{"title":"Remarks on SHD spaces and more divergence properties","authors":"","doi":"10.1016/j.topol.2024.109055","DOIUrl":"10.1016/j.topol.2024.109055","url":null,"abstract":"<div><p>The class of SHD spaces was recently introduced in <span><span>[12]</span></span>. The first part of this paper focuses on answering most of the questions presented in that article. For instance, we exhibit an example of a non-SHD Tychonoff space <em>X</em> such that <span><math><mi>F</mi><mo>[</mo><mi>X</mi><mo>]</mo></math></span>, the Pixley-Roy hyperspace of <em>X</em>, <em>βX</em>, the Stone-Čech compactification of <em>X</em>, and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span>, the ring of continuous functions over <em>X</em> equipped with the topology of pointwise convergence, are SHD.</p><p>In the second part of this work we will present some variations of the SHD notion, namely, the WSHD property and the OHD property. Furthermore, we will pay special attention to the relationships between <em>X</em> and <span><math><mi>F</mi><mo>[</mo><mi>X</mi><mo>]</mo></math></span> regarding these new concepts.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142164474","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 2022","authors":"","doi":"10.1016/j.topol.2024.109053","DOIUrl":"10.1016/j.topol.2024.109053","url":null,"abstract":"","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142136498","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":"Cofinal types and topological groups","authors":"","doi":"10.1016/j.topol.2024.109051","DOIUrl":"10.1016/j.topol.2024.109051","url":null,"abstract":"<div><p>The purpose of this note is to start the systematic analysis of cofinal types of topological groups.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142135999","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":"Quasi-uniform entropy vs topological entropy","authors":"","doi":"10.1016/j.topol.2024.109054","DOIUrl":"10.1016/j.topol.2024.109054","url":null,"abstract":"<div><p>In 2023 Haihambo and Olela Otafudu introduced and studied the notion of quasi-uniform entropy <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>Q</mi><mi>U</mi></mrow></msub><mo>(</mo><mi>ψ</mi><mo>)</mo></math></span> for a uniformly continuous self-map <em>ψ</em> of a quasi-metric or a quasi-uniform space <em>X</em>. In this paper, we discuss the connection between the topological entropy functions <span><math><mi>h</mi><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> and the quasi-uniform entropy function <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>Q</mi><mi>U</mi></mrow></msub></math></span> on a quasi-uniform space <em>X</em>, where <em>h</em> and <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> are the topological entropy functions defined using compact sets and finite open covers, respectively. In particular, we have shown that for a uniformly continuous self-map <em>ψ</em> of a <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-quasi-uniform space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>U</mi><mo>)</mo></math></span> we have <span><math><mi>h</mi><mo>(</mo><mi>ψ</mi><mo>)</mo><mo>≤</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>Q</mi><mi>U</mi></mrow></msub><mo>(</mo><mi>ψ</mi><mo>)</mo></math></span> when <em>X</em> is compact and <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>Q</mi><mi>U</mi></mrow></msub><mo>(</mo><mi>ψ</mi><mo>)</mo><mo>≤</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>ψ</mi><mo>)</mo></math></span> with equality if <em>X</em> is a compact <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> space.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142135996","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 aspherical configuration Lie groupoids","authors":"","doi":"10.1016/j.topol.2024.109052","DOIUrl":"10.1016/j.topol.2024.109052","url":null,"abstract":"<div><p>The complement of the hyperplanes <span><math><mo>{</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>}</mo></math></span>, for all <span><math><mi>i</mi><mo>≠</mo><mi>j</mi></math></span>, in <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, where <em>M</em> is an aspherical 2-manifold, is known to be aspherical. Here we consider the situation when <em>M</em> is a 2-dimensional orbifold. We prove this complement to be aspherical for a class of aspherical 2-dimensional orbifolds, and predict that it should be true in general also. We generalize this question in the category of Lie groupoids, as orbifolds can be identified with a certain kind of Lie groupoids.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142135997","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":"D-completion, well-filterification and sobrification","authors":"","doi":"10.1016/j.topol.2024.109050","DOIUrl":"10.1016/j.topol.2024.109050","url":null,"abstract":"<div><p>In this paper, we study the <em>D</em>-completion, well-filterification and sobrification of a <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> space. First, we present an example of a tapered closed set which is neither the closure of a directed set nor a closed <em>KF</em>-set. In 2020, Xu et al. asked whether closed <em>RD</em>-sets are exactly closed <em>WD</em>-sets for every <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> space. This example also gives a negative answer to the above problem, since each tapered closed set is a closed <em>WD</em>-set. Second, we provide a direct characterization for the <em>D</em>-completion of a poset by using the notion of pre-<em>C</em>-compact elements. Finally, for a given <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> space, we give some sufficient conditions which guarantee that each pair of its standard <em>D</em>-completion, standard well-filterification and standard sobrification agrees.