Topology and its Applications最新文献

{"title":"Cartesian product of combinatorially rich sets- algebraic, elementary and dynamical approaches","authors":"Pintu Debnath","doi":"10.1016/j.topol.2024.109148","DOIUrl":"10.1016/j.topol.2024.109148","url":null,"abstract":"<div><div>Using the methods of topological dynamics, H. Furstenberg introduced the notion of a central set and proved the famous Central Sets Theorem. In [Fund. Math 199 (2008)], D. De, N. Hindman, and D. Strauss introduced the notion of a <em>C</em>-set, satisfying the strong central sets theorem. In [Topology Proc. 35 (2010)], using the algebraic structure of the Stone-Čech compactification of a discrete semigroup, N. Hindman and D. Strauss proved that the Cartesian product of two <em>C</em>-sets is a <em>C</em>-set. S. Goswami has proved the same result using the elementary characterization of <em>C</em>-sets. In this article, we will prove that the product of two <em>C</em>-sets is a <em>C</em>-set, using the dynamical characterization of <em>C</em>-sets. Recently, S. Goswami has proved that the Cartesian product of two <em>CR</em>-sets is a <em>CR</em>-set, which was a question posed by N. Hindman, H. Hosseini, D. Strauss, and M. Tootkaboni in [Semigroup Forum 107 (2023)]. Here we also prove that the Cartesian product of two essential <em>CR</em>-sets is an essential <em>CR</em>-set.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109148"},"PeriodicalIF":0.6,"publicationDate":"2024-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743386","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 Rudin-Kiesler pre-order and the Pixley-Roy spaces over ultrafilters","authors":"Masami Sakai","doi":"10.1016/j.topol.2024.109136","DOIUrl":"10.1016/j.topol.2024.109136","url":null,"abstract":"&lt;div&gt;&lt;div&gt;For a &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-space &lt;em&gt;X&lt;/em&gt;, we denote by &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; the Pixley-Roy space over &lt;em&gt;X&lt;/em&gt;. For &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;mo&gt;∪&lt;/mo&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be the subspace of the Stone-Čech compactification &lt;em&gt;βω&lt;/em&gt; of the discrete space &lt;em&gt;ω&lt;/em&gt;. Motivated by Gul'ko's theorem (&lt;span&gt;&lt;span&gt;Theorem 1.1&lt;/span&gt;&lt;/span&gt;), we show: (1) &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are homeomorphic if and only if &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are homeomorphic (i.e., &lt;em&gt;p&lt;/em&gt; and &lt;em&gt;q&lt;/em&gt; are type-equivalent), (2) if &lt;em&gt;q&lt;/em&gt; is selective and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; can be embedded into &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, then &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are homeomorphic, (3) if &lt;em&gt;p&lt;/em&gt; is selective, then &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; contains copies of some &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; which are pairwise non-homeomorphic, and (4) &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⊕&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are pairwise non-homeomorphic, where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is the quotient space obtained by identifying the limit points of the topological sum &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⊕&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mro","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109136"},"PeriodicalIF":0.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706391","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 Dold-Whitney's parallelizability of 4-manifolds","authors":"Valentina Bais","doi":"10.1016/j.topol.2024.109144","DOIUrl":"10.1016/j.topol.2024.109144","url":null,"abstract":"<div><div>We present a proof of a theorem by Dold and Whitney, according to which a closed orientable 4-manifold is parallelizable if and only if its second Stiefel-Whitney class, first Pontryagin class and Euler characteristics vanish. This follows from a stronger result due to Dold and Whitney on the classification of oriented sphere bundles over a 4-complex. Our proof is based on an argument by R. Kirby on the classification of <span><math><mi>S</mi><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span>-principal bundles over the 4-sphere by means of their Euler and first Pontryagin classes.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109144"},"PeriodicalIF":0.