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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>The Δ₁-property of 𝑋 is equivalent to the Choquet property of 𝐵₁⁢(𝑋)</title>  <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> <base href="/html/2503.10491v1/"/></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/2503.10491v1#S1" title="In The Δ₁-property of 𝑋 is equivalent to the Choquet property of 𝐵₁⁢(𝑋)"><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/2503.10491v1#S2" title="In The Δ₁-property of 𝑋 is equivalent to the Choquet property of 𝐵₁⁢(𝑋)"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Main definitions and notation</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.10491v1#S3" title="In The Δ₁-property of 𝑋 is equivalent to the Choquet property of 𝐵₁⁢(𝑋)"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Main results</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.10491v1#S4" title="In The Δ₁-property of 𝑋 is equivalent to the Choquet property of 𝐵₁⁢(𝑋)"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Solution of open questions</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line ltx_fleqn"> <h1 class="ltx_title ltx_title_document">The <math alttext="\Delta_{1}" class="ltx_Math" display="inline" id="id1.m1.1"><semantics id="id1.m1.1b"><msub id="id1.m1.1.1" xref="id1.m1.1.1.cmml"><mi id="id1.m1.1.1.2" mathvariant="normal" xref="id1.m1.1.1.2.cmml">Δ</mi><mn id="id1.m1.1.1.3" xref="id1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="id1.m1.1c"><apply id="id1.m1.1.1.cmml" xref="id1.m1.1.1"><csymbol cd="ambiguous" id="id1.m1.1.1.1.cmml" xref="id1.m1.1.1">subscript</csymbol><ci id="id1.m1.1.1.2.cmml" xref="id1.m1.1.1.2">Δ</ci><cn id="id1.m1.1.1.3.cmml" type="integer" xref="id1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id1.m1.1d">\Delta_{1}</annotation><annotation encoding="application/x-llamapun" id="id1.m1.1e">roman_Δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-property of <math alttext="X" class="ltx_Math" display="inline" id="id2.m2.1"><semantics id="id2.m2.1b"><mi id="id2.m2.1.1" xref="id2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="id2.m2.1c"><ci id="id2.m2.1.1.cmml" xref="id2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="id2.m2.1d">X</annotation><annotation encoding="application/x-llamapun" id="id2.m2.1e">italic_X</annotation></semantics></math> is equivalent to the Choquet property of <math alttext="B_{1}(X)" class="ltx_Math" display="inline" id="id3.m3.1"><semantics id="id3.m3.1b"><mrow id="id3.m3.1.2" xref="id3.m3.1.2.cmml"><msub id="id3.m3.1.2.2" xref="id3.m3.1.2.2.cmml"><mi id="id3.m3.1.2.2.2" xref="id3.m3.1.2.2.2.cmml">B</mi><mn id="id3.m3.1.2.2.3" xref="id3.m3.1.2.2.3.cmml">1</mn></msub><mo id="id3.m3.1.2.1" xref="id3.m3.1.2.1.cmml">⁢</mo><mrow id="id3.m3.1.2.3.2" xref="id3.m3.1.2.cmml"><mo id="id3.m3.1.2.3.2.1" stretchy="false" xref="id3.m3.1.2.cmml">(</mo><mi id="id3.m3.1.1" xref="id3.m3.1.1.cmml">X</mi><mo id="id3.m3.1.2.3.2.2" stretchy="false" xref="id3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id3.m3.1c"><apply id="id3.m3.1.2.cmml" xref="id3.m3.1.2"><times id="id3.m3.1.2.1.cmml" xref="id3.m3.1.2.1"></times><apply id="id3.m3.1.2.2.cmml" xref="id3.m3.1.2.2"><csymbol cd="ambiguous" id="id3.m3.1.2.2.1.cmml" xref="id3.m3.1.2.2">subscript</csymbol><ci id="id3.m3.1.2.2.2.cmml" xref="id3.m3.1.2.2.2">𝐵</ci><cn id="id3.m3.1.2.2.3.cmml" type="integer" xref="id3.m3.1.2.2.3">1</cn></apply><ci id="id3.m3.1.1.cmml" xref="id3.m3.