{"title":"Products of consonant spaces and Čech complete spaces","authors":"Francis Jordan","doi":"10.1016/j.topol.2025.109495","DOIUrl":"10.1016/j.topol.2025.109495","url":null,"abstract":"<div><div>We characterize those consonant spaces whose product with any Čech complete is again consonant. We then show that in the class of completely regular spaces these spaces generalize spaces that are Menger at infinity. A perfect set theorem is established for these spaces. We also characterize those consonant spaces whose product with any Čech complete space of given hereditary Lindelöf number remains consonant. Various examples are given.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109495"},"PeriodicalIF":0.6,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144614192","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 extension problem for actions of compact Lie groups","authors":"Thales Fernando Vilamaior Paiva","doi":"10.1016/j.topol.2025.109496","DOIUrl":"10.1016/j.topol.2025.109496","url":null,"abstract":"<div><div>This article addresses the problem of extending a continuous action of a compact Lie group. To this end, we apply tools from equivariant cohomology theory, such as the Borel construction and the Leray-Serre spectral sequence, reformulating the initial problem as a morphism between appropriate spectral sequences, which enables us to establish criteria for determining whether a given action can be extended to a larger group.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109496"},"PeriodicalIF":0.6,"publicationDate":"2025-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570457","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}
José Gerardo Ahuatzi-Reyes , Norberto Ordoñez , Hugo Villanueva
{"title":"On increasing, locally persistent and persistent Whitney properties","authors":"José Gerardo Ahuatzi-Reyes , Norberto Ordoñez , Hugo Villanueva","doi":"10.1016/j.topol.2025.109492","DOIUrl":"10.1016/j.topol.2025.109492","url":null,"abstract":"<div><div>Let <em>X</em> be a metric continuum and let <span><math><mi>C</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> be the hyperspace of subcontinua of <em>X</em>. The problem of determining which topological properties are Whitney properties has been widely studied and has generated an ample line of research. This line has been enriched with new concepts, such as that of increasing Whitney property, which was studied in <span><span>[18]</span></span>. In order to extend these ideas in other directions, in this paper we introduce two new concepts: Whitney persistent property and locally Whitney persistent property (see <span><span>Definition 1.1</span></span>). We establish the relations that exist between these concepts and those of Whitney and increasing properties. Also, we determine, from a long list of topological properties, which ones are or are not increasing, locally persistent or persistent. For these purposes, we provided some general results and several examples. Part of this work extends the study given in <span><span>[18]</span></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109492"},"PeriodicalIF":0.6,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144549523","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":"Approach properties of probabilistic metrizable approach spaces","authors":"E. Colebunders , R. Lowen","doi":"10.1016/j.topol.2025.109494","DOIUrl":"10.1016/j.topol.2025.109494","url":null,"abstract":"<div><div>Our investigation of approach properties in probabilistic metrizable approach spaces is based on two faithful functors. The first one was introduced in <span><span>[9]</span></span> and goes from probabilistic metric spaces with respect to a continuous <em>t</em>-norm to the category of approach spaces. The second one goes from probabilistic metric spaces with respect to a continuous <em>t</em>-norm to the category of uniform gauge spaces. Using these functors we show that all probabilistic metrizable approach spaces are uniform. We characterize those probabilistic metrizable approach spaces that are associated with a certainly bounded probabilistic metric space or by one that is bounded in distribution. We show that precompactness of the probabilistic metric space is equivalent to the associated uniform gauge space having zero index of precompactness, and that completeness of the probabilistic metric space is equivalent to completeness of the associated uniform gauge space.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109494"},"PeriodicalIF":0.6,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144523357","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}
Batoul Yousefipour , S. Sajjad Gashti , Hassan Myrnouri
{"title":"Additive subgroups of real vector spaces and topologies on them","authors":"Batoul Yousefipour , S. Sajjad Gashti , Hassan Myrnouri","doi":"10.1016/j.topol.2025.109488","DOIUrl":"10.1016/j.topol.2025.109488","url":null,"abstract":"<div><div>In the present paper, we study group topologies on torsion free abelian groups and real vector spaces. We introduce the concept of <em>Q</em>-balanced subsets of a torsion free abelian group. Absorbed elements of a real vector space equipped with a group topology (with addition) is another concept that is presented.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109488"},"PeriodicalIF":0.6,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144523358","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":"A braidoid equivalence for spherical knotoids","authors":"Anastasios Kokkinakis","doi":"10.1016/j.topol.2025.109491","DOIUrl":"10.1016/j.topol.2025.109491","url":null,"abstract":"<div><div>Braidoids form a counterpart theory to the theory of planar knotoids, just as braids do for three-dimensional links. As such, planar knotoid diagrams represent the same knotoid in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> if and only if they can be presented as the closure of two labeled braidoid diagrams related by an equivalence relation, named <em>L</em>-equivalence. In this paper, we refine the notion of <em>L</em>-equivalence of braidoid diagrams in order to obtain an equivalence theorem for (multi)-knotoid diagrams in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> when represented as the closure of labeled braidoid diagrams.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109491"},"PeriodicalIF":0.6,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144580926","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":"Adding an uncountable discrete subspace by forcing","authors":"Akira Iwasa","doi":"10.1016/j.topol.2025.109489","DOIUrl":"10.1016/j.topol.2025.109489","url":null,"abstract":"<div><div>Suppose that a topological space <em>X</em> has no uncountable discrete subspace. We study whether <em>X</em> can obtain an uncountable discrete subspace in forcing extensions. We provide various such consistent examples.