Topology and its Applications最新文献

{"title":"On o-free sequences and compacta","authors":"Nathan Carlson","doi":"10.1016/j.topol.2025.109631","DOIUrl":"10.1016/j.topol.2025.109631","url":null,"abstract":"<div><div>We use the notion of an <em>o</em>-free sequence to give new bounds for the cardinality of Hausdorff spaces and regular spaces. There are several implications for compacta. One is that if <em>X</em> is a compactum then <span><math><mi>w</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mi>h</mi><mi>L</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>o</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span>, where <span><math><mi>o</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is the <em>o</em>-tightness introduced by Tkachenko. Another is that <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><mi>h</mi><mi>L</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>o</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo><mi>w</mi><msub><mrow><mi>ψ</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> if <em>X</em> is a compactum. This is shown to be a strict improvement of Arhangel'skiĭ's bound <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>ψ</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span>. Finally, we show <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><mi>h</mi><mi>L</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>o</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo><mi>π</mi><mi>χ</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> if <em>X</em> is a homogeneous compactum. We note <span><math><mi>h</mi><mi>L</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>o</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo><mi>π</mi><mi>χ</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> for such spaces, where <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> is de la Vega's bound for the cardinality of homogeneous compacta.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109631"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269842","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":"Guest Editorial for Special Issue","authors":"Bob Stephenson (Guest Editor),&nbsp;Alan Dow (Guest Editor),&nbsp;Paul Szeptycky (Guest Editor),&nbsp;Todd Eisworth (Guest Editor)","doi":"10.1016/j.topol.2025.109632","DOIUrl":"10.1016/j.topol.2025.109632","url":null,"abstract":"","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109632"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145322259","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":"Closed left ideal decompositions of βG ∖ G and wandering points","authors":"Valentin Keyantuo ,&nbsp;Yevhen Zelenyuk","doi":"10.1016/j.topol.2025.109646","DOIUrl":"10.1016/j.topol.2025.109646","url":null,"abstract":"<div><div>Let <em>G</em> be a countably infinite discrete group, let <em>βG</em> be the Stone-Čech compactification of <em>G</em>, and let <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>=</mo><mi>β</mi><mi>G</mi><mo>∖</mo><mi>G</mi></math></span>. For every closed left ideal <span><math><mi>X</mi><mo>⊆</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> of <em>βG</em>, there is a finest decomposition <span><math><mi>D</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of <em>X</em> into closed left ideals of <em>βG</em> with the property that the corresponding quotient space of <em>X</em> is Hausdorff. It is known that <span><math><mi>D</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></math></span> is nontrivial, and in fact <span><math><mo>|</mo><mi>D</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo><mo>|</mo><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>c</mi></mrow></msup></math></span>, and for some <span><math><mi>X</mi><mo>∈</mo><mi>D</mi><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></math></span>, <span><math><mi>D</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is nontrivial. We show that it is consistent with ZFC that, if <em>G</em> can be embedded algebraically into a compact group, then for every <span><math><mi>X</mi><mo>∈</mo><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><mi>D</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is nontrivial.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109646"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418006","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":"Hyperbolic handlebody complements in 3-manifolds","authors":"Colin Adams ,&nbsp;Francisco Gomez-Paz ,&nbsp;Jiachen Kang ,&nbsp;Lukas Krause ,&nbsp;Gregory Li ,&nbsp;Chloe Marple ,&nbsp;Ziwei Tan","doi":"10.1016/j.topol.2025.109642","DOIUrl":"10.1016/j.topol.2025.109642","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> be a compact and orientable 3-manifold. After capping off spherical boundaries with balls and removing any torus boundaries, we prove that the resulting manifold <em>M</em> contains handlebodies of arbitrary genus such that the closure of their complement is hyperbolic. We then extend the octahedral decomposition to obtain bounds on volume for some of these handlebody complements.