{"title":"No covering with nowhere dense P-sets in the Cohen model","authors":"Alan Dow , Osvaldo Guzmán","doi":"10.1016/j.topol.2025.109569","DOIUrl":"10.1016/j.topol.2025.109569","url":null,"abstract":"<div><div>We prove that if less than <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>-many Cohen reals are added to a model of <span>CH</span>, then <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> can not be covered by nowhere dense <span>P</span>-sets (equivalently, there is an ultrafilter on <em>ω</em> that does not contain a tower).</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"375 ","pages":"Article 109569"},"PeriodicalIF":0.5,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144932768","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":"Corrigendum to “Double homology and wedge-decomposable simplicial complexes” [Topology and its Applications 367 (2025) 109319]","authors":"Donald Stanley, Carlos Gabriel Valenzuela Ruiz","doi":"10.1016/j.topol.2025.109557","DOIUrl":"10.1016/j.topol.2025.109557","url":null,"abstract":"","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"375 ","pages":"Article 109557"},"PeriodicalIF":0.5,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144912538","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":"Embedding as a closed subspace in irreducible spaces","authors":"Yasushi Hirata, Yukinobu Yajima","doi":"10.1016/j.topol.2025.109568","DOIUrl":"10.1016/j.topol.2025.109568","url":null,"abstract":"<div><div>Irreducibility is a covering property which is not hereditary with respect to closed subspaces. It was shown in <span><span>[3]</span></span> that every (Tychonoff) space <em>X</em> can be embedded as a closed subspace in some irreducible (Tychonoff) space <em>Z</em>. Here we make the embedding theorem add <span><math><mi>L</mi><mo>(</mo><mi>Z</mi><mo>)</mo><mo>=</mo><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> or <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><mi>Z</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. Moreover, we consider some questions raised in <span><span>[3]</span></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"375 ","pages":"Article 109568"},"PeriodicalIF":0.5,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144926026","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":"Classes of Baire spaces defined by semi-neighborhoods of the diagonal","authors":"Evgenii Reznichenko","doi":"10.1016/j.topol.2025.109556","DOIUrl":"10.1016/j.topol.2025.109556","url":null,"abstract":"<div><div>With the help of semi-neighborhoods of the diagonal, classes of Baire spaces are defined: Δ, <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>h</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> Baire spaces. These classes of spaces are studied with the help of topological games. They are useful in studying continuity in groups: paratopological Δ-Baire groups, quasi-topological <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>h</mi></mrow></msub></math></span>-Baire groups, and semitopological <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>-Baire groups are topological groups.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"375 ","pages":"Article 109556"},"PeriodicalIF":0.5,"publicationDate":"2025-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144902631","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":"Recurrence and minimal sets of maps on local dendrite","authors":"Hafedh Abdelli","doi":"10.1016/j.topol.2025.109555","DOIUrl":"10.1016/j.topol.2025.109555","url":null,"abstract":"<div><div>Let <em>X</em> be a local dendrite with countably many endpoints and <span><math><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></math></span> be a continuous mapping. Denote by <span><math><mi>R</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span>, <span><math><mi>P</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span>, are the sets of recurrent, periodic points of <em>f</em> respectively and by <span><math><mi>M</mi><mo>(</mo><mi>f</mi><mo>)</mo><mo>:</mo><mo>=</mo><mrow><mo>{</mo><mi>M</mi><mo>:</mo><mtext>M is minimal of</mtext><mspace></mspace><mtext>f</mtext><mspace></mspace><mtext>and </mtext><mi>M</mi><mo>∩</mo><mover><mrow><mi>P</mi><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>‾</mo></mover><mo>=</mo><mo>∅</mo><mo>}</mo></mrow></math></span>. We prove that <span><math><mi>M</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span> is finite and <span><math><mover><mrow><mi>R</mi><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>‾</mo></mover><mo>=</mo><mover><mrow><mi>P</mi><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>‾</mo></mover><mo>∪</mo><mrow><mo>(</mo><munder><mo>⋃</mo><mrow><mi>M</mi><mo>∈</mo><mi>M</mi><mo>(</mo><mi>f</mi><mo>)</mo></mrow></munder><mi>M</mi><mo>)</mo></mrow></math></span>. As a consequence, <span><math><mover><mrow><mi>R</mi><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>‾</mo></mover><mo>=</mo><mrow><mi>SR</mi></mrow><mo>(</mo><mi>f</mi><mo>)</mo><mo>∪</mo><mover><mrow><mi>P</mi><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>‾</mo></mover></math></span>, where <span><math><mrow><mi>SR</mi></mrow><mo>(</mo><mi>f</mi><mo>)</mo></math></span> is the set of syndetically recurrent points of <em>f</em>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"375 ","pages":"Article 109555"},"PeriodicalIF":0.5,"publicationDate":"2025-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144902633","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 Gs-convergence and Gs-countable compactness","authors":"Li Liu , Shou Lin , Xiangeng Zhou","doi":"10.1016/j.topol.2025.109554","DOIUrl":"10.1016/j.topol.2025.109554","url":null,"abstract":"<div><div><em>G</em>-convergence and <em>G</em>-sequential convergence are two convergent methods of sequences on a set, but they do not imply each other. How to further find the connection between these two types of convergence on a set, and establish their relationships with certain topological spaces?</div><div>In this paper, <em>G</em>-sequential convergence is referred to as <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>-convergence. <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>-convergence on a set <em>X</em> is proved to be exactly the usual convergence of sequences in the <em>G</em>-open topological space <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span> and the <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>-open topological space <span><math><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow></msub></math></span>, which enriches our understanding of subset properties and prompts us to study <em>G</em>-convergence and <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>-convergence on topological spaces. Some relationships among <em>G</em>-hulls, <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>-hulls, <em>G</em>-closures and <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>-closures are discussed. Various types of countably compact-like properties in the sense of <em>G</em>-convergence or <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>-convergence are characterized and found to be interrelated. It is proved that for a method <em>G</em> on a set <em>X</em>, <em>X</em> is <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>-hull countably compact if and only if the space <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span> is sequentially compact, if and only if <span><math><msub><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow></msub></math></span> is countably compact. The <em>G</em>-product property of <em>G</em>-sequential compactness and <em>G</em>-countable compactness are studied, and the following question is negatively answered: Is <em>G</em>-sequential compactness preserved by the finite <em>G</em>-products?