The KKM theorem, due to Knaster, Kuratowski, and Mazurkiewicz in 1929, is a fundamental result in fixed-point theory, which has seen numerous extensions and applications. In this paper we survey old and recent generalizations of the KKM theorem and their applications in the areas of piercing numbers, mass partition, fair division, and matching theory. We also give a few new results utilizing KKM-type theorems, and discuss related open problems.
{"title":"Using the KKM theorem","authors":"Daniel McGinnis, Shira Zerbib","doi":"arxiv-2408.03921","DOIUrl":"https://doi.org/arxiv-2408.03921","url":null,"abstract":"The KKM theorem, due to Knaster, Kuratowski, and Mazurkiewicz in 1929, is a\u0000fundamental result in fixed-point theory, which has seen numerous extensions\u0000and applications. In this paper we survey old and recent generalizations of the\u0000KKM theorem and their applications in the areas of piercing numbers, mass\u0000partition, fair division, and matching theory. We also give a few new results\u0000utilizing KKM-type theorems, and discuss related open problems.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947745","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this paper is to introduce the notion of $a$-locally closed set by utilizing $a$-open sets defined by Ekici and to study some properties of this new notion. Also, some characterizations and many fundamental results regarding this new concept are obtained. Moreover, the relationships between the concepts defined within the scope of this study and some other types of local closed sets in the literature have been revealed.
{"title":"On $a$-locally closed sets","authors":"Bilge İzci, Murad Özkoç","doi":"arxiv-2408.03169","DOIUrl":"https://doi.org/arxiv-2408.03169","url":null,"abstract":"The aim of this paper is to introduce the notion of $a$-locally closed set by\u0000utilizing $a$-open sets defined by Ekici and to study some properties of this\u0000new notion. Also, some characterizations and many fundamental results regarding\u0000this new concept are obtained. Moreover, the relationships between the concepts\u0000defined within the scope of this study and some other types of local closed\u0000sets in the literature have been revealed.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $X$ and $Y$ be metrizable spaces and suppose that there exists a uniformly continuous surjection $T: C_{p}(X) to C_{p}(Y)$ (resp., $T: C_{p}^*(X) to C_{p}^*(Y)$), where $C_{p}(X)$ (resp., $C_{p}^*(X)$) denotes the space of all real-valued continuous (resp., continuous and bounded) functions on $X$ endowed with the pointwise convergence topology. We show that if additionally $T$ is an inversely bounded mapping and $X$ has some dimensional-like property $mathcal P$, then so does $Y$. For example, this is true if $mathcal P$ is one of the following properties: zero-dimensionality, countable-dimensionality or strong countable-dimensionality. Also, we consider other properties $mathcal P$: of being a scattered, or a strongly $sigma$-scattered space, or being a $Delta_1$-space (see [17]). Our results strengthen and extend several results from [6], [13], [17].
{"title":"On uniformly continuous surjections between $C_p$-spaces over metrizable spaces","authors":"A. Eysen, A. Leiderman, V. Valov","doi":"arxiv-2408.01870","DOIUrl":"https://doi.org/arxiv-2408.01870","url":null,"abstract":"Let $X$ and $Y$ be metrizable spaces and suppose that there exists a\u0000uniformly continuous surjection $T: C_{p}(X) to C_{p}(Y)$ (resp., $T:\u0000C_{p}^*(X) to C_{p}^*(Y)$), where $C_{p}(X)$ (resp., $C_{p}^*(X)$) denotes the\u0000space of all real-valued continuous (resp., continuous and bounded) functions\u0000on $X$ endowed with the pointwise convergence topology. We show that if additionally $T$ is an inversely bounded mapping and $X$ has\u0000some dimensional-like property $mathcal P$, then so does $Y$. For example,\u0000this is true if $mathcal P$ is one of the following properties:\u0000zero-dimensionality, countable-dimensionality or strong\u0000countable-dimensionality. Also, we consider other properties $mathcal P$: of being a scattered, or a\u0000strongly $sigma$-scattered space, or being a $Delta_1$-space (see [17]). Our\u0000results strengthen and extend several results from [6], [13], [17].","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper demonstrates that every ultrametric space is homeomorphic to a clade space of a pruned tree, i.e., a subspace of a tree's canopy. Furthermore, it characterizes several topological properties of ultrametrizable spaces through the features of their representing trees. This approach suggests that topological properties of ultrametrizable spaces should be studies via the study of naturally ordered pruned trees.
{"title":"Ultrametrizable spaces are homeomorphic to clade spaces of pruned trees","authors":"Itamar Bellaïche","doi":"arxiv-2407.21763","DOIUrl":"https://doi.org/arxiv-2407.21763","url":null,"abstract":"This paper demonstrates that every ultrametric space is homeomorphic to a\u0000clade space of a pruned tree, i.e., a subspace of a tree's canopy. Furthermore,\u0000it characterizes several topological properties of ultrametrizable spaces\u0000through the features of their representing trees. This approach suggests that\u0000topological properties of ultrametrizable spaces should be studies via the\u0000study of naturally ordered pruned trees.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869730","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the quasisymmetric packing minimality of homogeneous perfect sets, and obtain that a special class of homogeneous perfect sets with $operatorname{dim}_{P}E=1$ is quasisymmetrically packing minimal.
