Topology and its Applications最新文献

{"title":"One more cardinality bound involving the free set number","authors":"Angelo Bella","doi":"10.1016/j.topol.2025.109536","DOIUrl":"10.1016/j.topol.2025.109536","url":null,"abstract":"<div><div>We discuss the possibility to extend a theorem of Šapirovskiĭ on the cardinality of a Lindelöf sequential space to the more general setting of pseudoradial spaces. As a byproduct, this will lead to a cardinality bound involving the free set number.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109536"},"PeriodicalIF":0.5,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144830300","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":"Symplectic groups, mapping class groups and the stability of bounded cohomology","authors":"Thorben Kastenholz","doi":"10.1016/j.topol.2025.109531","DOIUrl":"10.1016/j.topol.2025.109531","url":null,"abstract":"<div><div>Mapping class groups satisfy cohomological stability. In this note we show how results by Bestvina and Fujiwara imply that their bounded cohomology does not stabilize, additionally we show that stabily polynomials in the Mumford-Morita-Miller classes are unbounded i.e. their norm tends to infinity as one increases the genus.</div><div>While the bounded cohomology of the symplectic group does stabilize, we show that it does not stabilize via isometries in degree 2. In order to establish this we calculate the norm of the signature class in <span><math><msub><mrow><mi>Sp</mi></mrow><mrow><mn>2</mn><mi>h</mi></mrow></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></math></span> and estimate the norm of the integral signature class.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109531"},"PeriodicalIF":0.5,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144827336","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 monotone version of countably paracompact frames","authors":"Er-Guang Yang","doi":"10.1016/j.topol.2025.109535","DOIUrl":"10.1016/j.topol.2025.109535","url":null,"abstract":"<div><div>In this paper, we introduce the notion of monotonically countably paracompact frames as the point-free extension of monotonically countably paracompact spaces as well as the monotonization of countably paracompact frames. Characterizations of such a frame in terms of real functions are also presented.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109535"},"PeriodicalIF":0.5,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144770727","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 selectivity and disjoint local π-bases","authors":"Hector Barriga-Acosta,&nbsp;Alan Dow","doi":"10.1016/j.topol.2025.109534","DOIUrl":"10.1016/j.topol.2025.109534","url":null,"abstract":"<div><div>We answer affirmatively two questions of Gruenhage and Tkachuk from <span><span>[3]</span></span>. The first result is that every compact space of countable tightness has a countable disjoint local <em>π</em>-base at every point. The second result is that a space <em>X</em> is discretely selective if it is hereditarily Lindelöf and has the property that the inequality <span><math><mi>π</mi><mi>χ</mi><mo>(</mo><mi>K</mi><mo>,</mo><mi>X</mi><mo>)</mo><mo>&gt;</mo><mi>ω</mi></math></span> holds for every compact set <em>K</em> of <em>X</em>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109534"},"PeriodicalIF":0.5,"publicationDate":"2025-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144750560","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 non-cut subcontinua of graphs","authors":"Alejandro Illanes ,&nbsp;Verónica Martínez-de-la-Vega ,&nbsp;Jorge E. Vega","doi":"10.1016/j.topol.2025.109533","DOIUrl":"10.1016/j.topol.2025.109533","url":null,"abstract":"<div><div>Given a continuum <em>X</em>, let <span><math><mi>C</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> be the hyperspace of all subcontinua of <em>X</em>. We consider the hyperspace <span><math><mi>N</mi><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>A</mi><mo>∈</mo><mi>C</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>:</mo><mi>X</mi><mo>∖</mo><mi>A</mi></math></span> is connected}. In this paper we prove that the only locally connected continua <em>X</em> for which <span><math><mi>N</mi><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is compact are the arcs and the simple closed curves. We also characterize the finite graphs <em>G</em> for which <span><math><mi>N</mi><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is connected.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109533"},"PeriodicalIF":0.5,"publicationDate":"2025-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144770726","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":"Weak Baumgartner axioms and universal spaces","authors":"Corey Bacal Switzer","doi":"10.1016/j.topol.2025.109530","DOIUrl":"10.1016/j.topol.2025.109530","url":null,"abstract":"<div><div>If <em>X</em> is a topological space and <em>κ</em> is a cardinal then <span><math><msub><mrow><mi>BA</mi></mrow><mrow><mi>κ</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is the statement that for each pair <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>⊆</mo><mi>X</mi></math></span> of <em>κ</em>-dense subsets there is an autohomeomorphism <span><math><mi>h</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></math></span> mapping <em>A</em> to <em>B</em>. In particular <span><math><msub><mrow><mi>BA</mi></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> is equivalent the celebrated Baumgartner axiom on isomorphism types of <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-dense linear orders. In this paper we consider two natural weakenings of <span><math><msub><mrow><mi>BA</mi></mrow><mrow><mi>κ</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> which we call <span><math><msubsup><mrow><mi>BA</mi></mrow><mrow><mi>κ</mi></mrow><mrow><mo>−</mo></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>κ</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> for arbitrary perfect Polish spaces <em>X</em>. We show that the first of these, though properly weaker, entails many of the more striking consequences of <span><math><msub><mrow><mi>BA</mi></mrow><mrow><mi>κ</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> while the second does not. Nevertheless the second is still independent of <span><math><mi>ZFC</mi></math></span> and we show in particular that it fails in the Cohen and random models. This motivates several new classes of pairs of spaces which are “very far from being homeomorphic” which we call “avoiding”, “strongly avoiding”, and “totally avoiding”. The paper concludes by studying these classes, particularly in the context of forcing theory, in an attempt to gauge how different weak Baumgartner axioms may be separated.