Pub Date : 2025-11-11DOI: 10.1016/j.topol.2025.109658
Danica Kosanović
Motivated by recent results on diffeomorphisms of 4-manifolds, this paper investigates fundamental groups of spaces of embeddings of in 4-manifolds. The majority of work goes into the case of framed immersed circles.
{"title":"On fundamental groups of spaces of framed embeddings of a circle in a 4-manifold","authors":"Danica Kosanović","doi":"10.1016/j.topol.2025.109658","DOIUrl":"10.1016/j.topol.2025.109658","url":null,"abstract":"<div><div>Motivated by recent results on diffeomorphisms of 4-manifolds, this paper investigates fundamental groups of spaces of embeddings of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>×</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> in 4-manifolds. The majority of work goes into the case of framed immersed circles.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109658"},"PeriodicalIF":0.5,"publicationDate":"2025-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145520225","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}
Pub Date : 2025-11-10DOI: 10.1016/j.topol.2025.109660
Daniel Jardón , Iván Sánchez , Manuel Sanchis
Given a uniform space , we denote by the family of all normal upper semicontinuous fuzzy sets with compact support. In this paper, we study transitivity on some uniformities on : the level-wise uniformity , the Skorokhod uniformity , and the sendograph uniformity . If is a continuous function, we mainly characterize when the induced dynamical systems , and are transitive, where is the Zadeh's extension of f.
给定一个一致空间（X，U），用F(X)表示具有紧支持的所有正规上半连续模糊集U:X→[0,1]的族。本文研究了F(X)上一些均匀性的可传递性：水平均匀性U∞、Skorokhod均匀性U0和传感器均匀性US。如果f:（X，U）→（X，U）是连续函数，我们主要刻画了f:（f (X),U∞）→（f (X),U∞）、f:（f (X),U0）→（f (X),U0）和f:（f (X)，US）→（f （X)，US）是可传递的，其中f: f (X)，US）→（f (X)，US）是f的Zadeh扩展。
{"title":"Transitivity of some uniformities on fuzzy sets","authors":"Daniel Jardón , Iván Sánchez , Manuel Sanchis","doi":"10.1016/j.topol.2025.109660","DOIUrl":"10.1016/j.topol.2025.109660","url":null,"abstract":"<div><div>Given a uniform space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>U</mi><mo>)</mo></math></span>, we denote by <span><math><mi>F</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> the family of all normal upper semicontinuous fuzzy sets <span><math><mi>u</mi><mo>:</mo><mi>X</mi><mo>→</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> with compact support. In this paper, we study transitivity on some uniformities on <span><math><mi>F</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>: the level-wise uniformity <span><math><msub><mrow><mi>U</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>, the Skorokhod uniformity <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, and the sendograph uniformity <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>S</mi></mrow></msub></math></span>. If <span><math><mi>f</mi><mo>:</mo><mo>(</mo><mi>X</mi><mo>,</mo><mi>U</mi><mo>)</mo><mo>→</mo><mo>(</mo><mi>X</mi><mo>,</mo><mi>U</mi><mo>)</mo></math></span> is a continuous function, we mainly characterize when the induced dynamical systems <span><math><mover><mrow><mi>f</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>:</mo><mo>(</mo><mi>F</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>U</mi></mrow><mrow><mo>∞</mo></mrow></msub><mo>)</mo><mo>→</mo><mo>(</mo><mi>F</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>U</mi></mrow><mrow><mo>∞</mo></mrow></msub><mo>)</mo></math></span>, <span><math><mover><mrow><mi>f</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>:</mo><mo>(</mo><mi>F</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>U</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo><mo>→</mo><mo>(</mo><mi>F</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>U</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> and <span><math><mover><mrow><mi>f</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>:</mo><mo>(</mo><mi>F</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>)</mo><mo>→</mo><mo>(</mo><mi>F</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>)</mo></math></span> are transitive, where <span><math><mover><mrow><mi>f</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> is the Zadeh's extension of <em>f</em>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109660"},"PeriodicalIF":0.5,"publicationDate":"2025-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145520223","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}
Pub Date : 2025-11-10DOI: 10.1016/j.topol.2025.109659
Ahmad Reza Haj Saeedi Sadegh, Jody Trout
We extend the deformation to the normal cone and tangent groupoid constructions from finite-dimensional manifolds to infinite-dimensional Banach and Fredholm manifolds. Next, we generalize the concept of Fredholm filtrations to get a more flexible and functorial theory. In particular, we show that if M is a Banach (or Fredholm) manifold with generalized filtration by finite-dimensional submanifolds, then there are induced generalized filtrations of the tangent bundle TM and of the tangent groupoid , which is not possible in the classical theory.
