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A note about dual representations of group actions on Lipschitz-free spaces 关于无 Lipschitz 空间上群作用的对偶表示的说明
arXiv - MATH - General Topology Pub Date : 2024-08-27 DOI: arxiv-2408.15208
Michael Megrelishvili
{"title":"A note about dual representations of group actions on Lipschitz-free spaces","authors":"Michael Megrelishvili","doi":"arxiv-2408.15208","DOIUrl":"https://doi.org/arxiv-2408.15208","url":null,"abstract":"Let $mathcal{F}(M)$ be the Lipschitz-free space of a pointed metric space\u0000$M$. For every isometric continuous group action of $G$ we have an induced\u0000continuous dual action on the weak-star compact unit ball\u0000$B_{mathcal{F}(M)^*}$ of the dual space $mathrm{Lip_0} (M)=mathcal{F}(M)^*$.\u0000We pose the question when a given abstract continuous action of $G$ on a\u0000topological space $X$ can be represented through a $G$-subspace of\u0000$B_{mathcal{F}(M)^*}$. One of such natural examples is the so-called metric\u0000compactification (of isometric $G$-spaces) for a pointed metric space. As well\u0000as the Gromov $G$-compactification of a bounded metric $G$-space. Note that\u0000there are sufficiently many representations of compact $G$-spaces on\u0000Lipschitz-free spaces.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181066","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
Generic Compacta from Relations between Finite Graphs: Theory Building and Examples 从有限图之间的关系看通用契约：理论构建与实例
arXiv - MATH - General Topology Pub Date : 2024-08-27 DOI: arxiv-2408.15228
Adam Bartoš, Tristan Bice, Alessandro Vignati
{"title":"Generic Compacta from Relations between Finite Graphs: Theory Building and Examples","authors":"Adam Bartoš, Tristan Bice, Alessandro Vignati","doi":"arxiv-2408.15228","DOIUrl":"https://doi.org/arxiv-2408.15228","url":null,"abstract":"In recent work, the authors developed a simple method of constructing\u0000topological spaces from certain well-behaved partially ordered sets -- those\u0000coming from sequences of relations between finite sets. This method associates\u0000a given poset with its spectrum, which is a compact T_1 topological space. In this paper, we focus on the case where such finite sets have a graph\u0000structure and the relations belong to a given graph category. We relate\u0000topological properties of the spectrum to combinatorial properties of the graph\u0000categories involved. We then utilise this to exhibit elementary combinatorial\u0000constructions of well-known continua as Fra\"iss'e limits of finite graphs in\u0000categories with relational morphisms.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181050","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
Sober $L$-convex spaces and $L$-join-semilattices 清醒的$L$凸空间和$L$连接半网格
arXiv - MATH - General Topology Pub Date : 2024-08-16 DOI: arxiv-2408.08520
Guojun Wu, Wei Yao
{"title":"Sober $L$-convex spaces and $L$-join-semilattices","authors":"Guojun Wu, Wei Yao","doi":"arxiv-2408.08520","DOIUrl":"https://doi.org/arxiv-2408.08520","url":null,"abstract":"With a complete residuated lattice $L$ as the truth value table, we extend\u0000the definition of sobriety of classical convex spaces to the framework of\u0000$L$-convex spaces. We provide a specific construction for the sobrification of\u0000an $L$-convex space, demonstrating that the full subcategory of sober\u0000$L$-convex spaces is reflective in the category of $L$-convex spaces with\u0000convexity-preserving mappings. Additionally, we introduce the concept of Scott\u0000$L$-convex structures on $L$-ordered sets. As an application of this type of\u0000sobriety, we obtain a characterization for the $L$-join-semilattice completion\u0000of an $L$-ordered set: an $L$-ordered set $Q$ is an $L$-join-semilattice\u0000completion of an $L$-ordered set $P$ if and only if the Scott $L$-convex space\u0000$(Q, sigma^{ast}(Q))$ is a sobrification of the Scott $L$-convex space $(P,\u0000sigma^{ast}(P))$.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181053","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
Products of two sober dcpo's need not be sober 两个清醒的 dcpo 的产品不必是清醒的
arXiv - MATH - General Topology Pub Date : 2024-08-16 DOI: arxiv-2408.08587
Hualin Miao, Xiaoyong Xi, Xiaodong Jia, Qingguo Li, Dongsheng Zhao
{"title":"Products of two sober dcpo's need not be sober","authors":"Hualin Miao, Xiaoyong Xi, Xiaodong Jia, Qingguo Li, Dongsheng Zhao","doi":"arxiv-2408.08587","DOIUrl":"https://doi.org/arxiv-2408.08587","url":null,"abstract":"We constructed two dcpo's whose Scott spaces are sober, but the Scott space\u0000of their order product is not sober. This answers an open problem on the\u0000sobriety of Scott spaces. Meantime, we show that if $M$ and $N$ are special\u0000type of sober complete lattices, then the Scott space of their order product\u0000$Mtimes N$ is sober.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181052","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