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142719800","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 uniform convergence topology on separable subsets","authors":"J.A. Cruz-Chapital ,&nbsp;A.D. Rojas-Sánchez ,&nbsp;Á. Tamariz-Mascarúa ,&nbsp;H. Villegas-Rodríguez","doi":"10.1016/j.topol.2024.109135","DOIUrl":"10.1016/j.topol.2024.109135","url":null,"abstract":"<div><div>For a topological space <em>X</em>, let <span><math><msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>X</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msub><mo>:</mo><mo>=</mo><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>X</mi></mrow></msup><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>)</mo></math></span> be the cartesian product of <span><math><mo>|</mo><mi>X</mi><mo>|</mo></math></span> copies of the real line <span><math><mi>R</mi></math></span> with the topology of the uniform convergence on separable subsets of <em>X</em>. In this article we analyze the subspace <span><math><mi>C</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of <span><math><msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>X</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msub></math></span> of all real-valued continuous functions on <em>X</em>, denoted by <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. We determine when <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is dense and when is closed in <span><math><msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>X</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msub></math></span>, and we obtain some results about the Baire property in <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. Finally, we determine the cellularity of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>α</mi><mo>]</mo><mo>)</mo></math></span> where <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mi>α</mi><mo>]</mo></math></span> is the space of ordinal numbers belonging to <span><math><mi>α</mi><mo>+</mo><mn>1</mn></math></span> with its usual order topology.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109135"},"PeriodicalIF":0.6,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706390","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":"Relatively functionally countable subsets of products","authors":"Anton E. Lipin","doi":"10.1016/j.topol.2024.109133","DOIUrl":"10.1016/j.topol.2024.109133","url":null,"abstract":"<div><div>A subset <em>A</em> of a topological space <em>X</em> is called <em>relatively functionally countable</em> (<em>RFC</em>) in <em>X</em>, if for each continuous function <span><math><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>R</mi></math></span> the set <span><math><mi>f</mi><mo>[</mo><mi>A</mi><mo>]</mo></math></span> is countable. We prove that all RFC subsets of a product <span><math><munder><mo>∏</mo><mrow><mi>n</mi><mo>∈</mo><mi>ω</mi></mrow></munder><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are countable, assuming that spaces <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are Tychonoff and all RFC subsets of every <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are countable. In particular, in a metrizable space every RFC subset is countable.</div><div>The main tool in the proof is the following result: for every Tychonoff space <em>X</em> and any countable set <span><math><mi>Q</mi><mo>⊆</mo><mi>X</mi></math></span> there is a continuous function <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> such that the restriction of <em>f</em> to <span><math><msup><mrow><mi>Q</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span> is injective.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109133"},"PeriodicalIF":0.6,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142654964","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":"Extendability to Marczewski-Burstin countably representable ideals","authors":"Marta Kwela,&nbsp;Jacek Tryba","doi":"10.1016/j.topol.2024.109134","DOIUrl":"10.1016/j.topol.2024.109134","url":null,"abstract":"<div><div>In the article we consider Marczewski-Burstin countably representable (in short: <span><math><mi>MBC</mi></math></span>) ideals. We propose a concept of extendability to <span><math><mi>MBC</mi></math></span> ideals and provide some of its properties like the fact that it lies between the notions of <em>ω</em>-+-diagonalizability and countable separability. We also answer the question posed in [Topology Appl. 248 (2018), 149–163], by showing that the ideal <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> is not <span><math><mi>MBC</mi></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109134"},"PeriodicalIF":0.6,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142654965","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":"MSNR spaces revisited","authors":"John E. Porter","doi":"10.1016/j.topol.2024.109132","DOIUrl":"10.1016/j.topol.2024.109132","url":null,"abstract":"<div><div>We revisit monotonically semi-neighborhood refining (MSNR) spaces which were introduced by Stares in 1996. MSNR spaces are shown to be lob-spaces with well-ordered (F). The relationships between MSNR spaces with other monotone covering properties are also explored. We show the existence of MSNR spaces that do not posses a monotone locally-finite refining operator and spaces with a monotone locally-finite refining operator that are not MSNR answering a question of Popvassilev and Porter. Compact MSNR spaces may not be metrizable in general, but compact MSNR LOTS are. GO-spaces whose underlying LOTS has a <em>σ</em>-closed-discrete dense subset are shown to have a monotone star-finite refining operator.