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109642"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418007","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":"Maximal ω1-compactness","authors":"Ofelia T. Alas ,&nbsp;L. Enrique Gutiérrez-Domínguez ,&nbsp;Richard G. Wilson","doi":"10.1016/j.topol.2025.109637","DOIUrl":"10.1016/j.topol.2025.109637","url":null,"abstract":"<div><div>A space is said to be <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-compact if every closed and discrete subset is countable. In this paper we study spaces whose topologies are maximal with respect to being <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-compact and we give characterisations of this property in a number of special cases. We also consider the preservation of <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-compactness in products.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109637"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145364653","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 note on the structural stability of almost one-to-one maps","authors":"María Isabel Cortez,&nbsp;Till Hauser","doi":"10.1016/j.topol.2025.109641","DOIUrl":"10.1016/j.topol.2025.109641","url":null,"abstract":"<div><div>A continuous surjection <span><math><mi>π</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></math></span> between compact Hausdorff spaces induces continuous surjections <span><math><mi>M</mi><mo>(</mo><mi>π</mi><mo>)</mo><mo>:</mo><mi>M</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>→</mo><mi>M</mi><mo>(</mo><mi>Y</mi><mo>)</mo></math></span> and <span><math><mi>H</mi><mo>(</mo><mi>π</mi><mo>)</mo><mo>:</mo><mi>H</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>→</mo><mi>H</mi><mo>(</mo><mi>Y</mi><mo>)</mo></math></span> between the spaces of regular Borel probability measures, and the spaces of closed subsets, respectively. It is well known that <span><math><mi>H</mi><mo>(</mo><mi>π</mi><mo>)</mo></math></span> is irreducible if and only if <em>π</em> is irreducible. We show that <span><math><mi>M</mi><mo>(</mo><mi>π</mi><mo>)</mo></math></span> is irreducible if and only if <em>π</em> is irreducible. Furthermore, we show that whenever <em>π</em> is almost one-to-one then <span><math><mi>M</mi><mo>(</mo><mi>π</mi><mo>)</mo></math></span> and <span><math><mi>H</mi><mo>(</mo><mi>π</mi><mo>)</mo></math></span> are almost one-to-one. In particular, we observe that continuous surjections between compact metric spaces are almost one-to-one if and only if <span><math><mi>H</mi><mo>(</mo><mi>π</mi><mo>)</mo></math></span> is almost one-to-one and a similar statement about <span><math><mi>M</mi><mo>(</mo><mi>π</mi><mo>)</mo></math></span>. Finally, we give alternative proofs for some results in <span><span>[8]</span></span> regarding semi-open maps.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109641"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145364656","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":"Pseudocompact versus countably compact in first countable spaces","authors":"István Juhász ,&nbsp;Lajos Soukup ,&nbsp;Zoltán Szentmiklóssy","doi":"10.1016/j.topol.2025.109650","DOIUrl":"10.1016/j.topol.2025.109650","url":null,"abstract":"<div><div>The primary objective of this work is to construct spaces that are “<em>pseudocompact but not countably compact</em>”, abbreviated as P-NC, while endowing them with additional properties.</div><div>First, motivated by an old problem of van Douwen concerning first countable P-NC spaces with countable extent, we construct from CH a locally compact and locally countable first countable P-NC space with countable spread.</div><div>A space is deemed <em>densely countably compact</em>, denoted as DCC for brevity, if it possesses a dense, countably compact subspace. Moreover, a space qualifies as <em>densely relatively countably compact</em>, abbreviated as DRC, if it contains a dense subset <em>D</em> such that every infinite subset of <em>D</em> has an accumulation point in <em>X</em>.</div><div>A countably compact space is DCC, a DCC space is DRC, and a DRC space is evidently pseudocompact. The Tychonoff plank is a DCC space but is not countably compact. A Ψ-space belongs to the class of DRC spaces but is ¬DCC. Lastly, if <span><math><mi>p</mi><mo>∈</mo><msup><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is not a P-point, then <span><math><mi>T</mi><mo>(</mo><mi>p</mi><mo>)</mo></math></span>, representing the type of <em>p</em> in <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, constitutes a pseudocompact subspace of <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> that is ¬DRC.