{"title":"On some kinds of ω-balancedness and (*) properties in certain semitopological groups","authors":"Liang-Xue Peng","doi":"10.1016/j.topol.2024.109001","DOIUrl":"https://doi.org/10.1016/j.topol.2024.109001","url":null,"abstract":"<div><p>In this article, we discuss some relationships of <em>ω</em>-balancedness and <span><math><mo>(</mo><mo>⁎</mo><mo>)</mo></math></span> properties which were introduced for giving characterizations of subgroups of topological products of certain para(semi)topological groups. We mainly get the following results.</p><p>If <em>G</em> is a regular <em>ω</em>-balanced locally <em>ω</em>-good semitopological group with a <em>q</em>-point, then <span><math><mi>I</mi><mi>r</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span> if and only if <span><math><mi>S</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span>. If <em>G</em> is a regular strongly paracompact semitopological group with a <em>q</em>-point and <span><math><mi>S</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span>, then <em>G</em> is completely <em>ω</em>-balanced if and only if <em>G</em> has property <span><math><mo>(</mo><msup><mrow></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></math></span>. If <em>G</em> is a regular paracompact <em>ω</em>-balanced locally good semitopological group with a <em>q</em>-point and <span><math><mi>S</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span>, then <em>G</em> has property <span><math><mo>(</mo><mi>w</mi><mo>⁎</mo><mo>)</mo></math></span> if and only if <em>G</em> has property (**). If <em>G</em> is a regular metacompact semitopological group with a <em>q</em>-point and <span><math><mi>S</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span>, then <em>G</em> is <em>MM</em>-<em>ω</em>-balanced if and only if <em>G</em> is <em>M</em>-<em>ω</em>-balanced.</p><p>We show that a semitopological group <em>G</em> admits a homeomorphic embedding as a subgroup of a product of metrizable semitopological groups if and only if <em>G</em> is topologically isomorphic to a subgroup of a product of semitopological groups which are first-countable paracompact regular <em>σ</em>-spaces and is topologically isomorphic to a subgroup of a product of Moore semitopological groups.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481108","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":"Positive links with arrangements of pseudocircles as shadows","authors":"Carolina Medina , Santino Ramírez , Jorge L. Ramírez-Alfonsín , Gelasio Salazar","doi":"10.1016/j.topol.2024.108999","DOIUrl":"https://doi.org/10.1016/j.topol.2024.108999","url":null,"abstract":"<div><p>An arrangement of pseudocircles <span><math><mi>A</mi></math></span> is a collection of Jordan curves in the plane that pairwise intersect (transversally) at exactly two points. How many non-equivalent links have <span><math><mi>A</mi></math></span> as their shadow? Motivated by this question, we study the number of non-equivalent positive oriented links that have an arrangement of pseudocircles as their shadow. We give sharp estimates on this number when <span><math><mi>A</mi></math></span> is one of the three unavoidable arrangements of pseudocircles.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481144","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":"Semi-proximal spaces and normality","authors":"Khulod Almontashery , Paul J. Szeptycki","doi":"10.1016/j.topol.2024.108990","DOIUrl":"10.1016/j.topol.2024.108990","url":null,"abstract":"<div><p>We consider the relationship between normality and semi-proximality. We give a consistent example of a first countable locally compact Dowker space that is not semi-proximal, and two ZFC examples of semi-proximal non-normal spaces. This answers a question of Nyikos. One of the examples is a subspace of <span><math><mo>(</mo><mi>ω</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>×</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. In contrast, we show that every normal subspace of a finite power of <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is semi-proximal.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0166864124001755/pdfft?md5=d3221f0dbf56e33825d4cf06ca7889f2&pid=1-s2.0-S0166864124001755-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141404894","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":"On points avoiding measures","authors":"Piotr Borodulin–Nadzieja , Artsiom Ranchynski","doi":"10.1016/j.topol.2024.108988","DOIUrl":"10.1016/j.topol.2024.108988","url":null,"abstract":"<div><p>We say that an element <em>x</em> of a topological space <em>X</em> avoids measures if for every Borel measure <em>μ</em> on <em>X</em> if <span><math><mi>μ</mi><mo>(</mo><mo>{</mo><mi>x</mi><mo>}</mo><mo>)</mo><mo>=</mo><mn>0</mn></math></span>, then there is an open <span><math><mi>U</mi><mo>∋</mo><mi>x</mi></math></span> such that <span><math><mi>μ</mi><mo>(</mo><mi>U</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. The negation of this property can viewed as a local version of the property of supporting a strictly positive measure. We study points avoiding measures in the general setting as well as in the context of <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, the remainder of Stone-Čech compactification of <em>ω</em>.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141404552","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 Erdős spaces","authors":"Alfredo Zaragoza","doi":"10.1016/j.topol.2024.108987","DOIUrl":"https://doi.org/10.1016/j.topol.2024.108987","url":null,"abstract":"<div><p>The objective of this work is to present some results related to some Erőds spaces. This paper answers a question made by the author in <span>[12]</span> proving that if <em>X</em> is a cohesive space then <span><math><mi>K</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is a cohesive space; we give a partial answer to question 7.3 of <span>[7]</span> providing an internal characterization of <span><math><mi>Q</mi><mo>×</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span>-factors for certain subsets of <span><math><mi>Q</mi><mo>×</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span>; and we give conditions under which a perfect or open image of the complete Erdős space is homeomorphic to the complete Erdős space.