4.9.2.p1.17.m17.1.1.1.1.1.3"><times id="S4.9.2.p1.17.m17.1.1.1.1.1.3.1.cmml" xref="S4.9.2.p1.17.m17.1.1.1.1.1.3.1"></times><ci id="S4.9.2.p1.17.m17.1.1.1.1.1.3.2.cmml" xref="S4.9.2.p1.17.m17.1.1.1.1.1.3.2">π½</ci><ci id="S4.9.2.p1.17.m17.1.1.1.1.1.3.3.cmml" xref="S4.9.2.p1.17.m17.1.1.1.1.1.3.3">π</ci></apply></apply><apply id="S4.9.2.p1.17.m17.2.2.2.2.2.1.cmml" xref="S4.9.2.p1.17.m17.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.9.2.p1.17.m17.2.2.2.2.2.1.1.cmml" xref="S4.9.2.p1.17.m17.2.2.2.2.2.1">subscript</csymbol><ci id="S4.9.2.p1.17.m17.2.2.2.2.2.1.2.cmml" xref="S4.9.2.p1.17.m17.2.2.2.2.2.1.2">πΌ</ci><ci id="S4.9.2.p1.17.m17.2.2.2.2.2.1.3.cmml" xref="S4.9.2.p1.17.m17.2.2.2.2.2.1.3">πΏ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.17.m17.2c">ab\in\operatorname{cl}_{\beta X}(I_{L})</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.17.m17.2d">italic_a italic_b β roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_X end_POSTSUBSCRIPT ( italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT )</annotation></semantics></math>, as required. Consider the Rees quotient semilattice <math alttext="S=\beta X/{\operatorname{cl}_{\beta X}(I_{L})}" class="ltx_Math" display="inline" id="S4.9.2.p1.18.m18.2"><semantics id="S4.9.2.p1.18.m18.2a"><mrow id="S4.9.2.p1.18.m18.2.2" xref="S4.9.2.p1.18.m18.2.2.cmml"><mi id="S4.9.2.p1.18.m18.2.2.4" xref="S4.9.2.p1.18.m18.2.2.4.cmml">S</mi><mo id="S4.9.2.p1.18.m18.2.2.3" xref="S4.9.2.p1.18.m18.2.2.3.cmml">=</mo><mrow id="S4.9.2.p1.18.m18.2.2.2" xref="S4.9.2.p1.18.m18.2.2.2.cmml"><mrow id="S4.9.2.p1.18.m18.2.2.2.4" xref="S4.9.2.p1.18.m18.2.2.2.4.cmml"><mi id="S4.9.2.p1.18.m18.2.2.2.4.2" xref="S4.9.2.p1.18.m18.2.2.2.4.2.cmml">Ξ²</mi><mo id="S4.9.2.p1.18.m18.2.2.2.4.1" xref="S4.9.2.p1.18.m18.2.2.2.4.1.cmml">β’</mo><mi id="S4.9.2.p1.18.m18.2.2.2.4.3" xref="S4.9.2.p1.18.m18.2.2.2.4.3.cmml">X</mi></mrow><mo id="S4.9.2.p1.18.m18.2.2.2.3" xref="S4.9.2.p1.18.m18.2.2.2.3.cmml">/</mo><mrow id="S4.9.2.p1.18.m18.2.2.2.2.2" xref="S4.9.2.p1.18.m18.2.2.2.2.3.cmml"><msub id="S4.9.2.p1.18.m18.1.1.1.1.1.1" xref="S4.9.2.p1.18.m18.1.1.1.1.1.1.cmml"><mi id="S4.9.2.p1.18.m18.1.1.1.1.1.1.2" xref="S4.9.2.p1.18.m18.1.1.1.1.1.1.2.cmml">cl</mi><mrow id="S4.9.2.p1.18.m18.1.1.1.1.1.1.3" xref="S4.9.2.p1.18.m18.1.1.1.1.1.1.3.cmml"><mi id="S4.9.2.p1.18.m18.1.1.1.1.1.1.3.2" xref="S4.9.2.p1.18.m18.1.1.1.1.1.1.3.2.cmml">Ξ²</mi><mo id="S4.9.2.p1.18.m18.1.1.1.1.1.1.3.1" xref="S4.9.2.p1.18.m18.1.1.1.1.1.1.3.1.cmml">β’</mo><mi id="S4.9.2.p1.18.m18.1.1.1.1.1.1.3.3" xref="S4.9.2.p1.18.m18.1.1.1.1.1.1.3.3.cmml">X</mi></mrow></msub><mo id="S4.9.2.p1.18.m18.2.2.2.2.2a" xref="S4.9.2.p1.18.m18.2.2.2.2.3.cmml">β‘</mo><mrow id="S4.9.2.p1.18.m18.2.2.2.2.2.2" xref="S4.9.2.p1.18.m18.2.2.2.2.3.cmml"><mo id="S4.9.2.p1.18.m18.2.2.2.2.2.2.2" stretchy="false" xref="S4.9.2.p1.18.m18.2.2.2.2.3.cmml">(</mo><msub id="S4.9.2.p1.18.m18.2.2.2.2.2.2.1" xref="S4.9.2.p1.18.m18.2.2.2.2.2.2.1.cmml"><mi id="S4.9.2.p1.18.m18.2.2.2.2.2.2.1.2" xref="S4.9.2.p1.18.m18.2.2.2.2.2.2.1.2.cmml">I</mi><mi id="S4.9.2.p1.18.m18.2.2.2.2.2.2.1.3" xref="S4.9.2.p1.18.m18.2.2.2.2.2.2.1.3.cmml">L</mi></msub><mo id="S4.9.2.p1.18.m18.2.2.2.2.2.2.3" stretchy="false" xref="S4.9.2.p1.18.m18.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.18.m18.2b"><apply id="S4.9.2.p1.18.m18.2.2.cmml" xref="S4.9.2.p1.18.m18.2.2"><eq id="S4.9.2.p1.18.m18.2.2.3.cmml" xref="S4.9.2.p1.18.m18.2.2.3"></eq><ci id="S4.9.2.p1.18.m18.2.2.4.cmml" xref="S4.9.2.p1.18.m18.2.2.4">π</ci><apply id="S4.9.2.p1.18.m18.2.2.2.cmml" xref="S4.9.2.p1.18.m18.2.2.2"><divide id="S4.9.2.p1.18.m18.2.2.2.3.cmml" xref="S4.9.2.p1.18.m18.2.2.2.3"></divide><apply id="S4.9.2.p1.18.m18.2.2.2.4.cmml" xref="S4.9.2.p1.18.m18.2.2.2.4"><times id="S4.9.2.p1.18.m18.2.2.2.4.1.cmml" xref="S4.9.2.p1.18.m18.2.2.2.4.1"></times><ci id="S4.9.2.p1.18.m18.2.2.2.4.2.cmml" xref="S4.9.2.p1.18.m18.2.2.2.4.2">π½</ci><ci id="S4.9.2.p1.18.m18.2.2.2.4.3.cmml" xref="S4.9.2.p1.18.m18.2.2.2.4.3">π</ci></apply><apply id="S4.9.2.p1.18.m18.2.2.2.2.3.cmml" xref="S4.9.2.p1.18.m18.2.2.2.2.2"><apply id="S4.9.2.p1.18.m18.1.1.1.1.1.1.cmml" xref="S4.9.2.p1.18.m18.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.9.2.p1.18.m18.1.1.1.1.1.1.1.cmml" xref="S4.9.2.p1.18.m18.1.1.1.1.1.1">subscript</csymbol><ci id="S4.9.2.p1.18.m18.1.1.1.1.1.1.2.cmml" xref="S4.9.2.p1.18.m18.1.1.1.1.1.1.2">cl</ci><apply id="S4.9.2.p1.18.m18.1.1.1.1.1.1.3.cmml" xref="S4.9.2.p1.18.m18.1.1.1.1.1.1.3"><times id="S4.9.2.p1.18.m18.1.1.1.1.1.1.3.1.cmml" xref="S4.9.2.p1.18.m18.1.1.1.1.1.1.3.1"></times><ci id="S4.9.2.p1.18.m18.1.1.1.1.1.1.3.2.cmml" xref="S4.9.2.p1.18.m18.1.1.1.1.1.1.3.2">π½</ci><ci id="S4.9.2.p1.18.m18.1.1.1.1.1.1.3.3.cmml" xref="S4.9.2.p1.18.m18.1.1.1.1.1.1.3.3">π</ci></apply></apply><apply id="S4.9.2.p1.18.m18.2.2.2.2.2.2.1.cmml" xref="S4.9.2.p1.18.m18.2.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.9.2.p1.18.m18.2.2.2.2.2.2.1.1.cmml" xref="S4.9.2.p1.18.m18.2.2.2.2.2.2.1">subscript</csymbol><ci id="S4.9.2.p1.18.m18.2.2.2.2.2.2.1.2.cmml" xref="S4.9.2.p1.18.m18.2.2.2.2.2.2.1.2">πΌ</ci><ci id="S4.9.2.p1.18.m18.2.2.2.2.2.2.1.3.cmml" xref="S4.9.2.p1.18.m18.2.2.2.2.2.2.1.3">πΏ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.18.m18.2c">S=\beta X/{\operatorname{cl}_{\beta X}(I_{L})}</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.18.m18.2d">italic_S = italic_Ξ² italic_X / roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_X end_POSTSUBSCRIPT ( italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT )</annotation></semantics></math> which is obtained by contracting the clopen ideal <math alttext="\operatorname{cl}_{\beta X}(I_{L})" class="ltx_Math" display="inline" id="S4.9.2.p1.19.m19.2"><semantics id="S4.9.2.p1.19.m19.2a"><mrow id="S4.9.2.p1.19.m19.2.2.2" xref="S4.9.2.p1.19.m19.2.2.3.cmml"><msub id="S4.9.2.p1.19.m19.1.1.1.1" xref="S4.9.2.p1.19.m19.1.1.1.1.cmml"><mi id="S4.9.2.p1.19.m19.1.1.1.1.2" xref="S4.9.2.p1.19.m19.1.1.1.1.2.cmml">cl</mi><mrow id="S4.9.2.p1.19.m19.1.1.1.1.3" xref="S4.9.2.p1.19.m19.1.1.1.1.3.cmml"><mi id="S4.9.2.p1.19.m19.1.1.1.1.3.2" xref="S4.9.2.p1.19.m19.1.1.1.1.3.2.cmml">Ξ²</mi><mo id="S4.9.2.p1.19.m19.1.1.1.1.3.1" xref="S4.9.2.p1.19.m19.1.1.1.1.3.1.cmml">β’</mo><mi id="S4.9.2.p1.19.m19.1.1.1.1.3.3" xref="S4.9.2.p1.19.m19.1.1.1.1.3.3.cmml">X</mi></mrow></msub><mo id="S4.9.2.p1.19.m19.2.2.2a" xref="S4.9.2.p1.19.m19.2.2.3.cmml">β‘</mo><mrow id="S4.9.2.p1.19.m19.2.2.2.2" xref="S4.9.2.p1.19.m19.2.2.3.cmml"><mo id="S4.9.2.p1.19.m19.2.2.2.2.2" stretchy="false" xref="S4.9.2.p1.19.m19.2.2.3.cmml">(</mo><msub id="S4.9.2.p1.19.m19.2.2.2.2.1" xref="S4.9.2.p1.19.m19.2.2.2.2.1.cmml"><mi id="S4.9.2.p1.19.m19.2.2.2.2.1.2" xref="S4.9.2.p1.19.m19.2.2.2.2.1.2.cmml">I</mi><mi id="S4.9.2.p1.19.m19.2.2.2.2.1.3" xref="S4.9.2.p1.19.m19.2.2.2.2.1.3.cmml">L</mi></msub><mo id="S4.9.2.p1.19.m19.2.2.2.2.3" stretchy="false" xref="S4.9.2.p1.19.m19.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.19.m19.2b"><apply id="S4.9.2.p1.19.m19.2.2.3.cmml" xref="S4.9.2.p1.19.m19.2.2.2"><apply id="S4.9.2.p1.19.m19.1.1.1.1.cmml" xref="S4.9.2.p1.19.m19.1.1.1.1"><csymbol cd="ambiguous" id="S4.9.2.p1.19.m19.1.1.1.1.1.cmml" xref="S4.9.2.p1.19.m19.1.1.1.1">subscript</csymbol><ci id="S4.9.2.p1.19.m19.1.1.1.1.2.cmml" xref="S4.9.2.p1.19.m19.1.1.1.1.2">cl</ci><apply id="S4.9.2.p1.19.m19.1.1.1.1.3.cmml" xref="S4.9.2.p1.19.m19.1.1.1.1.3"><times id="S4.9.2.p1.19.m19.1.1.1.1.3.1.cmml" xref="S4.9.2.p1.19.m19.1.1.1.1.3.1"></times><ci id="S4.9.2.p1.19.m19.1.1.1.1.3.2.cmml" xref="S4.9.2.p1.19.m19.1.1.1.1.3.2">π½</ci><ci id="S4.9.2.p1.19.m19.1.1.1.1.3.3.cmml" xref="S4.9.2.p1.19.m19.1.1.1.1.3.3">π</ci></apply></apply><apply id="S4.9.2.p1.19.m19.2.2.2.2.1.cmml" xref="S4.9.2.p1.19.m19.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.9.2.p1.19.m19.2.2.2.2.1.1.cmml" xref="S4.9.2.p1.19.m19.2.2.2.2.1">subscript</csymbol><ci id="S4.9.2.p1.19.m19.2.2.2.2.1.2.cmml" xref="S4.9.2.p1.19.m19.2.2.2.2.1.2">πΌ</ci><ci id="S4.9.2.p1.19.m19.2.2.2.2.1.3.cmml" xref="S4.9.2.p1.19.m19.2.2.2.2.1.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.19.m19.2c">\operatorname{cl}_{\beta X}(I_{L})</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.19.m19.2d">roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_X end_POSTSUBSCRIPT ( italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT )</annotation></semantics></math> to a point denoted by <math alttext="0" class="ltx_Math" display="inline" id="S4.9.2.p1.20.m20.1"><semantics id="S4.9.2.p1.20.m20.1a"><mn id="S4.9.2.p1.20.m20.1.1" xref="S4.9.2.p1.20.m20.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.20.m20.1b"><cn id="S4.9.2.p1.20.m20.1.1.cmml" type="integer" xref="S4.9.2.p1.20.m20.1.1">0</cn></annotation-xml></semantics></math>. Obviously, <math alttext="S" class="ltx_Math" display="inline" id="S4.9.2.p1.21.m21.1"><semantics id="S4.9.2.p1.21.m21.1a"><mi id="S4.9.2.p1.21.m21.1.1" xref="S4.9.2.p1.21.m21.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.21.m21.1b"><ci id="S4.9.2.p1.21.m21.1.1.cmml" xref="S4.9.2.p1.21.m21.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.21.m21.1c">S</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.21.m21.1d">italic_S</annotation></semantics></math> is a compact topological semilattice, <math alttext="0=\inf S" class="ltx_Math" display="inline" id="S4.9.2.p1.22.m22.1"><semantics id="S4.9.2.p1.22.m22.1a"><mrow id="S4.9.2.p1.22.m22.1.1" xref="S4.9.2.p1.22.m22.1.1.cmml"><mn id="S4.9.2.p1.22.m22.1.1.2" xref="S4.9.2.p1.22.m22.1.1.2.cmml">0</mn><mo id="S4.9.2.p1.22.m22.1.1.1" rspace="0.1389em" xref="S4.9.2.p1.22.m22.1.1.1.cmml">=</mo><mrow id="S4.9.2.p1.22.m22.1.1.3" xref="S4.9.2.p1.22.m22.1.1.3.cmml"><mo id="S4.9.2.p1.22.m22.1.1.3.1" lspace="0.1389em" rspace="0.167em" xref="S4.9.2.p1.22.m22.1.1.3.1.cmml">inf</mo><mi id="S4.9.2.p1.22.m22.1.1.3.2" xref="S4.9.2.p1.22.m22.1.1.3.2.cmml">S</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.22.m22.1b"><apply id="S4.9.2.p1.22.m22.1.1.cmml" xref="S4.9.2.p1.22.m22.1.1"><eq id="S4.9.2.p1.22.m22.1.1.1.cmml" xref="S4.9.2.p1.22.m22.1.1.1"></eq><cn id="S4.9.2.p1.22.m22.1.1.2.cmml" type="integer" xref="S4.9.2.p1.22.m22.1.1.2">0</cn><apply id="S4.9.2.p1.22.m22.1.1.3.cmml" xref="S4.9.2.p1.22.m22.1.1.3"><csymbol cd="latexml" id="S4.9.2.p1.22.m22.1.1.3.1.cmml" xref="S4.9.2.p1.22.m22.1.1.3.1">infimum</csymbol><ci id="S4.9.2.p1.22.m22.1.1.3.2.cmml" xref="S4.9.2.p1.22.m22.1.1.3.2">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.22.m22.1c">0=\inf S</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.22.m22.1d">0 = roman_inf italic_S</annotation></semantics></math> and <math alttext="0" class="ltx_Math" display="inline" id="S4.9.2.p1.23.m23.1"><semantics id="S4.9.2.p1.23.m23.1a"><mn id="S4.9.2.p1.23.m23.1.1" xref="S4.9.2.p1.23.m23.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.23.m23.1b"><cn id="S4.9.2.p1.23.m23.1.1.cmml" type="integer" xref="S4.9.2.p1.23.m23.1.1">0</cn></annotation-xml></semantics></math> is isolated in <math alttext="S" class="ltx_Math" display="inline" id="S4.9.2.p1.24.m24.1"><semantics id="S4.9.2.p1.24.m24.1a"><mi id="S4.9.2.p1.24.m24.1.1" xref="S4.9.2.p1.24.m24.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.24.m24.1b"><ci id="S4.9.2.p1.24.m24.1.1.cmml" xref="S4.9.2.p1.24.m24.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.24.m24.1c">S</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.24.m24.1d">italic_S</annotation></semantics></math>. Further we agree to identify the sets <math alttext="S\setminus\{0\}" class="ltx_Math" display="inline" id="S4.9.2.p1.25.m25.1"><semantics id="S4.9.2.p1.25.m25.1a"><mrow id="S4.9.2.p1.25.m25.1.2" xref="S4.9.2.p1.25.m25.1.2.cmml"><mi id="S4.9.2.p1.25.m25.1.2.2" xref="S4.9.2.p1.25.m25.1.2.2.cmml">S</mi><mo id="S4.9.2.p1.25.m25.1.2.1" xref="S4.9.2.p1.25.m25.1.2.1.cmml">β</mo><mrow id="S4.9.2.p1.25.m25.1.2.3.2" xref="S4.9.2.p1.25.m25.1.2.3.1.cmml"><mo id="S4.9.2.p1.25.m25.1.2.3.2.1" stretchy="false" xref="S4.9.2.p1.25.m25.1.2.3.1.cmml">{</mo><mn id="S4.9.2.p1.25.m25.1.1" xref="S4.9.2.p1.25.m25.1.1.cmml">0</mn><mo id="S4.9.2.p1.25.m25.1.2.3.2.2" stretchy="false" xref="S4.9.2.p1.25.m25.1.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.25.m25.1b"><apply id="S4.9.2.p1.25.m25.1.2.cmml" xref="S4.9.2.p1.25.m25.1.2"><setdiff id="S4.9.2.p1.25.m25.1.2.1.cmml" xref="S4.9.2.p1.25.m25.1.2.1"></setdiff><ci id="S4.9.2.p1.25.m25.1.2.2.cmml" xref="S4.9.2.p1.25.m25.1.2.2">π</ci><set id="S4.9.2.p1.25.m25.1.2.3.1.cmml" xref="S4.9.2.p1.25.m25.1.2.3.2"><cn id="S4.9.2.p1.25.m25.1.1.cmml" type="integer" xref="S4.9.2.p1.25.m25.1.1">0</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.25.m25.1c">S\setminus\{0\}</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.25.m25.1d">italic_S β { 0 }</annotation></semantics></math> and <math alttext="\beta X\setminus\operatorname{cl}_{\beta X}(I_{L})" class="ltx_Math" display="inline" id="S4.9.2.p1.26.m26.2"><semantics id="S4.9.2.p1.26.m26.2a"><mrow id="S4.9.2.p1.26.m26.2.2" xref="S4.9.2.p1.26.m26.2.2.cmml"><mrow id="S4.9.2.p1.26.m26.2.2.4" xref="S4.9.2.p1.26.m26.2.2.4.cmml"><mi id="S4.9.2.p1.26.m26.2.2.4.2" xref="S4.9.2.p1.26.m26.2.2.4.2.cmml">Ξ²</mi><mo id="S4.9.2.p1.26.m26.2.2.4.1" xref="S4.9.2.p1.26.m26.2.2.4.1.cmml">β’</mo><mi id="S4.9.2.p1.26.m26.2.2.4.3" xref="S4.9.2.p1.26.m26.2.2.4.3.cmml">X</mi></mrow><mo id="S4.9.2.p1.26.m26.2.2.3" xref="S4.9.2.p1.26.m26.2.2.3.cmml">β</mo><mrow id="S4.9.2.p1.26.m26.2.2.2.2" xref="S4.9.2.p1.26.m26.2.2.2.3.cmml"><msub id="S4.9.2.p1.26.m26.1.1.1.1.1" xref="S4.9.2.p1.26.m26.1.1.1.1.1.cmml"><mi id="S4.9.2.p1.26.m26.1.1.1.1.1.2" xref="S4.9.2.p1.26.m26.1.1.1.1.1.2.cmml">cl</mi><mrow id="S4.9.2.p1.26.m26.1.1.1.1.1.3" xref="S4.9.2.p1.26.m26.1.1.1.1.1.3.cmml"><mi id="S4.9.2.p1.26.m26.1.1.1.1.1.3.2" xref="S4.9.2.p1.26.m26.1.1.1.1.1.3.2.cmml">Ξ²</mi><mo id="S4.9.2.p1.26.m26.1.1.1.1.1.3.1" xref="S4.9.2.p1.26.m26.1.1.1.1.1.3.1.cmml">β’</mo><mi id="S4.9.2.p1.26.m26.1.1.1.1.1.3.3" xref="S4.9.2.p1.26.m26.1.1.1.1.1.3.3.cmml">X</mi></mrow></msub><mo id="S4.9.2.p1.26.m26.2.2.2.2a" xref="S4.9.2.p1.26.m26.2.2.2.3.cmml">β‘</mo><mrow id="S4.9.2.p1.26.m26.2.2.2.2.2" xref="S4.9.2.p1.26.m26.2.2.2.3.cmml"><mo id="S4.9.2.p1.26.m26.2.2.2.2.2.2" stretchy="false" xref="S4.9.2.p1.26.m26.2.2.2.3.cmml">(</mo><msub id="S4.9.2.p1.26.m26.2.2.2.2.2.1" xref="S4.9.2.p1.26.m26.2.2.2.2.2.1.cmml"><mi id="S4.9.2.p1.26.m26.2.2.2.2.2.1.2" xref="S4.9.2.p1.26.m26.2.2.2.2.2.1.2.cmml">I</mi><mi id="S4.9.2.p1.26.m26.2.2.2.2.2.1.3" xref="S4.9.2.p1.26.m26.2.2.2.2.2.1.3.cmml">L</mi></msub><mo id="S4.9.2.p1.26.m26.2.2.2.2.2.3" stretchy="false" xref="S4.9.2.p1.26.m26.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.26.m26.2b"><apply id="S4.9.2.p1.26.m26.2.2.cmml" xref="S4.9.2.p1.26.m26.2.2"><setdiff id="S4.9.2.p1.26.m26.2.2.3.cmml" xref="S4.9.2.p1.26.m26.2.2.3"></setdiff><apply id="S4.9.2.p1.26.m26.2.2.4.cmml" xref="S4.9.2.p1.26.m26.2.2.4"><times id="S4.9.2.p1.26.m26.2.2.4.1.cmml" xref="S4.9.2.p1.26.m26.2.2.4.1"></times><ci id="S4.9.2.p1.26.m26.2.2.4.2.cmml" xref="S4.9.2.p1.26.m26.2.2.4.2">π½</ci><ci id="S4.9.2.p1.26.m26.2.2.4.3.cmml" xref="S4.9.2.p1.26.m26.2.2.4.3">π</ci></apply><apply id="S4.9.2.p1.26.m26.2.2.2.3.cmml" xref="S4.9.2.p1.26.m26.2.2.2.2"><apply id="S4.9.2.p1.26.m26.1.1.1.1.1.cmml" xref="S4.9.2.p1.26.m26.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.9.2.p1.26.m26.1.1.1.1.1.1.cmml" xref="S4.9.2.p1.26.m26.1.1.1.1.1">subscript</csymbol><ci id="S4.9.2.p1.26.m26.1.1.1.1.1.2.cmml" xref="S4.9.2.p1.26.m26.1.1.1.1.1.2">cl</ci><apply id="S4.9.2.p1.26.m26.1.1.1.1.1.3.cmml" xref="S4.9.2.p1.26.m26.1.1.1.1.1.3"><times id="S4.9.2.p1.26.m26.1.1.1.1.1.3.1.cmml" xref="S4.9.2.p1.26.m26.1.1.1.1.1.3.1"></times><ci id="S4.9.2.p1.26.m26.1.1.1.1.1.3.2.cmml" xref="S4.9.2.p1.26.m26.1.1.1.1.1.3.2">π½</ci><ci id="S4.9.2.p1.26.m26.1.1.1.1.1.3.3.cmml" xref="S4.9.2.p1.26.m26.1.1.1.1.1.3.3">π</ci></apply></apply><apply id="S4.9.2.p1.26.m26.2.2.2.2.2.1.cmml" xref="S4.9.2.p1.26.m26.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.9.2.p1.26.m26.2.2.2.2.2.1.1.cmml" xref="S4.9.2.p1.26.m26.2.2.2.2.2.1">subscript</csymbol><ci id="S4.9.2.p1.26.m26.2.2.2.2.2.1.2.cmml" xref="S4.9.2.p1.26.m26.2.2.2.2.2.1.2">πΌ</ci><ci id="S4.9.2.p1.26.m26.2.2.2.2.2.1.3.cmml" xref="S4.9.2.p1.26.m26.2.2.2.2.2.1.3">πΏ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.26.m26.2c">\beta X\setminus\operatorname{cl}_{\beta X}(I_{L})</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.26.m26.2d">italic_Ξ² italic_X β roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_X end_POSTSUBSCRIPT ( italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT )</annotation></semantics></math>. Observe that the set <math alttext="\beta X\setminus\operatorname{cl}_{\beta X}(I_{L})" class="ltx_Math" display="inline" id="S4.9.2.p1.27.m27.2"><semantics id="S4.9.2.p1.27.m27.2a"><mrow id="S4.9.2.p1.27.m27.2.2" xref="S4.9.2.p1.27.m27.2.2.cmml"><mrow id="S4.9.2.p1.27.m27.2.2.4" xref="S4.9.2.p1.27.m27.2.2.4.cmml"><mi id="S4.9.2.p1.27.m27.2.2.4.2" xref="S4.9.2.p1.27.m27.2.2.4.2.cmml">Ξ²</mi><mo id="S4.9.2.p1.27.m27.2.2.4.1" xref="S4.9.2.p1.27.m27.2.2.4.1.cmml">β’</mo><mi id="S4.9.2.p1.27.m27.2.2.4.3" xref="S4.9.2.p1.27.m27.2.2.4.3.cmml">X</mi></mrow><mo id="S4.9.2.p1.27.m27.2.2.3" xref="S4.9.2.p1.27.m27.2.2.3.cmml">β</mo><mrow id="S4.9.2.p1.27.m27.2.2.2.2" xref="S4.9.2.p1.27.m27.2.2.2.3.cmml"><msub id="S4.9.2.p1.27.m27.1.1.1.1.1" xref="S4.9.2.p1.27.m27.1.1.1.1.1.cmml"><mi id="S4.9.2.p1.27.m27.1.1.1.1.1.2" xref="S4.9.2.p1.27.m27.1.1.1.1.1.2.cmml">cl</mi><mrow id="S4.9.2.p1.27.m27.1.1.1.1.1.3" xref="S4.9.2.p1.27.m27.1.1.1.1.1.3.cmml"><mi id="S4.9.2.p1.27.m27.1.1.1.1.1.3.2" xref="S4.9.2.p1.27.m27.1.1.1.1.1.3.2.cmml">Ξ²</mi><mo id="S4.9.2.p1.27.m27.1.1.1.1.1.3.1" xref="S4.9.2.p1.27.m27.1.1.1.1.1.3.1.cmml">β’</mo><mi id="S4.9.2.p1.27.m27.1.1.1.1.1.3.3" xref="S4.9.2.p1.27.m27.1.1.1.1.1.3.3.cmml">X</mi></mrow></msub><mo id="S4.9.2.p1.27.m27.2.2.2.2a" xref="S4.9.2.p1.27.m27.2.2.2.3.cmml">β‘</mo><mrow id="S4.9.2.p1.27.m27.2.2.2.2.2" xref="S4.9.2.p1.27.m27.2.2.2.3.cmml"><mo id="S4.9.2.p1.27.m27.2.2.2.2.2.2" stretchy="false" xref="S4.9.2.p1.27.m27.2.2.2.3.cmml">(</mo><msub id="S4.9.2.p1.27.m27.2.2.2.2.2.1" xref="S4.9.2.p1.27.m27.2.2.2.2.2.1.cmml"><mi id="S4.9.2.p1.27.m27.2.2.2.2.2.1.2" xref="S4.9.2.p1.27.m27.2.2.2.2.2.1.2.cmml">I</mi><mi id="S4.9.2.p1.27.m27.2.2.2.2.2.1.3" xref="S4.9.2.p1.27.m27.2.2.2.2.2.1.3.cmml">L</mi></msub><mo id="S4.9.2.p1.27.m27.2.2.2.2.2.3" stretchy="false" xref="S4.9.2.p1.27.m27.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.27.m27.2b"><apply id="S4.9.2.p1.27.m27.2.2.cmml" xref="S4.9.2.p1.27.m27.2.2"><setdiff id="S4.9.2.p1.27.m27.2.2.3.cmml" xref="S4.9.2.p1.27.m27.2.2.3"></setdiff><apply id="S4.9.2.p1.27.m27.2.2.4.cmml" xref="S4.9.2.p1.27.m27.2.2.4"><times id="S4.9.2.p1.27.m27.2.2.4.1.cmml" xref="S4.9.2.p1.27.m27.2.2.4.1"></times><ci id="S4.9.2.p1.27.m27.2.2.4.2.cmml" xref="S4.9.2.p1.27.m27.2.2.4.2">π½</ci><ci id="S4.9.2.p1.27.m27.2.2.4.3.cmml" xref="S4.9.2.p1.27.m27.2.2.4.3">π</ci></apply><apply id="S4.9.2.p1.27.m27.2.2.2.3.cmml" xref="S4.9.2.p1.27.m27.2.2.2.2"><apply id="S4.9.2.p1.27.m27.1.1.1.1.1.cmml" xref="S4.9.2.p1.27.m27.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.9.2.p1.27.m27.1.1.1.1.1.1.cmml" xref="S4.9.2.p1.27.m27.1.1.1.1.1">subscript</csymbol><ci id="S4.9.2.p1.27.m27.1.1.1.1.1.2.cmml" xref="S4.9.2.p1.27.m27.1.1.1.1.1.2">cl</ci><apply id="S4.9.2.p1.27.m27.1.1.1.1.1.3.cmml" xref="S4.9.2.p1.27.m27.1.1.1.1.1.3"><times id="S4.9.2.p1.27.m27.1.1.1.1.1.3.1.cmml" xref="S4.9.2.p1.27.m27.1.1.1.1.1.3.1"></times><ci id="S4.9.2.p1.27.m27.1.1.1.1.1.3.2.cmml" xref="S4.9.2.p1.27.m27.1.1.1.1.1.3.2">π½</ci><ci id="S4.9.2.p1.27.m27.1.1.1.1.1.3.3.cmml" xref="S4.9.2.p1.27.m27.1.1.1.1.1.3.3">π</ci></apply></apply><apply id="S4.9.2.p1.27.m27.2.2.2.2.2.1.cmml" xref="S4.9.2.p1.27.m27.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.9.2.p1.27.m27.2.2.2.2.2.1.1.cmml" xref="S4.9.2.p1.27.m27.2.2.2.2.2.1">subscript</csymbol><ci id="S4.9.2.p1.27.m27.2.2.2.2.2.1.2.cmml" xref="S4.9.2.p1.27.m27.2.2.2.2.2.1.2">πΌ</ci><ci id="S4.9.2.p1.27.m27.2.2.2.2.2.1.3.cmml" xref="S4.9.2.p1.27.m27.2.2.2.2.2.1.3">πΏ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.27.m27.2c">\beta X\setminus\operatorname{cl}_{\beta X}(I_{L})</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.27.m27.2d">italic_Ξ² italic_X β roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_X end_POSTSUBSCRIPT ( italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT )</annotation></semantics></math> is nonempty, as it contains <math alttext="L" class="ltx_Math" display="inline" id="S4.9.2.p1.28.m28.1"><semantics id="S4.9.2.p1.28.m28.1a"><mi id="S4.9.2.p1.28.m28.1.1" xref="S4.9.2.p1.28.m28.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.28.m28.1b"><ci id="S4.9.2.p1.28.m28.1.1.cmml" xref="S4.9.2.p1.28.m28.1.1">πΏ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.28.m28.1c">L</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.28.m28.1d">italic_L</annotation></semantics></math>. Let <math alttext="\mathfrak{C}" class="ltx_Math" display="inline" id="S4.9.2.p1.29.m29.1"><semantics id="S4.9.2.p1.29.m29.1a"><mi id="S4.9.2.p1.29.m29.1.1" xref="S4.9.2.p1.29.m29.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.29.m29.1b"><ci id="S4.9.2.p1.29.m29.1.1.cmml" xref="S4.9.2.p1.29.m29.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.29.m29.1c">\mathfrak{C}</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.29.m29.1d">fraktur_C</annotation></semantics></math> be the set of all maximal chains in <math alttext="S\setminus\{0\}" class="ltx_Math" display="inline" id="S4.9.2.p1.30.m30.1"><semantics id="S4.9.2.p1.30.m30.1a"><mrow id="S4.9.2.p1.30.m30.1.2" xref="S4.9.2.p1.30.m30.1.2.cmml"><mi id="S4.9.2.p1.30.m30.1.2.2" xref="S4.9.2.p1.30.m30.1.2.2.cmml">S</mi><mo id="S4.9.2.p1.30.m30.1.2.1" xref="S4.9.2.p1.30.m30.1.2.1.cmml">β</mo><mrow id="S4.9.2.p1.30.m30.1.2.3.2" xref="S4.9.2.p1.30.m30.1.2.3.1.cmml"><mo id="S4.9.2.p1.30.m30.1.2.3.2.1" stretchy="false" xref="S4.9.2.p1.30.m30.1.2.3.1.cmml">{</mo><mn id="S4.9.2.p1.30.m30.1.1" xref="S4.9.2.p1.30.m30.1.1.cmml">0</mn><mo id="S4.9.2.p1.30.m30.1.2.3.2.2" stretchy="false" xref="S4.9.2.p1.30.m30.1.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.30.m30.1b"><apply id="S4.9.2.p1.30.m30.1.2.cmml" xref="S4.9.2.p1.30.m30.1.2"><setdiff id="S4.9.2.p1.30.m30.1.2.1.cmml" xref="S4.9.2.p1.30.m30.1.2.1"></setdiff><ci id="S4.9.2.p1.30.m30.1.2.2.cmml" xref="S4.9.2.p1.30.m30.1.2.2">π</ci><set id="S4.9.2.p1.30.m30.1.2.3.1.cmml" xref="S4.9.2.p1.30.m30.1.2.3.2"><cn id="S4.9.2.p1.30.m30.1.1.cmml" type="integer" xref="S4.9.2.p1.30.m30.1.1">0</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.30.m30.1c">S\setminus\{0\}</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.30.m30.1d">italic_S β { 0 }</annotation></semantics></math>. By the compactness of <math alttext="S\setminus\{0\}" class="ltx_Math" display="inline" id="S4.9.2.p1.31.m31.1"><semantics id="S4.9.2.p1.31.m31.1a"><mrow id="S4.9.2.p1.31.m31.1.2" xref="S4.9.2.p1.31.m31.1.2.cmml"><mi id="S4.9.2.p1.31.m31.1.2.2" xref="S4.9.2.p1.31.m31.1.2.2.cmml">S</mi><mo id="S4.9.2.p1.31.m31.1.2.1" xref="S4.9.2.p1.31.m31.1.2.1.cmml">β</mo><mrow id="S4.9.2.p1.31.m31.1.2.3.2" xref="S4.9.2.p1.31.m31.1.2.3.1.cmml"><mo id="S4.9.2.p1.31.m31.1.2.3.2.1" stretchy="false" xref="S4.9.2.p1.31.m31.1.2.3.1.cmml">{</mo><mn id="S4.9.2.p1.31.m31.1.1" xref="S4.9.2.p1.31.m31.1.1.cmml">0</mn><mo id="S4.9.2.p1.31.m31.1.2.3.2.2" stretchy="false" xref="S4.9.2.p1.31.m31.1.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.31.m31.1b"><apply id="S4.9.2.p1.31.m31.1.2.cmml" xref="S4.9.2.p1.31.m31.1.2"><setdiff id="S4.9.2.p1.31.m31.1.2.1.cmml" xref="S4.9.2.p1.31.m31.1.2.1"></setdiff><ci id="S4.9.2.p1.31.m31.1.2.2.cmml" xref="S4.9.2.p1.31.m31.1.2.2">π</ci><set id="S4.9.2.p1.31.m31.1.2.3.1.cmml" xref="S4.9.2.p1.31.m31.1.2.3.2"><cn id="S4.9.2.p1.31.m31.1.1.cmml" type="integer" xref="S4.9.2.p1.31.m31.1.1">0</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.31.m31.1c">S\setminus\{0\}</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.31.m31.1d">italic_S β { 0 }</annotation></semantics></math>, each <math alttext="C\in\mathfrak{C}" class="ltx_Math" display="inline" id="S4.9.2.p1.32.m32.1"><semantics id="S4.9.2.p1.32.m32.1a"><mrow id="S4.9.2.p1.32.m32.1.1" xref="S4.9.2.p1.32.m32.1.1.cmml"><mi id="S4.9.2.p1.32.m32.1.1.2" xref="S4.9.2.p1.32.m32.1.1.2.cmml">C</mi><mo id="S4.9.2.p1.32.m32.1.1.1" xref="S4.9.2.p1.32.m32.1.1.1.cmml">β</mo><mi id="S4.9.2.p1.32.m32.1.1.3" xref="S4.9.2.p1.32.m32.1.1.3.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.32.m32.1b"><apply id="S4.9.2.p1.32.m32.1.1.cmml" xref="S4.9.2.p1.32.m32.1.1"><in id="S4.9.2.p1.32.m32.1.1.1.cmml" xref="S4.9.2.p1.32.m32.1.1.1"></in><ci id="S4.9.2.p1.32.m32.1.1.2.cmml" xref="S4.9.2.p1.32.m32.1.1.2">πΆ</ci><ci id="S4.9.2.p1.32.m32.1.1.3.cmml" xref="S4.9.2.p1.32.m32.1.1.3">β</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.32.m32.1c">C\in\mathfrak{C}</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.32.m32.1d">italic_C β fraktur_C</annotation></semantics></math> contains its minimum. To derive a contradiction, assume that the set</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex13"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="F=\{\min C:C\in\mathfrak{C}\}\subset\beta X\setminus\operatorname{cl}_{\beta X% }(I_{L})" class="ltx_Math" display="block" id="S4.Ex13.m1.4"><semantics id="S4.Ex13.m1.4a"><mrow id="S4.Ex13.m1.4.4" xref="S4.Ex13.m1.4.4.cmml"><mi id="S4.Ex13.m1.4.4.6" xref="S4.Ex13.m1.4.4.6.cmml">F</mi><mo id="S4.Ex13.m1.4.4.7" xref="S4.Ex13.m1.4.4.7.cmml">=</mo><mrow id="S4.Ex13.m1.2.2.2.2" xref="S4.Ex13.m1.2.2.2.3.cmml"><mo id="S4.Ex13.m1.2.2.2.2.3" stretchy="false" xref="S4.Ex13.m1.2.2.2.3.1.cmml">{</mo><mrow id="S4.Ex13.m1.1.1.1.1.1" xref="S4.Ex13.m1.1.1.1.1.1.cmml"><mi id="S4.Ex13.m1.1.1.1.1.1.1" xref="S4.Ex13.m1.1.1.1.1.1.1.cmml">min</mi><mo id="S4.Ex13.m1.1.1.1.1.1a" lspace="0.167em" xref="S4.Ex13.m1.1.1.1.1.1.cmml">β‘</mo><mi id="S4.Ex13.m1.1.1.1.1.1.2" xref="S4.Ex13.m1.1.1.1.1.1.2.cmml">C</mi></mrow><mo id="S4.Ex13.m1.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.Ex13.m1.2.2.2.3.1.cmml">:</mo><mrow id="S4.Ex13.m1.2.2.2.2.2" xref="S4.Ex13.m1.2.2.2.2.2.cmml"><mi id="S4.Ex13.m1.2.2.2.2.2.2" xref="S4.Ex13.m1.2.2.2.2.2.2.cmml">C</mi><mo id="S4.Ex13.m1.2.2.2.2.2.1" xref="S4.Ex13.m1.2.2.2.2.2.1.cmml">β</mo><mi id="S4.Ex13.m1.2.2.2.2.2.3" xref="S4.Ex13.m1.2.2.2.2.2.3.cmml">β</mi></mrow><mo id="S4.Ex13.m1.2.2.2.2.5" stretchy="false" xref="S4.Ex13.m1.2.2.2.3.1.cmml">}</mo></mrow><mo id="S4.Ex13.m1.4.4.8" xref="S4.Ex13.m1.4.4.8.cmml">β</mo><mrow id="S4.Ex13.m1.4.4.4" xref="S4.Ex13.m1.4.4.4.cmml"><mrow id="S4.Ex13.m1.4.4.4.4" xref="S4.Ex13.m1.4.4.4.4.cmml"><mi id="S4.Ex13.m1.4.4.4.4.2" xref="S4.Ex13.m1.4.4.4.4.2.cmml">Ξ²</mi><mo id="S4.Ex13.m1.4.4.4.4.1" xref="S4.Ex13.m1.4.4.4.4.1.cmml">β’</mo><mi id="S4.Ex13.m1.4.4.4.4.3" xref="S4.Ex13.m1.4.4.4.4.3.cmml">X</mi></mrow><mo id="S4.Ex13.m1.4.4.4.3" xref="S4.Ex13.m1.4.4.4.3.cmml">β</mo><mrow id="S4.Ex13.m1.4.4.4.2.2" xref="S4.Ex13.m1.4.4.4.2.3.cmml"><msub id="S4.Ex13.m1.3.3.3.1.1.1" xref="S4.Ex13.m1.3.3.3.1.1.1.cmml"><mi id="S4.Ex13.m1.3.3.3.1.1.1.2" xref="S4.Ex13.m1.3.3.3.1.1.1.2.cmml">cl</mi><mrow id="S4.Ex13.m1.3.3.3.1.1.1.3" xref="S4.Ex13.m1.3.3.3.1.1.1.3.cmml"><mi id="S4.Ex13.m1.3.3.3.1.1.1.3.2" xref="S4.Ex13.m1.3.3.3.1.1.1.3.2.cmml">Ξ²</mi><mo id="S4.Ex13.m1.3.3.3.1.1.1.3.1" xref="S4.Ex13.m1.3.3.3.1.1.1.3.1.cmml">β’</mo><mi id="S4.Ex13.m1.3.3.3.1.1.1.3.3" xref="S4.Ex13.m1.3.3.3.1.1.1.3.3.cmml">X</mi></mrow></msub><mo id="S4.Ex13.m1.4.4.4.2.2a" xref="S4.Ex13.m1.4.4.4.2.3.cmml">β‘</mo><mrow id="S4.Ex13.m1.4.4.4.2.2.2" xref="S4.Ex13.m1.4.4.4.2.3.cmml"><mo id="S4.Ex13.m1.4.4.4.2.2.2.2" stretchy="false" xref="S4.Ex13.m1.4.4.4.2.3.cmml">(</mo><msub id="S4.Ex13.m1.4.4.4.2.2.2.1" xref="S4.Ex13.m1.4.4.4.2.2.2.1.cmml"><mi id="S4.Ex13.m1.4.4.4.2.2.2.1.2" xref="S4.Ex13.m1.4.4.4.2.2.2.1.2.cmml">I</mi><mi id="S4.Ex13.m1.4.4.4.2.2.2.1.3" xref="S4.Ex13.m1.4.4.4.2.2.2.1.3.cmml">L</mi></msub><mo id="S4.Ex13.m1.4.4.4.2.2.2.3" stretchy="false" xref="S4.Ex13.m1.4.4.4.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex13.m1.4b"><apply id="S4.Ex13.m1.4.4.cmml" xref="S4.Ex13.m1.4.4"><and id="S4.Ex13.m1.4.4a.cmml" xref="S4.Ex13.m1.4.4"></and><apply id="S4.Ex13.m1.4.4b.cmml" xref="S4.Ex13.m1.4.4"><eq id="S4.Ex13.m1.4.4.7.cmml" xref="S4.Ex13.m1.4.4.7"></eq><ci id="S4.Ex13.m1.4.4.6.cmml" xref="S4.Ex13.m1.4.4.6">πΉ</ci><apply id="S4.Ex13.m1.2.2.2.3.cmml" xref="S4.Ex13.m1.2.2.2.2"><csymbol cd="latexml" id="S4.Ex13.m1.2.2.2.3.1.cmml" xref="S4.Ex13.m1.2.2.2.2.3">conditional-set</csymbol><apply id="S4.Ex13.m1.1.1.1.1.1.cmml" xref="S4.Ex13.m1.1.1.1.1.1"><min id="S4.Ex13.m1.1.1.1.1.1.1.cmml" xref="S4.Ex13.m1.1.1.1.1.1.1"></min><ci id="S4.Ex13.m1.1.1.1.1.1.2.cmml" xref="S4.Ex13.m1.1.1.1.1.1.2">πΆ</ci></apply><apply id="S4.Ex13.m1.2.2.2.2.2.cmml" xref="S4.Ex13.m1.2.2.2.2.2"><in id="S4.Ex13.m1.2.2.2.2.2.1.cmml" xref="S4.Ex13.m1.2.2.2.2.2.1"></in><ci id="S4.Ex13.m1.2.2.2.2.2.2.cmml" xref="S4.Ex13.m1.2.2.2.2.2.2">πΆ</ci><ci id="S4.Ex13.m1.2.2.2.2.2.3.cmml" xref="S4.Ex13.m1.2.2.2.2.2.3">β</ci></apply></apply></apply><apply id="S4.Ex13.m1.4.4c.cmml" xref="S4.Ex13.m1.4.4"><subset id="S4.Ex13.m1.4.4.8.cmml" xref="S4.Ex13.m1.4.4.8"></subset><share href="https://arxiv.org/html/2503.13666v1#S4.Ex13.m1.2.2.2.cmml" id="S4.Ex13.m1.4.4d.cmml" xref="S4.Ex13.m1.4.4"></share><apply id="S4.Ex13.m1.4.4.4.cmml" xref="S4.Ex13.m1.4.4.4"><setdiff id="S4.Ex13.m1.4.4.4.3.cmml" xref="S4.Ex13.m1.4.4.4.3"></setdiff><apply id="S4.Ex13.m1.4.4.4.4.cmml" xref="S4.Ex13.m1.4.4.4.4"><times id="S4.Ex13.m1.4.4.4.4.1.cmml" xref="S4.Ex13.m1.4.4.4.4.1"></times><ci id="S4.Ex13.m1.4.4.4.4.2.cmml" xref="S4.Ex13.m1.4.4.4.4.2">π½</ci><ci id="S4.Ex13.m1.4.4.4.4.3.cmml" xref="S4.Ex13.m1.4.4.4.4.3">π</ci></apply><apply id="S4.Ex13.m1.4.4.4.2.3.cmml" xref="S4.Ex13.m1.4.4.4.2.2"><apply id="S4.Ex13.m1.3.3.3.1.1.1.cmml" xref="S4.Ex13.m1.3.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.Ex13.m1.3.3.3.1.1.1.1.cmml" xref="S4.Ex13.m1.3.3.3.1.1.1">subscript</csymbol><ci id="S4.Ex13.m1.3.3.3.1.1.1.2.cmml" xref="S4.Ex13.m1.3.3.3.1.1.1.2">cl</ci><apply id="S4.Ex13.m1.3.3.3.1.1.1.3.cmml" xref="S4.Ex13.m1.3.3.3.1.1.1.3"><times id="S4.Ex13.m1.3.3.3.1.1.1.3.1.cmml" xref="S4.Ex13.m1.3.3.3.1.1.1.3.1"></times><ci id="S4.Ex13.m1.3.3.3.1.1.1.3.2.cmml" xref="S4.Ex13.m1.3.3.3.1.1.1.3.2">π½</ci><ci id="S4.Ex13.m1.3.3.3.1.1.1.3.3.cmml" xref="S4.Ex13.m1.3.3.3.1.1.1.3.3">π</ci></apply></apply><apply id="S4.Ex13.m1.4.4.4.2.2.2.1.cmml" xref="S4.Ex13.m1.4.4.4.2.2.2.1"><csymbol cd="ambiguous" id="S4.Ex13.m1.4.4.4.2.2.2.1.1.cmml" xref="S4.Ex13.m1.4.4.4.2.2.2.1">subscript</csymbol><ci id="S4.Ex13.m1.4.4.4.2.2.2.1.2.cmml" xref="S4.Ex13.m1.4.4.4.2.2.2.1.2">πΌ</ci><ci id="S4.Ex13.m1.4.4.4.2.2.2.1.3.cmml" xref="S4.Ex13.m1.4.4.4.2.2.2.1.3">πΏ</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex13.m1.4c">F=\{\min C:C\in\mathfrak{C}\}\subset\beta X\setminus\operatorname{cl}_{\beta X% }(I_{L})</annotation><annotation encoding="application/x-llamapun" id="S4.Ex13.m1.4d">italic_F = { roman_min italic_C : italic_C β fraktur_C } β italic_Ξ² italic_X β roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_X end_POSTSUBSCRIPT ( italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.9.2.p1.51">is infinite. It is easy to see that <math alttext="ab=0" class="ltx_Math" display="inline" id="S4.9.2.p1.33.m1.1"><semantics id="S4.9.2.p1.33.m1.1a"><mrow id="S4.9.2.p1.33.m1.1.1" xref="S4.9.2.p1.33.m1.1.1.cmml"><mrow id="S4.9.2.p1.33.m1.1.1.2" xref="S4.9.2.p1.33.m1.1.1.2.cmml"><mi id="S4.9.2.p1.33.m1.1.1.2.2" xref="S4.9.2.p1.33.m1.1.1.2.2.cmml">a</mi><mo id="S4.9.2.p1.33.m1.1.1.2.1" xref="S4.9.2.p1.33.m1.1.1.2.1.cmml">β’</mo><mi id="S4.9.2.p1.33.m1.1.1.2.3" xref="S4.9.2.p1.33.m1.1.1.2.3.cmml">b</mi></mrow><mo id="S4.9.2.p1.33.m1.1.1.1" xref="S4.9.2.p1.33.m1.1.1.1.cmml">=</mo><mn id="S4.9.2.p1.33.m1.1.1.3" xref="S4.9.2.p1.33.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.33.m1.1b"><apply id="S4.9.2.p1.33.m1.1.1.cmml" xref="S4.9.2.p1.33.m1.1.1"><eq id="S4.9.2.p1.33.m1.1.1.1.cmml" xref="S4.9.2.p1.33.m1.1.1.1"></eq><apply id="S4.9.2.p1.33.m1.1.1.2.cmml" xref="S4.9.2.p1.33.m1.1.1.2"><times id="S4.9.2.p1.33.m1.1.1.2.1.cmml" xref="S4.9.2.p1.33.m1.1.1.2.1"></times><ci id="S4.9.2.p1.33.m1.1.1.2.2.cmml" xref="S4.9.2.p1.33.m1.1.1.2.2">π</ci><ci id="S4.9.2.p1.33.m1.1.1.2.3.cmml" xref="S4.9.2.p1.33.m1.1.1.2.3">π</ci></apply><cn id="S4.9.2.p1.33.m1.1.1.3.cmml" type="integer" xref="S4.9.2.p1.33.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.33.m1.1c">ab=0</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.33.m1.1d">italic_a italic_b = 0</annotation></semantics></math> for any distinct <math alttext="a,b\in F" class="ltx_Math" display="inline" id="S4.9.2.p1.34.m2.2"><semantics id="S4.9.2.p1.34.m2.2a"><mrow id="S4.9.2.p1.34.m2.2.3" xref="S4.9.2.p1.34.m2.2.3.cmml"><mrow id="S4.9.2.p1.34.m2.2.3.2.2" xref="S4.9.2.p1.34.m2.2.3.2.1.cmml"><mi id="S4.9.2.p1.34.m2.1.1" xref="S4.9.2.p1.34.m2.1.1.cmml">a</mi><mo id="S4.9.2.p1.34.m2.2.3.2.2.1" xref="S4.9.2.p1.34.m2.2.3.2.1.cmml">,</mo><mi id="S4.9.2.p1.34.m2.2.2" xref="S4.9.2.p1.34.m2.2.2.cmml">b</mi></mrow><mo id="S4.9.2.p1.34.m2.2.3.1" xref="S4.9.2.p1.34.m2.2.3.1.cmml">β</mo><mi id="S4.9.2.p1.34.m2.2.3.3" xref="S4.9.2.p1.34.m2.2.3.3.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.34.m2.2b"><apply id="S4.9.2.p1.34.m2.2.3.cmml" xref="S4.9.2.p1.34.m2.2.3"><in id="S4.9.2.p1.34.m2.2.3.1.cmml" xref="S4.9.2.p1.34.m2.2.3.1"></in><list id="S4.9.2.p1.34.m2.2.3.2.1.cmml" xref="S4.9.2.p1.34.m2.2.3.2.2"><ci id="S4.9.2.p1.34.m2.1.1.cmml" xref="S4.9.2.p1.34.m2.1.1">π</ci><ci id="S4.9.2.p1.34.m2.2.2.cmml" xref="S4.9.2.p1.34.m2.2.2">π</ci></list><ci id="S4.9.2.p1.34.m2.2.3.3.cmml" xref="S4.9.2.p1.34.m2.2.3.3">πΉ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.34.m2.2c">a,b\in F</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.34.m2.2d">italic_a , italic_b β italic_F</annotation></semantics></math>. So, <math alttext="T=F\cup\{0\}" class="ltx_Math" display="inline" id="S4.9.2.p1.35.m3.1"><semantics id="S4.9.2.p1.35.m3.1a"><mrow id="S4.9.2.p1.35.m3.1.2" xref="S4.9.2.p1.35.m3.1.2.cmml"><mi id="S4.9.2.p1.35.m3.1.2.2" xref="S4.9.2.p1.35.m3.1.2.2.cmml">T</mi><mo id="S4.9.2.p1.35.m3.1.2.1" xref="S4.9.2.p1.35.m3.1.2.1.cmml">=</mo><mrow id="S4.9.2.p1.35.m3.1.2.3" xref="S4.9.2.p1.35.m3.1.2.3.cmml"><mi id="S4.9.2.p1.35.m3.1.2.3.2" xref="S4.9.2.p1.35.m3.1.2.3.2.cmml">F</mi><mo id="S4.9.2.p1.35.m3.1.2.3.1" xref="S4.9.2.p1.35.m3.1.2.3.1.cmml">βͺ</mo><mrow id="S4.9.2.p1.35.m3.1.2.3.3.2" xref="S4.9.2.p1.35.m3.1.2.3.3.1.cmml"><mo id="S4.9.2.p1.35.m3.1.2.3.3.2.1" stretchy="false" xref="S4.9.2.p1.35.m3.1.2.3.3.1.cmml">{</mo><mn id="S4.9.2.p1.35.m3.1.1" xref="S4.9.2.p1.35.m3.1.1.cmml">0</mn><mo id="S4.9.2.p1.35.m3.1.2.3.3.2.2" stretchy="false" xref="S4.9.2.p1.35.m3.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.35.m3.1b"><apply id="S4.9.2.p1.35.m3.1.2.cmml" xref="S4.9.2.p1.35.m3.1.2"><eq id="S4.9.2.p1.35.m3.1.2.1.cmml" xref="S4.9.2.p1.35.m3.1.2.1"></eq><ci id="S4.9.2.p1.35.m3.1.2.2.cmml" xref="S4.9.2.p1.35.m3.1.2.2">π</ci><apply id="S4.9.2.p1.35.m3.1.2.3.cmml" xref="S4.9.2.p1.35.m3.1.2.3"><union id="S4.9.2.p1.35.m3.1.2.3.1.cmml" xref="S4.9.2.p1.35.m3.1.2.3.1"></union><ci id="S4.9.2.p1.35.m3.1.2.3.2.cmml" xref="S4.9.2.p1.35.m3.1.2.3.2">πΉ</ci><set id="S4.9.2.p1.35.m3.1.2.3.3.1.cmml" xref="S4.9.2.p1.35.m3.1.2.3.3.2"><cn id="S4.9.2.p1.35.m3.1.1.cmml" type="integer" xref="S4.9.2.p1.35.m3.1.1">0</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.35.m3.1c">T=F\cup\{0\}</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.35.m3.1d">italic_T = italic_F βͺ { 0 }</annotation></semantics></math> is a subsemilattice of <math alttext="S" class="ltx_Math" display="inline" id="S4.9.2.p1.36.m4.1"><semantics id="S4.9.2.p1.36.m4.1a"><mi id="S4.9.2.p1.36.m4.1.1" xref="S4.9.2.p1.36.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.36.m4.1b"><ci id="S4.9.2.p1.36.m4.1.1.cmml" xref="S4.9.2.p1.36.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.36.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.36.m4.1d">italic_S</annotation></semantics></math>. Since each chain in <math alttext="T" class="ltx_Math" display="inline" id="S4.9.2.p1.37.m5.1"><semantics id="S4.9.2.p1.37.m5.1a"><mi id="S4.9.2.p1.37.m5.1.1" xref="S4.9.2.p1.37.m5.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.37.m5.1b"><ci id="S4.9.2.p1.37.m5.1.1.cmml" xref="S4.9.2.p1.37.m5.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.37.m5.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.37.m5.1d">italic_T</annotation></semantics></math> is finite, Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S3.Thmtheorem2" title="Theorem 3.2 (Banakh, Bardyla). β£ 3. Chains in topological semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">3.2</span></a> implies that <math alttext="T" class="ltx_Math" display="inline" id="S4.9.2.p1.38.m6.1"><semantics id="S4.9.2.p1.38.m6.1a"><mi id="S4.9.2.p1.38.m6.1.1" xref="S4.9.2.p1.38.m6.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.38.m6.1b"><ci id="S4.9.2.p1.38.m6.1.1.cmml" xref="S4.9.2.p1.38.m6.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.38.m6.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.38.m6.1d">italic_T</annotation></semantics></math> is closed in <math alttext="S" class="ltx_Math" display="inline" id="S4.9.2.p1.39.m7.1"><semantics id="S4.9.2.p1.39.m7.1a"><mi id="S4.9.2.p1.39.m7.1.1" xref="S4.9.2.p1.39.m7.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.39.m7.1b"><ci id="S4.9.2.p1.39.m7.1.1.cmml" xref="S4.9.2.p1.39.m7.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.39.m7.1c">S</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.39.m7.1d">italic_S</annotation></semantics></math> and thus compact. Consider an accumulation point <math alttext="y\in T" class="ltx_Math" display="inline" id="S4.9.2.p1.40.m8.1"><semantics id="S4.9.2.p1.40.m8.1a"><mrow id="S4.9.2.p1.40.m8.1.1" xref="S4.9.2.p1.40.m8.1.1.cmml"><mi id="S4.9.2.p1.40.m8.1.1.2" xref="S4.9.2.p1.40.m8.1.1.2.cmml">y</mi><mo id="S4.9.2.p1.40.m8.1.1.1" xref="S4.9.2.p1.40.m8.1.1.1.cmml">β</mo><mi id="S4.9.2.p1.40.m8.1.1.3" xref="S4.9.2.p1.40.m8.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.40.m8.1b"><apply id="S4.9.2.p1.40.m8.1.1.cmml" xref="S4.9.2.p1.40.m8.1.1"><in id="S4.9.2.p1.40.m8.1.1.1.cmml" xref="S4.9.2.p1.40.m8.1.1.1"></in><ci id="S4.9.2.p1.40.m8.1.1.2.cmml" xref="S4.9.2.p1.40.m8.1.1.2">π¦</ci><ci id="S4.9.2.p1.40.m8.1.1.3.cmml" xref="S4.9.2.p1.40.m8.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.40.m8.1c">y\in T</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.40.m8.1d">italic_y β italic_T</annotation></semantics></math> of <math alttext="F" class="ltx_Math" display="inline" id="S4.9.2.p1.41.m9.1"><semantics id="S4.9.2.p1.41.m9.1a"><mi id="S4.9.2.p1.41.m9.1.1" xref="S4.9.2.p1.41.m9.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.41.m9.1b"><ci id="S4.9.2.p1.41.m9.1.1.cmml" xref="S4.9.2.p1.41.m9.1.1">πΉ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.41.m9.1c">F</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.41.m9.1d">italic_F</annotation></semantics></math>. Observe that for each open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S4.9.2.p1.42.m10.1"><semantics id="S4.9.2.p1.42.m10.1a"><mi id="S4.9.2.p1.42.m10.1.1" xref="S4.9.2.p1.42.m10.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.42.m10.1b"><ci id="S4.9.2.p1.42.m10.1.1.cmml" xref="S4.9.2.p1.42.m10.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.42.m10.1c">U</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.42.m10.1d">italic_U</annotation></semantics></math> of <math alttext="y" class="ltx_Math" display="inline" id="S4.9.2.p1.43.m11.1"><semantics id="S4.9.2.p1.43.m11.1a"><mi id="S4.9.2.p1.43.m11.1.1" xref="S4.9.2.p1.43.m11.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.43.m11.1b"><ci id="S4.9.2.p1.43.m11.1.1.cmml" xref="S4.9.2.p1.43.m11.1.1">π¦</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.43.m11.1c">y</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.43.m11.1d">italic_y</annotation></semantics></math> there exist distinct <math alttext="a,b\in U\cap F" class="ltx_Math" display="inline" id="S4.9.2.p1.44.m12.2"><semantics id="S4.9.2.p1.44.m12.2a"><mrow id="S4.9.2.p1.44.m12.2.3" xref="S4.9.2.p1.44.m12.2.3.cmml"><mrow id="S4.9.2.p1.44.m12.2.3.2.2" xref="S4.9.2.p1.44.m12.2.3.2.1.cmml"><mi id="S4.9.2.p1.44.m12.1.1" xref="S4.9.2.p1.44.m12.1.1.cmml">a</mi><mo id="S4.9.2.p1.44.m12.2.3.2.2.1" xref="S4.9.2.p1.44.m12.2.3.2.1.cmml">,</mo><mi id="S4.9.2.p1.44.m12.2.2" xref="S4.9.2.p1.44.m12.2.2.cmml">b</mi></mrow><mo id="S4.9.2.p1.44.m12.2.3.1" xref="S4.9.2.p1.44.m12.2.3.1.cmml">β</mo><mrow id="S4.9.2.p1.44.m12.2.3.3" xref="S4.9.2.p1.44.m12.2.3.3.cmml"><mi id="S4.9.2.p1.44.m12.2.3.3.2" xref="S4.9.2.p1.44.m12.2.3.3.2.cmml">U</mi><mo id="S4.9.2.p1.44.m12.2.3.3.1" xref="S4.9.2.p1.44.m12.2.3.3.1.cmml">β©</mo><mi id="S4.9.2.p1.44.m12.2.3.3.3" xref="S4.9.2.p1.44.m12.2.3.3.3.cmml">F</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.44.m12.2b"><apply id="S4.9.2.p1.44.m12.2.3.cmml" xref="S4.9.2.p1.44.m12.2.3"><in id="S4.9.2.p1.44.m12.2.3.1.cmml" xref="S4.9.2.p1.44.m12.2.3.1"></in><list id="S4.9.2.p1.44.m12.2.3.2.1.cmml" xref="S4.9.2.p1.44.m12.2.3.2.2"><ci id="S4.9.2.p1.44.m12.1.1.cmml" xref="S4.9.2.p1.44.m12.1.1">π</ci><ci id="S4.9.2.p1.44.m12.2.2.cmml" xref="S4.9.2.p1.44.m12.2.2">π</ci></list><apply id="S4.9.2.p1.44.m12.2.3.3.cmml" xref="S4.9.2.p1.44.m12.2.3.3"><intersect id="S4.9.2.p1.44.m12.2.3.3.1.cmml" xref="S4.9.2.p1.44.m12.2.3.3.1"></intersect><ci id="S4.9.2.p1.44.m12.2.3.3.2.cmml" xref="S4.9.2.p1.44.m12.2.3.3.2">π</ci><ci id="S4.9.2.p1.44.m12.2.3.3.3.cmml" xref="S4.9.2.p1.44.m12.2.3.3.3">πΉ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.44.m12.2c">a,b\in U\cap F</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.44.m12.2d">italic_a , italic_b β italic_U β© italic_F</annotation></semantics></math> witnessing that <math alttext="0=ab\in UU" class="ltx_Math" display="inline" id="S4.9.2.p1.45.m13.1"><semantics id="S4.9.2.p1.45.m13.1a"><mrow id="S4.9.2.p1.45.m13.1.1" xref="S4.9.2.p1.45.m13.1.1.cmml"><mn id="S4.9.2.p1.45.m13.1.1.2" xref="S4.9.2.p1.45.m13.1.1.2.cmml">0</mn><mo id="S4.9.2.p1.45.m13.1.1.3" xref="S4.9.2.p1.45.m13.1.1.3.cmml">=</mo><mrow id="S4.9.2.p1.45.m13.1.1.4" xref="S4.9.2.p1.45.m13.1.1.4.cmml"><mi id="S4.9.2.p1.45.m13.1.1.4.2" xref="S4.9.2.p1.45.m13.1.1.4.2.cmml">a</mi><mo id="S4.9.2.p1.45.m13.1.1.4.1" xref="S4.9.2.p1.45.m13.1.1.4.1.cmml">β’</mo><mi id="S4.9.2.p1.45.m13.1.1.4.3" xref="S4.9.2.p1.45.m13.1.1.4.3.cmml">b</mi></mrow><mo id="S4.9.2.p1.45.m13.1.1.5" xref="S4.9.2.p1.45.m13.1.1.5.cmml">β</mo><mrow id="S4.9.2.p1.45.m13.1.1.6" xref="S4.9.2.p1.45.m13.1.1.6.cmml"><mi id="S4.9.2.p1.45.m13.1.1.6.2" xref="S4.9.2.p1.45.m13.1.1.6.2.cmml">U</mi><mo id="S4.9.2.p1.45.m13.1.1.6.1" xref="S4.9.2.p1.45.m13.1.1.6.1.cmml">β’</mo><mi id="S4.9.2.p1.45.m13.1.1.6.3" xref="S4.9.2.p1.45.m13.1.1.6.3.cmml">U</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.45.m13.1b"><apply id="S4.9.2.p1.45.m13.1.1.cmml" xref="S4.9.2.p1.45.m13.1.1"><and id="S4.9.2.p1.45.m13.1.1a.cmml" xref="S4.9.2.p1.45.m13.1.1"></and><apply id="S4.9.2.p1.45.m13.1.1b.cmml" xref="S4.9.2.p1.45.m13.1.1"><eq id="S4.9.2.p1.45.m13.1.1.3.cmml" xref="S4.9.2.p1.45.m13.1.1.3"></eq><cn id="S4.9.2.p1.45.m13.1.1.2.cmml" type="integer" xref="S4.9.2.p1.45.m13.1.1.2">0</cn><apply id="S4.9.2.p1.45.m13.1.1.4.cmml" xref="S4.9.2.p1.45.m13.1.1.4"><times id="S4.9.2.p1.45.m13.1.1.4.1.cmml" xref="S4.9.2.p1.45.m13.1.1.4.1"></times><ci id="S4.9.2.p1.45.m13.1.1.4.2.cmml" xref="S4.9.2.p1.45.m13.1.1.4.2">π</ci><ci id="S4.9.2.p1.45.m13.1.1.4.3.cmml" xref="S4.9.2.p1.45.m13.1.1.4.3">π</ci></apply></apply><apply id="S4.9.2.p1.45.m13.1.1c.cmml" xref="S4.9.2.p1.45.m13.1.1"><in id="S4.9.2.p1.45.m13.1.1.5.cmml" xref="S4.9.2.p1.45.m13.1.1.5"></in><share href="https://arxiv.org/html/2503.13666v1#S4.9.2.p1.45.m13.1.1.4.cmml" id="S4.9.2.p1.45.m13.1.1d.cmml" xref="S4.9.2.p1.45.m13.1.1"></share><apply id="S4.9.2.p1.45.m13.1.1.6.cmml" xref="S4.9.2.p1.45.m13.1.1.6"><times id="S4.9.2.p1.45.m13.1.1.6.1.cmml" xref="S4.9.2.p1.45.m13.1.1.6.1"></times><ci id="S4.9.2.p1.45.m13.1.1.6.2.cmml" xref="S4.9.2.p1.45.m13.1.1.6.2">π</ci><ci id="S4.9.2.p1.45.m13.1.1.6.3.cmml" xref="S4.9.2.p1.45.m13.1.1.6.3">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.45.m13.1c">0=ab\in UU</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.45.m13.1d">0 = italic_a italic_b β italic_U italic_U</annotation></semantics></math>. Since <math alttext="S" class="ltx_Math" display="inline" id="S4.9.2.p1.46.m14.1"><semantics id="S4.9.2.p1.46.m14.1a"><mi id="S4.9.2.p1.46.m14.1.1" xref="S4.9.2.p1.46.m14.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.46.m14.1b"><ci id="S4.9.2.p1.46.m14.1.1.cmml" xref="S4.9.2.p1.46.m14.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.46.m14.1c">S</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.46.m14.1d">italic_S</annotation></semantics></math> is a Hausdorff topological semilattice, we get that <math alttext="y=yy=0" class="ltx_Math" display="inline" id="S4.9.2.p1.47.m15.1"><semantics id="S4.9.2.p1.47.m15.1a"><mrow id="S4.9.2.p1.47.m15.1.1" xref="S4.9.2.p1.47.m15.1.1.cmml"><mi id="S4.9.2.p1.47.m15.1.1.2" xref="S4.9.2.p1.47.m15.1.1.2.cmml">y</mi><mo id="S4.9.2.p1.47.m15.1.1.3" xref="S4.9.2.p1.47.m15.1.1.3.cmml">=</mo><mrow id="S4.9.2.p1.47.m15.1.1.4" xref="S4.9.2.p1.47.m15.1.1.4.cmml"><mi id="S4.9.2.p1.47.m15.1.1.4.2" xref="S4.9.2.p1.47.m15.1.1.4.2.cmml">y</mi><mo id="S4.9.2.p1.47.m15.1.1.4.1" xref="S4.9.2.p1.47.m15.1.1.4.1.cmml">β’</mo><mi id="S4.9.2.p1.47.m15.1.1.4.3" xref="S4.9.2.p1.47.m15.1.1.4.3.cmml">y</mi></mrow><mo id="S4.9.2.p1.47.m15.1.1.5" xref="S4.9.2.p1.47.m15.1.1.5.cmml">=</mo><mn id="S4.9.2.p1.47.m15.1.1.6" xref="S4.9.2.p1.47.m15.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.47.m15.1b"><apply id="S4.9.2.p1.47.m15.1.1.cmml" xref="S4.9.2.p1.47.m15.1.1"><and id="S4.9.2.p1.47.m15.1.1a.cmml" xref="S4.9.2.p1.47.m15.1.1"></and><apply id="S4.9.2.p1.47.m15.1.1b.cmml" xref="S4.9.2.p1.47.m15.1.1"><eq id="S4.9.2.p1.47.m15.1.1.3.cmml" xref="S4.9.2.p1.47.m15.1.1.3"></eq><ci id="S4.9.2.p1.47.m15.1.1.2.cmml" xref="S4.9.2.p1.47.m15.1.1.2">π¦</ci><apply id="S4.9.2.p1.47.m15.1.1.4.cmml" xref="S4.9.2.p1.47.m15.1.1.4"><times id="S4.9.2.p1.47.m15.1.1.4.1.cmml" xref="S4.9.2.p1.47.m15.1.1.4.1"></times><ci id="S4.9.2.p1.47.m15.1.1.4.2.cmml" xref="S4.9.2.p1.47.m15.1.1.4.2">π¦</ci><ci id="S4.9.2.p1.47.m15.1.1.4.3.cmml" xref="S4.9.2.p1.47.m15.1.1.4.3">π¦</ci></apply></apply><apply id="S4.9.2.p1.47.m15.1.1c.cmml" xref="S4.9.2.p1.47.m15.1.1"><eq id="S4.9.2.p1.47.m15.1.1.5.cmml" xref="S4.9.2.p1.47.m15.1.1.5"></eq><share href="https://arxiv.org/html/2503.13666v1#S4.9.2.p1.47.m15.1.1.4.cmml" id="S4.9.2.p1.47.m15.1.1d.cmml" xref="S4.9.2.p1.47.m15.1.1"></share><cn id="S4.9.2.p1.47.m15.1.1.6.cmml" type="integer" xref="S4.9.2.p1.47.m15.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.47.m15.1c">y=yy=0</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.47.m15.1d">italic_y = italic_y italic_y = 0</annotation></semantics></math>. On the other hand, <math alttext="y\neq 0" class="ltx_Math" display="inline" id="S4.9.2.p1.48.m16.1"><semantics id="S4.9.2.p1.48.m16.1a"><mrow id="S4.9.2.p1.48.m16.1.1" xref="S4.9.2.p1.48.m16.1.1.cmml"><mi id="S4.9.2.p1.48.m16.1.1.2" xref="S4.9.2.p1.48.m16.1.1.2.cmml">y</mi><mo id="S4.9.2.p1.48.m16.1.1.1" xref="S4.9.2.p1.48.m16.1.1.1.cmml">β </mo><mn id="S4.9.2.p1.48.m16.1.1.3" xref="S4.9.2.p1.48.m16.