{"title":"On deformation spaces, tangent groupoids and generalized filtrations of Banach and Fredholm manifolds","authors":"Ahmad Reza Haj Saeedi Sadegh, Jody Trout","doi":"10.1016/j.topol.2025.109659","DOIUrl":"10.1016/j.topol.2025.109659","url":null,"abstract":"<div><div>We extend the deformation to the normal cone and tangent groupoid constructions from finite-dimensional manifolds to infinite-dimensional Banach and Fredholm manifolds. Next, we generalize the concept of Fredholm filtrations to get a more flexible and functorial theory. In particular, we show that if <em>M</em> is a Banach (or Fredholm) manifold with generalized filtration <span><math><mi>F</mi><mo>=</mo><msubsup><mrow><mo>{</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></mrow><mrow><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span> by finite-dimensional submanifolds, then there are induced generalized filtrations <span><math><mi>T</mi><mi>F</mi><mo>=</mo><msubsup><mrow><mo>{</mo><mi>T</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></mrow><mrow><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span> of the tangent bundle <em>TM</em> and <span><math><mi>T</mi><mi>F</mi><mo>=</mo><msubsup><mrow><mo>{</mo><mi>T</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></mrow><mrow><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span> of the tangent groupoid <span><math><mi>T</mi><mi>M</mi></math></span>, which is not possible in the classical theory.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109659"},"PeriodicalIF":0.5,"publicationDate":"2025-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145520224","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}
Pub Date : 2025-11-05DOI: 10.1016/j.topol.2025.109656
Louis H. Kauffman , Rama Mishra , Visakh Narayanan
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.
{"title":"Knots in RP3","authors":"Louis H. Kauffman , Rama Mishra , 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":"2025-11-05","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}
Pub Date : 2025-10-31DOI: 10.1016/j.topol.2025.109657
Kendall Heiney , Margaret Kipe , Samantha Pezzimenti , Kaelyn Pontes , Lực Ta
Knot mosaics were introduced by Kauffman and Lomonaco in the context of quantum knots, but have since been studied for their own right. A classical knot mosaic is formed on a square grid. In this work, we identify opposite edges of the square to form mosaics on the surface of a torus. We provide two algorithms for efficiently constructing toric mosaics of torus knots, providing upper bounds for the toric mosaic number. Using these results and a computer search, we provide a census of known toric mosaic numbers.
{"title":"Constructions of and bounds on the toric mosaic number","authors":"Kendall Heiney , Margaret Kipe , Samantha Pezzimenti , Kaelyn Pontes , Lực Ta","doi":"10.1016/j.topol.2025.109657","DOIUrl":"10.1016/j.topol.2025.109657","url":null,"abstract":"<div><div>Knot mosaics were introduced by Kauffman and Lomonaco in the context of quantum knots, but have since been studied for their own right. A classical knot mosaic is formed on a square grid. In this work, we identify opposite edges of the square to form mosaics on the surface of a torus. We provide two algorithms for efficiently constructing toric mosaics of torus knots, providing upper bounds for the toric mosaic number. Using these results and a computer search, we provide a census of known toric mosaic numbers.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109657"},"PeriodicalIF":0.5,"publicationDate":"2025-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466891","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}
Pub Date : 2025-10-31DOI: 10.1016/j.topol.2025.109655
Francis Jordan
We extend the characterization of the hereditary Baireness of the Vietoris hyperspace of separable metric spaces given by Gartside, Medini, and Zdomskyy [3] to the class of regular topological spaces of countable type. A theorem of Bouziad, Holá, and Zsilinszky [2] is also extended in a similar way.
{"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":"2025-10-31","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}
Pub Date : 2025-10-30DOI: 10.1016/j.topol.2025.109652
Ashwani K B, Ali Akbar K
A chaotic group action is a nonminimal, topologically transitive continuous group action with dense periodic points. In this paper, we discuss indecomposability for a continuous group action and prove that indecomposability is an equivalent definition of topological transitivity. Moreover, we prove that any infinite compact subset of the real line having a chaotic group action is homeomorphic to the middle third Cantor set.