{"title":"Quasisymmetric minimality on packing dimension for homogeneous perfect sets","authors":"Shishuang Liu, Yanzhe Li, Jiaojiao Yang","doi":"arxiv-2407.20562","DOIUrl":"https://doi.org/arxiv-2407.20562","url":null,"abstract":"In this paper, we study the quasisymmetric packing minimality of homogeneous\u0000perfect sets, and obtain that a special class of homogeneous perfect sets with\u0000$operatorname{dim}_{P}E=1$ is quasisymmetrically packing minimal.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article concerns the Herrlich-Chew theorem stating that a Hausdorff zero-dimensional space is $mathbb{N}$-compact if and only if every clopen ultrafilter with the countable intersection property in this space is fixed. It also concerns Hewitt's theorem stating that a Tychonoff space is realcompact if and only if every $z$-ultrafilter with the countable intersection property in this space is fixed. The axiom of choice was involved in the original proofs of these theorems. The aim of this article is to show that the Herrlich-Chew theorem is valid in $mathbf{ZF}$, but it is an open problem if Hewitt's theorem can be false in a model of $mathbf{ZF}$. It is proved that Hewitt's theorem is true in every model of $mathbf{ZF}$ in which the countable axiom of multiple choice is satisfied. A modification of Hewitt's theorem is given and proved true in $mathbf{ZF}$. Several applications of the results obtained are shown.
{"title":"Characterizations of $mathbb{N}$-compactness and realcompactness via ultrafilters in the absence of the axiom of choice","authors":"AliReza Olfati, Eliza Wajch","doi":"arxiv-2408.01461","DOIUrl":"https://doi.org/arxiv-2408.01461","url":null,"abstract":"This article concerns the Herrlich-Chew theorem stating that a Hausdorff\u0000zero-dimensional space is $mathbb{N}$-compact if and only if every clopen\u0000ultrafilter with the countable intersection property in this space is fixed. It\u0000also concerns Hewitt's theorem stating that a Tychonoff space is realcompact if\u0000and only if every $z$-ultrafilter with the countable intersection property in\u0000this space is fixed. The axiom of choice was involved in the original proofs of\u0000these theorems. The aim of this article is to show that the Herrlich-Chew\u0000theorem is valid in $mathbf{ZF}$, but it is an open problem if Hewitt's\u0000theorem can be false in a model of $mathbf{ZF}$. It is proved that Hewitt's\u0000theorem is true in every model of $mathbf{ZF}$ in which the countable axiom of\u0000multiple choice is satisfied. A modification of Hewitt's theorem is given and\u0000proved true in $mathbf{ZF}$. Several applications of the results obtained are\u0000shown.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article explores the topology of Pseudo-B"ottcher totally invariant connected components of the wandering set in dynamical systems generated by on-invertible inner (open surjective isolated) mappings of compact surfaces. We describe the possible topological types of these invariant connected subsets, which are more diverse then corresponding components of homeomorphisms.
本文探讨了由紧凑曲面的不可逆内映射（开放的投射孤立映射）所产生的动力系统中的游走集的完全不变量连接子集（Pseudo-B"otcher totally invariant connected components of the wandering set）的拓扑学。我们描述了这些不变连通子集的可能拓扑类型，它们比同态的相应分量更多样化。
{"title":"Pseudo-Böttcher components of the wandering set of inner mappings","authors":"Igor Yu. Vlasenko","doi":"arxiv-2407.19251","DOIUrl":"https://doi.org/arxiv-2407.19251","url":null,"abstract":"This article explores the topology of Pseudo-B\"ottcher totally invariant\u0000connected components of the wandering set in dynamical systems generated by\u0000on-invertible inner (open surjective isolated) mappings of compact surfaces. We\u0000describe the possible topological types of these invariant connected subsets,\u0000which are more diverse then corresponding components of homeomorphisms.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mikołaj Krupski, Kacper Kucharski, Witold Marciszewski
A topological space $Y$ has the property (B) of Banakh if there is a countable family ${A_n:nin mathbb{N}}$ of closed nowhere dense subsets of $Y$ absorbing all compact subsets of $Y$. In this note we show that the space $C_p(X)$ of continuous real-valued functions on a Tychonoff space $X$ with the topology of pointwise convergence, fails to satisfy the property (B) if and only if the space $X$ has the following property $(kappa)$: every sequence of disjoint finite subsets of $X$ has a subsequence with point--finite open expansion. Additionally, we provide an analogous characterization for the compact--open topology on $C(X)$. Finally, we give examples of Tychonoff spaces $X$ whose all bounded subsets are finite, yet $X$ fails to have the property $(kappa)$. This answers a question of Tkachuk.