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id3.m3.1d">B_{1}(X)</annotation><annotation encoding="application/x-llamapun" id="id3.m3.1e">italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_X )</annotation></semantics></math> </h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Alexander V. Osipov </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_address">Krasovskii Institute of Mathematics and Mechanics, <br class="ltx_break"/>Ural Federal University, Yekaterinburg, Russia </span> <span class="ltx_contact ltx_role_email"><a href="mailto:OAB@list.ru">OAB@list.ru</a> </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id16.13">We give a characterization of the <math alttext="\Delta_{1}" class="ltx_Math" display="inline" id="id4.1.m1.1"><semantics id="id4.1.m1.1a"><msub id="id4.1.m1.1.1" xref="id4.1.m1.1.1.cmml"><mi id="id4.1.m1.1.1.2" mathvariant="normal" xref="id4.1.m1.1.1.2.cmml">Δ</mi><mn id="id4.1.m1.1.1.3" xref="id4.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="id4.1.m1.1b"><apply id="id4.1.m1.1.1.cmml" xref="id4.1.m1.1.1"><csymbol cd="ambiguous" id="id4.1.m1.1.1.1.cmml" xref="id4.1.m1.1.1">subscript</csymbol><ci id="id4.1.m1.1.1.2.cmml" xref="id4.1.m1.1.1.2">Δ</ci><cn id="id4.1.m1.1.1.3.cmml" type="integer" xref="id4.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id4.1.m1.1c">\Delta_{1}</annotation><annotation encoding="application/x-llamapun" id="id4.1.m1.1d">roman_Δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-property of any Tychonoff space <math alttext="X" class="ltx_Math" display="inline" id="id5.2.m2.1"><semantics id="id5.2.m2.1a"><mi id="id5.2.m2.1.1" xref="id5.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="id5.2.m2.1b"><ci id="id5.2.m2.1.1.cmml" xref="id5.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="id5.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="id5.2.m2.1d">italic_X</annotation></semantics></math> in terms of the function space <math alttext="B_{1}(X)" class="ltx_Math" display="inline" id="id6.3.m3.1"><semantics id="id6.3.m3.1a"><mrow id="id6.3.m3.1.2" xref="id6.3.m3.1.2.cmml"><msub id="id6.3.m3.1.2.2" xref="id6.3.m3.1.2.2.cmml"><mi id="id6.3.m3.1.2.2.2" xref="id6.3.m3.1.2.2.2.cmml">B</mi><mn id="id6.3.m3.1.2.2.3" xref="id6.3.m3.1.2.2.3.cmml">1</mn></msub><mo id="id6.3.m3.1.2.1" xref="id6.3.m3.1.2.1.cmml">⁢</mo><mrow id="id6.3.m3.1.2.3.2" xref="id6.3.m3.1.2.cmml"><mo id="id6.3.m3.1.2.3.2.1" stretchy="false" xref="id6.3.m3.1.2.cmml">(</mo><mi id="id6.3.m3.1.1" xref="id6.3.m3.1.1.cmml">X</mi><mo id="id6.3.m3.1.2.3.2.2" stretchy="false" xref="id6.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id6.3.m3.1b"><apply id="id6.3.m3.1.2.cmml" xref="id6.3.m3.1.2"><times id="id6.3.m3.1.2.1.cmml" xref="id6.3.m3.1.2.1"></times><apply id="id6.3.m3.1.2.2.cmml" xref="id6.3.m3.1.2.2"><csymbol cd="ambiguous" id="id6.3.m3.1.2.2.1.cmml" xref="id6.3.m3.1.2.2">subscript</csymbol><ci id="id6.3.m3.1.2.2.2.cmml" xref="id6.3.m3.1.2.2.2">𝐵</ci><cn id="id6.3.m3.1.2.2.3.cmml" type="integer" xref="id6.3.m3.1.2.2.3">1</cn></apply><ci id="id6.3.m3.1.1.cmml" xref="id6.3.m3.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id6.3.m3.1c">B_{1}(X)</annotation><annotation encoding="application/x-llamapun" id="id6.3.m3.1d">italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_X )</annotation></semantics></math> of all Baire-one real-valued