The closure-complement-frontier problem in saturated polytopological spaces

<!DOCTYPE html> <html lang="en-US" xml:lang="en-US"> <head> <meta charset="utf-8"> <meta name="viewport" content="width=device-width, initial-scale=1.0"> <title> The closure-complement-frontier problem in saturated polytopological spaces | New Zealand Journal of Mathematics </title> <meta name="generator" content="Open Journal Systems 3.3.0.7"> <link rel="schema.DC" href="http://purl.org/dc/elements/1.1/" /> <meta name="DC.Creator.PersonalName" content="Sara Canilang"/> <meta name="DC.Creator.PersonalName" content="Michael P. Cohen"/> <meta name="DC.Creator.PersonalName" content="Nicolas Graese"/> <meta name="DC.Creator.PersonalName" content="Ian Seong"/> <meta name="DC.Date.created" scheme="ISO8601" content="2021-08-06"/> <meta name="DC.Date.dateSubmitted" scheme="ISO8601" content="2021-08-06"/> <meta name="DC.Date.issued" scheme="ISO8601" content="2021-07-29"/> <meta name="DC.Date.modified" scheme="ISO8601" content="2021-08-12"/> <meta name="DC.Description" xml:lang="en" content="Let $X$ be a space equipped with $n$ topologies $\tau_1,\ldots,\tau_n$ which are pairwise comparable and saturated, and for each $1\leq i\leq n$ let $k_i$ and $f_i$ be the associated topological closure and frontier operators, respectively. Inspired by the closure-complement theorem of Kuratowski, we prove that the monoid of set operators $\mathcal{KF}_n$ generated by $\{k_i,f_i:1\leq i\leq n\}\cup\{c\}$ (where $c$ denotes the set complement operator) has cardinality no more than $2p(n)$ where $p(n)=\frac{5}{24}n^4+\frac{37}{12}n^3+\frac{79}{24}n^2+\frac{101}{12}n+2$. The bound is sharp in the following sense: for each $n$ there exists a saturated polytopological space $(X,\tau_1,...,\tau_n)$ and a subset $A\subseteq X$ such that repeated application of the operators $k_i, f_i, c$ to $A$ will yield exactly $2p(n)$ distinct sets. In particular, following the tradition for Kuratowski-type problems, we exhibit an explicit initial set in $\mathbb{R}$, equipped with the usual and Sorgenfrey topologies, which yields $2p(2)=120$ distinct sets under the action of the monoid $\mathcal{KF}_2$."/> <meta name="DC.Format" scheme="IMT" content="application/pdf"/> <meta name="DC.Identifier" content="151"/> <meta name="DC.Identifier.pageNumber" content="3--27"/> <meta name="DC.Identifier.DOI" content="10.53733/151"/> <meta name="DC.Identifier.URI" content="https://nzjmath.org/index.php/NZJMATH/article/view/151"/> <meta name="DC.Language" scheme="ISO639-1" content="en"/> <meta name="DC.Rights" content="Copyright (c) 2021 the author"/> <meta name="DC.Rights" content=""/> <meta name="DC.Source" content="New Zealand Journal of Mathematics"/> <meta name="DC.Source.ISSN" content="1179-4984"/> <meta name="DC.Source.Volume" content="51"/> <meta name="DC.Source.URI" content="https://nzjmath.org/index.php/NZJMATH"/> <meta name="DC.Title" content="The closure-complement-frontier problem in saturated polytopological spaces"/> <meta name="DC.Type" content="Text.Serial.Journal"/> <meta name="DC.Type.articleType" content="Articles"/> <meta name="gs_meta_revision" content="1.1"/> <meta name="citation_journal_title" content="New Zealand Journal of Mathematics"/> <meta name="citation_journal_abbrev" content="NZ J Math"/> <meta name="citation_issn" content="1179-4984"/> <meta name="citation_author" content="Sara Canilang"/> <meta name="citation_author_institution" content="Carleton College"/> <meta name="citation_author" content="Michael P. Cohen"/> <meta name="citation_author_institution" content="Carleton College"/> <meta name="citation_author" content="Nicolas Graese"/> <meta name="citation_author_institution" content="Carleton College"/> <meta name="citation_author" content="Ian Seong"/> <meta name="citation_author_institution" content="Carleton College"/> <meta name="citation_title" content="The closure-complement-frontier problem in saturated polytopological spaces"/> <meta name="citation_language" content="en"/> <meta name="citation_date" content="2021/08/06"/> <meta name="citation_volume" content="51"/> <meta