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.48.m16.1b"><apply id="S4.9.2.p1.48.m16.1.1.cmml" xref="S4.9.2.p1.48.m16.1.1"><neq id="S4.9.2.p1.48.m16.1.1.1.cmml" xref="S4.9.2.p1.48.m16.1.1.1"></neq><ci id="S4.9.2.p1.48.m16.1.1.2.cmml" xref="S4.9.2.p1.48.m16.1.1.2">π¦</ci><cn id="S4.9.2.p1.48.m16.1.1.3.cmml" type="integer" xref="S4.9.2.p1.48.m16.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.48.m16.1c">y\neq 0</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.48.m16.1d">italic_y β 0</annotation></semantics></math>, as <math alttext="0" class="ltx_Math" display="inline" id="S4.9.2.p1.49.m17.1"><semantics id="S4.9.2.p1.49.m17.1a"><mn id="S4.9.2.p1.49.m17.1.1" xref="S4.9.2.p1.49.m17.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.49.m17.1b"><cn id="S4.9.2.p1.49.m17.1.1.cmml" type="integer" xref="S4.9.2.p1.49.m17.1.1">0</cn></annotation-xml></semantics></math> is an isolated point, a contradiction. So the set <math alttext="F" class="ltx_Math" display="inline" id="S4.9.2.p1.50.m18.1"><semantics id="S4.9.2.p1.50.m18.1a"><mi id="S4.9.2.p1.50.m18.1.1" xref="S4.9.2.p1.50.m18.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.50.m18.1b"><ci id="S4.9.2.p1.50.m18.1.1.cmml" xref="S4.9.2.p1.50.m18.1.1">πΉ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.50.m18.1c">F</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.50.m18.1d">italic_F</annotation></semantics></math> is finite. At this point it is straightforward to check that <math alttext="S\setminus\{0\}=\beta X\setminus\operatorname{cl}_{\beta X}(I_{L})=\bigcup_{x% \in F}{\uparrow}x" class="ltx_Math" display="inline" id="S4.9.2.p1.51.m19.3"><semantics id="S4.9.2.p1.51.m19.3a"><mrow id="S4.9.2.p1.51.m19.3.3" xref="S4.9.2.p1.51.m19.3.3.cmml"><mrow id="S4.9.2.p1.51.m19.3.3.4" xref="S4.9.2.p1.51.m19.3.3.4.cmml"><mi id="S4.9.2.p1.51.m19.3.3.4.2" xref="S4.9.2.p1.51.m19.3.3.4.2.cmml">S</mi><mo id="S4.9.2.p1.51.m19.3.3.4.1" xref="S4.9.2.p1.51.m19.3.3.4.1.cmml">β</mo><mrow id="S4.9.2.p1.51.m19.3.3.4.3.2" xref="S4.9.2.p1.51.m19.3.3.4.3.1.cmml"><mo id="S4.9.2.p1.51.m19.3.3.4.3.2.1" stretchy="false" xref="S4.9.2.p1.51.m19.3.3.4.3.1.cmml">{</mo><mn id="S4.9.2.p1.51.m19.1.1" xref="S4.9.2.p1.51.m19.1.1.cmml">0</mn><mo id="S4.9.2.p1.51.m19.3.3.4.3.2.2" stretchy="false" xref="S4.9.2.p1.51.m19.3.3.4.3.1.cmml">}</mo></mrow></mrow><mo id="S4.9.2.p1.51.m19.3.3.5" xref="S4.9.2.p1.51.m19.3.3.5.cmml">=</mo><mrow id="S4.9.2.p1.51.m19.3.3.2" xref="S4.9.2.p1.51.m19.3.3.2.cmml"><mrow id="S4.9.2.p1.51.m19.3.3.2.4" xref="S4.9.2.p1.51.m19.3.3.2.4.cmml"><mi id="S4.9.2.p1.51.m19.3.3.2.4.2" xref="S4.9.2.p1.51.m19.3.3.2.4.2.cmml">Ξ²</mi><mo id="S4.9.2.p1.51.m19.3.3.2.4.1" xref="S4.9.2.p1.51.m19.3.3.2.4.1.cmml">β’</mo><mi id="S4.9.2.p1.51.m19.3.3.2.4.3" xref="S4.9.2.p1.51.m19.3.3.2.4.3.cmml">X</mi></mrow><mo id="S4.9.2.p1.51.m19.3.3.2.3" xref="S4.9.2.p1.51.m19.3.3.2.3.cmml">β</mo><mrow id="S4.9.2.p1.51.m19.3.3.2.2.2" xref="S4.9.2.p1.51.m19.3.3.2.2.3.cmml"><msub id="S4.9.2.p1.51.m19.2.2.1.1.1.1" xref="S4.9.2.p1.51.m19.2.2.1.1.1.1.cmml"><mi id="S4.9.2.p1.51.m19.2.2.1.1.1.1.2" xref="S4.9.2.p1.51.m19.2.2.1.1.1.1.2.cmml">cl</mi><mrow id="S4.9.2.p1.51.m19.2.2.1.1.1.1.3" xref="S4.9.2.p1.51.m19.2.2.1.1.1.1.3.cmml"><mi id="S4.9.2.p1.51.m19.2.2.1.1.1.1.3.2" xref="S4.9.2.p1.51.m19.2.2.1.1.1.1.3.2.cmml">Ξ²</mi><mo id="S4.9.2.p1.51.m19.2.2.1.1.1.1.3.1" xref="S4.9.2.p1.51.m19.2.2.1.1.1.1.3.1.cmml">β’</mo><mi id="S4.9.2.p1.51.m19.2.2.1.1.1.1.3.3" xref="S4.9.2.p1.51.m19.2.2.1.1.1.1.3.3.cmml">X</mi></mrow></msub><mo id="S4.9.2.p1.51.m19.3.3.2.2.2a" xref="S4.9.2.p1.51.m19.3.3.2.2.3.cmml">β‘</mo><mrow id="S4.9.2.p1.51.m19.3.3.2.2.2.2" xref="S4.9.2.p1.51.m19.3.3.2.2.3.cmml"><mo id="S4.9.2.p1.51.m19.3.3.2.2.2.2.2" stretchy="false" xref="S4.9.2.p1.51.m19.3.3.2.2.3.cmml">(</mo><msub id="S4.9.2.p1.51.m19.3.3.2.2.2.2.1" xref="S4.9.2.p1.51.m19.3.3.2.2.2.2.1.cmml"><mi id="S4.9.2.p1.51.m19.3.3.2.2.2.2.1.2" xref="S4.9.2.p1.51.m19.3.3.2.2.2.2.1.2.cmml">I</mi><mi id="S4.9.2.p1.51.m19.3.3.2.2.2.2.1.3" xref="S4.9.2.p1.51.m19.3.3.2.2.2.2.1.3.cmml">L</mi></msub><mo id="S4.9.2.p1.51.m19.3.3.2.2.2.2.3" stretchy="false" xref="S4.9.2.p1.51.m19.3.3.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.9.2.p1.51.m19.3.3.6" rspace="0.111em" xref="S4.9.2.p1.51.m19.3.3.6.cmml">=</mo><msub id="S4.9.2.p1.51.m19.3.3.7" xref="S4.9.2.p1.51.m19.3.3.7.cmml"><mo id="S4.9.2.p1.51.m19.3.3.7.2" xref="S4.9.2.p1.51.m19.3.3.7.2.cmml">β</mo><mrow id="S4.9.2.p1.51.m19.3.3.7.3" xref="S4.9.2.p1.51.m19.3.3.7.3.cmml"><mi id="S4.9.2.p1.51.m19.3.3.7.3.2" xref="S4.9.2.p1.51.m19.3.3.7.3.2.cmml">x</mi><mo id="S4.9.2.p1.51.m19.3.3.7.3.1" xref="S4.9.2.p1.51.m19.3.3.7.3.1.cmml">β</mo><mi id="S4.9.2.p1.51.m19.3.3.7.3.3" xref="S4.9.2.p1.51.m19.3.3.7.3.3.cmml">F</mi></mrow></msub><mo id="S4.9.2.p1.51.m19.3.3.8" lspace="0.111em" stretchy="false" xref="S4.9.2.p1.51.m19.3.3.8.cmml">β</mo><mi id="S4.9.2.p1.51.m19.3.3.9" xref="S4.9.2.p1.51.m19.3.3.9.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.9.2.p1.51.m19.3b"><apply id="S4.9.2.p1.51.m19.3.3.cmml" xref="S4.9.2.p1.51.m19.3.3"><and id="S4.9.2.p1.51.m19.3.3a.cmml" xref="S4.9.2.p1.51.m19.3.3"></and><apply id="S4.9.2.p1.51.m19.3.3b.cmml" xref="S4.9.2.p1.51.m19.3.3"><eq id="S4.9.2.p1.51.m19.3.3.5.cmml" xref="S4.9.2.p1.51.m19.3.3.5"></eq><apply id="S4.9.2.p1.51.m19.3.3.4.cmml" xref="S4.9.2.p1.51.m19.3.3.4"><setdiff id="S4.9.2.p1.51.m19.3.3.4.1.cmml" xref="S4.9.2.p1.51.m19.3.3.4.1"></setdiff><ci id="S4.9.2.p1.51.m19.3.3.4.2.cmml" xref="S4.9.2.p1.51.m19.3.3.4.2">π</ci><set id="S4.9.2.p1.51.m19.3.3.4.3.1.cmml" xref="S4.9.2.p1.51.m19.3.3.4.3.2"><cn id="S4.9.2.p1.51.m19.1.1.cmml" type="integer" xref="S4.9.2.p1.51.m19.1.1">0</cn></set></apply><apply id="S4.9.2.p1.51.m19.3.3.2.cmml" xref="S4.9.2.p1.51.m19.3.3.2"><setdiff id="S4.9.2.p1.51.m19.3.3.2.3.cmml" xref="S4.9.2.p1.51.m19.3.3.2.3"></setdiff><apply id="S4.9.2.p1.51.m19.3.3.2.4.cmml" xref="S4.9.2.p1.51.m19.3.3.2.4"><times id="S4.9.2.p1.51.m19.3.3.2.4.1.cmml" xref="S4.9.2.p1.51.m19.3.3.2.4.1"></times><ci id="S4.9.2.p1.51.m19.3.3.2.4.2.cmml" xref="S4.9.2.p1.51.m19.3.3.2.4.2">π½</ci><ci id="S4.9.2.p1.51.m19.3.3.2.4.3.cmml" xref="S4.9.2.p1.51.m19.3.3.2.4.3">π</ci></apply><apply id="S4.9.2.p1.51.m19.3.3.2.2.3.cmml" xref="S4.9.2.p1.51.m19.3.3.2.2.2"><apply id="S4.9.2.p1.51.m19.2.2.1.1.1.1.cmml" xref="S4.9.2.p1.51.m19.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.9.2.p1.51.m19.2.2.1.1.1.1.1.cmml" xref="S4.9.2.p1.51.m19.2.2.1.1.1.1">subscript</csymbol><ci id="S4.9.2.p1.51.m19.2.2.1.1.1.1.2.cmml" xref="S4.9.2.p1.51.m19.2.2.1.1.1.1.2">cl</ci><apply id="S4.9.2.p1.51.m19.2.2.1.1.1.1.3.cmml" xref="S4.9.2.p1.51.m19.2.2.1.1.1.1.3"><times id="S4.9.2.p1.51.m19.2.2.1.1.1.1.3.1.cmml" xref="S4.9.2.p1.51.m19.2.2.1.1.1.1.3.1"></times><ci id="S4.9.2.p1.51.m19.2.2.1.1.1.1.3.2.cmml" xref="S4.9.2.p1.51.m19.2.2.1.1.1.1.3.2">π½</ci><ci id="S4.9.2.p1.51.m19.2.2.1.1.1.1.3.3.cmml" xref="S4.9.2.p1.51.m19.2.2.1.1.1.1.3.3">π</ci></apply></apply><apply id="S4.9.2.p1.51.m19.3.3.2.2.2.2.1.cmml" xref="S4.9.2.p1.51.m19.3.3.2.2.2.2.1"><csymbol cd="ambiguous" id="S4.9.2.p1.51.m19.3.3.2.2.2.2.1.1.cmml" xref="S4.9.2.p1.51.m19.3.3.2.2.2.2.1">subscript</csymbol><ci id="S4.9.2.p1.51.m19.3.3.2.2.2.2.1.2.cmml" xref="S4.9.2.p1.51.m19.3.3.2.2.2.2.1.2">πΌ</ci><ci id="S4.9.2.p1.51.m19.3.3.2.2.2.2.1.3.cmml" xref="S4.9.2.p1.51.m19.3.3.2.2.2.2.1.3">πΏ</ci></apply></apply></apply></apply><apply id="S4.9.2.p1.51.m19.3.3c.cmml" xref="S4.9.2.p1.51.m19.3.3"><eq id="S4.9.2.p1.51.m19.3.3.6.cmml" xref="S4.9.2.p1.51.m19.3.3.6"></eq><share href="https://arxiv.org/html/2503.13666v1#S4.9.2.p1.51.m19.3.3.2.cmml" id="S4.9.2.p1.51.m19.3.3d.cmml" xref="S4.9.2.p1.51.m19.3.3"></share><apply id="S4.9.2.p1.51.m19.3.3.7.cmml" xref="S4.9.2.p1.51.m19.3.3.7"><csymbol cd="ambiguous" id="S4.9.2.p1.51.m19.3.3.7.1.cmml" xref="S4.9.2.p1.51.m19.3.3.7">subscript</csymbol><union id="S4.9.2.p1.51.m19.3.3.7.2.cmml" xref="S4.9.2.p1.51.m19.3.3.7.2"></union><apply id="S4.9.2.p1.51.m19.3.3.7.3.cmml" xref="S4.9.2.p1.51.m19.3.3.7.3"><in id="S4.9.2.p1.51.m19.3.3.7.3.1.cmml" xref="S4.9.2.p1.51.m19.3.3.7.3.1"></in><ci id="S4.9.2.p1.51.m19.3.3.7.3.2.cmml" xref="S4.9.2.p1.51.m19.3.3.7.3.2">π₯</ci><ci id="S4.9.2.p1.51.m19.3.3.7.3.3.cmml" xref="S4.9.2.p1.51.m19.3.3.7.3.3">πΉ</ci></apply></apply></apply><apply id="S4.9.2.p1.51.m19.3.3e.cmml" xref="S4.9.2.p1.51.m19.3.3"><ci id="S4.9.2.p1.51.m19.3.3.8.cmml" xref="S4.9.2.p1.51.m19.3.3.8">β</ci><share href="https://arxiv.org/html/2503.13666v1#S4.9.2.p1.51.m19.3.3.7.cmml" id="S4.9.2.p1.51.m19.3.3f.cmml" xref="S4.9.2.p1.51.m19.3.3"></share><ci id="S4.9.2.p1.51.m19.3.3.9.cmml" xref="S4.9.2.p1.51.m19.3.3.9">π₯</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.9.2.p1.51.m19.3c">S\setminus\{0\}=\beta X\setminus\operatorname{cl}_{\beta X}(I_{L})=\bigcup_{x% \in F}{\uparrow}x</annotation><annotation encoding="application/x-llamapun" id="S4.9.2.p1.51.m19.3d">italic_S β { 0 } = italic_Ξ² italic_X β roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_X end_POSTSUBSCRIPT ( italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) = β start_POSTSUBSCRIPT italic_x β italic_F end_POSTSUBSCRIPT β italic_x</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.10.3.p2"> <p class="ltx_p" id="S4.10.3.p2.2">It remains to check that <math alttext="F\subset X" class="ltx_Math" display="inline" id="S4.10.3.p2.1.m1.1"><semantics id="S4.10.3.p2.1.m1.1a"><mrow id="S4.10.3.p2.1.m1.1.1" xref="S4.10.3.p2.1.m1.1.1.cmml"><mi id="S4.10.3.p2.1.m1.1.1.2" xref="S4.10.3.p2.1.m1.1.1.2.cmml">F</mi><mo id="S4.10.3.p2.1.m1.1.1.1" xref="S4.10.3.p2.1.m1.1.1.1.cmml">β</mo><mi id="S4.10.3.p2.1.m1.1.1.3" xref="S4.10.3.p2.1.m1.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.1.m1.1b"><apply id="S4.10.3.p2.1.m1.1.1.cmml" xref="S4.10.3.p2.1.m1.1.1"><subset id="S4.10.3.p2.1.m1.1.1.1.cmml" xref="S4.10.3.p2.1.m1.1.1.1"></subset><ci id="S4.10.3.p2.1.m1.1.1.2.cmml" xref="S4.10.3.p2.1.m1.1.1.2">πΉ</ci><ci id="S4.10.3.p2.1.m1.1.1.3.cmml" xref="S4.10.3.p2.1.m1.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.1.m1.1c">F\subset X</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.1.m1.1d">italic_F β italic_X</annotation></semantics></math>. Fix any <math alttext="f\in F" class="ltx_Math" display="inline" id="S4.10.3.p2.2.m2.1"><semantics id="S4.10.3.p2.2.m2.1a"><mrow id="S4.10.3.p2.2.m2.1.1" xref="S4.10.3.p2.2.m2.1.1.cmml"><mi id="S4.10.3.p2.2.m2.1.1.2" xref="S4.10.3.p2.2.m2.1.1.2.cmml">f</mi><mo id="S4.10.3.p2.2.m2.1.1.1" xref="S4.10.3.p2.2.m2.1.1.1.cmml">β</mo><mi id="S4.10.3.p2.2.m2.1.1.3" xref="S4.10.3.p2.2.m2.1.1.3.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.2.m2.1b"><apply id="S4.10.3.p2.2.m2.1.1.cmml" xref="S4.10.3.p2.2.m2.1.1"><in id="S4.10.3.p2.2.m2.1.1.1.cmml" xref="S4.10.3.p2.2.m2.1.1.1"></in><ci id="S4.10.3.p2.2.m2.1.1.2.cmml" xref="S4.10.3.p2.2.m2.1.1.2">π</ci><ci id="S4.10.3.p2.2.m2.1.1.3.cmml" xref="S4.10.3.p2.2.m2.1.1.3">πΉ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.2.m2.1c">f\in F</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.2.m2.1d">italic_f β italic_F</annotation></semantics></math>. Observe that</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="P:=S\setminus\big{(}\{0\}\cup\bigcup_{g\in F\setminus\{f\}}{\uparrow}g\big{)}=% \{x\in{\uparrow}f:x\notin\bigcup_{g\in F\setminus\{f\}}{\uparrow}g\}" class="ltx_math_unparsed" display="block" id="S4.Ex14.m1.3"><semantics id="S4.Ex14.m1.3a"><mrow id="S4.Ex14.m1.3b"><mi id="S4.Ex14.m1.3.4">P</mi><mo id="S4.Ex14.m1.3.5" lspace="0.278em" rspace="0.278em">:=</mo><mi id="S4.Ex14.m1.3.6">S</mi><mo id="S4.Ex14.m1.3.7">β</mo><mrow id="S4.Ex14.m1.3.8"><mo id="S4.Ex14.m1.3.8.1" maxsize="120%" minsize="120%">(</mo><mrow id="S4.Ex14.m1.3.8.2"><mo id="S4.Ex14.m1.3.8.2.1" stretchy="false">{</mo><mn id="S4.Ex14.m1.3.3">0</mn><mo id="S4.Ex14.m1.3.8.2.2" stretchy="false">}</mo></mrow><mo id="S4.Ex14.m1.3.8.3" rspace="0.055em">βͺ</mo><munder id="S4.Ex14.m1.3.8.4"><mo id="S4.Ex14.m1.3.8.4.2" movablelimits="false">β</mo><mrow id="S4.Ex14.m1.1.1.1"><mi id="S4.Ex14.m1.1.1.1.3">g</mi><mo id="S4.Ex14.m1.1.1.1.2">β</mo><mrow id="S4.Ex14.m1.1.1.1.4"><mi id="S4.Ex14.m1.1.1.1.4.2">F</mi><mo id="S4.Ex14.m1.1.1.1.4.1">β</mo><mrow id="S4.Ex14.m1.1.1.1.4.3.2"><mo id="S4.Ex14.m1.1.1.1.4.3.2.1" stretchy="false">{</mo><mi id="S4.Ex14.m1.1.1.1.1">f</mi><mo id="S4.Ex14.m1.1.1.1.4.3.2.2" stretchy="false">}</mo></mrow></mrow></mrow></munder><mo id="S4.Ex14.m1.3.8.5" lspace="0.111em" stretchy="false">β</mo><mi id="S4.Ex14.m1.3.8.6">g</mi><mo id="S4.Ex14.m1.3.8.7" maxsize="120%" minsize="120%">)</mo></mrow><mo id="S4.Ex14.m1.3.9">=</mo><mrow id="S4.Ex14.m1.3.10"><mo id="S4.Ex14.m1.3.10.1" stretchy="false">{</mo><mi id="S4.Ex14.m1.3.10.2">x</mi><mo id="S4.Ex14.m1.3.10.3" rspace="0em">β</mo><mo id="S4.Ex14.m1.3.10.4" lspace="0em" stretchy="false">β</mo><mi id="S4.Ex14.m1.3.10.5">f</mi><mo id="S4.Ex14.m1.3.10.6" lspace="0.278em" rspace="0.278em">:</mo><mi id="S4.Ex14.m1.3.10.7">x</mi><mo id="S4.Ex14.m1.3.10.8" rspace="0.111em">β</mo><munder id="S4.Ex14.m1.3.10.9"><mo id="S4.Ex14.m1.3.10.9.2" movablelimits="false">β</mo><mrow id="S4.Ex14.m1.2.2.1"><mi id="S4.Ex14.m1.2.2.1.3">g</mi><mo id="S4.Ex14.m1.2.2.1.2">β</mo><mrow id="S4.Ex14.m1.2.2.1.4"><mi id="S4.Ex14.m1.2.2.1.4.2">F</mi><mo id="S4.Ex14.m1.2.2.1.4.1">β</mo><mrow id="S4.Ex14.m1.2.2.1.4.3.2"><mo id="S4.Ex14.m1.2.2.1.4.3.2.1" stretchy="false">{</mo><mi id="S4.Ex14.m1.2.2.1.1">f</mi><mo id="S4.Ex14.m1.2.2.1.4.3.2.2" stretchy="false">}</mo></mrow></mrow></mrow></munder><mo id="S4.Ex14.m1.3.10.10" lspace="0.111em" stretchy="false">β</mo><mi id="S4.Ex14.m1.3.10.11">g</mi><mo id="S4.Ex14.m1.3.10.12" stretchy="false">}</mo></mrow></mrow><annotation encoding="application/x-tex" id="S4.Ex14.m1.3c">P:=S\setminus\big{(}\{0\}\cup\bigcup_{g\in F\setminus\{f\}}{\uparrow}g\big{)}=% \{x\in{\uparrow}f:x\notin\bigcup_{g\in F\setminus\{f\}}{\uparrow}g\}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex14.m1.3d">italic_P := italic_S β ( { 0 } βͺ β start_POSTSUBSCRIPT italic_g β italic_F β { italic_f } end_POSTSUBSCRIPT β italic_g ) = { italic_x β β italic_f : italic_x β β start_POSTSUBSCRIPT italic_g β italic_F β { italic_f } end_POSTSUBSCRIPT β italic_g }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.10.3.p2.25">is an open subsemilattice of <math alttext="S" class="ltx_Math" display="inline" id="S4.10.3.p2.3.m1.1"><semantics id="S4.10.3.p2.3.m1.1a"><mi id="S4.10.3.p2.3.m1.1.1" xref="S4.10.3.p2.3.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.3.m1.1b"><ci id="S4.10.3.p2.3.m1.1.1.cmml" xref="S4.10.3.p2.3.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.3.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.3.m1.1d">italic_S</annotation></semantics></math> such that <math alttext="\inf P=f\in P" class="ltx_Math" display="inline" id="S4.10.3.p2.4.m2.1"><semantics id="S4.10.3.p2.4.m2.1a"><mrow id="S4.10.3.p2.4.m2.1.1" xref="S4.10.3.p2.4.m2.1.1.cmml"><mrow id="S4.10.3.p2.4.m2.1.1.2" xref="S4.10.3.p2.4.m2.1.1.2.cmml"><mo id="S4.10.3.p2.4.m2.1.1.2.1" rspace="0.167em" xref="S4.10.3.p2.4.m2.1.1.2.1.cmml">inf</mo><mi id="S4.10.3.p2.4.m2.1.1.2.2" xref="S4.10.3.p2.4.m2.1.1.2.2.cmml">P</mi></mrow><mo id="S4.10.3.p2.4.m2.1.1.3" xref="S4.10.3.p2.4.m2.1.1.3.cmml">=</mo><mi id="S4.10.3.p2.4.m2.1.1.4" xref="S4.10.3.p2.4.m2.1.1.4.cmml">f</mi><mo id="S4.10.3.p2.4.m2.1.1.5" xref="S4.10.3.p2.4.m2.1.1.5.cmml">β</mo><mi id="S4.10.3.p2.4.m2.1.1.6" xref="S4.10.3.p2.4.m2.1.1.6.cmml">P</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.4.m2.1b"><apply id="S4.10.3.p2.4.m2.1.1.cmml" xref="S4.10.3.p2.4.m2.1.1"><and id="S4.10.3.p2.4.m2.1.1a.cmml" xref="S4.10.3.p2.4.m2.1.1"></and><apply id="S4.10.3.p2.4.m2.1.1b.cmml" xref="S4.10.3.p2.4.m2.1.1"><eq id="S4.10.3.p2.4.m2.1.1.3.cmml" xref="S4.10.3.p2.4.m2.1.1.3"></eq><apply id="S4.10.3.p2.4.m2.1.1.2.cmml" xref="S4.10.3.p2.4.m2.1.1.2"><csymbol cd="latexml" id="S4.10.3.p2.4.m2.1.1.2.1.cmml" xref="S4.10.3.p2.4.m2.1.1.2.1">infimum</csymbol><ci id="S4.10.3.p2.4.m2.1.1.2.2.cmml" xref="S4.10.3.p2.4.m2.1.1.2.2">π</ci></apply><ci id="S4.10.3.p2.4.m2.1.1.4.cmml" xref="S4.10.3.p2.4.m2.1.1.4">π</ci></apply><apply id="S4.10.3.p2.4.m2.1.1c.cmml" xref="S4.10.3.p2.4.m2.1.1"><in id="S4.10.3.p2.4.m2.1.1.5.cmml" xref="S4.10.3.p2.4.m2.1.1.5"></in><share href="https://arxiv.org/html/2503.13666v1#S4.10.3.p2.4.m2.1.1.4.cmml" id="S4.10.3.p2.4.m2.1.1d.cmml" xref="S4.10.3.p2.4.m2.1.1"></share><ci id="S4.10.3.p2.4.m2.1.1.6.cmml" xref="S4.10.3.p2.4.m2.1.1.6">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.4.m2.1c">\inf P=f\in P</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.4.m2.1d">roman_inf italic_P = italic_f β italic_P</annotation></semantics></math>. Then <math alttext="P\cap X" class="ltx_Math" display="inline" id="S4.10.3.p2.5.m3.1"><semantics id="S4.10.3.p2.5.m3.1a"><mrow id="S4.10.3.p2.5.m3.1.1" xref="S4.10.3.p2.5.m3.1.1.cmml"><mi id="S4.10.3.p2.5.m3.1.1.2" xref="S4.10.3.p2.5.m3.1.1.2.cmml">P</mi><mo id="S4.10.3.p2.5.m3.1.1.1" xref="S4.10.3.p2.5.m3.1.1.1.cmml">β©</mo><mi id="S4.10.3.p2.5.m3.1.1.3" xref="S4.10.3.p2.5.m3.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.5.m3.1b"><apply id="S4.10.3.p2.5.m3.1.1.cmml" xref="S4.10.3.p2.5.m3.1.1"><intersect id="S4.10.3.p2.5.m3.1.1.1.cmml" xref="S4.10.3.p2.5.m3.1.1.1"></intersect><ci id="S4.10.3.p2.5.m3.1.1.2.cmml" xref="S4.10.3.p2.5.m3.1.1.2">π</ci><ci id="S4.10.3.p2.5.m3.1.1.3.cmml" xref="S4.10.3.p2.5.m3.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.5.m3.1c">P\cap X</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.5.m3.1d">italic_P β© italic_X</annotation></semantics></math> is an open subsemilattice of <math alttext="X" class="ltx_Math" display="inline" id="S4.10.3.p2.6.m4.1"><semantics id="S4.10.3.p2.6.m4.1a"><mi id="S4.10.3.p2.6.m4.1.1" xref="S4.10.3.p2.6.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.6.m4.1b"><ci id="S4.10.3.p2.6.m4.1.1.cmml" xref="S4.10.3.p2.6.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.6.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.6.m4.1d">italic_X</annotation></semantics></math>. Taking into account that <math alttext="X" class="ltx_Math" display="inline" id="S4.10.3.p2.7.m5.1"><semantics id="S4.10.3.p2.7.m5.1a"><mi id="S4.10.3.p2.7.m5.1.1" xref="S4.10.3.p2.7.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.7.m5.1b"><ci id="S4.10.3.p2.7.m5.1.1.cmml" xref="S4.10.3.p2.7.m5.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.7.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.7.m5.1d">italic_X</annotation></semantics></math> is separable, Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S4.Thmtheorem1" title="Lemma 4.1. β£ 4. Nyikos semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">4.1</span></a> yields that the semilattice <math alttext="\operatorname{cl}_{X}(P\cap X)" class="ltx_Math" display="inline" id="S4.10.3.p2.8.m6.2"><semantics id="S4.10.3.p2.8.m6.2a"><mrow id="S4.10.3.p2.8.m6.2.2.2" xref="S4.10.3.p2.8.m6.2.2.3.cmml"><msub id="S4.10.3.p2.8.m6.1.1.1.1" xref="S4.10.3.p2.8.m6.1.1.1.1.cmml"><mi id="S4.10.3.p2.8.m6.1.1.1.1.2" xref="S4.10.3.p2.8.m6.1.1.1.1.2.cmml">cl</mi><mi id="S4.10.3.p2.8.m6.1.1.1.1.3" xref="S4.10.3.p2.8.m6.1.1.1.1.3.cmml">X</mi></msub><mo id="S4.10.3.p2.8.m6.2.2.2a" xref="S4.10.3.p2.8.m6.2.2.3.cmml">β‘</mo><mrow id="S4.10.3.p2.8.m6.2.2.2.2" xref="S4.10.3.p2.8.m6.2.2.3.cmml"><mo id="S4.10.3.p2.8.m6.2.2.2.2.2" stretchy="false" xref="S4.10.3.p2.8.m6.2.2.3.cmml">(</mo><mrow id="S4.10.3.p2.8.m6.2.2.2.2.1" xref="S4.10.3.p2.8.m6.2.2.2.2.1.cmml"><mi id="S4.10.3.p2.8.m6.2.2.2.2.1.2" xref="S4.10.3.p2.8.m6.2.2.2.2.1.2.cmml">P</mi><mo id="S4.10.3.p2.8.m6.2.2.2.2.1.1" xref="S4.10.3.p2.8.m6.2.2.2.2.1.1.cmml">β©</mo><mi id="S4.10.3.p2.8.m6.2.2.2.2.1.3" xref="S4.10.3.p2.8.m6.2.2.2.2.1.3.cmml">X</mi></mrow><mo id="S4.10.3.p2.8.m6.2.2.2.2.3" stretchy="false" xref="S4.10.3.p2.8.m6.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.8.m6.2b"><apply id="S4.10.3.p2.8.m6.2.2.3.cmml" xref="S4.10.3.p2.8.m6.2.2.2"><apply id="S4.10.3.p2.8.m6.1.1.1.1.cmml" xref="S4.10.3.p2.8.m6.1.1.1.1"><csymbol cd="ambiguous" id="S4.10.3.p2.8.m6.1.1.1.1.1.cmml" xref="S4.10.3.p2.8.m6.1.1.1.1">subscript</csymbol><ci id="S4.10.3.p2.8.m6.1.1.1.1.2.cmml" xref="S4.10.3.p2.8.m6.1.1.1.1.2">cl</ci><ci id="S4.10.3.p2.8.m6.1.1.1.1.3.cmml" xref="S4.10.3.p2.8.m6.1.1.1.1.3">π</ci></apply><apply id="S4.10.3.p2.8.m6.2.2.2.2.1.cmml" xref="S4.10.3.p2.8.m6.2.2.2.2.1"><intersect id="S4.10.3.p2.8.m6.2.2.2.2.1.1.cmml" xref="S4.10.3.p2.8.m6.2.2.2.2.1.1"></intersect><ci id="S4.10.3.p2.8.m6.2.2.2.2.1.2.cmml" xref="S4.10.3.p2.8.m6.2.2.2.2.1.2">π</ci><ci id="S4.10.3.p2.8.m6.2.2.2.2.1.3.cmml" xref="S4.10.3.p2.8.m6.2.2.2.2.1.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.8.m6.2c">\operatorname{cl}_{X}(P\cap X)</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.8.m6.2d">roman_cl start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_P β© italic_X )</annotation></semantics></math> contains the smallest element <math alttext="p" class="ltx_Math" display="inline" id="S4.10.3.p2.9.m7.1"><semantics id="S4.10.3.p2.9.m7.1a"><mi id="S4.10.3.p2.9.m7.1.1" xref="S4.10.3.p2.9.m7.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.9.m7.1b"><ci id="S4.10.3.p2.9.m7.1.1.cmml" xref="S4.10.3.p2.9.m7.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.9.m7.1c">p</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.9.m7.1d">italic_p</annotation></semantics></math>. Since <math alttext="\operatorname{cl}_{X}(P\cap X)\subseteq\operatorname{cl}_{\beta X}(P)\subseteq% {\uparrow}f" class="ltx_math_unparsed" display="inline" id="S4.10.3.p2.10.m8.1"><semantics id="S4.10.3.p2.10.m8.1a"><mrow id="S4.10.3.p2.10.m8.1b"><msub id="S4.10.3.p2.10.m8.1.2"><mi id="S4.10.3.p2.10.m8.1.2.2">cl</mi><mi id="S4.10.3.p2.10.m8.1.2.3">X</mi></msub><mrow id="S4.10.3.p2.10.m8.1.3"><mo id="S4.10.3.p2.10.m8.1.3.1" stretchy="false">(</mo><mi id="S4.10.3.p2.10.m8.1.3.2">P</mi><mo id="S4.10.3.p2.10.m8.1.3.3">β©</mo><mi id="S4.10.3.p2.10.m8.1.3.4">X</mi><mo id="S4.10.3.p2.10.m8.1.3.5" stretchy="false">)</mo></mrow><mo id="S4.10.3.p2.10.m8.1.4">β</mo><msub id="S4.10.3.p2.10.m8.1.5"><mi id="S4.10.3.p2.10.m8.1.5.2">cl</mi><mrow id="S4.10.3.p2.10.m8.1.5.3"><mi id="S4.10.3.p2.10.m8.1.5.3.2">Ξ²</mi><mo id="S4.10.3.p2.10.m8.1.5.3.1">β’</mo><mi id="S4.10.3.p2.10.m8.1.5.3.3">X</mi></mrow></msub><mrow id="S4.10.3.p2.10.m8.1.6"><mo id="S4.10.3.p2.10.m8.1.6.1" stretchy="false">(</mo><mi id="S4.10.3.p2.10.m8.1.1">P</mi><mo id="S4.10.3.p2.10.m8.1.6.2" stretchy="false">)</mo></mrow><mo id="S4.10.3.p2.10.m8.1.7" rspace="0em">β</mo><mo id="S4.10.3.p2.10.m8.1.8" lspace="0em" stretchy="false">β</mo><mi id="S4.10.3.p2.10.m8.1.9">f</mi></mrow><annotation encoding="application/x-tex" id="S4.10.3.p2.10.m8.1c">\operatorname{cl}_{X}(P\cap X)\subseteq\operatorname{cl}_{\beta X}(P)\subseteq% {\uparrow}f</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.10.m8.1d">roman_cl start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_P β© italic_X ) β roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_X end_POSTSUBSCRIPT ( italic_P ) β β italic_f</annotation></semantics></math>, we get <math alttext="f\leq p" class="ltx_Math" display="inline" id="S4.10.3.p2.11.m9.1"><semantics id="S4.10.3.p2.11.m9.1a"><mrow id="S4.10.3.p2.11.m9.1.1" xref="S4.10.3.p2.11.m9.1.1.cmml"><mi id="S4.10.3.p2.11.m9.1.1.2" xref="S4.10.3.p2.11.m9.1.1.2.cmml">f</mi><mo id="S4.10.3.p2.11.m9.1.1.1" xref="S4.10.3.p2.11.m9.1.1.1.cmml">β€</mo><mi id="S4.10.3.p2.11.m9.1.1.3" xref="S4.10.3.p2.11.m9.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.11.m9.1b"><apply id="S4.10.3.p2.11.m9.1.1.cmml" xref="S4.10.3.p2.11.m9.1.1"><leq id="S4.10.3.p2.11.m9.1.1.1.cmml" xref="S4.10.3.p2.11.m9.1.1.1"></leq><ci id="S4.10.3.p2.11.m9.1.1.2.cmml" xref="S4.10.3.p2.11.m9.1.1.2">π</ci><ci id="S4.10.3.p2.11.m9.1.1.3.cmml" xref="S4.10.3.p2.11.m9.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.11.m9.1c">f\leq p</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.11.m9.1d">italic_f β€ italic_p</annotation></semantics></math>. Since <math alttext="X" class="ltx_Math" display="inline" id="S4.10.3.p2.12.m10.1"><semantics id="S4.10.3.p2.12.m10.1a"><mi id="S4.10.3.p2.12.m10.1.1" xref="S4.10.3.p2.12.m10.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.12.m10.1b"><ci id="S4.10.3.p2.12.m10.1.1.cmml" xref="S4.10.3.p2.12.m10.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.12.m10.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.12.m10.1d">italic_X</annotation></semantics></math> is dense in <math alttext="\beta X" class="ltx_Math" display="inline" id="S4.10.3.p2.13.m11.1"><semantics id="S4.10.3.p2.13.m11.1a"><mrow id="S4.10.3.p2.13.m11.1.1" xref="S4.10.3.p2.13.m11.1.1.cmml"><mi id="S4.10.3.p2.13.m11.1.1.2" xref="S4.10.3.p2.13.m11.1.1.2.cmml">Ξ²</mi><mo id="S4.10.3.p2.13.m11.1.1.1" xref="S4.10.3.p2.13.m11.1.1.1.cmml">β’</mo><mi id="S4.10.3.p2.13.m11.1.1.3" xref="S4.10.3.p2.13.m11.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.13.m11.1b"><apply id="S4.10.3.p2.13.m11.1.1.cmml" xref="S4.10.3.p2.13.m11.1.1"><times id="S4.10.3.p2.13.m11.1.1.1.cmml" xref="S4.10.3.p2.13.m11.1.1.1"></times><ci id="S4.10.3.p2.13.m11.1.1.2.cmml" xref="S4.10.3.p2.13.m11.1.1.2">π½</ci><ci id="S4.10.3.p2.13.m11.1.1.3.cmml" xref="S4.10.3.p2.13.m11.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.13.m11.1c">\beta X</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.13.m11.1d">italic_Ξ² italic_X</annotation></semantics></math> and <math alttext="P" class="ltx_Math" display="inline" id="S4.10.3.p2.14.m12.1"><semantics id="S4.10.3.p2.14.m12.1a"><mi id="S4.10.3.p2.14.m12.1.1" xref="S4.10.3.p2.14.m12.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.14.m12.1b"><ci id="S4.10.3.p2.14.m12.1.1.cmml" xref="S4.10.3.p2.14.m12.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.14.m12.1c">P</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.14.m12.1d">italic_P</annotation></semantics></math> is open, for every open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S4.10.3.p2.15.m13.1"><semantics id="S4.10.3.p2.15.m13.1a"><mi id="S4.10.3.p2.15.m13.1.1" xref="S4.10.3.p2.15.m13.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.15.m13.1b"><ci id="S4.10.3.p2.15.m13.1.1.cmml" xref="S4.10.3.p2.15.m13.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.15.m13.1c">U</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.15.m13.1d">italic_U</annotation></semantics></math> of <math alttext="f" class="ltx_Math" display="inline" id="S4.10.3.p2.16.m14.1"><semantics id="S4.10.3.p2.16.m14.1a"><mi id="S4.10.3.p2.16.m14.1.1" xref="S4.10.3.p2.16.m14.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.16.m14.1b"><ci id="S4.10.3.p2.16.m14.1.1.cmml" xref="S4.10.3.p2.16.m14.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.16.m14.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.16.m14.1d">italic_f</annotation></semantics></math> the set <math alttext="U\cap P\cap X" class="ltx_Math" display="inline" id="S4.10.3.p2.17.m15.1"><semantics id="S4.10.3.p2.17.m15.1a"><mrow id="S4.10.3.p2.17.m15.1.1" xref="S4.10.3.p2.17.m15.1.1.cmml"><mi id="S4.10.3.p2.17.m15.1.1.2" xref="S4.10.3.p2.17.m15.1.1.2.cmml">U</mi><mo id="S4.10.3.p2.17.m15.1.1.1" xref="S4.10.3.p2.17.m15.1.1.1.cmml">β©</mo><mi id="S4.10.3.p2.17.m15.1.1.3" xref="S4.10.3.p2.17.m15.1.1.3.cmml">P</mi><mo id="S4.10.3.p2.17.m15.1.1.1a" xref="S4.10.3.p2.17.m15.1.1.1.cmml">β©</mo><mi id="S4.10.3.p2.17.m15.1.1.4" xref="S4.10.3.p2.17.m15.1.1.4.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.17.m15.1b"><apply id="S4.10.3.p2.17.m15.1.1.cmml" xref="S4.10.3.p2.17.m15.1.1"><intersect id="S4.10.3.p2.17.m15.1.1.1.cmml" xref="S4.10.3.p2.17.m15.1.1.1"></intersect><ci id="S4.10.3.p2.17.m15.1.1.2.cmml" xref="S4.10.3.p2.17.m15.1.1.2">π</ci><ci id="S4.10.3.p2.17.m15.1.1.3.cmml" xref="S4.10.3.p2.17.m15.1.1.3">π</ci><ci id="S4.10.3.p2.17.m15.1.1.4.cmml" xref="S4.10.3.p2.17.m15.1.1.4">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.17.m15.1c">U\cap P\cap X</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.17.m15.1d">italic_U β© italic_P β© italic_X</annotation></semantics></math> is not empty. Thus <math alttext="p\in Up" class="ltx_Math" display="inline" id="S4.10.3.p2.18.m16.1"><semantics id="S4.10.3.p2.18.m16.1a"><mrow id="S4.10.3.p2.18.m16.1.1" xref="S4.10.3.p2.18.m16.1.1.cmml"><mi id="S4.10.3.p2.18.m16.1.1.2" xref="S4.10.3.p2.18.m16.1.1.2.cmml">p</mi><mo id="S4.10.3.p2.18.m16.1.1.1" xref="S4.10.3.p2.18.m16.1.1.1.cmml">β</mo><mrow id="S4.10.3.p2.18.m16.1.1.3" xref="S4.10.3.p2.18.m16.1.1.3.cmml"><mi id="S4.10.3.p2.18.m16.1.1.3.2" xref="S4.10.3.p2.18.m16.1.1.3.2.cmml">U</mi><mo id="S4.10.3.p2.18.m16.1.1.3.1" xref="S4.10.3.p2.18.m16.1.1.3.1.cmml">β’</mo><mi id="S4.10.3.p2.18.m16.1.1.3.3" xref="S4.10.3.p2.18.m16.1.1.3.3.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.18.m16.1b"><apply id="S4.10.3.p2.18.m16.1.1.cmml" xref="S4.10.3.p2.18.m16.1.1"><in id="S4.10.3.p2.18.m16.1.1.1.cmml" xref="S4.10.3.p2.18.m16.1.1.1"></in><ci id="S4.10.3.p2.18.m16.1.1.2.cmml" xref="S4.10.3.p2.18.m16.1.1.2">π</ci><apply id="S4.10.3.p2.18.m16.1.1.3.cmml" xref="S4.10.3.p2.18.m16.1.1.3"><times id="S4.10.3.p2.18.m16.1.1.3.1.cmml" xref="S4.10.3.p2.18.m16.1.1.3.1"></times><ci id="S4.10.3.p2.18.m16.1.1.3.2.cmml" xref="S4.10.3.p2.18.m16.1.1.3.2">π</ci><ci id="S4.10.3.p2.18.m16.1.1.3.3.cmml" xref="S4.10.3.p2.18.m16.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.18.m16.1c">p\in Up</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.18.m16.1d">italic_p β italic_U italic_p</annotation></semantics></math> for each open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S4.10.3.p2.19.m17.1"><semantics id="S4.10.3.p2.19.m17.1a"><mi id="S4.10.3.p2.19.m17.1.1" xref="S4.10.3.p2.19.m17.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.19.m17.1b"><ci id="S4.10.3.p2.19.m17.1.1.cmml" xref="S4.10.3.p2.19.m17.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.19.m17.1c">U</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.19.m17.1d">italic_U</annotation></semantics></math> of <math alttext="f" class="ltx_Math" display="inline" id="S4.10.3.p2.20.m18.1"><semantics id="S4.10.3.p2.20.m18.1a"><mi id="S4.10.3.p2.20.m18.1.1" xref="S4.10.3.p2.20.m18.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.20.m18.1b"><ci id="S4.10.3.p2.20.m18.1.1.cmml" xref="S4.10.3.p2.20.m18.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.20.m18.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.20.m18.1d">italic_f</annotation></semantics></math>. Since <math alttext="S" class="ltx_Math" display="inline" id="S4.10.3.p2.21.m19.1"><semantics id="S4.10.3.p2.21.m19.1a"><mi id="S4.10.3.p2.21.m19.1.1" xref="S4.10.3.p2.21.m19.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.21.m19.1b"><ci id="S4.10.3.p2.21.m19.1.1.cmml" xref="S4.10.3.p2.21.m19.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.21.m19.1c">S</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.21.m19.1d">italic_S</annotation></semantics></math> is a Hausdorff topological semilattice, <math alttext="fp=p" class="ltx_Math" display="inline" id="S4.10.3.p2.22.m20.1"><semantics id="S4.10.3.p2.22.m20.1a"><mrow id="S4.10.3.p2.22.m20.1.1" xref="S4.10.3.p2.22.m20.1.1.cmml"><mrow id="S4.10.3.p2.22.m20.1.1.2" xref="S4.10.3.p2.22.m20.1.1.2.cmml"><mi id="S4.10.3.p2.22.m20.1.1.2.2" xref="S4.10.3.p2.22.m20.1.1.2.2.cmml">f</mi><mo id="S4.10.3.p2.22.m20.1.1.2.1" xref="S4.10.3.p2.22.m20.1.1.2.1.cmml">β’</mo><mi id="S4.10.3.p2.22.m20.1.1.2.3" xref="S4.10.3.p2.22.m20.1.1.2.3.cmml">p</mi></mrow><mo id="S4.10.3.p2.22.m20.1.1.1" xref="S4.10.3.p2.22.m20.1.1.1.cmml">=</mo><mi id="S4.10.3.p2.22.m20.1.1.3" xref="S4.10.3.p2.22.m20.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.22.m20.1b"><apply id="S4.10.3.p2.22.m20.1.1.cmml" xref="S4.10.3.p2.22.m20.1.1"><eq id="S4.10.3.p2.22.m20.1.1.1.cmml" xref="S4.10.3.p2.22.m20.1.1.1"></eq><apply id="S4.10.3.p2.22.m20.1.1.2.cmml" xref="S4.10.3.p2.22.m20.1.1.2"><times id="S4.10.3.p2.22.m20.1.1.2.1.cmml" xref="S4.10.3.p2.22.m20.1.1.2.1"></times><ci id="S4.10.3.p2.22.m20.1.1.2.2.cmml" xref="S4.10.3.p2.22.m20.1.1.2.2">π</ci><ci id="S4.10.3.p2.22.m20.1.1.2.3.cmml" xref="S4.10.3.p2.22.m20.1.1.2.3">π</ci></apply><ci id="S4.10.3.p2.22.m20.1.1.3.cmml" xref="S4.10.3.p2.22.m20.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.22.m20.1c">fp=p</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.22.m20.1d">italic_f italic_p = italic_p</annotation></semantics></math>, i.e. <math alttext="p\leq f" class="ltx_Math" display="inline" id="S4.10.3.p2.23.m21.1"><semantics id="S4.10.3.p2.23.m21.1a"><mrow id="S4.10.3.p2.23.m21.1.1" xref="S4.10.3.p2.23.m21.1.1.cmml"><mi id="S4.10.3.p2.23.m21.1.1.2" xref="S4.10.3.p2.23.m21.1.1.2.cmml">p</mi><mo id="S4.10.3.p2.23.m21.1.1.1" xref="S4.10.3.p2.23.m21.1.1.1.cmml">β€</mo><mi id="S4.10.3.p2.23.m21.1.1.3" xref="S4.10.3.p2.23.m21.1.1.3.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.23.m21.1b"><apply id="S4.10.3.p2.23.m21.1.1.cmml" xref="S4.10.3.p2.23.m21.1.1"><leq id="S4.10.3.p2.23.m21.1.1.1.cmml" xref="S4.10.3.p2.23.m21.1.1.1"></leq><ci id="S4.10.3.p2.23.m21.1.1.2.cmml" xref="S4.10.3.p2.23.m21.1.1.2">π</ci><ci id="S4.10.3.p2.23.m21.1.1.3.cmml" xref="S4.10.3.p2.23.m21.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.23.m21.1c">p\leq f</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.23.m21.1d">italic_p β€ italic_f</annotation></semantics></math>. Hence <math alttext="p=f" class="ltx_Math" display="inline" id="S4.10.3.p2.24.m22.1"><semantics id="S4.10.3.p2.24.m22.1a"><mrow id="S4.10.3.p2.24.m22.1.1" xref="S4.10.3.p2.24.m22.1.1.cmml"><mi id="S4.10.3.p2.24.m22.1.1.2" xref="S4.10.3.p2.24.m22.1.1.2.cmml">p</mi><mo id="S4.10.3.p2.24.m22.1.1.1" xref="S4.10.3.p2.24.m22.1.1.1.cmml">=</mo><mi id="S4.10.3.p2.24.m22.1.1.3" xref="S4.10.3.p2.24.m22.1.1.3.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.24.m22.1b"><apply id="S4.10.3.p2.24.m22.1.1.cmml" xref="S4.10.3.p2.24.m22.1.1"><eq id="S4.10.3.p2.24.m22.1.1.1.cmml" xref="S4.10.3.p2.24.m22.1.1.1"></eq><ci id="S4.10.3.p2.24.m22.1.1.2.cmml" xref="S4.10.3.p2.24.m22.1.1.2">π</ci><ci id="S4.10.3.p2.24.m22.1.1.3.cmml" xref="S4.10.3.p2.24.m22.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.24.m22.1c">p=f</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.24.m22.1d">italic_p = italic_f</annotation></semantics></math> and <math alttext="f\in\operatorname{cl}_{X}(P\cap X)\subset X" class="ltx_Math" display="inline" id="S4.10.3.p2.25.m23.2"><semantics id="S4.10.3.p2.25.m23.2a"><mrow id="S4.10.3.p2.25.m23.2.2" xref="S4.10.3.p2.25.m23.2.2.cmml"><mi id="S4.10.3.p2.25.m23.2.2.4" xref="S4.10.3.p2.25.m23.2.2.4.cmml">f</mi><mo id="S4.10.3.p2.25.m23.2.2.5" xref="S4.10.3.p2.25.m23.2.2.5.cmml">β</mo><mrow id="S4.10.3.p2.25.m23.2.2.2.2" xref="S4.10.3.p2.25.m23.2.2.2.3.cmml"><msub id="S4.10.3.p2.25.m23.1.1.1.1.1" xref="S4.10.3.p2.25.m23.1.1.1.1.1.cmml"><mi id="S4.10.3.p2.25.m23.1.1.1.1.1.2" xref="S4.10.3.p2.25.m23.1.1.1.1.1.2.cmml">cl</mi><mi id="S4.10.3.p2.25.m23.1.1.1.1.1.3" xref="S4.10.3.p2.25.m23.1.1.1.1.1.3.cmml">X</mi></msub><mo id="S4.10.3.p2.25.m23.2.2.2.2a" xref="S4.10.3.p2.25.m23.2.2.2.3.cmml">β‘</mo><mrow id="S4.10.3.p2.25.m23.2.2.2.2.2" xref="S4.10.3.p2.25.m23.2.2.2.3.cmml"><mo id="S4.10.3.p2.25.m23.2.2.2.2.2.2" stretchy="false" xref="S4.10.3.p2.25.m23.2.2.2.3.cmml">(</mo><mrow id="S4.10.3.p2.25.m23.2.2.2.2.2.1" xref="S4.10.3.p2.25.m23.2.2.2.2.2.1.cmml"><mi id="S4.10.3.p2.25.m23.2.2.2.2.2.1.2" xref="S4.10.3.p2.25.m23.2.2.2.2.2.1.2.cmml">P</mi><mo id="S4.10.3.p2.25.m23.2.2.2.2.2.1.1" xref="S4.10.3.p2.25.m23.2.2.2.2.2.1.1.cmml">β©</mo><mi id="S4.10.3.p2.25.m23.2.2.2.2.2.1.3" xref="S4.10.3.p2.25.m23.2.2.2.2.2.1.3.cmml">X</mi></mrow><mo id="S4.10.3.p2.25.m23.2.2.2.2.2.3" stretchy="false" xref="S4.10.3.p2.25.m23.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.10.3.p2.25.m23.2.2.6" xref="S4.10.3.p2.25.m23.2.2.6.cmml">β</mo><mi id="S4.10.3.p2.25.m23.2.2.7" xref="S4.10.3.p2.25.m23.2.2.7.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.10.3.p2.25.m23.2b"><apply id="S4.10.3.p2.25.m23.2.2.cmml" xref="S4.10.3.p2.25.m23.2.2"><and id="S4.10.3.p2.25.m23.2.2a.cmml" xref="S4.10.3.p2.25.m23.2.2"></and><apply id="S4.10.3.p2.25.m23.2.2b.cmml" xref="S4.10.3.p2.25.m23.2.2"><in id="S4.10.3.p2.25.m23.2.2.5.cmml" xref="S4.10.3.p2.25.m23.2.2.5"></in><ci id="S4.10.3.p2.25.m23.2.2.4.cmml" xref="S4.10.3.p2.25.m23.2.2.4">π</ci><apply id="S4.10.3.p2.25.m23.2.2.2.3.cmml" xref="S4.10.3.p2.25.m23.2.2.2.2"><apply id="S4.10.3.p2.25.m23.1.1.1.1.1.cmml" xref="S4.10.3.p2.25.m23.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.10.3.p2.25.m23.1.1.1.1.1.1.cmml" xref="S4.10.3.p2.25.m23.1.1.1.1.1">subscript</csymbol><ci id="S4.10.3.p2.25.m23.1.1.1.1.1.2.cmml" xref="S4.10.3.p2.25.m23.1.1.1.1.1.2">cl</ci><ci id="S4.10.3.p2.25.m23.1.1.1.1.1.3.cmml" xref="S4.10.3.p2.25.m23.1.1.1.1.1.3">π</ci></apply><apply id="S4.10.3.p2.25.m23.2.2.2.2.2.1.cmml" xref="S4.10.3.p2.25.m23.2.2.2.2.2.1"><intersect id="S4.10.3.p2.25.m23.2.2.2.2.2.1.1.cmml" xref="S4.10.3.p2.25.m23.2.2.2.2.2.1.1"></intersect><ci id="S4.10.3.p2.25.m23.2.2.2.2.2.1.2.cmml" xref="S4.10.3.p2.25.m23.2.2.2.2.2.1.2">π</ci><ci id="S4.10.3.p2.25.m23.2.2.2.2.2.1.3.cmml" xref="S4.10.3.p2.25.m23.2.2.2.2.2.1.3">π</ci></apply></apply></apply><apply id="S4.10.3.p2.25.m23.2.2c.cmml" xref="S4.10.3.p2.25.m23.2.2"><subset id="S4.10.3.p2.25.m23.2.2.6.cmml" xref="S4.10.3.p2.25.m23.2.2.6"></subset><share href="https://arxiv.org/html/2503.13666v1#S4.10.3.p2.25.m23.2.2.2.cmml" id="S4.10.3.p2.25.m23.2.2d.cmml" xref="S4.10.3.p2.25.m23.2.2"></share><ci id="S4.10.3.p2.25.m23.2.2.7.cmml" xref="S4.10.3.p2.25.m23.2.2.7">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.10.3.p2.25.m23.2c">f\in\operatorname{cl}_{X}(P\cap X)\subset X</annotation><annotation encoding="application/x-llamapun" id="S4.10.3.p2.25.m23.2d">italic_f β roman_cl start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_P β© italic_X ) β italic_X</annotation></semantics></math>. β</p> </div> </div> <div class="ltx_para" id="S4.11.p3"> <p class="ltx_p" id="S4.11.p3.2">By Claim <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S4.Thmtheorem10" title="Claim 4.10. β£ Proof. β£ 4. Nyikos semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">4.10</span></a>, the sets <math alttext="I_{L}" class="ltx_Math" display="inline" id="S4.11.p3.1.m1.1"><semantics id="S4.11.p3.1.m1.1a"><msub id="S4.11.p3.1.m1.1.1" xref="S4.11.p3.1.m1.1.1.cmml"><mi id="S4.11.p3.1.m1.1.1.2" xref="S4.11.p3.1.m1.1.1.2.cmml">I</mi><mi id="S4.11.p3.1.m1.1.1.3" xref="S4.11.p3.1.m1.1.1.3.cmml">L</mi></msub><annotation-xml encoding="MathML-Content" id="S4.11.p3.1.m1.1b"><apply id="S4.11.p3.1.m1.1.1.cmml" xref="S4.11.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.11.p3.1.m1.1.1.1.cmml" xref="S4.11.p3.1.m1.1.1">subscript</csymbol><ci id="S4.11.p3.1.m1.1.1.2.cmml" xref="S4.11.p3.1.m1.1.1.2">πΌ</ci><ci id="S4.11.p3.1.m1.1.1.3.cmml" xref="S4.11.p3.1.m1.1.1.3">πΏ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p3.1.m1.1c">I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.11.p3.1.m1.1d">italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="X\setminus I_{L}" class="ltx_Math" display="inline" id="S4.11.p3.2.m2.1"><semantics id="S4.11.p3.2.m2.1a"><mrow id="S4.11.p3.2.m2.1.1" xref="S4.11.p3.2.m2.1.1.cmml"><mi id="S4.11.p3.2.m2.1.1.2" xref="S4.11.p3.2.m2.1.1.2.cmml">X</mi><mo id="S4.11.p3.2.m2.1.1.1" xref="S4.11.p3.2.m2.1.1.1.cmml">β</mo><msub id="S4.11.p3.2.m2.1.1.3" xref="S4.11.p3.2.m2.1.1.3.cmml"><mi id="S4.11.p3.2.m2.1.1.3.2" xref="S4.11.p3.2.m2.1.1.3.2.cmml">I</mi><mi id="S4.11.p3.2.m2.1.1.3.3" xref="S4.11.p3.2.m2.1.1.3.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p3.2.m2.1b"><apply id="S4.11.p3.2.m2.1.1.cmml" xref="S4.11.p3.2.m2.1.1"><setdiff id="S4.11.p3.2.m2.1.1.1.cmml" xref="S4.11.p3.2.m2.1.1.1"></setdiff><ci id="S4.11.p3.2.m2.1.1.2.cmml" xref="S4.11.p3.2.m2.1.1.2">π</ci><apply id="S4.11.p3.2.m2.1.1.3.cmml" xref="S4.11.p3.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.11.p3.2.m2.1.1.3.1.cmml" xref="S4.11.p3.2.m2.1.1.3">subscript</csymbol><ci id="S4.11.p3.2.m2.1.1.3.2.cmml" xref="S4.11.p3.2.m2.1.1.3.2">πΌ</ci><ci id="S4.11.p3.2.m2.1.1.3.3.cmml" xref="S4.11.p3.2.m2.1.1.3.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p3.2.m2.1c">X\setminus I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.11.p3.2.m2.1d">italic_X β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> are clopen. Corollary 3.6.2 from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib22" title="">22</a>]</cite> implies that</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex15"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{cl}_{\beta X}(X\setminus I_{L})\cap\operatorname{cl}_{\beta X}(I% _{L})=\varnothing." class="ltx_Math" display="block" id="S4.Ex15.m1.1"><semantics id="S4.Ex15.m1.1a"><mrow id="S4.Ex15.m1.1.1.1" xref="S4.Ex15.m1.1.1.1.1.cmml"><mrow id="S4.Ex15.m1.1.1.1.1" xref="S4.Ex15.m1.1.1.1.1.cmml"><mrow id="S4.Ex15.m1.1.1.1.1.4" xref="S4.Ex15.m1.1.1.1.1.4.cmml"><mrow id="S4.Ex15.m1.1.1.1.1.2.2.2" xref="S4.Ex15.m1.1.1.1.1.2.2.3.cmml"><msub id="S4.Ex15.m1.1.1.1.1.1.1.1.1" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex15.m1.1.1.1.1.1.1.1.1.2" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.2.cmml">cl</mi><mrow id="S4.Ex15.m1.1.1.1.1.1.1.1.1.3" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S4.Ex15.m1.1.1.1.1.1.1.1.1.3.2" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.3.2.cmml">Ξ²</mi><mo id="S4.Ex15.m1.1.1.1.1.1.1.1.1.3.1" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.3.1.cmml">β’</mo><mi id="S4.Ex15.m1.1.1.1.1.1.1.1.1.3.3" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.3.3.cmml">X</mi></mrow></msub><mo id="S4.Ex15.m1.1.1.1.1.2.2.2a" xref="S4.Ex15.m1.1.1.1.1.2.2.3.cmml">β‘</mo><mrow id="S4.Ex15.m1.1.1.1.1.2.2.2.2" xref="S4.Ex15.m1.1.1.1.1.2.2.3.cmml"><mo id="S4.Ex15.m1.1.1.1.1.2.2.2.2.2" stretchy="false" xref="S4.Ex15.m1.1.1.1.1.2.2.3.cmml">(</mo><mrow id="S4.Ex15.m1.1.1.1.1.2.2.2.2.1" xref="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.cmml"><mi id="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.2" xref="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.2.cmml">X</mi><mo id="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.1" xref="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.1.cmml">β</mo><msub id="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.3" xref="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.3.cmml"><mi id="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.3.2" xref="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.3.2.cmml">I</mi><mi id="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.3.3" xref="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.3.3.cmml">L</mi></msub></mrow><mo id="S4.Ex15.m1.1.1.1.1.2.2.2.2.3" stretchy="false" xref="S4.Ex15.m1.1.1.1.1.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.Ex15.m1.1.1.1.1.4.5" xref="S4.Ex15.m1.1.1.1.1.4.5.cmml">β©</mo><mrow id="S4.Ex15.m1.1.1.1.1.4.4.2" xref="S4.Ex15.m1.1.1.1.1.4.4.3.cmml"><msub id="S4.Ex15.m1.1.1.1.1.3.3.1.1" xref="S4.Ex15.m1.1.1.1.1.3.3.1.1.cmml"><mi id="S4.Ex15.m1.1.1.1.1.3.3.1.1.2" xref="S4.Ex15.m1.1.1.1.1.3.3.1.1.2.cmml">cl</mi><mrow id="S4.Ex15.m1.1.1.1.1.3.3.1.1.3" xref="S4.Ex15.m1.1.1.1.1.3.3.1.1.3.cmml"><mi id="S4.Ex15.m1.1.1.1.1.3.3.1.1.3.2" xref="S4.Ex15.m1.1.1.1.1.3.3.1.1.3.2.cmml">Ξ²</mi><mo id="S4.Ex15.m1.1.1.1.1.3.3.1.1.3.1" xref="S4.Ex15.m1.1.1.1.1.3.3.1.1.3.1.cmml">β’</mo><mi id="S4.Ex15.m1.1.1.1.1.3.3.1.1.3.3" xref="S4.Ex15.m1.1.1.1.1.3.3.1.1.3.3.cmml">X</mi></mrow></msub><mo id="S4.Ex15.m1.1.1.1.1.4.4.2a" xref="S4.Ex15.m1.1.1.1.1.4.4.3.cmml">β‘</mo><mrow id="S4.Ex15.m1.1.1.1.1.4.4.2.2" xref="S4.Ex15.m1.1.1.1.1.4.4.3.cmml"><mo id="S4.Ex15.m1.1.1.1.1.4.4.2.2.2" stretchy="false" xref="S4.Ex15.m1.1.1.1.1.4.4.3.cmml">(</mo><msub id="S4.Ex15.m1.1.1.1.1.4.4.2.2.1" xref="S4.Ex15.m1.1.1.1.1.4.4.2.2.1.cmml"><mi id="S4.Ex15.m1.1.1.1.1.4.4.2.2.1.2" xref="S4.Ex15.m1.1.1.1.1.4.4.2.2.1.2.cmml">I</mi><mi id="S4.Ex15.m1.1.1.1.1.4.4.2.2.1.3" xref="S4.Ex15.m1.1.1.1.1.4.4.2.2.1.3.cmml">L</mi></msub><mo id="S4.Ex15.m1.1.1.1.1.4.4.2.2.3" stretchy="false" xref="S4.Ex15.m1.1.1.1.1.4.4.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex15.m1.1.1.1.1.5" xref="S4.Ex15.m1.1.1.1.1.5.cmml">=</mo><mi id="S4.Ex15.m1.1.1.1.1.6" mathvariant="normal" xref="S4.Ex15.m1.1.1.1.1.6.cmml">β </mi></mrow><mo id="S4.Ex15.m1.1.1.1.2" lspace="0em" xref="S4.Ex15.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex15.m1.1b"><apply id="S4.Ex15.m1.1.1.1.1.cmml" xref="S4.Ex15.m1.1.1.1"><eq id="S4.Ex15.m1.1.1.1.1.5.cmml" xref="S4.Ex15.m1.1.1.1.1.5"></eq><apply id="S4.Ex15.m1.1.1.1.1.4.cmml" xref="S4.Ex15.m1.1.1.1.1.4"><intersect id="S4.Ex15.m1.1.1.1.1.4.5.cmml" xref="S4.Ex15.m1.1.1.1.1.4.5"></intersect><apply id="S4.Ex15.m1.1.1.1.1.2.2.3.cmml" xref="S4.Ex15.m1.1.1.1.1.2.2.2"><apply id="S4.Ex15.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex15.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Ex15.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.2">cl</ci><apply id="S4.Ex15.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.3"><times id="S4.Ex15.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.3.1"></times><ci id="S4.Ex15.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.3.2">π½</ci><ci id="S4.Ex15.m1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.3.3">π</ci></apply></apply><apply id="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.cmml" xref="S4.Ex15.m1.1.1.1.1.2.2.2.2.1"><setdiff id="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.1.cmml" xref="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.1"></setdiff><ci id="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.2.cmml" xref="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.2">π</ci><apply id="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.3.cmml" xref="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.3"><csymbol cd="ambiguous" id="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.3.1.cmml" xref="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.3">subscript</csymbol><ci id="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.3.2.cmml" xref="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.3.2">πΌ</ci><ci id="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.3.3.cmml" xref="S4.Ex15.m1.1.1.1.1.2.2.2.2.1.3.3">πΏ</ci></apply></apply></apply><apply id="S4.Ex15.m1.1.1.1.1.4.4.3.cmml" xref="S4.Ex15.m1.1.1.1.1.4.4.2"><apply id="S4.Ex15.m1.1.1.1.1.3.3.1.1.cmml" xref="S4.Ex15.m1.1.1.1.1.3.3.1.1"><csymbol cd="ambiguous" id="S4.Ex15.m1.1.1.1.1.3.3.1.1.1.cmml" 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xref="S4.Ex15.m1.1.1.1.1.6"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex15.m1.1c">\operatorname{cl}_{\beta X}(X\setminus I_{L})\cap\operatorname{cl}_{\beta X}(I% _{L})=\varnothing.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex15.m1.1d">roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_X end_POSTSUBSCRIPT ( italic_X β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) β© roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_X end_POSTSUBSCRIPT ( italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) = β .