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Using the KKM theorem 利用 KKM 定理
arXiv - MATH - General Topology Pub Date : 2024-08-07 DOI: arxiv-2408.03921
Daniel McGinnis, Shira Zerbib
{"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}
引用次数: 0
On $a$-locally closed sets 关于$a$局部封闭集
arXiv - MATH - General Topology Pub Date : 2024-08-06 DOI: arxiv-2408.03169
Bilge İzci, Murad Özkoç
{"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}
引用次数: 0
On uniformly continuous surjections between $C_p$-spaces over metrizable spaces 论可元空间上 $C_p$ 空间之间的均匀连续投射
arXiv - MATH - General Topology Pub Date : 2024-08-03 DOI: arxiv-2408.01870
A. Eysen, A. Leiderman, V. Valov
{"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}
引用次数: 0
Ultrametrizable spaces are homeomorphic to clade spaces of pruned trees 超三维空间与剪枝树支系空间同构
arXiv - MATH - General Topology Pub Date : 2024-07-31 DOI: arxiv-2407.21763
Itamar Bellaïche
{"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}
引用次数: 0
Quasisymmetric minimality on packing dimension for homogeneous perfect sets 同质完全集包装维度上的准对称最小性
arXiv - MATH - General Topology Pub Date : 2024-07-30 DOI: arxiv-2407.20562
Shishuang Liu, Yanzhe Li, Jiaojiao Yang
{"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}
引用次数: 0
Characterizations of $mathbb{N}$-compactness and realcompactness via ultrafilters in the absence of the axiom of choice 在没有选择公理的情况下通过超滤波器对$mathbb{N}$紧凑性和实紧凑性的描述
arXiv - MATH - General Topology Pub Date : 2024-07-27 DOI: arxiv-2408.01461
AliReza Olfati, Eliza Wajch
{"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}
引用次数: 0
Pseudo-Böttcher components of the wandering set of inner mappings 内映射徘徊集的伪博歇尔成分
arXiv - MATH - General Topology Pub Date : 2024-07-27 DOI: arxiv-2407.19251
Igor Yu. Vlasenko
{"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}
引用次数: 0
Characterizing function spaces which have the property (B) of Banakh 表征具有 Banakh 属性 (B) 的函数空间
arXiv - MATH - General Topology Pub Date : 2024-07-26 DOI: arxiv-2407.18618
Mikołaj Krupski, Kacper Kucharski, Witold Marciszewski
{"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}
引用次数: 0
On some recent selective properties involving networks 关于最近涉及网络的一些选择性特性
arXiv - MATH - General Topology Pub Date : 2024-07-26 DOI: arxiv-2407.18713
Maddalena Bonanzinga, Davide Giacopello, Santi Spadaro, Lyubomyr Zdomskyy
{"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}
引用次数: 0
On stratifications and poset-stratified spaces 论分层和正方体分层空间
arXiv - MATH - General Topology Pub Date : 2024-07-25 DOI: arxiv-2407.17690
Lukas Waas, Jon Woolf, Shoji Yokura
{"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}
引用次数: 0
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