name="citation_firstpage" content="3"/> <meta name="citation_lastpage" content="27"/> <meta name="citation_doi" content="10.53733/151"/> <meta name="citation_abstract_html_url" content="https://nzjmath.org/index.php/NZJMATH/article/view/151"/> <meta name="citation_pdf_url" content="https://nzjmath.org/index.php/NZJMATH/article/download/151/39"/> <script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/MathJax.js?config=TeX-MML-AM_CHTML"></script> <link rel="stylesheet" href="https://nzjmath.org/index.php/NZJMATH/$$$call$$$/page/page/css?name=stylesheet" type="text/css" /><link rel="stylesheet" href="https://nzjmath.org/index.php/NZJMATH/$$$call$$$/page/page/css?name=font" type="text/css" /><link rel="stylesheet" href="https://nzjmath.org/lib/pkp/styles/fontawesome/fontawesome.css?v=3.3.0.7" type="text/css" /> </head> <body class="pkp_page_article pkp_op_view" dir="ltr"> <div class="pkp_structure_page"> <header class="pkp_structure_head" id="headerNavigationContainer" role="banner"> <nav class="cmp_skip_to_content" aria-label="Jump to content links"> <a href="#pkp_content_main">Skip to main content</a> <a href="#siteNav">Skip to main navigation menu</a> <a href="#pkp_content_footer">Skip to site footer</a> </nav> <div class="pkp_head_wrapper"> <div class="pkp_site_name_wrapper"> <button class="pkp_site_nav_toggle"> <span>Open Menu</span> </button> <div class="pkp_site_name"> <a href=" https://nzjmath.org/index.php/NZJMATH/index " class="is_text">New Zealand Journal of Mathematics</a> </div> </div> <nav class="pkp_site_nav_menu" aria-label="Site Navigation"> <a id="siteNav"></a> <div class="pkp_navigation_primary_row"> <div class="pkp_navigation_primary_wrapper"> <ul id="navigationPrimary" class="pkp_navigation_primary pkp_nav_list"> <li class=""> <a href="https://nzjmath.org/index.php/NZJMATH/about"> Information </a> <ul> <li class=""> <a href="https://nzjmath.org/index.php/NZJMATH/about"> About the journal </a> </li> <li class=""> <a href="https://nzjmath.org/index.php/NZJMATH/about/submissions"> How to submit </a> </li> <li class=""> <a href="https://nzjmath.org/index.php/NZJMATH/about/editorialTeam"> Editorial Team </a> </li> <li class=""> <a href="https://nzjmath.org/index.php/NZJMATH/about/privacy"> Privacy Statement </a> </li> </ul> </li> <li class=""> <a href="https://nzjmath.org/index.php/NZJMATH/issue/current"> Current issue </a> </li> <li class=""> <a href="https://nzjmath.org/index.php/NZJMATH/issue/archive"> Recent back issues </a> </li> <li class=""> <a href="https://nzjmath.org/index.php/NZJMATH/old-archives"> Historical issues </a> </li> <li class=""> <a href="https://nzjmath.org/index.php/NZJMATH/about/contact"> Contact </a> </li> </ul> <div class="pkp_navigation_search_wrapper"> <a href="https://nzjmath.org/index.php/NZJMATH/search" class="pkp_search pkp_search_desktop"> <span class="fa fa-search" aria-hidden="true"></span> Search </a> </div> </div> </div> <div class="pkp_navigation_user_wrapper" id="navigationUserWrapper"> <ul id="navigationUser" class="pkp_navigation_user pkp_nav_list"> <li class="profile"> <a href="https://nzjmath.org/index.php/NZJMATH/user/register"> Register </a> </li> <li class="profile"> <a href="https://nzjmath.org/index.php/NZJMATH/login"> Login </a> </li> </ul> </div> </nav> </div> </header> <div class="pkp_structure_content has_sidebar"> <div class="pkp_structure_main" role="main"> <a id="pkp_content_main"></a> <div class="page page_article"> <nav class="cmp_breadcrumbs" role="navigation" aria-label="You are here:"> <ol> <li> <a href="https://nzjmath.org/index.php/NZJMATH/index"> Home </a> <span class="separator">/</span> </li> <li> <a href="https://nzjmath.org/index.php/NZJMATH/issue/archive"> Archives </a> <span class="separator">/</span> </li> <li> <a href="https://nzjmath.org/index.php/NZJMATH/issue/view/6"> Vol. 51 (2021) </a> <span class="separator">/</span> </li> <li class="current" aria-current="page"> <span aria-current="page"> Articles </span> </li> </ol> </nav> <article class="obj_article_details"> <h1 class="page_title"> The closure-complement-frontier problem in saturated polytopological spaces </h1> <div class="row"> <div