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.11.p3.3">Since <math alttext="L\subset X\setminus I_{L}" class="ltx_Math" display="inline" id="S4.11.p3.3.m1.1"><semantics id="S4.11.p3.3.m1.1a"><mrow id="S4.11.p3.3.m1.1.1" xref="S4.11.p3.3.m1.1.1.cmml"><mi id="S4.11.p3.3.m1.1.1.2" xref="S4.11.p3.3.m1.1.1.2.cmml">L</mi><mo id="S4.11.p3.3.m1.1.1.1" xref="S4.11.p3.3.m1.1.1.1.cmml">β</mo><mrow id="S4.11.p3.3.m1.1.1.3" xref="S4.11.p3.3.m1.1.1.3.cmml"><mi id="S4.11.p3.3.m1.1.1.3.2" xref="S4.11.p3.3.m1.1.1.3.2.cmml">X</mi><mo id="S4.11.p3.3.m1.1.1.3.1" xref="S4.11.p3.3.m1.1.1.3.1.cmml">β</mo><msub id="S4.11.p3.3.m1.1.1.3.3" xref="S4.11.p3.3.m1.1.1.3.3.cmml"><mi id="S4.11.p3.3.m1.1.1.3.3.2" xref="S4.11.p3.3.m1.1.1.3.3.2.cmml">I</mi><mi id="S4.11.p3.3.m1.1.1.3.3.3" xref="S4.11.p3.3.m1.1.1.3.3.3.cmml">L</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p3.3.m1.1b"><apply id="S4.11.p3.3.m1.1.1.cmml" xref="S4.11.p3.3.m1.1.1"><subset id="S4.11.p3.3.m1.1.1.1.cmml" xref="S4.11.p3.3.m1.1.1.1"></subset><ci id="S4.11.p3.3.m1.1.1.2.cmml" xref="S4.11.p3.3.m1.1.1.2">πΏ</ci><apply id="S4.11.p3.3.m1.1.1.3.cmml" xref="S4.11.p3.3.m1.1.1.3"><setdiff id="S4.11.p3.3.m1.1.1.3.1.cmml" xref="S4.11.p3.3.m1.1.1.3.1"></setdiff><ci id="S4.11.p3.3.m1.1.1.3.2.cmml" xref="S4.11.p3.3.m1.1.1.3.2">π</ci><apply id="S4.11.p3.3.m1.1.1.3.3.cmml" xref="S4.11.p3.3.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.11.p3.3.m1.1.1.3.3.1.cmml" xref="S4.11.p3.3.m1.1.1.3.3">subscript</csymbol><ci id="S4.11.p3.3.m1.1.1.3.3.2.cmml" xref="S4.11.p3.3.m1.1.1.3.3.2">πΌ</ci><ci id="S4.11.p3.3.m1.1.1.3.3.3.cmml" xref="S4.11.p3.3.m1.1.1.3.3.3">πΏ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p3.3.m1.1c">L\subset X\setminus I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.11.p3.3.m1.1d">italic_L β italic_X β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>, we obtain</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex16"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="z\in\operatorname{cl}_{\beta X}(L)\subseteq\operatorname{cl}_{\beta X}(X% \setminus I_{L})\subseteq\beta X\setminus\operatorname{cl}_{\beta X}(I_{L})." class="ltx_Math" display="block" id="S4.Ex16.m1.2"><semantics id="S4.Ex16.m1.2a"><mrow id="S4.Ex16.m1.2.2.1" xref="S4.Ex16.m1.2.2.1.1.cmml"><mrow id="S4.Ex16.m1.2.2.1.1" xref="S4.Ex16.m1.2.2.1.1.cmml"><mi id="S4.Ex16.m1.2.2.1.1.7" xref="S4.Ex16.m1.2.2.1.1.7.cmml">z</mi><mo id="S4.Ex16.m1.2.2.1.1.8" xref="S4.Ex16.m1.2.2.1.1.8.cmml">β</mo><mrow id="S4.Ex16.m1.2.2.1.1.1.1" xref="S4.Ex16.m1.2.2.1.1.1.2.cmml"><msub id="S4.Ex16.m1.2.2.1.1.1.1.1" xref="S4.Ex16.m1.2.2.1.1.1.1.1.cmml"><mi id="S4.Ex16.m1.2.2.1.1.1.1.1.2" xref="S4.Ex16.m1.2.2.1.1.1.1.1.2.cmml">cl</mi><mrow id="S4.Ex16.m1.2.2.1.1.1.1.1.3" xref="S4.Ex16.m1.2.2.1.1.1.1.1.3.cmml"><mi id="S4.Ex16.m1.2.2.1.1.1.1.1.3.2" xref="S4.Ex16.m1.2.2.1.1.1.1.1.3.2.cmml">Ξ²</mi><mo id="S4.Ex16.m1.2.2.1.1.1.1.1.3.1" xref="S4.Ex16.m1.2.2.1.1.1.1.1.3.1.cmml">β’</mo><mi id="S4.Ex16.m1.2.2.1.1.1.1.1.3.3" xref="S4.Ex16.m1.2.2.1.1.1.1.1.3.3.cmml">X</mi></mrow></msub><mo id="S4.Ex16.m1.2.2.1.1.1.1a" xref="S4.Ex16.m1.2.2.1.1.1.2.cmml">β‘</mo><mrow id="S4.Ex16.m1.2.2.1.1.1.1.2" xref="S4.Ex16.m1.2.2.1.1.1.2.cmml"><mo id="S4.Ex16.m1.2.2.1.1.1.1.2.1" stretchy="false" xref="S4.Ex16.m1.2.2.1.1.1.2.cmml">(</mo><mi id="S4.Ex16.m1.1.1" xref="S4.Ex16.m1.1.1.cmml">L</mi><mo id="S4.Ex16.m1.2.2.1.1.1.1.2.2" stretchy="false" xref="S4.Ex16.m1.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex16.m1.2.2.1.1.9" xref="S4.Ex16.m1.2.2.1.1.9.cmml">β</mo><mrow id="S4.Ex16.m1.2.2.1.1.3.2" xref="S4.Ex16.m1.2.2.1.1.3.3.cmml"><msub id="S4.Ex16.m1.2.2.1.1.2.1.1" xref="S4.Ex16.m1.2.2.1.1.2.1.1.cmml"><mi id="S4.Ex16.m1.2.2.1.1.2.1.1.2" xref="S4.Ex16.m1.2.2.1.1.2.1.1.2.cmml">cl</mi><mrow id="S4.Ex16.m1.2.2.1.1.2.1.1.3" xref="S4.Ex16.m1.2.2.1.1.2.1.1.3.cmml"><mi id="S4.Ex16.m1.2.2.1.1.2.1.1.3.2" xref="S4.Ex16.m1.2.2.1.1.2.1.1.3.2.cmml">Ξ²</mi><mo id="S4.Ex16.m1.2.2.1.1.2.1.1.3.1" xref="S4.Ex16.m1.2.2.1.1.2.1.1.3.1.cmml">β’</mo><mi id="S4.Ex16.m1.2.2.1.1.2.1.1.3.3" xref="S4.Ex16.m1.2.2.1.1.2.1.1.3.3.cmml">X</mi></mrow></msub><mo id="S4.Ex16.m1.2.2.1.1.3.2a" xref="S4.Ex16.m1.2.2.1.1.3.3.cmml">β‘</mo><mrow id="S4.Ex16.m1.2.2.1.1.3.2.2" xref="S4.Ex16.m1.2.2.1.1.3.3.cmml"><mo id="S4.Ex16.m1.2.2.1.1.3.2.2.2" stretchy="false" xref="S4.Ex16.m1.2.2.1.1.3.3.cmml">(</mo><mrow id="S4.Ex16.m1.2.2.1.1.3.2.2.1" xref="S4.Ex16.m1.2.2.1.1.3.2.2.1.cmml"><mi id="S4.Ex16.m1.2.2.1.1.3.2.2.1.2" xref="S4.Ex16.m1.2.2.1.1.3.2.2.1.2.cmml">X</mi><mo id="S4.Ex16.m1.2.2.1.1.3.2.2.1.1" xref="S4.Ex16.m1.2.2.1.1.3.2.2.1.1.cmml">β</mo><msub id="S4.Ex16.m1.2.2.1.1.3.2.2.1.3" xref="S4.Ex16.m1.2.2.1.1.3.2.2.1.3.cmml"><mi id="S4.Ex16.m1.2.2.1.1.3.2.2.1.3.2" xref="S4.Ex16.m1.2.2.1.1.3.2.2.1.3.2.cmml">I</mi><mi id="S4.Ex16.m1.2.2.1.1.3.2.2.1.3.3" xref="S4.Ex16.m1.2.2.1.1.3.2.2.1.3.3.cmml">L</mi></msub></mrow><mo id="S4.Ex16.m1.2.2.1.1.3.2.2.3" stretchy="false" xref="S4.Ex16.m1.2.2.1.1.3.3.cmml">)</mo></mrow></mrow><mo id="S4.Ex16.m1.2.2.1.1.10" xref="S4.Ex16.m1.2.2.1.1.10.cmml">β</mo><mrow id="S4.Ex16.m1.2.2.1.1.5" xref="S4.Ex16.m1.2.2.1.1.5.cmml"><mrow id="S4.Ex16.m1.2.2.1.1.5.4" xref="S4.Ex16.m1.2.2.1.1.5.4.cmml"><mi id="S4.Ex16.m1.2.2.1.1.5.4.2" xref="S4.Ex16.m1.2.2.1.1.5.4.2.cmml">Ξ²</mi><mo id="S4.Ex16.m1.2.2.1.1.5.4.1" xref="S4.Ex16.m1.2.2.1.1.5.4.1.cmml">β’</mo><mi id="S4.Ex16.m1.2.2.1.1.5.4.3" xref="S4.Ex16.m1.2.2.1.1.5.4.3.cmml">X</mi></mrow><mo id="S4.Ex16.m1.2.2.1.1.5.3" xref="S4.Ex16.m1.2.2.1.1.5.3.cmml">β</mo><mrow id="S4.Ex16.m1.2.2.1.1.5.2.2" xref="S4.Ex16.m1.2.2.1.1.5.2.3.cmml"><msub id="S4.Ex16.m1.2.2.1.1.4.1.1.1" xref="S4.Ex16.m1.2.2.1.1.4.1.1.1.cmml"><mi id="S4.Ex16.m1.2.2.1.1.4.1.1.1.2" xref="S4.Ex16.m1.2.2.1.1.4.1.1.1.2.cmml">cl</mi><mrow id="S4.Ex16.m1.2.2.1.1.4.1.1.1.3" xref="S4.Ex16.m1.2.2.1.1.4.1.1.1.3.cmml"><mi id="S4.Ex16.m1.2.2.1.1.4.1.1.1.3.2" xref="S4.Ex16.m1.2.2.1.1.4.1.1.1.3.2.cmml">Ξ²</mi><mo id="S4.Ex16.m1.2.2.1.1.4.1.1.1.3.1" xref="S4.Ex16.m1.2.2.1.1.4.1.1.1.3.1.cmml">β’</mo><mi id="S4.Ex16.m1.2.2.1.1.4.1.1.1.3.3" xref="S4.Ex16.m1.2.2.1.1.4.1.1.1.3.3.cmml">X</mi></mrow></msub><mo id="S4.Ex16.m1.2.2.1.1.5.2.2a" xref="S4.Ex16.m1.2.2.1.1.5.2.3.cmml">β‘</mo><mrow id="S4.Ex16.m1.2.2.1.1.5.2.2.2" xref="S4.Ex16.m1.2.2.1.1.5.2.3.cmml"><mo id="S4.Ex16.m1.2.2.1.1.5.2.2.2.2" stretchy="false" xref="S4.Ex16.m1.2.2.1.1.5.2.3.cmml">(</mo><msub id="S4.Ex16.m1.2.2.1.1.5.2.2.2.1" xref="S4.Ex16.m1.2.2.1.1.5.2.2.2.1.cmml"><mi id="S4.Ex16.m1.2.2.1.1.5.2.2.2.1.2" xref="S4.Ex16.m1.2.2.1.1.5.2.2.2.1.2.cmml">I</mi><mi id="S4.Ex16.m1.2.2.1.1.5.2.2.2.1.3" xref="S4.Ex16.m1.2.2.1.1.5.2.2.2.1.3.cmml">L</mi></msub><mo id="S4.Ex16.m1.2.2.1.1.5.2.2.2.3" stretchy="false" xref="S4.Ex16.m1.2.2.1.1.5.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S4.Ex16.m1.2.2.1.2" lspace="0em" xref="S4.Ex16.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex16.m1.2b"><apply id="S4.Ex16.m1.2.2.1.1.cmml" xref="S4.Ex16.m1.2.2.1"><and id="S4.Ex16.m1.2.2.1.1a.cmml" xref="S4.Ex16.m1.2.2.1"></and><apply id="S4.Ex16.m1.2.2.1.1b.cmml" xref="S4.Ex16.m1.2.2.1"><in id="S4.Ex16.m1.2.2.1.1.8.cmml" xref="S4.Ex16.m1.2.2.1.1.8"></in><ci id="S4.Ex16.m1.2.2.1.1.7.cmml" xref="S4.Ex16.m1.2.2.1.1.7">π§</ci><apply id="S4.Ex16.m1.2.2.1.1.1.2.cmml" xref="S4.Ex16.m1.2.2.1.1.1.1"><apply id="S4.Ex16.m1.2.2.1.1.1.1.1.cmml" xref="S4.Ex16.m1.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex16.m1.2.2.1.1.1.1.1.1.cmml" xref="S4.Ex16.m1.2.2.1.1.1.1.1">subscript</csymbol><ci id="S4.Ex16.m1.2.2.1.1.1.1.1.2.cmml" xref="S4.Ex16.m1.2.2.1.1.1.1.1.2">cl</ci><apply id="S4.Ex16.m1.2.2.1.1.1.1.1.3.cmml" xref="S4.Ex16.m1.2.2.1.1.1.1.1.3"><times id="S4.Ex16.m1.2.2.1.1.1.1.1.3.1.cmml" xref="S4.Ex16.m1.2.2.1.1.1.1.1.3.1"></times><ci id="S4.Ex16.m1.2.2.1.1.1.1.1.3.2.cmml" xref="S4.Ex16.m1.2.2.1.1.1.1.1.3.2">π½</ci><ci id="S4.Ex16.m1.2.2.1.1.1.1.1.3.3.cmml" xref="S4.Ex16.m1.2.2.1.1.1.1.1.3.3">π</ci></apply></apply><ci id="S4.Ex16.m1.1.1.cmml" xref="S4.Ex16.m1.1.1">πΏ</ci></apply></apply><apply id="S4.Ex16.m1.2.2.1.1c.cmml" xref="S4.Ex16.m1.2.2.1"><subset id="S4.Ex16.m1.2.2.1.1.9.cmml" xref="S4.Ex16.m1.2.2.1.1.9"></subset><share href="https://arxiv.org/html/2503.13666v1#S4.Ex16.m1.2.2.1.1.1.cmml" id="S4.Ex16.m1.2.2.1.1d.cmml" xref="S4.Ex16.m1.2.2.1"></share><apply id="S4.Ex16.m1.2.2.1.1.3.3.cmml" xref="S4.Ex16.m1.2.2.1.1.3.2"><apply id="S4.Ex16.m1.2.2.1.1.2.1.1.cmml" xref="S4.Ex16.m1.2.2.1.1.2.1.1"><csymbol cd="ambiguous" id="S4.Ex16.m1.2.2.1.1.2.1.1.1.cmml" xref="S4.Ex16.m1.2.2.1.1.2.1.1">subscript</csymbol><ci id="S4.Ex16.m1.2.2.1.1.2.1.1.2.cmml" xref="S4.Ex16.m1.2.2.1.1.2.1.1.2">cl</ci><apply id="S4.Ex16.m1.2.2.1.1.2.1.1.3.cmml" xref="S4.Ex16.m1.2.2.1.1.2.1.1.3"><times id="S4.Ex16.m1.2.2.1.1.2.1.1.3.1.cmml" xref="S4.Ex16.m1.2.2.1.1.2.1.1.3.1"></times><ci id="S4.Ex16.m1.2.2.1.1.2.1.1.3.2.cmml" xref="S4.Ex16.m1.2.2.1.1.2.1.1.3.2">π½</ci><ci id="S4.Ex16.m1.2.2.1.1.2.1.1.3.3.cmml" xref="S4.Ex16.m1.2.2.1.1.2.1.1.3.3">π</ci></apply></apply><apply id="S4.Ex16.m1.2.2.1.1.3.2.2.1.cmml" xref="S4.Ex16.m1.2.2.1.1.3.2.2.1"><setdiff id="S4.Ex16.m1.2.2.1.1.3.2.2.1.1.cmml" xref="S4.Ex16.m1.2.2.1.1.3.2.2.1.1"></setdiff><ci id="S4.Ex16.m1.2.2.1.1.3.2.2.1.2.cmml" xref="S4.Ex16.m1.2.2.1.1.3.2.2.1.2">π</ci><apply id="S4.Ex16.m1.2.2.1.1.3.2.2.1.3.cmml" xref="S4.Ex16.m1.2.2.1.1.3.2.2.1.3"><csymbol cd="ambiguous" id="S4.Ex16.m1.2.2.1.1.3.2.2.1.3.1.cmml" xref="S4.Ex16.m1.2.2.1.1.3.2.2.1.3">subscript</csymbol><ci id="S4.Ex16.m1.2.2.1.1.3.2.2.1.3.2.cmml" xref="S4.Ex16.m1.2.2.1.1.3.2.2.1.3.2">πΌ</ci><ci id="S4.Ex16.m1.2.2.1.1.3.2.2.1.3.3.cmml" xref="S4.Ex16.m1.2.2.1.1.3.2.2.1.3.3">πΏ</ci></apply></apply></apply></apply><apply id="S4.Ex16.m1.2.2.1.1e.cmml" xref="S4.Ex16.m1.2.2.1"><subset id="S4.Ex16.m1.2.2.1.1.10.cmml" xref="S4.Ex16.m1.2.2.1.1.10"></subset><share href="https://arxiv.org/html/2503.13666v1#S4.Ex16.m1.2.2.1.1.3.cmml" id="S4.Ex16.m1.2.2.1.1f.cmml" xref="S4.Ex16.m1.2.2.1"></share><apply id="S4.Ex16.m1.2.2.1.1.5.cmml" xref="S4.Ex16.m1.2.2.1.1.5"><setdiff id="S4.Ex16.m1.2.2.1.1.5.3.cmml" xref="S4.Ex16.m1.2.2.1.1.5.3"></setdiff><apply id="S4.Ex16.m1.2.2.1.1.5.4.cmml" xref="S4.Ex16.m1.2.2.1.1.5.4"><times id="S4.Ex16.m1.2.2.1.1.5.4.1.cmml" xref="S4.Ex16.m1.2.2.1.1.5.4.1"></times><ci id="S4.Ex16.m1.2.2.1.1.5.4.2.cmml" xref="S4.Ex16.m1.2.2.1.1.5.4.2">π½</ci><ci id="S4.Ex16.m1.2.2.1.1.5.4.3.cmml" xref="S4.Ex16.m1.2.2.1.1.5.4.3">π</ci></apply><apply id="S4.Ex16.m1.2.2.1.1.5.2.3.cmml" xref="S4.Ex16.m1.2.2.1.1.5.2.2"><apply id="S4.Ex16.m1.2.2.1.1.4.1.1.1.cmml" xref="S4.Ex16.m1.2.2.1.1.4.1.1.1"><csymbol cd="ambiguous" id="S4.Ex16.m1.2.2.1.1.4.1.1.1.1.cmml" xref="S4.Ex16.m1.2.2.1.1.4.1.1.1">subscript</csymbol><ci id="S4.Ex16.m1.2.2.1.1.4.1.1.1.2.cmml" xref="S4.Ex16.m1.2.2.1.1.4.1.1.1.2">cl</ci><apply id="S4.Ex16.m1.2.2.1.1.4.1.1.1.3.cmml" xref="S4.Ex16.m1.2.2.1.1.4.1.1.1.3"><times id="S4.Ex16.m1.2.2.1.1.4.1.1.1.3.1.cmml" xref="S4.Ex16.m1.2.2.1.1.4.1.1.1.3.1"></times><ci id="S4.Ex16.m1.2.2.1.1.4.1.1.1.3.2.cmml" xref="S4.Ex16.m1.2.2.1.1.4.1.1.1.3.2">π½</ci><ci id="S4.Ex16.m1.2.2.1.1.4.1.1.1.3.3.cmml" xref="S4.Ex16.m1.2.2.1.1.4.1.1.1.3.3">π</ci></apply></apply><apply id="S4.Ex16.m1.2.2.1.1.5.2.2.2.1.cmml" xref="S4.Ex16.m1.2.2.1.1.5.2.2.2.1"><csymbol cd="ambiguous" id="S4.Ex16.m1.2.2.1.1.5.2.2.2.1.1.cmml" xref="S4.Ex16.m1.2.2.1.1.5.2.2.2.1">subscript</csymbol><ci id="S4.Ex16.m1.2.2.1.1.5.2.2.2.1.2.cmml" xref="S4.Ex16.m1.2.2.1.1.5.2.2.2.1.2">πΌ</ci><ci id="S4.Ex16.m1.2.2.1.1.5.2.2.2.1.3.cmml" xref="S4.Ex16.m1.2.2.1.1.5.2.2.2.1.3">πΏ</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex16.m1.2c">z\in\operatorname{cl}_{\beta X}(L)\subseteq\operatorname{cl}_{\beta X}(X% \setminus I_{L})\subseteq\beta X\setminus\operatorname{cl}_{\beta X}(I_{L}).</annotation><annotation encoding="application/x-llamapun" id="S4.Ex16.m1.2d">italic_z β roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_X end_POSTSUBSCRIPT ( italic_L ) β roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_X end_POSTSUBSCRIPT ( italic_X β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) β italic_Ξ² italic_X β roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_X end_POSTSUBSCRIPT ( italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.11.p3.7">By Claim <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S4.Thmtheorem11" title="Claim 4.11. β£ Proof. β£ 4. Nyikos semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">4.11</span></a>, there exists <math alttext="f\in F\subset X\setminus I_{L}" class="ltx_Math" display="inline" id="S4.11.p3.4.m1.1"><semantics id="S4.11.p3.4.m1.1a"><mrow id="S4.11.p3.4.m1.1.1" xref="S4.11.p3.4.m1.1.1.cmml"><mi id="S4.11.p3.4.m1.1.1.2" xref="S4.11.p3.4.m1.1.1.2.cmml">f</mi><mo id="S4.11.p3.4.m1.1.1.3" xref="S4.11.p3.4.m1.1.1.3.cmml">β</mo><mi id="S4.11.p3.4.m1.1.1.4" xref="S4.11.p3.4.m1.1.1.4.cmml">F</mi><mo id="S4.11.p3.4.m1.1.1.5" xref="S4.11.p3.4.m1.1.1.5.cmml">β</mo><mrow id="S4.11.p3.4.m1.1.1.6" xref="S4.11.p3.4.m1.1.1.6.cmml"><mi id="S4.11.p3.4.m1.1.1.6.2" xref="S4.11.p3.4.m1.1.1.6.2.cmml">X</mi><mo id="S4.11.p3.4.m1.1.1.6.1" xref="S4.11.p3.4.m1.1.1.6.1.cmml">β</mo><msub id="S4.11.p3.4.m1.1.1.6.3" xref="S4.11.p3.4.m1.1.1.6.3.cmml"><mi id="S4.11.p3.4.m1.1.1.6.3.2" xref="S4.11.p3.4.m1.1.1.6.3.2.cmml">I</mi><mi id="S4.11.p3.4.m1.1.1.6.3.3" xref="S4.11.p3.4.m1.1.1.6.3.3.cmml">L</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p3.4.m1.1b"><apply id="S4.11.p3.4.m1.1.1.cmml" xref="S4.11.p3.4.m1.1.1"><and id="S4.11.p3.4.m1.1.1a.cmml" xref="S4.11.p3.4.m1.1.1"></and><apply id="S4.11.p3.4.m1.1.1b.cmml" xref="S4.11.p3.4.m1.1.1"><in id="S4.11.p3.4.m1.1.1.3.cmml" xref="S4.11.p3.4.m1.1.1.3"></in><ci id="S4.11.p3.4.m1.1.1.2.cmml" xref="S4.11.p3.4.m1.1.1.2">π</ci><ci id="S4.11.p3.4.m1.1.1.4.cmml" xref="S4.11.p3.4.m1.1.1.4">πΉ</ci></apply><apply id="S4.11.p3.4.m1.1.1c.cmml" xref="S4.11.p3.4.m1.1.1"><subset id="S4.11.p3.4.m1.1.1.5.cmml" xref="S4.11.p3.4.m1.1.1.5"></subset><share href="https://arxiv.org/html/2503.13666v1#S4.11.p3.4.m1.1.1.4.cmml" id="S4.11.p3.4.m1.1.1d.cmml" xref="S4.11.p3.4.m1.1.1"></share><apply id="S4.11.p3.4.m1.1.1.6.cmml" xref="S4.11.p3.4.m1.1.1.6"><setdiff id="S4.11.p3.4.m1.1.1.6.1.cmml" xref="S4.11.p3.4.m1.1.1.6.1"></setdiff><ci id="S4.11.p3.4.m1.1.1.6.2.cmml" xref="S4.11.p3.4.m1.1.1.6.2">π</ci><apply id="S4.11.p3.4.m1.1.1.6.3.cmml" xref="S4.11.p3.4.m1.1.1.6.3"><csymbol cd="ambiguous" id="S4.11.p3.4.m1.1.1.6.3.1.cmml" xref="S4.11.p3.4.m1.1.1.6.3">subscript</csymbol><ci id="S4.11.p3.4.m1.1.1.6.3.2.cmml" xref="S4.11.p3.4.m1.1.1.6.3.2">πΌ</ci><ci id="S4.11.p3.4.m1.1.1.6.3.3.cmml" xref="S4.11.p3.4.m1.1.1.6.3.3">πΏ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p3.4.m1.1c">f\in F\subset X\setminus I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.11.p3.4.m1.1d">italic_f β italic_F β italic_X β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="f\leq z" class="ltx_Math" display="inline" id="S4.11.p3.5.m2.1"><semantics id="S4.11.p3.5.m2.1a"><mrow id="S4.11.p3.5.m2.1.1" xref="S4.11.p3.5.m2.1.1.cmml"><mi id="S4.11.p3.5.m2.1.1.2" xref="S4.11.p3.5.m2.1.1.2.cmml">f</mi><mo id="S4.11.p3.5.m2.1.1.1" xref="S4.11.p3.5.m2.1.1.1.cmml">β€</mo><mi id="S4.11.p3.5.m2.1.1.3" xref="S4.11.p3.5.m2.1.1.3.cmml">z</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p3.5.m2.1b"><apply id="S4.11.p3.5.m2.1.1.cmml" xref="S4.11.p3.5.m2.1.1"><leq id="S4.11.p3.5.m2.1.1.1.cmml" xref="S4.11.p3.5.m2.1.1.1"></leq><ci id="S4.11.p3.5.m2.1.1.2.cmml" xref="S4.11.p3.5.m2.1.1.2">π</ci><ci id="S4.11.p3.5.m2.1.1.3.cmml" xref="S4.11.p3.5.m2.1.1.3">π§</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p3.5.m2.1c">f\leq z</annotation><annotation encoding="application/x-llamapun" id="S4.11.p3.5.m2.1d">italic_f β€ italic_z</annotation></semantics></math>. But then <math alttext="fL=(fz)L=f(zL)=\{fz\}=\{f\}" class="ltx_Math" display="inline" id="S4.11.p3.6.m3.4"><semantics id="S4.11.p3.6.m3.4a"><mrow id="S4.11.p3.6.m3.4.4" xref="S4.11.p3.6.m3.4.4.cmml"><mrow id="S4.11.p3.6.m3.4.4.5" xref="S4.11.p3.6.m3.4.4.5.cmml"><mi id="S4.11.p3.6.m3.4.4.5.2" xref="S4.11.p3.6.m3.4.4.5.2.cmml">f</mi><mo id="S4.11.p3.6.m3.4.4.5.1" xref="S4.11.p3.6.m3.4.4.5.1.cmml">β’</mo><mi id="S4.11.p3.6.m3.4.4.5.3" xref="S4.11.p3.6.m3.4.4.5.3.cmml">L</mi></mrow><mo id="S4.11.p3.6.m3.4.4.6" xref="S4.11.p3.6.m3.4.4.6.cmml">=</mo><mrow id="S4.11.p3.6.m3.2.2.1" xref="S4.11.p3.6.m3.2.2.1.cmml"><mrow id="S4.11.p3.6.m3.2.2.1.1.1" xref="S4.11.p3.6.m3.2.2.1.1.1.1.cmml"><mo id="S4.11.p3.6.m3.2.2.1.1.1.2" stretchy="false" xref="S4.11.p3.6.m3.2.2.1.1.1.1.cmml">(</mo><mrow id="S4.11.p3.6.m3.2.2.1.1.1.1" xref="S4.11.p3.6.m3.2.2.1.1.1.1.cmml"><mi id="S4.11.p3.6.m3.2.2.1.1.1.1.2" xref="S4.11.p3.6.m3.2.2.1.1.1.1.2.cmml">f</mi><mo id="S4.11.p3.6.m3.2.2.1.1.1.1.1" xref="S4.11.p3.6.m3.2.2.1.1.1.1.1.cmml">β’</mo><mi id="S4.11.p3.6.m3.2.2.1.1.1.1.3" xref="S4.11.p3.6.m3.2.2.1.1.1.1.3.cmml">z</mi></mrow><mo id="S4.11.p3.6.m3.2.2.1.1.1.3" stretchy="false" xref="S4.11.p3.6.m3.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S4.11.p3.6.m3.2.2.1.2" xref="S4.11.p3.6.m3.2.2.1.2.cmml">β’</mo><mi id="S4.11.p3.6.m3.2.2.1.3" xref="S4.11.p3.6.m3.2.2.1.3.cmml">L</mi></mrow><mo id="S4.11.p3.6.m3.4.4.7" xref="S4.11.p3.6.m3.4.4.7.cmml">=</mo><mrow id="S4.11.p3.6.m3.3.3.2" xref="S4.11.p3.6.m3.3.3.2.cmml"><mi id="S4.11.p3.6.m3.3.3.2.3" xref="S4.11.p3.6.m3.3.3.2.3.cmml">f</mi><mo id="S4.11.p3.6.m3.3.3.2.2" xref="S4.11.p3.6.m3.3.3.2.2.cmml">β’</mo><mrow id="S4.11.p3.6.m3.3.3.2.1.1" xref="S4.11.p3.6.m3.3.3.2.1.1.1.cmml"><mo id="S4.11.p3.6.m3.3.3.2.1.1.2" stretchy="false" xref="S4.11.p3.6.m3.3.3.2.1.1.1.cmml">(</mo><mrow id="S4.11.p3.6.m3.3.3.2.1.1.1" xref="S4.11.p3.6.m3.3.3.2.1.1.1.cmml"><mi id="S4.11.p3.6.m3.3.3.2.1.1.1.2" xref="S4.11.p3.6.m3.3.3.2.1.1.1.2.cmml">z</mi><mo id="S4.11.p3.6.m3.3.3.2.1.1.1.1" xref="S4.11.p3.6.m3.3.3.2.1.1.1.1.cmml">β’</mo><mi id="S4.11.p3.6.m3.3.3.2.1.1.1.3" xref="S4.11.p3.6.m3.3.3.2.1.1.1.3.cmml">L</mi></mrow><mo id="S4.11.p3.6.m3.3.3.2.1.1.3" stretchy="false" xref="S4.11.p3.6.m3.3.3.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.11.p3.6.m3.4.4.8" xref="S4.11.p3.6.m3.4.4.8.cmml">=</mo><mrow id="S4.11.p3.6.m3.4.4.3.1" xref="S4.11.p3.6.m3.4.4.3.2.cmml"><mo id="S4.11.p3.6.m3.4.4.3.1.2" stretchy="false" xref="S4.11.p3.6.m3.4.4.3.2.cmml">{</mo><mrow id="S4.11.p3.6.m3.4.4.3.1.1" xref="S4.11.p3.6.m3.4.4.3.1.1.cmml"><mi id="S4.11.p3.6.m3.4.4.3.1.1.2" xref="S4.11.p3.6.m3.4.4.3.1.1.2.cmml">f</mi><mo id="S4.11.p3.6.m3.4.4.3.1.1.1" xref="S4.11.p3.6.m3.4.4.3.1.1.1.cmml">β’</mo><mi id="S4.11.p3.6.m3.4.4.3.1.1.3" xref="S4.11.p3.6.m3.4.4.3.1.1.3.cmml">z</mi></mrow><mo id="S4.11.p3.6.m3.4.4.3.1.3" stretchy="false" xref="S4.11.p3.6.m3.4.4.3.2.cmml">}</mo></mrow><mo id="S4.11.p3.6.m3.4.4.9" xref="S4.11.p3.6.m3.4.4.9.cmml">=</mo><mrow id="S4.11.p3.6.m3.4.4.10.2" xref="S4.11.p3.6.m3.4.4.10.1.cmml"><mo id="S4.11.p3.6.m3.4.4.10.2.1" stretchy="false" xref="S4.11.p3.6.m3.4.4.10.1.cmml">{</mo><mi id="S4.11.p3.6.m3.1.1" xref="S4.11.p3.6.m3.1.1.cmml">f</mi><mo id="S4.11.p3.6.m3.4.4.10.2.2" stretchy="false" xref="S4.11.p3.6.m3.4.4.10.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p3.6.m3.4b"><apply id="S4.11.p3.6.m3.4.4.cmml" xref="S4.11.p3.6.m3.4.4"><and id="S4.11.p3.6.m3.4.4a.cmml" xref="S4.11.p3.6.m3.4.4"></and><apply id="S4.11.p3.6.m3.4.4b.cmml" xref="S4.11.p3.6.m3.4.4"><eq id="S4.11.p3.6.m3.4.4.6.cmml" xref="S4.11.p3.6.m3.4.4.6"></eq><apply id="S4.11.p3.6.m3.4.4.5.cmml" xref="S4.11.p3.6.m3.4.4.5"><times id="S4.11.p3.6.m3.4.4.5.1.cmml" xref="S4.11.p3.6.m3.4.4.5.1"></times><ci id="S4.11.p3.6.m3.4.4.5.2.cmml" xref="S4.11.p3.6.m3.4.4.5.2">π</ci><ci id="S4.11.p3.6.m3.4.4.5.3.cmml" xref="S4.11.p3.6.m3.4.4.5.3">πΏ</ci></apply><apply id="S4.11.p3.6.m3.2.2.1.cmml" xref="S4.11.p3.6.m3.2.2.1"><times id="S4.11.p3.6.m3.2.2.1.2.cmml" xref="S4.11.p3.6.m3.2.2.1.2"></times><apply id="S4.11.p3.6.m3.2.2.1.1.1.1.cmml" xref="S4.11.p3.6.m3.2.2.1.1.1"><times id="S4.11.p3.6.m3.2.2.1.1.1.1.1.cmml" xref="S4.11.p3.6.m3.2.2.1.1.1.1.1"></times><ci id="S4.11.p3.6.m3.2.2.1.1.1.1.2.cmml" xref="S4.11.p3.6.m3.2.2.1.1.1.1.2">π</ci><ci id="S4.11.p3.6.m3.2.2.1.1.1.1.3.cmml" xref="S4.11.p3.6.m3.2.2.1.1.1.1.3">π§</ci></apply><ci id="S4.11.p3.6.m3.2.2.1.3.cmml" xref="S4.11.p3.6.m3.2.2.1.3">πΏ</ci></apply></apply><apply id="S4.11.p3.6.m3.4.4c.cmml" xref="S4.11.p3.6.m3.4.4"><eq id="S4.11.p3.6.m3.4.4.7.cmml" xref="S4.11.p3.6.m3.4.4.7"></eq><share href="https://arxiv.org/html/2503.13666v1#S4.11.p3.6.m3.2.2.1.cmml" id="S4.11.p3.6.m3.4.4d.cmml" xref="S4.11.p3.6.m3.4.4"></share><apply id="S4.11.p3.6.m3.3.3.2.cmml" xref="S4.11.p3.6.m3.3.3.2"><times id="S4.11.p3.6.m3.3.3.2.2.cmml" xref="S4.11.p3.6.m3.3.3.2.2"></times><ci id="S4.11.p3.6.m3.3.3.2.3.cmml" xref="S4.11.p3.6.m3.3.3.2.3">π</ci><apply id="S4.11.p3.6.m3.3.3.2.1.1.1.cmml" xref="S4.11.p3.6.m3.3.3.2.1.1"><times id="S4.11.p3.6.m3.3.3.2.1.1.1.1.cmml" xref="S4.11.p3.6.m3.3.3.2.1.1.1.1"></times><ci id="S4.11.p3.6.m3.3.3.2.1.1.1.2.cmml" xref="S4.11.p3.6.m3.3.3.2.1.1.1.2">π§</ci><ci id="S4.11.p3.6.m3.3.3.2.1.1.1.3.cmml" xref="S4.11.p3.6.m3.3.3.2.1.1.1.3">πΏ</ci></apply></apply></apply><apply id="S4.11.p3.6.m3.4.4e.cmml" xref="S4.11.p3.6.m3.4.4"><eq id="S4.11.p3.6.m3.4.4.8.cmml" xref="S4.11.p3.6.m3.4.4.8"></eq><share href="https://arxiv.org/html/2503.13666v1#S4.11.p3.6.m3.3.3.2.cmml" id="S4.11.p3.6.m3.4.4f.cmml" xref="S4.11.p3.6.m3.4.4"></share><set id="S4.11.p3.6.m3.4.4.3.2.cmml" xref="S4.11.p3.6.m3.4.4.3.1"><apply id="S4.11.p3.6.m3.4.4.3.1.1.cmml" xref="S4.11.p3.6.m3.4.4.3.1.1"><times id="S4.11.p3.6.m3.4.4.3.1.1.1.cmml" xref="S4.11.p3.6.m3.4.4.3.1.1.1"></times><ci id="S4.11.p3.6.m3.4.4.3.1.1.2.cmml" xref="S4.11.p3.6.m3.4.4.3.1.1.2">π</ci><ci id="S4.11.p3.6.m3.4.4.3.1.1.3.cmml" xref="S4.11.p3.6.m3.4.4.3.1.1.3">π§</ci></apply></set></apply><apply id="S4.11.p3.6.m3.4.4g.cmml" xref="S4.11.p3.6.m3.4.4"><eq id="S4.11.p3.6.m3.4.4.9.cmml" xref="S4.11.p3.6.m3.4.4.9"></eq><share href="https://arxiv.org/html/2503.13666v1#S4.11.p3.6.m3.4.4.3.cmml" id="S4.11.p3.6.m3.4.4h.cmml" xref="S4.11.p3.6.m3.4.4"></share><set id="S4.11.p3.6.m3.4.4.10.1.cmml" xref="S4.11.p3.6.m3.4.4.10.2"><ci id="S4.11.p3.6.m3.1.1.cmml" xref="S4.11.p3.6.m3.1.1">π</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p3.6.m3.4c">fL=(fz)L=f(zL)=\{fz\}=\{f\}</annotation><annotation encoding="application/x-llamapun" id="S4.11.p3.6.m3.4d">italic_f italic_L = ( italic_f italic_z ) italic_L = italic_f ( italic_z italic_L ) = { italic_f italic_z } = { italic_f }</annotation></semantics></math>. Hence <math alttext="f\in I_{L}" class="ltx_Math" display="inline" id="S4.11.p3.7.m4.1"><semantics id="S4.11.p3.7.m4.1a"><mrow id="S4.11.p3.7.m4.1.1" xref="S4.11.p3.7.m4.1.1.cmml"><mi id="S4.11.p3.7.m4.1.1.2" xref="S4.11.p3.7.m4.1.1.2.cmml">f</mi><mo id="S4.11.p3.7.m4.1.1.1" xref="S4.11.p3.7.m4.1.1.1.cmml">β</mo><msub id="S4.11.p3.7.m4.1.1.3" xref="S4.11.p3.7.m4.1.1.3.cmml"><mi id="S4.11.p3.7.m4.1.1.3.2" xref="S4.11.p3.7.m4.1.1.3.2.cmml">I</mi><mi id="S4.11.p3.7.m4.1.1.3.3" xref="S4.11.p3.7.m4.1.1.3.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.11.p3.7.m4.1b"><apply id="S4.11.p3.7.m4.1.1.cmml" xref="S4.11.p3.7.m4.1.1"><in id="S4.11.p3.7.m4.1.1.1.cmml" xref="S4.11.p3.7.m4.1.1.1"></in><ci id="S4.11.p3.7.m4.1.1.2.cmml" xref="S4.11.p3.7.m4.1.1.2">π</ci><apply id="S4.11.p3.7.m4.1.1.3.cmml" xref="S4.11.p3.7.m4.1.1.3"><csymbol cd="ambiguous" id="S4.11.p3.7.m4.1.1.3.1.cmml" xref="S4.11.p3.7.m4.1.1.3">subscript</csymbol><ci id="S4.11.p3.7.m4.1.1.3.2.cmml" xref="S4.11.p3.7.m4.1.1.3.2">πΌ</ci><ci id="S4.11.p3.7.m4.1.1.3.3.cmml" xref="S4.11.p3.7.m4.1.1.3.3">πΏ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.11.p3.7.m4.1c">f\in I_{L}</annotation><annotation encoding="application/x-llamapun" id="S4.11.p3.7.m4.1d">italic_f β italic_I start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>, a contradiction. β</p> </div> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S4.Thmtheorem12"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem12.1.1.1">Theorem 4.12</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem12.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem12.p1"> <p class="ltx_p" id="S4.Thmtheorem12.p1.1"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem12.p1.1.1">A locally compact Nyikos topological semilattice is compact.</span></p> </div> </div> <div class="ltx_proof" id="S4.12"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.12.p1"> <p class="ltx_p" id="S4.12.p1.4">Propositions <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S4.Thmtheorem8" title="Proposition 4.8. β£ 4. Nyikos semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">4.8</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S4.Thmtheorem9" title="Proposition 4.9. β£ 4. Nyikos semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">4.9</span></a> imply that each locally compact Nyikos topological semilattice <math alttext="X" class="ltx_Math" display="inline" id="S4.12.p1.1.m1.1"><semantics id="S4.12.p1.1.m1.1a"><mi id="S4.12.p1.1.m1.1.1" xref="S4.12.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.12.p1.1.m1.1b"><ci id="S4.12.p1.1.m1.1.1.cmml" xref="S4.12.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.1.m1.1d">italic_X</annotation></semantics></math> contains no isomorphic copies of <math alttext="(\omega_{1},\min)" class="ltx_Math" display="inline" id="S4.12.p1.2.m2.2"><semantics id="S4.12.p1.2.m2.2a"><mrow id="S4.12.p1.2.m2.2.2.1" xref="S4.12.p1.2.m2.2.2.2.cmml"><mo id="S4.12.p1.2.m2.2.2.1.2" stretchy="false" xref="S4.12.p1.2.m2.2.2.2.cmml">(</mo><msub id="S4.12.p1.2.m2.2.2.1.1" xref="S4.12.p1.2.m2.2.2.1.1.cmml"><mi id="S4.12.p1.2.m2.2.2.1.1.2" xref="S4.12.p1.2.m2.2.2.1.1.2.cmml">Ο</mi><mn id="S4.12.p1.2.m2.2.2.1.1.3" xref="S4.12.p1.2.m2.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.12.p1.2.m2.2.2.1.3" xref="S4.12.p1.2.m2.2.2.2.cmml">,</mo><mi id="S4.12.p1.2.m2.1.1" xref="S4.12.p1.2.m2.1.1.cmml">min</mi><mo id="S4.12.p1.2.m2.2.2.1.4" stretchy="false" xref="S4.12.p1.2.m2.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p1.2.m2.2b"><interval closure="open" id="S4.12.p1.2.m2.2.2.2.cmml" xref="S4.12.p1.2.m2.2.2.1"><apply id="S4.12.p1.2.m2.2.2.1.1.cmml" xref="S4.12.p1.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S4.12.p1.2.m2.2.2.1.1.1.cmml" xref="S4.12.p1.2.m2.2.2.1.1">subscript</csymbol><ci id="S4.12.p1.2.m2.2.2.1.1.2.cmml" xref="S4.12.p1.2.m2.2.2.1.1.2">π</ci><cn id="S4.12.p1.2.m2.2.2.1.1.3.cmml" type="integer" xref="S4.12.p1.2.m2.2.2.1.1.3">1</cn></apply><min id="S4.12.p1.2.m2.1.1.cmml" xref="S4.12.p1.2.m2.1.1"></min></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.2.m2.2c">(\omega_{1},\min)</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.2.m2.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_min )</annotation></semantics></math> or <math alttext="(\omega_{1},\max)" class="ltx_Math" display="inline" id="S4.12.p1.3.m3.2"><semantics id="S4.12.p1.3.m3.2a"><mrow id="S4.12.p1.3.m3.2.2.1" xref="S4.12.p1.3.m3.2.2.2.cmml"><mo id="S4.12.p1.3.m3.2.2.1.2" stretchy="false" xref="S4.12.p1.3.m3.2.2.2.cmml">(</mo><msub id="S4.12.p1.3.m3.2.2.1.1" xref="S4.12.p1.3.m3.2.2.1.1.cmml"><mi id="S4.12.p1.3.m3.2.2.1.1.2" xref="S4.12.p1.3.m3.2.2.1.1.2.cmml">Ο</mi><mn id="S4.12.p1.3.m3.2.2.1.1.3" xref="S4.12.p1.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.12.p1.3.m3.2.2.1.3" xref="S4.12.p1.3.m3.2.2.2.cmml">,</mo><mi id="S4.12.p1.3.m3.1.1" xref="S4.12.p1.3.m3.1.1.cmml">max</mi><mo id="S4.12.p1.3.m3.2.2.1.4" stretchy="false" xref="S4.12.p1.3.m3.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.12.p1.3.m3.2b"><interval closure="open" id="S4.12.p1.3.m3.2.2.2.cmml" xref="S4.12.p1.3.m3.2.2.1"><apply id="S4.12.p1.3.m3.2.2.1.1.cmml" xref="S4.12.p1.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S4.12.p1.3.m3.2.2.1.1.1.cmml" xref="S4.12.p1.3.m3.2.2.1.1">subscript</csymbol><ci id="S4.12.p1.3.m3.2.2.1.1.2.cmml" xref="S4.12.p1.3.m3.2.2.1.1.2">π</ci><cn id="S4.12.p1.3.m3.2.2.1.1.3.cmml" type="integer" xref="S4.12.p1.3.m3.2.2.1.1.3">1</cn></apply><max id="S4.12.p1.3.m3.1.1.cmml" xref="S4.12.p1.3.m3.1.1"></max></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.3.m3.2c">(\omega_{1},\max)</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.3.m3.2d">( italic_Ο start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_max )</annotation></semantics></math>. By Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem2" title="Theorem B. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">B</span></a>, <math alttext="X" class="ltx_Math" display="inline" id="S4.12.p1.4.m4.1"><semantics id="S4.12.p1.4.m4.1a"><mi id="S4.12.p1.4.m4.1.1" xref="S4.12.p1.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.12.p1.4.m4.1b"><ci id="S4.12.p1.4.m4.1.1.cmml" xref="S4.12.p1.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.12.p1.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.12.p1.4.m4.1d">italic_X</annotation></semantics></math> is compact. β</p> </div> </div> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5. </span>Proof of the main result and final remarks</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.12">A partially ordered space <math alttext="(X,\leq)" class="ltx_Math" display="inline" id="S5.p1.1.m1.2"><semantics id="S5.p1.1.m1.2a"><mrow id="S5.p1.1.m1.2.3.2" xref="S5.p1.1.m1.2.3.1.cmml"><mo id="S5.p1.1.m1.2.3.2.1" stretchy="false" xref="S5.p1.1.m1.2.3.1.cmml">(</mo><mi id="S5.p1.1.m1.1.1" xref="S5.p1.1.m1.1.1.cmml">X</mi><mo id="S5.p1.1.m1.2.3.2.2" xref="S5.p1.1.m1.2.3.1.cmml">,</mo><mo id="S5.p1.1.m1.2.2" lspace="0em" rspace="0em" xref="S5.p1.1.m1.2.2.cmml">β€</mo><mo id="S5.p1.1.m1.2.3.2.3" stretchy="false" xref="S5.p1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.1.m1.2b"><interval closure="open" id="S5.p1.1.m1.2.3.1.cmml" xref="S5.p1.1.m1.2.3.2"><ci id="S5.p1.1.m1.1.1.cmml" xref="S5.p1.1.m1.1.1">π</ci><leq id="S5.p1.1.m1.2.2.cmml" xref="S5.p1.1.m1.2.2"></leq></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.1.m1.2c">(X,\leq)</annotation><annotation encoding="application/x-llamapun" id="S5.p1.1.m1.2d">( italic_X , β€ )</annotation></semantics></math> endowed with a topology is called a <span class="ltx_text ltx_font_italic" id="S5.p1.12.1">pospace</span> if <math alttext="\leq" class="ltx_Math" display="inline" id="S5.p1.2.m2.1"><semantics id="S5.p1.2.m2.1a"><mo id="S5.p1.2.m2.1.1" xref="S5.p1.2.m2.1.1.cmml">β€</mo><annotation-xml encoding="MathML-Content" id="S5.p1.2.m2.1b"><leq id="S5.p1.2.m2.1.1.cmml" xref="S5.p1.2.m2.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.2.m2.1c">\leq</annotation><annotation encoding="application/x-llamapun" id="S5.p1.2.m2.1d">β€</annotation></semantics></math> is a closed subset of <math alttext="X{\times}X" class="ltx_Math" display="inline" id="S5.p1.3.m3.1"><semantics id="S5.p1.3.m3.1a"><mrow id="S5.p1.3.m3.1.1" xref="S5.p1.3.m3.1.1.cmml"><mi id="S5.p1.3.m3.1.1.2" xref="S5.p1.3.m3.1.1.2.cmml">X</mi><mo id="S5.p1.3.m3.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.p1.3.m3.1.1.1.cmml">Γ</mo><mi id="S5.p1.3.m3.1.1.3" xref="S5.p1.3.m3.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.3.m3.1b"><apply id="S5.p1.3.m3.1.1.cmml" xref="S5.p1.3.m3.1.1"><times id="S5.p1.3.m3.1.1.1.cmml" xref="S5.p1.3.m3.1.1.1"></times><ci id="S5.p1.3.m3.1.1.2.cmml" xref="S5.p1.3.m3.1.1.2">π</ci><ci id="S5.p1.3.m3.1.1.3.cmml" xref="S5.p1.3.m3.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.3.m3.1c">X{\times}X</annotation><annotation encoding="application/x-llamapun" id="S5.p1.3.m3.1d">italic_X Γ italic_X</annotation></semantics></math>. Each inverse semigroup <math alttext="S" class="ltx_Math" display="inline" id="S5.p1.4.m4.1"><semantics id="S5.p1.4.m4.1a"><mi id="S5.p1.4.m4.1.1" xref="S5.p1.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.p1.4.m4.1b"><ci id="S5.p1.4.m4.1.1.cmml" xref="S5.p1.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.p1.4.m4.1d">italic_S</annotation></semantics></math> carries the natural partial order <math alttext="\leq" class="ltx_Math" display="inline" id="S5.p1.5.m5.1"><semantics id="S5.p1.5.m5.1a"><mo id="S5.p1.5.m5.1.1" xref="S5.p1.5.m5.1.1.cmml">β€</mo><annotation-xml encoding="MathML-Content" id="S5.p1.5.m5.1b"><leq id="S5.p1.5.m5.1.1.cmml" xref="S5.p1.5.m5.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.5.m5.1c">\leq</annotation><annotation encoding="application/x-llamapun" id="S5.p1.5.m5.1d">β€</annotation></semantics></math> defined by <math alttext="x\leq y" class="ltx_Math" display="inline" id="S5.p1.6.m6.1"><semantics id="S5.p1.6.m6.1a"><mrow id="S5.p1.6.m6.1.1" xref="S5.p1.6.m6.1.1.cmml"><mi id="S5.p1.6.m6.1.1.2" xref="S5.p1.6.m6.1.1.2.cmml">x</mi><mo id="S5.p1.6.m6.1.1.1" xref="S5.p1.6.m6.1.1.1.cmml">β€</mo><mi id="S5.p1.6.m6.1.1.3" xref="S5.p1.6.m6.1.1.3.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.6.m6.1b"><apply id="S5.p1.6.m6.1.1.cmml" xref="S5.p1.6.m6.1.1"><leq id="S5.p1.6.m6.1.1.1.cmml" xref="S5.p1.6.m6.1.1.1"></leq><ci id="S5.p1.6.m6.1.1.2.cmml" xref="S5.p1.6.m6.1.1.2">π₯</ci><ci id="S5.p1.6.m6.1.1.3.cmml" xref="S5.p1.6.m6.1.1.3">π¦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.6.m6.1c">x\leq y</annotation><annotation encoding="application/x-llamapun" id="S5.p1.6.m6.1d">italic_x β€ italic_y</annotation></semantics></math> if and only if <math alttext="x=xx^{-1}y" class="ltx_Math" display="inline" id="S5.p1.7.m7.1"><semantics id="S5.p1.7.m7.1a"><mrow id="S5.p1.7.m7.1.1" xref="S5.p1.7.m7.1.1.cmml"><mi id="S5.p1.7.m7.1.1.2" xref="S5.p1.7.m7.1.1.2.cmml">x</mi><mo id="S5.p1.7.m7.1.1.1" xref="S5.p1.7.m7.1.1.1.cmml">=</mo><mrow id="S5.p1.7.m7.1.1.3" xref="S5.p1.7.m7.1.1.3.cmml"><mi id="S5.p1.7.m7.1.1.3.2" xref="S5.p1.7.m7.1.1.3.2.cmml">x</mi><mo id="S5.p1.7.m7.1.1.3.1" xref="S5.p1.7.m7.1.1.3.1.cmml">β’</mo><msup id="S5.p1.7.m7.1.1.3.3" xref="S5.p1.7.m7.1.1.3.3.cmml"><mi id="S5.p1.7.m7.1.1.3.3.2" xref="S5.p1.7.m7.1.1.3.3.2.cmml">x</mi><mrow id="S5.p1.7.m7.1.1.3.3.3" xref="S5.p1.7.m7.1.1.3.3.3.cmml"><mo id="S5.p1.7.m7.1.1.3.3.3a" xref="S5.p1.7.m7.1.1.3.3.3.cmml">β</mo><mn id="S5.p1.7.m7.1.1.3.3.3.2" xref="S5.p1.7.m7.1.1.3.3.3.2.cmml">1</mn></mrow></msup><mo id="S5.p1.7.m7.1.1.3.1a" xref="S5.p1.7.m7.1.1.3.1.cmml">β’</mo><mi id="S5.p1.7.m7.1.1.3.4" xref="S5.p1.7.m7.1.1.3.4.cmml">y</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.7.m7.1b"><apply id="S5.p1.7.m7.1.1.cmml" xref="S5.p1.7.m7.1.1"><eq id="S5.p1.7.m7.1.1.1.cmml" xref="S5.p1.7.m7.1.1.1"></eq><ci id="S5.p1.7.m7.1.1.2.cmml" xref="S5.p1.7.m7.1.1.2">π₯</ci><apply id="S5.p1.7.m7.1.1.3.cmml" xref="S5.p1.7.m7.1.1.3"><times id="S5.p1.7.m7.1.1.3.1.cmml" xref="S5.p1.7.m7.1.1.3.1"></times><ci id="S5.p1.7.m7.1.1.3.2.cmml" xref="S5.p1.7.m7.1.1.3.2">π₯</ci><apply id="S5.p1.7.m7.1.1.3.3.cmml" xref="S5.p1.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S5.p1.7.m7.1.1.3.3.1.cmml" xref="S5.p1.7.m7.1.1.3.3">superscript</csymbol><ci id="S5.p1.7.m7.1.1.3.3.2.cmml" xref="S5.p1.7.m7.1.1.3.3.2">π₯</ci><apply id="S5.p1.7.m7.1.1.3.3.3.cmml" xref="S5.p1.7.m7.1.1.3.3.3"><minus id="S5.p1.7.m7.1.1.3.3.3.1.cmml" xref="S5.p1.7.m7.1.1.3.3.3"></minus><cn id="S5.p1.7.m7.1.1.3.3.3.2.cmml" type="integer" xref="S5.p1.7.m7.1.1.3.3.3.2">1</cn></apply></apply><ci id="S5.p1.7.m7.1.1.3.4.cmml" xref="S5.p1.7.m7.1.1.3.4">π¦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.7.m7.1c">x=xx^{-1}y</annotation><annotation encoding="application/x-llamapun" id="S5.p1.7.m7.1d">italic_x = italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_y</annotation></semantics></math>. The Greenβs relations <math alttext="\mathcal{L}" class="ltx_Math" display="inline" id="S5.p1.8.m8.1"><semantics id="S5.p1.8.m8.1a"><mi class="ltx_font_mathcaligraphic" id="S5.p1.8.m8.1.1" xref="S5.p1.8.m8.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S5.p1.8.m8.1b"><ci id="S5.p1.8.m8.1.1.cmml" xref="S5.p1.8.m8.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.8.m8.1c">\mathcal{L}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.8.m8.1d">caligraphic_L</annotation></semantics></math>, <math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S5.p1.9.m9.1"><semantics id="S5.p1.9.m9.1a"><mi class="ltx_font_mathcaligraphic" id="S5.p1.9.m9.1.1" xref="S5.p1.9.m9.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S5.p1.9.m9.1b"><ci id="S5.p1.9.m9.1.1.cmml" xref="S5.p1.9.m9.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.9.m9.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.9.m9.1d">caligraphic_R</annotation></semantics></math>, <math alttext="\mathcal{H}" class="ltx_Math" display="inline" id="S5.p1.10.m10.1"><semantics id="S5.p1.10.m10.1a"><mi class="ltx_font_mathcaligraphic" id="S5.p1.10.m10.1.1" xref="S5.p1.10.m10.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S5.p1.10.m10.1b"><ci id="S5.p1.10.m10.1.1.cmml" xref="S5.p1.10.m10.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.10.m10.1c">\mathcal{H}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.10.m10.1d">caligraphic_H</annotation></semantics></math> and <math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="S5.p1.11.m11.1"><semantics id="S5.p1.11.m11.1a"><mi class="ltx_font_mathcaligraphic" id="S5.p1.11.m11.1.1" xref="S5.p1.11.m11.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S5.p1.11.m11.1b"><ci id="S5.p1.11.m11.1.1.cmml" xref="S5.p1.11.m11.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.11.m11.1c">\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.11.m11.1d">caligraphic_D</annotation></semantics></math> on an inverse semigroup <math alttext="S" class="ltx_Math" display="inline" id="S5.p1.12.m12.1"><semantics id="S5.p1.12.m12.1a"><mi id="S5.p1.12.m12.1.1" xref="S5.p1.12.m12.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.p1.12.m12.1b"><ci id="S5.p1.12.m12.1.1.cmml" xref="S5.p1.12.m12.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.12.m12.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.p1.12.m12.1d">italic_S</annotation></semantics></math> are defined as follows:</p> </div> <div class="ltx_para" id="S5.p2"> <ol class="ltx_enumerate" id="S5.I1"> <li class="ltx_item" id="S5.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S5.I1.i1.p1"> <p class="ltx_p" id="S5.I1.i1.p1.2"><math alttext="(x,y)\in\mathcal{L}" class="ltx_Math" display="inline" id="S5.I1.i1.p1.1.m1.2"><semantics id="S5.I1.i1.p1.1.m1.2a"><mrow id="S5.I1.i1.p1.1.m1.2.3" xref="S5.I1.i1.p1.1.m1.2.3.cmml"><mrow id="S5.I1.i1.p1.1.m1.2.3.2.2" xref="S5.I1.i1.p1.1.m1.2.3.2.1.cmml"><mo id="S5.I1.i1.p1.1.m1.2.3.2.2.1" stretchy="false" xref="S5.I1.i1.p1.1.m1.2.3.2.1.cmml">(</mo><mi id="S5.I1.i1.p1.1.m1.1.1" xref="S5.I1.i1.p1.1.m1.1.1.cmml">x</mi><mo id="S5.I1.i1.p1.1.m1.2.3.2.2.2" xref="S5.I1.i1.p1.1.m1.2.3.2.1.cmml">,</mo><mi id="S5.I1.i1.p1.1.m1.2.2" xref="S5.I1.i1.p1.1.m1.2.2.cmml">y</mi><mo id="S5.I1.i1.p1.1.m1.2.3.2.2.3" stretchy="false" xref="S5.I1.i1.p1.1.m1.2.3.2.1.cmml">)</mo></mrow><mo id="S5.I1.i1.p1.1.m1.2.3.1" xref="S5.I1.i1.p1.1.m1.2.3.1.cmml">β</mo><mi class="ltx_font_mathcaligraphic" id="S5.I1.i1.p1.1.m1.2.3.3" xref="S5.I1.i1.p1.1.m1.2.3.3.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.1.m1.2b"><apply id="S5.I1.i1.p1.1.m1.2.3.cmml" xref="S5.I1.i1.p1.1.m1.2.3"><in id="S5.I1.i1.p1.1.m1.2.3.1.cmml" xref="S5.I1.i1.p1.1.m1.2.3.1"></in><interval closure="open" id="S5.I1.i1.p1.1.m1.2.3.2.1.cmml" xref="S5.I1.i1.p1.1.m1.2.3.2.2"><ci id="S5.I1.i1.p1.1.m1.1.1.cmml" xref="S5.I1.i1.p1.1.m1.1.1">π₯</ci><ci id="S5.I1.i1.p1.1.m1.2.2.cmml" xref="S5.I1.i1.p1.1.m1.2.2">π¦</ci></interval><ci id="S5.I1.i1.p1.1.m1.2.3.3.cmml" xref="S5.I1.i1.p1.1.m1.2.3.3">β</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.1.m1.2c">(x,y)\in\mathcal{L}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.1.m1.2d">( italic_x , italic_y ) β caligraphic_L</annotation></semantics></math> if and only if <math alttext="x^{-1}x=y^{-1}y" class="ltx_Math" display="inline" id="S5.I1.i1.p1.2.m2.1"><semantics id="S5.I1.i1.p1.2.m2.1a"><mrow id="S5.I1.i1.p1.2.m2.1.1" xref="S5.I1.i1.p1.2.m2.1.1.cmml"><mrow id="S5.I1.i1.p1.2.m2.1.1.2" xref="S5.I1.i1.p1.2.m2.1.1.2.cmml"><msup id="S5.I1.i1.p1.2.m2.1.1.2.2" xref="S5.I1.i1.p1.2.m2.1.1.2.2.cmml"><mi id="S5.I1.i1.p1.2.m2.1.1.2.2.2" xref="S5.I1.i1.p1.2.m2.1.1.2.2.2.cmml">x</mi><mrow id="S5.I1.i1.p1.2.m2.1.1.2.2.3" xref="S5.I1.i1.p1.2.m2.1.1.2.2.3.cmml"><mo id="S5.I1.i1.p1.2.m2.1.1.2.2.3a" xref="S5.I1.i1.p1.2.m2.1.1.2.2.3.cmml">β</mo><mn id="S5.I1.i1.p1.2.m2.1.1.2.2.3.2" xref="S5.I1.i1.p1.2.m2.1.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S5.I1.i1.p1.2.m2.1.1.2.1" xref="S5.I1.i1.p1.2.m2.1.1.2.1.cmml">β’</mo><mi id="S5.I1.i1.p1.2.m2.1.1.2.3" xref="S5.I1.i1.p1.2.m2.1.1.2.3.cmml">x</mi></mrow><mo id="S5.I1.i1.p1.2.m2.1.1.1" xref="S5.I1.i1.p1.2.m2.1.1.1.cmml">=</mo><mrow id="S5.I1.i1.p1.2.m2.1.1.3" xref="S5.I1.i1.p1.2.m2.1.1.3.cmml"><msup id="S5.I1.i1.p1.2.m2.1.1.3.2" xref="S5.I1.i1.p1.2.m2.1.1.3.2.cmml"><mi id="S5.I1.i1.p1.2.m2.1.1.3.2.2" xref="S5.I1.i1.p1.2.m2.1.1.3.2.2.cmml">y</mi><mrow id="S5.I1.i1.p1.2.m2.1.1.3.2.3" xref="S5.I1.i1.p1.2.m2.1.1.3.2.3.cmml"><mo id="S5.I1.i1.p1.2.m2.1.1.3.2.3a" xref="S5.I1.i1.p1.2.m2.1.1.3.2.3.cmml">β</mo><mn id="S5.I1.i1.p1.2.m2.1.1.3.2.3.2" xref="S5.I1.i1.p1.2.m2.1.1.3.2.3.2.cmml">1</mn></mrow></msup><mo id="S5.I1.i1.p1.2.m2.1.1.3.1" xref="S5.I1.i1.p1.2.m2.1.1.3.1.cmml">β’</mo><mi id="S5.I1.i1.p1.2.m2.1.1.3.3" xref="S5.I1.i1.p1.2.m2.1.1.3.3.cmml">y</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.2.m2.1b"><apply id="S5.I1.i1.p1.2.m2.1.1.cmml" xref="S5.I1.i1.p1.2.m2.1.1"><eq id="S5.I1.i1.p1.2.m2.1.1.1.cmml" xref="S5.I1.i1.p1.2.m2.1.1.1"></eq><apply id="S5.I1.i1.p1.2.m2.1.1.2.cmml" xref="S5.I1.i1.p1.2.m2.1.1.2"><times id="S5.I1.i1.p1.2.m2.1.1.2.1.cmml" xref="S5.I1.i1.p1.2.m2.1.1.2.1"></times><apply id="S5.I1.i1.p1.2.m2.1.1.2.2.cmml" xref="S5.I1.i1.p1.2.m2.1.1.2.2"><csymbol cd="ambiguous" id="S5.I1.i1.p1.2.m2.1.1.2.2.1.cmml" xref="S5.I1.i1.p1.2.m2.1.1.2.2">superscript</csymbol><ci id="S5.I1.i1.p1.2.m2.1.1.2.2.2.cmml" xref="S5.I1.i1.p1.2.m2.1.1.2.2.2">π₯</ci><apply id="S5.I1.i1.p1.2.m2.1.1.2.2.3.cmml" xref="S5.I1.i1.p1.2.m2.1.1.2.2.3"><minus id="S5.I1.i1.p1.2.m2.1.1.2.2.3.1.cmml" xref="S5.I1.i1.p1.2.m2.1.1.2.2.3"></minus><cn id="S5.I1.i1.p1.2.m2.1.1.2.2.3.2.cmml" type="integer" xref="S5.I1.i1.p1.2.m2.1.1.2.2.3.2">1</cn></apply></apply><ci id="S5.I1.i1.p1.2.m2.1.1.2.3.cmml" xref="S5.I1.i1.p1.2.m2.1.1.2.3">π₯</ci></apply><apply id="S5.I1.i1.p1.2.m2.1.1.3.cmml" xref="S5.I1.i1.p1.2.m2.1.1.3"><times id="S5.I1.i1.p1.2.m2.1.1.3.1.cmml" xref="S5.I1.i1.p1.2.m2.1.1.3.1"></times><apply id="S5.I1.i1.p1.2.m2.1.1.3.2.cmml" xref="S5.I1.i1.p1.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.I1.i1.p1.2.m2.1.1.3.2.1.cmml" xref="S5.I1.i1.p1.2.m2.1.1.3.2">superscript</csymbol><ci id="S5.I1.i1.p1.2.m2.1.1.3.2.2.cmml" xref="S5.I1.i1.p1.2.m2.1.1.3.2.2">π¦</ci><apply id="S5.I1.i1.p1.2.m2.1.1.3.2.3.cmml" xref="S5.I1.i1.p1.2.m2.1.1.3.2.3"><minus id="S5.I1.i1.p1.2.m2.1.1.3.2.3.1.cmml" xref="S5.I1.i1.p1.2.m2.1.1.3.2.3"></minus><cn id="S5.I1.i1.p1.2.m2.1.1.3.2.3.2.cmml" type="integer" xref="S5.I1.i1.p1.2.m2.1.1.3.2.3.2">1</cn></apply></apply><ci id="S5.I1.i1.p1.2.m2.1.1.3.3.cmml" xref="S5.I1.i1.p1.2.m2.1.1.3.3">π¦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.2.m2.1c">x^{-1}x=y^{-1}y</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.2.m2.1d">italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x = italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_y</annotation></semantics></math>;</p> </div> </li> <li class="ltx_item" id="S5.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S5.I1.i2.p1"> <p class="ltx_p" id="S5.I1.i2.p1.2"><math alttext="(x,y)\in\mathcal{R}" class="ltx_Math" display="inline" id="S5.I1.i2.p1.1.m1.2"><semantics id="S5.I1.i2.p1.1.m1.2a"><mrow id="S5.I1.i2.p1.1.m1.2.3" xref="S5.I1.i2.p1.1.m1.2.3.cmml"><mrow id="S5.I1.i2.p1.1.m1.2.3.2.2" xref="S5.I1.i2.p1.1.m1.2.3.2.1.cmml"><mo id="S5.I1.i2.p1.1.m1.2.3.2.2.1" stretchy="false" xref="S5.I1.i2.p1.1.m1.2.3.2.1.cmml">(</mo><mi id="S5.I1.i2.p1.1.m1.1.1" xref="S5.I1.i2.p1.1.m1.1.1.cmml">x</mi><mo id="S5.I1.i2.p1.1.m1.2.3.2.2.2" xref="S5.I1.i2.p1.1.m1.2.3.2.1.cmml">,</mo><mi id="S5.I1.i2.p1.1.m1.2.2" xref="S5.I1.i2.p1.1.m1.2.2.cmml">y</mi><mo id="S5.I1.i2.p1.1.m1.2.3.2.2.3" stretchy="false" xref="S5.I1.i2.p1.1.m1.2.3.2.1.cmml">)</mo></mrow><mo id="S5.I1.i2.p1.1.m1.2.3.1" xref="S5.I1.i2.p1.1.m1.2.3.1.cmml">β</mo><mi class="ltx_font_mathcaligraphic" id="S5.I1.i2.p1.1.m1.2.3.3" xref="S5.I1.i2.p1.1.m1.2.3.3.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.1.m1.2b"><apply id="S5.I1.i2.p1.1.m1.2.3.cmml" xref="S5.I1.i2.p1.1.m1.2.3"><in id="S5.I1.i2.p1.1.m1.2.3.1.cmml" xref="S5.I1.i2.p1.1.m1.2.3.1"></in><interval closure="open" id="S5.I1.i2.p1.1.m1.2.3.2.1.cmml" xref="S5.I1.i2.p1.1.m1.2.3.2.2"><ci id="S5.I1.i2.p1.1.m1.1.1.cmml" xref="S5.I1.i2.p1.1.m1.1.1">π₯</ci><ci id="S5.I1.i2.p1.1.m1.2.2.cmml" xref="S5.I1.i2.p1.1.m1.2.2">π¦</ci></interval><ci id="S5.I1.i2.p1.1.m1.2.3.3.cmml" xref="S5.I1.i2.p1.1.m1.2.3.3">β</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.1.m1.2c">(x,y)\in\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.1.m1.2d">( italic_x , italic_y ) β caligraphic_R</annotation></semantics></math> if and only if <math alttext="xx^{-1}=yy^{-1}" class="ltx_Math" display="inline" id="S5.I1.i2.p1.2.m2.1"><semantics id="S5.I1.i2.p1.2.m2.1a"><mrow id="S5.I1.i2.p1.2.m2.1.1" xref="S5.I1.i2.p1.2.m2.1.1.cmml"><mrow id="S5.I1.i2.p1.2.m2.1.1.2" xref="S5.I1.i2.p1.2.m2.1.1.2.cmml"><mi id="S5.I1.i2.p1.2.m2.1.1.2.2" xref="S5.I1.i2.p1.2.m2.1.1.2.2.cmml">x</mi><mo id="S5.I1.i2.p1.2.m2.1.1.2.1" xref="S5.I1.i2.p1.2.m2.1.1.2.1.cmml">β’</mo><msup id="S5.I1.i2.p1.2.m2.1.1.2.3" xref="S5.I1.i2.p1.2.m2.1.1.2.3.cmml"><mi id="S5.I1.i2.p1.2.m2.1.1.2.3.2" xref="S5.I1.i2.p1.2.m2.1.1.2.3.2.cmml">x</mi><mrow id="S5.I1.i2.p1.2.m2.1.1.2.3.3" xref="S5.I1.i2.p1.2.m2.1.1.2.3.3.cmml"><mo id="S5.I1.i2.p1.2.m2.1.1.2.3.3a" xref="S5.I1.i2.p1.2.m2.1.1.2.3.3.cmml">β</mo><mn id="S5.I1.i2.p1.2.m2.1.1.2.3.3.2" xref="S5.I1.i2.p1.2.m2.1.1.2.3.3.2.cmml">1</mn></mrow></msup></mrow><mo id="S5.I1.i2.p1.2.m2.1.1.1" xref="S5.I1.i2.p1.2.m2.1.1.1.cmml">=</mo><mrow id="S5.I1.i2.p1.2.m2.1.1.3" xref="S5.I1.i2.p1.2.m2.1.1.3.cmml"><mi id="S5.I1.i2.p1.2.m2.1.1.3.2" xref="S5.I1.i2.p1.2.m2.1.1.3.2.cmml">y</mi><mo id="S5.I1.i2.p1.2.m2.1.1.3.1" xref="S5.I1.i2.p1.2.m2.1.1.3.1.cmml">β’</mo><msup id="S5.I1.i2.p1.2.m2.1.1.3.3" xref="S5.I1.i2.p1.2.m2.1.1.3.3.cmml"><mi id="S5.I1.i2.p1.2.m2.1.1.3.3.2" xref="S5.I1.i2.p1.2.m2.1.1.3.3.2.cmml">y</mi><mrow id="S5.I1.i2.p1.2.m2.1.1.3.3.3" xref="S5.I1.i2.p1.2.m2.1.1.3.3.3.cmml"><mo id="S5.I1.i2.p1.2.m2.1.1.3.3.3a" xref="S5.I1.i2.p1.2.m2.1.1.3.3.3.cmml">β</mo><mn id="S5.I1.i2.p1.2.m2.1.1.3.3.3.2" xref="S5.I1.i2.p1.2.m2.1.1.3.3.3.2.cmml">1</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.2.m2.1b"><apply id="S5.I1.i2.p1.2.m2.1.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1"><eq id="S5.I1.i2.p1.2.m2.1.1.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1.1"></eq><apply id="S5.I1.i2.p1.2.m2.1.1.2.cmml" xref="S5.I1.i2.p1.2.m2.1.1.2"><times id="S5.I1.i2.p1.2.m2.1.1.2.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1.2.1"></times><ci id="S5.I1.i2.p1.2.m2.1.1.2.2.cmml" xref="S5.I1.i2.p1.2.m2.1.1.2.2">π₯</ci><apply id="S5.I1.i2.p1.2.m2.1.1.2.3.cmml" xref="S5.I1.i2.p1.2.m2.1.1.2.3"><csymbol cd="ambiguous" id="S5.I1.i2.p1.2.m2.1.1.2.3.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1.2.3">superscript</csymbol><ci id="S5.I1.i2.p1.2.m2.1.1.2.3.2.cmml" xref="S5.I1.i2.p1.2.m2.1.1.2.3.2">π₯</ci><apply id="S5.I1.i2.p1.2.m2.1.1.2.3.3.cmml" xref="S5.I1.i2.p1.2.m2.1.1.2.3.3"><minus id="S5.I1.i2.p1.2.m2.1.1.2.3.3.