{"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":"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":"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}
{"title":"Some properties involving feeble compactness, IV: Compactness-like properties defined via dense subspaces","authors":"J.A. Martínez-Cadena , Á. Tamariz-Mascarúa","doi":"10.1016/j.topol.2025.109487","DOIUrl":"10.1016/j.topol.2025.109487","url":null,"abstract":"<div><div>We study several compactness-like properties arising from the existence of dense subspaces satisfying relative compactness conditions, namely: <em>countably pracompact</em>, <em>totally countably pracompact</em>, <em>densely ω-bounded</em>, and <em>sequentially pracompact</em> spaces. These classes refine classical notions such as sequential compactness and <em>ω</em>-boundedness and admit a natural hierarchy. We establish various preservation results for these properties under perfect open mappings and product spaces. In the context of topological groups, we prove that if <em>H</em> is a locally compact subgroup of a topological group <em>G</em>, then the corresponding quotient mapping <span><math><mi>π</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>G</mi><mo>/</mo><mi>H</mi></math></span> allows the transfer of local versions of these properties from <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> to <em>G</em>. We also analyze the extent to which these properties satisfy the <em>three-space property</em> and introduce the class of <em>PC-spaces</em> to characterize when such transfer is possible. Finally, we address structural questions on densely <em>ω</em>-bounded paratopological groups and provide conditions under which they are topological.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109487"},"PeriodicalIF":0.6,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144523355","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}
Omar Antolín-Camarena, Luis Eduardo García-Hernández, Luis Jorge Sánchez Saldaña
{"title":"The classifying space for commutativity of geometric orientable 3-manifold groups","authors":"Omar Antolín-Camarena, Luis Eduardo García-Hernández, Luis Jorge Sánchez Saldaña","doi":"10.1016/j.topol.2025.109484","DOIUrl":"10.1016/j.topol.2025.109484","url":null,"abstract":"<div><div>For a topological group <em>G</em> let <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>com</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the total space of the universal transitionally commutative principal <em>G</em>-bundle as defined by Adem–Cohen–Torres-Giese. So far this space has been most studied in the case of compact Lie groups; but in this paper we focus on the case of infinite discrete groups.</div><div>For a discrete group <em>G</em>, the space <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>com</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is homotopy equivalent to the geometric realization of the order complex of the poset of cosets of abelian subgroups of <em>G</em>. We show that for fundamental groups of closed orientable geometric 3-manifolds, this space is always homotopy equivalent to a wedge of circles.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109484"},"PeriodicalIF":0.6,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144517786","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 the openness of the idempotent barycenter map related to a t-norm","authors":"Dawid Krasiński , Taras Radul","doi":"10.1016/j.topol.2025.109485","DOIUrl":"10.1016/j.topol.2025.109485","url":null,"abstract":"<div><div>We demonstrate that the idempotent barycenter map, associated with a t-norm ⁎, is open if and only if the map of max-⁎ convex combination is open. Nevertheless, we illustrate that the characteristics of the idempotent barycenter map, in general, depend on the specific t-norm being employed.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109485"},"PeriodicalIF":0.6,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144480511","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":"Involutions on the product of projective space and sphere","authors":"Dimpi , Hemant Kumar Singh","doi":"10.1016/j.topol.2025.109481","DOIUrl":"10.1016/j.topol.2025.109481","url":null,"abstract":"<div><div>This paper focuses on the actions of <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> on a finite CW-complex <em>X</em>, whose mod 2 cohomology is isomorphic to that of a product of projective space and sphere, <span><math><mi>F</mi><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span>, where <span><math><mi>F</mi></math></span> is either <span><math><mi>R</mi></math></span> or <span><math><mi>C</mi></math></span>. For the case when <em>X</em> is totally nonhomologous to zero in <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span>, we determine the possible connected fixed point sets and construct examples illustrating these possibilities. We also address the case when <em>X</em> is not totally nonhomologous to zero in <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span> under certain assumptions.</div><div>Furthermore, we compute the cohomology ring structure of the orbit spaces of free involutions on <em>X</em>. As an application, we derive the Borsuk-Ulam type results.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109481"},"PeriodicalIF":0.6,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491641","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":"An extension of the class σE and complete Erdős space","authors":"Alfredo Zaragoza","doi":"10.1016/j.topol.2025.109483","DOIUrl":"10.1016/j.topol.2025.109483","url":null,"abstract":"<div><div>In <span><span>[6]</span></span>, a topological characterization of the product of the rational numbers and complete Erdős space was found introducing the class <span><math><mi>σ</mi><mi>E</mi></math></span>. The objective of this work is to introduce the class of spaces <span><math><mi>σ</mi><msup><mrow><mi>E</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span> and study some of its properties. We will also show that for certain subsets of complete Erdős space, the classes <span><math><mi>σ</mi><mi>E</mi></math></span> and <span><math><mi>σ</mi><msup><mrow><mi>E</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span> coincide. This fact gives a partial answer to some questions of <span><span>[6]</span></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109483"},"PeriodicalIF":0.6,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144338271","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}
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