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109530"},"PeriodicalIF":0.5,"publicationDate":"2025-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144722241","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":"Topologically independent sets in topological groups and vector spaces","authors":"Jan Spěvák","doi":"10.1016/j.topol.2025.109527","DOIUrl":"10.1016/j.topol.2025.109527","url":null,"abstract":"<div><div>We study topological versions of an independent set in an abelian group and a linearly independent set in a vector space, a <em>topologically independent set</em> in a topological group and a <em>topologically linearly independent set</em> in a topological vector space. These counterparts of their algebraic versions are defined analogously and possess similar properties.</div><div>Let <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>×</mo></mrow></msup></math></span> be the multiplicative group of the field of complex numbers with its usual topology. We prove that a subset <em>A</em> of an arbitrary Tychonoff power of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>×</mo></mrow></msup></math></span> is topologically independent if and only if the topological subgroup <span><math><mo>〈</mo><mi>A</mi><mo>〉</mo></math></span> that it generates is the Tychonoff direct sum <span><math><msub><mrow><mo>⨁</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></msub><mo>〈</mo><mi>a</mi><mo>〉</mo></math></span>.</div><div>This theorem substantially generalizes an earlier result of the author, who has proved this for Abelian precompact groups.</div><div>Further, we show that topologically independent and topologically linearly independent sets coincide in vector spaces with weak topologies, although they are different in general.</div><div>We characterize topologically linearly independent sets in vector spaces with weak topologies and normed spaces. In a weak topology, a set <em>A</em> is topologically linearly independent if and only if its linear span is the Tychonoff direct sum <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow></msup></math></span>. In normed spaces <em>A</em> is topologically linearly independent if and only if it is uniformly minimal. Thus, from the point of view of topological linear independence, the Tychonoff direct sums <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow></msup></math></span> and (linear spans of) uniformly minimal sets, which are closely related to bounded biorthogonal systems, are of the same essence.</div><div>We also provide an application of topological linear independence to Lipschitz-free spaces.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109527"},"PeriodicalIF":0.5,"publicationDate":"2025-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144722436","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 non-R-factorizable product of R-factorizable groups","authors":"Ol'ga Sipacheva","doi":"10.1016/j.topol.2025.109529","DOIUrl":"10.1016/j.topol.2025.109529","url":null,"abstract":"<div><div>An example of two zero-dimensional <span><math><mi>R</mi></math></span>-factorizable groups whose product is not <span><math><mi>R</mi></math></span>-factorizable is constructed. One of these groups is second-countable and the other Lindelöf to any finite power.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109529"},"PeriodicalIF":0.6,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144696427","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 kR-spaces and sR-spaces","authors":"Saak Gabriyelyan ,&nbsp;Evgenii Reznichenko","doi":"10.1016/j.topol.2025.109528","DOIUrl":"10.1016/j.topol.2025.109528","url":null,"abstract":"<div><div>We give new characterizations of spaces <em>X</em> which are <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span>-spaces or <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span>-spaces. Applying the obtained results we provide some sufficient and necessary conditions on <em>X</em> for which <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is a <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span>-space or an <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span>-space. It is proved that <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is a <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span>-space for any space <em>X</em> with one non-isolated point; if, in addition, <span><math><mo>|</mo><mi>X</mi><mo>|</mo></math></span> is not sequential, then <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is even an <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span>-space. Under <span><math><mo>(</mo><mi>C</mi><mi>H</mi><mo>)</mo></math></span>, it is shown that there exists a separable metrizable space <em>X</em> such that <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is an Ascoli space but not a <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span>-space.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109528"},"PeriodicalIF":0.6,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144687180","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":"Strong uniform and strong Whitney convergences on C(X,Y)","authors":"Tarun Kumar Chauhan","doi":"10.1016/j.topol.2025.109526","DOIUrl":"10.1016/j.topol.2025.109526","url":null,"abstract":"<div><div>For any two metric spaces <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span>, <span><math><mo>(</mo><mi>Y</mi><mo>,</mo><mi>ρ</mi><mo>)</mo></math></span> and a bornology <span><math><mi>B</mi></math></span> on <em>X</em>, we investigate particular subsets of <span><math><mi>C</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span> which are clopen under the topology <span><math><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>B</mi></mrow><mrow><mi>s</mi><mi>w</mi></mrow></msubsup></math></span> of strong Whitney convergence on bornology and the topology <span><math><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>B</mi></mrow><mrow><mi>s</mi></mrow></msubsup></math></span> of strong uniform convergence on bornology. We show that the space <span><math><mi>C</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span> endowed with these topologies is not generally connected. In the process, we also provide new characterizations for the notions of shields and bornology that are shielded from closed sets, using these particular subsets of <span><math><mi>C</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109526"},"PeriodicalIF":0.6,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144655135","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}