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1.2.3.3"></minus><cn id="S5.I1.i2.p1.2.m2.1.1.2.3.3.2.cmml" type="integer" xref="S5.I1.i2.p1.2.m2.1.1.2.3.3.2">1</cn></apply></apply></apply><apply id="S5.I1.i2.p1.2.m2.1.1.3.cmml" xref="S5.I1.i2.p1.2.m2.1.1.3"><times id="S5.I1.i2.p1.2.m2.1.1.3.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1.3.1"></times><ci id="S5.I1.i2.p1.2.m2.1.1.3.2.cmml" xref="S5.I1.i2.p1.2.m2.1.1.3.2">π¦</ci><apply id="S5.I1.i2.p1.2.m2.1.1.3.3.cmml" xref="S5.I1.i2.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.I1.i2.p1.2.m2.1.1.3.3.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1.3.3">superscript</csymbol><ci id="S5.I1.i2.p1.2.m2.1.1.3.3.2.cmml" xref="S5.I1.i2.p1.2.m2.1.1.3.3.2">π¦</ci><apply id="S5.I1.i2.p1.2.m2.1.1.3.3.3.cmml" xref="S5.I1.i2.p1.2.m2.1.1.3.3.3"><minus id="S5.I1.i2.p1.2.m2.1.1.3.3.3.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1.3.3.3"></minus><cn id="S5.I1.i2.p1.2.m2.1.1.3.3.3.2.cmml" type="integer" xref="S5.I1.i2.p1.2.m2.1.1.3.3.3.2">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.2.m2.1c">xx^{-1}=yy^{-1}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.2.m2.1d">italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_y italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>;</p> </div> </li> <li class="ltx_item" id="S5.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(iii)</span> <div class="ltx_para" id="S5.I1.i3.p1"> <p class="ltx_p" id="S5.I1.i3.p1.1"><math alttext="\mathcal{H}=\mathcal{L}\cap\mathcal{R}" class="ltx_Math" display="inline" id="S5.I1.i3.p1.1.m1.1"><semantics id="S5.I1.i3.p1.1.m1.1a"><mrow id="S5.I1.i3.p1.1.m1.1.1" xref="S5.I1.i3.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I1.i3.p1.1.m1.1.1.2" xref="S5.I1.i3.p1.1.m1.1.1.2.cmml">β</mi><mo id="S5.I1.i3.p1.1.m1.1.1.1" xref="S5.I1.i3.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S5.I1.i3.p1.1.m1.1.1.3" xref="S5.I1.i3.p1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I1.i3.p1.1.m1.1.1.3.2" xref="S5.I1.i3.p1.1.m1.1.1.3.2.cmml">β</mi><mo id="S5.I1.i3.p1.1.m1.1.1.3.1" xref="S5.I1.i3.p1.1.m1.1.1.3.1.cmml">β©</mo><mi class="ltx_font_mathcaligraphic" id="S5.I1.i3.p1.1.m1.1.1.3.3" xref="S5.I1.i3.p1.1.m1.1.1.3.3.cmml">β</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.1.m1.1b"><apply id="S5.I1.i3.p1.1.m1.1.1.cmml" xref="S5.I1.i3.p1.1.m1.1.1"><eq id="S5.I1.i3.p1.1.m1.1.1.1.cmml" xref="S5.I1.i3.p1.1.m1.1.1.1"></eq><ci id="S5.I1.i3.p1.1.m1.1.1.2.cmml" xref="S5.I1.i3.p1.1.m1.1.1.2">β</ci><apply id="S5.I1.i3.p1.1.m1.1.1.3.cmml" xref="S5.I1.i3.p1.1.m1.1.1.3"><intersect id="S5.I1.i3.p1.1.m1.1.1.3.1.cmml" xref="S5.I1.i3.p1.1.m1.1.1.3.1"></intersect><ci id="S5.I1.i3.p1.1.m1.1.1.3.2.cmml" xref="S5.I1.i3.p1.1.m1.1.1.3.2">β</ci><ci id="S5.I1.i3.p1.1.m1.1.1.3.3.cmml" xref="S5.I1.i3.p1.1.m1.1.1.3.3">β</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.1.m1.1c">\mathcal{H}=\mathcal{L}\cap\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.1.m1.1d">caligraphic_H = caligraphic_L β© caligraphic_R</annotation></semantics></math>;</p> </div> </li> <li class="ltx_item" id="S5.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(iv)</span> <div class="ltx_para" id="S5.I1.i4.p1"> <p class="ltx_p" id="S5.I1.i4.p1.1"><math alttext="\mathcal{D}=\mathcal{L}\circ\mathcal{R}=\mathcal{R}\circ\mathcal{L}" class="ltx_Math" display="inline" id="S5.I1.i4.p1.1.m1.1"><semantics id="S5.I1.i4.p1.1.m1.1a"><mrow id="S5.I1.i4.p1.1.m1.1.1" xref="S5.I1.i4.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I1.i4.p1.1.m1.1.1.2" xref="S5.I1.i4.p1.1.m1.1.1.2.cmml">π</mi><mo id="S5.I1.i4.p1.1.m1.1.1.3" xref="S5.I1.i4.p1.1.m1.1.1.3.cmml">=</mo><mrow id="S5.I1.i4.p1.1.m1.1.1.4" xref="S5.I1.i4.p1.1.m1.1.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I1.i4.p1.1.m1.1.1.4.2" xref="S5.I1.i4.p1.1.m1.1.1.4.2.cmml">β</mi><mo id="S5.I1.i4.p1.1.m1.1.1.4.1" lspace="0.222em" rspace="0.222em" xref="S5.I1.i4.p1.1.m1.1.1.4.1.cmml">β</mo><mi class="ltx_font_mathcaligraphic" id="S5.I1.i4.p1.1.m1.1.1.4.3" xref="S5.I1.i4.p1.1.m1.1.1.4.3.cmml">β</mi></mrow><mo id="S5.I1.i4.p1.1.m1.1.1.5" xref="S5.I1.i4.p1.1.m1.1.1.5.cmml">=</mo><mrow id="S5.I1.i4.p1.1.m1.1.1.6" xref="S5.I1.i4.p1.1.m1.1.1.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I1.i4.p1.1.m1.1.1.6.2" xref="S5.I1.i4.p1.1.m1.1.1.6.2.cmml">β</mi><mo id="S5.I1.i4.p1.1.m1.1.1.6.1" lspace="0.222em" rspace="0.222em" xref="S5.I1.i4.p1.1.m1.1.1.6.1.cmml">β</mo><mi class="ltx_font_mathcaligraphic" id="S5.I1.i4.p1.1.m1.1.1.6.3" xref="S5.I1.i4.p1.1.m1.1.1.6.3.cmml">β</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i4.p1.1.m1.1b"><apply id="S5.I1.i4.p1.1.m1.1.1.cmml" xref="S5.I1.i4.p1.1.m1.1.1"><and id="S5.I1.i4.p1.1.m1.1.1a.cmml" xref="S5.I1.i4.p1.1.m1.1.1"></and><apply id="S5.I1.i4.p1.1.m1.1.1b.cmml" xref="S5.I1.i4.p1.1.m1.1.1"><eq id="S5.I1.i4.p1.1.m1.1.1.3.cmml" xref="S5.I1.i4.p1.1.m1.1.1.3"></eq><ci id="S5.I1.i4.p1.1.m1.1.1.2.cmml" xref="S5.I1.i4.p1.1.m1.1.1.2">π</ci><apply id="S5.I1.i4.p1.1.m1.1.1.4.cmml" xref="S5.I1.i4.p1.1.m1.1.1.4"><compose id="S5.I1.i4.p1.1.m1.1.1.4.1.cmml" xref="S5.I1.i4.p1.1.m1.1.1.4.1"></compose><ci id="S5.I1.i4.p1.1.m1.1.1.4.2.cmml" xref="S5.I1.i4.p1.1.m1.1.1.4.2">β</ci><ci id="S5.I1.i4.p1.1.m1.1.1.4.3.cmml" xref="S5.I1.i4.p1.1.m1.1.1.4.3">β</ci></apply></apply><apply id="S5.I1.i4.p1.1.m1.1.1c.cmml" xref="S5.I1.i4.p1.1.m1.1.1"><eq id="S5.I1.i4.p1.1.m1.1.1.5.cmml" xref="S5.I1.i4.p1.1.m1.1.1.5"></eq><share href="https://arxiv.org/html/2503.13666v1#S5.I1.i4.p1.1.m1.1.1.4.cmml" id="S5.I1.i4.p1.1.m1.1.1d.cmml" xref="S5.I1.i4.p1.1.m1.1.1"></share><apply id="S5.I1.i4.p1.1.m1.1.1.6.cmml" xref="S5.I1.i4.p1.1.m1.1.1.6"><compose id="S5.I1.i4.p1.1.m1.1.1.6.1.cmml" xref="S5.I1.i4.p1.1.m1.1.1.6.1"></compose><ci id="S5.I1.i4.p1.1.m1.1.1.6.2.cmml" xref="S5.I1.i4.p1.1.m1.1.1.6.2">β</ci><ci id="S5.I1.i4.p1.1.m1.1.1.6.3.cmml" xref="S5.I1.i4.p1.1.m1.1.1.6.3">β</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i4.p1.1.m1.1c">\mathcal{D}=\mathcal{L}\circ\mathcal{R}=\mathcal{R}\circ\mathcal{L}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i4.p1.1.m1.1d">caligraphic_D = caligraphic_L β caligraphic_R = caligraphic_R β caligraphic_L</annotation></semantics></math>.</p> </div> </li> </ol> <p class="ltx_p" id="S5.p2.1">For more about Greenβs relations on inverse semigroups see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib36" title="">36</a>, Chapter 3.2]</cite>. The following two results can be considered folklore.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S5.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem1.1.1.1">Proposition 5.1</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem1.2.2"> </span>(Folklore)<span class="ltx_text ltx_font_bold" id="S5.Thmtheorem1.3.3">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem1.p1"> <p class="ltx_p" id="S5.Thmtheorem1.p1.1"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem1.p1.1.1">Every nonempty compact subset of a pospace contains a minimal element.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S5.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem2.1.1.1">Proposition 5.2</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem2.2.2"> </span>(Folklore)<span class="ltx_text ltx_font_bold" id="S5.Thmtheorem2.3.3">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem2.p1"> <p class="ltx_p" id="S5.Thmtheorem2.p1.3"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem2.p1.3.3">If <math alttext="S" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.1.1.m1.1"><semantics id="S5.Thmtheorem2.p1.1.1.m1.1a"><mi id="S5.Thmtheorem2.p1.1.1.m1.1.1" xref="S5.Thmtheorem2.p1.1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.1.1.m1.1b"><ci id="S5.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.1.1.m1.1d">italic_S</annotation></semantics></math> is a topological inverse semigroup, then each two <math alttext="\mathcal{H}" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.2.2.m2.1"><semantics id="S5.Thmtheorem2.p1.2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem2.p1.2.2.m2.1.1" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.2.2.m2.1b"><ci id="S5.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.2.2.m2.1c">\mathcal{H}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.2.2.m2.1d">caligraphic_H</annotation></semantics></math>-classes within the one <math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.3.3.m3.1"><semantics id="S5.Thmtheorem2.p1.3.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem2.p1.3.3.m3.1.1" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.3.3.m3.1b"><ci id="S5.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.3.3.m3.1c">\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.3.3.m3.1d">caligraphic_D</annotation></semantics></math>-class are homeomorphic.</span></p> </div> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.1">In fact, the proof of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S5.Thmtheorem2" title="Proposition 5.2 (Folklore). β£ 5. Proof of the main result and final remarks β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">5.2</span></a> follows from the proof of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib31" title="">31</a>, Lemma 2.2.3]</cite>. We are in a position to prove Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem1" title="Theorem A. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">A</span></a>, which states that every locally compact Nyikos inverse topological semigroup is compact.</p> </div> <div class="ltx_proof" id="S5.6"> <h6 class="ltx_title ltx_runin ltx_font_bold ltx_title_proof">Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem1" title="Theorem A. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">A</span></a>.</h6> <div class="ltx_para" id="S5.1.p1"> <p class="ltx_p" id="S5.1.p1.13">Consider a locally compact Nyikos inverse topological semigroup <math alttext="S" class="ltx_Math" display="inline" id="S5.1.p1.1.m1.1"><semantics id="S5.1.p1.1.m1.1a"><mi id="S5.1.p1.1.m1.1.1" xref="S5.1.p1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.1.m1.1b"><ci id="S5.1.p1.1.m1.1.1.cmml" xref="S5.1.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.1.m1.1d">italic_S</annotation></semantics></math>. By Corollary <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S1.Thmtheorem4" title="Corollary 1.4. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">1.4</span></a>, <math alttext="S" class="ltx_Math" display="inline" id="S5.1.p1.2.m2.1"><semantics id="S5.1.p1.2.m2.1a"><mi id="S5.1.p1.2.m2.1.1" xref="S5.1.p1.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.2.m2.1b"><ci id="S5.1.p1.2.m2.1.1.cmml" xref="S5.1.p1.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.2.m2.1d">italic_S</annotation></semantics></math> is a topological inverse semigroup. Note that the semilattice of idempotents <math alttext="E(S)" class="ltx_Math" display="inline" id="S5.1.p1.3.m3.1"><semantics id="S5.1.p1.3.m3.1a"><mrow id="S5.1.p1.3.m3.1.2" xref="S5.1.p1.3.m3.1.2.cmml"><mi id="S5.1.p1.3.m3.1.2.2" xref="S5.1.p1.3.m3.1.2.2.cmml">E</mi><mo id="S5.1.p1.3.m3.1.2.1" xref="S5.1.p1.3.m3.1.2.1.cmml">β’</mo><mrow id="S5.1.p1.3.m3.1.2.3.2" xref="S5.1.p1.3.m3.1.2.cmml"><mo id="S5.1.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S5.1.p1.3.m3.1.2.cmml">(</mo><mi id="S5.1.p1.3.m3.1.1" xref="S5.1.p1.3.m3.1.1.cmml">S</mi><mo id="S5.1.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S5.1.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.3.m3.1b"><apply id="S5.1.p1.3.m3.1.2.cmml" xref="S5.1.p1.3.m3.1.2"><times id="S5.1.p1.3.m3.1.2.1.cmml" xref="S5.1.p1.3.m3.1.2.1"></times><ci id="S5.1.p1.3.m3.1.2.2.cmml" xref="S5.1.p1.3.m3.1.2.2">πΈ</ci><ci id="S5.1.p1.3.m3.1.1.cmml" xref="S5.1.p1.3.m3.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.3.m3.1c">E(S)</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.3.m3.1d">italic_E ( italic_S )</annotation></semantics></math> is a retract of <math alttext="S" class="ltx_Math" display="inline" id="S5.1.p1.4.m4.1"><semantics id="S5.1.p1.4.m4.1a"><mi id="S5.1.p1.4.m4.1.1" xref="S5.1.p1.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.4.m4.1b"><ci id="S5.1.p1.4.m4.1.1.cmml" xref="S5.1.p1.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.4.m4.1d">italic_S</annotation></semantics></math> under the continuous map <math alttext="\pi:x\mapsto xx^{-1}" class="ltx_Math" display="inline" id="S5.1.p1.5.m5.1"><semantics id="S5.1.p1.5.m5.1a"><mrow id="S5.1.p1.5.m5.1.1" xref="S5.1.p1.5.m5.1.1.cmml"><mi id="S5.1.p1.5.m5.1.1.2" xref="S5.1.p1.5.m5.1.1.2.cmml">Ο</mi><mo id="S5.1.p1.5.m5.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.1.p1.5.m5.1.1.1.cmml">:</mo><mrow id="S5.1.p1.5.m5.1.1.3" xref="S5.1.p1.5.m5.1.1.3.cmml"><mi id="S5.1.p1.5.m5.1.1.3.2" xref="S5.1.p1.5.m5.1.1.3.2.cmml">x</mi><mo id="S5.1.p1.5.m5.1.1.3.1" stretchy="false" xref="S5.1.p1.5.m5.1.1.3.1.cmml">β¦</mo><mrow id="S5.1.p1.5.m5.1.1.3.3" xref="S5.1.p1.5.m5.1.1.3.3.cmml"><mi id="S5.1.p1.5.m5.1.1.3.3.2" xref="S5.1.p1.5.m5.1.1.3.3.2.cmml">x</mi><mo id="S5.1.p1.5.m5.1.1.3.3.1" xref="S5.1.p1.5.m5.1.1.3.3.1.cmml">β’</mo><msup id="S5.1.p1.5.m5.1.1.3.3.3" xref="S5.1.p1.5.m5.1.1.3.3.3.cmml"><mi id="S5.1.p1.5.m5.1.1.3.3.3.2" xref="S5.1.p1.5.m5.1.1.3.3.3.2.cmml">x</mi><mrow id="S5.1.p1.5.m5.1.1.3.3.3.3" xref="S5.1.p1.5.m5.1.1.3.3.3.3.cmml"><mo id="S5.1.p1.5.m5.1.1.3.3.3.3a" xref="S5.1.p1.5.m5.1.1.3.3.3.3.cmml">β</mo><mn id="S5.1.p1.5.m5.1.1.3.3.3.3.2" xref="S5.1.p1.5.m5.1.1.3.3.3.3.2.cmml">1</mn></mrow></msup></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.5.m5.1b"><apply id="S5.1.p1.5.m5.1.1.cmml" xref="S5.1.p1.5.m5.1.1"><ci id="S5.1.p1.5.m5.1.1.1.cmml" xref="S5.1.p1.5.m5.1.1.1">:</ci><ci id="S5.1.p1.5.m5.1.1.2.cmml" xref="S5.1.p1.5.m5.1.1.2">π</ci><apply id="S5.1.p1.5.m5.1.1.3.cmml" xref="S5.1.p1.5.m5.1.1.3"><csymbol cd="latexml" id="S5.1.p1.5.m5.1.1.3.1.cmml" xref="S5.1.p1.5.m5.1.1.3.1">maps-to</csymbol><ci id="S5.1.p1.5.m5.1.1.3.2.cmml" xref="S5.1.p1.5.m5.1.1.3.2">π₯</ci><apply id="S5.1.p1.5.m5.1.1.3.3.cmml" xref="S5.1.p1.5.m5.1.1.3.3"><times id="S5.1.p1.5.m5.1.1.3.3.1.cmml" xref="S5.1.p1.5.m5.1.1.3.3.1"></times><ci id="S5.1.p1.5.m5.1.1.3.3.2.cmml" xref="S5.1.p1.5.m5.1.1.3.3.2">π₯</ci><apply id="S5.1.p1.5.m5.1.1.3.3.3.cmml" xref="S5.1.p1.5.m5.1.1.3.3.3"><csymbol cd="ambiguous" id="S5.1.p1.5.m5.1.1.3.3.3.1.cmml" xref="S5.1.p1.5.m5.1.1.3.3.3">superscript</csymbol><ci id="S5.1.p1.5.m5.1.1.3.3.3.2.cmml" xref="S5.1.p1.5.m5.1.1.3.3.3.2">π₯</ci><apply id="S5.1.p1.5.m5.1.1.3.3.3.3.cmml" xref="S5.1.p1.5.m5.1.1.3.3.3.3"><minus id="S5.1.p1.5.m5.1.1.3.3.3.3.1.cmml" xref="S5.1.p1.5.m5.1.1.3.3.3.3"></minus><cn id="S5.1.p1.5.m5.1.1.3.3.3.3.2.cmml" type="integer" xref="S5.1.p1.5.m5.1.1.3.3.3.3.2">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.5.m5.1c">\pi:x\mapsto xx^{-1}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.5.m5.1d">italic_Ο : italic_x β¦ italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. It follows that <math alttext="E(S)" class="ltx_Math" display="inline" id="S5.1.p1.6.m6.1"><semantics id="S5.1.p1.6.m6.1a"><mrow id="S5.1.p1.6.m6.1.2" xref="S5.1.p1.6.m6.1.2.cmml"><mi id="S5.1.p1.6.m6.1.2.2" xref="S5.1.p1.6.m6.1.2.2.cmml">E</mi><mo id="S5.1.p1.6.m6.1.2.1" xref="S5.1.p1.6.m6.1.2.1.cmml">β’</mo><mrow id="S5.1.p1.6.m6.1.2.3.2" xref="S5.1.p1.6.m6.1.2.cmml"><mo id="S5.1.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S5.1.p1.6.m6.1.2.cmml">(</mo><mi id="S5.1.p1.6.m6.1.1" xref="S5.1.p1.6.m6.1.1.cmml">S</mi><mo id="S5.1.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S5.1.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.6.m6.1b"><apply id="S5.1.p1.6.m6.1.2.cmml" xref="S5.1.p1.6.m6.1.2"><times id="S5.1.p1.6.m6.1.2.1.cmml" xref="S5.1.p1.6.m6.1.2.1"></times><ci id="S5.1.p1.6.m6.1.2.2.cmml" xref="S5.1.p1.6.m6.1.2.2">πΈ</ci><ci id="S5.1.p1.6.m6.1.1.cmml" xref="S5.1.p1.6.m6.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.6.m6.1c">E(S)</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.6.m6.1d">italic_E ( italic_S )</annotation></semantics></math> is a locally compact Nyikos topological semilattice. Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S4.Thmtheorem12" title="Theorem 4.12. β£ 4. Nyikos semilattices β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">4.12</span></a> implies that <math alttext="E(S)" class="ltx_Math" display="inline" id="S5.1.p1.7.m7.1"><semantics id="S5.1.p1.7.m7.1a"><mrow id="S5.1.p1.7.m7.1.2" xref="S5.1.p1.7.m7.1.2.cmml"><mi id="S5.1.p1.7.m7.1.2.2" xref="S5.1.p1.7.m7.1.2.2.cmml">E</mi><mo id="S5.1.p1.7.m7.1.2.1" xref="S5.1.p1.7.m7.1.2.1.cmml">β’</mo><mrow id="S5.1.p1.7.m7.1.2.3.2" xref="S5.1.p1.7.m7.1.2.cmml"><mo id="S5.1.p1.7.m7.1.2.3.2.1" stretchy="false" xref="S5.1.p1.7.m7.1.2.cmml">(</mo><mi id="S5.1.p1.7.m7.1.1" xref="S5.1.p1.7.m7.1.1.cmml">S</mi><mo id="S5.1.p1.7.m7.1.2.3.2.2" stretchy="false" xref="S5.1.p1.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.7.m7.1b"><apply id="S5.1.p1.7.m7.1.2.cmml" xref="S5.1.p1.7.m7.1.2"><times id="S5.1.p1.7.m7.1.2.1.cmml" xref="S5.1.p1.7.m7.1.2.1"></times><ci id="S5.1.p1.7.m7.1.2.2.cmml" xref="S5.1.p1.7.m7.1.2.2">πΈ</ci><ci id="S5.1.p1.7.m7.1.1.cmml" xref="S5.1.p1.7.m7.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.7.m7.1c">E(S)</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.7.m7.1d">italic_E ( italic_S )</annotation></semantics></math> is compact. By Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem3" title="Theorem C. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">C</span></a>, <math alttext="\beta S" class="ltx_Math" display="inline" id="S5.1.p1.8.m8.1"><semantics id="S5.1.p1.8.m8.1a"><mrow id="S5.1.p1.8.m8.1.1" xref="S5.1.p1.8.m8.1.1.cmml"><mi id="S5.1.p1.8.m8.1.1.2" xref="S5.1.p1.8.m8.1.1.2.cmml">Ξ²</mi><mo id="S5.1.p1.8.m8.1.1.1" xref="S5.1.p1.8.m8.1.1.1.cmml">β’</mo><mi id="S5.1.p1.8.m8.1.1.3" xref="S5.1.p1.8.m8.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.8.m8.1b"><apply id="S5.1.p1.8.m8.1.1.cmml" xref="S5.1.p1.8.m8.1.1"><times id="S5.1.p1.8.m8.1.1.1.cmml" xref="S5.1.p1.8.m8.1.1.1"></times><ci id="S5.1.p1.8.m8.1.1.2.cmml" xref="S5.1.p1.8.m8.1.1.2">π½</ci><ci id="S5.1.p1.8.m8.1.1.3.cmml" xref="S5.1.p1.8.m8.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.8.m8.1c">\beta S</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.8.m8.1d">italic_Ξ² italic_S</annotation></semantics></math> is a topological inverse semigroup. Since <math alttext="S" class="ltx_Math" display="inline" id="S5.1.p1.9.m9.1"><semantics id="S5.1.p1.9.m9.1a"><mi id="S5.1.p1.9.m9.1.1" xref="S5.1.p1.9.m9.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.9.m9.1b"><ci id="S5.1.p1.9.m9.1.1.cmml" xref="S5.1.p1.9.m9.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.9.m9.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.9.m9.1d">italic_S</annotation></semantics></math> is dense in <math alttext="\beta S" class="ltx_Math" display="inline" id="S5.1.p1.10.m10.1"><semantics id="S5.1.p1.10.m10.1a"><mrow id="S5.1.p1.10.m10.1.1" xref="S5.1.p1.10.m10.1.1.cmml"><mi id="S5.1.p1.10.m10.1.1.2" xref="S5.1.p1.10.m10.1.1.2.cmml">Ξ²</mi><mo id="S5.1.p1.10.m10.1.1.1" xref="S5.1.p1.10.m10.1.1.1.cmml">β’</mo><mi id="S5.1.p1.10.m10.1.1.3" xref="S5.1.p1.10.m10.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.10.m10.1b"><apply id="S5.1.p1.10.m10.1.1.cmml" xref="S5.1.p1.10.m10.1.1"><times id="S5.1.p1.10.m10.1.1.1.cmml" xref="S5.1.p1.10.m10.1.1.1"></times><ci id="S5.1.p1.10.m10.1.1.2.cmml" xref="S5.1.p1.10.m10.1.1.2">π½</ci><ci id="S5.1.p1.10.m10.1.1.3.cmml" xref="S5.1.p1.10.m10.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.10.m10.1c">\beta S</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.10.m10.1d">italic_Ξ² italic_S</annotation></semantics></math>, we get that <math alttext="E(S)" class="ltx_Math" display="inline" id="S5.1.p1.11.m11.1"><semantics id="S5.1.p1.11.m11.1a"><mrow id="S5.1.p1.11.m11.1.2" xref="S5.1.p1.11.m11.1.2.cmml"><mi id="S5.1.p1.11.m11.1.2.2" xref="S5.1.p1.11.m11.1.2.2.cmml">E</mi><mo id="S5.1.p1.11.m11.1.2.1" xref="S5.1.p1.11.m11.1.2.1.cmml">β’</mo><mrow id="S5.1.p1.11.m11.1.2.3.2" xref="S5.1.p1.11.m11.1.2.cmml"><mo id="S5.1.p1.11.m11.1.2.3.2.1" stretchy="false" xref="S5.1.p1.11.m11.1.2.cmml">(</mo><mi id="S5.1.p1.11.m11.1.1" xref="S5.1.p1.11.m11.1.1.cmml">S</mi><mo id="S5.1.p1.11.m11.1.2.3.2.2" stretchy="false" xref="S5.1.p1.11.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.11.m11.1b"><apply id="S5.1.p1.11.m11.1.2.cmml" xref="S5.1.p1.11.m11.1.2"><times id="S5.1.p1.11.m11.1.2.1.cmml" xref="S5.1.p1.11.m11.1.2.1"></times><ci id="S5.1.p1.11.m11.1.2.2.cmml" xref="S5.1.p1.11.m11.1.2.2">πΈ</ci><ci id="S5.1.p1.11.m11.1.1.cmml" xref="S5.1.p1.11.m11.1.1">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.11.m11.1c">E(S)</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.11.m11.1d">italic_E ( italic_S )</annotation></semantics></math> is dense in <math alttext="E(\beta S)" class="ltx_Math" display="inline" id="S5.1.p1.12.m12.1"><semantics id="S5.1.p1.12.m12.1a"><mrow id="S5.1.p1.12.m12.1.1" xref="S5.1.p1.12.m12.1.1.cmml"><mi id="S5.1.p1.12.m12.1.1.3" xref="S5.1.p1.12.m12.1.1.3.cmml">E</mi><mo id="S5.1.p1.12.m12.1.1.2" xref="S5.1.p1.12.m12.1.1.2.cmml">β’</mo><mrow id="S5.1.p1.12.m12.1.1.1.1" xref="S5.1.p1.12.m12.1.1.1.1.1.cmml"><mo id="S5.1.p1.12.m12.1.1.1.1.2" stretchy="false" xref="S5.1.p1.12.m12.1.1.1.1.1.cmml">(</mo><mrow id="S5.1.p1.12.m12.1.1.1.1.1" xref="S5.1.p1.12.m12.1.1.1.1.1.cmml"><mi id="S5.1.p1.12.m12.1.1.1.1.1.2" xref="S5.1.p1.12.m12.1.1.1.1.1.2.cmml">Ξ²</mi><mo id="S5.1.p1.12.m12.1.1.1.1.1.1" xref="S5.1.p1.12.m12.1.1.1.1.1.1.cmml">β’</mo><mi id="S5.1.p1.12.m12.1.1.1.1.1.3" xref="S5.1.p1.12.m12.1.1.1.1.1.3.cmml">S</mi></mrow><mo id="S5.1.p1.12.m12.1.1.1.1.3" stretchy="false" xref="S5.1.p1.12.m12.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.12.m12.1b"><apply id="S5.1.p1.12.m12.1.1.cmml" xref="S5.1.p1.12.m12.1.1"><times id="S5.1.p1.12.m12.1.1.2.cmml" xref="S5.1.p1.12.m12.1.1.2"></times><ci id="S5.1.p1.12.m12.1.1.3.cmml" xref="S5.1.p1.12.m12.1.1.3">πΈ</ci><apply id="S5.1.p1.12.m12.1.1.1.1.1.cmml" xref="S5.1.p1.12.m12.1.1.1.1"><times id="S5.1.p1.12.m12.1.1.1.1.1.1.cmml" xref="S5.1.p1.12.m12.1.1.1.1.1.1"></times><ci id="S5.1.p1.12.m12.1.1.1.1.1.2.cmml" xref="S5.1.p1.12.m12.1.1.1.1.1.2">π½</ci><ci id="S5.1.p1.12.m12.1.1.1.1.1.3.cmml" xref="S5.1.p1.12.m12.1.1.1.1.1.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.12.m12.1c">E(\beta S)</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.12.m12.1d">italic_E ( italic_Ξ² italic_S )</annotation></semantics></math>. Thus <math alttext="E(S)=E(\beta S)" class="ltx_Math" display="inline" id="S5.1.p1.13.m13.2"><semantics id="S5.1.p1.13.m13.2a"><mrow id="S5.1.p1.13.m13.2.2" xref="S5.1.p1.13.m13.2.2.cmml"><mrow id="S5.1.p1.13.m13.2.2.3" xref="S5.1.p1.13.m13.2.2.3.cmml"><mi id="S5.1.p1.13.m13.2.2.3.2" xref="S5.1.p1.13.m13.2.2.3.2.cmml">E</mi><mo id="S5.1.p1.13.m13.2.2.3.1" xref="S5.1.p1.13.m13.2.2.3.1.cmml">β’</mo><mrow id="S5.1.p1.13.m13.2.2.3.3.2" xref="S5.1.p1.13.m13.2.2.3.cmml"><mo id="S5.1.p1.13.m13.2.2.3.3.2.1" stretchy="false" xref="S5.1.p1.13.m13.2.2.3.cmml">(</mo><mi id="S5.1.p1.13.m13.1.1" xref="S5.1.p1.13.m13.1.1.cmml">S</mi><mo id="S5.1.p1.13.m13.2.2.3.3.2.2" stretchy="false" xref="S5.1.p1.13.m13.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.1.p1.13.m13.2.2.2" xref="S5.1.p1.13.m13.2.2.2.cmml">=</mo><mrow id="S5.1.p1.13.m13.2.2.1" xref="S5.1.p1.13.m13.2.2.1.cmml"><mi id="S5.1.p1.13.m13.2.2.1.3" xref="S5.1.p1.13.m13.2.2.1.3.cmml">E</mi><mo id="S5.1.p1.13.m13.2.2.1.2" xref="S5.1.p1.13.m13.2.2.1.2.cmml">β’</mo><mrow id="S5.1.p1.13.m13.2.2.1.1.1" xref="S5.1.p1.13.m13.2.2.1.1.1.1.cmml"><mo id="S5.1.p1.13.m13.2.2.1.1.1.2" stretchy="false" xref="S5.1.p1.13.m13.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.1.p1.13.m13.2.2.1.1.1.1" xref="S5.1.p1.13.m13.2.2.1.1.1.1.cmml"><mi id="S5.1.p1.13.m13.2.2.1.1.1.1.2" xref="S5.1.p1.13.m13.2.2.1.1.1.1.2.cmml">Ξ²</mi><mo id="S5.1.p1.13.m13.2.2.1.1.1.1.1" xref="S5.1.p1.13.m13.2.2.1.1.1.1.1.cmml">β’</mo><mi id="S5.1.p1.13.m13.2.2.1.1.1.1.3" xref="S5.1.p1.13.m13.2.2.1.1.1.1.3.cmml">S</mi></mrow><mo id="S5.1.p1.13.m13.2.2.1.1.1.3" stretchy="false" xref="S5.1.p1.13.m13.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.13.m13.2b"><apply id="S5.1.p1.13.m13.2.2.cmml" xref="S5.1.p1.13.m13.2.2"><eq id="S5.1.p1.13.m13.2.2.2.cmml" xref="S5.1.p1.13.m13.2.2.2"></eq><apply id="S5.1.p1.13.m13.2.2.3.cmml" xref="S5.1.p1.13.m13.2.2.3"><times id="S5.1.p1.13.m13.2.2.3.1.cmml" xref="S5.1.p1.13.m13.2.2.3.1"></times><ci id="S5.1.p1.13.m13.2.2.3.2.cmml" xref="S5.1.p1.13.m13.2.2.3.2">πΈ</ci><ci id="S5.1.p1.13.m13.1.1.cmml" xref="S5.1.p1.13.m13.1.1">π</ci></apply><apply id="S5.1.p1.13.m13.2.2.1.cmml" xref="S5.1.p1.13.m13.2.2.1"><times id="S5.1.p1.13.m13.2.2.1.2.cmml" xref="S5.1.p1.13.m13.2.2.1.2"></times><ci id="S5.1.p1.13.m13.2.2.1.3.cmml" xref="S5.1.p1.13.m13.2.2.1.3">πΈ</ci><apply id="S5.1.p1.13.m13.2.2.1.1.1.1.cmml" xref="S5.1.p1.13.m13.2.2.1.1.1"><times id="S5.1.p1.13.m13.2.2.1.1.1.1.1.cmml" xref="S5.1.p1.13.m13.2.2.1.1.1.1.1"></times><ci id="S5.1.p1.13.m13.2.2.1.1.1.1.2.cmml" xref="S5.1.p1.13.m13.2.2.1.1.1.1.2">π½</ci><ci id="S5.1.p1.13.m13.2.2.1.1.1.1.3.cmml" xref="S5.1.p1.13.m13.2.2.1.1.1.1.3">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.13.m13.2c">E(S)=E(\beta S)</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.13.m13.2d">italic_E ( italic_S ) = italic_E ( italic_Ξ² italic_S )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.2.p2"> <p class="ltx_p" id="S5.2.p2.11">Seeking a contradiction, assume that <math alttext="\beta S\setminus S\neq\varnothing" class="ltx_Math" display="inline" id="S5.2.p2.1.m1.1"><semantics id="S5.2.p2.1.m1.1a"><mrow id="S5.2.p2.1.m1.1.1" xref="S5.2.p2.1.m1.1.1.cmml"><mrow id="S5.2.p2.1.m1.1.1.2" xref="S5.2.p2.1.m1.1.1.2.cmml"><mrow id="S5.2.p2.1.m1.1.1.2.2" xref="S5.2.p2.1.m1.1.1.2.2.cmml"><mi id="S5.2.p2.1.m1.1.1.2.2.2" xref="S5.2.p2.1.m1.1.1.2.2.2.cmml">Ξ²</mi><mo id="S5.2.p2.1.m1.1.1.2.2.1" xref="S5.2.p2.1.m1.1.1.2.2.1.cmml">β’</mo><mi id="S5.2.p2.1.m1.1.1.2.2.3" xref="S5.2.p2.1.m1.1.1.2.2.3.cmml">S</mi></mrow><mo id="S5.2.p2.1.m1.1.1.2.1" xref="S5.2.p2.1.m1.1.1.2.1.cmml">β</mo><mi id="S5.2.p2.1.m1.1.1.2.3" xref="S5.2.p2.1.m1.1.1.2.3.cmml">S</mi></mrow><mo id="S5.2.p2.1.m1.1.1.1" xref="S5.2.p2.1.m1.1.1.1.cmml">β </mo><mi id="S5.2.p2.1.m1.1.1.3" mathvariant="normal" xref="S5.2.p2.1.m1.1.1.3.cmml">β </mi></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.1.m1.1b"><apply id="S5.2.p2.1.m1.1.1.cmml" xref="S5.2.p2.1.m1.1.1"><neq id="S5.2.p2.1.m1.1.1.1.cmml" xref="S5.2.p2.1.m1.1.1.1"></neq><apply id="S5.2.p2.1.m1.1.1.2.cmml" xref="S5.2.p2.1.m1.1.1.2"><setdiff id="S5.2.p2.1.m1.1.1.2.1.cmml" xref="S5.2.p2.1.m1.1.1.2.1"></setdiff><apply id="S5.2.p2.1.m1.1.1.2.2.cmml" xref="S5.2.p2.1.m1.1.1.2.2"><times id="S5.2.p2.1.m1.1.1.2.2.1.cmml" xref="S5.2.p2.1.m1.1.1.2.2.1"></times><ci id="S5.2.p2.1.m1.1.1.2.2.2.cmml" xref="S5.2.p2.1.m1.1.1.2.2.2">π½</ci><ci id="S5.2.p2.1.m1.1.1.2.2.3.cmml" xref="S5.2.p2.1.m1.1.1.2.2.3">π</ci></apply><ci id="S5.2.p2.1.m1.1.1.2.3.cmml" xref="S5.2.p2.1.m1.1.1.2.3">π</ci></apply><emptyset id="S5.2.p2.1.m1.1.1.3.cmml" xref="S5.2.p2.1.m1.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.1.m1.1c">\beta S\setminus S\neq\varnothing</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.1.m1.1d">italic_Ξ² italic_S β italic_S β β </annotation></semantics></math>. Since <math alttext="S" class="ltx_Math" display="inline" id="S5.2.p2.2.m2.1"><semantics id="S5.2.p2.2.m2.1a"><mi id="S5.2.p2.2.m2.1.1" xref="S5.2.p2.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.2.m2.1b"><ci id="S5.2.p2.2.m2.1.1.cmml" xref="S5.2.p2.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.2.m2.1d">italic_S</annotation></semantics></math> is locally compact, the remainder <math alttext="\beta S\setminus S" class="ltx_Math" display="inline" id="S5.2.p2.3.m3.1"><semantics id="S5.2.p2.3.m3.1a"><mrow id="S5.2.p2.3.m3.1.1" xref="S5.2.p2.3.m3.1.1.cmml"><mrow id="S5.2.p2.3.m3.1.1.2" xref="S5.2.p2.3.m3.1.1.2.cmml"><mi id="S5.2.p2.3.m3.1.1.2.2" xref="S5.2.p2.3.m3.1.1.2.2.cmml">Ξ²</mi><mo id="S5.2.p2.3.m3.1.1.2.1" xref="S5.2.p2.3.m3.1.1.2.1.cmml">β’</mo><mi id="S5.2.p2.3.m3.1.1.2.3" xref="S5.2.p2.3.m3.1.1.2.3.cmml">S</mi></mrow><mo id="S5.2.p2.3.m3.1.1.1" xref="S5.2.p2.3.m3.1.1.1.cmml">β</mo><mi id="S5.2.p2.3.m3.1.1.3" xref="S5.2.p2.3.m3.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.3.m3.1b"><apply id="S5.2.p2.3.m3.1.1.cmml" xref="S5.2.p2.3.m3.1.1"><setdiff id="S5.2.p2.3.m3.1.1.1.cmml" xref="S5.2.p2.3.m3.1.1.1"></setdiff><apply id="S5.2.p2.3.m3.1.1.2.cmml" xref="S5.2.p2.3.m3.1.1.2"><times id="S5.2.p2.3.m3.1.1.2.1.cmml" xref="S5.2.p2.3.m3.1.1.2.1"></times><ci id="S5.2.p2.3.m3.1.1.2.2.cmml" xref="S5.2.p2.3.m3.1.1.2.2">π½</ci><ci id="S5.2.p2.3.m3.1.1.2.3.cmml" xref="S5.2.p2.3.m3.1.1.2.3">π</ci></apply><ci id="S5.2.p2.3.m3.1.1.3.cmml" xref="S5.2.p2.3.m3.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.3.m3.1c">\beta S\setminus S</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.3.m3.1d">italic_Ξ² italic_S β italic_S</annotation></semantics></math> is closed and, consequently, compact. As <math alttext="\beta S" class="ltx_Math" display="inline" id="S5.2.p2.4.m4.1"><semantics id="S5.2.p2.4.m4.1a"><mrow id="S5.2.p2.4.m4.1.1" xref="S5.2.p2.4.m4.1.1.cmml"><mi id="S5.2.p2.4.m4.1.1.2" xref="S5.2.p2.4.m4.1.1.2.cmml">Ξ²</mi><mo id="S5.2.p2.4.m4.1.1.1" xref="S5.2.p2.4.m4.1.1.1.cmml">β’</mo><mi id="S5.2.p2.4.m4.1.1.3" xref="S5.2.p2.4.m4.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.4.m4.1b"><apply id="S5.2.p2.4.m4.1.1.cmml" xref="S5.2.p2.4.m4.1.1"><times id="S5.2.p2.4.m4.1.1.1.cmml" xref="S5.2.p2.4.m4.1.1.1"></times><ci id="S5.2.p2.4.m4.1.1.2.cmml" xref="S5.2.p2.4.m4.1.1.2">π½</ci><ci id="S5.2.p2.4.m4.1.1.3.cmml" xref="S5.2.p2.4.m4.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.4.m4.1c">\beta S</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.4.m4.1d">italic_Ξ² italic_S</annotation></semantics></math> is a topological inverse semigroup, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib37" title="">37</a>, Proposition 3.8]</cite> implies that <math alttext="\beta S" class="ltx_Math" display="inline" id="S5.2.p2.5.m5.1"><semantics id="S5.2.p2.5.m5.1a"><mrow id="S5.2.p2.5.m5.1.1" xref="S5.2.p2.5.m5.1.1.cmml"><mi id="S5.2.p2.5.m5.1.1.2" xref="S5.2.p2.5.m5.1.1.2.cmml">Ξ²</mi><mo id="S5.2.p2.5.m5.1.1.1" xref="S5.2.p2.5.m5.1.1.1.cmml">β’</mo><mi id="S5.2.p2.5.m5.1.1.3" xref="S5.2.p2.5.m5.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.5.m5.1b"><apply id="S5.2.p2.5.m5.1.1.cmml" xref="S5.2.p2.5.m5.1.1"><times id="S5.2.p2.5.m5.1.1.1.cmml" xref="S5.2.p2.5.m5.1.1.1"></times><ci id="S5.2.p2.5.m5.1.1.2.cmml" xref="S5.2.p2.5.m5.1.1.2">π½</ci><ci id="S5.2.p2.5.m5.1.1.3.cmml" xref="S5.2.p2.5.m5.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.5.m5.1c">\beta S</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.5.m5.1d">italic_Ξ² italic_S</annotation></semantics></math> is a pospace with respect to the natural partial order. By Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S5.Thmtheorem1" title="Proposition 5.1 (Folklore). β£ 5. Proof of the main result and final remarks β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">5.1</span></a>, <math alttext="\beta S\setminus S" class="ltx_Math" display="inline" id="S5.2.p2.6.m6.1"><semantics id="S5.2.p2.6.m6.1a"><mrow id="S5.2.p2.6.m6.1.1" xref="S5.2.p2.6.m6.1.1.cmml"><mrow id="S5.2.p2.6.m6.1.1.2" xref="S5.2.p2.6.m6.1.1.2.cmml"><mi id="S5.2.p2.6.m6.1.1.2.2" xref="S5.2.p2.6.m6.1.1.2.2.cmml">Ξ²</mi><mo id="S5.2.p2.6.m6.1.1.2.1" xref="S5.2.p2.6.m6.1.1.2.1.cmml">β’</mo><mi id="S5.2.p2.6.m6.1.1.2.3" xref="S5.2.p2.6.m6.1.1.2.3.cmml">S</mi></mrow><mo id="S5.2.p2.6.m6.1.1.1" xref="S5.2.p2.6.m6.1.1.1.cmml">β</mo><mi id="S5.2.p2.6.m6.1.1.3" xref="S5.2.p2.6.m6.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.6.m6.1b"><apply id="S5.2.p2.6.m6.1.1.cmml" xref="S5.2.p2.6.m6.1.1"><setdiff id="S5.2.p2.6.m6.1.1.1.cmml" xref="S5.2.p2.6.m6.1.1.1"></setdiff><apply id="S5.2.p2.6.m6.1.1.2.cmml" xref="S5.2.p2.6.m6.1.1.2"><times id="S5.2.p2.6.m6.1.1.2.1.cmml" xref="S5.2.p2.6.m6.1.1.2.1"></times><ci id="S5.2.p2.6.m6.1.1.2.2.cmml" xref="S5.2.p2.6.m6.1.1.2.2">π½</ci><ci id="S5.2.p2.6.m6.1.1.2.3.cmml" xref="S5.2.p2.6.m6.1.1.2.3">π</ci></apply><ci id="S5.2.p2.6.m6.1.1.3.cmml" xref="S5.2.p2.6.m6.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.6.m6.1c">\beta S\setminus S</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.6.m6.1d">italic_Ξ² italic_S β italic_S</annotation></semantics></math> contains a minimal element <math alttext="h" class="ltx_Math" display="inline" id="S5.2.p2.7.m7.1"><semantics id="S5.2.p2.7.m7.1a"><mi id="S5.2.p2.7.m7.1.1" xref="S5.2.p2.7.m7.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.7.m7.1b"><ci id="S5.2.p2.7.m7.1.1.cmml" xref="S5.2.p2.7.m7.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.7.m7.1c">h</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.7.m7.1d">italic_h</annotation></semantics></math>. Let <math alttext="e=hh^{-1}" class="ltx_Math" display="inline" id="S5.2.p2.8.m8.1"><semantics id="S5.2.p2.8.m8.1a"><mrow id="S5.2.p2.8.m8.1.1" xref="S5.2.p2.8.m8.1.1.cmml"><mi id="S5.2.p2.8.m8.1.1.2" xref="S5.2.p2.8.m8.1.1.2.cmml">e</mi><mo id="S5.2.p2.8.m8.1.1.1" xref="S5.2.p2.8.m8.1.1.1.cmml">=</mo><mrow id="S5.2.p2.8.m8.1.1.3" xref="S5.2.p2.8.m8.1.1.3.cmml"><mi id="S5.2.p2.8.m8.1.1.3.2" xref="S5.2.p2.8.m8.1.1.3.2.cmml">h</mi><mo id="S5.2.p2.8.m8.1.1.3.1" xref="S5.2.p2.8.m8.1.1.3.1.cmml">β’</mo><msup id="S5.2.p2.8.m8.1.1.3.3" xref="S5.2.p2.8.m8.1.1.3.3.cmml"><mi id="S5.2.p2.8.m8.1.1.3.3.2" xref="S5.2.p2.8.m8.1.1.3.3.2.cmml">h</mi><mrow id="S5.2.p2.8.m8.1.1.3.3.3" xref="S5.2.p2.8.m8.1.1.3.3.3.cmml"><mo id="S5.2.p2.8.m8.1.1.3.3.3a" xref="S5.2.p2.8.m8.1.1.3.3.3.cmml">β</mo><mn id="S5.2.p2.8.m8.1.1.3.3.3.2" xref="S5.2.p2.8.m8.1.1.3.3.3.2.cmml">1</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.8.m8.1b"><apply id="S5.2.p2.8.m8.1.1.cmml" xref="S5.2.p2.8.m8.1.1"><eq id="S5.2.p2.8.m8.1.1.1.cmml" xref="S5.2.p2.8.m8.1.1.1"></eq><ci id="S5.2.p2.8.m8.1.1.2.cmml" xref="S5.2.p2.8.m8.1.1.2">π</ci><apply id="S5.2.p2.8.m8.1.1.3.cmml" xref="S5.2.p2.8.m8.1.1.3"><times id="S5.2.p2.8.m8.1.1.3.1.cmml" xref="S5.2.p2.8.m8.1.1.3.1"></times><ci id="S5.2.p2.8.m8.1.1.3.2.cmml" xref="S5.2.p2.8.m8.1.1.3.2">β</ci><apply id="S5.2.p2.8.m8.1.1.3.3.cmml" xref="S5.2.p2.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="S5.2.p2.8.m8.1.1.3.3.1.cmml" xref="S5.2.p2.8.m8.1.1.3.3">superscript</csymbol><ci id="S5.2.p2.8.m8.1.1.3.3.2.cmml" xref="S5.2.p2.8.m8.1.1.3.3.2">β</ci><apply id="S5.2.p2.8.m8.1.1.3.3.3.cmml" xref="S5.2.p2.8.m8.1.1.3.3.3"><minus id="S5.2.p2.8.m8.1.1.3.3.3.1.cmml" xref="S5.2.p2.8.m8.1.1.3.3.3"></minus><cn id="S5.2.p2.8.m8.1.1.3.3.3.2.cmml" type="integer" xref="S5.2.p2.8.m8.1.1.3.3.3.2">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.8.m8.1c">e=hh^{-1}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.8.m8.1d">italic_e = italic_h italic_h start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="f=h^{-1}h" class="ltx_Math" display="inline" id="S5.2.p2.9.m9.1"><semantics id="S5.2.p2.9.m9.1a"><mrow id="S5.2.p2.9.m9.1.1" xref="S5.2.p2.9.m9.1.1.cmml"><mi id="S5.2.p2.9.m9.1.1.2" xref="S5.2.p2.9.m9.1.1.2.cmml">f</mi><mo id="S5.2.p2.9.m9.1.1.1" xref="S5.2.p2.9.m9.1.1.1.cmml">=</mo><mrow id="S5.2.p2.9.m9.1.1.3" xref="S5.2.p2.9.m9.1.1.3.cmml"><msup id="S5.2.p2.9.m9.1.1.3.2" xref="S5.2.p2.9.m9.1.1.3.2.cmml"><mi id="S5.2.p2.9.m9.1.1.3.2.2" xref="S5.2.p2.9.m9.1.1.3.2.2.cmml">h</mi><mrow id="S5.2.p2.9.m9.1.1.3.2.3" xref="S5.2.p2.9.m9.1.1.3.2.3.cmml"><mo id="S5.2.p2.9.m9.1.1.3.2.3a" xref="S5.2.p2.9.m9.1.1.3.2.3.cmml">β</mo><mn id="S5.2.p2.9.m9.1.1.3.2.3.2" xref="S5.2.p2.9.m9.1.1.3.2.3.2.cmml">1</mn></mrow></msup><mo id="S5.2.p2.9.m9.1.1.3.1" xref="S5.2.p2.9.m9.1.1.3.1.cmml">β’</mo><mi id="S5.2.p2.9.m9.1.1.3.3" xref="S5.2.p2.9.m9.1.1.3.3.cmml">h</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.9.m9.1b"><apply id="S5.2.p2.9.m9.1.1.cmml" xref="S5.2.p2.9.m9.1.1"><eq id="S5.2.p2.9.m9.1.1.1.cmml" xref="S5.2.p2.9.m9.1.1.1"></eq><ci id="S5.2.p2.9.m9.1.1.2.cmml" xref="S5.2.p2.9.m9.1.1.2">π</ci><apply id="S5.2.p2.9.m9.1.1.3.cmml" xref="S5.2.p2.9.m9.1.1.3"><times id="S5.2.p2.9.m9.1.1.3.1.cmml" xref="S5.2.p2.9.m9.1.1.3.1"></times><apply id="S5.2.p2.9.m9.1.1.3.2.cmml" xref="S5.2.p2.9.m9.1.1.3.2"><csymbol cd="ambiguous" id="S5.2.p2.9.m9.1.1.3.2.1.cmml" xref="S5.2.p2.9.m9.1.1.3.2">superscript</csymbol><ci id="S5.2.p2.9.m9.1.1.3.2.2.cmml" xref="S5.2.p2.9.m9.1.1.3.2.2">β</ci><apply id="S5.2.p2.9.m9.1.1.3.2.3.cmml" xref="S5.2.p2.9.m9.1.1.3.2.3"><minus id="S5.2.p2.9.m9.1.1.3.2.3.1.cmml" xref="S5.2.p2.9.m9.1.1.3.2.3"></minus><cn id="S5.2.p2.9.m9.1.1.3.2.3.2.cmml" type="integer" xref="S5.2.p2.9.m9.1.1.3.2.3.2">1</cn></apply></apply><ci id="S5.2.p2.9.m9.1.1.3.3.cmml" xref="S5.2.p2.9.m9.1.1.3.3">β</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.9.m9.1c">f=h^{-1}h</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.9.m9.1d">italic_f = italic_h start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_h</annotation></semantics></math>. Since <math alttext="E(S)=E(\beta S)" class="ltx_Math" display="inline" id="S5.2.p2.10.m10.2"><semantics id="S5.2.p2.10.m10.2a"><mrow id="S5.2.p2.10.m10.2.2" xref="S5.2.p2.10.m10.2.2.cmml"><mrow id="S5.2.p2.10.m10.2.2.3" xref="S5.2.p2.10.m10.2.2.3.cmml"><mi id="S5.2.p2.10.m10.2.2.3.2" xref="S5.2.p2.10.m10.2.2.3.2.cmml">E</mi><mo id="S5.2.p2.10.m10.2.2.3.1" xref="S5.2.p2.10.m10.2.2.3.1.cmml">β’</mo><mrow id="S5.2.p2.10.m10.2.2.3.3.2" xref="S5.2.p2.10.m10.2.2.3.cmml"><mo id="S5.2.p2.10.m10.2.2.3.3.2.1" stretchy="false" xref="S5.2.p2.10.m10.2.2.3.cmml">(</mo><mi id="S5.2.p2.10.m10.1.1" xref="S5.2.p2.10.m10.1.1.cmml">S</mi><mo id="S5.2.p2.10.m10.2.2.3.3.2.2" stretchy="false" xref="S5.2.p2.10.m10.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.2.p2.10.m10.2.2.2" xref="S5.2.p2.10.m10.2.2.2.cmml">=</mo><mrow id="S5.2.p2.10.m10.2.2.1" xref="S5.2.p2.10.m10.2.2.1.cmml"><mi id="S5.2.p2.10.m10.2.2.1.3" xref="S5.2.p2.10.m10.2.2.1.3.cmml">E</mi><mo id="S5.2.p2.10.m10.2.2.1.2" xref="S5.2.p2.10.m10.2.2.1.2.cmml">β’</mo><mrow id="S5.2.p2.10.m10.2.2.1.1.1" xref="S5.2.p2.10.m10.2.2.1.1.1.1.cmml"><mo id="S5.2.p2.10.m10.2.2.1.1.1.2" stretchy="false" xref="S5.2.p2.10.m10.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.2.p2.10.m10.2.2.1.1.1.1" xref="S5.2.p2.10.m10.2.2.1.1.1.1.cmml"><mi id="S5.2.p2.10.m10.2.2.1.1.1.1.2" xref="S5.2.p2.10.m10.2.2.1.1.1.1.2.cmml">Ξ²</mi><mo id="S5.2.p2.10.m10.2.2.1.1.1.1.1" xref="S5.2.p2.10.m10.2.2.1.1.1.1.1.cmml">β’</mo><mi id="S5.2.p2.10.m10.2.2.1.1.1.1.3" xref="S5.2.p2.10.m10.2.2.1.1.1.1.3.cmml">S</mi></mrow><mo id="S5.2.p2.10.m10.2.2.1.1.1.3" stretchy="false" xref="S5.2.p2.10.m10.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.10.m10.2b"><apply id="S5.2.p2.10.m10.2.2.cmml" xref="S5.2.p2.10.m10.2.2"><eq id="S5.2.p2.10.m10.2.2.2.cmml" xref="S5.2.p2.10.m10.2.2.2"></eq><apply id="S5.2.p2.10.m10.2.2.3.cmml" xref="S5.2.p2.10.m10.2.2.3"><times id="S5.2.p2.10.m10.2.2.3.1.cmml" xref="S5.2.p2.10.m10.2.2.3.1"></times><ci id="S5.2.p2.10.m10.2.2.3.2.cmml" xref="S5.2.p2.10.m10.2.2.3.2">πΈ</ci><ci id="S5.2.p2.10.m10.1.1.cmml" xref="S5.2.p2.10.m10.1.1">π</ci></apply><apply id="S5.2.p2.10.m10.2.2.1.cmml" xref="S5.2.p2.10.m10.2.2.1"><times id="S5.2.p2.10.m10.2.2.1.2.cmml" xref="S5.2.p2.10.m10.2.2.1.2"></times><ci id="S5.2.p2.10.m10.2.2.1.3.cmml" xref="S5.2.p2.10.m10.2.2.1.3">πΈ</ci><apply id="S5.2.p2.10.m10.2.2.1.1.1.1.cmml" xref="S5.2.p2.10.m10.2.2.1.1.1"><times id="S5.2.p2.10.m10.2.2.1.1.1.1.1.cmml" xref="S5.2.p2.10.m10.2.2.1.1.1.1.1"></times><ci id="S5.2.p2.10.m10.2.2.1.1.1.1.2.cmml" xref="S5.2.p2.10.m10.2.2.1.1.1.1.2">π½</ci><ci id="S5.2.p2.10.m10.2.2.1.1.1.1.3.cmml" xref="S5.2.p2.10.m10.2.2.1.1.1.1.3">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.10.m10.2c">E(S)=E(\beta S)</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.10.m10.2d">italic_E ( italic_S ) = italic_E ( italic_Ξ² italic_S )</annotation></semantics></math>, we have <math alttext="e,f\in S" class="ltx_Math" display="inline" id="S5.2.p2.11.m11.2"><semantics id="S5.2.p2.11.m11.2a"><mrow id="S5.2.p2.11.m11.2.3" xref="S5.2.p2.11.m11.2.3.cmml"><mrow id="S5.2.p2.11.m11.2.3.2.2" xref="S5.2.p2.11.m11.2.3.2.1.cmml"><mi id="S5.2.p2.11.m11.1.1" xref="S5.2.p2.11.m11.1.1.cmml">e</mi><mo id="S5.2.p2.11.m11.2.3.2.2.1" xref="S5.2.p2.11.m11.2.3.2.1.cmml">,</mo><mi id="S5.2.p2.11.m11.2.2" xref="S5.2.p2.11.m11.2.2.cmml">f</mi></mrow><mo id="S5.2.p2.11.m11.2.3.1" xref="S5.2.p2.11.m11.2.3.1.cmml">β</mo><mi id="S5.2.p2.11.m11.2.3.3" xref="S5.2.p2.11.m11.2.3.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.11.m11.2b"><apply id="S5.2.p2.11.m11.2.3.cmml" xref="S5.2.p2.11.m11.2.3"><in id="S5.2.p2.11.m11.2.3.1.cmml" xref="S5.2.p2.11.m11.2.3.1"></in><list id="S5.2.p2.11.m11.2.3.2.1.cmml" xref="S5.2.p2.11.m11.2.3.2.2"><ci id="S5.2.p2.11.m11.1.1.cmml" xref="S5.2.p2.11.m11.1.1">π</ci><ci id="S5.2.p2.11.m11.2.2.cmml" xref="S5.2.p2.11.m11.2.2">π</ci></list><ci id="S5.2.p2.11.m11.2.3.3.cmml" xref="S5.2.p2.11.m11.2.3.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.11.m11.2c">e,f\in S</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.11.m11.2d">italic_e , italic_f β italic_S</annotation></semantics></math>. Let</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex17"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="T=\{x\in S:xx^{-1}\leq e\hbox{ and }x^{-1}x\leq f\}." class="ltx_Math" display="block" id="S5.Ex17.m1.1"><semantics id="S5.Ex17.m1.1a"><mrow id="S5.Ex17.m1.1.1.1" xref="S5.Ex17.m1.1.1.1.1.cmml"><mrow id="S5.Ex17.m1.1.1.1.1" xref="S5.Ex17.m1.1.1.1.1.cmml"><mi id="S5.Ex17.m1.1.1.1.1.4" xref="S5.Ex17.m1.1.1.1.1.4.cmml">T</mi><mo id="S5.Ex17.m1.1.1.1.1.3" xref="S5.Ex17.m1.1.1.1.1.3.cmml">=</mo><mrow id="S5.Ex17.m1.1.1.1.1.2.2" xref="S5.Ex17.m1.1.1.1.1.2.3.cmml"><mo id="S5.Ex17.m1.1.1.1.1.2.2.3" stretchy="false" xref="S5.Ex17.m1.1.1.1.1.2.3.1.cmml">{</mo><mrow id="S5.Ex17.m1.1.1.1.1.1.1.1" xref="S5.Ex17.m1.1.1.1.1.1.1.1.cmml"><mi id="S5.Ex17.m1.1.1.1.1.1.1.1.2" xref="S5.Ex17.m1.1.1.1.1.1.1.1.2.cmml">x</mi><mo id="S5.Ex17.m1.1.1.1.1.1.1.1.1" xref="S5.Ex17.m1.1.1.1.1.1.1.1.1.cmml">β</mo><mi id="S5.Ex17.m1.1.1.1.1.1.1.1.3" xref="S5.Ex17.m1.1.1.1.1.1.1.1.3.cmml">S</mi></mrow><mo id="S5.Ex17.m1.1.1.1.1.2.2.4" lspace="0.278em" rspace="0.278em" xref="S5.Ex17.m1.1.1.1.1.2.3.1.cmml">:</mo><mrow id="S5.Ex17.m1.1.1.1.1.2.2.2" xref="S5.Ex17.m1.1.1.1.1.2.2.2.cmml"><mrow id="S5.Ex17.m1.1.1.1.1.2.2.2.2" xref="S5.Ex17.m1.1.1.1.1.2.2.2.2.cmml"><mi id="S5.Ex17.m1.1.1.1.1.2.2.2.2.2" xref="S5.Ex17.m1.1.1.1.1.2.2.2.2.2.cmml">x</mi><mo id="S5.Ex17.m1.1.1.1.1.2.2.2.2.1" xref="S5.Ex17.m1.1.1.1.1.2.2.2.2.1.cmml">β’</mo><msup id="S5.Ex17.m1.1.1.1.1.2.2.2.2.3" xref="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.cmml"><mi id="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.2" xref="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.2.cmml">x</mi><mrow id="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.3" xref="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.3.cmml"><mo id="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.3a" xref="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.3.cmml">β</mo><mn id="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.3.2" xref="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.3.2.cmml">1</mn></mrow></msup></mrow><mo id="S5.Ex17.m1.1.1.1.1.2.2.2.3" xref="S5.Ex17.m1.1.1.1.1.2.2.2.3.cmml">β€</mo><mrow id="S5.Ex17.m1.1.1.1.1.2.2.2.4" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.cmml"><mi id="S5.Ex17.m1.1.1.1.1.2.2.2.4.2" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.2.cmml">e</mi><mo id="S5.Ex17.m1.1.1.1.1.2.2.2.4.1" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.1.cmml">β’</mo><mtext id="S5.Ex17.m1.1.1.1.1.2.2.2.4.3" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.3a.cmml"> and </mtext><mo id="S5.Ex17.m1.1.1.1.1.2.2.2.4.1a" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.1.cmml">β’</mo><msup id="S5.Ex17.m1.1.1.1.1.2.2.2.4.4" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.cmml"><mi id="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.2" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.2.cmml">x</mi><mrow id="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.3" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.3.cmml"><mo id="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.3a" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.3.cmml">β</mo><mn id="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.3.2" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.3.2.cmml">1</mn></mrow></msup><mo id="S5.Ex17.m1.1.1.1.1.2.2.2.4.1b" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.1.cmml">β’</mo><mi id="S5.Ex17.m1.1.1.1.1.2.2.2.4.5" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.5.cmml">x</mi></mrow><mo id="S5.Ex17.m1.1.1.1.1.2.2.2.5" xref="S5.Ex17.m1.1.1.1.1.2.2.2.5.cmml">β€</mo><mi id="S5.Ex17.m1.1.1.1.1.2.2.2.6" xref="S5.Ex17.m1.1.1.1.1.2.2.2.6.cmml">f</mi></mrow><mo id="S5.Ex17.m1.1.1.1.1.2.2.5" stretchy="false" xref="S5.Ex17.m1.1.1.1.1.2.3.1.cmml">}</mo></mrow></mrow><mo id="S5.Ex17.m1.1.1.1.2" lspace="0em" xref="S5.Ex17.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex17.m1.1b"><apply id="S5.Ex17.m1.1.1.1.1.cmml" xref="S5.Ex17.m1.1.1.1"><eq id="S5.Ex17.m1.1.1.1.1.3.cmml" xref="S5.Ex17.m1.1.1.1.1.3"></eq><ci id="S5.Ex17.m1.1.1.1.1.4.cmml" xref="S5.Ex17.m1.1.1.1.1.4">π</ci><apply id="S5.Ex17.m1.1.1.1.1.2.3.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2"><csymbol cd="latexml" id="S5.Ex17.m1.1.1.1.1.2.3.1.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.3">conditional-set</csymbol><apply id="S5.Ex17.m1.1.1.1.1.1.1.1.cmml" xref="S5.Ex17.m1.1.1.1.1.1.1.1"><in id="S5.Ex17.m1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex17.m1.1.1.1.1.1.1.1.1"></in><ci id="S5.Ex17.m1.1.1.1.1.1.1.1.2.cmml" xref="S5.Ex17.m1.1.1.1.1.1.1.1.2">π₯</ci><ci id="S5.Ex17.m1.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex17.m1.1.1.1.1.1.1.1.3">π</ci></apply><apply id="S5.Ex17.m1.1.1.1.1.2.2.2.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2"><and id="S5.Ex17.m1.1.1.1.1.2.2.2a.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2"></and><apply id="S5.Ex17.m1.1.1.1.1.2.2.2b.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2"><leq id="S5.Ex17.m1.1.1.1.1.2.2.2.3.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.3"></leq><apply id="S5.Ex17.m1.1.1.1.1.2.2.2.2.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.2"><times id="S5.Ex17.m1.1.1.1.1.2.2.2.2.1.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.2.1"></times><ci id="S5.Ex17.m1.1.1.1.1.2.2.2.2.2.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.2.2">π₯</ci><apply id="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.2.3"><csymbol cd="ambiguous" id="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.1.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.2.3">superscript</csymbol><ci id="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.2.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.2">π₯</ci><apply id="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.3.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.3"><minus id="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.3.1.