{"title":"Characterizing function spaces which have the property (B) of Banakh","authors":"Mikołaj Krupski, Kacper Kucharski, Witold Marciszewski","doi":"arxiv-2407.18618","DOIUrl":"https://doi.org/arxiv-2407.18618","url":null,"abstract":"A topological space $Y$ has the property (B) of Banakh if there is a\u0000countable family ${A_n:nin mathbb{N}}$ of closed nowhere dense subsets of\u0000$Y$ absorbing all compact subsets of $Y$. In this note we show that the space\u0000$C_p(X)$ of continuous real-valued functions on a Tychonoff space $X$ with the\u0000topology of pointwise convergence, fails to satisfy the property (B) if and\u0000only if the space $X$ has the following property $(kappa)$: every sequence of\u0000disjoint finite subsets of $X$ has a subsequence with point--finite open\u0000expansion. Additionally, we provide an analogous characterization for the\u0000compact--open topology on $C(X)$. Finally, we give examples of Tychonoff spaces\u0000$X$ whose all bounded subsets are finite, yet $X$ fails to have the property\u0000$(kappa)$. This answers a question of Tkachuk.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"80 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we investigate R-,H-, and M-{it nw}-selective properties introduced in cite{BG}. In particular, we provide consistent uncountable examples of such spaces and we define textit{trivial} R-,H-, and M-{it nw}-selective spaces the ones with countable net weight having, additionally, the cardinality and the weight strictly less then $cov({cal M})$, $frak b$, and $frak d$, respectively. Since we establish that spaces having cardinalities more than $cov({cal M})$, $frak b$, and $frak d$, fail to have the R-,H-, and M-{it nw}-selective properties, respectively, non-trivial examples should eventually have weight greater than or equal to these small cardinals. Using forcing methods, we construct consistent countable non-trivial examples of R-{it nw}-selective and H-{it nw}-selective spaces and we establish some limitations to constructions of non-trivial examples. Moreover, we consistently prove the existence of two H-{it nw}-selective spaces whose product fails to be M-{it nw}-selective. Finally, we study some relations between {it nw}-selective properties and a strong version of the HFD property.
{"title":"On some recent selective properties involving networks","authors":"Maddalena Bonanzinga, Davide Giacopello, Santi Spadaro, Lyubomyr Zdomskyy","doi":"arxiv-2407.18713","DOIUrl":"https://doi.org/arxiv-2407.18713","url":null,"abstract":"In this paper we investigate R-,H-, and M-{it nw}-selective properties\u0000introduced in cite{BG}. In particular, we provide consistent uncountable\u0000examples of such spaces and we define textit{trivial} R-,H-, and M-{it\u0000nw}-selective spaces the ones with countable net weight having, additionally,\u0000the cardinality and the weight strictly less then $cov({cal M})$, $frak b$,\u0000and $frak d$, respectively. Since we establish that spaces having\u0000cardinalities more than $cov({cal M})$, $frak b$, and $frak d$, fail to have\u0000the R-,H-, and M-{it nw}-selective properties, respectively, non-trivial\u0000examples should eventually have weight greater than or equal to these small\u0000cardinals. Using forcing methods, we construct consistent countable non-trivial\u0000examples of R-{it nw}-selective and H-{it nw}-selective spaces and we\u0000establish some limitations to constructions of non-trivial examples. Moreover,\u0000we consistently prove the existence of two H-{it nw}-selective spaces whose\u0000product fails to be M-{it nw}-selective. Finally, we study some relations\u0000between {it nw}-selective properties and a strong version of the HFD property.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"56 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A stratified space is a topological space equipped with a emph{stratification}, which is a decomposition or partition of the topological space satisfying certain extra conditions. More recently, the notion of poset-stratified space, i.e., topological space endowed with a continuous map to a poset with its Alexandrov topology, has been popularized. Both notions of stratified spaces are ubiquitous in mathematics, ranging from investigations of singular structures in algebraic geometry to extensions of the homotopy hypothesis in higher category theory. In this article we study the precise mathematical relation between these different approaches to stratified spaces.
分层空间是一个拓扑空间，它具有一个分层映射，是拓扑空间满足某些额外条件的分解或分割。最近，"poset-stratified space "的概念得到了推广，即拓扑空间被赋予了一个连续的映射到一个具有亚历山德罗夫拓扑的poset。这两个分层空间概念在数学中无处不在，从代数几何中对星状结构的研究到高范畴理论中对同调假说的扩展，不一而足。在本文中，我们将研究这些不同的分层空间方法之间的精确数学关系。
{"title":"On stratifications and poset-stratified spaces","authors":"Lukas Waas, Jon Woolf, Shoji Yokura","doi":"arxiv-2407.17690","DOIUrl":"https://doi.org/arxiv-2407.17690","url":null,"abstract":"A stratified space is a topological space equipped with a\u0000emph{stratification}, which is a decomposition or partition of the topological\u0000space satisfying certain extra conditions. More recently, the notion of\u0000poset-stratified space, i.e., topological space endowed with a continuous map\u0000to a poset with its Alexandrov topology, has been popularized. Both notions of\u0000stratified spaces are ubiquitous in mathematics, ranging from investigations of\u0000singular structures in algebraic geometry to extensions of the homotopy\u0000hypothesis in higher category theory. In this article we study the precise\u0000mathematical relation between these different approaches to stratified spaces.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"162 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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