Topology and its Applications最新文献

{"title":"On sequential versions of distributional topological complexity","authors":"Ekansh Jauhari","doi":"10.1016/j.topol.2025.109271","DOIUrl":"10.1016/j.topol.2025.109271","url":null,"abstract":"<div><div>We define a (non-decreasing) sequence <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>dTC</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>}</mo></mrow><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow></msub></math></span> of sequential versions of distributional topological complexity (<span><math><mi>dTC</mi></math></span>) of a space <em>X</em> introduced by Dranishnikov and Jauhari <span><span>[5]</span></span>. This sequence generalizes <span><math><mi>dTC</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> in the sense that <span><math><msub><mrow><mi>dTC</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mi>dTC</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>, and is a direct analog to the well-known sequence <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>TC</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>}</mo></mrow><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow></msub></math></span>. We show that like <span><math><msub><mrow><mi>TC</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> and <span><math><mi>dTC</mi></math></span>, the sequential versions <span><math><msub><mrow><mi>dTC</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> are also homotopy invariants. Furthermore, <span><math><msub><mrow><mi>dTC</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> relates with the distributional LS-category (<span><math><mi>dcat</mi></math></span>) of products of <em>X</em> in the same way as <span><math><msub><mrow><mi>TC</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> relates with the classical LS-category (<span><math><mi>cat</mi></math></span>) of products of <em>X</em>. On one hand, we show that in general, <span><math><msub><mrow><mi>dTC</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> is a different concept than <span><math><msub><mrow><mi>TC</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> for each <span><math><mi>m</mi><mo>≥</mo><mn>2</mn></math></span>. On the other hand, by finding sharp cohomological lower bounds to <span><math><msub><mrow><mi>dTC</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span>, we provide various examples of closed manifolds <em>X</em> for which the sequences <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>TC</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>}</mo></mrow><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>dTC</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>}</mo></mrow><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow></msub></math></span> coincide.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"363 ","pages":"Article 109271"},"PeriodicalIF":0.6,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143378592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The separating disk complex for a handlebody","authors":"Sangbum Cho ,&nbsp;Jung Hoon Lee","doi":"10.1016/j.topol.2025.109272","DOIUrl":"10.1016/j.topol.2025.109272","url":null,"abstract":"<div><div>We prove that the separating disk complex for a handlebody is connected. We present two proofs, one is based on the properties of primitive curves while the other one uses the action of the handlebody group on the complex. We also show that the separating reducing sphere complex for a double handlebody is connected.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"363 ","pages":"Article 109272"},"PeriodicalIF":0.6,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Metric duality for Abelian groups","authors":"Piotr Niemiec","doi":"10.1016/j.topol.2024.109155","DOIUrl":"10.1016/j.topol.2024.109155","url":null,"abstract":"<div><div>The main aim of the paper is to introduce the concept of metric duality in the category of topological Abelian groups that extends the classical notion of duality for normed vector spaces and behaves quite nicely for LCA groups (equipped with <em>nice</em> metrics). In particular, it is shown that each Polish LCA group admits a reflexive proper metric and, more generally, all LCA groups possess reflexive (proper) metric structures.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"360 ","pages":"Article 109155"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Editorial on the Mary Ellen Rudin Young Researcher Award competition 2023","authors":"Alan Dow,&nbsp;Jan van Mill (Editors-in-Chief Topology and its Application)","doi":"10.1016/j.topol.2024.109152","DOIUrl":"10.1016/j.topol.2024.109152","url":null,"abstract":"","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"360 ","pages":"Article 109152"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Anti-absorbing ternary operations on metric spaces","authors":"Leonid V. Kovalev","doi":"10.1016/j.topol.2024.109153","DOIUrl":"10.1016/j.topol.2024.109153","url":null,"abstract":"<div><div>The existence of a median-type ternary operation on a metric space is known to have a number of implications for the geometry of the space. For such operations, if two of the three arguments coincide, they also coincide with the output of the operation. We consider ternary operations with the opposite property: if two of the arguments coincide, the output is equal to the third one. The existence of such an operation is a necessary condition for the space to be an absolute retract.