class="main_entry"> <section class="item authors"> <h2 class="pkp_screen_reader">Authors</h2> <ul class="authors"> <li> <span class="name"> Sara Canilang </span> <span class="affiliation"> Carleton College </span> </li> <li> <span class="name"> Michael P. Cohen </span> <span class="affiliation"> Carleton College </span> </li> <li> <span class="name"> Nicolas Graese </span> <span class="affiliation"> Carleton College </span> </li> <li> <span class="name"> Ian Seong </span> <span class="affiliation"> Carleton College </span> </li> </ul> </section> <section class="item doi"> <h2 class="label"> DOI: </h2> <span class="value"> <a href="https://doi.org/10.53733/151"> https://doi.org/10.53733/151 </a> </span> </section> <section class="item abstract"> <h2 class="label">Abstract</h2> <p>Let $X$ be a space equipped with $n$ topologies $\tau_1,\ldots,\tau_n$ which are pairwise comparable and saturated, and for each $1\leq i\leq n$ let $k_i$ and $f_i$ be the associated topological closure and frontier operators, respectively. Inspired by the closure-complement theorem of Kuratowski, we prove that the monoid of set operators $\mathcal{KF}_n$ generated by $\{k_i,f_i:1\leq i\leq n\}\cup\{c\}$ (where $c$ denotes the set complement operator) has cardinality no more than $2p(n)$ where $p(n)=\frac{5}{24}n^4+\frac{37}{12}n^3+\frac{79}{24}n^2+\frac{101}{12}n+2$. The bound is sharp in the following sense: for each $n$ there exists a saturated polytopological space $(X,\tau_1,...,\tau_n)$ and a subset $A\subseteq X$ such that repeated application of the operators $k_i, f_i, c$ to $A$ will yield exactly $2p(n)$ distinct sets. In particular, following the tradition for Kuratowski-type problems, we exhibit an explicit initial set in $\mathbb{R}$, equipped with the usual and Sorgenfrey topologies, which yields $2p(2)=120$ distinct sets under the action of the monoid $\mathcal{KF}_2$.</p> </section> <div class="item downloads_chart"> <h3 class="label"> Downloads </h3> <div class="value"> <canvas class="usageStatsGraph" data-object-type="Submission" data-object-id="151"></canvas> <div class="usageStatsUnavailable" data-object-type="Submission" data-object-id="151"> Download data is not yet available. </div> </div> </div> <section class="item author_bios"> <h2 class="label"> Author Biographies </h2> <section class="sub_item"> <h3 class="label"> Sara Canilang, <span class="affiliation">Carleton College</span> </h3> <div class="value"> <p>Department of Mathematics and Statistics,<br>Carleton College,<br>Northfield, MN 55057<br>USA</p> </div> </section> <section class="sub_item"> <h3 class="label"> Michael P. Cohen, <span class="affiliation">Carleton College</span> </h3> <div class="value"> <p>Department of Mathematics and Statistics,<br>Carleton College,<br>Northfield, MN 55057<br>USA</p> </div> </section> <section class="sub_item"> <h3 class="label"> Nicolas Graese, <span class="affiliation">Carleton College</span> </h3> <div class="value"> <p>Department of Mathematics and Statistics,<br>Carleton College,<br>Northfield, MN 55057<br>USA</p> </div> </section> <section class="sub_item"> <h3 class="label"> Ian Seong, <span class="affiliation">Carleton College</span> </h3> <div class="value"> <p>Department of Mathematics and Statistics,<br>Carleton College,<br>Northfield, MN 55057 <br>USA</p> </div> </section> </section> </div> <div class="entry_details"> <div class="item galleys"> <h2 class="pkp_screen_reader"> Downloads </h2> <ul class="value galleys_links"> <li> <a class="obj_galley_link pdf" href="https://nzjmath.org/index.php/NZJMATH/article/view/151/39"> pdf final </a> </li> </ul> </div> <div class="item published"> <section class="sub_item"> <h2 class="label"> Published </h2> <div class="value"> <span>06-08-2021</span> </div> </section> </div> <div class="item citation"> <section class="sub_item citation_display"> <h2 class="label"> How to Cite </h2> <div class="value"> <div id="citationOutput" role="region" aria-live="polite"> <div class="csl-bib-body"> <div class="csl-entry">Canilang, S., Cohen, M. P., Graese, N., & Seong, I. (2021). The closure-complement-frontier problem in