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141323269","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":"Connectedness of certain graph coloring complexes","authors":"Nandini Nilakantan , Samir Shukla","doi":"10.1016/j.topol.2024.108985","DOIUrl":"https://doi.org/10.1016/j.topol.2024.108985","url":null,"abstract":"<div><p>In this article, we consider the bipartite graphs <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>×</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We prove that the connectedness of the complex <span><math><mtext>Hom</mtext><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>×</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></math></span> is <span><math><mi>m</mi><mo>−</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span> if <span><math><mi>m</mi><mo>≥</mo><mi>n</mi></math></span> and <span><math><mi>m</mi><mo>−</mo><mn>3</mn></math></span> in all the other cases. Therefore, we show that for this class of graphs, <span><math><mtext>Hom</mtext><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></math></span> is exactly <span><math><mo>(</mo><mi>m</mi><mo>−</mo><mi>d</mi><mo>−</mo><mn>2</mn><mo>)</mo></math></span>-connected, <span><math><mi>m</mi><mo>≥</mo><mi>n</mi></math></span>, where <em>d</em> is the maximal degree of the graph <em>G</em>.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141434837","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":"Vector bundles over non-Hausdorff manifolds","authors":"David O'Connell","doi":"10.1016/j.topol.2024.108982","DOIUrl":"https://doi.org/10.1016/j.topol.2024.108982","url":null,"abstract":"<div><p>In this paper we generalise the theory of real vector bundles to a certain class of non-Hausdorff manifolds. In particular, it is shown that every vector bundle fibred over these non-Hausdorff manifolds can be constructed as a colimit of standard vector bundles. We then use this description to introduce various formulas that express non-Hausdorff structures in terms of data defined on certain Hausdorff submanifolds. Finally, we use Čech cohomology to classify the real non-Hausdorff line bundles.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0166864124001676/pdfft?md5=c27d5e13349e4e55067738836fba7460&pid=1-s2.0-S0166864124001676-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141323268","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":"On idempotent convexities and idempotent barycenter maps","authors":"Dawid Krasiński , Taras Radul","doi":"10.1016/j.topol.2024.108974","DOIUrl":"https://doi.org/10.1016/j.topol.2024.108974","url":null,"abstract":"<div><p>We consider an isomorphism between the idempotent convexity based on the maximum and the addition operations and the idempotent measure convexity on the maximum and the multiplication operations. We use this isomorphism to investigate topological properties of the barycenter map related to the maximum and the multiplication operations.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141264091","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 arc-wise connection relation in the plane","authors":"Gabriel Debs, Jean Saint Raymond","doi":"10.1016/j.topol.2024.108975","DOIUrl":"https://doi.org/10.1016/j.topol.2024.108975","url":null,"abstract":"<div><p>We prove that the arc-wise connection relation in a <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>δ</mi></mrow></msub></math></span> subset of the plane is Borel.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141263956","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}
Florencio Corona-Vázquez, José A. Martínez-Cortez, Russell-Aarón Quiñones-Estrella, Javier Sánchez-Martínez
{"title":"About the hyperspace H(X)/H(X;K)","authors":"Florencio Corona-Vázquez, José A. Martínez-Cortez, Russell-Aarón Quiñones-Estrella, Javier Sánchez-Martínez","doi":"10.1016/j.topol.2024.108972","DOIUrl":"https://doi.org/10.1016/j.topol.2024.108972","url":null,"abstract":"<div><p>Let <em>X</em> be a continuum, <em>K</em> a nonempty closed subset of <em>X</em>, and let <em>n</em> be a positive integer. In this paper, we consider the hyperspaces <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>X</mi></mrow></msup></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span>, consisting of all nonempty closed subsets of <em>X</em> and of all nonempty closed subsets of <em>X</em> having at most <em>n</em> components, respectively. If <span><math><mi>H</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>∈</mo><mo>{</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>X</mi></mrow></msup><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>}</mo></math></span>, <span><math><mi>H</mi><mo>(</mo><mi>X</mi><mo>;</mo><mi>K</mi><mo>)</mo></math></span> denotes the hyperspace of all elements in <span><math><mi>H</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> intersecting <em>K</em>. In this paper we present some topological properties of the quotient space <span><math><mi>H</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>/</mo><mi>H</mi><mo>(</mo><mi>X</mi><mo>;</mo><mi>K</mi><mo>)</mo></math></span>, going forward in its study in the available literature. In the class of finite graphs, we study the problem of determining conditions on <em>X</em> and <em>K</em> such that <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>/</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>;</mo><mi>K</mi><mo>)</mo></math></span> are homeomorphic, obtaining in this direction some characterizations.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141263960","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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