Pub Date : 2024-06-18DOI: 10.1016/j.topol.2024.109001
Liang-Xue Peng
In this article, we discuss some relationships of ω-balancedness and properties which were introduced for giving characterizations of subgroups of topological products of certain para(semi)topological groups. We mainly get the following results.
If G is a regular ω-balanced locally ω-good semitopological group with a q-point, then if and only if . If G is a regular strongly paracompact semitopological group with a q-point and , then G is completely ω-balanced if and only if G has property . If G is a regular paracompact ω-balanced locally good semitopological group with a q-point and , then G has property if and only if G has property (**). If G is a regular metacompact semitopological group with a q-point and , then G is MM-ω-balanced if and only if G is M-ω-balanced.
We show that a semitopological group G admits a homeomorphic embedding as a subgroup of a product of metrizable semitopological groups if and only if G is topologically isomorphic to a subgroup of a product of semitopological groups which are first-countable paracompact regular σ-spaces and is topologically isomorphic to a subgroup of a product of Moore semitopological groups.
本文讨论了ω平衡性和(⁎)性质的一些关系,这些关系是为了给出某些副(半)拓扑群的拓扑积的子群的特征而引入的。如果 G 是一个有 q 点的正则 ω 平衡局部 ω 好半拓扑群,那么当且仅当 Sm(G)≤ω 时,Ir(G)≤ω。如果 G 是一个有 q 点的正则强准紧密半坡群,且 Sm(G)≤ω,那么当且仅当 G 具有(⁎)性质时,G 才是完全ω平衡的。若 G 是一个有 q 点的正则准圆锥ω平衡局部良好半坡群,且 Sm(G)≤ω,则当且仅当 G 具有性质 (**) 时,G 才具有性质 (w⁎)。如果 G 是具有 q 点的正则元紧密半坡群,且 Sm(G)≤ω ,那么只有当 G 是 M-ω 平衡时,G 才是 MM-ω 平衡的。我们证明,当且仅当 G 在拓扑上同构于第一可数paracompact正则σ空间的半坡群积的一个子群,并且在拓扑上同构于摩尔半坡群积的一个子群时,半坡群 G 可以同构嵌入为可元半坡群积的一个子群。
{"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}
An arrangement of pseudocircles is a collection of Jordan curves in the plane that pairwise intersect (transversally) at exactly two points. How many non-equivalent links have 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 is one of the three unavoidable arrangements of pseudocircles.
假圆的排列 A 是平面内恰好两点成对相交(横交)的约旦曲线的集合。有多少非等价链接以 A 为影?受这一问题的启发,我们研究了以伪圆排列作为其阴影的非等价正向链接的数量。当 A 是三种不可避免的伪圆排列之一时,我们给出了关于这一数目的精确估计。
{"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}
Pub Date : 2024-06-11DOI: 10.1016/j.topol.2024.108990
Khulod Almontashery , Paul J. Szeptycki
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 . In contrast, we show that every normal subspace of a finite power of is semi-proximal.
{"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}
Pub Date : 2024-06-07DOI: 10.1016/j.topol.2024.108988
Piotr Borodulin–Nadzieja , Artsiom Ranchynski
We say that an element x of a topological space X avoids measures if for every Borel measure μ on X if , then there is an open such that . 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 , the remainder of Stone-Čech compactification of ω.
我们说拓扑空间 X 的元素 x 避开度量的条件是:对于 X 上的每一个伯勒度量 μ,如果 μ({x})=0 则存在一个开放的 U∋x,使得 μ(U)=0。这个性质的否定可以看作是支持严格正度量性质的局部版本。我们将研究在一般情况下以及在ω⁎(ω的斯通切赫剩余紧凑化)的背景下避免度量的点。
{"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}
Pub Date : 2024-06-07DOI: 10.1016/j.topol.2024.108987
Alfredo Zaragoza
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 [12] proving that if X is a cohesive space then is a cohesive space; we give a partial answer to question 7.3 of [7] providing an internal characterization of -factors for certain subsets of and ; and we give conditions under which a perfect or open image of the complete Erdős space is homeomorphic to the complete Erdős space.
{"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}
Pub Date : 2024-06-05DOI: 10.1016/j.topol.2024.108985
Nandini Nilakantan , Samir Shukla
In this article, we consider the bipartite graphs . We prove that the connectedness of the complex is if and in all the other cases. Therefore, we show that for this class of graphs, is exactly -connected, , where d is the maximal degree of the graph G.
在本文中,我们考虑的是双方图 K2×Kn。我们证明,复数 Hom(K2×Kn,Km) 的连通性在 m≥n 时为 m-n-1,在所有其他情况下为 m-3。因此,我们证明了对于这一类图,Hom(G,Km) 恰好是 (m-d-2)- 连接的,m≥n,其中 d 是图 G 的最大度。
{"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}
Pub Date : 2024-06-04DOI: 10.1016/j.topol.2024.108982
David O'Connell
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.
在本文中,我们将实向量束理论推广到某一类非豪斯多夫流形。特别是,本文证明了纤维在这些非豪斯多夫流形上的每个向量束都可以构造为标准向量束的集合。然后,我们利用这一描述引入各种公式,用定义在某些豪斯多夫子流形上的数据来表达非豪斯多夫结构。最后,我们使用 Čech cohomology 对实非豪斯多夫线束进行分类。
{"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}
Pub Date : 2024-05-31DOI: 10.1016/j.topol.2024.108974
Dawid Krasiński , Taras Radul
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.
{"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}
Pub Date : 2024-05-31DOI: 10.1016/j.topol.2024.108975
Gabriel Debs, Jean Saint Raymond
We prove that the arc-wise connection relation in a subset of the plane is Borel.
我们证明了平面的 Gδ 子集中的弧向连接关系是 Borel 的。
{"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}
Pub Date : 2024-05-29DOI: 10.1016/j.topol.2024.108972
Florencio Corona-Vázquez, José A. Martínez-Cortez, Russell-Aarón Quiñones-Estrella, Javier Sánchez-Martínez
Let X be a continuum, K a nonempty closed subset of X, and let n be a positive integer. In this paper, we consider the hyperspaces and , consisting of all nonempty closed subsets of X and of all nonempty closed subsets of X having at most n components, respectively. If , denotes the hyperspace of all elements in intersecting K. In this paper we present some topological properties of the quotient space , going forward in its study in the available literature. In the class of finite graphs, we study the problem of determining conditions on X and K such that and are homeomorphic, obtaining in this direction some characterizations.
设 X 是连续体,K 是 X 的非空封闭子集,n 是正整数。在本文中,我们考虑超空间 2X 和 Cn(X),它们分别由 X 的所有非空封闭子集和 X 的所有最多有 n 个分量的非空封闭子集组成。如果 H(X)∈{2X,Cn(X)},H(X;K) 表示 H(X) 中所有元素与 K 相交的超空间。在本文中,我们将介绍商空间 H(X)/H(X;K) 的一些拓扑性质,并在现有文献中继续对其进行研究。在有限图类中,我们研究了如何确定 X 和 K 的条件,从而使 Cn(X) 和 Cn(X)/Cn(X;K) 同构,并在此方向上获得了一些特征。
{"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}