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142130076","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":"Sequences with increasing subsequence","authors":"","doi":"10.1016/j.topol.2024.109049","DOIUrl":"10.1016/j.topol.2024.109049","url":null,"abstract":"<div><p>We study analytic and Borel subsets defined similarly to the old example of analytic complete set given by Luzin. Luzin's example, which is essentially a subset of the Baire space, is based on the natural partial order on naturals, i.e. division. It consists of sequences which contain increasing subsequence in given order.</p><p>We consider a variety of sets defined in a similar way. Some of them occurs to be Borel subsets of the Baire space, while others are analytic complete, hence not Borel.</p><p>In particular, we show that an analogon of Luzin example based on the natural linear order on rationals is analytic complete. We also characterize all countable linear orders having such property.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142122766","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":"Dynamic properties of the dynamical system (FnK(X),FnK(f))","authors":"","doi":"10.1016/j.topol.2024.109048","DOIUrl":"10.1016/j.topol.2024.109048","url":null,"abstract":"<div><p>Let <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> be a dynamical system, where <em>X</em> is a nondegenerate continuum and <em>f</em> is a map. For any positive integer <em>n</em>, we consider the hyperspace <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> with the Vietoris topology. For <span><math><mi>n</mi><mo>></mo><mn>1</mn></math></span> and <span><math><mi>K</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> the subset <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>,</mo><mi>X</mi><mo>)</mo></math></span> of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is defined as the collection of elements of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> containing <em>K</em>. We consider the quotient hyperspace <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mi>⧸</mi><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>,</mo><mi>X</mi><mo>)</mo></math></span>, which is obtained from <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> by shrinking <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>,</mo><mi>X</mi><mo>)</mo></math></span> to one point set. Furthermore, we consider the induced maps <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo><mo>:</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>→</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>f</mi><mo>)</mo><mo>:</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo><mo>→</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. In this paper, we introduce the dynamical system <span><math><mo>(</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>f</mi><mo>)</mo><mo>)</mo></math></span> and we study relationships between the conditions <span><math><mi>f</mi><mo>∈</mo><mi>M</mi></math></span>, <span><math><msub><mrow><mi>F</mi></mr","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142098390","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":"Compact subspaces of the space of separately continuous functions with the cross-uniform topology","authors":"","doi":"10.1016/j.topol.2024.109047","DOIUrl":"10.1016/j.topol.2024.109047","url":null,"abstract":"<div><p>We consider two natural topologies on the space <span><math><mi>S</mi><mo>(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo>)</mo></math></span> of all separately continuous functions defined on the product of two topological spaces <em>X</em> and <em>Y</em> and ranged into a topological or metric space <em>Z</em>. These topologies are the cross-open topology and the cross-uniform topology. We show that these topologies coincides if <em>X</em> and <em>Y</em> are pseudocompacts and <em>Z</em> is a metric space. We prove that a compact space <em>K</em> embeds into <span><math><mi>S</mi><mo>(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo>)</mo></math></span> for infinite compacts <em>X</em>, <em>Y</em> and a metrizable space <span><math><mi>Z</mi><mo>⊇</mo><mi>R</mi></math></span> if and only if the weight of <em>K</em> is less than the sharp cellularity of both spaces <em>X</em> and <em>Y</em>.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0166864124002323/pdfft?md5=5bdb3be46d45742e443d43a7c083ed49&pid=1-s2.0-S0166864124002323-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142098389","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 generalized metric property in strongly topological gyrogroups","authors":"","doi":"10.1016/j.topol.2024.109046","DOIUrl":"10.1016/j.topol.2024.109046","url":null,"abstract":"<div><p>A topological gyrogroup is a gyrogroup endowed with a topology such that the binary operation is jointly continuous and the inverse mapping is also continuous. It is shown that a strongly topological gyrogroup is a <em>q</em>-space if and only if it is an <em>M</em>-space. Then a characterization about weakly feathered strongly topological gyrogroups is given, that is, a strongly topological gyrogroup <em>G</em> is weakly feathered if and only if it contains a compact strong subgyrogroup <em>H</em> such that the quotient space <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> is submetrizable.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084147","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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