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.3"></minus><cn id="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.3.2.cmml" type="integer" xref="S5.Ex17.m1.1.1.1.1.2.2.2.2.3.3.2">1</cn></apply></apply></apply><apply id="S5.Ex17.m1.1.1.1.1.2.2.2.4.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4"><times id="S5.Ex17.m1.1.1.1.1.2.2.2.4.1.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.1"></times><ci id="S5.Ex17.m1.1.1.1.1.2.2.2.4.2.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.2">π</ci><ci id="S5.Ex17.m1.1.1.1.1.2.2.2.4.3a.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.3"><mtext id="S5.Ex17.m1.1.1.1.1.2.2.2.4.3.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.3"> and </mtext></ci><apply id="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.4"><csymbol cd="ambiguous" id="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.1.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.4">superscript</csymbol><ci id="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.2.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.2">π₯</ci><apply id="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.3.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.3"><minus id="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.3.1.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.3"></minus><cn id="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.3.2.cmml" type="integer" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.4.3.2">1</cn></apply></apply><ci id="S5.Ex17.m1.1.1.1.1.2.2.2.4.5.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.4.5">π₯</ci></apply></apply><apply id="S5.Ex17.m1.1.1.1.1.2.2.2c.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2"><leq id="S5.Ex17.m1.1.1.1.1.2.2.2.5.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.5"></leq><share href="https://arxiv.org/html/2503.13666v1#S5.Ex17.m1.1.1.1.1.2.2.2.4.cmml" id="S5.Ex17.m1.1.1.1.1.2.2.2d.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2"></share><ci id="S5.Ex17.m1.1.1.1.1.2.2.2.6.cmml" xref="S5.Ex17.m1.1.1.1.1.2.2.2.6">π</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex17.m1.1c">T=\{x\in S:xx^{-1}\leq e\hbox{ and }x^{-1}x\leq f\}.</annotation><annotation encoding="application/x-llamapun" id="S5.Ex17.m1.1d">italic_T = { italic_x β italic_S : italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT β€ italic_e and italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x β€ italic_f } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.2.p2.22">Fix an open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S5.2.p2.12.m1.1"><semantics id="S5.2.p2.12.m1.1a"><mi id="S5.2.p2.12.m1.1.1" xref="S5.2.p2.12.m1.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.12.m1.1b"><ci id="S5.2.p2.12.m1.1.1.cmml" xref="S5.2.p2.12.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.12.m1.1c">U</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.12.m1.1d">italic_U</annotation></semantics></math> of <math alttext="h" class="ltx_Math" display="inline" id="S5.2.p2.13.m2.1"><semantics id="S5.2.p2.13.m2.1a"><mi id="S5.2.p2.13.m2.1.1" xref="S5.2.p2.13.m2.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.13.m2.1b"><ci id="S5.2.p2.13.m2.1.1.cmml" xref="S5.2.p2.13.m2.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.13.m2.1c">h</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.13.m2.1d">italic_h</annotation></semantics></math>. Since <math alttext="ehf=h" class="ltx_Math" display="inline" id="S5.2.p2.14.m3.1"><semantics id="S5.2.p2.14.m3.1a"><mrow id="S5.2.p2.14.m3.1.1" xref="S5.2.p2.14.m3.1.1.cmml"><mrow id="S5.2.p2.14.m3.1.1.2" xref="S5.2.p2.14.m3.1.1.2.cmml"><mi id="S5.2.p2.14.m3.1.1.2.2" xref="S5.2.p2.14.m3.1.1.2.2.cmml">e</mi><mo id="S5.2.p2.14.m3.1.1.2.1" xref="S5.2.p2.14.m3.1.1.2.1.cmml">β’</mo><mi id="S5.2.p2.14.m3.1.1.2.3" xref="S5.2.p2.14.m3.1.1.2.3.cmml">h</mi><mo id="S5.2.p2.14.m3.1.1.2.1a" xref="S5.2.p2.14.m3.1.1.2.1.cmml">β’</mo><mi id="S5.2.p2.14.m3.1.1.2.4" xref="S5.2.p2.14.m3.1.1.2.4.cmml">f</mi></mrow><mo id="S5.2.p2.14.m3.1.1.1" xref="S5.2.p2.14.m3.1.1.1.cmml">=</mo><mi id="S5.2.p2.14.m3.1.1.3" xref="S5.2.p2.14.m3.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.14.m3.1b"><apply id="S5.2.p2.14.m3.1.1.cmml" xref="S5.2.p2.14.m3.1.1"><eq id="S5.2.p2.14.m3.1.1.1.cmml" xref="S5.2.p2.14.m3.1.1.1"></eq><apply id="S5.2.p2.14.m3.1.1.2.cmml" xref="S5.2.p2.14.m3.1.1.2"><times id="S5.2.p2.14.m3.1.1.2.1.cmml" xref="S5.2.p2.14.m3.1.1.2.1"></times><ci id="S5.2.p2.14.m3.1.1.2.2.cmml" xref="S5.2.p2.14.m3.1.1.2.2">π</ci><ci id="S5.2.p2.14.m3.1.1.2.3.cmml" xref="S5.2.p2.14.m3.1.1.2.3">β</ci><ci id="S5.2.p2.14.m3.1.1.2.4.cmml" xref="S5.2.p2.14.m3.1.1.2.4">π</ci></apply><ci id="S5.2.p2.14.m3.1.1.3.cmml" xref="S5.2.p2.14.m3.1.1.3">β</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.14.m3.1c">ehf=h</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.14.m3.1d">italic_e italic_h italic_f = italic_h</annotation></semantics></math>, the continuity of the semigroup operation in <math alttext="\beta S" class="ltx_Math" display="inline" id="S5.2.p2.15.m4.1"><semantics id="S5.2.p2.15.m4.1a"><mrow id="S5.2.p2.15.m4.1.1" xref="S5.2.p2.15.m4.1.1.cmml"><mi id="S5.2.p2.15.m4.1.1.2" xref="S5.2.p2.15.m4.1.1.2.cmml">Ξ²</mi><mo id="S5.2.p2.15.m4.1.1.1" xref="S5.2.p2.15.m4.1.1.1.cmml">β’</mo><mi id="S5.2.p2.15.m4.1.1.3" xref="S5.2.p2.15.m4.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.15.m4.1b"><apply id="S5.2.p2.15.m4.1.1.cmml" xref="S5.2.p2.15.m4.1.1"><times id="S5.2.p2.15.m4.1.1.1.cmml" xref="S5.2.p2.15.m4.1.1.1"></times><ci id="S5.2.p2.15.m4.1.1.2.cmml" xref="S5.2.p2.15.m4.1.1.2">π½</ci><ci id="S5.2.p2.15.m4.1.1.3.cmml" xref="S5.2.p2.15.m4.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.15.m4.1c">\beta S</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.15.m4.1d">italic_Ξ² italic_S</annotation></semantics></math> yields an open neighborhood <math alttext="V" class="ltx_Math" display="inline" id="S5.2.p2.16.m5.1"><semantics id="S5.2.p2.16.m5.1a"><mi id="S5.2.p2.16.m5.1.1" xref="S5.2.p2.16.m5.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.16.m5.1b"><ci id="S5.2.p2.16.m5.1.1.cmml" xref="S5.2.p2.16.m5.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.16.m5.1c">V</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.16.m5.1d">italic_V</annotation></semantics></math> of <math alttext="h" class="ltx_Math" display="inline" id="S5.2.p2.17.m6.1"><semantics id="S5.2.p2.17.m6.1a"><mi id="S5.2.p2.17.m6.1.1" xref="S5.2.p2.17.m6.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.17.m6.1b"><ci id="S5.2.p2.17.m6.1.1.cmml" xref="S5.2.p2.17.m6.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.17.m6.1c">h</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.17.m6.1d">italic_h</annotation></semantics></math> such that <math alttext="eVf\subseteq U" class="ltx_Math" display="inline" id="S5.2.p2.18.m7.1"><semantics id="S5.2.p2.18.m7.1a"><mrow id="S5.2.p2.18.m7.1.1" xref="S5.2.p2.18.m7.1.1.cmml"><mrow id="S5.2.p2.18.m7.1.1.2" xref="S5.2.p2.18.m7.1.1.2.cmml"><mi id="S5.2.p2.18.m7.1.1.2.2" xref="S5.2.p2.18.m7.1.1.2.2.cmml">e</mi><mo id="S5.2.p2.18.m7.1.1.2.1" xref="S5.2.p2.18.m7.1.1.2.1.cmml">β’</mo><mi id="S5.2.p2.18.m7.1.1.2.3" xref="S5.2.p2.18.m7.1.1.2.3.cmml">V</mi><mo id="S5.2.p2.18.m7.1.1.2.1a" xref="S5.2.p2.18.m7.1.1.2.1.cmml">β’</mo><mi id="S5.2.p2.18.m7.1.1.2.4" xref="S5.2.p2.18.m7.1.1.2.4.cmml">f</mi></mrow><mo id="S5.2.p2.18.m7.1.1.1" xref="S5.2.p2.18.m7.1.1.1.cmml">β</mo><mi id="S5.2.p2.18.m7.1.1.3" xref="S5.2.p2.18.m7.1.1.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.18.m7.1b"><apply id="S5.2.p2.18.m7.1.1.cmml" xref="S5.2.p2.18.m7.1.1"><subset id="S5.2.p2.18.m7.1.1.1.cmml" xref="S5.2.p2.18.m7.1.1.1"></subset><apply id="S5.2.p2.18.m7.1.1.2.cmml" xref="S5.2.p2.18.m7.1.1.2"><times id="S5.2.p2.18.m7.1.1.2.1.cmml" xref="S5.2.p2.18.m7.1.1.2.1"></times><ci id="S5.2.p2.18.m7.1.1.2.2.cmml" xref="S5.2.p2.18.m7.1.1.2.2">π</ci><ci id="S5.2.p2.18.m7.1.1.2.3.cmml" xref="S5.2.p2.18.m7.1.1.2.3">π</ci><ci id="S5.2.p2.18.m7.1.1.2.4.cmml" xref="S5.2.p2.18.m7.1.1.2.4">π</ci></apply><ci id="S5.2.p2.18.m7.1.1.3.cmml" xref="S5.2.p2.18.m7.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.18.m7.1c">eVf\subseteq U</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.18.m7.1d">italic_e italic_V italic_f β italic_U</annotation></semantics></math>. Observe that for every <math alttext="x\in S\cap V" class="ltx_Math" display="inline" id="S5.2.p2.19.m8.1"><semantics id="S5.2.p2.19.m8.1a"><mrow id="S5.2.p2.19.m8.1.1" xref="S5.2.p2.19.m8.1.1.cmml"><mi id="S5.2.p2.19.m8.1.1.2" xref="S5.2.p2.19.m8.1.1.2.cmml">x</mi><mo id="S5.2.p2.19.m8.1.1.1" xref="S5.2.p2.19.m8.1.1.1.cmml">β</mo><mrow id="S5.2.p2.19.m8.1.1.3" xref="S5.2.p2.19.m8.1.1.3.cmml"><mi id="S5.2.p2.19.m8.1.1.3.2" xref="S5.2.p2.19.m8.1.1.3.2.cmml">S</mi><mo id="S5.2.p2.19.m8.1.1.3.1" xref="S5.2.p2.19.m8.1.1.3.1.cmml">β©</mo><mi id="S5.2.p2.19.m8.1.1.3.3" xref="S5.2.p2.19.m8.1.1.3.3.cmml">V</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.19.m8.1b"><apply id="S5.2.p2.19.m8.1.1.cmml" xref="S5.2.p2.19.m8.1.1"><in id="S5.2.p2.19.m8.1.1.1.cmml" xref="S5.2.p2.19.m8.1.1.1"></in><ci id="S5.2.p2.19.m8.1.1.2.cmml" xref="S5.2.p2.19.m8.1.1.2">π₯</ci><apply id="S5.2.p2.19.m8.1.1.3.cmml" xref="S5.2.p2.19.m8.1.1.3"><intersect id="S5.2.p2.19.m8.1.1.3.1.cmml" xref="S5.2.p2.19.m8.1.1.3.1"></intersect><ci id="S5.2.p2.19.m8.1.1.3.2.cmml" xref="S5.2.p2.19.m8.1.1.3.2">π</ci><ci id="S5.2.p2.19.m8.1.1.3.3.cmml" xref="S5.2.p2.19.m8.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.19.m8.1c">x\in S\cap V</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.19.m8.1d">italic_x β italic_S β© italic_V</annotation></semantics></math> we have <math alttext="exf\in T\cap U" class="ltx_Math" display="inline" id="S5.2.p2.20.m9.1"><semantics id="S5.2.p2.20.m9.1a"><mrow id="S5.2.p2.20.m9.1.1" xref="S5.2.p2.20.m9.1.1.cmml"><mrow id="S5.2.p2.20.m9.1.1.2" xref="S5.2.p2.20.m9.1.1.2.cmml"><mi id="S5.2.p2.20.m9.1.1.2.2" xref="S5.2.p2.20.m9.1.1.2.2.cmml">e</mi><mo id="S5.2.p2.20.m9.1.1.2.1" xref="S5.2.p2.20.m9.1.1.2.1.cmml">β’</mo><mi id="S5.2.p2.20.m9.1.1.2.3" xref="S5.2.p2.20.m9.1.1.2.3.cmml">x</mi><mo id="S5.2.p2.20.m9.1.1.2.1a" xref="S5.2.p2.20.m9.1.1.2.1.cmml">β’</mo><mi id="S5.2.p2.20.m9.1.1.2.4" xref="S5.2.p2.20.m9.1.1.2.4.cmml">f</mi></mrow><mo id="S5.2.p2.20.m9.1.1.1" xref="S5.2.p2.20.m9.1.1.1.cmml">β</mo><mrow id="S5.2.p2.20.m9.1.1.3" xref="S5.2.p2.20.m9.1.1.3.cmml"><mi id="S5.2.p2.20.m9.1.1.3.2" xref="S5.2.p2.20.m9.1.1.3.2.cmml">T</mi><mo id="S5.2.p2.20.m9.1.1.3.1" xref="S5.2.p2.20.m9.1.1.3.1.cmml">β©</mo><mi id="S5.2.p2.20.m9.1.1.3.3" xref="S5.2.p2.20.m9.1.1.3.3.cmml">U</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.20.m9.1b"><apply id="S5.2.p2.20.m9.1.1.cmml" xref="S5.2.p2.20.m9.1.1"><in id="S5.2.p2.20.m9.1.1.1.cmml" xref="S5.2.p2.20.m9.1.1.1"></in><apply id="S5.2.p2.20.m9.1.1.2.cmml" xref="S5.2.p2.20.m9.1.1.2"><times id="S5.2.p2.20.m9.1.1.2.1.cmml" xref="S5.2.p2.20.m9.1.1.2.1"></times><ci id="S5.2.p2.20.m9.1.1.2.2.cmml" xref="S5.2.p2.20.m9.1.1.2.2">π</ci><ci id="S5.2.p2.20.m9.1.1.2.3.cmml" xref="S5.2.p2.20.m9.1.1.2.3">π₯</ci><ci id="S5.2.p2.20.m9.1.1.2.4.cmml" xref="S5.2.p2.20.m9.1.1.2.4">π</ci></apply><apply id="S5.2.p2.20.m9.1.1.3.cmml" xref="S5.2.p2.20.m9.1.1.3"><intersect id="S5.2.p2.20.m9.1.1.3.1.cmml" xref="S5.2.p2.20.m9.1.1.3.1"></intersect><ci id="S5.2.p2.20.m9.1.1.3.2.cmml" xref="S5.2.p2.20.m9.1.1.3.2">π</ci><ci id="S5.2.p2.20.m9.1.1.3.3.cmml" xref="S5.2.p2.20.m9.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.20.m9.1c">exf\in T\cap U</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.20.m9.1d">italic_e italic_x italic_f β italic_T β© italic_U</annotation></semantics></math>. Since <math alttext="U" class="ltx_Math" display="inline" id="S5.2.p2.21.m10.1"><semantics id="S5.2.p2.21.m10.1a"><mi id="S5.2.p2.21.m10.1.1" xref="S5.2.p2.21.m10.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.21.m10.1b"><ci id="S5.2.p2.21.m10.1.1.cmml" xref="S5.2.p2.21.m10.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.21.m10.1c">U</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.21.m10.1d">italic_U</annotation></semantics></math> was chosen arbitrarily, <math alttext="h\in\operatorname{cl}_{\beta S}(T)" class="ltx_Math" display="inline" id="S5.2.p2.22.m11.2"><semantics id="S5.2.p2.22.m11.2a"><mrow id="S5.2.p2.22.m11.2.2" xref="S5.2.p2.22.m11.2.2.cmml"><mi id="S5.2.p2.22.m11.2.2.3" xref="S5.2.p2.22.m11.2.2.3.cmml">h</mi><mo id="S5.2.p2.22.m11.2.2.2" xref="S5.2.p2.22.m11.2.2.2.cmml">β</mo><mrow id="S5.2.p2.22.m11.2.2.1.1" xref="S5.2.p2.22.m11.2.2.1.2.cmml"><msub id="S5.2.p2.22.m11.2.2.1.1.1" xref="S5.2.p2.22.m11.2.2.1.1.1.cmml"><mi id="S5.2.p2.22.m11.2.2.1.1.1.2" xref="S5.2.p2.22.m11.2.2.1.1.1.2.cmml">cl</mi><mrow id="S5.2.p2.22.m11.2.2.1.1.1.3" xref="S5.2.p2.22.m11.2.2.1.1.1.3.cmml"><mi id="S5.2.p2.22.m11.2.2.1.1.1.3.2" xref="S5.2.p2.22.m11.2.2.1.1.1.3.2.cmml">Ξ²</mi><mo id="S5.2.p2.22.m11.2.2.1.1.1.3.1" xref="S5.2.p2.22.m11.2.2.1.1.1.3.1.cmml">β’</mo><mi id="S5.2.p2.22.m11.2.2.1.1.1.3.3" xref="S5.2.p2.22.m11.2.2.1.1.1.3.3.cmml">S</mi></mrow></msub><mo id="S5.2.p2.22.m11.2.2.1.1a" xref="S5.2.p2.22.m11.2.2.1.2.cmml">β‘</mo><mrow id="S5.2.p2.22.m11.2.2.1.1.2" xref="S5.2.p2.22.m11.2.2.1.2.cmml"><mo id="S5.2.p2.22.m11.2.2.1.1.2.1" stretchy="false" xref="S5.2.p2.22.m11.2.2.1.2.cmml">(</mo><mi id="S5.2.p2.22.m11.1.1" xref="S5.2.p2.22.m11.1.1.cmml">T</mi><mo id="S5.2.p2.22.m11.2.2.1.1.2.2" stretchy="false" xref="S5.2.p2.22.m11.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.22.m11.2b"><apply id="S5.2.p2.22.m11.2.2.cmml" xref="S5.2.p2.22.m11.2.2"><in id="S5.2.p2.22.m11.2.2.2.cmml" xref="S5.2.p2.22.m11.2.2.2"></in><ci id="S5.2.p2.22.m11.2.2.3.cmml" xref="S5.2.p2.22.m11.2.2.3">β</ci><apply id="S5.2.p2.22.m11.2.2.1.2.cmml" xref="S5.2.p2.22.m11.2.2.1.1"><apply id="S5.2.p2.22.m11.2.2.1.1.1.cmml" xref="S5.2.p2.22.m11.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.2.p2.22.m11.2.2.1.1.1.1.cmml" xref="S5.2.p2.22.m11.2.2.1.1.1">subscript</csymbol><ci id="S5.2.p2.22.m11.2.2.1.1.1.2.cmml" xref="S5.2.p2.22.m11.2.2.1.1.1.2">cl</ci><apply id="S5.2.p2.22.m11.2.2.1.1.1.3.cmml" xref="S5.2.p2.22.m11.2.2.1.1.1.3"><times id="S5.2.p2.22.m11.2.2.1.1.1.3.1.cmml" xref="S5.2.p2.22.m11.2.2.1.1.1.3.1"></times><ci id="S5.2.p2.22.m11.2.2.1.1.1.3.2.cmml" xref="S5.2.p2.22.m11.2.2.1.1.1.3.2">π½</ci><ci id="S5.2.p2.22.m11.2.2.1.1.1.3.3.cmml" xref="S5.2.p2.22.m11.2.2.1.1.1.3.3">π</ci></apply></apply><ci id="S5.2.p2.22.m11.1.1.cmml" xref="S5.2.p2.22.m11.1.1">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.22.m11.2c">h\in\operatorname{cl}_{\beta S}(T)</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.22.m11.2d">italic_h β roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_S end_POSTSUBSCRIPT ( italic_T )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.3.p3"> <p class="ltx_p" id="S5.3.p3.22">Since <math alttext="S" class="ltx_Math" display="inline" id="S5.3.p3.1.m1.1"><semantics id="S5.3.p3.1.m1.1a"><mi id="S5.3.p3.1.m1.1.1" xref="S5.3.p3.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.3.p3.1.m1.1b"><ci id="S5.3.p3.1.m1.1.1.cmml" xref="S5.3.p3.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.1.m1.1d">italic_S</annotation></semantics></math> is a topological inverse semigroup, the maximal subgroups <math alttext="H_{e}=\{x\in S\colon xx^{-1}=e=x^{-1}x\}" class="ltx_Math" display="inline" id="S5.3.p3.2.m2.2"><semantics id="S5.3.p3.2.m2.2a"><mrow id="S5.3.p3.2.m2.2.2" xref="S5.3.p3.2.m2.2.2.cmml"><msub id="S5.3.p3.2.m2.2.2.4" xref="S5.3.p3.2.m2.2.2.4.cmml"><mi id="S5.3.p3.2.m2.2.2.4.2" xref="S5.3.p3.2.m2.2.2.4.2.cmml">H</mi><mi id="S5.3.p3.2.m2.2.2.4.3" xref="S5.3.p3.2.m2.2.2.4.3.cmml">e</mi></msub><mo id="S5.3.p3.2.m2.2.2.3" xref="S5.3.p3.2.m2.2.2.3.cmml">=</mo><mrow id="S5.3.p3.2.m2.2.2.2.2" xref="S5.3.p3.2.m2.2.2.2.3.cmml"><mo id="S5.3.p3.2.m2.2.2.2.2.3" stretchy="false" xref="S5.3.p3.2.m2.2.2.2.3.1.cmml">{</mo><mrow id="S5.3.p3.2.m2.1.1.1.1.1" xref="S5.3.p3.2.m2.1.1.1.1.1.cmml"><mi id="S5.3.p3.2.m2.1.1.1.1.1.2" xref="S5.3.p3.2.m2.1.1.1.1.1.2.cmml">x</mi><mo id="S5.3.p3.2.m2.1.1.1.1.1.1" xref="S5.3.p3.2.m2.1.1.1.1.1.1.cmml">β</mo><mi id="S5.3.p3.2.m2.1.1.1.1.1.3" xref="S5.3.p3.2.m2.1.1.1.1.1.3.cmml">S</mi></mrow><mo id="S5.3.p3.2.m2.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S5.3.p3.2.m2.2.2.2.3.1.cmml">:</mo><mrow id="S5.3.p3.2.m2.2.2.2.2.2" xref="S5.3.p3.2.m2.2.2.2.2.2.cmml"><mrow id="S5.3.p3.2.m2.2.2.2.2.2.2" xref="S5.3.p3.2.m2.2.2.2.2.2.2.cmml"><mi id="S5.3.p3.2.m2.2.2.2.2.2.2.2" xref="S5.3.p3.2.m2.2.2.2.2.2.2.2.cmml">x</mi><mo id="S5.3.p3.2.m2.2.2.2.2.2.2.1" xref="S5.3.p3.2.m2.2.2.2.2.2.2.1.cmml">β’</mo><msup id="S5.3.p3.2.m2.2.2.2.2.2.2.3" xref="S5.3.p3.2.m2.2.2.2.2.2.2.3.cmml"><mi id="S5.3.p3.2.m2.2.2.2.2.2.2.3.2" xref="S5.3.p3.2.m2.2.2.2.2.2.2.3.2.cmml">x</mi><mrow id="S5.3.p3.2.m2.2.2.2.2.2.2.3.3" xref="S5.3.p3.2.m2.2.2.2.2.2.2.3.3.cmml"><mo id="S5.3.p3.2.m2.2.2.2.2.2.2.3.3a" xref="S5.3.p3.2.m2.2.2.2.2.2.2.3.3.cmml">β</mo><mn id="S5.3.p3.2.m2.2.2.2.2.2.2.3.3.2" xref="S5.3.p3.2.m2.2.2.2.2.2.2.3.3.2.cmml">1</mn></mrow></msup></mrow><mo id="S5.3.p3.2.m2.2.2.2.2.2.3" xref="S5.3.p3.2.m2.2.2.2.2.2.3.cmml">=</mo><mi id="S5.3.p3.2.m2.2.2.2.2.2.4" xref="S5.3.p3.2.m2.2.2.2.2.2.4.cmml">e</mi><mo id="S5.3.p3.2.m2.2.2.2.2.2.5" xref="S5.3.p3.2.m2.2.2.2.2.2.5.cmml">=</mo><mrow id="S5.3.p3.2.m2.2.2.2.2.2.6" 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id="S5.3.p3.3.m3.2.2.2.2.2d.cmml" xref="S5.3.p3.3.m3.2.2.2.2.2"></share><apply id="S5.3.p3.3.m3.2.2.2.2.2.6.cmml" xref="S5.3.p3.3.m3.2.2.2.2.2.6"><times id="S5.3.p3.3.m3.2.2.2.2.2.6.1.cmml" xref="S5.3.p3.3.m3.2.2.2.2.2.6.1"></times><apply id="S5.3.p3.3.m3.2.2.2.2.2.6.2.cmml" xref="S5.3.p3.3.m3.2.2.2.2.2.6.2"><csymbol cd="ambiguous" id="S5.3.p3.3.m3.2.2.2.2.2.6.2.1.cmml" xref="S5.3.p3.3.m3.2.2.2.2.2.6.2">superscript</csymbol><ci id="S5.3.p3.3.m3.2.2.2.2.2.6.2.2.cmml" xref="S5.3.p3.3.m3.2.2.2.2.2.6.2.2">π₯</ci><apply id="S5.3.p3.3.m3.2.2.2.2.2.6.2.3.cmml" xref="S5.3.p3.3.m3.2.2.2.2.2.6.2.3"><minus id="S5.3.p3.3.m3.2.2.2.2.2.6.2.3.1.cmml" xref="S5.3.p3.3.m3.2.2.2.2.2.6.2.3"></minus><cn id="S5.3.p3.3.m3.2.2.2.2.2.6.2.3.2.cmml" type="integer" xref="S5.3.p3.3.m3.2.2.2.2.2.6.2.3.2">1</cn></apply></apply><ci id="S5.3.p3.3.m3.2.2.2.2.2.6.3.cmml" xref="S5.3.p3.3.m3.2.2.2.2.2.6.3">π₯</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.3.m3.2c">H_{f}=\{x\in S\colon xx^{-1}=f=x^{-1}x\}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.3.m3.2d">italic_H start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT = { italic_x β italic_S : italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_f = italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x }</annotation></semantics></math> are closed in <math alttext="S" class="ltx_Math" display="inline" id="S5.3.p3.4.m4.1"><semantics id="S5.3.p3.4.m4.1a"><mi id="S5.3.p3.4.m4.1.1" xref="S5.3.p3.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.3.p3.4.m4.1b"><ci id="S5.3.p3.4.m4.1.1.cmml" xref="S5.3.p3.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.4.m4.1d">italic_S</annotation></semantics></math>. Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S1.Thmtheorem3" title="Theorem 1.3 (Mukherjea, Tserpes). β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">1.3</span></a> implies that each closed subgroup of <math alttext="S" class="ltx_Math" display="inline" id="S5.3.p3.5.m5.1"><semantics id="S5.3.p3.5.m5.1a"><mi id="S5.3.p3.5.m5.1.1" xref="S5.3.p3.5.m5.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.3.p3.5.m5.1b"><ci id="S5.3.p3.5.m5.1.1.cmml" xref="S5.3.p3.5.m5.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.5.m5.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.5.m5.1d">italic_S</annotation></semantics></math> is compact. Thus, the sets <math alttext="H_{e}" class="ltx_Math" display="inline" id="S5.3.p3.6.m6.1"><semantics id="S5.3.p3.6.m6.1a"><msub id="S5.3.p3.6.m6.1.1" xref="S5.3.p3.6.m6.1.1.cmml"><mi id="S5.3.p3.6.m6.1.1.2" xref="S5.3.p3.6.m6.1.1.2.cmml">H</mi><mi id="S5.3.p3.6.m6.1.1.3" xref="S5.3.p3.6.m6.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="S5.3.p3.6.m6.1b"><apply id="S5.3.p3.6.m6.1.1.cmml" xref="S5.3.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S5.3.p3.6.m6.1.1.1.cmml" xref="S5.3.p3.6.m6.1.1">subscript</csymbol><ci id="S5.3.p3.6.m6.1.1.2.cmml" xref="S5.3.p3.6.m6.1.1.2">π»</ci><ci id="S5.3.p3.6.m6.1.1.3.cmml" xref="S5.3.p3.6.m6.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.6.m6.1c">H_{e}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.6.m6.1d">italic_H start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="H_{f}" class="ltx_Math" display="inline" id="S5.3.p3.7.m7.1"><semantics id="S5.3.p3.7.m7.1a"><msub id="S5.3.p3.7.m7.1.1" xref="S5.3.p3.7.m7.1.1.cmml"><mi id="S5.3.p3.7.m7.1.1.2" xref="S5.3.p3.7.m7.1.1.2.cmml">H</mi><mi id="S5.3.p3.7.m7.1.1.3" xref="S5.3.p3.7.m7.1.1.3.cmml">f</mi></msub><annotation-xml encoding="MathML-Content" id="S5.3.p3.7.m7.1b"><apply id="S5.3.p3.7.m7.1.1.cmml" xref="S5.3.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S5.3.p3.7.m7.1.1.1.cmml" xref="S5.3.p3.7.m7.1.1">subscript</csymbol><ci id="S5.3.p3.7.m7.1.1.2.cmml" xref="S5.3.p3.7.m7.1.1.2">π»</ci><ci id="S5.3.p3.7.m7.1.1.3.cmml" xref="S5.3.p3.7.m7.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.7.m7.1c">H_{f}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.7.m7.1d">italic_H start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT</annotation></semantics></math> are compact. Observe that <math alttext="H_{e}" class="ltx_Math" display="inline" id="S5.3.p3.8.m8.1"><semantics id="S5.3.p3.8.m8.1a"><msub id="S5.3.p3.8.m8.1.1" xref="S5.3.p3.8.m8.1.1.cmml"><mi id="S5.3.p3.8.m8.1.1.2" xref="S5.3.p3.8.m8.1.1.2.cmml">H</mi><mi id="S5.3.p3.8.m8.1.1.3" xref="S5.3.p3.8.m8.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="S5.3.p3.8.m8.1b"><apply id="S5.3.p3.8.m8.1.1.cmml" xref="S5.3.p3.8.m8.1.1"><csymbol cd="ambiguous" id="S5.3.p3.8.m8.1.1.1.cmml" xref="S5.3.p3.8.m8.1.1">subscript</csymbol><ci id="S5.3.p3.8.m8.1.1.2.cmml" xref="S5.3.p3.8.m8.1.1.2">π»</ci><ci id="S5.3.p3.8.m8.1.1.3.cmml" xref="S5.3.p3.8.m8.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.8.m8.1c">H_{e}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.8.m8.1d">italic_H start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math> is the <math alttext="\mathcal{H}" class="ltx_Math" display="inline" id="S5.3.p3.9.m9.1"><semantics id="S5.3.p3.9.m9.1a"><mi class="ltx_font_mathcaligraphic" id="S5.3.p3.9.m9.1.1" xref="S5.3.p3.9.m9.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S5.3.p3.9.m9.1b"><ci id="S5.3.p3.9.m9.1.1.cmml" xref="S5.3.p3.9.m9.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.9.m9.1c">\mathcal{H}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.9.m9.1d">caligraphic_H</annotation></semantics></math>-class of the element <math alttext="e" class="ltx_Math" display="inline" id="S5.3.p3.10.m10.1"><semantics id="S5.3.p3.10.m10.1a"><mi id="S5.3.p3.10.m10.1.1" xref="S5.3.p3.10.m10.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S5.3.p3.10.m10.1b"><ci id="S5.3.p3.10.m10.1.1.cmml" xref="S5.3.p3.10.m10.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.10.m10.1c">e</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.10.m10.1d">italic_e</annotation></semantics></math> and <math alttext="H_{f}" class="ltx_Math" display="inline" id="S5.3.p3.11.m11.1"><semantics id="S5.3.p3.11.m11.1a"><msub id="S5.3.p3.11.m11.1.1" xref="S5.3.p3.11.m11.1.1.cmml"><mi id="S5.3.p3.11.m11.1.1.2" xref="S5.3.p3.11.m11.1.1.2.cmml">H</mi><mi id="S5.3.p3.11.m11.1.1.3" xref="S5.3.p3.11.m11.1.1.3.cmml">f</mi></msub><annotation-xml encoding="MathML-Content" id="S5.3.p3.11.m11.1b"><apply id="S5.3.p3.11.m11.1.1.cmml" xref="S5.3.p3.11.m11.1.1"><csymbol cd="ambiguous" id="S5.3.p3.11.m11.1.1.1.cmml" xref="S5.3.p3.11.m11.1.1">subscript</csymbol><ci id="S5.3.p3.11.m11.1.1.2.cmml" xref="S5.3.p3.11.m11.1.1.2">π»</ci><ci id="S5.3.p3.11.m11.1.1.3.cmml" xref="S5.3.p3.11.m11.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.11.m11.1c">H_{f}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.11.m11.1d">italic_H start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT</annotation></semantics></math> is the <math alttext="\mathcal{H}" class="ltx_Math" display="inline" id="S5.3.p3.12.m12.1"><semantics id="S5.3.p3.12.m12.1a"><mi class="ltx_font_mathcaligraphic" id="S5.3.p3.12.m12.1.1" xref="S5.3.p3.12.m12.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S5.3.p3.12.m12.1b"><ci id="S5.3.p3.12.m12.1.1.cmml" xref="S5.3.p3.12.m12.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.12.m12.1c">\mathcal{H}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.12.m12.1d">caligraphic_H</annotation></semantics></math>-class of the element <math alttext="f" class="ltx_Math" display="inline" id="S5.3.p3.13.m13.1"><semantics id="S5.3.p3.13.m13.1a"><mi id="S5.3.p3.13.m13.1.1" xref="S5.3.p3.13.m13.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S5.3.p3.13.m13.1b"><ci id="S5.3.p3.13.m13.1.1.cmml" xref="S5.3.p3.13.m13.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.13.m13.1c">f</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.13.m13.1d">italic_f</annotation></semantics></math>. Consider the <math alttext="\mathcal{H}" class="ltx_Math" display="inline" id="S5.3.p3.14.m14.1"><semantics id="S5.3.p3.14.m14.1a"><mi class="ltx_font_mathcaligraphic" id="S5.3.p3.14.m14.1.1" xref="S5.3.p3.14.m14.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S5.3.p3.14.m14.1b"><ci id="S5.3.p3.14.m14.1.1.cmml" xref="S5.3.p3.14.m14.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.14.m14.1c">\mathcal{H}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.14.m14.1d">caligraphic_H</annotation></semantics></math>-class <math alttext="H_{e,f}:=\{x\in S\colon xx^{-1}=e\hbox{ and }x^{-1}x=f\}" class="ltx_Math" display="inline" id="S5.3.p3.15.m15.4"><semantics id="S5.3.p3.15.m15.4a"><mrow id="S5.3.p3.15.m15.4.4" xref="S5.3.p3.15.m15.4.4.cmml"><msub id="S5.3.p3.15.m15.4.4.4" xref="S5.3.p3.15.m15.4.4.4.cmml"><mi id="S5.3.p3.15.m15.4.4.4.2" xref="S5.3.p3.15.m15.4.4.4.2.cmml">H</mi><mrow id="S5.3.p3.15.m15.2.2.2.4" xref="S5.3.p3.15.m15.2.2.2.3.cmml"><mi id="S5.3.p3.15.m15.1.1.1.1" xref="S5.3.p3.15.m15.1.1.1.1.cmml">e</mi><mo id="S5.3.p3.15.m15.2.2.2.4.1" xref="S5.3.p3.15.m15.2.2.2.3.cmml">,</mo><mi id="S5.3.p3.15.m15.2.2.2.2" xref="S5.3.p3.15.m15.2.2.2.2.cmml">f</mi></mrow></msub><mo id="S5.3.p3.15.m15.4.4.3" lspace="0.278em" rspace="0.278em" xref="S5.3.p3.15.m15.4.4.3.cmml">:=</mo><mrow id="S5.3.p3.15.m15.4.4.2.2" xref="S5.3.p3.15.m15.4.4.2.3.cmml"><mo id="S5.3.p3.15.m15.4.4.2.2.3" stretchy="false" xref="S5.3.p3.15.m15.4.4.2.3.1.cmml">{</mo><mrow id="S5.3.p3.15.m15.3.3.1.1.1" xref="S5.3.p3.15.m15.3.3.1.1.1.cmml"><mi id="S5.3.p3.15.m15.3.3.1.1.1.2" xref="S5.3.p3.15.m15.3.3.1.1.1.2.cmml">x</mi><mo id="S5.3.p3.15.m15.3.3.1.1.1.1" xref="S5.3.p3.15.m15.3.3.1.1.1.1.cmml">β</mo><mi id="S5.3.p3.15.m15.3.3.1.1.1.3" xref="S5.3.p3.15.m15.3.3.1.1.1.3.cmml">S</mi></mrow><mo id="S5.3.p3.15.m15.4.4.2.2.4" lspace="0.278em" rspace="0.278em" xref="S5.3.p3.15.m15.4.4.2.3.1.cmml">:</mo><mrow id="S5.3.p3.15.m15.4.4.2.2.2" xref="S5.3.p3.15.m15.4.4.2.2.2.cmml"><mrow id="S5.3.p3.15.m15.4.4.2.2.2.2" xref="S5.3.p3.15.m15.4.4.2.2.2.2.cmml"><mi id="S5.3.p3.15.m15.4.4.2.2.2.2.2" xref="S5.3.p3.15.m15.4.4.2.2.2.2.2.cmml">x</mi><mo id="S5.3.p3.15.m15.4.4.2.2.2.2.1" xref="S5.3.p3.15.m15.4.4.2.2.2.2.1.cmml">β’</mo><msup id="S5.3.p3.15.m15.4.4.2.2.2.2.3" xref="S5.3.p3.15.m15.4.4.2.2.2.2.3.cmml"><mi id="S5.3.p3.15.m15.4.4.2.2.2.2.3.2" xref="S5.3.p3.15.m15.4.4.2.2.2.2.3.2.cmml">x</mi><mrow id="S5.3.p3.15.m15.4.4.2.2.2.2.3.3" xref="S5.3.p3.15.m15.4.4.2.2.2.2.3.3.cmml"><mo id="S5.3.p3.15.m15.4.4.2.2.2.2.3.3a" xref="S5.3.p3.15.m15.4.4.2.2.2.2.3.3.cmml">β</mo><mn id="S5.3.p3.15.m15.4.4.2.2.2.2.3.3.2" xref="S5.3.p3.15.m15.4.4.2.2.2.2.3.3.2.cmml">1</mn></mrow></msup></mrow><mo id="S5.3.p3.15.m15.4.4.2.2.2.3" xref="S5.3.p3.15.m15.4.4.2.2.2.3.cmml">=</mo><mrow id="S5.3.p3.15.m15.4.4.2.2.2.4" xref="S5.3.p3.15.m15.4.4.2.2.2.4.cmml"><mi id="S5.3.p3.15.m15.4.4.2.2.2.4.2" 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xref="S5.3.p3.15.m15.4.4.2.2.2.5.cmml">=</mo><mi id="S5.3.p3.15.m15.4.4.2.2.2.6" xref="S5.3.p3.15.m15.4.4.2.2.2.6.cmml">f</mi></mrow><mo id="S5.3.p3.15.m15.4.4.2.2.5" stretchy="false" xref="S5.3.p3.15.m15.4.4.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.15.m15.4b"><apply id="S5.3.p3.15.m15.4.4.cmml" xref="S5.3.p3.15.m15.4.4"><csymbol cd="latexml" id="S5.3.p3.15.m15.4.4.3.cmml" xref="S5.3.p3.15.m15.4.4.3">assign</csymbol><apply id="S5.3.p3.15.m15.4.4.4.cmml" xref="S5.3.p3.15.m15.4.4.4"><csymbol cd="ambiguous" id="S5.3.p3.15.m15.4.4.4.1.cmml" xref="S5.3.p3.15.m15.4.4.4">subscript</csymbol><ci id="S5.3.p3.15.m15.4.4.4.2.cmml" xref="S5.3.p3.15.m15.4.4.4.2">π»</ci><list id="S5.3.p3.15.m15.2.2.2.3.cmml" xref="S5.3.p3.15.m15.2.2.2.4"><ci id="S5.3.p3.15.m15.1.1.1.1.cmml" xref="S5.3.p3.15.m15.1.1.1.1">π</ci><ci id="S5.3.p3.15.m15.2.2.2.2.cmml" xref="S5.3.p3.15.m15.2.2.2.2">π</ci></list></apply><apply id="S5.3.p3.15.m15.4.4.2.3.cmml" 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xref="S5.3.p3.15.m15.4.4.2.2.2.2.2">π₯</ci><apply id="S5.3.p3.15.m15.4.4.2.2.2.2.3.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2.2.3"><csymbol cd="ambiguous" id="S5.3.p3.15.m15.4.4.2.2.2.2.3.1.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2.2.3">superscript</csymbol><ci id="S5.3.p3.15.m15.4.4.2.2.2.2.3.2.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2.2.3.2">π₯</ci><apply id="S5.3.p3.15.m15.4.4.2.2.2.2.3.3.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2.2.3.3"><minus id="S5.3.p3.15.m15.4.4.2.2.2.2.3.3.1.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2.2.3.3"></minus><cn id="S5.3.p3.15.m15.4.4.2.2.2.2.3.3.2.cmml" type="integer" xref="S5.3.p3.15.m15.4.4.2.2.2.2.3.3.2">1</cn></apply></apply></apply><apply id="S5.3.p3.15.m15.4.4.2.2.2.4.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2.4"><times id="S5.3.p3.15.m15.4.4.2.2.2.4.1.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2.4.1"></times><ci id="S5.3.p3.15.m15.4.4.2.2.2.4.2.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2.4.2">π</ci><ci id="S5.3.p3.15.m15.4.4.2.2.2.4.3a.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2.4.3"><mtext id="S5.3.p3.15.m15.4.4.2.2.2.4.3.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2.4.3"> and </mtext></ci><apply id="S5.3.p3.15.m15.4.4.2.2.2.4.4.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2.4.4"><csymbol cd="ambiguous" id="S5.3.p3.15.m15.4.4.2.2.2.4.4.1.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2.4.4">superscript</csymbol><ci id="S5.3.p3.15.m15.4.4.2.2.2.4.4.2.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2.4.4.2">π₯</ci><apply id="S5.3.p3.15.m15.4.4.2.2.2.4.4.3.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2.4.4.3"><minus id="S5.3.p3.15.m15.4.4.2.2.2.4.4.3.1.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2.4.4.3"></minus><cn id="S5.3.p3.15.m15.4.4.2.2.2.4.4.3.2.cmml" type="integer" xref="S5.3.p3.15.m15.4.4.2.2.2.4.4.3.2">1</cn></apply></apply><ci id="S5.3.p3.15.m15.4.4.2.2.2.4.5.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2.4.5">π₯</ci></apply></apply><apply id="S5.3.p3.15.m15.4.4.2.2.2c.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2"><eq id="S5.3.p3.15.m15.4.4.2.2.2.5.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2.5"></eq><share href="https://arxiv.org/html/2503.13666v1#S5.3.p3.15.m15.4.4.2.2.2.4.cmml" id="S5.3.p3.15.m15.4.4.2.2.2d.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2"></share><ci id="S5.3.p3.15.m15.4.4.2.2.2.6.cmml" xref="S5.3.p3.15.m15.4.4.2.2.2.6">π</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.15.m15.4c">H_{e,f}:=\{x\in S\colon xx^{-1}=e\hbox{ and }x^{-1}x=f\}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.15.m15.4d">italic_H start_POSTSUBSCRIPT italic_e , italic_f end_POSTSUBSCRIPT := { italic_x β italic_S : italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_e and italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x = italic_f }</annotation></semantics></math>. If <math alttext="H_{e,f}\neq\varnothing" class="ltx_Math" display="inline" id="S5.3.p3.16.m16.2"><semantics id="S5.3.p3.16.m16.2a"><mrow id="S5.3.p3.16.m16.2.3" xref="S5.3.p3.16.m16.2.3.cmml"><msub id="S5.3.p3.16.m16.2.3.2" xref="S5.3.p3.16.m16.2.3.2.cmml"><mi id="S5.3.p3.16.m16.2.3.2.2" xref="S5.3.p3.16.m16.2.3.2.2.cmml">H</mi><mrow id="S5.3.p3.16.m16.2.2.2.4" xref="S5.3.p3.16.m16.2.2.2.3.cmml"><mi id="S5.3.p3.16.m16.1.1.1.1" xref="S5.3.p3.16.m16.1.1.1.1.cmml">e</mi><mo id="S5.3.p3.16.m16.2.2.2.4.1" xref="S5.3.p3.16.m16.2.2.2.3.cmml">,</mo><mi id="S5.3.p3.16.m16.2.2.2.2" xref="S5.3.p3.16.m16.2.2.2.2.cmml">f</mi></mrow></msub><mo id="S5.3.p3.16.m16.2.3.1" xref="S5.3.p3.16.m16.2.3.1.cmml">β </mo><mi id="S5.3.p3.16.m16.2.3.3" mathvariant="normal" xref="S5.3.p3.16.m16.2.3.3.cmml">β </mi></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.16.m16.2b"><apply id="S5.3.p3.16.m16.2.3.cmml" xref="S5.3.p3.16.m16.2.3"><neq id="S5.3.p3.16.m16.2.3.1.cmml" xref="S5.3.p3.16.m16.2.3.1"></neq><apply id="S5.3.p3.16.m16.2.3.2.cmml" xref="S5.3.p3.16.m16.2.3.2"><csymbol cd="ambiguous" id="S5.3.p3.16.m16.2.3.2.1.cmml" xref="S5.3.p3.16.m16.2.3.2">subscript</csymbol><ci id="S5.3.p3.16.m16.2.3.2.2.cmml" xref="S5.3.p3.16.m16.2.3.2.2">π»</ci><list id="S5.3.p3.16.m16.2.2.2.3.cmml" xref="S5.3.p3.16.m16.2.2.2.4"><ci id="S5.3.p3.16.m16.1.1.1.1.cmml" xref="S5.3.p3.16.m16.1.1.1.1">π</ci><ci id="S5.3.p3.16.m16.2.2.2.2.cmml" xref="S5.3.p3.16.m16.2.2.2.2">π</ci></list></apply><emptyset id="S5.3.p3.16.m16.2.3.3.cmml" xref="S5.3.p3.16.m16.2.3.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.16.m16.2c">H_{e,f}\neq\varnothing</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.16.m16.2d">italic_H start_POSTSUBSCRIPT italic_e , italic_f end_POSTSUBSCRIPT β β </annotation></semantics></math>, then <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib36" title="">36</a>, Proposition 5]</cite> implies that <math alttext="H_{e,f}" class="ltx_Math" display="inline" id="S5.3.p3.17.m17.2"><semantics id="S5.3.p3.17.m17.2a"><msub id="S5.3.p3.17.m17.2.3" xref="S5.3.p3.17.m17.2.3.cmml"><mi id="S5.3.p3.17.m17.2.3.2" xref="S5.3.p3.17.m17.2.3.2.cmml">H</mi><mrow id="S5.3.p3.17.m17.2.2.2.4" xref="S5.3.p3.17.m17.2.2.2.3.cmml"><mi id="S5.3.p3.17.m17.1.1.1.1" xref="S5.3.p3.17.m17.1.1.1.1.cmml">e</mi><mo id="S5.3.p3.17.m17.2.2.2.4.1" xref="S5.3.p3.17.m17.2.2.2.3.cmml">,</mo><mi id="S5.3.p3.17.m17.2.2.2.2" xref="S5.3.p3.17.m17.2.2.2.2.cmml">f</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.3.p3.17.m17.2b"><apply id="S5.3.p3.17.m17.2.3.cmml" xref="S5.3.p3.17.m17.2.3"><csymbol cd="ambiguous" id="S5.3.p3.17.m17.2.3.1.cmml" xref="S5.3.p3.17.m17.2.3">subscript</csymbol><ci id="S5.3.p3.17.m17.2.3.2.cmml" xref="S5.3.p3.17.m17.2.3.2">π»</ci><list id="S5.3.p3.17.m17.2.2.2.3.cmml" xref="S5.3.p3.17.m17.2.2.2.4"><ci id="S5.3.p3.17.m17.1.1.1.1.cmml" xref="S5.3.p3.17.m17.1.1.1.1">π</ci><ci id="S5.3.p3.17.m17.2.2.2.2.cmml" xref="S5.3.p3.17.m17.2.2.2.2">π</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.17.m17.2c">H_{e,f}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.17.m17.2d">italic_H start_POSTSUBSCRIPT italic_e , italic_f end_POSTSUBSCRIPT</annotation></semantics></math> lies in the same <math alttext="\mathcal{D}" class="ltx_Math" display="inline" id="S5.3.p3.18.m18.1"><semantics id="S5.3.p3.18.m18.1a"><mi class="ltx_font_mathcaligraphic" id="S5.3.p3.18.m18.1.1" xref="S5.3.p3.18.m18.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S5.3.p3.18.m18.1b"><ci id="S5.3.p3.18.m18.1.1.cmml" xref="S5.3.p3.18.m18.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.18.m18.1c">\mathcal{D}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.18.m18.1d">caligraphic_D</annotation></semantics></math>-class with <math alttext="H_{e}" class="ltx_Math" display="inline" id="S5.3.p3.19.m19.1"><semantics id="S5.3.p3.19.m19.1a"><msub id="S5.3.p3.19.m19.1.1" xref="S5.3.p3.19.m19.1.1.cmml"><mi id="S5.3.p3.19.m19.1.1.2" xref="S5.3.p3.19.m19.1.1.2.cmml">H</mi><mi id="S5.3.p3.19.m19.1.1.3" xref="S5.3.p3.19.m19.1.1.3.cmml">e</mi></msub><annotation-xml encoding="MathML-Content" id="S5.3.p3.19.m19.1b"><apply id="S5.3.p3.19.m19.1.1.cmml" xref="S5.3.p3.19.m19.1.1"><csymbol cd="ambiguous" id="S5.3.p3.19.m19.1.1.1.cmml" xref="S5.3.p3.19.m19.1.1">subscript</csymbol><ci id="S5.3.p3.19.m19.1.1.2.cmml" xref="S5.3.p3.19.m19.1.1.2">π»</ci><ci id="S5.3.p3.19.m19.1.1.3.cmml" xref="S5.3.p3.19.m19.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.19.m19.1c">H_{e}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.19.m19.1d">italic_H start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="H_{f}" class="ltx_Math" display="inline" id="S5.3.p3.20.m20.1"><semantics id="S5.3.p3.20.m20.1a"><msub id="S5.3.p3.20.m20.1.1" xref="S5.3.p3.20.m20.1.1.cmml"><mi id="S5.3.p3.20.m20.1.1.2" xref="S5.3.p3.20.m20.1.1.2.cmml">H</mi><mi id="S5.3.p3.20.m20.1.1.3" xref="S5.3.p3.20.m20.1.1.3.cmml">f</mi></msub><annotation-xml encoding="MathML-Content" id="S5.3.p3.20.m20.1b"><apply id="S5.3.p3.20.m20.1.1.cmml" xref="S5.3.p3.20.m20.1.1"><csymbol cd="ambiguous" id="S5.3.p3.20.m20.1.1.1.cmml" xref="S5.3.p3.20.m20.1.1">subscript</csymbol><ci id="S5.3.p3.20.m20.1.1.2.cmml" xref="S5.3.p3.20.m20.1.1.2">π»</ci><ci id="S5.3.p3.20.m20.1.1.3.cmml" xref="S5.3.p3.20.m20.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.20.m20.1c">H_{f}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.20.m20.1d">italic_H start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT</annotation></semantics></math>. By Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S5.Thmtheorem2" title="Proposition 5.2 (Folklore). β£ 5. Proof of the main result and final remarks β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">5.2</span></a>, the set <math alttext="H_{e,f}" class="ltx_Math" display="inline" id="S5.3.p3.21.m21.2"><semantics id="S5.3.p3.21.m21.2a"><msub id="S5.3.p3.21.m21.2.3" xref="S5.3.p3.21.m21.2.3.cmml"><mi id="S5.3.p3.21.m21.2.3.2" xref="S5.3.p3.21.m21.2.3.2.cmml">H</mi><mrow id="S5.3.p3.21.m21.2.2.2.4" xref="S5.3.p3.21.m21.2.2.2.3.cmml"><mi id="S5.3.p3.21.m21.1.1.1.1" xref="S5.3.p3.21.m21.1.1.1.1.cmml">e</mi><mo id="S5.3.p3.21.m21.2.2.2.4.1" xref="S5.3.p3.21.m21.2.2.2.3.cmml">,</mo><mi id="S5.3.p3.21.m21.2.2.2.2" xref="S5.3.p3.21.m21.2.2.2.2.cmml">f</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.3.p3.21.m21.2b"><apply id="S5.3.p3.21.m21.2.3.cmml" xref="S5.3.p3.21.m21.2.3"><csymbol cd="ambiguous" id="S5.3.p3.21.m21.2.3.1.cmml" xref="S5.3.p3.21.m21.2.3">subscript</csymbol><ci id="S5.3.p3.21.m21.2.3.2.cmml" xref="S5.3.p3.21.m21.2.3.2">π»</ci><list id="S5.3.p3.21.m21.2.2.2.3.cmml" xref="S5.3.p3.21.m21.2.2.2.4"><ci id="S5.3.p3.21.m21.1.1.1.1.cmml" xref="S5.3.p3.21.m21.1.1.1.1">π</ci><ci id="S5.3.p3.21.m21.2.2.2.2.cmml" xref="S5.3.p3.21.m21.2.2.2.2">π</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.21.m21.2c">H_{e,f}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.21.m21.2d">italic_H start_POSTSUBSCRIPT italic_e , italic_f end_POSTSUBSCRIPT</annotation></semantics></math> is compact. Thus <math alttext="h\in\operatorname{cl}_{\beta S}(T\setminus H_{e,f})" class="ltx_Math" display="inline" id="S5.3.p3.22.m22.4"><semantics id="S5.3.p3.22.m22.4a"><mrow id="S5.3.p3.22.m22.4.4" xref="S5.3.p3.22.m22.4.4.cmml"><mi id="S5.3.p3.22.m22.4.4.4" xref="S5.3.p3.22.m22.4.4.4.cmml">h</mi><mo id="S5.3.p3.22.m22.4.4.3" xref="S5.3.p3.22.m22.4.4.3.cmml">β</mo><mrow id="S5.3.p3.22.m22.4.4.2.2" xref="S5.3.p3.22.m22.4.4.2.3.cmml"><msub id="S5.3.p3.22.m22.3.3.1.1.1" xref="S5.3.p3.22.m22.3.3.1.1.1.cmml"><mi id="S5.3.p3.22.m22.3.3.1.1.1.2" xref="S5.3.p3.22.m22.3.3.1.1.1.2.cmml">cl</mi><mrow id="S5.3.p3.22.m22.3.3.1.1.1.3" xref="S5.3.p3.22.m22.3.3.1.1.1.3.cmml"><mi id="S5.3.p3.22.m22.3.3.1.1.1.3.2" xref="S5.3.p3.22.m22.3.3.1.1.1.3.2.cmml">Ξ²</mi><mo id="S5.3.p3.22.m22.3.3.1.1.1.3.1" xref="S5.3.p3.22.m22.3.3.1.1.1.3.1.cmml">β’</mo><mi id="S5.3.p3.22.m22.3.3.1.1.1.3.3" xref="S5.3.p3.22.m22.3.3.1.1.1.3.3.cmml">S</mi></mrow></msub><mo id="S5.3.p3.22.m22.4.4.2.2a" xref="S5.3.p3.22.m22.4.4.2.3.cmml">β‘</mo><mrow id="S5.3.p3.22.m22.4.4.2.2.2" xref="S5.3.p3.22.m22.4.4.2.3.cmml"><mo id="S5.3.p3.22.m22.4.4.2.2.2.2" stretchy="false" xref="S5.3.p3.22.m22.4.4.2.3.cmml">(</mo><mrow id="S5.3.p3.22.m22.4.4.2.2.2.1" xref="S5.3.p3.22.m22.4.4.2.2.2.1.cmml"><mi id="S5.3.p3.22.m22.4.4.2.2.2.1.2" xref="S5.3.p3.22.m22.4.4.2.2.2.1.2.cmml">T</mi><mo id="S5.3.p3.22.m22.4.4.2.2.2.1.1" xref="S5.3.p3.22.m22.4.4.2.2.2.1.1.cmml">β</mo><msub id="S5.3.p3.22.m22.4.4.2.2.2.1.3" xref="S5.3.p3.22.m22.4.4.2.2.2.1.3.cmml"><mi id="S5.3.p3.22.m22.4.4.2.2.2.1.3.2" xref="S5.3.p3.22.m22.4.4.2.2.2.1.3.2.cmml">H</mi><mrow id="S5.3.p3.22.m22.2.2.2.4" xref="S5.3.p3.22.m22.2.2.2.3.cmml"><mi id="S5.3.p3.22.m22.1.1.1.1" xref="S5.3.p3.22.m22.1.1.1.1.cmml">e</mi><mo id="S5.3.p3.22.m22.2.2.2.4.1" xref="S5.3.p3.22.m22.2.2.2.3.cmml">,</mo><mi id="S5.3.p3.22.m22.2.2.2.2" xref="S5.3.p3.22.m22.2.2.2.2.cmml">f</mi></mrow></msub></mrow><mo id="S5.3.p3.22.m22.4.4.2.2.2.3" stretchy="false" xref="S5.3.p3.22.m22.4.4.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p3.22.m22.4b"><apply id="S5.3.p3.22.m22.4.4.cmml" xref="S5.3.p3.22.m22.4.4"><in id="S5.3.p3.22.m22.4.4.3.cmml" xref="S5.3.p3.22.m22.4.4.3"></in><ci id="S5.3.p3.22.m22.4.4.4.cmml" xref="S5.3.p3.22.m22.4.4.4">β</ci><apply id="S5.3.p3.22.m22.4.4.2.3.cmml" xref="S5.3.p3.22.m22.4.4.2.2"><apply id="S5.3.p3.22.m22.3.3.1.1.1.cmml" xref="S5.3.p3.22.m22.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.3.p3.22.m22.3.3.1.1.1.1.cmml" xref="S5.3.p3.22.m22.3.3.1.1.1">subscript</csymbol><ci id="S5.3.p3.22.m22.3.3.1.1.1.2.cmml" xref="S5.3.p3.22.m22.3.3.1.1.1.2">cl</ci><apply id="S5.3.p3.22.m22.3.3.1.1.1.3.cmml" xref="S5.3.p3.22.m22.3.3.1.1.1.3"><times id="S5.3.p3.22.m22.3.3.1.1.1.3.1.cmml" xref="S5.3.p3.22.m22.3.3.1.1.1.3.1"></times><ci id="S5.3.p3.22.m22.3.3.1.1.1.3.2.cmml" xref="S5.3.p3.22.m22.3.3.1.1.1.3.2">π½</ci><ci id="S5.3.p3.22.m22.3.3.1.1.1.3.3.cmml" xref="S5.3.p3.22.m22.3.3.1.1.1.3.3">π</ci></apply></apply><apply id="S5.3.p3.22.m22.4.4.2.2.2.1.cmml" xref="S5.3.p3.22.m22.4.4.2.2.2.1"><setdiff id="S5.3.p3.22.m22.4.4.2.2.2.1.1.cmml" xref="S5.3.p3.22.m22.4.4.2.2.2.1.1"></setdiff><ci id="S5.3.p3.22.m22.4.4.2.2.2.1.2.cmml" xref="S5.3.p3.22.m22.4.4.2.2.2.1.2">π</ci><apply id="S5.3.p3.22.m22.4.4.2.2.2.1.3.cmml" xref="S5.3.p3.22.m22.4.4.2.2.2.1.3"><csymbol cd="ambiguous" id="S5.3.p3.22.m22.4.4.2.2.2.1.3.1.cmml" xref="S5.3.p3.22.m22.4.4.2.2.2.1.3">subscript</csymbol><ci id="S5.3.p3.22.m22.4.4.2.2.2.1.3.2.cmml" xref="S5.3.p3.22.m22.4.4.2.2.2.1.3.2">π»</ci><list id="S5.3.p3.22.m22.2.2.2.3.cmml" xref="S5.3.p3.22.m22.2.2.2.4"><ci id="S5.3.p3.22.m22.1.1.1.1.cmml" xref="S5.3.p3.22.m22.1.1.1.1">π</ci><ci id="S5.3.p3.22.m22.2.2.2.2.cmml" xref="S5.3.p3.22.m22.2.2.2.2">π</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p3.22.m22.4c">h\in\operatorname{cl}_{\beta S}(T\setminus H_{e,f})</annotation><annotation encoding="application/x-llamapun" id="S5.3.p3.22.m22.4d">italic_h β roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_S end_POSTSUBSCRIPT ( italic_T β italic_H start_POSTSUBSCRIPT italic_e , italic_f end_POSTSUBSCRIPT )</annotation></semantics></math>. Two cases are possible:</p> <ol class="ltx_enumerate" id="S5.I2"> <li class="ltx_item" id="S5.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S5.I2.i1.p1"> <p class="ltx_p" id="S5.I2.i1.p1.1"><math alttext="h\in\operatorname{cl}_{\beta S}(\{x\in S:xx^{-1}<e\hbox{ and }x^{-1}x\leq f\})" class="ltx_Math" display="inline" id="S5.I2.i1.p1.1.m1.2"><semantics id="S5.I2.i1.p1.1.m1.2a"><mrow id="S5.I2.i1.p1.1.m1.2.2" xref="S5.I2.i1.p1.1.m1.2.2.cmml"><mi id="S5.I2.i1.p1.1.m1.2.2.4" xref="S5.I2.i1.p1.1.m1.2.2.4.cmml">h</mi><mo id="S5.I2.i1.p1.1.m1.2.2.3" xref="S5.I2.i1.p1.1.m1.2.2.3.cmml">β</mo><mrow id="S5.I2.i1.p1.1.m1.2.2.2.2" xref="S5.I2.i1.p1.1.m1.2.2.2.3.cmml"><msub id="S5.I2.i1.p1.1.m1.1.1.1.1.1" xref="S5.I2.i1.p1.1.m1.1.1.1.1.1.cmml"><mi id="S5.I2.i1.p1.1.m1.1.1.1.1.1.2" xref="S5.I2.i1.p1.1.m1.1.1.1.1.1.2.cmml">cl</mi><mrow id="S5.I2.i1.p1.1.m1.1.1.1.1.1.3" xref="S5.I2.i1.p1.1.m1.1.1.1.1.1.3.cmml"><mi id="S5.I2.i1.p1.1.m1.1.1.1.1.1.3.2" xref="S5.I2.i1.p1.1.m1.1.1.1.1.1.3.2.cmml">Ξ²</mi><mo id="S5.I2.i1.p1.1.m1.1.1.1.1.1.3.1" xref="S5.I2.i1.p1.1.m1.1.1.1.1.1.3.1.cmml">β’</mo><mi id="S5.I2.i1.p1.1.m1.1.1.1.1.1.3.3" xref="S5.I2.i1.p1.1.m1.1.1.1.1.1.3.3.cmml">S</mi></mrow></msub><mo id="S5.I2.i1.p1.1.m1.2.2.2.2a" xref="S5.I2.i1.p1.1.m1.2.2.2.3.cmml">β‘</mo><mrow id="S5.I2.i1.p1.1.m1.2.2.2.2.2" xref="S5.I2.i1.p1.1.m1.2.2.2.3.cmml"><mo id="S5.I2.i1.p1.1.m1.2.2.2.2.2.2" stretchy="false" xref="S5.I2.i1.p1.1.m1.2.2.2.3.cmml">(</mo><mrow id="S5.I2.i1.p1.1.m1.2.2.2.2.2.1.2" xref="S5.I2.i1.p1.1.m1.2.2.2.2.2.1.3.cmml"><mo id="S5.I2.i1.p1.1.m1.2.2.2.2.2.1.2.3" stretchy="false" xref="S5.I2.i1.p1.1.m1.2.2.2.2.2.1.3.1.cmml">{</mo><mrow id="S5.I2.i1.p1.1.m1.2.2.2.2.2.1.1.1" xref="S5.I2.i1.p1.1.m1.2.2.2.2.2.1.1.1.cmml"><mi id="S5.I2.i1.p1.1.m1.2.2.2.2.2.1.1.1.2" xref="S5.I2.i1.p1.1.m1.2.2.2.2.2.1.1.1.2.cmml">x</mi><mo id="S5.I2.i1.p1.1.m1.2.2.2.2.2.1.1.1.1" 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xref="S5.I2.i1.p1.1.m1.2.2.2.2.2.1.2.2.6">π</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.1.m1.2c">h\in\operatorname{cl}_{\beta S}(\{x\in S:xx^{-1}<e\hbox{ and }x^{-1}x\leq f\})</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.1.m1.2d">italic_h β roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_S end_POSTSUBSCRIPT ( { italic_x β italic_S : italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT < italic_e and italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x β€ italic_f } )</annotation></semantics></math>;</p> </div> </li> <li class="ltx_item" id="S5.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S5.I2.i2.p1"> <p class="ltx_p" id="S5.I2.i2.p1.1"><math alttext="h\in\operatorname{cl}_{\beta S}(\{x\in S:xx^{-1}\leq e\hbox{ and }x^{-1}x<f\})" class="ltx_Math" display="inline" id="S5.I2.i2.p1.1.m1.2"><semantics 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id="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.3a.cmml" xref="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.3"><mtext id="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.3.cmml" xref="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.3"> and </mtext></ci><apply id="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.4.cmml" xref="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.4"><csymbol cd="ambiguous" id="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.4.1.cmml" xref="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.4">superscript</csymbol><ci id="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.4.2.cmml" xref="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.4.2">π₯</ci><apply id="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.4.3.cmml" xref="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.4.3"><minus id="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.4.3.1.cmml" xref="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.4.3"></minus><cn id="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.4.3.2.cmml" type="integer" xref="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.4.3.2">1</cn></apply></apply><ci id="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.5.cmml" xref="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.5">π₯</ci></apply></apply><apply id="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2c.cmml" xref="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2"><lt id="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.5.cmml" xref="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.5"></lt><share href="https://arxiv.org/html/2503.13666v1#S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.4.cmml" id="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2d.cmml" xref="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2"></share><ci id="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.6.cmml" xref="S5.I2.i2.p1.1.m1.2.2.2.2.2.1.2.2.6">π</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.1.m1.2c">h\in\operatorname{cl}_{\beta S}(\{x\in S:xx^{-1}\leq e\hbox{ and }x^{-1}x<f\})</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.1.m1.2d">italic_h β roman_cl start_POSTSUBSCRIPT italic_Ξ² italic_S end_POSTSUBSCRIPT ( { italic_x β italic_S : italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT β€ italic_e and italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x < italic_f } )</annotation></semantics></math>.