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"360 ","pages":"Article 109153"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Some properties involving feeble compactness, III: (Weakly) compact-bounded topological groups","authors":"J.A. Martínez-Cadena,&nbsp;Á. Tamariz-Mascarúa","doi":"10.1016/j.topol.2024.109149","DOIUrl":"10.1016/j.topol.2024.109149","url":null,"abstract":"<div><div>We study two topological properties weaker than feeble compactness in the class of (para)topological groups, the compact-boundedness and weak compact-boundedness, both introduced by Angoa, Ortiz-Castillo and Tamariz-Mascarúa in <span><span>[2]</span></span>. First, given a subgroup <em>H</em> of a topological group <em>G</em>, we show how to extend these properties from the quotient space <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> to <em>G</em>; this, in the cases when <em>H</em> is a compact, locally compact or (weakly) compact-bounded subgroup. Secondly, we prove the main result of this article: if a Tychonoff space <em>X</em> is compact-bounded and not scattered, then the free topological group <span><math><mi>F</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and the free Abelian topological group <span><math><mi>A</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> admit a non-trivial metrizable quotient group; thus extending Theorem 4.7 by Leiderman and Tkachenko in <span><span>[15]</span></span>. Finally, we study the <em>r</em>-weakly compact-bounded subsets of a topological space <em>X</em>. We show that <em>r</em>-weak compact-boundedness is a productive property. Moreover, sufficient conditions are given in order for a <em>C</em>-compact subset of a paratopological group <em>G</em> to become an <em>r</em>-weakly compact-bounded subset. This article is part of a larger work developed in <span><span>[16]</span></span> and <span><span>[17]</span></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"360 ","pages":"Article 109149"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Discrete Morse theory on ΩS2","authors":"Lacey Johnson ,&nbsp;Kevin Knudson","doi":"10.1016/j.topol.2024.109185","DOIUrl":"10.1016/j.topol.2024.109185","url":null,"abstract":"<div><div>One classical consequence of Morse theory is that it provides a description of a cell structure for a CW-complex having the homotopy type of the loop space of a manifold. In this paper, we study this result through the lens of discrete Morse theory. This requires a suitable simplicial model for the loop space. Here, we use Milnor's <span><math><msup><mrow><mtext>F</mtext></mrow><mrow><mo>+</mo></mrow></msup><mtext>K</mtext></math></span> construction to model the loop space of the sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, describe a discrete gradient on it, and identify a collection of critical cells. We also compute the action of the boundary operator in the Morse complex on these critical cells, showing that they are potential homology generators. A careful analysis allows us to recover the calculation of the first homology of <span><math><mi>Ω</mi><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"360 ","pages":"Article 109185"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The hyperspace of k-dimensional closed convex sets","authors":"Adriana Escobedo-Bustamante ,&nbsp;Natalia Jonard-Pérez","doi":"10.1016/j.topol.2024.109154","DOIUrl":"10.1016/j.topol.2024.109154","url":null,"abstract":"<div><div>For every <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, let <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> denote the hyperspace of all <em>k</em>-dimensional closed convex subsets of the Euclidean space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> endowed with the Atouch-Wets topology. Let <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> be the subset of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> consisting of all <em>k</em>-dimensional compact convex subsets. In this paper we explore the topology of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> and the relation of these hyperspaces with the Grassmann manifold <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. We prove that both <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> are Hilbert cube manifolds with a fiber bundle structure over <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. We also show that the fiber of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> with respect to this fiber bundle structure is homeomorphic with <span><math><msup><mrow><mi>R</mi></mrow><mrow><mfrac><mrow><mi>k</mi><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>2</mn><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>×</mo><mi>Q</mi></math></span>, where <em>Q</em> stands for the Hilbert cube.