{"title":"Indecomposability of group actions","authors":"Ashwani K B, Ali Akbar K","doi":"10.1016/j.topol.2025.109652","DOIUrl":"10.1016/j.topol.2025.109652","url":null,"abstract":"<div><div>A chaotic group action is a nonminimal, topologically transitive continuous group action with dense periodic points. In this paper, we discuss indecomposability for a continuous group action and prove that indecomposability is an equivalent definition of topological transitivity. Moreover, we prove that any infinite compact subset of the real line having a chaotic group action is homeomorphic to the middle third Cantor set.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109652"},"PeriodicalIF":0.5,"publicationDate":"2025-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466892","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}
Pub Date : 2025-10-30DOI: 10.1016/j.topol.2025.109651
Conrad Plaut
We introduce “weakly chained spaces”, which in the metric case can be defined in a single paragraph using only the definition of “metric space”. Using this simple notion we more or less completely resolve the question of when a metrizable space X has a generalized universal covering map, which we call the uniform universal cover (UU-cover): if and only if it is weakly chained. These concepts are defined for uniform spaces, but one may extend all results to metrizable topological spaces via the fine uniformity, and we describe the relationship between this work and that of Fischer-Zastrow on generalized universal covers. We also show that the UU-cover has the analogous properties to those of the traditional universal cover: universal, lifting, uniqueness and functorial.
One of our main results concerns conditions under which an inverse limit of metric spaces is weakly chained. This theorem, in turn, has applications (in another paper) to boundaries of geodesically complete, co-compact, proper CAT(0) spaces, which may be regarded as inverse limits of the (weakly chained) metric spheres at a basepoint.
{"title":"Weakly chained spaces","authors":"Conrad Plaut","doi":"10.1016/j.topol.2025.109651","DOIUrl":"10.1016/j.topol.2025.109651","url":null,"abstract":"<div><div>We introduce “weakly chained spaces”, which in the metric case can be defined in a single paragraph using only the definition of “metric space”. Using this simple notion we more or less completely resolve the question of when a metrizable space <em>X</em> has a generalized universal covering map, which we call the uniform universal cover (UU-cover): if and only if it is weakly chained. These concepts are defined for uniform spaces, but one may extend all results to metrizable topological spaces via the fine uniformity, and we describe the relationship between this work and that of Fischer-Zastrow on generalized universal covers. We also show that the UU-cover has the analogous properties to those of the traditional universal cover: universal, lifting, uniqueness and functorial.</div><div>One of our main results concerns conditions under which an inverse limit of metric spaces is weakly chained. This theorem, in turn, has applications (in another paper) to boundaries of geodesically complete, co-compact, proper CAT(0) spaces, which may be regarded as inverse limits of the (weakly chained) metric spheres at a basepoint.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109651"},"PeriodicalIF":0.5,"publicationDate":"2025-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466921","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}
Pub Date : 2025-10-30DOI: 10.1016/j.topol.2025.109654
Xiaoquan Xu
The main purpose of this paper is to reveal some finer links between d-spaces and -spaces by introducing and studying a new class of -spaces — strongly well-filtered spaces. The relationships among -spaces, -spaces, sober spaces, (strongly) well-filtered spaces and (strong) d-spaces are discussed. It is shown that if and is closed for any nonempty closed set A and saturated compact set K of a -space X, then X is strongly well-filtered. An unexpected result is proved which states that for any poset P, the Scott space ΣP is a strong d-space iff it is strongly well-filtered. So the Scott space of a complete lattice is always strongly well-filtered. Some basic properties of strongly well-filtered spaces are investigated. It is shown that the strong well-filteredness is closed-hereditary and saturated-hereditary, and every retract of a strongly well-filtered space is strongly well-filtered. We give two Scott spaces which are strongly well-filtered and an R-space but their product space is not a strong d-space. This answers an open question posed by Lawson and Xu. Hence the category S- of strongly well-filtered spaces and continuous mappings is not reflective in the category of -spaces and continuous mappings. Finally, we investigate the conditions under which the Smyth power space and Scott power space of a -space is strongly well-filtered. Several such conditions are given.