saturated polytopological spaces. <i>New Zealand Journal of Mathematics</i>, <i>51</i>, 3–27. https://doi.org/10.53733/151</div> </div> </div> <div class="citation_formats"> <button class="cmp_button citation_formats_button" aria-controls="cslCitationFormats" aria-expanded="false" data-csl-dropdown="true"> More Citation Formats </button> <div id="cslCitationFormats" class="citation_formats_list" aria-hidden="true"> <ul class="citation_formats_styles"> <li> <a aria-controls="citationOutput" href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/acm-sig-proceedings?submissionId=151&publicationId=156" data-load-citation data-json-href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/acm-sig-proceedings?submissionId=151&publicationId=156&return=json" > ACM </a> </li> <li> <a aria-controls="citationOutput" href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/acs-nano?submissionId=151&publicationId=156" data-load-citation data-json-href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/acs-nano?submissionId=151&publicationId=156&return=json" > ACS </a> </li> <li> <a aria-controls="citationOutput" href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/apa?submissionId=151&publicationId=156" data-load-citation data-json-href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/apa?submissionId=151&publicationId=156&return=json" > APA </a> </li> <li> <a aria-controls="citationOutput" href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/associacao-brasileira-de-normas-tecnicas?submissionId=151&publicationId=156" data-load-citation data-json-href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/associacao-brasileira-de-normas-tecnicas?submissionId=151&publicationId=156&return=json" > ABNT </a> </li> <li> <a aria-controls="citationOutput" href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/chicago-author-date?submissionId=151&publicationId=156" data-load-citation data-json-href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/chicago-author-date?submissionId=151&publicationId=156&return=json" > Chicago </a> </li> <li> <a aria-controls="citationOutput" href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/harvard-cite-them-right?submissionId=151&publicationId=156" data-load-citation data-json-href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/harvard-cite-them-right?submissionId=151&publicationId=156&return=json" > Harvard </a> </li> <li> <a aria-controls="citationOutput" href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/ieee?submissionId=151&publicationId=156" data-load-citation data-json-href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/ieee?submissionId=151&publicationId=156&return=json" > IEEE </a> </li> <li> <a aria-controls="citationOutput" href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/modern-language-association?submissionId=151&publicationId=156" data-load-citation data-json-href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/modern-language-association?submissionId=151&publicationId=156&return=json" > MLA </a> </li> <li> <a aria-controls="citationOutput" href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/turabian-fullnote-bibliography?submissionId=151&publicationId=156" data-load-citation data-json-href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/turabian-fullnote-bibliography?submissionId=151&publicationId=156&return=json" > Turabian </a> </li> <li> <a aria-controls="citationOutput" href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/vancouver?submissionId=151&publicationId=156" data-load-citation data-json-href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/get/vancouver?submissionId=151&publicationId=156&return=json" > Vancouver </a> </li> </ul> <div class="label"> Download Citation </div> <ul class="citation_formats_styles"> <li> <a href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/download/ris?submissionId=151&publicationId=156"> <span class="fa fa-download"></span> Endnote/Zotero/Mendeley (RIS) </a> </li> <li> <a href="https://nzjmath.org/index.php/NZJMATH/citationstylelanguage/download/bibtex?submissionId=151&publicationId=156"> <span class="fa