</p> </div> </li> </ol> </div> <div class="ltx_para" id="S5.4.p4"> <p class="ltx_p" id="S5.4.p4.32">(i) Since <math alttext="e=hh^{-1}" class="ltx_Math" display="inline" id="S5.4.p4.1.m1.1"><semantics id="S5.4.p4.1.m1.1a"><mrow id="S5.4.p4.1.m1.1.1" xref="S5.4.p4.1.m1.1.1.cmml"><mi id="S5.4.p4.1.m1.1.1.2" xref="S5.4.p4.1.m1.1.1.2.cmml">e</mi><mo id="S5.4.p4.1.m1.1.1.1" xref="S5.4.p4.1.m1.1.1.1.cmml">=</mo><mrow id="S5.4.p4.1.m1.1.1.3" xref="S5.4.p4.1.m1.1.1.3.cmml"><mi id="S5.4.p4.1.m1.1.1.3.2" xref="S5.4.p4.1.m1.1.1.3.2.cmml">h</mi><mo id="S5.4.p4.1.m1.1.1.3.1" xref="S5.4.p4.1.m1.1.1.3.1.cmml">β’</mo><msup id="S5.4.p4.1.m1.1.1.3.3" xref="S5.4.p4.1.m1.1.1.3.3.cmml"><mi id="S5.4.p4.1.m1.1.1.3.3.2" xref="S5.4.p4.1.m1.1.1.3.3.2.cmml">h</mi><mrow id="S5.4.p4.1.m1.1.1.3.3.3" xref="S5.4.p4.1.m1.1.1.3.3.3.cmml"><mo id="S5.4.p4.1.m1.1.1.3.3.3a" xref="S5.4.p4.1.m1.1.1.3.3.3.cmml">β</mo><mn id="S5.4.p4.1.m1.1.1.3.3.3.2" xref="S5.4.p4.1.m1.1.1.3.3.3.2.cmml">1</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.1.m1.1b"><apply id="S5.4.p4.1.m1.1.1.cmml" xref="S5.4.p4.1.m1.1.1"><eq id="S5.4.p4.1.m1.1.1.1.cmml" xref="S5.4.p4.1.m1.1.1.1"></eq><ci id="S5.4.p4.1.m1.1.1.2.cmml" xref="S5.4.p4.1.m1.1.1.2">π</ci><apply id="S5.4.p4.1.m1.1.1.3.cmml" xref="S5.4.p4.1.m1.1.1.3"><times id="S5.4.p4.1.m1.1.1.3.1.cmml" xref="S5.4.p4.1.m1.1.1.3.1"></times><ci id="S5.4.p4.1.m1.1.1.3.2.cmml" xref="S5.4.p4.1.m1.1.1.3.2">β</ci><apply id="S5.4.p4.1.m1.1.1.3.3.cmml" xref="S5.4.p4.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.4.p4.1.m1.1.1.3.3.1.cmml" xref="S5.4.p4.1.m1.1.1.3.3">superscript</csymbol><ci id="S5.4.p4.1.m1.1.1.3.3.2.cmml" xref="S5.4.p4.1.m1.1.1.3.3.2">β</ci><apply id="S5.4.p4.1.m1.1.1.3.3.3.cmml" xref="S5.4.p4.1.m1.1.1.3.3.3"><minus id="S5.4.p4.1.m1.1.1.3.3.3.1.cmml" xref="S5.4.p4.1.m1.1.1.3.3.3"></minus><cn id="S5.4.p4.1.m1.1.1.3.3.3.2.cmml" type="integer" xref="S5.4.p4.1.m1.1.1.3.3.3.2">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.1.m1.1c">e=hh^{-1}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.1.m1.1d">italic_e = italic_h italic_h start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\beta S" class="ltx_Math" display="inline" id="S5.4.p4.2.m2.1"><semantics id="S5.4.p4.2.m2.1a"><mrow id="S5.4.p4.2.m2.1.1" xref="S5.4.p4.2.m2.1.1.cmml"><mi id="S5.4.p4.2.m2.1.1.2" xref="S5.4.p4.2.m2.1.1.2.cmml">Ξ²</mi><mo id="S5.4.p4.2.m2.1.1.1" xref="S5.4.p4.2.m2.1.1.1.cmml">β’</mo><mi id="S5.4.p4.2.m2.1.1.3" xref="S5.4.p4.2.m2.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.2.m2.1b"><apply id="S5.4.p4.2.m2.1.1.cmml" xref="S5.4.p4.2.m2.1.1"><times id="S5.4.p4.2.m2.1.1.1.cmml" xref="S5.4.p4.2.m2.1.1.1"></times><ci id="S5.4.p4.2.m2.1.1.2.cmml" xref="S5.4.p4.2.m2.1.1.2">π½</ci><ci id="S5.4.p4.2.m2.1.1.3.cmml" xref="S5.4.p4.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.2.m2.1c">\beta S</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.2.m2.1d">italic_Ξ² italic_S</annotation></semantics></math> is a topological inverse semigroup, for each open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S5.4.p4.3.m3.1"><semantics id="S5.4.p4.3.m3.1a"><mi id="S5.4.p4.3.m3.1.1" xref="S5.4.p4.3.m3.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S5.4.p4.3.m3.1b"><ci id="S5.4.p4.3.m3.1.1.cmml" xref="S5.4.p4.3.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.3.m3.1c">U</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.3.m3.1d">italic_U</annotation></semantics></math> of <math alttext="e" class="ltx_Math" display="inline" id="S5.4.p4.4.m4.1"><semantics id="S5.4.p4.4.m4.1a"><mi id="S5.4.p4.4.m4.1.1" xref="S5.4.p4.4.m4.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S5.4.p4.4.m4.1b"><ci id="S5.4.p4.4.m4.1.1.cmml" xref="S5.4.p4.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.4.m4.1c">e</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.4.m4.1d">italic_e</annotation></semantics></math> there exists an open neighborhood <math alttext="V" class="ltx_Math" display="inline" id="S5.4.p4.5.m5.1"><semantics id="S5.4.p4.5.m5.1a"><mi id="S5.4.p4.5.m5.1.1" xref="S5.4.p4.5.m5.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S5.4.p4.5.m5.1b"><ci id="S5.4.p4.5.m5.1.1.cmml" xref="S5.4.p4.5.m5.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.5.m5.1c">V</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.5.m5.1d">italic_V</annotation></semantics></math> of <math alttext="h" class="ltx_Math" display="inline" id="S5.4.p4.6.m6.1"><semantics id="S5.4.p4.6.m6.1a"><mi id="S5.4.p4.6.m6.1.1" xref="S5.4.p4.6.m6.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S5.4.p4.6.m6.1b"><ci id="S5.4.p4.6.m6.1.1.cmml" xref="S5.4.p4.6.m6.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.6.m6.1c">h</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.6.m6.1d">italic_h</annotation></semantics></math> such that <math alttext="VV^{-1}\subseteq U" class="ltx_Math" display="inline" id="S5.4.p4.7.m7.1"><semantics id="S5.4.p4.7.m7.1a"><mrow id="S5.4.p4.7.m7.1.1" xref="S5.4.p4.7.m7.1.1.cmml"><mrow id="S5.4.p4.7.m7.1.1.2" xref="S5.4.p4.7.m7.1.1.2.cmml"><mi id="S5.4.p4.7.m7.1.1.2.2" xref="S5.4.p4.7.m7.1.1.2.2.cmml">V</mi><mo id="S5.4.p4.7.m7.1.1.2.1" xref="S5.4.p4.7.m7.1.1.2.1.cmml">β’</mo><msup id="S5.4.p4.7.m7.1.1.2.3" xref="S5.4.p4.7.m7.1.1.2.3.cmml"><mi id="S5.4.p4.7.m7.1.1.2.3.2" xref="S5.4.p4.7.m7.1.1.2.3.2.cmml">V</mi><mrow id="S5.4.p4.7.m7.1.1.2.3.3" xref="S5.4.p4.7.m7.1.1.2.3.3.cmml"><mo id="S5.4.p4.7.m7.1.1.2.3.3a" xref="S5.4.p4.7.m7.1.1.2.3.3.cmml">β</mo><mn id="S5.4.p4.7.m7.1.1.2.3.3.2" xref="S5.4.p4.7.m7.1.1.2.3.3.2.cmml">1</mn></mrow></msup></mrow><mo id="S5.4.p4.7.m7.1.1.1" xref="S5.4.p4.7.m7.1.1.1.cmml">β</mo><mi id="S5.4.p4.7.m7.1.1.3" xref="S5.4.p4.7.m7.1.1.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.7.m7.1b"><apply id="S5.4.p4.7.m7.1.1.cmml" xref="S5.4.p4.7.m7.1.1"><subset id="S5.4.p4.7.m7.1.1.1.cmml" xref="S5.4.p4.7.m7.1.1.1"></subset><apply id="S5.4.p4.7.m7.1.1.2.cmml" xref="S5.4.p4.7.m7.1.1.2"><times id="S5.4.p4.7.m7.1.1.2.1.cmml" xref="S5.4.p4.7.m7.1.1.2.1"></times><ci id="S5.4.p4.7.m7.1.1.2.2.cmml" xref="S5.4.p4.7.m7.1.1.2.2">π</ci><apply id="S5.4.p4.7.m7.1.1.2.3.cmml" xref="S5.4.p4.7.m7.1.1.2.3"><csymbol cd="ambiguous" id="S5.4.p4.7.m7.1.1.2.3.1.cmml" xref="S5.4.p4.7.m7.1.1.2.3">superscript</csymbol><ci id="S5.4.p4.7.m7.1.1.2.3.2.cmml" xref="S5.4.p4.7.m7.1.1.2.3.2">π</ci><apply id="S5.4.p4.7.m7.1.1.2.3.3.cmml" xref="S5.4.p4.7.m7.1.1.2.3.3"><minus id="S5.4.p4.7.m7.1.1.2.3.3.1.cmml" xref="S5.4.p4.7.m7.1.1.2.3.3"></minus><cn id="S5.4.p4.7.m7.1.1.2.3.3.2.cmml" type="integer" xref="S5.4.p4.7.m7.1.1.2.3.3.2">1</cn></apply></apply></apply><ci id="S5.4.p4.7.m7.1.1.3.cmml" xref="S5.4.p4.7.m7.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.7.m7.1c">VV^{-1}\subseteq U</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.7.m7.1d">italic_V italic_V start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT β italic_U</annotation></semantics></math>. By the assumption, there is <math alttext="x\in V" class="ltx_Math" display="inline" id="S5.4.p4.8.m8.1"><semantics id="S5.4.p4.8.m8.1a"><mrow id="S5.4.p4.8.m8.1.1" xref="S5.4.p4.8.m8.1.1.cmml"><mi id="S5.4.p4.8.m8.1.1.2" xref="S5.4.p4.8.m8.1.1.2.cmml">x</mi><mo id="S5.4.p4.8.m8.1.1.1" xref="S5.4.p4.8.m8.1.1.1.cmml">β</mo><mi id="S5.4.p4.8.m8.1.1.3" xref="S5.4.p4.8.m8.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.8.m8.1b"><apply id="S5.4.p4.8.m8.1.1.cmml" xref="S5.4.p4.8.m8.1.1"><in id="S5.4.p4.8.m8.1.1.1.cmml" xref="S5.4.p4.8.m8.1.1.1"></in><ci id="S5.4.p4.8.m8.1.1.2.cmml" xref="S5.4.p4.8.m8.1.1.2">π₯</ci><ci id="S5.4.p4.8.m8.1.1.3.cmml" xref="S5.4.p4.8.m8.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.8.m8.1c">x\in V</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.8.m8.1d">italic_x β italic_V</annotation></semantics></math> such that <math alttext="xx^{-1}<e" class="ltx_Math" display="inline" id="S5.4.p4.9.m9.1"><semantics id="S5.4.p4.9.m9.1a"><mrow id="S5.4.p4.9.m9.1.1" xref="S5.4.p4.9.m9.1.1.cmml"><mrow id="S5.4.p4.9.m9.1.1.2" xref="S5.4.p4.9.m9.1.1.2.cmml"><mi id="S5.4.p4.9.m9.1.1.2.2" xref="S5.4.p4.9.m9.1.1.2.2.cmml">x</mi><mo id="S5.4.p4.9.m9.1.1.2.1" xref="S5.4.p4.9.m9.1.1.2.1.cmml">β’</mo><msup id="S5.4.p4.9.m9.1.1.2.3" xref="S5.4.p4.9.m9.1.1.2.3.cmml"><mi id="S5.4.p4.9.m9.1.1.2.3.2" xref="S5.4.p4.9.m9.1.1.2.3.2.cmml">x</mi><mrow id="S5.4.p4.9.m9.1.1.2.3.3" xref="S5.4.p4.9.m9.1.1.2.3.3.cmml"><mo id="S5.4.p4.9.m9.1.1.2.3.3a" xref="S5.4.p4.9.m9.1.1.2.3.3.cmml">β</mo><mn id="S5.4.p4.9.m9.1.1.2.3.3.2" xref="S5.4.p4.9.m9.1.1.2.3.3.2.cmml">1</mn></mrow></msup></mrow><mo id="S5.4.p4.9.m9.1.1.1" xref="S5.4.p4.9.m9.1.1.1.cmml"><</mo><mi id="S5.4.p4.9.m9.1.1.3" xref="S5.4.p4.9.m9.1.1.3.cmml">e</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.9.m9.1b"><apply id="S5.4.p4.9.m9.1.1.cmml" xref="S5.4.p4.9.m9.1.1"><lt id="S5.4.p4.9.m9.1.1.1.cmml" xref="S5.4.p4.9.m9.1.1.1"></lt><apply id="S5.4.p4.9.m9.1.1.2.cmml" xref="S5.4.p4.9.m9.1.1.2"><times id="S5.4.p4.9.m9.1.1.2.1.cmml" xref="S5.4.p4.9.m9.1.1.2.1"></times><ci id="S5.4.p4.9.m9.1.1.2.2.cmml" xref="S5.4.p4.9.m9.1.1.2.2">π₯</ci><apply id="S5.4.p4.9.m9.1.1.2.3.cmml" xref="S5.4.p4.9.m9.1.1.2.3"><csymbol cd="ambiguous" id="S5.4.p4.9.m9.1.1.2.3.1.cmml" xref="S5.4.p4.9.m9.1.1.2.3">superscript</csymbol><ci id="S5.4.p4.9.m9.1.1.2.3.2.cmml" xref="S5.4.p4.9.m9.1.1.2.3.2">π₯</ci><apply id="S5.4.p4.9.m9.1.1.2.3.3.cmml" xref="S5.4.p4.9.m9.1.1.2.3.3"><minus id="S5.4.p4.9.m9.1.1.2.3.3.1.cmml" xref="S5.4.p4.9.m9.1.1.2.3.3"></minus><cn id="S5.4.p4.9.m9.1.1.2.3.3.2.cmml" type="integer" xref="S5.4.p4.9.m9.1.1.2.3.3.2">1</cn></apply></apply></apply><ci id="S5.4.p4.9.m9.1.1.3.cmml" xref="S5.4.p4.9.m9.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.9.m9.1c">xx^{-1}<e</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.9.m9.1d">italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT < italic_e</annotation></semantics></math> and <math alttext="xx^{-1}\in U" class="ltx_Math" display="inline" id="S5.4.p4.10.m10.1"><semantics id="S5.4.p4.10.m10.1a"><mrow id="S5.4.p4.10.m10.1.1" xref="S5.4.p4.10.m10.1.1.cmml"><mrow id="S5.4.p4.10.m10.1.1.2" xref="S5.4.p4.10.m10.1.1.2.cmml"><mi id="S5.4.p4.10.m10.1.1.2.2" xref="S5.4.p4.10.m10.1.1.2.2.cmml">x</mi><mo id="S5.4.p4.10.m10.1.1.2.1" xref="S5.4.p4.10.m10.1.1.2.1.cmml">β’</mo><msup id="S5.4.p4.10.m10.1.1.2.3" xref="S5.4.p4.10.m10.1.1.2.3.cmml"><mi id="S5.4.p4.10.m10.1.1.2.3.2" xref="S5.4.p4.10.m10.1.1.2.3.2.cmml">x</mi><mrow id="S5.4.p4.10.m10.1.1.2.3.3" xref="S5.4.p4.10.m10.1.1.2.3.3.cmml"><mo id="S5.4.p4.10.m10.1.1.2.3.3a" xref="S5.4.p4.10.m10.1.1.2.3.3.cmml">β</mo><mn id="S5.4.p4.10.m10.1.1.2.3.3.2" xref="S5.4.p4.10.m10.1.1.2.3.3.2.cmml">1</mn></mrow></msup></mrow><mo id="S5.4.p4.10.m10.1.1.1" xref="S5.4.p4.10.m10.1.1.1.cmml">β</mo><mi id="S5.4.p4.10.m10.1.1.3" xref="S5.4.p4.10.m10.1.1.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.10.m10.1b"><apply id="S5.4.p4.10.m10.1.1.cmml" xref="S5.4.p4.10.m10.1.1"><in id="S5.4.p4.10.m10.1.1.1.cmml" xref="S5.4.p4.10.m10.1.1.1"></in><apply id="S5.4.p4.10.m10.1.1.2.cmml" xref="S5.4.p4.10.m10.1.1.2"><times id="S5.4.p4.10.m10.1.1.2.1.cmml" xref="S5.4.p4.10.m10.1.1.2.1"></times><ci id="S5.4.p4.10.m10.1.1.2.2.cmml" xref="S5.4.p4.10.m10.1.1.2.2">π₯</ci><apply id="S5.4.p4.10.m10.1.1.2.3.cmml" xref="S5.4.p4.10.m10.1.1.2.3"><csymbol cd="ambiguous" id="S5.4.p4.10.m10.1.1.2.3.1.cmml" xref="S5.4.p4.10.m10.1.1.2.3">superscript</csymbol><ci id="S5.4.p4.10.m10.1.1.2.3.2.cmml" xref="S5.4.p4.10.m10.1.1.2.3.2">π₯</ci><apply id="S5.4.p4.10.m10.1.1.2.3.3.cmml" xref="S5.4.p4.10.m10.1.1.2.3.3"><minus id="S5.4.p4.10.m10.1.1.2.3.3.1.cmml" xref="S5.4.p4.10.m10.1.1.2.3.3"></minus><cn id="S5.4.p4.10.m10.1.1.2.3.3.2.cmml" type="integer" xref="S5.4.p4.10.m10.1.1.2.3.3.2">1</cn></apply></apply></apply><ci id="S5.4.p4.10.m10.1.1.3.cmml" xref="S5.4.p4.10.m10.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.10.m10.1c">xx^{-1}\in U</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.10.m10.1d">italic_x italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT β italic_U</annotation></semantics></math>. It follows that <math alttext="e\in\operatorname{cl}_{S}({\downarrow}e\setminus\{e\})" class="ltx_Math" display="inline" id="S5.4.p4.11.m11.2"><semantics id="S5.4.p4.11.m11.2a"><mrow id="S5.4.p4.11.m11.2.2" xref="S5.4.p4.11.m11.2.2.cmml"><mi id="S5.4.p4.11.m11.2.2.3" xref="S5.4.p4.11.m11.2.2.3.cmml">e</mi><mo id="S5.4.p4.11.m11.2.2.2" xref="S5.4.p4.11.m11.2.2.2.cmml">β</mo><mrow id="S5.4.p4.11.m11.2.2.1" xref="S5.4.p4.11.m11.2.2.1.cmml"><msub id="S5.4.p4.11.m11.2.2.1.3" xref="S5.4.p4.11.m11.2.2.1.3.cmml"><mi id="S5.4.p4.11.m11.2.2.1.3.2" xref="S5.4.p4.11.m11.2.2.1.3.2.cmml">cl</mi><mi id="S5.4.p4.11.m11.2.2.1.3.3" xref="S5.4.p4.11.m11.2.2.1.3.3.cmml">S</mi></msub><mspace id="S5.4.p4.11.m11.2.2.1a" width="0.556em" xref="S5.4.p4.11.m11.2.2.1.cmml"></mspace><mrow id="S5.4.p4.11.m11.2.2.1.1.1" xref="S5.4.p4.11.m11.2.2.1.1.1.1.cmml"><mo id="S5.4.p4.11.m11.2.2.1.1.1.2" stretchy="false" xref="S5.4.p4.11.m11.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.4.p4.11.m11.2.2.1.1.1.1" xref="S5.4.p4.11.m11.2.2.1.1.1.1.cmml"><mi id="S5.4.p4.11.m11.2.2.1.1.1.1.2" xref="S5.4.p4.11.m11.2.2.1.1.1.1.2.cmml"></mi><mo id="S5.4.p4.11.m11.2.2.1.1.1.1.1" stretchy="false" xref="S5.4.p4.11.m11.2.2.1.1.1.1.1.cmml">β</mo><mrow id="S5.4.p4.11.m11.2.2.1.1.1.1.3" xref="S5.4.p4.11.m11.2.2.1.1.1.1.3.cmml"><mi id="S5.4.p4.11.m11.2.2.1.1.1.1.3.2" xref="S5.4.p4.11.m11.2.2.1.1.1.1.3.2.cmml">e</mi><mo id="S5.4.p4.11.m11.2.2.1.1.1.1.3.1" xref="S5.4.p4.11.m11.2.2.1.1.1.1.3.1.cmml">β</mo><mrow id="S5.4.p4.11.m11.2.2.1.1.1.1.3.3.2" xref="S5.4.p4.11.m11.2.2.1.1.1.1.3.3.1.cmml"><mo id="S5.4.p4.11.m11.2.2.1.1.1.1.3.3.2.1" stretchy="false" xref="S5.4.p4.11.m11.2.2.1.1.1.1.3.3.1.cmml">{</mo><mi id="S5.4.p4.11.m11.1.1" xref="S5.4.p4.11.m11.1.1.cmml">e</mi><mo id="S5.4.p4.11.m11.2.2.1.1.1.1.3.3.2.2" stretchy="false" xref="S5.4.p4.11.m11.2.2.1.1.1.1.3.3.1.cmml">}</mo></mrow></mrow></mrow><mo id="S5.4.p4.11.m11.2.2.1.1.1.3" stretchy="false" xref="S5.4.p4.11.m11.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.11.m11.2b"><apply id="S5.4.p4.11.m11.2.2.cmml" xref="S5.4.p4.11.m11.2.2"><in id="S5.4.p4.11.m11.2.2.2.cmml" xref="S5.4.p4.11.m11.2.2.2"></in><ci id="S5.4.p4.11.m11.2.2.3.cmml" xref="S5.4.p4.11.m11.2.2.3">π</ci><apply id="S5.4.p4.11.m11.2.2.1.cmml" xref="S5.4.p4.11.m11.2.2.1"><csymbol cd="latexml" id="S5.4.p4.11.m11.2.2.1.2.cmml" xref="S5.4.p4.11.m11.2.2.1">annotated</csymbol><apply id="S5.4.p4.11.m11.2.2.1.3.cmml" xref="S5.4.p4.11.m11.2.2.1.3"><csymbol cd="ambiguous" id="S5.4.p4.11.m11.2.2.1.3.1.cmml" xref="S5.4.p4.11.m11.2.2.1.3">subscript</csymbol><ci id="S5.4.p4.11.m11.2.2.1.3.2.cmml" xref="S5.4.p4.11.m11.2.2.1.3.2">cl</ci><ci id="S5.4.p4.11.m11.2.2.1.3.3.cmml" xref="S5.4.p4.11.m11.2.2.1.3.3">π</ci></apply><apply id="S5.4.p4.11.m11.2.2.1.1.1.1.cmml" xref="S5.4.p4.11.m11.2.2.1.1.1"><ci id="S5.4.p4.11.m11.2.2.1.1.1.1.1.cmml" xref="S5.4.p4.11.m11.2.2.1.1.1.1.1">β</ci><csymbol cd="latexml" id="S5.4.p4.11.m11.2.2.1.1.1.1.2.cmml" xref="S5.4.p4.11.m11.2.2.1.1.1.1.2">absent</csymbol><apply id="S5.4.p4.11.m11.2.2.1.1.1.1.3.cmml" xref="S5.4.p4.11.m11.2.2.1.1.1.1.3"><setdiff id="S5.4.p4.11.m11.2.2.1.1.1.1.3.1.cmml" xref="S5.4.p4.11.m11.2.2.1.1.1.1.3.1"></setdiff><ci id="S5.4.p4.11.m11.2.2.1.1.1.1.3.2.cmml" xref="S5.4.p4.11.m11.2.2.1.1.1.1.3.2">π</ci><set id="S5.4.p4.11.m11.2.2.1.1.1.1.3.3.1.cmml" xref="S5.4.p4.11.m11.2.2.1.1.1.1.3.3.2"><ci id="S5.4.p4.11.m11.1.1.cmml" xref="S5.4.p4.11.m11.1.1">π</ci></set></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.11.m11.2c">e\in\operatorname{cl}_{S}({\downarrow}e\setminus\{e\})</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.11.m11.2d">italic_e β roman_cl start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( β italic_e β { italic_e } )</annotation></semantics></math>. Since <math alttext="S" class="ltx_Math" display="inline" id="S5.4.p4.12.m12.1"><semantics id="S5.4.p4.12.m12.1a"><mi id="S5.4.p4.12.m12.1.1" xref="S5.4.p4.12.m12.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.4.p4.12.m12.1b"><ci id="S5.4.p4.12.m12.1.1.cmml" xref="S5.4.p4.12.m12.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.12.m12.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.12.m12.1d">italic_S</annotation></semantics></math> is first-countable, there exists a sequence <math alttext="\{e_{n}:n\in\omega\}\subseteq{\downarrow}e\setminus\{e\}" class="ltx_math_unparsed" display="inline" id="S5.4.p4.13.m13.1"><semantics id="S5.4.p4.13.m13.1a"><mrow id="S5.4.p4.13.m13.1b"><mrow id="S5.4.p4.13.m13.1.1"><mo id="S5.4.p4.13.m13.1.1.1" stretchy="false">{</mo><msub id="S5.4.p4.13.m13.1.1.2"><mi id="S5.4.p4.13.m13.1.1.2.2">e</mi><mi id="S5.4.p4.13.m13.1.1.2.3">n</mi></msub><mo id="S5.4.p4.13.m13.1.1.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S5.4.p4.13.m13.1.1.4">n</mi><mo id="S5.4.p4.13.m13.1.1.5">β</mo><mi id="S5.4.p4.13.m13.1.1.6">Ο</mi><mo id="S5.4.p4.13.m13.1.1.7" stretchy="false">}</mo></mrow><mo id="S5.4.p4.13.m13.1.2" rspace="0em">β</mo><mo id="S5.4.p4.13.m13.1.3" lspace="0em" stretchy="false">β</mo><mi id="S5.4.p4.13.m13.1.4">e</mi><mo id="S5.4.p4.13.m13.1.5">β</mo><mrow id="S5.4.p4.13.m13.1.6"><mo id="S5.4.p4.13.m13.1.6.1" stretchy="false">{</mo><mi id="S5.4.p4.13.m13.1.6.2">e</mi><mo id="S5.4.p4.13.m13.1.6.3" stretchy="false">}</mo></mrow></mrow><annotation encoding="application/x-tex" id="S5.4.p4.13.m13.1c">\{e_{n}:n\in\omega\}\subseteq{\downarrow}e\setminus\{e\}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.13.m13.1d">{ italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n β italic_Ο } β β italic_e β { italic_e }</annotation></semantics></math> that converges to <math alttext="e" class="ltx_Math" display="inline" id="S5.4.p4.14.m14.1"><semantics id="S5.4.p4.14.m14.1a"><mi id="S5.4.p4.14.m14.1.1" xref="S5.4.p4.14.m14.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S5.4.p4.14.m14.1b"><ci id="S5.4.p4.14.m14.1.1.cmml" xref="S5.4.p4.14.m14.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.14.m14.1c">e</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.14.m14.1d">italic_e</annotation></semantics></math>. Since <math alttext="eh=h" class="ltx_Math" display="inline" id="S5.4.p4.15.m15.1"><semantics id="S5.4.p4.15.m15.1a"><mrow id="S5.4.p4.15.m15.1.1" xref="S5.4.p4.15.m15.1.1.cmml"><mrow id="S5.4.p4.15.m15.1.1.2" xref="S5.4.p4.15.m15.1.1.2.cmml"><mi id="S5.4.p4.15.m15.1.1.2.2" xref="S5.4.p4.15.m15.1.1.2.2.cmml">e</mi><mo id="S5.4.p4.15.m15.1.1.2.1" xref="S5.4.p4.15.m15.1.1.2.1.cmml">β’</mo><mi id="S5.4.p4.15.m15.1.1.2.3" xref="S5.4.p4.15.m15.1.1.2.3.cmml">h</mi></mrow><mo id="S5.4.p4.15.m15.1.1.1" xref="S5.4.p4.15.m15.1.1.1.cmml">=</mo><mi id="S5.4.p4.15.m15.1.1.3" xref="S5.4.p4.15.m15.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.15.m15.1b"><apply id="S5.4.p4.15.m15.1.1.cmml" xref="S5.4.p4.15.m15.1.1"><eq id="S5.4.p4.15.m15.1.1.1.cmml" xref="S5.4.p4.15.m15.1.1.1"></eq><apply id="S5.4.p4.15.m15.1.1.2.cmml" xref="S5.4.p4.15.m15.1.1.2"><times id="S5.4.p4.15.m15.1.1.2.1.cmml" xref="S5.4.p4.15.m15.1.1.2.1"></times><ci id="S5.4.p4.15.m15.1.1.2.2.cmml" xref="S5.4.p4.15.m15.1.1.2.2">π</ci><ci id="S5.4.p4.15.m15.1.1.2.3.cmml" xref="S5.4.p4.15.m15.1.1.2.3">β</ci></apply><ci id="S5.4.p4.15.m15.1.1.3.cmml" xref="S5.4.p4.15.m15.1.1.3">β</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.15.m15.1c">eh=h</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.15.m15.1d">italic_e italic_h = italic_h</annotation></semantics></math> we get that the sequence <math alttext="\{e_{n}h:n\in\omega\}" class="ltx_Math" display="inline" id="S5.4.p4.16.m16.2"><semantics id="S5.4.p4.16.m16.2a"><mrow id="S5.4.p4.16.m16.2.2.2" xref="S5.4.p4.16.m16.2.2.3.cmml"><mo id="S5.4.p4.16.m16.2.2.2.3" stretchy="false" xref="S5.4.p4.16.m16.2.2.3.1.cmml">{</mo><mrow id="S5.4.p4.16.m16.1.1.1.1" xref="S5.4.p4.16.m16.1.1.1.1.cmml"><msub id="S5.4.p4.16.m16.1.1.1.1.2" xref="S5.4.p4.16.m16.1.1.1.1.2.cmml"><mi id="S5.4.p4.16.m16.1.1.1.1.2.2" xref="S5.4.p4.16.m16.1.1.1.1.2.2.cmml">e</mi><mi id="S5.4.p4.16.m16.1.1.1.1.2.3" xref="S5.4.p4.16.m16.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S5.4.p4.16.m16.1.1.1.1.1" xref="S5.4.p4.16.m16.1.1.1.1.1.cmml">β’</mo><mi id="S5.4.p4.16.m16.1.1.1.1.3" xref="S5.4.p4.16.m16.1.1.1.1.3.cmml">h</mi></mrow><mo id="S5.4.p4.16.m16.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S5.4.p4.16.m16.2.2.3.1.cmml">:</mo><mrow id="S5.4.p4.16.m16.2.2.2.2" xref="S5.4.p4.16.m16.2.2.2.2.cmml"><mi id="S5.4.p4.16.m16.2.2.2.2.2" xref="S5.4.p4.16.m16.2.2.2.2.2.cmml">n</mi><mo id="S5.4.p4.16.m16.2.2.2.2.1" xref="S5.4.p4.16.m16.2.2.2.2.1.cmml">β</mo><mi id="S5.4.p4.16.m16.2.2.2.2.3" xref="S5.4.p4.16.m16.2.2.2.2.3.cmml">Ο</mi></mrow><mo id="S5.4.p4.16.m16.2.2.2.5" stretchy="false" xref="S5.4.p4.16.m16.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.16.m16.2b"><apply id="S5.4.p4.16.m16.2.2.3.cmml" xref="S5.4.p4.16.m16.2.2.2"><csymbol cd="latexml" id="S5.4.p4.16.m16.2.2.3.1.cmml" xref="S5.4.p4.16.m16.2.2.2.3">conditional-set</csymbol><apply id="S5.4.p4.16.m16.1.1.1.1.cmml" xref="S5.4.p4.16.m16.1.1.1.1"><times id="S5.4.p4.16.m16.1.1.1.1.1.cmml" xref="S5.4.p4.16.m16.1.1.1.1.1"></times><apply id="S5.4.p4.16.m16.1.1.1.1.2.cmml" xref="S5.4.p4.16.m16.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.4.p4.16.m16.1.1.1.1.2.1.cmml" xref="S5.4.p4.16.m16.1.1.1.1.2">subscript</csymbol><ci id="S5.4.p4.16.m16.1.1.1.1.2.2.cmml" xref="S5.4.p4.16.m16.1.1.1.1.2.2">π</ci><ci id="S5.4.p4.16.m16.1.1.1.1.2.3.cmml" xref="S5.4.p4.16.m16.1.1.1.1.2.3">π</ci></apply><ci id="S5.4.p4.16.m16.1.1.1.1.3.cmml" xref="S5.4.p4.16.m16.1.1.1.1.3">β</ci></apply><apply id="S5.4.p4.16.m16.2.2.2.2.cmml" xref="S5.4.p4.16.m16.2.2.2.2"><in id="S5.4.p4.16.m16.2.2.2.2.1.cmml" xref="S5.4.p4.16.m16.2.2.2.2.1"></in><ci id="S5.4.p4.16.m16.2.2.2.2.2.cmml" xref="S5.4.p4.16.m16.2.2.2.2.2">π</ci><ci id="S5.4.p4.16.m16.2.2.2.2.3.cmml" xref="S5.4.p4.16.m16.2.2.2.2.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.16.m16.2c">\{e_{n}h:n\in\omega\}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.16.m16.2d">{ italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_h : italic_n β italic_Ο }</annotation></semantics></math> converges to <math alttext="h" class="ltx_Math" display="inline" id="S5.4.p4.17.m17.1"><semantics id="S5.4.p4.17.m17.1a"><mi id="S5.4.p4.17.m17.1.1" xref="S5.4.p4.17.m17.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S5.4.p4.17.m17.1b"><ci id="S5.4.p4.17.m17.1.1.cmml" xref="S5.4.p4.17.m17.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.17.m17.1c">h</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.17.m17.1d">italic_h</annotation></semantics></math>. Note that <math alttext="e_{n}h\neq h" class="ltx_Math" display="inline" id="S5.4.p4.18.m18.1"><semantics id="S5.4.p4.18.m18.1a"><mrow id="S5.4.p4.18.m18.1.1" xref="S5.4.p4.18.m18.1.1.cmml"><mrow id="S5.4.p4.18.m18.1.1.2" xref="S5.4.p4.18.m18.1.1.2.cmml"><msub id="S5.4.p4.18.m18.1.1.2.2" xref="S5.4.p4.18.m18.1.1.2.2.cmml"><mi id="S5.4.p4.18.m18.1.1.2.2.2" xref="S5.4.p4.18.m18.1.1.2.2.2.cmml">e</mi><mi id="S5.4.p4.18.m18.1.1.2.2.3" xref="S5.4.p4.18.m18.1.1.2.2.3.cmml">n</mi></msub><mo id="S5.4.p4.18.m18.1.1.2.1" xref="S5.4.p4.18.m18.1.1.2.1.cmml">β’</mo><mi id="S5.4.p4.18.m18.1.1.2.3" xref="S5.4.p4.18.m18.1.1.2.3.cmml">h</mi></mrow><mo id="S5.4.p4.18.m18.1.1.1" xref="S5.4.p4.18.m18.1.1.1.cmml">β </mo><mi id="S5.4.p4.18.m18.1.1.3" xref="S5.4.p4.18.m18.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.18.m18.1b"><apply id="S5.4.p4.18.m18.1.1.cmml" xref="S5.4.p4.18.m18.1.1"><neq id="S5.4.p4.18.m18.1.1.1.cmml" xref="S5.4.p4.18.m18.1.1.1"></neq><apply id="S5.4.p4.18.m18.1.1.2.cmml" xref="S5.4.p4.18.m18.1.1.2"><times id="S5.4.p4.18.m18.1.1.2.1.cmml" xref="S5.4.p4.18.m18.1.1.2.1"></times><apply id="S5.4.p4.18.m18.1.1.2.2.cmml" xref="S5.4.p4.18.m18.1.1.2.2"><csymbol cd="ambiguous" id="S5.4.p4.18.m18.1.1.2.2.1.cmml" xref="S5.4.p4.18.m18.1.1.2.2">subscript</csymbol><ci id="S5.4.p4.18.m18.1.1.2.2.2.cmml" xref="S5.4.p4.18.m18.1.1.2.2.2">π</ci><ci id="S5.4.p4.18.m18.1.1.2.2.3.cmml" xref="S5.4.p4.18.m18.1.1.2.2.3">π</ci></apply><ci id="S5.4.p4.18.m18.1.1.2.3.cmml" xref="S5.4.p4.18.m18.1.1.2.3">β</ci></apply><ci id="S5.4.p4.18.m18.1.1.3.cmml" xref="S5.4.p4.18.m18.1.1.3">β</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.18.m18.1c">e_{n}h\neq h</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.18.m18.1d">italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_h β italic_h</annotation></semantics></math> for each <math alttext="n\in\omega" class="ltx_Math" display="inline" id="S5.4.p4.19.m19.1"><semantics id="S5.4.p4.19.m19.1a"><mrow id="S5.4.p4.19.m19.1.1" xref="S5.4.p4.19.m19.1.1.cmml"><mi id="S5.4.p4.19.m19.1.1.2" xref="S5.4.p4.19.m19.1.1.2.cmml">n</mi><mo id="S5.4.p4.19.m19.1.1.1" xref="S5.4.p4.19.m19.1.1.1.cmml">β</mo><mi id="S5.4.p4.19.m19.1.1.3" xref="S5.4.p4.19.m19.1.1.3.cmml">Ο</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.19.m19.1b"><apply id="S5.4.p4.19.m19.1.1.cmml" xref="S5.4.p4.19.m19.1.1"><in id="S5.4.p4.19.m19.1.1.1.cmml" xref="S5.4.p4.19.m19.1.1.1"></in><ci id="S5.4.p4.19.m19.1.1.2.cmml" xref="S5.4.p4.19.m19.1.1.2">π</ci><ci id="S5.4.p4.19.m19.1.1.3.cmml" xref="S5.4.p4.19.m19.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.19.m19.1c">n\in\omega</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.19.m19.1d">italic_n β italic_Ο</annotation></semantics></math>, as otherwise <math alttext="e=hh^{-1}=e_{n}hh^{-1}e_{n}=e_{n}ee_{n}=e_{n}" class="ltx_Math" display="inline" id="S5.4.p4.20.m20.1"><semantics id="S5.4.p4.20.m20.1a"><mrow id="S5.4.p4.20.m20.1.1" xref="S5.4.p4.20.m20.1.1.cmml"><mi id="S5.4.p4.20.m20.1.1.2" xref="S5.4.p4.20.m20.1.1.2.cmml">e</mi><mo id="S5.4.p4.20.m20.1.1.3" xref="S5.4.p4.20.m20.1.1.3.cmml">=</mo><mrow id="S5.4.p4.20.m20.1.1.4" xref="S5.4.p4.20.m20.1.1.4.cmml"><mi id="S5.4.p4.20.m20.1.1.4.2" xref="S5.4.p4.20.m20.1.1.4.2.cmml">h</mi><mo id="S5.4.p4.20.m20.1.1.4.1" xref="S5.4.p4.20.m20.1.1.4.1.cmml">β’</mo><msup id="S5.4.p4.20.m20.1.1.4.3" xref="S5.4.p4.20.m20.1.1.4.3.cmml"><mi id="S5.4.p4.20.m20.1.1.4.3.2" xref="S5.4.p4.20.m20.1.1.4.3.2.cmml">h</mi><mrow id="S5.4.p4.20.m20.1.1.4.3.3" xref="S5.4.p4.20.m20.1.1.4.3.3.cmml"><mo id="S5.4.p4.20.m20.1.1.4.3.3a" xref="S5.4.p4.20.m20.1.1.4.3.3.cmml">β</mo><mn id="S5.4.p4.20.m20.1.1.4.3.3.2" xref="S5.4.p4.20.m20.1.1.4.3.3.2.cmml">1</mn></mrow></msup></mrow><mo id="S5.4.p4.20.m20.1.1.5" xref="S5.4.p4.20.m20.1.1.5.cmml">=</mo><mrow id="S5.4.p4.20.m20.1.1.6" xref="S5.4.p4.20.m20.1.1.6.cmml"><msub id="S5.4.p4.20.m20.1.1.6.2" xref="S5.4.p4.20.m20.1.1.6.2.cmml"><mi id="S5.4.p4.20.m20.1.1.6.2.2" xref="S5.4.p4.20.m20.1.1.6.2.2.cmml">e</mi><mi id="S5.4.p4.20.m20.1.1.6.2.3" xref="S5.4.p4.20.m20.1.1.6.2.3.cmml">n</mi></msub><mo id="S5.4.p4.20.m20.1.1.6.1" xref="S5.4.p4.20.m20.1.1.6.1.cmml">β’</mo><mi id="S5.4.p4.20.m20.1.1.6.3" xref="S5.4.p4.20.m20.1.1.6.3.cmml">h</mi><mo id="S5.4.p4.20.m20.1.1.6.1a" xref="S5.4.p4.20.m20.1.1.6.1.cmml">β’</mo><msup id="S5.4.p4.20.m20.1.1.6.4" xref="S5.4.p4.20.m20.1.1.6.4.cmml"><mi id="S5.4.p4.20.m20.1.1.6.4.2" xref="S5.4.p4.20.m20.1.1.6.4.2.cmml">h</mi><mrow id="S5.4.p4.20.m20.1.1.6.4.3" xref="S5.4.p4.20.m20.1.1.6.4.3.cmml"><mo id="S5.4.p4.20.m20.1.1.6.4.3a" xref="S5.4.p4.20.m20.1.1.6.4.3.cmml">β</mo><mn id="S5.4.p4.20.m20.1.1.6.4.3.2" xref="S5.4.p4.20.m20.1.1.6.4.3.2.cmml">1</mn></mrow></msup><mo id="S5.4.p4.20.m20.1.1.6.1b" xref="S5.4.p4.20.m20.1.1.6.1.cmml">β’</mo><msub id="S5.4.p4.20.m20.1.1.6.5" xref="S5.4.p4.20.m20.1.1.6.5.cmml"><mi id="S5.4.p4.20.m20.1.1.6.5.2" xref="S5.4.p4.20.m20.1.1.6.5.2.cmml">e</mi><mi id="S5.4.p4.20.m20.1.1.6.5.3" xref="S5.4.p4.20.m20.1.1.6.5.3.cmml">n</mi></msub></mrow><mo id="S5.4.p4.20.m20.1.1.7" xref="S5.4.p4.20.m20.1.1.7.cmml">=</mo><mrow id="S5.4.p4.20.m20.1.1.8" xref="S5.4.p4.20.m20.1.1.8.cmml"><msub id="S5.4.p4.20.m20.1.1.8.2" xref="S5.4.p4.20.m20.1.1.8.2.cmml"><mi id="S5.4.p4.20.m20.1.1.8.2.2" xref="S5.4.p4.20.m20.1.1.8.2.2.cmml">e</mi><mi id="S5.4.p4.20.m20.1.1.8.2.3" xref="S5.4.p4.20.m20.1.1.8.2.3.cmml">n</mi></msub><mo id="S5.4.p4.20.m20.1.1.8.1" xref="S5.4.p4.20.m20.1.1.8.1.cmml">β’</mo><mi id="S5.4.p4.20.m20.1.1.8.3" xref="S5.4.p4.20.m20.1.1.8.3.cmml">e</mi><mo id="S5.4.p4.20.m20.1.1.8.1a" xref="S5.4.p4.20.m20.1.1.8.1.cmml">β’</mo><msub id="S5.4.p4.20.m20.1.1.8.4" xref="S5.4.p4.20.m20.1.1.8.4.cmml"><mi id="S5.4.p4.20.m20.1.1.8.4.2" xref="S5.4.p4.20.m20.1.1.8.4.2.cmml">e</mi><mi id="S5.4.p4.20.m20.1.1.8.4.3" xref="S5.4.p4.20.m20.1.1.8.4.3.cmml">n</mi></msub></mrow><mo id="S5.4.p4.20.m20.1.1.9" xref="S5.4.p4.20.m20.1.1.9.cmml">=</mo><msub id="S5.4.p4.20.m20.1.1.10" xref="S5.4.p4.20.m20.1.1.10.cmml"><mi id="S5.4.p4.20.m20.1.1.10.2" xref="S5.4.p4.20.m20.1.1.10.2.cmml">e</mi><mi id="S5.4.p4.20.m20.1.1.10.3" xref="S5.4.p4.20.m20.1.1.10.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.20.m20.1b"><apply id="S5.4.p4.20.m20.1.1.cmml" xref="S5.4.p4.20.m20.1.1"><and id="S5.4.p4.20.m20.1.1a.cmml" xref="S5.4.p4.20.m20.1.1"></and><apply id="S5.4.p4.20.m20.1.1b.cmml" xref="S5.4.p4.20.m20.1.1"><eq id="S5.4.p4.20.m20.1.1.3.cmml" xref="S5.4.p4.20.m20.1.1.3"></eq><ci id="S5.4.p4.20.m20.1.1.2.cmml" xref="S5.4.p4.20.m20.1.1.2">π</ci><apply id="S5.4.p4.20.m20.1.1.4.cmml" xref="S5.4.p4.20.m20.1.1.4"><times id="S5.4.p4.20.m20.1.1.4.1.cmml" xref="S5.4.p4.20.m20.1.1.4.1"></times><ci id="S5.4.p4.20.m20.1.1.4.2.cmml" xref="S5.4.p4.20.m20.1.1.4.2">β</ci><apply id="S5.4.p4.20.m20.1.1.4.3.cmml" xref="S5.4.p4.20.m20.1.1.4.3"><csymbol cd="ambiguous" id="S5.4.p4.20.m20.1.1.4.3.1.cmml" xref="S5.4.p4.20.m20.1.1.4.3">superscript</csymbol><ci id="S5.4.p4.20.m20.1.1.4.3.2.cmml" xref="S5.4.p4.20.m20.1.1.4.3.2">β</ci><apply id="S5.4.p4.20.m20.1.1.4.3.3.cmml" xref="S5.4.p4.20.m20.1.1.4.3.3"><minus id="S5.4.p4.20.m20.1.1.4.3.3.1.cmml" xref="S5.4.p4.20.m20.1.1.4.3.3"></minus><cn id="S5.4.p4.20.m20.1.1.4.3.3.2.cmml" type="integer" xref="S5.4.p4.20.m20.1.1.4.3.3.2">1</cn></apply></apply></apply></apply><apply id="S5.4.p4.20.m20.1.1c.cmml" xref="S5.4.p4.20.m20.1.1"><eq id="S5.4.p4.20.m20.1.1.5.cmml" xref="S5.4.p4.20.m20.1.1.5"></eq><share href="https://arxiv.org/html/2503.13666v1#S5.4.p4.20.m20.1.1.4.cmml" id="S5.4.p4.20.m20.1.1d.cmml" xref="S5.4.p4.20.m20.1.1"></share><apply id="S5.4.p4.20.m20.1.1.6.cmml" xref="S5.4.p4.20.m20.1.1.6"><times id="S5.4.p4.20.m20.1.1.6.1.cmml" xref="S5.4.p4.20.m20.1.1.6.1"></times><apply id="S5.4.p4.20.m20.1.1.6.2.cmml" xref="S5.4.p4.20.m20.1.1.6.2"><csymbol cd="ambiguous" id="S5.4.p4.20.m20.1.1.6.2.1.cmml" xref="S5.4.p4.20.m20.1.1.6.2">subscript</csymbol><ci id="S5.4.p4.20.m20.1.1.6.2.2.cmml" xref="S5.4.p4.20.m20.1.1.6.2.2">π</ci><ci id="S5.4.p4.20.m20.1.1.6.2.3.cmml" xref="S5.4.p4.20.m20.1.1.6.2.3">π</ci></apply><ci id="S5.4.p4.20.m20.1.1.6.3.cmml" xref="S5.4.p4.20.m20.1.1.6.3">β</ci><apply id="S5.4.p4.20.m20.1.1.6.4.cmml" xref="S5.4.p4.20.m20.1.1.6.4"><csymbol cd="ambiguous" id="S5.4.p4.20.m20.1.1.6.4.1.cmml" xref="S5.4.p4.20.m20.1.1.6.4">superscript</csymbol><ci id="S5.4.p4.20.m20.1.1.6.4.2.cmml" xref="S5.4.p4.20.m20.1.1.6.4.2">β</ci><apply id="S5.4.p4.20.m20.1.1.6.4.3.cmml" xref="S5.4.p4.20.m20.1.1.6.4.3"><minus id="S5.4.p4.20.m20.1.1.6.4.3.1.cmml" xref="S5.4.p4.20.m20.1.1.6.4.3"></minus><cn id="S5.4.p4.20.m20.1.1.6.4.3.2.cmml" type="integer" xref="S5.4.p4.20.m20.1.1.6.4.3.2">1</cn></apply></apply><apply id="S5.4.p4.20.m20.1.1.6.5.cmml" xref="S5.4.p4.20.m20.1.1.6.5"><csymbol cd="ambiguous" id="S5.4.p4.20.m20.1.1.6.5.1.cmml" xref="S5.4.p4.20.m20.1.1.6.5">subscript</csymbol><ci id="S5.4.p4.20.m20.1.1.6.5.2.cmml" xref="S5.4.p4.20.m20.1.1.6.5.2">π</ci><ci id="S5.4.p4.20.m20.1.1.6.5.3.cmml" xref="S5.4.p4.20.m20.1.1.6.5.3">π</ci></apply></apply></apply><apply id="S5.4.p4.20.m20.1.1e.cmml" xref="S5.4.p4.20.m20.1.1"><eq id="S5.4.p4.20.m20.1.1.7.cmml" xref="S5.4.p4.20.m20.1.1.7"></eq><share href="https://arxiv.org/html/2503.13666v1#S5.4.p4.20.m20.1.1.6.cmml" id="S5.4.p4.20.m20.1.1f.cmml" xref="S5.4.p4.20.m20.1.1"></share><apply id="S5.4.p4.20.m20.1.1.8.cmml" xref="S5.4.p4.20.m20.1.1.8"><times id="S5.4.p4.20.m20.1.1.8.1.cmml" xref="S5.4.p4.20.m20.1.1.8.1"></times><apply id="S5.4.p4.20.m20.1.1.8.2.cmml" xref="S5.4.p4.20.m20.1.1.8.2"><csymbol cd="ambiguous" id="S5.4.p4.20.m20.1.1.8.2.1.cmml" xref="S5.4.p4.20.m20.1.1.8.2">subscript</csymbol><ci id="S5.4.p4.20.m20.1.1.8.2.2.cmml" xref="S5.4.p4.20.m20.1.1.8.2.2">π</ci><ci id="S5.4.p4.20.m20.1.1.8.2.3.cmml" xref="S5.4.p4.20.m20.1.1.8.2.3">π</ci></apply><ci id="S5.4.p4.20.m20.1.1.8.3.cmml" xref="S5.4.p4.20.m20.1.1.8.3">π</ci><apply id="S5.4.p4.20.m20.1.1.8.4.cmml" xref="S5.4.p4.20.m20.1.1.8.4"><csymbol cd="ambiguous" id="S5.4.p4.20.m20.1.1.8.4.1.cmml" xref="S5.4.p4.20.m20.1.1.8.4">subscript</csymbol><ci id="S5.4.p4.20.m20.1.1.8.4.2.cmml" xref="S5.4.p4.20.m20.1.1.8.4.2">π</ci><ci id="S5.4.p4.20.m20.1.1.8.4.3.cmml" xref="S5.4.p4.20.m20.1.1.8.4.3">π</ci></apply></apply></apply><apply id="S5.4.p4.20.m20.1.1g.cmml" xref="S5.4.p4.20.m20.1.1"><eq id="S5.4.p4.20.m20.1.1.9.cmml" xref="S5.4.p4.20.m20.1.1.9"></eq><share href="https://arxiv.org/html/2503.13666v1#S5.4.p4.20.m20.1.1.8.cmml" id="S5.4.p4.20.m20.1.1h.cmml" xref="S5.4.p4.20.m20.1.1"></share><apply id="S5.4.p4.20.m20.1.1.10.cmml" xref="S5.4.p4.20.m20.1.1.10"><csymbol cd="ambiguous" id="S5.4.p4.20.m20.1.1.10.1.cmml" xref="S5.4.p4.20.m20.1.1.10">subscript</csymbol><ci id="S5.4.p4.20.m20.1.1.10.2.cmml" xref="S5.4.p4.20.m20.1.1.10.2">π</ci><ci id="S5.4.p4.20.m20.1.1.10.3.cmml" xref="S5.4.p4.20.m20.1.1.10.3">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.20.m20.1c">e=hh^{-1}=e_{n}hh^{-1}e_{n}=e_{n}ee_{n}=e_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.20.m20.1d">italic_e = italic_h italic_h start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_h italic_h start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_e italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> for some <math alttext="n\in\omega" class="ltx_Math" display="inline" id="S5.4.p4.21.m21.1"><semantics id="S5.4.p4.21.m21.1a"><mrow id="S5.4.p4.21.m21.1.1" xref="S5.4.p4.21.m21.1.1.cmml"><mi id="S5.4.p4.21.m21.1.1.2" xref="S5.4.p4.21.m21.1.1.2.cmml">n</mi><mo id="S5.4.p4.21.m21.1.1.1" xref="S5.4.p4.21.m21.1.1.1.cmml">β</mo><mi id="S5.4.p4.21.m21.1.1.3" xref="S5.4.p4.21.m21.1.1.3.cmml">Ο</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.21.m21.1b"><apply id="S5.4.p4.21.m21.1.1.cmml" xref="S5.4.p4.21.m21.1.1"><in id="S5.4.p4.21.m21.1.1.1.cmml" xref="S5.4.p4.21.m21.1.1.1"></in><ci id="S5.4.p4.21.m21.1.1.2.cmml" xref="S5.4.p4.21.m21.1.1.2">π</ci><ci id="S5.4.p4.21.m21.1.1.3.cmml" xref="S5.4.p4.21.m21.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.21.m21.1c">n\in\omega</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.21.m21.1d">italic_n β italic_Ο</annotation></semantics></math>, which contradicts the choice of <math alttext="e_{n}" class="ltx_Math" display="inline" id="S5.4.p4.22.m22.1"><semantics id="S5.4.p4.22.m22.1a"><msub id="S5.4.p4.22.m22.1.1" xref="S5.4.p4.22.m22.1.1.cmml"><mi id="S5.4.p4.22.m22.1.1.2" xref="S5.4.p4.22.m22.1.1.2.cmml">e</mi><mi id="S5.4.p4.22.m22.1.1.3" xref="S5.4.p4.22.m22.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S5.4.p4.22.m22.1b"><apply id="S5.4.p4.22.m22.1.1.cmml" xref="S5.4.p4.22.m22.1.1"><csymbol cd="ambiguous" id="S5.4.p4.22.m22.1.1.1.cmml" xref="S5.4.p4.22.m22.1.1">subscript</csymbol><ci id="S5.4.p4.22.m22.1.1.2.cmml" xref="S5.4.p4.22.m22.1.1.2">π</ci><ci id="S5.4.p4.22.m22.1.1.3.cmml" xref="S5.4.p4.22.m22.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.22.m22.1c">e_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.22.m22.1d">italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="e_{n}hh^{-1}e_{n}h=e_{n}h" class="ltx_Math" display="inline" id="S5.4.p4.23.m23.1"><semantics id="S5.4.p4.23.m23.1a"><mrow id="S5.4.p4.23.m23.1.1" xref="S5.4.p4.23.m23.1.1.cmml"><mrow id="S5.4.p4.23.m23.1.1.2" xref="S5.4.p4.23.m23.1.1.2.cmml"><msub id="S5.4.p4.23.m23.1.1.2.2" xref="S5.4.p4.23.m23.1.1.2.2.cmml"><mi id="S5.4.p4.23.m23.1.1.2.2.2" xref="S5.4.p4.23.m23.1.1.2.2.2.cmml">e</mi><mi id="S5.4.p4.23.m23.1.1.2.2.3" xref="S5.4.p4.23.m23.1.1.2.2.3.cmml">n</mi></msub><mo id="S5.4.p4.23.m23.1.1.2.1" xref="S5.4.p4.23.m23.1.1.2.1.cmml">β’</mo><mi id="S5.4.p4.23.m23.1.1.2.3" xref="S5.4.p4.23.m23.1.1.2.3.cmml">h</mi><mo id="S5.4.p4.23.m23.1.1.2.1a" xref="S5.4.p4.23.m23.1.1.2.1.cmml">β’</mo><msup id="S5.4.p4.23.m23.1.1.2.4" xref="S5.4.p4.23.m23.1.1.2.4.cmml"><mi id="S5.4.p4.23.m23.1.1.2.4.2" xref="S5.4.p4.23.m23.1.1.2.4.2.cmml">h</mi><mrow id="S5.4.p4.23.m23.1.1.2.4.3" xref="S5.4.p4.23.m23.1.1.2.4.3.cmml"><mo id="S5.4.p4.23.m23.1.1.2.4.3a" xref="S5.4.p4.23.m23.1.1.2.4.3.cmml">β</mo><mn id="S5.4.p4.23.m23.1.1.2.4.3.2" xref="S5.4.p4.23.m23.1.1.2.4.3.2.cmml">1</mn></mrow></msup><mo id="S5.4.p4.23.m23.1.1.2.1b" xref="S5.4.p4.23.m23.1.1.2.1.cmml">β’</mo><msub id="S5.4.p4.23.m23.1.1.2.5" xref="S5.4.p4.23.m23.1.1.2.5.cmml"><mi id="S5.4.p4.23.m23.1.1.2.5.2" xref="S5.4.p4.23.m23.1.1.2.5.2.cmml">e</mi><mi id="S5.4.p4.23.m23.1.1.2.5.3" xref="S5.4.p4.23.m23.1.1.2.5.3.cmml">n</mi></msub><mo id="S5.4.p4.23.m23.1.1.2.1c" xref="S5.4.p4.23.m23.1.1.2.1.cmml">β’</mo><mi id="S5.4.p4.23.m23.1.1.2.6" xref="S5.4.p4.23.m23.1.1.2.6.cmml">h</mi></mrow><mo id="S5.4.p4.23.m23.1.1.1" xref="S5.4.p4.23.m23.1.1.1.cmml">=</mo><mrow id="S5.4.p4.23.m23.1.1.3" xref="S5.4.p4.23.m23.1.1.3.cmml"><msub id="S5.4.p4.23.m23.1.1.3.2" xref="S5.4.p4.23.m23.1.1.3.2.cmml"><mi id="S5.4.p4.23.m23.1.1.3.2.2" xref="S5.4.p4.23.m23.1.1.3.2.2.cmml">e</mi><mi id="S5.4.p4.23.m23.1.1.3.2.3" xref="S5.4.p4.23.m23.1.1.3.2.3.cmml">n</mi></msub><mo id="S5.4.p4.23.m23.1.1.3.1" xref="S5.4.p4.23.m23.1.1.3.1.cmml">β’</mo><mi id="S5.4.p4.23.m23.1.1.3.3" xref="S5.4.p4.23.m23.1.1.3.3.cmml">h</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.23.m23.1b"><apply id="S5.4.p4.23.m23.1.1.cmml" xref="S5.4.p4.23.m23.1.1"><eq id="S5.4.p4.23.m23.1.1.1.cmml" xref="S5.4.p4.23.m23.1.1.1"></eq><apply id="S5.4.p4.23.m23.1.1.2.cmml" xref="S5.4.p4.23.m23.1.1.2"><times id="S5.4.p4.23.m23.1.1.2.1.cmml" xref="S5.4.p4.23.m23.1.1.2.1"></times><apply id="S5.4.p4.23.m23.1.1.2.2.cmml" xref="S5.4.p4.23.m23.1.1.2.2"><csymbol cd="ambiguous" id="S5.4.p4.23.m23.1.1.2.2.1.cmml" xref="S5.4.p4.23.m23.1.1.2.2">subscript</csymbol><ci id="S5.4.p4.23.m23.1.1.2.2.2.cmml" xref="S5.4.p4.23.m23.1.1.2.2.2">π</ci><ci id="S5.4.p4.23.m23.1.1.2.2.3.cmml" xref="S5.4.p4.23.m23.1.1.2.2.3">π</ci></apply><ci id="S5.4.p4.23.m23.1.1.2.3.cmml" xref="S5.4.p4.23.m23.1.1.2.3">β</ci><apply id="S5.4.p4.23.m23.1.1.2.4.cmml" xref="S5.4.p4.23.m23.1.1.2.4"><csymbol cd="ambiguous" id="S5.4.p4.23.m23.1.1.2.4.1.cmml" xref="S5.4.p4.23.m23.1.1.2.4">superscript</csymbol><ci id="S5.4.p4.23.m23.1.1.2.4.2.cmml" xref="S5.4.p4.23.m23.1.1.2.4.2">β</ci><apply id="S5.4.p4.23.m23.1.1.2.4.3.cmml" xref="S5.4.p4.23.m23.1.1.2.4.3"><minus id="S5.4.p4.23.m23.1.1.2.4.3.1.cmml" xref="S5.4.p4.23.m23.1.1.2.4.3"></minus><cn id="S5.4.p4.23.m23.1.1.2.4.3.2.cmml" type="integer" xref="S5.4.p4.23.m23.1.1.2.4.3.2">1</cn></apply></apply><apply id="S5.4.p4.23.m23.1.1.2.5.cmml" xref="S5.4.p4.23.m23.1.1.2.5"><csymbol cd="ambiguous" id="S5.4.p4.23.m23.1.1.2.5.1.cmml" xref="S5.4.p4.23.m23.1.1.2.5">subscript</csymbol><ci id="S5.4.p4.23.m23.1.1.2.5.2.cmml" xref="S5.4.p4.23.m23.1.1.2.5.2">π</ci><ci id="S5.4.p4.23.m23.1.1.2.5.3.cmml" xref="S5.4.p4.23.m23.1.1.2.5.3">π</ci></apply><ci id="S5.4.p4.23.m23.1.1.2.6.cmml" xref="S5.4.p4.23.m23.1.1.2.6">β</ci></apply><apply id="S5.4.p4.23.m23.1.1.3.cmml" xref="S5.4.p4.23.m23.1.1.3"><times id="S5.4.p4.23.m23.1.1.3.1.cmml" xref="S5.4.p4.23.m23.1.1.3.1"></times><apply id="S5.4.p4.23.m23.1.1.3.2.cmml" xref="S5.4.p4.23.m23.1.1.3.2"><csymbol cd="ambiguous" id="S5.4.p4.23.m23.1.1.3.2.1.cmml" xref="S5.4.p4.23.m23.1.1.3.2">subscript</csymbol><ci id="S5.4.p4.23.m23.1.1.3.2.2.cmml" xref="S5.4.p4.23.m23.1.1.3.2.2">π</ci><ci id="S5.4.p4.23.m23.1.1.3.2.3.cmml" xref="S5.4.p4.23.m23.1.1.3.2.3">π</ci></apply><ci id="S5.4.p4.23.m23.1.1.3.3.cmml" xref="S5.4.p4.23.m23.1.1.3.3">β</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.23.m23.1c">e_{n}hh^{-1}e_{n}h=e_{n}h</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.23.m23.1d">italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_h italic_h start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_h = italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_h</annotation></semantics></math>, we get that <math alttext="e_{n}h\leq h" class="ltx_Math" display="inline" id="S5.4.p4.24.m24.1"><semantics id="S5.4.p4.24.m24.1a"><mrow id="S5.4.p4.24.m24.1.1" xref="S5.4.p4.24.m24.1.1.cmml"><mrow id="S5.4.p4.24.m24.1.1.2" xref="S5.4.p4.24.m24.1.1.2.cmml"><msub id="S5.4.p4.24.m24.1.1.2.2" xref="S5.4.p4.24.m24.1.1.2.2.cmml"><mi id="S5.4.p4.24.m24.1.1.2.2.2" xref="S5.4.p4.24.m24.1.1.2.2.2.cmml">e</mi><mi id="S5.4.p4.24.m24.1.1.2.2.3" xref="S5.4.p4.24.m24.1.1.2.2.3.cmml">n</mi></msub><mo id="S5.4.p4.24.m24.1.1.2.1" xref="S5.4.p4.24.m24.1.1.2.1.cmml">β’</mo><mi id="S5.4.p4.24.m24.1.1.2.3" xref="S5.4.p4.24.m24.1.1.2.3.cmml">h</mi></mrow><mo id="S5.4.p4.24.m24.1.1.1" xref="S5.4.p4.24.m24.1.1.1.cmml">β€</mo><mi id="S5.4.p4.24.m24.1.1.3" xref="S5.4.p4.24.m24.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.24.m24.1b"><apply id="S5.4.p4.24.m24.1.1.cmml" xref="S5.4.p4.24.m24.1.1"><leq id="S5.4.p4.24.m24.1.1.1.cmml" xref="S5.4.p4.24.m24.1.1.1"></leq><apply id="S5.4.p4.24.m24.1.1.2.cmml" xref="S5.4.p4.24.m24.1.1.2"><times id="S5.4.p4.24.m24.1.1.2.1.cmml" xref="S5.4.p4.24.m24.1.1.2.1"></times><apply id="S5.4.p4.24.m24.1.1.2.2.cmml" xref="S5.4.p4.24.m24.1.1.2.2"><csymbol cd="ambiguous" id="S5.4.p4.24.m24.1.1.2.2.1.cmml" xref="S5.4.p4.24.m24.1.1.2.2">subscript</csymbol><ci id="S5.4.p4.24.m24.1.1.2.2.2.cmml" xref="S5.4.p4.24.m24.1.1.2.2.2">π</ci><ci id="S5.4.p4.24.m24.1.1.2.2.3.cmml" xref="S5.4.p4.24.m24.1.1.2.2.3">π</ci></apply><ci id="S5.4.p4.24.m24.1.1.2.3.cmml" xref="S5.4.p4.24.m24.1.1.2.3">β</ci></apply><ci id="S5.4.p4.24.m24.1.1.3.cmml" xref="S5.4.p4.24.m24.1.1.3">β</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.24.m24.1c">e_{n}h\leq h</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.24.m24.1d">italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_h β€ italic_h</annotation></semantics></math> with respect to the natural partial order on <math alttext="\beta S" class="ltx_Math" display="inline" id="S5.4.p4.25.m25.1"><semantics id="S5.4.p4.25.m25.1a"><mrow id="S5.4.p4.25.m25.1.1" xref="S5.4.p4.25.m25.1.1.cmml"><mi id="S5.4.p4.25.m25.1.1.2" xref="S5.4.p4.25.m25.1.1.2.cmml">Ξ²</mi><mo id="S5.4.p4.25.m25.1.1.1" xref="S5.4.p4.25.m25.1.1.1.cmml">β’</mo><mi id="S5.4.p4.25.m25.1.1.3" xref="S5.4.p4.25.m25.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.25.m25.1b"><apply id="S5.4.p4.25.m25.1.1.cmml" xref="S5.4.p4.25.m25.1.1"><times id="S5.4.p4.25.m25.1.1.1.cmml" xref="S5.4.p4.25.m25.1.1.1"></times><ci id="S5.4.p4.25.m25.1.1.2.cmml" xref="S5.4.p4.25.m25.1.1.2">π½</ci><ci id="S5.4.p4.25.m25.1.1.3.cmml" xref="S5.4.p4.25.m25.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.25.m25.1c">\beta S</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.25.m25.1d">italic_Ξ² italic_S</annotation></semantics></math>. Since <math alttext="h" class="ltx_Math" display="inline" id="S5.4.p4.26.m26.1"><semantics id="S5.4.p4.26.m26.1a"><mi id="S5.4.p4.26.m26.1.1" xref="S5.4.p4.26.m26.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S5.4.p4.26.m26.1b"><ci id="S5.4.p4.26.m26.1.1.cmml" xref="S5.4.p4.26.m26.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.26.m26.1c">h</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.26.m26.1d">italic_h</annotation></semantics></math> is a minimal element in <math alttext="\beta S\setminus S" class="ltx_Math" display="inline" id="S5.4.p4.27.m27.1"><semantics id="S5.4.p4.27.m27.1a"><mrow id="S5.4.p4.27.m27.1.1" xref="S5.4.p4.27.m27.1.1.cmml"><mrow id="S5.4.p4.27.m27.1.1.2" xref="S5.4.p4.27.m27.1.1.2.cmml"><mi id="S5.4.p4.27.m27.1.1.2.2" xref="S5.4.p4.27.m27.1.1.2.2.cmml">Ξ²</mi><mo id="S5.4.p4.27.m27.1.1.2.1" xref="S5.4.p4.27.m27.1.1.2.1.cmml">β’</mo><mi id="S5.4.p4.27.m27.1.1.2.3" xref="S5.4.p4.27.m27.1.1.2.3.cmml">S</mi></mrow><mo id="S5.4.p4.27.m27.1.1.1" xref="S5.4.p4.27.m27.1.1.1.cmml">β</mo><mi id="S5.4.p4.27.m27.1.1.3" xref="S5.4.p4.27.m27.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.27.m27.1b"><apply id="S5.4.p4.27.m27.1.1.cmml" xref="S5.4.p4.27.m27.1.1"><setdiff id="S5.4.p4.27.m27.1.1.1.cmml" xref="S5.4.p4.27.m27.1.1.1"></setdiff><apply id="S5.4.p4.27.m27.1.1.2.cmml" xref="S5.4.p4.27.m27.1.1.2"><times id="S5.4.p4.27.m27.1.1.2.1.cmml" xref="S5.4.p4.27.m27.1.1.2.1"></times><ci id="S5.4.p4.27.m27.1.1.2.2.cmml" xref="S5.4.p4.27.m27.1.1.2.2">π½</ci><ci id="S5.4.p4.27.m27.1.1.2.3.cmml" xref="S5.4.p4.27.m27.1.1.2.3">π</ci></apply><ci id="S5.4.p4.27.m27.1.1.3.cmml" xref="S5.4.p4.27.m27.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.27.m27.1c">\beta S\setminus S</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.27.m27.1d">italic_Ξ² italic_S β italic_S</annotation></semantics></math>, we get that <math alttext="e_{n}h\in S" class="ltx_Math" display="inline" id="S5.4.p4.28.m28.1"><semantics id="S5.4.p4.28.m28.1a"><mrow id="S5.4.p4.28.m28.1.1" xref="S5.4.p4.28.m28.1.1.cmml"><mrow id="S5.4.p4.28.m28.1.1.2" xref="S5.4.p4.28.m28.1.1.2.cmml"><msub id="S5.4.p4.28.m28.1.1.2.2" xref="S5.4.p4.28.m28.1.1.2.2.cmml"><mi id="S5.4.p4.28.m28.1.1.2.2.2" xref="S5.4.p4.28.m28.1.1.2.2.2.cmml">e</mi><mi id="S5.4.p4.28.m28.1.1.2.2.3" xref="S5.4.p4.28.m28.1.1.2.2.3.cmml">n</mi></msub><mo id="S5.4.p4.28.m28.1.1.2.1" xref="S5.4.p4.28.m28.1.1.2.1.cmml">β’</mo><mi id="S5.4.p4.28.m28.1.1.2.3" xref="S5.4.p4.28.m28.1.1.2.3.cmml">h</mi></mrow><mo id="S5.4.p4.28.m28.1.1.1" xref="S5.4.p4.28.m28.1.1.1.cmml">β</mo><mi id="S5.4.p4.28.m28.1.1.3" xref="S5.4.p4.28.m28.