{"title":"Strongly well-filtered spaces and strong d-spaces","authors":"Xiaoquan Xu","doi":"10.1016/j.topol.2025.109654","DOIUrl":"10.1016/j.topol.2025.109654","url":null,"abstract":"<div><div>The main purpose of this paper is to reveal some finer links between <em>d</em>-spaces and <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-spaces by introducing and studying a new class of <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-spaces — strongly well-filtered spaces. The relationships among <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-spaces, <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-spaces, sober spaces, (strongly) well-filtered spaces and (strong) <em>d</em>-spaces are discussed. It is shown that if <span><math><mi>m</mi><mi>a</mi><mi>x</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>≠</mo><mo>∅</mo></math></span> and <span><math><mo>↓</mo><mo>(</mo><mi>A</mi><mo>∩</mo><mi>K</mi><mo>)</mo></math></span> is closed for any nonempty closed set <em>A</em> and saturated compact set <em>K</em> of a <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-space <em>X</em>, then <em>X</em> is strongly well-filtered. An unexpected result is proved which states that for any poset <em>P</em>, the Scott space Σ<em>P</em> is a strong <em>d</em>-space iff it is strongly well-filtered. So the Scott space of a complete lattice is always strongly well-filtered. Some basic properties of strongly well-filtered spaces are investigated. It is shown that the strong well-filteredness is closed-hereditary and saturated-hereditary, and every retract of a strongly well-filtered space is strongly well-filtered. We give two Scott spaces which are strongly well-filtered and an R-space but their product space is not a strong <em>d</em>-space. This answers an open question posed by Lawson and Xu. Hence the category <strong>S</strong>-<span><math><msub><mrow><mi>Top</mi></mrow><mrow><mi>w</mi></mrow></msub></math></span> of strongly well-filtered spaces and continuous mappings is not reflective in the category <span><math><msub><mrow><mi>Top</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> of <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-spaces and continuous mappings. Finally, we investigate the conditions under which the Smyth power space and Scott power space of a <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-space is strongly well-filtered. Several such conditions are given.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109654"},"PeriodicalIF":0.5,"publicationDate":"2025-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418011","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}
Pub Date : 2025-10-28DOI: 10.1016/j.topol.2025.109653
Nathan Carlson
We give a new bound for the cardinality of a Tychonoff homogeneous space using cozero sets. This leads to improved cardinal inequalities for compact homogeneous spaces that generalize to the locally compact setting. In this connection it is also shown that for any Hausdorff space X, where is the point-wise compactness type of X. This extends Arhangel′skiĭ's result that when X is compact Hausdorff. In addition pseudocompactness is investigated in connection with homogeneity. Among other results, we show that if X is a ccc locally compact noncompact space such that the one-point compactification of X is homogeneous and has character , then X is pseudocompact. It follows that if X is either or and then is pseudocompact.
{"title":"On local compactness, pseudocompactness, and homogeneity","authors":"Nathan Carlson","doi":"10.1016/j.topol.2025.109653","DOIUrl":"10.1016/j.topol.2025.109653","url":null,"abstract":"<div><div>We give a new bound for the cardinality of a Tychonoff homogeneous space using cozero sets. This leads to improved cardinal inequalities for compact homogeneous spaces that generalize to the locally compact setting. In this connection it is also shown that <span><math><mi>w</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mi>n</mi><mi>w</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>p</mi><mi>c</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> for any Hausdorff space <em>X</em>, where <span><math><mi>p</mi><mi>c</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is the point-wise compactness type of <em>X</em>. This extends Arhangel′skiĭ's result that <span><math><mi>w</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mi>n</mi><mi>w</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> when <em>X</em> is compact Hausdorff. In addition pseudocompactness is investigated in connection with homogeneity. Among other results, we show that if <em>X</em> is a ccc locally compact noncompact space such that the one-point compactification of <em>X</em> is homogeneous and has character <span><math><mi>c</mi></math></span>, then <em>X</em> is pseudocompact. It follows that if <em>X</em> is either <span><math><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>c</mi></mrow></msup></math></span> or <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>c</mi></mrow></msup></math></span> and <span><math><mi>p</mi><mo>∈</mo><mi>X</mi></math></span> then <span><math><mi>X</mi><mo>﹨</mo><mo>{</mo><mi>p</mi><mo>}</mo></math></span> is pseudocompact.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109653"},"PeriodicalIF":0.5,"publicationDate":"2025-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418097","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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