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109489"},"PeriodicalIF":0.6,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144523356","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":"A study on some versions of mi-spaces","authors":"Liang-Xue Peng","doi":"10.1016/j.topol.2025.109493","DOIUrl":"10.1016/j.topol.2025.109493","url":null,"abstract":"<div><div>We introduce notions of <em>c</em>-<em>σ</em>-<span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>-spaces (<em>c</em>-<span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>-spaces) for <span><math><mi>i</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>}</mo></math></span>. A space <em>X</em> is called a <em>c</em>-<span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-space (<em>c</em>-<span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-space, <em>c</em>-<span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-space) if <em>X</em> has a closure-preserving local base (closure-preserving local quasi-base, cushioned local pair-base) at every compact subset <em>F</em> of <em>X</em>. We give some characterizations of <em>σ</em>-<span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>-spaces (<span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>-spaces), <em>s</em>-<em>σ</em>-<span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>-spaces (<em>s</em>-<span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>-spaces), and <em>c</em>-<em>σ</em>-<span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>-spaces (<em>c</em>-<span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>-spaces). Thus some known conclusions can be obtained by these characterizations. We also get the following results. Every stratifiable <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-space is a <em>c</em>-<span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-space. Every ordinal is hereditarily a <em>c</em>-<span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-space. If <em>X</em> is a generalized ordered (GO-) space and a sequence <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> of points in <em>X</em> has a limit point <em>x</em> in <em>X</em>, then the set <span><math><mi>C</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>}</mo><mo>∪</mo><mo>{</mo><mi>x</mi><mo>}</mo></math></span> has a closure-preserving local base in <em>X</em>. If <em>X</em> is a monotonically (countably) metacompact regular space, then <em>X</em> is a <span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-space. If <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is a regular space which has a <em>σ</em>-<em>NSR</em> pair-base at every point of <span><math><msub><mrow><mi>X</mi","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109493"},"PeriodicalIF":0.6,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144549379","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":"Small compacts","authors":"Angel Calderón-Villalobos , Iván Sánchez","doi":"10.1016/j.topol.2025.109490","DOIUrl":"10.1016/j.topol.2025.109490","url":null,"abstract":"<div><div>For a subset <em>A</em> of an almost topological group <em>G</em>, the Hattori space <span><math><mi>H</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is a topological space whose underlying set is <em>G</em> and whose topology <span><math><mi>τ</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is defined as follows: if <span><math><mi>x</mi><mo>∈</mo><mi>A</mi></math></span> (respectively, <span><math><mi>x</mi><mo>∉</mo><mi>A</mi></math></span>), then the neighborhoods of <em>x</em> in <span><math><mi>H</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> are the same neighborhoods of <em>x</em> in the reflection group (respectively, <em>G</em>). In this paper, we show the following:<ul><li><span>i)</span><span><div><em>G</em> is an almost topological group if and only if the Hattori topology <span><math><mi>τ</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> can be defined on <em>G</em> for each subset <em>A</em> of <em>G</em>.</div></span></li><li><span>ii)</span><span><div>If <em>A</em> is a subset of a proper almost topological group <em>G</em>, then <span><math><mi>H</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is locally compact if and only if <span><math><msub><mrow><mi>G</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span> is locally compact, <span><math><mi>G</mi><mo>∖</mo><mi>A</mi></math></span> is closed in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span> and for each <span><math><mi>x</mi><mo>∈</mo><mi>G</mi><mo>∖</mo><mi>A</mi></math></span>, there exists <span><math><mi>U</mi><mo>∈</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>e</mi></mrow></msub></math></span> such that <span><math><msup><mrow><mover><mrow><mi>U</mi><mi>x</mi></mrow><mo>‾</mo></mover></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mo>⁎</mo></mrow></msub></mrow></msup><mo>∩</mo><mo>(</mo><mi>G</mi><mo>∖</mo><mi>A</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>x</mi><mo>}</mo></math></span> and <span><math><msup><mrow><mover><mrow><mi>U</mi><mi>x</mi></mrow><mo>‾</mo></mover></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mo>⁎</mo></mrow></msub></mrow></msup><mo>∖</mo><mi>V</mi><mi>x</mi></math></span> is closed in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span>, for each <span><math><mi>V</mi><mo>∈</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>e</mi></mrow></msub></math></span>.</div></span></li><li><span>iii)</span><span><div>If <em>A</em> is a subset of a proper almost topological group <em>G</em> such that <span><math><msub><mrow><mi>G</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span> has countable pseudocharacter, then <span><math><mi>H</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> has small compacts if and only if <em>A</em> has small compacts.</div></span></li></ul> Moreover, we study the property of being <em>σ</em>-compact in Hattori spaces <span><math><mi>H</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, where <em>A</em> is a subset of an almost topological group <em>G</","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109490"},"PeriodicalIF":0.6,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144517813","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":"Homology of graph burnings","authors":"Yuri Muranov, Anna Muranova","doi":"10.1016/j.topol.2025.109486","DOIUrl":"10.1016/j.topol.2025.109486","url":null,"abstract":"<div><div>In this paper we study graph burnings using methods of algebraic topology. We prove that the time function of a burning is a graph map to a path graph. We use this fact to define a category whose objects are graph burnings and morphisms are graph maps which commute with the time functions of the burnings. In this category we study relations between burnings of different graphs and, in particular, between burnings of a graph and its subgraphs. For every graph, we define a simplicial complex, arising from the set of all the burnings, which we call a configuration space of the burnings. The simplicial structure of the configuration space defines burning homology of the graph. We describe properties of the configuration space and the burning homology theory. We prove that the one-dimensional skeleton of the configuration space of a graph <em>G</em> coincides with the complement graph of <em>G</em>. The results are illustrated with numerous examples.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109486"},"PeriodicalIF":0.6,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144517811","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}
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