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.28.m28.1b"><apply id="S5.4.p4.28.m28.1.1.cmml" xref="S5.4.p4.28.m28.1.1"><in id="S5.4.p4.28.m28.1.1.1.cmml" xref="S5.4.p4.28.m28.1.1.1"></in><apply id="S5.4.p4.28.m28.1.1.2.cmml" xref="S5.4.p4.28.m28.1.1.2"><times id="S5.4.p4.28.m28.1.1.2.1.cmml" xref="S5.4.p4.28.m28.1.1.2.1"></times><apply id="S5.4.p4.28.m28.1.1.2.2.cmml" xref="S5.4.p4.28.m28.1.1.2.2"><csymbol cd="ambiguous" id="S5.4.p4.28.m28.1.1.2.2.1.cmml" xref="S5.4.p4.28.m28.1.1.2.2">subscript</csymbol><ci id="S5.4.p4.28.m28.1.1.2.2.2.cmml" xref="S5.4.p4.28.m28.1.1.2.2.2">π</ci><ci id="S5.4.p4.28.m28.1.1.2.2.3.cmml" xref="S5.4.p4.28.m28.1.1.2.2.3">π</ci></apply><ci id="S5.4.p4.28.m28.1.1.2.3.cmml" xref="S5.4.p4.28.m28.1.1.2.3">β</ci></apply><ci id="S5.4.p4.28.m28.1.1.3.cmml" xref="S5.4.p4.28.m28.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.28.m28.1c">e_{n}h\in S</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.28.m28.1d">italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_h β italic_S</annotation></semantics></math> for all <math alttext="n\in\omega" class="ltx_Math" display="inline" id="S5.4.p4.29.m29.1"><semantics id="S5.4.p4.29.m29.1a"><mrow id="S5.4.p4.29.m29.1.1" xref="S5.4.p4.29.m29.1.1.cmml"><mi id="S5.4.p4.29.m29.1.1.2" xref="S5.4.p4.29.m29.1.1.2.cmml">n</mi><mo id="S5.4.p4.29.m29.1.1.1" xref="S5.4.p4.29.m29.1.1.1.cmml">β</mo><mi id="S5.4.p4.29.m29.1.1.3" xref="S5.4.p4.29.m29.1.1.3.cmml">Ο</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.29.m29.1b"><apply id="S5.4.p4.29.m29.1.1.cmml" xref="S5.4.p4.29.m29.1.1"><in id="S5.4.p4.29.m29.1.1.1.cmml" xref="S5.4.p4.29.m29.1.1.1"></in><ci id="S5.4.p4.29.m29.1.1.2.cmml" xref="S5.4.p4.29.m29.1.1.2">π</ci><ci id="S5.4.p4.29.m29.1.1.3.cmml" xref="S5.4.p4.29.m29.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.29.m29.1c">n\in\omega</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.29.m29.1d">italic_n β italic_Ο</annotation></semantics></math>. It follows that <math alttext="\{e_{n}h:n\in\omega\}" class="ltx_Math" display="inline" id="S5.4.p4.30.m30.2"><semantics id="S5.4.p4.30.m30.2a"><mrow id="S5.4.p4.30.m30.2.2.2" xref="S5.4.p4.30.m30.2.2.3.cmml"><mo id="S5.4.p4.30.m30.2.2.2.3" stretchy="false" xref="S5.4.p4.30.m30.2.2.3.1.cmml">{</mo><mrow id="S5.4.p4.30.m30.1.1.1.1" xref="S5.4.p4.30.m30.1.1.1.1.cmml"><msub id="S5.4.p4.30.m30.1.1.1.1.2" xref="S5.4.p4.30.m30.1.1.1.1.2.cmml"><mi id="S5.4.p4.30.m30.1.1.1.1.2.2" xref="S5.4.p4.30.m30.1.1.1.1.2.2.cmml">e</mi><mi id="S5.4.p4.30.m30.1.1.1.1.2.3" xref="S5.4.p4.30.m30.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S5.4.p4.30.m30.1.1.1.1.1" xref="S5.4.p4.30.m30.1.1.1.1.1.cmml">β’</mo><mi id="S5.4.p4.30.m30.1.1.1.1.3" xref="S5.4.p4.30.m30.1.1.1.1.3.cmml">h</mi></mrow><mo id="S5.4.p4.30.m30.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S5.4.p4.30.m30.2.2.3.1.cmml">:</mo><mrow id="S5.4.p4.30.m30.2.2.2.2" xref="S5.4.p4.30.m30.2.2.2.2.cmml"><mi id="S5.4.p4.30.m30.2.2.2.2.2" xref="S5.4.p4.30.m30.2.2.2.2.2.cmml">n</mi><mo id="S5.4.p4.30.m30.2.2.2.2.1" xref="S5.4.p4.30.m30.2.2.2.2.1.cmml">β</mo><mi id="S5.4.p4.30.m30.2.2.2.2.3" xref="S5.4.p4.30.m30.2.2.2.2.3.cmml">Ο</mi></mrow><mo id="S5.4.p4.30.m30.2.2.2.5" stretchy="false" xref="S5.4.p4.30.m30.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p4.30.m30.2b"><apply id="S5.4.p4.30.m30.2.2.3.cmml" xref="S5.4.p4.30.m30.2.2.2"><csymbol cd="latexml" id="S5.4.p4.30.m30.2.2.3.1.cmml" xref="S5.4.p4.30.m30.2.2.2.3">conditional-set</csymbol><apply id="S5.4.p4.30.m30.1.1.1.1.cmml" xref="S5.4.p4.30.m30.1.1.1.1"><times id="S5.4.p4.30.m30.1.1.1.1.1.cmml" xref="S5.4.p4.30.m30.1.1.1.1.1"></times><apply id="S5.4.p4.30.m30.1.1.1.1.2.cmml" xref="S5.4.p4.30.m30.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.4.p4.30.m30.1.1.1.1.2.1.cmml" xref="S5.4.p4.30.m30.1.1.1.1.2">subscript</csymbol><ci id="S5.4.p4.30.m30.1.1.1.1.2.2.cmml" xref="S5.4.p4.30.m30.1.1.1.1.2.2">π</ci><ci id="S5.4.p4.30.m30.1.1.1.1.2.3.cmml" xref="S5.4.p4.30.m30.1.1.1.1.2.3">π</ci></apply><ci id="S5.4.p4.30.m30.1.1.1.1.3.cmml" xref="S5.4.p4.30.m30.1.1.1.1.3">β</ci></apply><apply id="S5.4.p4.30.m30.2.2.2.2.cmml" xref="S5.4.p4.30.m30.2.2.2.2"><in id="S5.4.p4.30.m30.2.2.2.2.1.cmml" xref="S5.4.p4.30.m30.2.2.2.2.1"></in><ci id="S5.4.p4.30.m30.2.2.2.2.2.cmml" xref="S5.4.p4.30.m30.2.2.2.2.2">π</ci><ci id="S5.4.p4.30.m30.2.2.2.2.3.cmml" xref="S5.4.p4.30.m30.2.2.2.2.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.30.m30.2c">\{e_{n}h:n\in\omega\}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.30.m30.2d">{ italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_h : italic_n β italic_Ο }</annotation></semantics></math> is an infinite closed discrete subset of <math alttext="S" class="ltx_Math" display="inline" id="S5.4.p4.31.m31.1"><semantics id="S5.4.p4.31.m31.1a"><mi id="S5.4.p4.31.m31.1.1" xref="S5.4.p4.31.m31.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.4.p4.31.m31.1b"><ci id="S5.4.p4.31.m31.1.1.cmml" xref="S5.4.p4.31.m31.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.31.m31.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.31.m31.1d">italic_S</annotation></semantics></math>, which contradicts the countable compactness of <math alttext="S" class="ltx_Math" display="inline" id="S5.4.p4.32.m32.1"><semantics id="S5.4.p4.32.m32.1a"><mi id="S5.4.p4.32.m32.1.1" xref="S5.4.p4.32.m32.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.4.p4.32.m32.1b"><ci id="S5.4.p4.32.m32.1.1.cmml" xref="S5.4.p4.32.m32.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p4.32.m32.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.4.p4.32.m32.1d">italic_S</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.5.p5"> <p class="ltx_p" id="S5.5.p5.26">(ii) Since <math alttext="f=h^{-1}h" class="ltx_Math" display="inline" id="S5.5.p5.1.m1.1"><semantics id="S5.5.p5.1.m1.1a"><mrow id="S5.5.p5.1.m1.1.1" xref="S5.5.p5.1.m1.1.1.cmml"><mi id="S5.5.p5.1.m1.1.1.2" xref="S5.5.p5.1.m1.1.1.2.cmml">f</mi><mo id="S5.5.p5.1.m1.1.1.1" xref="S5.5.p5.1.m1.1.1.1.cmml">=</mo><mrow id="S5.5.p5.1.m1.1.1.3" xref="S5.5.p5.1.m1.1.1.3.cmml"><msup id="S5.5.p5.1.m1.1.1.3.2" xref="S5.5.p5.1.m1.1.1.3.2.cmml"><mi id="S5.5.p5.1.m1.1.1.3.2.2" xref="S5.5.p5.1.m1.1.1.3.2.2.cmml">h</mi><mrow id="S5.5.p5.1.m1.1.1.3.2.3" xref="S5.5.p5.1.m1.1.1.3.2.3.cmml"><mo id="S5.5.p5.1.m1.1.1.3.2.3a" xref="S5.5.p5.1.m1.1.1.3.2.3.cmml">β</mo><mn id="S5.5.p5.1.m1.1.1.3.2.3.2" xref="S5.5.p5.1.m1.1.1.3.2.3.2.cmml">1</mn></mrow></msup><mo id="S5.5.p5.1.m1.1.1.3.1" xref="S5.5.p5.1.m1.1.1.3.1.cmml">β’</mo><mi id="S5.5.p5.1.m1.1.1.3.3" xref="S5.5.p5.1.m1.1.1.3.3.cmml">h</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.1.m1.1b"><apply id="S5.5.p5.1.m1.1.1.cmml" xref="S5.5.p5.1.m1.1.1"><eq id="S5.5.p5.1.m1.1.1.1.cmml" xref="S5.5.p5.1.m1.1.1.1"></eq><ci id="S5.5.p5.1.m1.1.1.2.cmml" xref="S5.5.p5.1.m1.1.1.2">π</ci><apply id="S5.5.p5.1.m1.1.1.3.cmml" xref="S5.5.p5.1.m1.1.1.3"><times id="S5.5.p5.1.m1.1.1.3.1.cmml" xref="S5.5.p5.1.m1.1.1.3.1"></times><apply id="S5.5.p5.1.m1.1.1.3.2.cmml" xref="S5.5.p5.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.5.p5.1.m1.1.1.3.2.1.cmml" xref="S5.5.p5.1.m1.1.1.3.2">superscript</csymbol><ci id="S5.5.p5.1.m1.1.1.3.2.2.cmml" xref="S5.5.p5.1.m1.1.1.3.2.2">β</ci><apply id="S5.5.p5.1.m1.1.1.3.2.3.cmml" xref="S5.5.p5.1.m1.1.1.3.2.3"><minus id="S5.5.p5.1.m1.1.1.3.2.3.1.cmml" xref="S5.5.p5.1.m1.1.1.3.2.3"></minus><cn id="S5.5.p5.1.m1.1.1.3.2.3.2.cmml" type="integer" xref="S5.5.p5.1.m1.1.1.3.2.3.2">1</cn></apply></apply><ci id="S5.5.p5.1.m1.1.1.3.3.cmml" xref="S5.5.p5.1.m1.1.1.3.3">β</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.1.m1.1c">f=h^{-1}h</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.1.m1.1d">italic_f = italic_h start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_h</annotation></semantics></math> and <math alttext="\beta S" class="ltx_Math" display="inline" id="S5.5.p5.2.m2.1"><semantics id="S5.5.p5.2.m2.1a"><mrow id="S5.5.p5.2.m2.1.1" xref="S5.5.p5.2.m2.1.1.cmml"><mi id="S5.5.p5.2.m2.1.1.2" xref="S5.5.p5.2.m2.1.1.2.cmml">Ξ²</mi><mo id="S5.5.p5.2.m2.1.1.1" xref="S5.5.p5.2.m2.1.1.1.cmml">β’</mo><mi id="S5.5.p5.2.m2.1.1.3" xref="S5.5.p5.2.m2.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.2.m2.1b"><apply id="S5.5.p5.2.m2.1.1.cmml" xref="S5.5.p5.2.m2.1.1"><times id="S5.5.p5.2.m2.1.1.1.cmml" xref="S5.5.p5.2.m2.1.1.1"></times><ci id="S5.5.p5.2.m2.1.1.2.cmml" xref="S5.5.p5.2.m2.1.1.2">π½</ci><ci id="S5.5.p5.2.m2.1.1.3.cmml" xref="S5.5.p5.2.m2.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.2.m2.1c">\beta S</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.2.m2.1d">italic_Ξ² italic_S</annotation></semantics></math> is a topological inverse semigroup, for each open neighborhood <math alttext="U" class="ltx_Math" display="inline" id="S5.5.p5.3.m3.1"><semantics id="S5.5.p5.3.m3.1a"><mi id="S5.5.p5.3.m3.1.1" xref="S5.5.p5.3.m3.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S5.5.p5.3.m3.1b"><ci id="S5.5.p5.3.m3.1.1.cmml" xref="S5.5.p5.3.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.3.m3.1c">U</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.3.m3.1d">italic_U</annotation></semantics></math> of <math alttext="f" class="ltx_Math" display="inline" id="S5.5.p5.4.m4.1"><semantics id="S5.5.p5.4.m4.1a"><mi id="S5.5.p5.4.m4.1.1" xref="S5.5.p5.4.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S5.5.p5.4.m4.1b"><ci id="S5.5.p5.4.m4.1.1.cmml" xref="S5.5.p5.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.4.m4.1c">f</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.4.m4.1d">italic_f</annotation></semantics></math> there exists an open neighborhood <math alttext="V" class="ltx_Math" display="inline" id="S5.5.p5.5.m5.1"><semantics id="S5.5.p5.5.m5.1a"><mi id="S5.5.p5.5.m5.1.1" xref="S5.5.p5.5.m5.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S5.5.p5.5.m5.1b"><ci id="S5.5.p5.5.m5.1.1.cmml" xref="S5.5.p5.5.m5.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.5.m5.1c">V</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.5.m5.1d">italic_V</annotation></semantics></math> of <math alttext="h" class="ltx_Math" display="inline" id="S5.5.p5.6.m6.1"><semantics id="S5.5.p5.6.m6.1a"><mi id="S5.5.p5.6.m6.1.1" xref="S5.5.p5.6.m6.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S5.5.p5.6.m6.1b"><ci id="S5.5.p5.6.m6.1.1.cmml" xref="S5.5.p5.6.m6.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.6.m6.1c">h</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.6.m6.1d">italic_h</annotation></semantics></math> such that <math alttext="V^{-1}V\subseteq U" class="ltx_Math" display="inline" id="S5.5.p5.7.m7.1"><semantics id="S5.5.p5.7.m7.1a"><mrow id="S5.5.p5.7.m7.1.1" xref="S5.5.p5.7.m7.1.1.cmml"><mrow id="S5.5.p5.7.m7.1.1.2" xref="S5.5.p5.7.m7.1.1.2.cmml"><msup id="S5.5.p5.7.m7.1.1.2.2" xref="S5.5.p5.7.m7.1.1.2.2.cmml"><mi id="S5.5.p5.7.m7.1.1.2.2.2" xref="S5.5.p5.7.m7.1.1.2.2.2.cmml">V</mi><mrow id="S5.5.p5.7.m7.1.1.2.2.3" xref="S5.5.p5.7.m7.1.1.2.2.3.cmml"><mo id="S5.5.p5.7.m7.1.1.2.2.3a" xref="S5.5.p5.7.m7.1.1.2.2.3.cmml">β</mo><mn id="S5.5.p5.7.m7.1.1.2.2.3.2" xref="S5.5.p5.7.m7.1.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S5.5.p5.7.m7.1.1.2.1" xref="S5.5.p5.7.m7.1.1.2.1.cmml">β’</mo><mi id="S5.5.p5.7.m7.1.1.2.3" xref="S5.5.p5.7.m7.1.1.2.3.cmml">V</mi></mrow><mo id="S5.5.p5.7.m7.1.1.1" xref="S5.5.p5.7.m7.1.1.1.cmml">β</mo><mi id="S5.5.p5.7.m7.1.1.3" xref="S5.5.p5.7.m7.1.1.3.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.7.m7.1b"><apply id="S5.5.p5.7.m7.1.1.cmml" xref="S5.5.p5.7.m7.1.1"><subset id="S5.5.p5.7.m7.1.1.1.cmml" xref="S5.5.p5.7.m7.1.1.1"></subset><apply id="S5.5.p5.7.m7.1.1.2.cmml" xref="S5.5.p5.7.m7.1.1.2"><times id="S5.5.p5.7.m7.1.1.2.1.cmml" xref="S5.5.p5.7.m7.1.1.2.1"></times><apply id="S5.5.p5.7.m7.1.1.2.2.cmml" xref="S5.5.p5.7.m7.1.1.2.2"><csymbol cd="ambiguous" id="S5.5.p5.7.m7.1.1.2.2.1.cmml" xref="S5.5.p5.7.m7.1.1.2.2">superscript</csymbol><ci id="S5.5.p5.7.m7.1.1.2.2.2.cmml" xref="S5.5.p5.7.m7.1.1.2.2.2">π</ci><apply id="S5.5.p5.7.m7.1.1.2.2.3.cmml" xref="S5.5.p5.7.m7.1.1.2.2.3"><minus id="S5.5.p5.7.m7.1.1.2.2.3.1.cmml" xref="S5.5.p5.7.m7.1.1.2.2.3"></minus><cn id="S5.5.p5.7.m7.1.1.2.2.3.2.cmml" type="integer" xref="S5.5.p5.7.m7.1.1.2.2.3.2">1</cn></apply></apply><ci id="S5.5.p5.7.m7.1.1.2.3.cmml" xref="S5.5.p5.7.m7.1.1.2.3">π</ci></apply><ci id="S5.5.p5.7.m7.1.1.3.cmml" xref="S5.5.p5.7.m7.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.7.m7.1c">V^{-1}V\subseteq U</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.7.m7.1d">italic_V start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_V β italic_U</annotation></semantics></math>. By the assumption, there is <math alttext="x\in V" class="ltx_Math" display="inline" id="S5.5.p5.8.m8.1"><semantics id="S5.5.p5.8.m8.1a"><mrow id="S5.5.p5.8.m8.1.1" xref="S5.5.p5.8.m8.1.1.cmml"><mi id="S5.5.p5.8.m8.1.1.2" xref="S5.5.p5.8.m8.1.1.2.cmml">x</mi><mo id="S5.5.p5.8.m8.1.1.1" xref="S5.5.p5.8.m8.1.1.1.cmml">β</mo><mi id="S5.5.p5.8.m8.1.1.3" xref="S5.5.p5.8.m8.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.8.m8.1b"><apply id="S5.5.p5.8.m8.1.1.cmml" xref="S5.5.p5.8.m8.1.1"><in id="S5.5.p5.8.m8.1.1.1.cmml" xref="S5.5.p5.8.m8.1.1.1"></in><ci id="S5.5.p5.8.m8.1.1.2.cmml" xref="S5.5.p5.8.m8.1.1.2">π₯</ci><ci id="S5.5.p5.8.m8.1.1.3.cmml" xref="S5.5.p5.8.m8.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.8.m8.1c">x\in V</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.8.m8.1d">italic_x β italic_V</annotation></semantics></math> such that <math alttext="f>x^{-1}x\in U" class="ltx_Math" display="inline" id="S5.5.p5.9.m9.1"><semantics id="S5.5.p5.9.m9.1a"><mrow id="S5.5.p5.9.m9.1.1" xref="S5.5.p5.9.m9.1.1.cmml"><mi id="S5.5.p5.9.m9.1.1.2" xref="S5.5.p5.9.m9.1.1.2.cmml">f</mi><mo id="S5.5.p5.9.m9.1.1.3" xref="S5.5.p5.9.m9.1.1.3.cmml">></mo><mrow id="S5.5.p5.9.m9.1.1.4" xref="S5.5.p5.9.m9.1.1.4.cmml"><msup id="S5.5.p5.9.m9.1.1.4.2" xref="S5.5.p5.9.m9.1.1.4.2.cmml"><mi id="S5.5.p5.9.m9.1.1.4.2.2" xref="S5.5.p5.9.m9.1.1.4.2.2.cmml">x</mi><mrow id="S5.5.p5.9.m9.1.1.4.2.3" xref="S5.5.p5.9.m9.1.1.4.2.3.cmml"><mo id="S5.5.p5.9.m9.1.1.4.2.3a" xref="S5.5.p5.9.m9.1.1.4.2.3.cmml">β</mo><mn id="S5.5.p5.9.m9.1.1.4.2.3.2" xref="S5.5.p5.9.m9.1.1.4.2.3.2.cmml">1</mn></mrow></msup><mo id="S5.5.p5.9.m9.1.1.4.1" xref="S5.5.p5.9.m9.1.1.4.1.cmml">β’</mo><mi id="S5.5.p5.9.m9.1.1.4.3" xref="S5.5.p5.9.m9.1.1.4.3.cmml">x</mi></mrow><mo id="S5.5.p5.9.m9.1.1.5" xref="S5.5.p5.9.m9.1.1.5.cmml">β</mo><mi id="S5.5.p5.9.m9.1.1.6" xref="S5.5.p5.9.m9.1.1.6.cmml">U</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.9.m9.1b"><apply id="S5.5.p5.9.m9.1.1.cmml" xref="S5.5.p5.9.m9.1.1"><and id="S5.5.p5.9.m9.1.1a.cmml" xref="S5.5.p5.9.m9.1.1"></and><apply id="S5.5.p5.9.m9.1.1b.cmml" xref="S5.5.p5.9.m9.1.1"><gt id="S5.5.p5.9.m9.1.1.3.cmml" xref="S5.5.p5.9.m9.1.1.3"></gt><ci id="S5.5.p5.9.m9.1.1.2.cmml" xref="S5.5.p5.9.m9.1.1.2">π</ci><apply id="S5.5.p5.9.m9.1.1.4.cmml" xref="S5.5.p5.9.m9.1.1.4"><times id="S5.5.p5.9.m9.1.1.4.1.cmml" xref="S5.5.p5.9.m9.1.1.4.1"></times><apply id="S5.5.p5.9.m9.1.1.4.2.cmml" xref="S5.5.p5.9.m9.1.1.4.2"><csymbol cd="ambiguous" id="S5.5.p5.9.m9.1.1.4.2.1.cmml" xref="S5.5.p5.9.m9.1.1.4.2">superscript</csymbol><ci id="S5.5.p5.9.m9.1.1.4.2.2.cmml" xref="S5.5.p5.9.m9.1.1.4.2.2">π₯</ci><apply id="S5.5.p5.9.m9.1.1.4.2.3.cmml" xref="S5.5.p5.9.m9.1.1.4.2.3"><minus id="S5.5.p5.9.m9.1.1.4.2.3.1.cmml" xref="S5.5.p5.9.m9.1.1.4.2.3"></minus><cn id="S5.5.p5.9.m9.1.1.4.2.3.2.cmml" type="integer" xref="S5.5.p5.9.m9.1.1.4.2.3.2">1</cn></apply></apply><ci id="S5.5.p5.9.m9.1.1.4.3.cmml" xref="S5.5.p5.9.m9.1.1.4.3">π₯</ci></apply></apply><apply id="S5.5.p5.9.m9.1.1c.cmml" xref="S5.5.p5.9.m9.1.1"><in id="S5.5.p5.9.m9.1.1.5.cmml" xref="S5.5.p5.9.m9.1.1.5"></in><share href="https://arxiv.org/html/2503.13666v1#S5.5.p5.9.m9.1.1.4.cmml" id="S5.5.p5.9.m9.1.1d.cmml" xref="S5.5.p5.9.m9.1.1"></share><ci id="S5.5.p5.9.m9.1.1.6.cmml" xref="S5.5.p5.9.m9.1.1.6">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.9.m9.1c">f>x^{-1}x\in U</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.9.m9.1d">italic_f > italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x β italic_U</annotation></semantics></math>. It follows that <math alttext="f\in\operatorname{cl}_{S}({\downarrow}f\setminus\{f\})" class="ltx_Math" display="inline" id="S5.5.p5.10.m10.2"><semantics id="S5.5.p5.10.m10.2a"><mrow id="S5.5.p5.10.m10.2.2" xref="S5.5.p5.10.m10.2.2.cmml"><mi id="S5.5.p5.10.m10.2.2.3" xref="S5.5.p5.10.m10.2.2.3.cmml">f</mi><mo id="S5.5.p5.10.m10.2.2.2" xref="S5.5.p5.10.m10.2.2.2.cmml">β</mo><mrow id="S5.5.p5.10.m10.2.2.1" xref="S5.5.p5.10.m10.2.2.1.cmml"><msub id="S5.5.p5.10.m10.2.2.1.3" xref="S5.5.p5.10.m10.2.2.1.3.cmml"><mi id="S5.5.p5.10.m10.2.2.1.3.2" xref="S5.5.p5.10.m10.2.2.1.3.2.cmml">cl</mi><mi id="S5.5.p5.10.m10.2.2.1.3.3" xref="S5.5.p5.10.m10.2.2.1.3.3.cmml">S</mi></msub><mspace id="S5.5.p5.10.m10.2.2.1a" width="0.556em" xref="S5.5.p5.10.m10.2.2.1.cmml"></mspace><mrow id="S5.5.p5.10.m10.2.2.1.1.1" xref="S5.5.p5.10.m10.2.2.1.1.1.1.cmml"><mo id="S5.5.p5.10.m10.2.2.1.1.1.2" stretchy="false" xref="S5.5.p5.10.m10.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.5.p5.10.m10.2.2.1.1.1.1" xref="S5.5.p5.10.m10.2.2.1.1.1.1.cmml"><mi id="S5.5.p5.10.m10.2.2.1.1.1.1.2" xref="S5.5.p5.10.m10.2.2.1.1.1.1.2.cmml"></mi><mo id="S5.5.p5.10.m10.2.2.1.1.1.1.1" stretchy="false" xref="S5.5.p5.10.m10.2.2.1.1.1.1.1.cmml">β</mo><mrow id="S5.5.p5.10.m10.2.2.1.1.1.1.3" xref="S5.5.p5.10.m10.2.2.1.1.1.1.3.cmml"><mi id="S5.5.p5.10.m10.2.2.1.1.1.1.3.2" xref="S5.5.p5.10.m10.2.2.1.1.1.1.3.2.cmml">f</mi><mo id="S5.5.p5.10.m10.2.2.1.1.1.1.3.1" xref="S5.5.p5.10.m10.2.2.1.1.1.1.3.1.cmml">β</mo><mrow id="S5.5.p5.10.m10.2.2.1.1.1.1.3.3.2" xref="S5.5.p5.10.m10.2.2.1.1.1.1.3.3.1.cmml"><mo id="S5.5.p5.10.m10.2.2.1.1.1.1.3.3.2.1" stretchy="false" xref="S5.5.p5.10.m10.2.2.1.1.1.1.3.3.1.cmml">{</mo><mi id="S5.5.p5.10.m10.1.1" xref="S5.5.p5.10.m10.1.1.cmml">f</mi><mo id="S5.5.p5.10.m10.2.2.1.1.1.1.3.3.2.2" stretchy="false" xref="S5.5.p5.10.m10.2.2.1.1.1.1.3.3.1.cmml">}</mo></mrow></mrow></mrow><mo id="S5.5.p5.10.m10.2.2.1.1.1.3" stretchy="false" xref="S5.5.p5.10.m10.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.10.m10.2b"><apply id="S5.5.p5.10.m10.2.2.cmml" xref="S5.5.p5.10.m10.2.2"><in id="S5.5.p5.10.m10.2.2.2.cmml" xref="S5.5.p5.10.m10.2.2.2"></in><ci id="S5.5.p5.10.m10.2.2.3.cmml" xref="S5.5.p5.10.m10.2.2.3">π</ci><apply id="S5.5.p5.10.m10.2.2.1.cmml" xref="S5.5.p5.10.m10.2.2.1"><csymbol cd="latexml" id="S5.5.p5.10.m10.2.2.1.2.cmml" xref="S5.5.p5.10.m10.2.2.1">annotated</csymbol><apply id="S5.5.p5.10.m10.2.2.1.3.cmml" xref="S5.5.p5.10.m10.2.2.1.3"><csymbol cd="ambiguous" id="S5.5.p5.10.m10.2.2.1.3.1.cmml" xref="S5.5.p5.10.m10.2.2.1.3">subscript</csymbol><ci id="S5.5.p5.10.m10.2.2.1.3.2.cmml" xref="S5.5.p5.10.m10.2.2.1.3.2">cl</ci><ci id="S5.5.p5.10.m10.2.2.1.3.3.cmml" xref="S5.5.p5.10.m10.2.2.1.3.3">π</ci></apply><apply id="S5.5.p5.10.m10.2.2.1.1.1.1.cmml" xref="S5.5.p5.10.m10.2.2.1.1.1"><ci id="S5.5.p5.10.m10.2.2.1.1.1.1.1.cmml" xref="S5.5.p5.10.m10.2.2.1.1.1.1.1">β</ci><csymbol cd="latexml" id="S5.5.p5.10.m10.2.2.1.1.1.1.2.cmml" xref="S5.5.p5.10.m10.2.2.1.1.1.1.2">absent</csymbol><apply id="S5.5.p5.10.m10.2.2.1.1.1.1.3.cmml" xref="S5.5.p5.10.m10.2.2.1.1.1.1.3"><setdiff id="S5.5.p5.10.m10.2.2.1.1.1.1.3.1.cmml" xref="S5.5.p5.10.m10.2.2.1.1.1.1.3.1"></setdiff><ci id="S5.5.p5.10.m10.2.2.1.1.1.1.3.2.cmml" xref="S5.5.p5.10.m10.2.2.1.1.1.1.3.2">π</ci><set id="S5.5.p5.10.m10.2.2.1.1.1.1.3.3.1.cmml" xref="S5.5.p5.10.m10.2.2.1.1.1.1.3.3.2"><ci id="S5.5.p5.10.m10.1.1.cmml" xref="S5.5.p5.10.m10.1.1">π</ci></set></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.10.m10.2c">f\in\operatorname{cl}_{S}({\downarrow}f\setminus\{f\})</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.10.m10.2d">italic_f β roman_cl start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( β italic_f β { italic_f } )</annotation></semantics></math>. Since <math alttext="X" class="ltx_Math" display="inline" id="S5.5.p5.11.m11.1"><semantics id="S5.5.p5.11.m11.1a"><mi id="S5.5.p5.11.m11.1.1" xref="S5.5.p5.11.m11.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.5.p5.11.m11.1b"><ci id="S5.5.p5.11.m11.1.1.cmml" xref="S5.5.p5.11.m11.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.11.m11.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.11.m11.1d">italic_X</annotation></semantics></math> is first-countable, there exists a sequence <math alttext="\{f_{n}:n\in\omega\}\subseteq{\downarrow}f\setminus\{f\}" class="ltx_math_unparsed" display="inline" id="S5.5.p5.12.m12.1"><semantics id="S5.5.p5.12.m12.1a"><mrow id="S5.5.p5.12.m12.1b"><mrow id="S5.5.p5.12.m12.1.1"><mo id="S5.5.p5.12.m12.1.1.1" stretchy="false">{</mo><msub id="S5.5.p5.12.m12.1.1.2"><mi id="S5.5.p5.12.m12.1.1.2.2">f</mi><mi id="S5.5.p5.12.m12.1.1.2.3">n</mi></msub><mo id="S5.5.p5.12.m12.1.1.3" lspace="0.278em" rspace="0.278em">:</mo><mi id="S5.5.p5.12.m12.1.1.4">n</mi><mo id="S5.5.p5.12.m12.1.1.5">β</mo><mi id="S5.5.p5.12.m12.1.1.6">Ο</mi><mo id="S5.5.p5.12.m12.1.1.7" stretchy="false">}</mo></mrow><mo id="S5.5.p5.12.m12.1.2" rspace="0em">β</mo><mo id="S5.5.p5.12.m12.1.3" lspace="0em" stretchy="false">β</mo><mi id="S5.5.p5.12.m12.1.4">f</mi><mo id="S5.5.p5.12.m12.1.5">β</mo><mrow id="S5.5.p5.12.m12.1.6"><mo id="S5.5.p5.12.m12.1.6.1" stretchy="false">{</mo><mi id="S5.5.p5.12.m12.1.6.2">f</mi><mo id="S5.5.p5.12.m12.1.6.3" stretchy="false">}</mo></mrow></mrow><annotation encoding="application/x-tex" id="S5.5.p5.12.m12.1c">\{f_{n}:n\in\omega\}\subseteq{\downarrow}f\setminus\{f\}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.12.m12.1d">{ italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n β italic_Ο } β β italic_f β { italic_f }</annotation></semantics></math> that converges to <math alttext="f" class="ltx_Math" display="inline" id="S5.5.p5.13.m13.1"><semantics id="S5.5.p5.13.m13.1a"><mi id="S5.5.p5.13.m13.1.1" xref="S5.5.p5.13.m13.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S5.5.p5.13.m13.1b"><ci id="S5.5.p5.13.m13.1.1.cmml" xref="S5.5.p5.13.m13.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.13.m13.1c">f</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.13.m13.1d">italic_f</annotation></semantics></math>. Since <math alttext="hf=h" class="ltx_Math" display="inline" id="S5.5.p5.14.m14.1"><semantics id="S5.5.p5.14.m14.1a"><mrow id="S5.5.p5.14.m14.1.1" xref="S5.5.p5.14.m14.1.1.cmml"><mrow id="S5.5.p5.14.m14.1.1.2" xref="S5.5.p5.14.m14.1.1.2.cmml"><mi id="S5.5.p5.14.m14.1.1.2.2" xref="S5.5.p5.14.m14.1.1.2.2.cmml">h</mi><mo id="S5.5.p5.14.m14.1.1.2.1" xref="S5.5.p5.14.m14.1.1.2.1.cmml">β’</mo><mi id="S5.5.p5.14.m14.1.1.2.3" xref="S5.5.p5.14.m14.1.1.2.3.cmml">f</mi></mrow><mo id="S5.5.p5.14.m14.1.1.1" xref="S5.5.p5.14.m14.1.1.1.cmml">=</mo><mi id="S5.5.p5.14.m14.1.1.3" xref="S5.5.p5.14.m14.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.14.m14.1b"><apply id="S5.5.p5.14.m14.1.1.cmml" xref="S5.5.p5.14.m14.1.1"><eq id="S5.5.p5.14.m14.1.1.1.cmml" xref="S5.5.p5.14.m14.1.1.1"></eq><apply id="S5.5.p5.14.m14.1.1.2.cmml" xref="S5.5.p5.14.m14.1.1.2"><times id="S5.5.p5.14.m14.1.1.2.1.cmml" xref="S5.5.p5.14.m14.1.1.2.1"></times><ci id="S5.5.p5.14.m14.1.1.2.2.cmml" xref="S5.5.p5.14.m14.1.1.2.2">β</ci><ci id="S5.5.p5.14.m14.1.1.2.3.cmml" xref="S5.5.p5.14.m14.1.1.2.3">π</ci></apply><ci id="S5.5.p5.14.m14.1.1.3.cmml" xref="S5.5.p5.14.m14.1.1.3">β</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.14.m14.1c">hf=h</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.14.m14.1d">italic_h italic_f = italic_h</annotation></semantics></math> we get that the sequence <math alttext="\{hf_{n}:n\in\omega\}" class="ltx_Math" display="inline" id="S5.5.p5.15.m15.2"><semantics id="S5.5.p5.15.m15.2a"><mrow id="S5.5.p5.15.m15.2.2.2" xref="S5.5.p5.15.m15.2.2.3.cmml"><mo id="S5.5.p5.15.m15.2.2.2.3" stretchy="false" xref="S5.5.p5.15.m15.2.2.3.1.cmml">{</mo><mrow id="S5.5.p5.15.m15.1.1.1.1" xref="S5.5.p5.15.m15.1.1.1.1.cmml"><mi id="S5.5.p5.15.m15.1.1.1.1.2" xref="S5.5.p5.15.m15.1.1.1.1.2.cmml">h</mi><mo id="S5.5.p5.15.m15.1.1.1.1.1" xref="S5.5.p5.15.m15.1.1.1.1.1.cmml">β’</mo><msub id="S5.5.p5.15.m15.1.1.1.1.3" xref="S5.5.p5.15.m15.1.1.1.1.3.cmml"><mi id="S5.5.p5.15.m15.1.1.1.1.3.2" xref="S5.5.p5.15.m15.1.1.1.1.3.2.cmml">f</mi><mi id="S5.5.p5.15.m15.1.1.1.1.3.3" xref="S5.5.p5.15.m15.1.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S5.5.p5.15.m15.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S5.5.p5.15.m15.2.2.3.1.cmml">:</mo><mrow id="S5.5.p5.15.m15.2.2.2.2" xref="S5.5.p5.15.m15.2.2.2.2.cmml"><mi id="S5.5.p5.15.m15.2.2.2.2.2" xref="S5.5.p5.15.m15.2.2.2.2.2.cmml">n</mi><mo id="S5.5.p5.15.m15.2.2.2.2.1" xref="S5.5.p5.15.m15.2.2.2.2.1.cmml">β</mo><mi id="S5.5.p5.15.m15.2.2.2.2.3" xref="S5.5.p5.15.m15.2.2.2.2.3.cmml">Ο</mi></mrow><mo id="S5.5.p5.15.m15.2.2.2.5" stretchy="false" xref="S5.5.p5.15.m15.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.15.m15.2b"><apply id="S5.5.p5.15.m15.2.2.3.cmml" xref="S5.5.p5.15.m15.2.2.2"><csymbol cd="latexml" id="S5.5.p5.15.m15.2.2.3.1.cmml" xref="S5.5.p5.15.m15.2.2.2.3">conditional-set</csymbol><apply id="S5.5.p5.15.m15.1.1.1.1.cmml" xref="S5.5.p5.15.m15.1.1.1.1"><times id="S5.5.p5.15.m15.1.1.1.1.1.cmml" xref="S5.5.p5.15.m15.1.1.1.1.1"></times><ci id="S5.5.p5.15.m15.1.1.1.1.2.cmml" xref="S5.5.p5.15.m15.1.1.1.1.2">β</ci><apply id="S5.5.p5.15.m15.1.1.1.1.3.cmml" xref="S5.5.p5.15.m15.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.5.p5.15.m15.1.1.1.1.3.1.cmml" xref="S5.5.p5.15.m15.1.1.1.1.3">subscript</csymbol><ci id="S5.5.p5.15.m15.1.1.1.1.3.2.cmml" xref="S5.5.p5.15.m15.1.1.1.1.3.2">π</ci><ci id="S5.5.p5.15.m15.1.1.1.1.3.3.cmml" xref="S5.5.p5.15.m15.1.1.1.1.3.3">π</ci></apply></apply><apply id="S5.5.p5.15.m15.2.2.2.2.cmml" xref="S5.5.p5.15.m15.2.2.2.2"><in id="S5.5.p5.15.m15.2.2.2.2.1.cmml" xref="S5.5.p5.15.m15.2.2.2.2.1"></in><ci id="S5.5.p5.15.m15.2.2.2.2.2.cmml" xref="S5.5.p5.15.m15.2.2.2.2.2">π</ci><ci id="S5.5.p5.15.m15.2.2.2.2.3.cmml" xref="S5.5.p5.15.m15.2.2.2.2.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.15.m15.2c">\{hf_{n}:n\in\omega\}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.15.m15.2d">{ italic_h italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n β italic_Ο }</annotation></semantics></math> converges to <math alttext="h" class="ltx_Math" display="inline" id="S5.5.p5.16.m16.1"><semantics id="S5.5.p5.16.m16.1a"><mi id="S5.5.p5.16.m16.1.1" xref="S5.5.p5.16.m16.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S5.5.p5.16.m16.1b"><ci id="S5.5.p5.16.m16.1.1.cmml" xref="S5.5.p5.16.m16.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.16.m16.1c">h</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.16.m16.1d">italic_h</annotation></semantics></math>. Similarly as above it can be checked that <math alttext="h\notin\{hf_{n}:n\in\omega\}" class="ltx_Math" display="inline" id="S5.5.p5.17.m17.2"><semantics id="S5.5.p5.17.m17.2a"><mrow id="S5.5.p5.17.m17.2.2" xref="S5.5.p5.17.m17.2.2.cmml"><mi id="S5.5.p5.17.m17.2.2.4" xref="S5.5.p5.17.m17.2.2.4.cmml">h</mi><mo id="S5.5.p5.17.m17.2.2.3" xref="S5.5.p5.17.m17.2.2.3.cmml">β</mo><mrow id="S5.5.p5.17.m17.2.2.2.2" xref="S5.5.p5.17.m17.2.2.2.3.cmml"><mo id="S5.5.p5.17.m17.2.2.2.2.3" stretchy="false" xref="S5.5.p5.17.m17.2.2.2.3.1.cmml">{</mo><mrow id="S5.5.p5.17.m17.1.1.1.1.1" xref="S5.5.p5.17.m17.1.1.1.1.1.cmml"><mi id="S5.5.p5.17.m17.1.1.1.1.1.2" xref="S5.5.p5.17.m17.1.1.1.1.1.2.cmml">h</mi><mo id="S5.5.p5.17.m17.1.1.1.1.1.1" xref="S5.5.p5.17.m17.1.1.1.1.1.1.cmml">β’</mo><msub id="S5.5.p5.17.m17.1.1.1.1.1.3" xref="S5.5.p5.17.m17.1.1.1.1.1.3.cmml"><mi id="S5.5.p5.17.m17.1.1.1.1.1.3.2" xref="S5.5.p5.17.m17.1.1.1.1.1.3.2.cmml">f</mi><mi id="S5.5.p5.17.m17.1.1.1.1.1.3.3" xref="S5.5.p5.17.m17.1.1.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S5.5.p5.17.m17.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S5.5.p5.17.m17.2.2.2.3.1.cmml">:</mo><mrow id="S5.5.p5.17.m17.2.2.2.2.2" xref="S5.5.p5.17.m17.2.2.2.2.2.cmml"><mi id="S5.5.p5.17.m17.2.2.2.2.2.2" xref="S5.5.p5.17.m17.2.2.2.2.2.2.cmml">n</mi><mo id="S5.5.p5.17.m17.2.2.2.2.2.1" xref="S5.5.p5.17.m17.2.2.2.2.2.1.cmml">β</mo><mi id="S5.5.p5.17.m17.2.2.2.2.2.3" xref="S5.5.p5.17.m17.2.2.2.2.2.3.cmml">Ο</mi></mrow><mo id="S5.5.p5.17.m17.2.2.2.2.5" stretchy="false" xref="S5.5.p5.17.m17.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.17.m17.2b"><apply id="S5.5.p5.17.m17.2.2.cmml" xref="S5.5.p5.17.m17.2.2"><notin id="S5.5.p5.17.m17.2.2.3.cmml" xref="S5.5.p5.17.m17.2.2.3"></notin><ci id="S5.5.p5.17.m17.2.2.4.cmml" xref="S5.5.p5.17.m17.2.2.4">β</ci><apply id="S5.5.p5.17.m17.2.2.2.3.cmml" xref="S5.5.p5.17.m17.2.2.2.2"><csymbol cd="latexml" id="S5.5.p5.17.m17.2.2.2.3.1.cmml" xref="S5.5.p5.17.m17.2.2.2.2.3">conditional-set</csymbol><apply id="S5.5.p5.17.m17.1.1.1.1.1.cmml" xref="S5.5.p5.17.m17.1.1.1.1.1"><times id="S5.5.p5.17.m17.1.1.1.1.1.1.cmml" xref="S5.5.p5.17.m17.1.1.1.1.1.1"></times><ci id="S5.5.p5.17.m17.1.1.1.1.1.2.cmml" xref="S5.5.p5.17.m17.1.1.1.1.1.2">β</ci><apply id="S5.5.p5.17.m17.1.1.1.1.1.3.cmml" xref="S5.5.p5.17.m17.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.5.p5.17.m17.1.1.1.1.1.3.1.cmml" xref="S5.5.p5.17.m17.1.1.1.1.1.3">subscript</csymbol><ci id="S5.5.p5.17.m17.1.1.1.1.1.3.2.cmml" xref="S5.5.p5.17.m17.1.1.1.1.1.3.2">π</ci><ci id="S5.5.p5.17.m17.1.1.1.1.1.3.3.cmml" xref="S5.5.p5.17.m17.1.1.1.1.1.3.3">π</ci></apply></apply><apply id="S5.5.p5.17.m17.2.2.2.2.2.cmml" xref="S5.5.p5.17.m17.2.2.2.2.2"><in id="S5.5.p5.17.m17.2.2.2.2.2.1.cmml" xref="S5.5.p5.17.m17.2.2.2.2.2.1"></in><ci id="S5.5.p5.17.m17.2.2.2.2.2.2.cmml" xref="S5.5.p5.17.m17.2.2.2.2.2.2">π</ci><ci id="S5.5.p5.17.m17.2.2.2.2.2.3.cmml" xref="S5.5.p5.17.m17.2.2.2.2.2.3">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.17.m17.2c">h\notin\{hf_{n}:n\in\omega\}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.17.m17.2d">italic_h β { italic_h italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n β italic_Ο }</annotation></semantics></math> and <math alttext="hf_{n}\leq h" class="ltx_Math" display="inline" id="S5.5.p5.18.m18.1"><semantics id="S5.5.p5.18.m18.1a"><mrow id="S5.5.p5.18.m18.1.1" xref="S5.5.p5.18.m18.1.1.cmml"><mrow id="S5.5.p5.18.m18.1.1.2" xref="S5.5.p5.18.m18.1.1.2.cmml"><mi id="S5.5.p5.18.m18.1.1.2.2" xref="S5.5.p5.18.m18.1.1.2.2.cmml">h</mi><mo id="S5.5.p5.18.m18.1.1.2.1" xref="S5.5.p5.18.m18.1.1.2.1.cmml">β’</mo><msub id="S5.5.p5.18.m18.1.1.2.3" xref="S5.5.p5.18.m18.1.1.2.3.cmml"><mi id="S5.5.p5.18.m18.1.1.2.3.2" xref="S5.5.p5.18.m18.1.1.2.3.2.cmml">f</mi><mi id="S5.5.p5.18.m18.1.1.2.3.3" xref="S5.5.p5.18.m18.1.1.2.3.3.cmml">n</mi></msub></mrow><mo id="S5.5.p5.18.m18.1.1.1" xref="S5.5.p5.18.m18.1.1.1.cmml">β€</mo><mi id="S5.5.p5.18.m18.1.1.3" xref="S5.5.p5.18.m18.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.18.m18.1b"><apply id="S5.5.p5.18.m18.1.1.cmml" xref="S5.5.p5.18.m18.1.1"><leq id="S5.5.p5.18.m18.1.1.1.cmml" xref="S5.5.p5.18.m18.1.1.1"></leq><apply id="S5.5.p5.18.m18.1.1.2.cmml" xref="S5.5.p5.18.m18.1.1.2"><times id="S5.5.p5.18.m18.1.1.2.1.cmml" xref="S5.5.p5.18.m18.1.1.2.1"></times><ci id="S5.5.p5.18.m18.1.1.2.2.cmml" xref="S5.5.p5.18.m18.1.1.2.2">β</ci><apply id="S5.5.p5.18.m18.1.1.2.3.cmml" xref="S5.5.p5.18.m18.1.1.2.3"><csymbol cd="ambiguous" id="S5.5.p5.18.m18.1.1.2.3.1.cmml" xref="S5.5.p5.18.m18.1.1.2.3">subscript</csymbol><ci id="S5.5.p5.18.m18.1.1.2.3.2.cmml" xref="S5.5.p5.18.m18.1.1.2.3.2">π</ci><ci id="S5.5.p5.18.m18.1.1.2.3.3.cmml" xref="S5.5.p5.18.m18.1.1.2.3.3">π</ci></apply></apply><ci id="S5.5.p5.18.m18.1.1.3.cmml" xref="S5.5.p5.18.m18.1.1.3">β</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.18.m18.1c">hf_{n}\leq h</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.18.m18.1d">italic_h italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT β€ italic_h</annotation></semantics></math> for every <math alttext="n\in\omega" class="ltx_Math" display="inline" id="S5.5.p5.19.m19.1"><semantics id="S5.5.p5.19.m19.1a"><mrow id="S5.5.p5.19.m19.1.1" xref="S5.5.p5.19.m19.1.1.cmml"><mi id="S5.5.p5.19.m19.1.1.2" xref="S5.5.p5.19.m19.1.1.2.cmml">n</mi><mo id="S5.5.p5.19.m19.1.1.1" xref="S5.5.p5.19.m19.1.1.1.cmml">β</mo><mi id="S5.5.p5.19.m19.1.1.3" xref="S5.5.p5.19.m19.1.1.3.cmml">Ο</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.19.m19.1b"><apply id="S5.5.p5.19.m19.1.1.cmml" xref="S5.5.p5.19.m19.1.1"><in id="S5.5.p5.19.m19.1.1.1.cmml" xref="S5.5.p5.19.m19.1.1.1"></in><ci id="S5.5.p5.19.m19.1.1.2.cmml" xref="S5.5.p5.19.m19.1.1.2">π</ci><ci id="S5.5.p5.19.m19.1.1.3.cmml" xref="S5.5.p5.19.m19.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.19.m19.1c">n\in\omega</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.19.m19.1d">italic_n β italic_Ο</annotation></semantics></math>. Since <math alttext="h" class="ltx_Math" display="inline" id="S5.5.p5.20.m20.1"><semantics id="S5.5.p5.20.m20.1a"><mi id="S5.5.p5.20.m20.1.1" xref="S5.5.p5.20.m20.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S5.5.p5.20.m20.1b"><ci id="S5.5.p5.20.m20.1.1.cmml" xref="S5.5.p5.20.m20.1.1">β</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.20.m20.1c">h</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.20.m20.1d">italic_h</annotation></semantics></math> is a minimal element in <math alttext="\beta S\setminus S" class="ltx_Math" display="inline" id="S5.5.p5.21.m21.1"><semantics id="S5.5.p5.21.m21.1a"><mrow id="S5.5.p5.21.m21.1.1" xref="S5.5.p5.21.m21.1.1.cmml"><mrow id="S5.5.p5.21.m21.1.1.2" xref="S5.5.p5.21.m21.1.1.2.cmml"><mi id="S5.5.p5.21.m21.1.1.2.2" xref="S5.5.p5.21.m21.1.1.2.2.cmml">Ξ²</mi><mo id="S5.5.p5.21.m21.1.1.2.1" xref="S5.5.p5.21.m21.1.1.2.1.cmml">β’</mo><mi id="S5.5.p5.21.m21.1.1.2.3" xref="S5.5.p5.21.m21.1.1.2.3.cmml">S</mi></mrow><mo id="S5.5.p5.21.m21.1.1.1" xref="S5.5.p5.21.m21.1.1.1.cmml">β</mo><mi id="S5.5.p5.21.m21.1.1.3" xref="S5.5.p5.21.m21.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.21.m21.1b"><apply id="S5.5.p5.21.m21.1.1.cmml" xref="S5.5.p5.21.m21.1.1"><setdiff id="S5.5.p5.21.m21.1.1.1.cmml" xref="S5.5.p5.21.m21.1.1.1"></setdiff><apply id="S5.5.p5.21.m21.1.1.2.cmml" xref="S5.5.p5.21.m21.1.1.2"><times id="S5.5.p5.21.m21.1.1.2.1.cmml" xref="S5.5.p5.21.m21.1.1.2.1"></times><ci id="S5.5.p5.21.m21.1.1.2.2.cmml" xref="S5.5.p5.21.m21.1.1.2.2">π½</ci><ci id="S5.5.p5.21.m21.1.1.2.3.cmml" xref="S5.5.p5.21.m21.1.1.2.3">π</ci></apply><ci id="S5.5.p5.21.m21.1.1.3.cmml" xref="S5.5.p5.21.m21.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.21.m21.1c">\beta S\setminus S</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.21.m21.1d">italic_Ξ² italic_S β italic_S</annotation></semantics></math>, we get that <math alttext="hf_{n}\in S" class="ltx_Math" display="inline" id="S5.5.p5.22.m22.1"><semantics id="S5.5.p5.22.m22.1a"><mrow id="S5.5.p5.22.m22.1.1" xref="S5.5.p5.22.m22.1.1.cmml"><mrow id="S5.5.p5.22.m22.1.1.2" xref="S5.5.p5.22.m22.1.1.2.cmml"><mi id="S5.5.p5.22.m22.1.1.2.2" xref="S5.5.p5.22.m22.1.1.2.2.cmml">h</mi><mo id="S5.5.p5.22.m22.1.1.2.1" xref="S5.5.p5.22.m22.1.1.2.1.cmml">β’</mo><msub id="S5.5.p5.22.m22.1.1.2.3" xref="S5.5.p5.22.m22.1.1.2.3.cmml"><mi id="S5.5.p5.22.m22.1.1.2.3.2" xref="S5.5.p5.22.m22.1.1.2.3.2.cmml">f</mi><mi id="S5.5.p5.22.m22.1.1.2.3.3" xref="S5.5.p5.22.m22.1.1.2.3.3.cmml">n</mi></msub></mrow><mo id="S5.5.p5.22.m22.1.1.1" xref="S5.5.p5.22.m22.1.1.1.cmml">β</mo><mi id="S5.5.p5.22.m22.1.1.3" xref="S5.5.p5.22.m22.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.22.m22.1b"><apply id="S5.5.p5.22.m22.1.1.cmml" xref="S5.5.p5.22.m22.1.1"><in id="S5.5.p5.22.m22.1.1.1.cmml" xref="S5.5.p5.22.m22.1.1.1"></in><apply id="S5.5.p5.22.m22.1.1.2.cmml" xref="S5.5.p5.22.m22.1.1.2"><times id="S5.5.p5.22.m22.1.1.2.1.cmml" xref="S5.5.p5.22.m22.1.1.2.1"></times><ci id="S5.5.p5.22.m22.1.1.2.2.cmml" xref="S5.5.p5.22.m22.1.1.2.2">β</ci><apply id="S5.5.p5.22.m22.1.1.2.3.cmml" xref="S5.5.p5.22.m22.1.1.2.3"><csymbol cd="ambiguous" id="S5.5.p5.22.m22.1.1.2.3.1.cmml" xref="S5.5.p5.22.m22.1.1.2.3">subscript</csymbol><ci id="S5.5.p5.22.m22.1.1.2.3.2.cmml" xref="S5.5.p5.22.m22.1.1.2.3.2">π</ci><ci id="S5.5.p5.22.m22.1.1.2.3.3.cmml" xref="S5.5.p5.22.m22.1.1.2.3.3">π</ci></apply></apply><ci id="S5.5.p5.22.m22.1.1.3.cmml" xref="S5.5.p5.22.m22.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.22.m22.1c">hf_{n}\in S</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.22.m22.1d">italic_h italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT β italic_S</annotation></semantics></math> for all <math alttext="n\in\omega" class="ltx_Math" display="inline" id="S5.5.p5.23.m23.1"><semantics id="S5.5.p5.23.m23.1a"><mrow id="S5.5.p5.23.m23.1.1" xref="S5.5.p5.23.m23.1.1.cmml"><mi id="S5.5.p5.23.m23.1.1.2" xref="S5.5.p5.23.m23.1.1.2.cmml">n</mi><mo id="S5.5.p5.23.m23.1.1.1" xref="S5.5.p5.23.m23.1.1.1.cmml">β</mo><mi id="S5.5.p5.23.m23.1.1.3" xref="S5.5.p5.23.m23.1.1.3.cmml">Ο</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.23.m23.1b"><apply id="S5.5.p5.23.m23.1.1.cmml" xref="S5.5.p5.23.m23.1.1"><in id="S5.5.p5.23.m23.1.1.1.cmml" xref="S5.5.p5.23.m23.1.1.1"></in><ci id="S5.5.p5.23.m23.1.1.2.cmml" xref="S5.5.p5.23.m23.1.1.2">π</ci><ci id="S5.5.p5.23.m23.1.1.3.cmml" xref="S5.5.p5.23.m23.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.23.m23.1c">n\in\omega</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.23.m23.1d">italic_n β italic_Ο</annotation></semantics></math>. It follows that <math alttext="\{hf_{n}:n\in\omega\}" class="ltx_Math" display="inline" id="S5.5.p5.24.m24.2"><semantics id="S5.5.p5.24.m24.2a"><mrow id="S5.5.p5.24.m24.2.2.2" xref="S5.5.p5.24.m24.2.2.3.cmml"><mo id="S5.5.p5.24.m24.2.2.2.3" stretchy="false" xref="S5.5.p5.24.m24.2.2.3.1.cmml">{</mo><mrow id="S5.5.p5.24.m24.1.1.1.1" xref="S5.5.p5.24.m24.1.1.1.1.cmml"><mi id="S5.5.p5.24.m24.1.1.1.1.2" xref="S5.5.p5.24.m24.1.1.1.1.2.cmml">h</mi><mo id="S5.5.p5.24.m24.1.1.1.1.1" xref="S5.5.p5.24.m24.1.1.1.1.1.cmml">β’</mo><msub id="S5.5.p5.24.m24.1.1.1.1.3" xref="S5.5.p5.24.m24.1.1.1.1.3.cmml"><mi id="S5.5.p5.24.m24.1.1.1.1.3.2" xref="S5.5.p5.24.m24.1.1.1.1.3.2.cmml">f</mi><mi id="S5.5.p5.24.m24.1.1.1.1.3.3" xref="S5.5.p5.24.m24.1.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S5.5.p5.24.m24.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S5.5.p5.24.m24.2.2.3.1.cmml">:</mo><mrow id="S5.5.p5.24.m24.2.2.2.2" xref="S5.5.p5.24.m24.2.2.2.2.cmml"><mi id="S5.5.p5.24.m24.2.2.2.2.2" xref="S5.5.p5.24.m24.2.2.2.2.2.cmml">n</mi><mo id="S5.5.p5.24.m24.2.2.2.2.1" xref="S5.5.p5.24.m24.2.2.2.2.1.cmml">β</mo><mi id="S5.5.p5.24.m24.2.2.2.2.3" xref="S5.5.p5.24.m24.2.2.2.2.3.cmml">Ο</mi></mrow><mo id="S5.5.p5.24.m24.2.2.2.5" stretchy="false" xref="S5.5.p5.24.m24.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p5.24.m24.2b"><apply id="S5.5.p5.24.m24.2.2.3.cmml" xref="S5.5.p5.24.m24.2.2.2"><csymbol cd="latexml" id="S5.5.p5.24.m24.2.2.3.1.cmml" xref="S5.5.p5.24.m24.2.2.2.3">conditional-set</csymbol><apply id="S5.5.p5.24.m24.1.1.1.1.cmml" xref="S5.5.p5.24.m24.1.1.1.1"><times id="S5.5.p5.24.m24.1.1.1.1.1.cmml" xref="S5.5.p5.24.m24.1.1.1.1.1"></times><ci id="S5.5.p5.24.m24.1.1.1.1.2.cmml" xref="S5.5.p5.24.m24.1.1.1.1.2">β</ci><apply id="S5.5.p5.24.m24.1.1.1.1.3.cmml" xref="S5.5.p5.24.m24.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.5.p5.24.m24.1.1.1.1.3.1.cmml" xref="S5.5.p5.24.m24.1.1.1.1.3">subscript</csymbol><ci id="S5.5.p5.24.m24.1.1.1.1.3.2.cmml" xref="S5.5.p5.24.m24.1.1.1.1.3.2">π</ci><ci id="S5.5.p5.24.m24.1.1.1.1.3.3.cmml" xref="S5.5.p5.24.m24.1.1.1.1.3.3">π</ci></apply></apply><apply id="S5.5.p5.24.m24.2.2.2.2.cmml" xref="S5.5.p5.24.m24.2.2.2.2"><in id="S5.5.p5.24.m24.2.2.2.2.1.cmml" xref="S5.5.p5.24.m24.2.2.2.2.1"></in><ci id="S5.5.p5.24.m24.2.2.2.2.2.cmml" xref="S5.5.p5.24.m24.2.2.2.2.2">π</ci><ci id="S5.5.p5.24.m24.2.2.2.2.3.cmml" xref="S5.5.p5.24.m24.2.2.2.2.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.24.m24.2c">\{hf_{n}:n\in\omega\}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.24.m24.2d">{ italic_h italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT : italic_n β italic_Ο }</annotation></semantics></math> is an infinite closed discrete subset of <math alttext="S" class="ltx_Math" display="inline" id="S5.5.p5.25.m25.1"><semantics id="S5.5.p5.25.m25.1a"><mi id="S5.5.p5.25.m25.1.1" xref="S5.5.p5.25.m25.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.5.p5.25.m25.1b"><ci id="S5.5.p5.25.m25.1.1.cmml" xref="S5.5.p5.25.m25.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.25.m25.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.25.m25.1d">italic_S</annotation></semantics></math>, which contradicts the countable compactness of <math alttext="S" class="ltx_Math" display="inline" id="S5.5.p5.26.m26.1"><semantics id="S5.5.p5.26.m26.1a"><mi id="S5.5.p5.26.m26.1.1" xref="S5.5.p5.26.m26.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.5.p5.26.m26.1b"><ci id="S5.5.p5.26.m26.1.1.cmml" xref="S5.5.p5.26.m26.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p5.26.m26.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.5.p5.26.m26.1d">italic_S</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.6.p6"> <p class="ltx_p" id="S5.6.p6.1">The obtained contradictions imply <math alttext="S=\beta S" class="ltx_Math" display="inline" id="S5.6.p6.1.m1.1"><semantics id="S5.6.p6.1.m1.1a"><mrow id="S5.6.p6.1.m1.1.1" xref="S5.6.p6.1.m1.1.1.cmml"><mi id="S5.6.p6.1.m1.1.1.2" xref="S5.6.p6.1.m1.1.1.2.cmml">S</mi><mo id="S5.6.p6.1.m1.1.1.1" xref="S5.6.p6.1.m1.1.1.1.cmml">=</mo><mrow id="S5.6.p6.1.m1.1.1.3" xref="S5.6.p6.1.m1.1.1.3.cmml"><mi id="S5.6.p6.1.m1.1.1.3.2" xref="S5.6.p6.1.m1.1.1.3.2.cmml">Ξ²</mi><mo id="S5.6.p6.1.m1.1.1.3.1" xref="S5.6.p6.1.m1.1.1.3.1.cmml">β’</mo><mi id="S5.6.p6.1.m1.1.1.3.3" xref="S5.6.p6.1.m1.1.1.3.3.cmml">S</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p6.1.m1.1b"><apply id="S5.6.p6.1.m1.1.1.cmml" xref="S5.6.p6.1.m1.1.1"><eq id="S5.6.p6.1.m1.1.1.1.cmml" xref="S5.6.p6.1.m1.1.1.1"></eq><ci id="S5.6.p6.1.m1.1.1.2.cmml" xref="S5.6.p6.1.m1.1.1.2">π</ci><apply id="S5.6.p6.1.m1.1.1.3.cmml" xref="S5.6.p6.1.m1.1.1.3"><times id="S5.6.p6.1.m1.1.1.3.1.cmml" xref="S5.6.p6.1.m1.1.1.3.1"></times><ci id="S5.6.p6.1.m1.1.1.3.2.cmml" xref="S5.6.p6.1.m1.1.1.3.2">π½</ci><ci id="S5.6.p6.1.m1.1.1.3.3.cmml" xref="S5.6.p6.1.m1.1.1.3.3">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p6.1.m1.1c">S=\beta S</annotation><annotation encoding="application/x-llamapun" id="S5.6.p6.1.m1.1d">italic_S = italic_Ξ² italic_S</annotation></semantics></math>. β</p> </div> </div> <div class="ltx_para" id="S5.p4"> <p class="ltx_p" id="S5.p4.4">A <span class="ltx_text ltx_font_italic" id="S5.p4.4.1">band</span> is a semigroup consisting of idempotents. A semigroup <math alttext="S" class="ltx_Math" display="inline" id="S5.p4.1.m1.1"><semantics id="S5.p4.1.m1.1a"><mi id="S5.p4.1.m1.1.1" xref="S5.p4.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.p4.1.m1.1b"><ci id="S5.p4.1.m1.1.1.cmml" xref="S5.p4.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p4.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.p4.1.m1.1d">italic_S</annotation></semantics></math> is called <span class="ltx_text ltx_font_italic" id="S5.p4.4.2">simple</span> if <math alttext="S" class="ltx_Math" display="inline" id="S5.p4.2.m2.1"><semantics id="S5.p4.2.m2.1a"><mi id="S5.p4.2.m2.1.1" xref="S5.p4.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.p4.2.m2.1b"><ci id="S5.p4.2.m2.1.1.cmml" xref="S5.p4.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p4.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.p4.2.m2.1d">italic_S</annotation></semantics></math> contains no proper two-sided ideals or, in other words, <math alttext="SxS=S" class="ltx_Math" display="inline" id="S5.p4.3.m3.1"><semantics id="S5.p4.3.m3.1a"><mrow id="S5.p4.3.m3.1.1" xref="S5.p4.3.m3.1.1.cmml"><mrow id="S5.p4.3.m3.1.1.2" xref="S5.p4.3.m3.1.1.2.cmml"><mi id="S5.p4.3.m3.1.1.2.2" xref="S5.p4.3.m3.1.1.2.2.cmml">S</mi><mo id="S5.p4.3.m3.1.1.2.1" xref="S5.p4.3.m3.1.1.2.1.cmml">β’</mo><mi id="S5.p4.3.m3.1.1.2.3" xref="S5.p4.3.m3.1.1.2.3.cmml">x</mi><mo id="S5.p4.3.m3.1.1.2.1a" xref="S5.p4.3.m3.1.1.2.1.cmml">β’</mo><mi id="S5.p4.3.m3.1.1.2.4" xref="S5.p4.3.m3.1.1.2.4.cmml">S</mi></mrow><mo id="S5.p4.3.m3.1.1.1" xref="S5.p4.3.m3.1.1.1.cmml">=</mo><mi id="S5.p4.3.m3.1.1.3" xref="S5.p4.3.m3.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p4.3.m3.1b"><apply id="S5.p4.3.m3.1.1.cmml" xref="S5.p4.3.m3.1.1"><eq id="S5.p4.3.m3.1.1.1.cmml" xref="S5.p4.3.m3.1.1.1"></eq><apply id="S5.p4.3.m3.1.1.2.cmml" xref="S5.p4.3.m3.1.1.2"><times id="S5.p4.3.m3.1.1.2.1.cmml" xref="S5.p4.3.m3.1.1.2.1"></times><ci id="S5.p4.3.m3.1.1.2.2.cmml" xref="S5.p4.3.m3.1.1.2.2">π</ci><ci id="S5.p4.3.m3.1.1.2.3.cmml" xref="S5.p4.3.m3.1.1.2.3">π₯</ci><ci id="S5.p4.3.m3.1.1.2.4.cmml" xref="S5.p4.3.m3.1.1.2.4">π</ci></apply><ci id="S5.p4.3.m3.1.1.3.cmml" xref="S5.p4.3.m3.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p4.3.m3.1c">SxS=S</annotation><annotation encoding="application/x-llamapun" id="S5.p4.3.m3.1d">italic_S italic_x italic_S = italic_S</annotation></semantics></math> for any <math alttext="x\in S" class="ltx_Math" display="inline" id="S5.p4.4.m4.1"><semantics id="S5.p4.4.m4.1a"><mrow id="S5.p4.4.m4.1.1" xref="S5.p4.4.m4.1.1.cmml"><mi id="S5.p4.4.m4.1.1.2" xref="S5.p4.4.m4.1.1.2.cmml">x</mi><mo id="S5.p4.4.m4.1.1.1" xref="S5.p4.4.m4.1.1.1.cmml">β</mo><mi id="S5.p4.4.m4.1.1.3" xref="S5.p4.4.m4.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p4.4.m4.1b"><apply id="S5.p4.4.m4.1.1.cmml" xref="S5.p4.4.m4.1.1"><in id="S5.p4.4.m4.1.1.1.cmml" xref="S5.p4.4.m4.1.1.1"></in><ci id="S5.p4.4.m4.1.1.2.cmml" xref="S5.p4.4.m4.1.1.2">π₯</ci><ci id="S5.p4.4.m4.1.1.3.cmml" xref="S5.p4.4.m4.1.1.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p4.4.m4.1c">x\in S</annotation><annotation encoding="application/x-llamapun" id="S5.p4.4.m4.1d">italic_x β italic_S</annotation></semantics></math>. The following example shows that Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem1" title="Theorem A. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">A</span></a> does not generalize over simple bands.</p> </div> <div class="ltx_theorem ltx_theorem_example" id="S5.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem3.1.1.1">Example 5.3</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem3.p1"> <p class="ltx_p" id="S5.Thmtheorem3.p1.21">Let <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.1.m1.1"><semantics id="S5.Thmtheorem3.p1.1.m1.1a"><mi id="S5.Thmtheorem3.p1.1.m1.1.1" xref="S5.Thmtheorem3.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.1.m1.1b"><ci id="S5.Thmtheorem3.p1.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.1.m1.1d">italic_X</annotation></semantics></math> be one of the consistent examples of a locally compact noncompact Nyikos space discussed in the introduction. Let <math alttext="X_{1}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.2.m2.1"><semantics id="S5.Thmtheorem3.p1.2.m2.1a"><msub id="S5.Thmtheorem3.p1.2.m2.1.1" xref="S5.Thmtheorem3.p1.2.m2.1.1.cmml"><mi id="S5.Thmtheorem3.p1.2.m2.1.1.2" xref="S5.Thmtheorem3.p1.2.m2.1.1.2.cmml">X</mi><mn id="S5.Thmtheorem3.p1.2.m2.1.1.3" xref="S5.Thmtheorem3.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.2.m2.1b"><apply id="S5.Thmtheorem3.p1.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.2.m2.1.1.1.cmml" xref="S5.Thmtheorem3.p1.2.m2.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p1.2.m2.1.1.2.cmml" xref="S5.Thmtheorem3.p1.2.m2.1.1.2">π</ci><cn id="S5.Thmtheorem3.p1.2.m2.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.2.m2.1c">X_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.2.m2.1d">italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> be the space <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.3.m3.1"><semantics id="S5.Thmtheorem3.p1.3.m3.1a"><mi id="S5.Thmtheorem3.p1.3.m3.1.1" xref="S5.Thmtheorem3.p1.