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"360 ","pages":"Article 109154"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The first-countability in generalizations of topological groups with ideal convergence","authors":"Xin Liu,&nbsp;Shou Lin,&nbsp;Xiangeng Zhou","doi":"10.1016/j.topol.2024.109150","DOIUrl":"10.1016/j.topol.2024.109150","url":null,"abstract":"<div><div>The study of convergence in topological groups has become a frontier research subject. In many cases, the first-countability is an important and strong condition. Based on <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>s</mi><mi>n</mi></mrow></msub></math></span>-continuity, the present paper discusses how topology and algebra are related through a notion of continuity generated by ideal convergence. We introduce the classes of generalizations of topological groups, give the structures of <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>s</mi><mi>n</mi></mrow></msub></math></span>-topological groups by certain sequential coreflections, and obtain generalized metric properties of <span><math><mi>I</mi></math></span>-<em>snf</em>-countable para-<span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>s</mi><mi>n</mi></mrow></msub></math></span>-topological groups.</div><div>Let <span><math><mi>I</mi></math></span> be an admissible ideal on the set <span><math><mi>N</mi></math></span> of natural numbers. The following results are obtained.<ul><li><span>(1)</span><span><div>Every T₂, <span><math><mi>I</mi></math></span>-<em>snf</em>-countable para-<span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>s</mi><mi>n</mi></mrow></msub></math></span>-topological group is an <em>sn</em>-quasi-metrizable and <em>cs</em>-submetrizable space.</div></span></li><li><span>(2)</span><span><div>A T₀, <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>s</mi><mi>n</mi></mrow></msub></math></span>-topological group is an <span><math><mi>I</mi></math></span>-<em>snf</em>-countable space if and only if it is a <em>cs</em>-metrizable space satisfying that each sequentially open subset is <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>s</mi><mi>n</mi></mrow></msub></math></span>-open.</div></span></li></ul></div><div>These show the unique role of <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>s</mi><mi>n</mi></mrow></msub></math></span>-continuity in the study of topological groups and related structures, and present a version of topological algebra using the notion of ideals.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"360 ","pages":"Article 109150"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On MM-ω-balancedness and FR(Fm)-factorizable semi(para)topological groups","authors":"Liang-Xue Peng,&nbsp;Yu-Ming Deng","doi":"10.1016/j.topol.2024.109183","DOIUrl":"10.1016/j.topol.2024.109183","url":null,"abstract":"<div><div>In the second part of this article, we introduce a notion which is called <em>MM</em>-<em>ω</em>-balancedness in the class of semitopological groups. We show that if <em>G</em> is a semitopological (paratopological) group, then <em>G</em> is topologically isomorphic to a subgroup of the product of a family of metacompact Moore semitopological (paratopological) groups if and only if <em>G</em> is regular <em>MM</em>-<em>ω</em>-balanced and <span><math><mi>I</mi><mi>r</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span>. If <em>G</em> is a <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> <em>bM</em>-<em>ω</em>-balanced semitopological group and <span><math><mi>f</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>H</mi></math></span> is an open continuous homomorphism of <em>G</em> onto a first-countable semitopological group <em>H</em> such that <span><math><mi>ker</mi><mo>⁡</mo><mo>(</mo><mi>f</mi><mo>)</mo></math></span> is a countably compact subgroup of <em>G</em>, then <em>H</em> is a metacompact developable space.</div><div>In the third part of this article, we introduce notions of <span><math><mi>F</mi><mi>R</mi></math></span>-factorizability and <span><math><mi>F</mi><mi>m</mi></math></span>-factorizability. We give some equivalent conditions that a semitopological (paratopological) group is <span><math><mi>F</mi><mi>R</mi></math></span>-factorizable or <span><math><mi>F</mi><mi>m</mi></math></span>-factorizable. If <em>G</em> is a Tychonoff <span><math><mi>F</mi><mi>R</mi></math></span> (<span><math><mi>F</mi><mi>m</mi></math></span>)-factorizable semitopological group and <span><math><mi>f</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>H</mi></math></span> is a continuous open homomorphism of <em>G</em> onto a semitopological group <em>H</em>, then <em>H</em> is <span><math><mi>F</mi><mi>R</mi></math></span> (<span><math><mi>F</mi><mi>m</mi></math></span>)-factorizable. If <em>G</em> is a <span><math><mi>F</mi><mi>R</mi></math></span> (<span><math><mi>F</mi><mi>m</mi></math></span>)-factorizable paratopological group and <span><math><mi>f</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>H</mi></math></span> is a continuous <em>d</em>-open homomorphism of <em>G</em> onto a paratopological group <em>H</em>, then <em>H</em> is <span><math><mi>F</mi><mi>R</mi></math></span> (<span><math><mi>F</mi><mi>m</mi></math></span>)-factorizable.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"360 ","pages":"Article 109183"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}