</div><div>This study establishes the position of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>-convergence in <em>G</em>-convergence and the usual convergence of sequences, and also shows that <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>-convergence provides a feasible approach for revealing richer properties of <em>G</em>-convergence.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"375 ","pages":"Article 109554"},"PeriodicalIF":0.5,"publicationDate":"2025-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144908047","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":"Crossing change maps in filtered grid homology","authors":"Matthew Kendall","doi":"10.1016/j.topol.2025.109552","DOIUrl":"10.1016/j.topol.2025.109552","url":null,"abstract":"<div><div>We extend the crossing-change maps between grid complexes, defined by Ozsváth–Szabó–Stipsicz, to filtered grid complexes and give a combinatorial formulation of the Alishahi–Eftekhary knot invariant.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"375 ","pages":"Article 109552"},"PeriodicalIF":0.5,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144902621","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":"Metrizability and certain generalized metrizability on hyperspaces with the locally finite topology","authors":"Zhangyong Cai , Chuan Liu","doi":"10.1016/j.topol.2025.109551","DOIUrl":"10.1016/j.topol.2025.109551","url":null,"abstract":"<div><div>For a topological space <em>X</em>, let <span><math><mi>C</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> be the set of all nonempty closed of <em>X</em>, and denote the set <span><math><mi>C</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> with the locally finite topology by <span><math><mo>(</mo><mi>C</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><mtext>LF</mtext><mo>)</mo></math></span>.</div><div>In the first part of this paper, generalized metric spaces properties such as semi-stratifiability and Lašnev space property etc on hyperspaces with the locally finite topology are investigated and some metrization theorems are established.</div><div>(1) If <em>X</em> is a normal <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-space, then <span><math><mo>(</mo><mi>C</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><mtext>LF</mtext><mo>)</mo></math></span> is semi-stratifiable if and only if <span><math><mo>(</mo><mi>C</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><mtext>LF</mtext><mo>)</mo></math></span> is metrizable.</div><div>(2) If <em>X</em> is a metrizable space, then <span><math><mo>(</mo><mi>C</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><mtext>LF</mtext><mo>)</mo></math></span> is of countable tightness if and only if <span><math><mo>(</mo><mi>C</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><mtext>LF</mtext><mo>)</mo></math></span> is metrizable.</div><div>(3) <span><math><mo>(</mo><mi>C</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><mtext>LF</mtext><mo>)</mo></math></span> is a Lašnev space if and only if it is metrizable.</div><div>(4) <span><math><mo>(</mo><mi>M</mi><mi>A</mi><mo>+</mo><mo>¬</mo><mi>C</mi><mi>H</mi><mo>)</mo></math></span> <span><math><mo>(</mo><mi>C</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><mtext>LF</mtext><mo>)</mo></math></span> is a <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-space if and only if it is metrizable.</div><div>In the second part, the first step is made towards the investigation of hyperspaces with the locally finite topology with an <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span>-base. The following results are obtained.</div><div>(5) If <span><math><mo>(</mo><mi>C</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><mtext>LF</mtext><mo>)</mo></math></span> is a Fréchet-Urysohn space with an <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span>-base, then <span><math><mo>(</mo><mi>C</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><mtext>LF</mtext><mo>)</mo></math></span> is first-countable.</div><div>(6) Let <em>X</em> be metrizable. If <span><math><mo>(</mo><mi>C</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><mtext>LF</mtext><mo>)</mo></math></span> has an <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span>-base, then ","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"375 ","pages":"Article 109551"},"PeriodicalIF":0.5,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144893693","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":"Pointwise doubling indices of uniform Bernoulli measures on Bedford-McMullen carpets","authors":"Hui Rao, Yan-Li Xu, Yuan Zhang","doi":"10.1016/j.topol.2025.109549","DOIUrl":"10.1016/j.topol.2025.109549","url":null,"abstract":"<div><div>Doubling measures were introduced by Beurling and Ahlfors in 1956 and they have since become a basic concept in analysis on metric spaces. In this paper, for a measure which is not doubling, we introduce a notion of pointwise doubling index, and calculate the pointwise doubling indices of uniform Bernoulli measures on Bedford-McMullen carpets. As an application, we obtain several new Lipschitz invariants of Bedford-McMullen carpets.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"375 ","pages":"Article 109549"},"PeriodicalIF":0.5,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144895814","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 self-homeomorphisms of the divisor topological space","authors":"Jhixon Macías","doi":"10.1016/j.topol.2025.109550","DOIUrl":"10.1016/j.topol.2025.109550","url":null,"abstract":"<div><div>Let <em>D</em> be the topological space <span><math><mo>(</mo><msub><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>τ</mi><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> is the set of integers ≥2 and <em>τ</em> is the divisor topology. In this work (among other results), we characterize the self-homeomorphisms on <em>D</em>, and show that the group of homeomorphisms of <em>D</em> is isomorphic to the symmetric group of the integers.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"375 ","pages":"Article 109550"},"PeriodicalIF":0.5,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144886987","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}