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.3.m3.1b"><ci id="S5.Thmtheorem3.p1.3.m3.1.1.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.3.m3.1d">italic_X</annotation></semantics></math> endowed with the left zero operation, i.e., <math alttext="xy=x" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.4.m4.1"><semantics id="S5.Thmtheorem3.p1.4.m4.1a"><mrow id="S5.Thmtheorem3.p1.4.m4.1.1" xref="S5.Thmtheorem3.p1.4.m4.1.1.cmml"><mrow id="S5.Thmtheorem3.p1.4.m4.1.1.2" xref="S5.Thmtheorem3.p1.4.m4.1.1.2.cmml"><mi id="S5.Thmtheorem3.p1.4.m4.1.1.2.2" xref="S5.Thmtheorem3.p1.4.m4.1.1.2.2.cmml">x</mi><mo id="S5.Thmtheorem3.p1.4.m4.1.1.2.1" xref="S5.Thmtheorem3.p1.4.m4.1.1.2.1.cmml">β’</mo><mi id="S5.Thmtheorem3.p1.4.m4.1.1.2.3" xref="S5.Thmtheorem3.p1.4.m4.1.1.2.3.cmml">y</mi></mrow><mo id="S5.Thmtheorem3.p1.4.m4.1.1.1" xref="S5.Thmtheorem3.p1.4.m4.1.1.1.cmml">=</mo><mi id="S5.Thmtheorem3.p1.4.m4.1.1.3" xref="S5.Thmtheorem3.p1.4.m4.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.4.m4.1b"><apply id="S5.Thmtheorem3.p1.4.m4.1.1.cmml" xref="S5.Thmtheorem3.p1.4.m4.1.1"><eq id="S5.Thmtheorem3.p1.4.m4.1.1.1.cmml" xref="S5.Thmtheorem3.p1.4.m4.1.1.1"></eq><apply id="S5.Thmtheorem3.p1.4.m4.1.1.2.cmml" xref="S5.Thmtheorem3.p1.4.m4.1.1.2"><times id="S5.Thmtheorem3.p1.4.m4.1.1.2.1.cmml" xref="S5.Thmtheorem3.p1.4.m4.1.1.2.1"></times><ci id="S5.Thmtheorem3.p1.4.m4.1.1.2.2.cmml" xref="S5.Thmtheorem3.p1.4.m4.1.1.2.2">π₯</ci><ci id="S5.Thmtheorem3.p1.4.m4.1.1.2.3.cmml" xref="S5.Thmtheorem3.p1.4.m4.1.1.2.3">π¦</ci></apply><ci id="S5.Thmtheorem3.p1.4.m4.1.1.3.cmml" xref="S5.Thmtheorem3.p1.4.m4.1.1.3">π₯</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.4.m4.1c">xy=x</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.4.m4.1d">italic_x italic_y = italic_x</annotation></semantics></math> for each <math alttext="x,y\in X" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.5.m5.2"><semantics id="S5.Thmtheorem3.p1.5.m5.2a"><mrow id="S5.Thmtheorem3.p1.5.m5.2.3" xref="S5.Thmtheorem3.p1.5.m5.2.3.cmml"><mrow id="S5.Thmtheorem3.p1.5.m5.2.3.2.2" xref="S5.Thmtheorem3.p1.5.m5.2.3.2.1.cmml"><mi id="S5.Thmtheorem3.p1.5.m5.1.1" xref="S5.Thmtheorem3.p1.5.m5.1.1.cmml">x</mi><mo id="S5.Thmtheorem3.p1.5.m5.2.3.2.2.1" xref="S5.Thmtheorem3.p1.5.m5.2.3.2.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.5.m5.2.2" xref="S5.Thmtheorem3.p1.5.m5.2.2.cmml">y</mi></mrow><mo id="S5.Thmtheorem3.p1.5.m5.2.3.1" xref="S5.Thmtheorem3.p1.5.m5.2.3.1.cmml">β</mo><mi id="S5.Thmtheorem3.p1.5.m5.2.3.3" xref="S5.Thmtheorem3.p1.5.m5.2.3.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.5.m5.2b"><apply id="S5.Thmtheorem3.p1.5.m5.2.3.cmml" xref="S5.Thmtheorem3.p1.5.m5.2.3"><in id="S5.Thmtheorem3.p1.5.m5.2.3.1.cmml" xref="S5.Thmtheorem3.p1.5.m5.2.3.1"></in><list id="S5.Thmtheorem3.p1.5.m5.2.3.2.1.cmml" xref="S5.Thmtheorem3.p1.5.m5.2.3.2.2"><ci id="S5.Thmtheorem3.p1.5.m5.1.1.cmml" xref="S5.Thmtheorem3.p1.5.m5.1.1">π₯</ci><ci id="S5.Thmtheorem3.p1.5.m5.2.2.cmml" xref="S5.Thmtheorem3.p1.5.m5.2.2">π¦</ci></list><ci id="S5.Thmtheorem3.p1.5.m5.2.3.3.cmml" xref="S5.Thmtheorem3.p1.5.m5.2.3.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.5.m5.2c">x,y\in X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.5.m5.2d">italic_x , italic_y β italic_X</annotation></semantics></math>. It is easy to see that this operation is continuous and associative, that is, <math alttext="X_{1}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.6.m6.1"><semantics id="S5.Thmtheorem3.p1.6.m6.1a"><msub id="S5.Thmtheorem3.p1.6.m6.1.1" xref="S5.Thmtheorem3.p1.6.m6.1.1.cmml"><mi id="S5.Thmtheorem3.p1.6.m6.1.1.2" xref="S5.Thmtheorem3.p1.6.m6.1.1.2.cmml">X</mi><mn id="S5.Thmtheorem3.p1.6.m6.1.1.3" xref="S5.Thmtheorem3.p1.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.6.m6.1b"><apply id="S5.Thmtheorem3.p1.6.m6.1.1.cmml" xref="S5.Thmtheorem3.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.6.m6.1.1.1.cmml" xref="S5.Thmtheorem3.p1.6.m6.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p1.6.m6.1.1.2.cmml" xref="S5.Thmtheorem3.p1.6.m6.1.1.2">π</ci><cn id="S5.Thmtheorem3.p1.6.m6.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.6.m6.1c">X_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.6.m6.1d">italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is a topological semigroup. Let <math alttext="X_{2}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.7.m7.1"><semantics id="S5.Thmtheorem3.p1.7.m7.1a"><msub id="S5.Thmtheorem3.p1.7.m7.1.1" xref="S5.Thmtheorem3.p1.7.m7.1.1.cmml"><mi id="S5.Thmtheorem3.p1.7.m7.1.1.2" xref="S5.Thmtheorem3.p1.7.m7.1.1.2.cmml">X</mi><mn id="S5.Thmtheorem3.p1.7.m7.1.1.3" xref="S5.Thmtheorem3.p1.7.m7.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.7.m7.1b"><apply id="S5.Thmtheorem3.p1.7.m7.1.1.cmml" xref="S5.Thmtheorem3.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.7.m7.1.1.1.cmml" xref="S5.Thmtheorem3.p1.7.m7.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p1.7.m7.1.1.2.cmml" xref="S5.Thmtheorem3.p1.7.m7.1.1.2">π</ci><cn id="S5.Thmtheorem3.p1.7.m7.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.7.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.7.m7.1c">X_{2}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.7.m7.1d">italic_X start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> be the space <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.8.m8.1"><semantics id="S5.Thmtheorem3.p1.8.m8.1a"><mi id="S5.Thmtheorem3.p1.8.m8.1.1" xref="S5.Thmtheorem3.p1.8.m8.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.8.m8.1b"><ci id="S5.Thmtheorem3.p1.8.m8.1.1.cmml" xref="S5.Thmtheorem3.p1.8.m8.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.8.m8.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.8.m8.1d">italic_X</annotation></semantics></math> endowed with the right zero operation, i.e., <math alttext="xy=y" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.9.m9.1"><semantics id="S5.Thmtheorem3.p1.9.m9.1a"><mrow id="S5.Thmtheorem3.p1.9.m9.1.1" xref="S5.Thmtheorem3.p1.9.m9.1.1.cmml"><mrow id="S5.Thmtheorem3.p1.9.m9.1.1.2" xref="S5.Thmtheorem3.p1.9.m9.1.1.2.cmml"><mi id="S5.Thmtheorem3.p1.9.m9.1.1.2.2" xref="S5.Thmtheorem3.p1.9.m9.1.1.2.2.cmml">x</mi><mo id="S5.Thmtheorem3.p1.9.m9.1.1.2.1" xref="S5.Thmtheorem3.p1.9.m9.1.1.2.1.cmml">β’</mo><mi id="S5.Thmtheorem3.p1.9.m9.1.1.2.3" xref="S5.Thmtheorem3.p1.9.m9.1.1.2.3.cmml">y</mi></mrow><mo id="S5.Thmtheorem3.p1.9.m9.1.1.1" xref="S5.Thmtheorem3.p1.9.m9.1.1.1.cmml">=</mo><mi id="S5.Thmtheorem3.p1.9.m9.1.1.3" xref="S5.Thmtheorem3.p1.9.m9.1.1.3.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.9.m9.1b"><apply id="S5.Thmtheorem3.p1.9.m9.1.1.cmml" xref="S5.Thmtheorem3.p1.9.m9.1.1"><eq id="S5.Thmtheorem3.p1.9.m9.1.1.1.cmml" xref="S5.Thmtheorem3.p1.9.m9.1.1.1"></eq><apply id="S5.Thmtheorem3.p1.9.m9.1.1.2.cmml" xref="S5.Thmtheorem3.p1.9.m9.1.1.2"><times id="S5.Thmtheorem3.p1.9.m9.1.1.2.1.cmml" xref="S5.Thmtheorem3.p1.9.m9.1.1.2.1"></times><ci id="S5.Thmtheorem3.p1.9.m9.1.1.2.2.cmml" xref="S5.Thmtheorem3.p1.9.m9.1.1.2.2">π₯</ci><ci id="S5.Thmtheorem3.p1.9.m9.1.1.2.3.cmml" xref="S5.Thmtheorem3.p1.9.m9.1.1.2.3">π¦</ci></apply><ci id="S5.Thmtheorem3.p1.9.m9.1.1.3.cmml" xref="S5.Thmtheorem3.p1.9.m9.1.1.3">π¦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.9.m9.1c">xy=y</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.9.m9.1d">italic_x italic_y = italic_y</annotation></semantics></math> for each <math alttext="x,y\in X" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.10.m10.2"><semantics id="S5.Thmtheorem3.p1.10.m10.2a"><mrow id="S5.Thmtheorem3.p1.10.m10.2.3" xref="S5.Thmtheorem3.p1.10.m10.2.3.cmml"><mrow id="S5.Thmtheorem3.p1.10.m10.2.3.2.2" xref="S5.Thmtheorem3.p1.10.m10.2.3.2.1.cmml"><mi id="S5.Thmtheorem3.p1.10.m10.1.1" xref="S5.Thmtheorem3.p1.10.m10.1.1.cmml">x</mi><mo id="S5.Thmtheorem3.p1.10.m10.2.3.2.2.1" xref="S5.Thmtheorem3.p1.10.m10.2.3.2.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.10.m10.2.2" xref="S5.Thmtheorem3.p1.10.m10.2.2.cmml">y</mi></mrow><mo id="S5.Thmtheorem3.p1.10.m10.2.3.1" xref="S5.Thmtheorem3.p1.10.m10.2.3.1.cmml">β</mo><mi id="S5.Thmtheorem3.p1.10.m10.2.3.3" xref="S5.Thmtheorem3.p1.10.m10.2.3.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.10.m10.2b"><apply id="S5.Thmtheorem3.p1.10.m10.2.3.cmml" xref="S5.Thmtheorem3.p1.10.m10.2.3"><in id="S5.Thmtheorem3.p1.10.m10.2.3.1.cmml" xref="S5.Thmtheorem3.p1.10.m10.2.3.1"></in><list id="S5.Thmtheorem3.p1.10.m10.2.3.2.1.cmml" xref="S5.Thmtheorem3.p1.10.m10.2.3.2.2"><ci id="S5.Thmtheorem3.p1.10.m10.1.1.cmml" xref="S5.Thmtheorem3.p1.10.m10.1.1">π₯</ci><ci id="S5.Thmtheorem3.p1.10.m10.2.2.cmml" xref="S5.Thmtheorem3.p1.10.m10.2.2">π¦</ci></list><ci id="S5.Thmtheorem3.p1.10.m10.2.3.3.cmml" xref="S5.Thmtheorem3.p1.10.m10.2.3.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.10.m10.2c">x,y\in X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.10.m10.2d">italic_x , italic_y β italic_X</annotation></semantics></math>. Similarly, one can check that <math alttext="X_{2}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.11.m11.1"><semantics id="S5.Thmtheorem3.p1.11.m11.1a"><msub id="S5.Thmtheorem3.p1.11.m11.1.1" xref="S5.Thmtheorem3.p1.11.m11.1.1.cmml"><mi id="S5.Thmtheorem3.p1.11.m11.1.1.2" xref="S5.Thmtheorem3.p1.11.m11.1.1.2.cmml">X</mi><mn id="S5.Thmtheorem3.p1.11.m11.1.1.3" xref="S5.Thmtheorem3.p1.11.m11.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.11.m11.1b"><apply id="S5.Thmtheorem3.p1.11.m11.1.1.cmml" xref="S5.Thmtheorem3.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.11.m11.1.1.1.cmml" xref="S5.Thmtheorem3.p1.11.m11.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p1.11.m11.1.1.2.cmml" xref="S5.Thmtheorem3.p1.11.m11.1.1.2">π</ci><cn id="S5.Thmtheorem3.p1.11.m11.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.11.m11.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.11.m11.1c">X_{2}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.11.m11.1d">italic_X start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is a topological semigroup. Since the finite product of Nyikos spaces remains Nyikos, the direct product <math alttext="S:=X_{1}{\times}X_{2}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.12.m12.1"><semantics id="S5.Thmtheorem3.p1.12.m12.1a"><mrow id="S5.Thmtheorem3.p1.12.m12.1.1" xref="S5.Thmtheorem3.p1.12.m12.1.1.cmml"><mi id="S5.Thmtheorem3.p1.12.m12.1.1.2" xref="S5.Thmtheorem3.p1.12.m12.1.1.2.cmml">S</mi><mo id="S5.Thmtheorem3.p1.12.m12.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.Thmtheorem3.p1.12.m12.1.1.1.cmml">:=</mo><mrow id="S5.Thmtheorem3.p1.12.m12.1.1.3" xref="S5.Thmtheorem3.p1.12.m12.1.1.3.cmml"><msub id="S5.Thmtheorem3.p1.12.m12.1.1.3.2" xref="S5.Thmtheorem3.p1.12.m12.1.1.3.2.cmml"><mi id="S5.Thmtheorem3.p1.12.m12.1.1.3.2.2" xref="S5.Thmtheorem3.p1.12.m12.1.1.3.2.2.cmml">X</mi><mn id="S5.Thmtheorem3.p1.12.m12.1.1.3.2.3" xref="S5.Thmtheorem3.p1.12.m12.1.1.3.2.3.cmml">1</mn></msub><mo id="S5.Thmtheorem3.p1.12.m12.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S5.Thmtheorem3.p1.12.m12.1.1.3.1.cmml">Γ</mo><msub id="S5.Thmtheorem3.p1.12.m12.1.1.3.3" xref="S5.Thmtheorem3.p1.12.m12.1.1.3.3.cmml"><mi id="S5.Thmtheorem3.p1.12.m12.1.1.3.3.2" xref="S5.Thmtheorem3.p1.12.m12.1.1.3.3.2.cmml">X</mi><mn id="S5.Thmtheorem3.p1.12.m12.1.1.3.3.3" xref="S5.Thmtheorem3.p1.12.m12.1.1.3.3.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.12.m12.1b"><apply id="S5.Thmtheorem3.p1.12.m12.1.1.cmml" xref="S5.Thmtheorem3.p1.12.m12.1.1"><csymbol cd="latexml" id="S5.Thmtheorem3.p1.12.m12.1.1.1.cmml" xref="S5.Thmtheorem3.p1.12.m12.1.1.1">assign</csymbol><ci id="S5.Thmtheorem3.p1.12.m12.1.1.2.cmml" xref="S5.Thmtheorem3.p1.12.m12.1.1.2">π</ci><apply id="S5.Thmtheorem3.p1.12.m12.1.1.3.cmml" xref="S5.Thmtheorem3.p1.12.m12.1.1.3"><times id="S5.Thmtheorem3.p1.12.m12.1.1.3.1.cmml" xref="S5.Thmtheorem3.p1.12.m12.1.1.3.1"></times><apply id="S5.Thmtheorem3.p1.12.m12.1.1.3.2.cmml" xref="S5.Thmtheorem3.p1.12.m12.1.1.3.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.12.m12.1.1.3.2.1.cmml" xref="S5.Thmtheorem3.p1.12.m12.1.1.3.2">subscript</csymbol><ci id="S5.Thmtheorem3.p1.12.m12.1.1.3.2.2.cmml" xref="S5.Thmtheorem3.p1.12.m12.1.1.3.2.2">π</ci><cn id="S5.Thmtheorem3.p1.12.m12.1.1.3.2.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.12.m12.1.1.3.2.3">1</cn></apply><apply id="S5.Thmtheorem3.p1.12.m12.1.1.3.3.cmml" xref="S5.Thmtheorem3.p1.12.m12.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.12.m12.1.1.3.3.1.cmml" xref="S5.Thmtheorem3.p1.12.m12.1.1.3.3">subscript</csymbol><ci id="S5.Thmtheorem3.p1.12.m12.1.1.3.3.2.cmml" xref="S5.Thmtheorem3.p1.12.m12.1.1.3.3.2">π</ci><cn id="S5.Thmtheorem3.p1.12.m12.1.1.3.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.12.m12.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.12.m12.1c">S:=X_{1}{\times}X_{2}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.12.m12.1d">italic_S := italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT Γ italic_X start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is a noncompact Nyikos topological semigroup. Note that <math alttext="(a,b)(a,b)=(a,b)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.13.m13.6"><semantics id="S5.Thmtheorem3.p1.13.m13.6a"><mrow id="S5.Thmtheorem3.p1.13.m13.6.7" xref="S5.Thmtheorem3.p1.13.m13.6.7.cmml"><mrow id="S5.Thmtheorem3.p1.13.m13.6.7.2" xref="S5.Thmtheorem3.p1.13.m13.6.7.2.cmml"><mrow id="S5.Thmtheorem3.p1.13.m13.6.7.2.2.2" xref="S5.Thmtheorem3.p1.13.m13.6.7.2.2.1.cmml"><mo id="S5.Thmtheorem3.p1.13.m13.6.7.2.2.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.13.m13.6.7.2.2.1.cmml">(</mo><mi id="S5.Thmtheorem3.p1.13.m13.1.1" xref="S5.Thmtheorem3.p1.13.m13.1.1.cmml">a</mi><mo id="S5.Thmtheorem3.p1.13.m13.6.7.2.2.2.2" xref="S5.Thmtheorem3.p1.13.m13.6.7.2.2.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.13.m13.2.2" xref="S5.Thmtheorem3.p1.13.m13.2.2.cmml">b</mi><mo id="S5.Thmtheorem3.p1.13.m13.6.7.2.2.2.3" stretchy="false" xref="S5.Thmtheorem3.p1.13.m13.6.7.2.2.1.cmml">)</mo></mrow><mo id="S5.Thmtheorem3.p1.13.m13.6.7.2.1" xref="S5.Thmtheorem3.p1.13.m13.6.7.2.1.cmml">β’</mo><mrow id="S5.Thmtheorem3.p1.13.m13.6.7.2.3.2" xref="S5.Thmtheorem3.p1.13.m13.6.7.2.3.1.cmml"><mo id="S5.Thmtheorem3.p1.13.m13.6.7.2.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.13.m13.6.7.2.3.1.cmml">(</mo><mi id="S5.Thmtheorem3.p1.13.m13.3.3" xref="S5.Thmtheorem3.p1.13.m13.3.3.cmml">a</mi><mo id="S5.Thmtheorem3.p1.13.m13.6.7.2.3.2.2" xref="S5.Thmtheorem3.p1.13.m13.6.7.2.3.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.13.m13.4.4" xref="S5.Thmtheorem3.p1.13.m13.4.4.cmml">b</mi><mo id="S5.Thmtheorem3.p1.13.m13.6.7.2.3.2.3" stretchy="false" xref="S5.Thmtheorem3.p1.13.m13.6.7.2.3.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p1.13.m13.6.7.1" xref="S5.Thmtheorem3.p1.13.m13.6.7.1.cmml">=</mo><mrow id="S5.Thmtheorem3.p1.13.m13.6.7.3.2" xref="S5.Thmtheorem3.p1.13.m13.6.7.3.1.cmml"><mo id="S5.Thmtheorem3.p1.13.m13.6.7.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.13.m13.6.7.3.1.cmml">(</mo><mi id="S5.Thmtheorem3.p1.13.m13.5.5" xref="S5.Thmtheorem3.p1.13.m13.5.5.cmml">a</mi><mo id="S5.Thmtheorem3.p1.13.m13.6.7.3.2.2" xref="S5.Thmtheorem3.p1.13.m13.6.7.3.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.13.m13.6.6" xref="S5.Thmtheorem3.p1.13.m13.6.6.cmml">b</mi><mo id="S5.Thmtheorem3.p1.13.m13.6.7.3.2.3" stretchy="false" xref="S5.Thmtheorem3.p1.13.m13.6.7.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.13.m13.6b"><apply id="S5.Thmtheorem3.p1.13.m13.6.7.cmml" xref="S5.Thmtheorem3.p1.13.m13.6.7"><eq id="S5.Thmtheorem3.p1.13.m13.6.7.1.cmml" xref="S5.Thmtheorem3.p1.13.m13.6.7.1"></eq><apply id="S5.Thmtheorem3.p1.13.m13.6.7.2.cmml" xref="S5.Thmtheorem3.p1.13.m13.6.7.2"><times id="S5.Thmtheorem3.p1.13.m13.6.7.2.1.cmml" xref="S5.Thmtheorem3.p1.13.m13.6.7.2.1"></times><interval closure="open" id="S5.Thmtheorem3.p1.13.m13.6.7.2.2.1.cmml" xref="S5.Thmtheorem3.p1.13.m13.6.7.2.2.2"><ci id="S5.Thmtheorem3.p1.13.m13.1.1.cmml" xref="S5.Thmtheorem3.p1.13.m13.1.1">π</ci><ci id="S5.Thmtheorem3.p1.13.m13.2.2.cmml" xref="S5.Thmtheorem3.p1.13.m13.2.2">π</ci></interval><interval closure="open" id="S5.Thmtheorem3.p1.13.m13.6.7.2.3.1.cmml" xref="S5.Thmtheorem3.p1.13.m13.6.7.2.3.2"><ci id="S5.Thmtheorem3.p1.13.m13.3.3.cmml" xref="S5.Thmtheorem3.p1.13.m13.3.3">π</ci><ci id="S5.Thmtheorem3.p1.13.m13.4.4.cmml" xref="S5.Thmtheorem3.p1.13.m13.4.4">π</ci></interval></apply><interval closure="open" id="S5.Thmtheorem3.p1.13.m13.6.7.3.1.cmml" xref="S5.Thmtheorem3.p1.13.m13.6.7.3.2"><ci id="S5.Thmtheorem3.p1.13.m13.5.5.cmml" xref="S5.Thmtheorem3.p1.13.m13.5.5">π</ci><ci id="S5.Thmtheorem3.p1.13.m13.6.6.cmml" xref="S5.Thmtheorem3.p1.13.m13.6.6">π</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.13.m13.6c">(a,b)(a,b)=(a,b)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.13.m13.6d">( italic_a , italic_b ) ( italic_a , italic_b ) = ( italic_a , italic_b )</annotation></semantics></math> for every <math alttext="(a,b)\in S" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.14.m14.2"><semantics id="S5.Thmtheorem3.p1.14.m14.2a"><mrow id="S5.Thmtheorem3.p1.14.m14.2.3" xref="S5.Thmtheorem3.p1.14.m14.2.3.cmml"><mrow id="S5.Thmtheorem3.p1.14.m14.2.3.2.2" xref="S5.Thmtheorem3.p1.14.m14.2.3.2.1.cmml"><mo id="S5.Thmtheorem3.p1.14.m14.2.3.2.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.14.m14.2.3.2.1.cmml">(</mo><mi id="S5.Thmtheorem3.p1.14.m14.1.1" xref="S5.Thmtheorem3.p1.14.m14.1.1.cmml">a</mi><mo id="S5.Thmtheorem3.p1.14.m14.2.3.2.2.2" xref="S5.Thmtheorem3.p1.14.m14.2.3.2.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.14.m14.2.2" xref="S5.Thmtheorem3.p1.14.m14.2.2.cmml">b</mi><mo id="S5.Thmtheorem3.p1.14.m14.2.3.2.2.3" stretchy="false" xref="S5.Thmtheorem3.p1.14.m14.2.3.2.1.cmml">)</mo></mrow><mo id="S5.Thmtheorem3.p1.14.m14.2.3.1" xref="S5.Thmtheorem3.p1.14.m14.2.3.1.cmml">β</mo><mi id="S5.Thmtheorem3.p1.14.m14.2.3.3" xref="S5.Thmtheorem3.p1.14.m14.2.3.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.14.m14.2b"><apply id="S5.Thmtheorem3.p1.14.m14.2.3.cmml" xref="S5.Thmtheorem3.p1.14.m14.2.3"><in id="S5.Thmtheorem3.p1.14.m14.2.3.1.cmml" xref="S5.Thmtheorem3.p1.14.m14.2.3.1"></in><interval closure="open" id="S5.Thmtheorem3.p1.14.m14.2.3.2.1.cmml" xref="S5.Thmtheorem3.p1.14.m14.2.3.2.2"><ci id="S5.Thmtheorem3.p1.14.m14.1.1.cmml" xref="S5.Thmtheorem3.p1.14.m14.1.1">π</ci><ci id="S5.Thmtheorem3.p1.14.m14.2.2.cmml" xref="S5.Thmtheorem3.p1.14.m14.2.2">π</ci></interval><ci id="S5.Thmtheorem3.p1.14.m14.2.3.3.cmml" xref="S5.Thmtheorem3.p1.14.m14.2.3.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.14.m14.2c">(a,b)\in S</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.14.m14.2d">( italic_a , italic_b ) β italic_S</annotation></semantics></math>, which makes <math alttext="S" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.15.m15.1"><semantics id="S5.Thmtheorem3.p1.15.m15.1a"><mi id="S5.Thmtheorem3.p1.15.m15.1.1" xref="S5.Thmtheorem3.p1.15.m15.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.15.m15.1b"><ci id="S5.Thmtheorem3.p1.15.m15.1.1.cmml" xref="S5.Thmtheorem3.p1.15.m15.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.15.m15.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.15.m15.1d">italic_S</annotation></semantics></math> a band. Fix <math alttext="(a,b)\in S" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.16.m16.2"><semantics id="S5.Thmtheorem3.p1.16.m16.2a"><mrow id="S5.Thmtheorem3.p1.16.m16.2.3" xref="S5.Thmtheorem3.p1.16.m16.2.3.cmml"><mrow id="S5.Thmtheorem3.p1.16.m16.2.3.2.2" xref="S5.Thmtheorem3.p1.16.m16.2.3.2.1.cmml"><mo id="S5.Thmtheorem3.p1.16.m16.2.3.2.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.16.m16.2.3.2.1.cmml">(</mo><mi id="S5.Thmtheorem3.p1.16.m16.1.1" xref="S5.Thmtheorem3.p1.16.m16.1.1.cmml">a</mi><mo id="S5.Thmtheorem3.p1.16.m16.2.3.2.2.2" xref="S5.Thmtheorem3.p1.16.m16.2.3.2.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.16.m16.2.2" xref="S5.Thmtheorem3.p1.16.m16.2.2.cmml">b</mi><mo id="S5.Thmtheorem3.p1.16.m16.2.3.2.2.3" stretchy="false" xref="S5.Thmtheorem3.p1.16.m16.2.3.2.1.cmml">)</mo></mrow><mo id="S5.Thmtheorem3.p1.16.m16.2.3.1" xref="S5.Thmtheorem3.p1.16.m16.2.3.1.cmml">β</mo><mi id="S5.Thmtheorem3.p1.16.m16.2.3.3" xref="S5.Thmtheorem3.p1.16.m16.2.3.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.16.m16.2b"><apply id="S5.Thmtheorem3.p1.16.m16.2.3.cmml" xref="S5.Thmtheorem3.p1.16.m16.2.3"><in id="S5.Thmtheorem3.p1.16.m16.2.3.1.cmml" xref="S5.Thmtheorem3.p1.16.m16.2.3.1"></in><interval closure="open" id="S5.Thmtheorem3.p1.16.m16.2.3.2.1.cmml" xref="S5.Thmtheorem3.p1.16.m16.2.3.2.2"><ci id="S5.Thmtheorem3.p1.16.m16.1.1.cmml" xref="S5.Thmtheorem3.p1.16.m16.1.1">π</ci><ci id="S5.Thmtheorem3.p1.16.m16.2.2.cmml" xref="S5.Thmtheorem3.p1.16.m16.2.2">π</ci></interval><ci id="S5.Thmtheorem3.p1.16.m16.2.3.3.cmml" xref="S5.Thmtheorem3.p1.16.m16.2.3.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.16.m16.2c">(a,b)\in S</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.16.m16.2d">( italic_a , italic_b ) β italic_S</annotation></semantics></math>. Then for any <math alttext="(c,d)\in S" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.17.m17.2"><semantics id="S5.Thmtheorem3.p1.17.m17.2a"><mrow id="S5.Thmtheorem3.p1.17.m17.2.3" xref="S5.Thmtheorem3.p1.17.m17.2.3.cmml"><mrow id="S5.Thmtheorem3.p1.17.m17.2.3.2.2" xref="S5.Thmtheorem3.p1.17.m17.2.3.2.1.cmml"><mo id="S5.Thmtheorem3.p1.17.m17.2.3.2.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.17.m17.2.3.2.1.cmml">(</mo><mi id="S5.Thmtheorem3.p1.17.m17.1.1" xref="S5.Thmtheorem3.p1.17.m17.1.1.cmml">c</mi><mo id="S5.Thmtheorem3.p1.17.m17.2.3.2.2.2" xref="S5.Thmtheorem3.p1.17.m17.2.3.2.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.17.m17.2.2" xref="S5.Thmtheorem3.p1.17.m17.2.2.cmml">d</mi><mo id="S5.Thmtheorem3.p1.17.m17.2.3.2.2.3" stretchy="false" xref="S5.Thmtheorem3.p1.17.m17.2.3.2.1.cmml">)</mo></mrow><mo id="S5.Thmtheorem3.p1.17.m17.2.3.1" xref="S5.Thmtheorem3.p1.17.m17.2.3.1.cmml">β</mo><mi id="S5.Thmtheorem3.p1.17.m17.2.3.3" xref="S5.Thmtheorem3.p1.17.m17.2.3.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.17.m17.2b"><apply id="S5.Thmtheorem3.p1.17.m17.2.3.cmml" xref="S5.Thmtheorem3.p1.17.m17.2.3"><in id="S5.Thmtheorem3.p1.17.m17.2.3.1.cmml" xref="S5.Thmtheorem3.p1.17.m17.2.3.1"></in><interval closure="open" id="S5.Thmtheorem3.p1.17.m17.2.3.2.1.cmml" xref="S5.Thmtheorem3.p1.17.m17.2.3.2.2"><ci id="S5.Thmtheorem3.p1.17.m17.1.1.cmml" xref="S5.Thmtheorem3.p1.17.m17.1.1">π</ci><ci id="S5.Thmtheorem3.p1.17.m17.2.2.cmml" xref="S5.Thmtheorem3.p1.17.m17.2.2">π</ci></interval><ci id="S5.Thmtheorem3.p1.17.m17.2.3.3.cmml" xref="S5.Thmtheorem3.p1.17.m17.2.3.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.17.m17.2c">(c,d)\in S</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.17.m17.2d">( italic_c , italic_d ) β italic_S</annotation></semantics></math> we have <math alttext="(c,d)=(c,a)(a,b)(b,d)\in S(a,b)S" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.18.m18.10"><semantics id="S5.Thmtheorem3.p1.18.m18.10a"><mrow id="S5.Thmtheorem3.p1.18.m18.10.11" xref="S5.Thmtheorem3.p1.18.m18.10.11.cmml"><mrow id="S5.Thmtheorem3.p1.18.m18.10.11.2.2" xref="S5.Thmtheorem3.p1.18.m18.10.11.2.1.cmml"><mo id="S5.Thmtheorem3.p1.18.m18.10.11.2.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.18.m18.10.11.2.1.cmml">(</mo><mi id="S5.Thmtheorem3.p1.18.m18.1.1" xref="S5.Thmtheorem3.p1.18.m18.1.1.cmml">c</mi><mo id="S5.Thmtheorem3.p1.18.m18.10.11.2.2.2" xref="S5.Thmtheorem3.p1.18.m18.10.11.2.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.18.m18.2.2" xref="S5.Thmtheorem3.p1.18.m18.2.2.cmml">d</mi><mo id="S5.Thmtheorem3.p1.18.m18.10.11.2.2.3" stretchy="false" xref="S5.Thmtheorem3.p1.18.m18.10.11.2.1.cmml">)</mo></mrow><mo id="S5.Thmtheorem3.p1.18.m18.10.11.3" xref="S5.Thmtheorem3.p1.18.m18.10.11.3.cmml">=</mo><mrow id="S5.Thmtheorem3.p1.18.m18.10.11.4" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.cmml"><mrow id="S5.Thmtheorem3.p1.18.m18.10.11.4.2.2" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.2.1.cmml"><mo id="S5.Thmtheorem3.p1.18.m18.10.11.4.2.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.2.1.cmml">(</mo><mi id="S5.Thmtheorem3.p1.18.m18.3.3" xref="S5.Thmtheorem3.p1.18.m18.3.3.cmml">c</mi><mo id="S5.Thmtheorem3.p1.18.m18.10.11.4.2.2.2" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.2.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.18.m18.4.4" xref="S5.Thmtheorem3.p1.18.m18.4.4.cmml">a</mi><mo id="S5.Thmtheorem3.p1.18.m18.10.11.4.2.2.3" stretchy="false" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.2.1.cmml">)</mo></mrow><mo id="S5.Thmtheorem3.p1.18.m18.10.11.4.1" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.1.cmml">β’</mo><mrow id="S5.Thmtheorem3.p1.18.m18.10.11.4.3.2" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.3.1.cmml"><mo id="S5.Thmtheorem3.p1.18.m18.10.11.4.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.3.1.cmml">(</mo><mi id="S5.Thmtheorem3.p1.18.m18.5.5" xref="S5.Thmtheorem3.p1.18.m18.5.5.cmml">a</mi><mo id="S5.Thmtheorem3.p1.18.m18.10.11.4.3.2.2" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.3.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.18.m18.6.6" xref="S5.Thmtheorem3.p1.18.m18.6.6.cmml">b</mi><mo id="S5.Thmtheorem3.p1.18.m18.10.11.4.3.2.3" stretchy="false" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.3.1.cmml">)</mo></mrow><mo id="S5.Thmtheorem3.p1.18.m18.10.11.4.1a" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.1.cmml">β’</mo><mrow id="S5.Thmtheorem3.p1.18.m18.10.11.4.4.2" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.4.1.cmml"><mo id="S5.Thmtheorem3.p1.18.m18.10.11.4.4.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.4.1.cmml">(</mo><mi id="S5.Thmtheorem3.p1.18.m18.7.7" xref="S5.Thmtheorem3.p1.18.m18.7.7.cmml">b</mi><mo id="S5.Thmtheorem3.p1.18.m18.10.11.4.4.2.2" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.4.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.18.m18.8.8" xref="S5.Thmtheorem3.p1.18.m18.8.8.cmml">d</mi><mo id="S5.Thmtheorem3.p1.18.m18.10.11.4.4.2.3" stretchy="false" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.4.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p1.18.m18.10.11.5" xref="S5.Thmtheorem3.p1.18.m18.10.11.5.cmml">β</mo><mrow id="S5.Thmtheorem3.p1.18.m18.10.11.6" xref="S5.Thmtheorem3.p1.18.m18.10.11.6.cmml"><mi id="S5.Thmtheorem3.p1.18.m18.10.11.6.2" xref="S5.Thmtheorem3.p1.18.m18.10.11.6.2.cmml">S</mi><mo id="S5.Thmtheorem3.p1.18.m18.10.11.6.1" xref="S5.Thmtheorem3.p1.18.m18.10.11.6.1.cmml">β’</mo><mrow id="S5.Thmtheorem3.p1.18.m18.10.11.6.3.2" xref="S5.Thmtheorem3.p1.18.m18.10.11.6.3.1.cmml"><mo id="S5.Thmtheorem3.p1.18.m18.10.11.6.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.18.m18.10.11.6.3.1.cmml">(</mo><mi id="S5.Thmtheorem3.p1.18.m18.9.9" xref="S5.Thmtheorem3.p1.18.m18.9.9.cmml">a</mi><mo id="S5.Thmtheorem3.p1.18.m18.10.11.6.3.2.2" xref="S5.Thmtheorem3.p1.18.m18.10.11.6.3.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.18.m18.10.10" xref="S5.Thmtheorem3.p1.18.m18.10.10.cmml">b</mi><mo id="S5.Thmtheorem3.p1.18.m18.10.11.6.3.2.3" stretchy="false" xref="S5.Thmtheorem3.p1.18.m18.10.11.6.3.1.cmml">)</mo></mrow><mo id="S5.Thmtheorem3.p1.18.m18.10.11.6.1a" xref="S5.Thmtheorem3.p1.18.m18.10.11.6.1.cmml">β’</mo><mi id="S5.Thmtheorem3.p1.18.m18.10.11.6.4" xref="S5.Thmtheorem3.p1.18.m18.10.11.6.4.cmml">S</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.18.m18.10b"><apply id="S5.Thmtheorem3.p1.18.m18.10.11.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.11"><and id="S5.Thmtheorem3.p1.18.m18.10.11a.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.11"></and><apply id="S5.Thmtheorem3.p1.18.m18.10.11b.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.11"><eq id="S5.Thmtheorem3.p1.18.m18.10.11.3.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.11.3"></eq><interval closure="open" id="S5.Thmtheorem3.p1.18.m18.10.11.2.1.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.11.2.2"><ci id="S5.Thmtheorem3.p1.18.m18.1.1.cmml" xref="S5.Thmtheorem3.p1.18.m18.1.1">π</ci><ci id="S5.Thmtheorem3.p1.18.m18.2.2.cmml" xref="S5.Thmtheorem3.p1.18.m18.2.2">π</ci></interval><apply id="S5.Thmtheorem3.p1.18.m18.10.11.4.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.11.4"><times id="S5.Thmtheorem3.p1.18.m18.10.11.4.1.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.1"></times><interval closure="open" id="S5.Thmtheorem3.p1.18.m18.10.11.4.2.1.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.2.2"><ci id="S5.Thmtheorem3.p1.18.m18.3.3.cmml" xref="S5.Thmtheorem3.p1.18.m18.3.3">π</ci><ci id="S5.Thmtheorem3.p1.18.m18.4.4.cmml" xref="S5.Thmtheorem3.p1.18.m18.4.4">π</ci></interval><interval closure="open" id="S5.Thmtheorem3.p1.18.m18.10.11.4.3.1.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.3.2"><ci id="S5.Thmtheorem3.p1.18.m18.5.5.cmml" xref="S5.Thmtheorem3.p1.18.m18.5.5">π</ci><ci id="S5.Thmtheorem3.p1.18.m18.6.6.cmml" xref="S5.Thmtheorem3.p1.18.m18.6.6">π</ci></interval><interval closure="open" id="S5.Thmtheorem3.p1.18.m18.10.11.4.4.1.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.11.4.4.2"><ci id="S5.Thmtheorem3.p1.18.m18.7.7.cmml" xref="S5.Thmtheorem3.p1.18.m18.7.7">π</ci><ci id="S5.Thmtheorem3.p1.18.m18.8.8.cmml" xref="S5.Thmtheorem3.p1.18.m18.8.8">π</ci></interval></apply></apply><apply id="S5.Thmtheorem3.p1.18.m18.10.11c.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.11"><in id="S5.Thmtheorem3.p1.18.m18.10.11.5.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.11.5"></in><share href="https://arxiv.org/html/2503.13666v1#S5.Thmtheorem3.p1.18.m18.10.11.4.cmml" id="S5.Thmtheorem3.p1.18.m18.10.11d.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.11"></share><apply id="S5.Thmtheorem3.p1.18.m18.10.11.6.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.11.6"><times id="S5.Thmtheorem3.p1.18.m18.10.11.6.1.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.11.6.1"></times><ci id="S5.Thmtheorem3.p1.18.m18.10.11.6.2.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.11.6.2">π</ci><interval closure="open" id="S5.Thmtheorem3.p1.18.m18.10.11.6.3.1.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.11.6.3.2"><ci id="S5.Thmtheorem3.p1.18.m18.9.9.cmml" xref="S5.Thmtheorem3.p1.18.m18.9.9">π</ci><ci id="S5.Thmtheorem3.p1.18.m18.10.10.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.10">π</ci></interval><ci id="S5.Thmtheorem3.p1.18.m18.10.11.6.4.cmml" xref="S5.Thmtheorem3.p1.18.m18.10.11.6.4">π</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.18.m18.10c">(c,d)=(c,a)(a,b)(b,d)\in S(a,b)S</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.18.m18.10d">( italic_c , italic_d ) = ( italic_c , italic_a ) ( italic_a , italic_b ) ( italic_b , italic_d ) β italic_S ( italic_a , italic_b ) italic_S</annotation></semantics></math>, witnessing that <math alttext="S=S(a,b)S" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.19.m19.2"><semantics id="S5.Thmtheorem3.p1.19.m19.2a"><mrow id="S5.Thmtheorem3.p1.19.m19.2.3" xref="S5.Thmtheorem3.p1.19.m19.2.3.cmml"><mi id="S5.Thmtheorem3.p1.19.m19.2.3.2" xref="S5.Thmtheorem3.p1.19.m19.2.3.2.cmml">S</mi><mo id="S5.Thmtheorem3.p1.19.m19.2.3.1" xref="S5.Thmtheorem3.p1.19.m19.2.3.1.cmml">=</mo><mrow id="S5.Thmtheorem3.p1.19.m19.2.3.3" xref="S5.Thmtheorem3.p1.19.m19.2.3.3.cmml"><mi id="S5.Thmtheorem3.p1.19.m19.2.3.3.2" xref="S5.Thmtheorem3.p1.19.m19.2.3.3.2.cmml">S</mi><mo id="S5.Thmtheorem3.p1.19.m19.2.3.3.1" xref="S5.Thmtheorem3.p1.19.m19.2.3.3.1.cmml">β’</mo><mrow id="S5.Thmtheorem3.p1.19.m19.2.3.3.3.2" xref="S5.Thmtheorem3.p1.19.m19.2.3.3.3.1.cmml"><mo id="S5.Thmtheorem3.p1.19.m19.2.3.3.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.19.m19.2.3.3.3.1.cmml">(</mo><mi id="S5.Thmtheorem3.p1.19.m19.1.1" xref="S5.Thmtheorem3.p1.19.m19.1.1.cmml">a</mi><mo id="S5.Thmtheorem3.p1.19.m19.2.3.3.3.2.2" xref="S5.Thmtheorem3.p1.19.m19.2.3.3.3.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.19.m19.2.2" xref="S5.Thmtheorem3.p1.19.m19.2.2.cmml">b</mi><mo id="S5.Thmtheorem3.p1.19.m19.2.3.3.3.2.3" stretchy="false" xref="S5.Thmtheorem3.p1.19.m19.2.3.3.3.1.cmml">)</mo></mrow><mo id="S5.Thmtheorem3.p1.19.m19.2.3.3.1a" xref="S5.Thmtheorem3.p1.19.m19.2.3.3.1.cmml">β’</mo><mi id="S5.Thmtheorem3.p1.19.m19.2.3.3.4" xref="S5.Thmtheorem3.p1.19.m19.2.3.3.4.cmml">S</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.19.m19.2b"><apply id="S5.Thmtheorem3.p1.19.m19.2.3.cmml" xref="S5.Thmtheorem3.p1.19.m19.2.3"><eq id="S5.Thmtheorem3.p1.19.m19.2.3.1.cmml" xref="S5.Thmtheorem3.p1.19.m19.2.3.1"></eq><ci id="S5.Thmtheorem3.p1.19.m19.2.3.2.cmml" xref="S5.Thmtheorem3.p1.19.m19.2.3.2">π</ci><apply id="S5.Thmtheorem3.p1.19.m19.2.3.3.cmml" xref="S5.Thmtheorem3.p1.19.m19.2.3.3"><times id="S5.Thmtheorem3.p1.19.m19.2.3.3.1.cmml" xref="S5.Thmtheorem3.p1.19.m19.2.3.3.1"></times><ci id="S5.Thmtheorem3.p1.19.m19.2.3.3.2.cmml" xref="S5.Thmtheorem3.p1.19.m19.2.3.3.2">π</ci><interval closure="open" id="S5.Thmtheorem3.p1.19.m19.2.3.3.3.1.cmml" xref="S5.Thmtheorem3.p1.19.m19.2.3.3.3.2"><ci id="S5.Thmtheorem3.p1.19.m19.1.1.cmml" xref="S5.Thmtheorem3.p1.19.m19.1.1">π</ci><ci id="S5.Thmtheorem3.p1.19.m19.2.2.cmml" xref="S5.Thmtheorem3.p1.19.m19.2.2">π</ci></interval><ci id="S5.Thmtheorem3.p1.19.m19.2.3.3.4.cmml" xref="S5.Thmtheorem3.p1.19.m19.2.3.3.4">π</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.19.m19.2c">S=S(a,b)S</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.19.m19.2d">italic_S = italic_S ( italic_a , italic_b ) italic_S</annotation></semantics></math>. Since the point <math alttext="(a,b)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.20.m20.2"><semantics id="S5.Thmtheorem3.p1.20.m20.2a"><mrow id="S5.Thmtheorem3.p1.20.m20.2.3.2" xref="S5.Thmtheorem3.p1.20.m20.2.3.1.cmml"><mo id="S5.Thmtheorem3.p1.20.m20.2.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.20.m20.2.3.1.cmml">(</mo><mi id="S5.Thmtheorem3.p1.20.m20.1.1" xref="S5.Thmtheorem3.p1.20.m20.1.1.cmml">a</mi><mo id="S5.Thmtheorem3.p1.20.m20.2.3.2.2" xref="S5.Thmtheorem3.p1.20.m20.2.3.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.20.m20.2.2" xref="S5.Thmtheorem3.p1.20.m20.2.2.cmml">b</mi><mo id="S5.Thmtheorem3.p1.20.m20.2.3.2.3" stretchy="false" xref="S5.Thmtheorem3.p1.20.m20.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.20.m20.2b"><interval closure="open" id="S5.Thmtheorem3.p1.20.m20.2.3.1.cmml" xref="S5.Thmtheorem3.p1.20.m20.2.3.2"><ci id="S5.Thmtheorem3.p1.20.m20.1.1.cmml" xref="S5.Thmtheorem3.p1.20.m20.1.1">π</ci><ci id="S5.Thmtheorem3.p1.20.m20.2.2.cmml" xref="S5.Thmtheorem3.p1.20.m20.2.2">π</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.20.m20.2c">(a,b)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.20.m20.2d">( italic_a , italic_b )</annotation></semantics></math> was chosen arbitrarily, the band <math alttext="S" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.21.m21.1"><semantics id="S5.Thmtheorem3.p1.21.m21.1a"><mi id="S5.Thmtheorem3.p1.21.m21.1.1" xref="S5.Thmtheorem3.p1.21.m21.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.21.m21.1b"><ci id="S5.Thmtheorem3.p1.21.m21.1.1.cmml" xref="S5.Thmtheorem3.p1.21.m21.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.21.m21.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.21.m21.1d">italic_S</annotation></semantics></math> is simple.</p> </div> </div> <div class="ltx_para" id="S5.p5"> <p class="ltx_p" id="S5.p5.1">The <span class="ltx_text ltx_font_italic" id="S5.p5.1.1">Ostaszewski space</span> is constructed under (<math alttext="\diamondsuit" class="ltx_Math" display="inline" id="S5.p5.1.m1.1"><semantics id="S5.p5.1.m1.1a"><mi id="S5.p5.1.m1.1.1" mathvariant="normal" xref="S5.p5.1.m1.1.1.cmml">β’</mi><annotation-xml encoding="MathML-Content" id="S5.p5.1.m1.1b"><ci id="S5.p5.1.m1.1.1.cmml" xref="S5.p5.1.m1.1.1">β’</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p5.1.m1.1c">\diamondsuit</annotation><annotation encoding="application/x-llamapun" id="S5.p5.1.m1.1d">β’</annotation></semantics></math>) in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#bib.bib43" title="">43</a>]</cite>. It is a Nyikos space and possesses (among others) the following properties: locally compact, hereditary separable, each compact subspace is countable. The results of this paper establish another peculiar property of the Ostaszewski space.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S5.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem4.1.1.1">Proposition 5.4</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem4.p1"> <p class="ltx_p" id="S5.Thmtheorem4.p1.2"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem4.p1.2.2">Let <math alttext="S" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.1.1.m1.1"><semantics id="S5.Thmtheorem4.p1.1.1.m1.1a"><mi id="S5.Thmtheorem4.p1.1.1.m1.1.1" xref="S5.Thmtheorem4.p1.1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.1.1.m1.1b"><ci id="S5.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem4.p1.1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.1.1.m1.1d">italic_S</annotation></semantics></math> be the Ostaszewski space endowed with a continuous semigroup operation. Then each inverse subsemigroup of <math alttext="S" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.2.2.m2.1"><semantics id="S5.Thmtheorem4.p1.2.2.m2.1a"><mi id="S5.Thmtheorem4.p1.2.2.m2.1.1" xref="S5.Thmtheorem4.p1.2.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.2.2.m2.1b"><ci id="S5.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem4.p1.2.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.2.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.2.2.m2.1d">italic_S</annotation></semantics></math> is countable.</span></p> </div> </div> <div class="ltx_proof" id="S5.7"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.7.p1"> <p class="ltx_p" id="S5.7.p1.10">Let <math alttext="Y" class="ltx_Math" display="inline" id="S5.7.p1.1.m1.1"><semantics id="S5.7.p1.1.m1.1a"><mi id="S5.7.p1.1.m1.1.1" xref="S5.7.p1.1.m1.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S5.7.p1.1.m1.1b"><ci id="S5.7.p1.1.m1.1.1.cmml" xref="S5.7.p1.1.m1.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.7.p1.1.m1.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S5.7.p1.1.m1.1d">italic_Y</annotation></semantics></math> be an inverse subsemigroup of <math alttext="S" class="ltx_Math" display="inline" id="S5.7.p1.2.m2.1"><semantics id="S5.7.p1.2.m2.1a"><mi id="S5.7.p1.2.m2.1.1" xref="S5.7.p1.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.7.p1.2.m2.1b"><ci id="S5.7.p1.2.m2.1.1.cmml" xref="S5.7.p1.2.m2.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.7.p1.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.7.p1.2.m2.1d">italic_S</annotation></semantics></math>. Note that <math alttext="\overline{Y}" class="ltx_Math" display="inline" id="S5.7.p1.3.m3.1"><semantics id="S5.7.p1.3.m3.1a"><mover accent="true" id="S5.7.p1.3.m3.1.1" xref="S5.7.p1.3.m3.1.1.cmml"><mi id="S5.7.p1.3.m3.1.1.2" xref="S5.7.p1.3.m3.1.1.2.cmml">Y</mi><mo id="S5.7.p1.3.m3.1.1.1" xref="S5.7.p1.3.m3.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S5.7.p1.3.m3.1b"><apply id="S5.7.p1.3.m3.1.1.cmml" xref="S5.7.p1.3.m3.1.1"><ci id="S5.7.p1.3.m3.1.1.1.cmml" xref="S5.7.p1.3.m3.1.1.1">Β―</ci><ci id="S5.7.p1.3.m3.1.1.2.cmml" xref="S5.7.p1.3.m3.1.1.2">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.7.p1.3.m3.1c">\overline{Y}</annotation><annotation encoding="application/x-llamapun" id="S5.7.p1.3.m3.1d">overΒ― start_ARG italic_Y end_ARG</annotation></semantics></math> is a locally compact first-countable countably compact topological subsemigroup of <math alttext="S" class="ltx_Math" display="inline" id="S5.7.p1.4.m4.1"><semantics id="S5.7.p1.4.m4.1a"><mi id="S5.7.p1.4.m4.1.1" xref="S5.7.p1.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.7.p1.4.m4.1b"><ci id="S5.7.p1.4.m4.1.1.cmml" xref="S5.7.p1.4.m4.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.7.p1.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.7.p1.4.m4.1d">italic_S</annotation></semantics></math>. Since <math alttext="S" class="ltx_Math" display="inline" id="S5.7.p1.5.m5.1"><semantics id="S5.7.p1.5.m5.1a"><mi id="S5.7.p1.5.m5.1.1" xref="S5.7.p1.5.m5.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.7.p1.5.m5.1b"><ci id="S5.7.p1.5.m5.1.1.cmml" xref="S5.7.p1.5.m5.1.1">π</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.7.p1.5.m5.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.7.p1.5.m5.1d">italic_S</annotation></semantics></math> is hereditary separable, <math alttext="\overline{Y}" class="ltx_Math" display="inline" id="S5.7.p1.6.m6.1"><semantics id="S5.7.p1.6.m6.1a"><mover accent="true" id="S5.7.p1.6.m6.1.1" xref="S5.7.p1.6.m6.1.1.cmml"><mi id="S5.7.p1.6.m6.1.1.2" xref="S5.7.p1.6.m6.1.1.2.cmml">Y</mi><mo id="S5.7.p1.6.m6.1.1.1" xref="S5.7.p1.6.m6.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S5.7.p1.6.m6.1b"><apply id="S5.7.p1.6.m6.1.1.cmml" xref="S5.7.p1.6.m6.1.1"><ci id="S5.7.p1.6.m6.1.1.1.cmml" xref="S5.7.p1.6.m6.1.1.1">Β―</ci><ci id="S5.7.p1.6.m6.1.1.2.cmml" xref="S5.7.p1.6.m6.1.1.2">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.7.p1.6.m6.1c">\overline{Y}</annotation><annotation encoding="application/x-llamapun" id="S5.7.p1.6.m6.1d">overΒ― start_ARG italic_Y end_ARG</annotation></semantics></math> is a locally compact Nyikos topological semigroup. Corollary <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#S1.Thmtheorem4" title="Corollary 1.4. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">1.4</span></a> implies that <math alttext="\overline{Y}" class="ltx_Math" display="inline" id="S5.7.p1.7.m7.1"><semantics id="S5.7.p1.7.m7.1a"><mover accent="true" id="S5.7.p1.7.m7.1.1" xref="S5.7.p1.7.m7.1.1.cmml"><mi id="S5.7.p1.7.m7.1.1.2" xref="S5.7.p1.7.m7.1.1.2.cmml">Y</mi><mo id="S5.7.p1.7.m7.1.1.1" xref="S5.7.p1.7.m7.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S5.7.p1.7.m7.1b"><apply id="S5.7.p1.7.m7.1.1.cmml" xref="S5.7.p1.7.m7.1.1"><ci id="S5.7.p1.7.m7.1.1.1.cmml" xref="S5.7.p1.7.m7.1.1.1">Β―</ci><ci id="S5.7.p1.7.m7.1.1.2.cmml" xref="S5.7.p1.7.m7.1.1.2">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.7.p1.7.m7.1c">\overline{Y}</annotation><annotation encoding="application/x-llamapun" id="S5.7.p1.7.m7.1d">overΒ― start_ARG italic_Y end_ARG</annotation></semantics></math> is a topological inverse semigroup. By Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.13666v1#Thmltheorem1" title="Theorem A. β£ 1. Introduction β£ Countably compact inverse semigroups and Nyikos problem"><span class="ltx_text ltx_ref_tag">A</span></a>, <math alttext="\overline{Y}" class="ltx_Math" display="inline" id="S5.7.p1.8.m8.1"><semantics id="S5.7.p1.8.m8.1a"><mover accent="true" id="S5.7.p1.8.m8.1.1" xref="S5.7.p1.8.m8.1.1.cmml"><mi id="S5.7.p1.8.m8.1.1.2" xref="S5.7.p1.8.m8.1.1.2.cmml">Y</mi><mo id="S5.7.p1.8.m8.1.1.1" xref="S5.7.p1.8.m8.1.1.1.cmml">Β―</mo></mover><annotation-xml encoding="MathML-Content" id="S5.7.p1.8.m8.1b"><apply id="S5.7.p1.8.m8.1.1.cmml" xref="S5.7.p1.8.m8.1.1"><ci id="S5.7.p1.8.m8.1.1.1.cmml" xref="S5.7.p1.8.m8.1.1.1">Β―</ci><ci id="S5.7.p1.8.m8.1.1.2.cmml" xref="S5.7.p1.8.m8.1.1.2">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.7.p1.8.m8.1c">\overline{Y}</annotation><annotation encoding="application/x-llamapun" id="S5.7.p1.8.m8.1d">overΒ― start_ARG italic_Y end_ARG</annotation></semantics></math> is compact. Since each compact subspace of the Ostaszewski space is countable, we get <math alttext="|\overline{Y}|\leq\omega" class="ltx_Math" display="inline" id="S5.7.p1.9.m9.1"><semantics id="S5.7.p1.9.m9.1a"><mrow id="S5.7.p1.9.m9.1.2" xref="S5.7.p1.9.m9.1.2.cmml"><mrow id="S5.7.p1.9.m9.1.2.2.2" xref="S5.7.p1.9.m9.1.2.2.1.cmml"><mo id="S5.7.p1.9.m9.1.2.2.2.1" stretchy="false" xref="S5.7.p1.9.m9.1.2.2.1.1.cmml">|</mo><mover accent="true" id="S5.7.p1.9.m9.1.1" xref="S5.7.p1.9.m9.1.1.cmml"><mi id="S5.7.p1.9.m9.1.1.2" xref="S5.7.p1.9.m9.1.1.2.cmml">Y</mi><mo id="S5.7.p1.9.m9.1.1.1" xref="S5.7.p1.9.m9.1.1.1.cmml">Β―</mo></mover><mo id="S5.7.p1.9.m9.1.2.2.2.2" stretchy="false" xref="S5.7.p1.9.m9.1.2.2.1.1.cmml">|</mo></mrow><mo id="S5.7.p1.9.m9.1.2.1" xref="S5.7.p1.9.m9.1.2.1.cmml">β€</mo><mi id="S5.7.p1.9.m9.1.2.3" xref="S5.7.p1.9.m9.1.2.3.cmml">Ο</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.7.p1.9.m9.1b"><apply id="S5.7.p1.9.m9.1.2.cmml" xref="S5.7.p1.9.m9.1.2"><leq id="S5.7.p1.9.m9.1.2.1.cmml" xref="S5.7.p1.9.m9.1.2.1"></leq><apply id="S5.7.p1.9.m9.1.2.2.1.cmml" xref="S5.7.p1.9.m9.1.2.2.2"><abs id="S5.7.p1.9.m9.1.2.2.1.1.cmml" xref="S5.7.p1.9.m9.1.2.2.2.1"></abs><apply id="S5.7.p1.9.m9.1.1.cmml" xref="S5.7.p1.9.m9.1.1"><ci id="S5.7.p1.9.m9.1.1.1.cmml" xref="S5.7.p1.9.m9.1.1.1">Β―</ci><ci id="S5.7.p1.9.m9.1.1.2.cmml" xref="S5.7.p1.9.m9.1.1.2">π</ci></apply></apply><ci id="S5.7.p1.9.m9.1.2.3.cmml" xref="S5.7.p1.9.m9.1.2.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.7.p1.9.m9.1c">|\overline{Y}|\leq\omega</annotation><annotation encoding="application/x-llamapun" id="S5.7.p1.9.m9.1d">| overΒ― start_ARG italic_Y end_ARG | β€ italic_Ο</annotation></semantics></math>. Hence <math alttext="|Y|\leq\omega" class="ltx_Math" display="inline" id="S5.7.p1.10.m10.1"><semantics id="S5.7.p1.10.m10.1a"><mrow id="S5.7.p1.10.m10.1.2" xref="S5.7.p1.10.m10.1.2.cmml"><mrow id="S5.7.p1.10.m10.1.2.2.2" xref="S5.7.p1.10.m10.1.2.2.1.cmml"><mo id="S5.7.p1.10.m10.1.2.2.2.1" stretchy="false" xref="S5.7.p1.10.m10.1.2.2.1.1.cmml">|</mo><mi id="S5.7.p1.10.m10.1.1" xref="S5.7.p1.10.m10.1.1.cmml">Y</mi><mo id="S5.7.p1.10.m10.1.2.2.2.2" stretchy="false" xref="S5.7.p1.10.m10.1.2.2.1.1.cmml">|</mo></mrow><mo id="S5.7.p1.10.m10.1.2.1" xref="S5.7.p1.10.m10.1.2.1.cmml">β€</mo><mi id="S5.7.p1.10.m10.1.2.3" xref="S5.7.p1.10.m10.1.2.3.cmml">Ο</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.7.p1.10.m10.1b"><apply id="S5.7.p1.10.m10.1.2.cmml" xref="S5.7.p1.10.m10.1.2"><leq id="S5.7.p1.10.m10.1.2.1.cmml" xref="S5.7.p1.10.m10.1.2.1"></leq><apply id="S5.7.p1.10.m10.1.2.2.1.cmml" xref="S5.7.p1.10.m10.1.2.2.2"><abs id="S5.7.p1.10.m10.1.2.2.1.1.cmml" xref="S5.7.p1.10.m10.1.2.2.2.1"></abs><ci id="S5.7.p1.10.m10.1.1.cmml" xref="S5.7.p1.10.m10.1.1">π</ci></apply><ci id="S5.7.p1.10.m10.1.2.3.cmml" xref="S5.7.p1.10.m10.1.2.3">π</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.7.p1.10.m10.1c">|Y|\leq\omega</annotation><annotation encoding="application/x-llamapun" id="S5.7.p1.10.m10.1d">| italic_Y | β€ italic_Ο</annotation></semantics></math>, as required. β</p> </div> </div> <div class="ltx_para" id="S5.p6"> <p class="ltx_p" id="S5.p6.1">We finish this paper with the following conjecture.</p> </div> <div class="ltx_theorem ltx_theorem_conjecture" id="S5.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem5.1.1.1">Conjecture 5.5</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem5.p1"> <p class="ltx_p" id="S5.Thmtheorem5.p1.1">Each Nyikos inverse topological semigroup is compact.</p> </div> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_tag_bibitem">[1]</span> <span class="ltx_bibblock"> T. 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