</div><div>When considering a topological property denoted as <em>Q</em>, we define a space <em>X</em> as “<em>R-hereditarily Q</em>” if every regular closed subspace of <em>X</em> also possesses property <em>Q</em>. The Tychonoff plank and the Ψ-space are not R-hereditary examples for separating the above-mentioned properties. However, the aforementioned space <span><math><mi>T</mi><mo>(</mo><mi>p</mi><mo>)</mo></math></span> is an R-hereditary example, albeit not being first countable.</div><div>In this paper we want to find (first countable) examples which separate these properties R-hereditarily. We have obtained the following result.<ul><li><span>(1)</span><span><div>There is a R-hereditarily “DCC, but not countably compact” space.</div></span></li><li><span>(2)</span><span><div>If CH holds, then there is a R-hereditarily “DRC, but ¬DCC” space.</div></span></li><li><span>(3)</span><span><div>If <span><math><mi>s</mi><mo>=</mo><mi>c</mi></math></span>, then there is a first countable, R-hereditarily “pseudocompact, but ¬DRC” space.</div></span></li></ul> In contrast to (2), it is unknown whether a first countable, R-hereditarily “DRC, but ¬DCC” space <em>X</em> can exist.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109650"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466896","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 covering dimension of topological spaces","authors":"Chunxu Ji ,&nbsp;Kaiyun Wang","doi":"10.1016/j.topol.2025.109661","DOIUrl":"10.1016/j.topol.2025.109661","url":null,"abstract":"<div><div>This paper focuses on the quasi covering dimension of topological spaces. We prove that a topological space and its Kolmogorov quotient space have the same quasi covering dimension. Subsequently, for a finite <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math></span>, we show that its quasi covering dimension <span><math><mi>d</mi><mi>i</mi><msub><mrow><mi>m</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo>)</mo><mo>=</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><mi>w</mi><mi>i</mi><mi>d</mi><mi>t</mi><mi>h</mi><mo>(</mo><mo>↓</mo><mspace></mspace><mi>x</mi><mo>)</mo><mo>|</mo><mi>x</mi><mo>∈</mo><mrow><mi>Max</mi></mrow><mo>(</mo><mi>X</mi><mo>)</mo><mo>}</mo><mo>−</mo><mn>1</mn></math></span>, where <span><math><mrow><mi>Max</mi></mrow><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is the set of all maximal elements of <em>X</em> under the specialization order. Furthermore, we investigate the subspace theorems of quasi covering dimension of finite topological spaces and give an example presenting that the Cartesian product theorem does not hold in general. Finally, we discuss the relationships between the quasi covering dimension of topological spaces and that of frames.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109661"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145571264","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":"Spaces of countable type with hereditarily Baire Vietoris hyperspace","authors":"Francis Jordan","doi":"10.1016/j.topol.2025.109655","DOIUrl":"10.1016/j.topol.2025.109655","url":null,"abstract":"<div><div>We extend the characterization of the hereditary Baireness of the Vietoris hyperspace of separable metric spaces given by Gartside, Medini, and Zdomskyy <span><span>[3]</span></span> to the class of regular topological spaces of countable type. A theorem of Bouziad, Holá, and Zsilinszky <span><span>[2]</span></span> is also extended in a similar way.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109655"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145520222","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":"Knots in RP3","authors":"Louis H. Kauffman ,&nbsp;Rama Mishra ,&nbsp;Visakh Narayanan","doi":"10.1016/j.topol.2025.109656","DOIUrl":"10.1016/j.topol.2025.109656","url":null,"abstract":"<div><div>This paper studies knots in three dimensional projective space. Techniques in virtual knot theory are applied to obtain a Jones polynomial for projective links and it is shown that this is equivalent to the Jones polynomial defined by Drobotukhina. A Khovanov homology theory for projective knots is constructed by using virtual Khovanov homology and the virtual Rasmussen invariant of Dye, Kaestner, and Kauffman. This homology theory is compared with the Khovanov theory developed by Manolescu and Willis for projective knots. It is shown that these theories are essentially equivalent, giving new viewpoints for both methods. The paper ends with problems about these approaches and an example of multiple projectivizations of the figure-8 knot whose equivalence is unknown at this time.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109656"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466920","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}