fa-download"></span> BibTeX </a> </li> </ul> </div> </div> </div> </section> </div> <div class="item issue"> <section class="sub_item"> <h2 class="label"> Issue </h2> <div class="value"> <a class="title" href="https://nzjmath.org/index.php/NZJMATH/issue/view/6"> Vol. 51 (2021) </a> </div> </section> <section class="sub_item"> <h2 class="label"> Section </h2> <div class="value"> Articles </div> </section> </div> </div> </div> </article> </div> </div> <div class="pkp_structure_sidebar left" role="complementary" aria-label="Sidebar"> <div class="pkp_block block_developed_by"> <h2 class="pkp_screen_reader"> Developed By </h2> <div class="content"> <a href="http://pkp.sfu.ca/ojs/"> Open Journal Systems </a> </div> </div> <div class="pkp_block block_information"> <h2 class="title">Information</h2> <div class="content"> <ul> <li> <a href="https://nzjmath.org/index.php/NZJMATH/information/readers"> For Readers </a> </li> <li> <a href="https://nzjmath.org/index.php/NZJMATH/information/authors"> For Authors </a> </li> <li> <a href="https://nzjmath.org/index.php/NZJMATH/information/librarians"> For Librarians </a> </li> </ul> </div> </div> </div> </div> <div class="pkp_structure_footer_wrapper" role="contentinfo"> <a id="pkp_content_footer"></a> <div class="pkp_structure_footer"> <div class="pkp_footer_content"> <p>In Memoriam, Editorial Board member Professor Sir Vaughan F. R. Jones 1952-2020.</p> <p>See Volume 52 for the Vaughan Jones Memorial Special Issue.</p> </div> <div class="pkp_brand_footer" role="complementary"> <a href="https://nzjmath.org/index.php/NZJMATH/about/aboutThisPublishingSystem"> <img alt="More information about the publishing system, Platform and Workflow by OJS/PKP." src="https://nzjmath.org/templates/images/ojs_brand.png"> </a> </div> </div> </div> </div> <script src="https://nzjmath.org/lib/pkp/lib/vendor/components/jquery/jquery.min.js?v=3.3.0.7" type="text/javascript"></script><script src="https://nzjmath.org/lib/pkp/lib/vendor/components/jqueryui/jquery-ui.min.js?v=3.3.0.7" type="text/javascript"></script><script src="https://nzjmath.org/plugins/themes/default/js/lib/popper/popper.js?v=3.3.0.7" type="text/javascript"></script><script src="https://nzjmath.org/plugins/themes/default/js/lib/bootstrap/util.js?v=3.3.0.7" type="text/javascript"></script><script src="https://nzjmath.org/plugins/themes/default/js/lib/bootstrap/dropdown.js?v=3.3.0.7" type="text/javascript"></script><script src="https://nzjmath.org/plugins/themes/default/js/main.js?v=3.3.0.7" type="text/javascript"></script><script src="https://nzjmath.org/plugins/generic/citationStyleLanguage/js/articleCitation.js?v=3.3.0.7" type="text/javascript"></script><script type="text/javascript">var pkpUsageStats = pkpUsageStats || {};pkpUsageStats.data = pkpUsageStats.data || {};pkpUsageStats.data.Submission = pkpUsageStats.data.Submission || {};pkpUsageStats.data.Submission[151] = {"data":{"2021":{"1":0,"2":0,"3":0,"4":0,"5":0,"6":0,"7":0,"8":19,"9":14,"10":21,"11":4,"12":4},"2022":{"1":2,"2":11,"3":3,"4":9,"5":12,"6":4,"7":2,"8":8,"9":4,"10":4,"11":10,"12":3},"2023":{"1":7,"2":1,"3":6,"4":3,"5":5,"6":4,"7":10,"8":6,"9":4,"10":10,"11":8,"12":4},"2024":{"1":3,"2":5,"3":14,"4":11,"5":7,"6":11,"7":11,"8":12,"9":5,"10":9,"11":7,"12":8},"2025":{"1":4,"2":3,"3":22,"4":15,"5":0,"6":0,"7":0,"8":0,"9":0,"10":0,"11":0,"12":0}},"label":"All Downloads","color":"79,181,217","total":349};</script><script src="https://cdnjs.cloudflare.com/ajax/libs/Chart.js/2.0.1/Chart.js?v=3.3.0.7" type="text/javascript"></script><script type="text/javascript">var pkpUsageStats = pkpUsageStats || {};pkpUsageStats.locale = pkpUsageStats.locale || {};pkpUsageStats.locale.months = ["Jan","Feb","Mar","Apr","May","Jun","Jul","Aug","Sep","Oct","Nov","Dec"];pkpUsageStats.config = pkpUsageStats.config || {};pkpUsageStats.config.chartType = "bar";</script><script src="https://nzjmath.org/plugins/generic/usageStats/js/UsageStatsFrontendHandler.js?v=3.3.0.7" type="text/javascript"></script> </body> </html>

CINXE.COM

The closure-complement-frontier problem in saturated polytopological spaces | New Zealand Journal of Mathematics