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109132"},"PeriodicalIF":0.6,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142656836","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 Ψω-factorizable groups","authors":"Heng Zhang ,&nbsp;Wenfei Xi ,&nbsp;Yaoqiang Wu ,&nbsp;Hongling Li","doi":"10.1016/j.topol.2024.109129","DOIUrl":"10.1016/j.topol.2024.109129","url":null,"abstract":"<div><div>A topological group <em>G</em> is called <span><math><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>-factorizable (resp. <span><math><mi>M</mi></math></span>-factorizable) if every continuous real-valued function on <em>G</em> admits a factorization via a continuous homomorphism onto a topological group <em>H</em> with <span><math><mi>ψ</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span> (resp. a first-countable group). The first purpose of this article is to discuss some characterizations of <span><math><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>-factorizable groups. It is shown that a topological group <em>G</em> is <span><math><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>-factorizable if and only if every continuous real-valued function on <em>G</em> is <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>δ</mi></mrow></msub></math></span>-uniformly continuous, if and only if for every cozero-set <em>U</em> of <em>G</em>, there exists a <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>δ</mi></mrow></msub></math></span>-subgroup <em>N</em> of <em>G</em> such that <span><math><mi>U</mi><mi>N</mi><mo>=</mo><mi>U</mi></math></span>. Sufficient conditions on the <span><math><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>-factorizable group <em>G</em> to be <span><math><mi>M</mi></math></span>-factorizable are that <em>G</em> is <em>τ</em>-fine and <em>τ</em>-steady for a cardinal <em>τ</em>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109129"},"PeriodicalIF":0.6,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142656835","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 functor of comonotonically maxitive functionals","authors":"Taras Radul","doi":"10.1016/j.topol.2024.109131","DOIUrl":"10.1016/j.topol.2024.109131","url":null,"abstract":"<div><div>We introduce a functor of functionals that preserve the maximum of comonotone functions and the addition of constants. This functor is a subfunctor of the functor of order-preserving functionals and includes the idempotent measure functor as a subfunctor. The main aim of this paper is to demonstrate that this functor is isomorphic to the capacity functor. We establish this isomorphism using the fuzzy max-plus integral. In essence, this result can be viewed as an idempotent analogue of the Riesz Theorem, which establishes a correspondence between the set of <em>σ</em>-additive regular Borel measures and the set of positive linear functionals.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109131"},"PeriodicalIF":0.6,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592851","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 remarks on (a)-characterized subgroups of the circle","authors":"Nikola Bogdanovic","doi":"10.1016/j.topol.2024.109130","DOIUrl":"10.1016/j.topol.2024.109130","url":null,"abstract":"<div><div>In recent years, Barbieri, Dikranjan, Giordano Bruno and Weber have made progress on the problem of determining which characterized subgroups of the circle group are <em>(a-)factorizable</em>, that is, can be written as the sum of two proper (<em>a</em>-)characterized subgroups. We correct an imprecision in one of their results, <span><span>[2, Theorem 5.9]</span></span> from 2017, determining the countable <em>a</em>-characterized subgroups of <span><math><mi>T</mi></math></span> which are also <em>a</em>-factorizable. We also provide a revised proof of <span><span>[11, Proposition 1.3]</span></span> (Dikranjan, Kunen, 2007), asserting that <span><math><mi>Q</mi><mo>/</mo><mi>Z</mi></math></span> is characterized.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109130"},"PeriodicalIF":0.6,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142654963","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}