A Dichotomy for Finite Abstract Simplicial Complexes 有限抽象简单复合物的二分法
arXiv - MATH - General Topology Pub Date : 2024-08-15 DOI: arxiv-2408.08199
Sebastian Meyer
{"title":"A Dichotomy for Finite Abstract Simplicial Complexes","authors":"Sebastian Meyer","doi":"arxiv-2408.08199","DOIUrl":"https://doi.org/arxiv-2408.08199","url":null,"abstract":"Given two finite abstract simplicial complexes A and B, one can define a new\u0000simplicial complex on the set of simplicial maps from A to B. After adding two\u0000technicalities, we call this complex Homsc(A, B). We prove the following dichotomy: For a fixed finite abstract simplicial\u0000complex B, either Homsc(A, B) is always a disjoint union of contractible spaces\u0000or every finite CW-complex can be obtained up to a homotopy equivalence as\u0000Homsc(A, B) by choosing A in a right way. We furthermore show that the first case is equivalent to the existence of a\u0000nontrivial social choice function and that in this case, the space itself is\u0000homotopy equivalent to a discrete set. Secondly, we give a generalization to finite relational structures and show\u0000that this dichotomy coincides with a complexity theoretic dichotomy for\u0000constraint satisfaction problems, namely in the first case, the problem is in P\u0000and in the second case NP-complete. This generalizes a result from [SW24]\u0000respectively arXiv:2307.03446 [cs.CC]","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181058","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
Local and global properties of spaces of minimal usco maps 最小 usco 映射空间的局部和全局特性
arXiv - MATH - General Topology Pub Date : 2024-08-14 DOI: arxiv-2408.07409
Serhii Bardyla, Branislav Novotný, Jaroslav Šupina
{"title":"Local and global properties of spaces of minimal usco maps","authors":"Serhii Bardyla, Branislav Novotný, Jaroslav Šupina","doi":"arxiv-2408.07409","DOIUrl":"https://doi.org/arxiv-2408.07409","url":null,"abstract":"In this paper, we study an interplay between local and global properties of\u0000spaces of minimal usco maps equipped with the topology of uniform convergence\u0000on compact sets. In particular, for each locally compact space $X$ and metric\u0000space $Y$, we characterize the space of minimal usco maps from $X$ to $Y$,\u0000satisfying one of the following properties: (i) compact, (ii) locally compact,\u0000(iii) $sigma$-compact, (iv) locally $sigma$-compact, (v) metrizable, (vi)\u0000ccc, (vii) locally ccc, where in the last two items we additionally assumed\u0000that $Y$ is separable and non-discrete. Some of the aforementioned results\u0000complement ones of v{L}ubica Hol'a and Duv{s}an Hol'y. Also, we obtain\u0000analogical characterizations for spaces of minimal cusco maps.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181055","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
Relative sectional number and the coincidence property 相对截面数和重合特性
arXiv - MATH - General Topology Pub Date : 2024-08-14 DOI: arxiv-2408.07316
Cesar A. Ipanaque Zapata, Felipe A. Torres Estrella
{"title":"Relative sectional number and the coincidence property","authors":"Cesar A. Ipanaque Zapata, Felipe A. Torres Estrella","doi":"arxiv-2408.07316","DOIUrl":"https://doi.org/arxiv-2408.07316","url":null,"abstract":"For a Hausdorff space $Y$, a topological space $X$ and a map $g:Xto Y$, we\u0000present a connection between the relative sectional number of the first\u0000coordinate projection $pi_{2,1}^Y:F(Y,2)to Y$ with respect to $g$, and the\u0000coincidence property (CP) for $(X,Y;g)$, where $(X,Y;g)$ has the coincidence\u0000property (CP) if, for every map $f:Xto Y$, there is a point $x$ of $X$ such\u0000that $f(x)=g(x)$. Explicitly, we demonstrate that $(X,Y;g)$ has the CP if and\u0000only if 2 is the minimal cardinality of open covers ${U_i}$ of $X$ such that\u0000each $U_i$ admits a local lifting for $g$ with respect to $pi_{2,1}^Y$. This\u0000characterisation connects a standard problem in coincidence theory to current\u0000research trends in sectional category and topological robotics. Motivated by\u0000this connection, we introduce the notion of relative topological complexity of\u0000a map.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142180957","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 the probabilistic metrizability of approach spaces 论方法空间的概率元可操作性
arXiv - MATH - General Topology Pub Date : 2024-08-14 DOI: arxiv-2408.07548
Hongliang Lai, Lili Shen, Junche Yu
{"title":"On the probabilistic metrizability of approach spaces","authors":"Hongliang Lai, Lili Shen, Junche Yu","doi":"arxiv-2408.07548","DOIUrl":"https://doi.org/arxiv-2408.07548","url":null,"abstract":"We investigate approach spaces generated by probabilistic metric spaces with\u0000respect to a continuous t-norm $*$ on the unit interval $[0,1]$. Let $k^*$ be\u0000the supremum of the idempotent elements of $*$ in $[0,1)$. It is shown that if\u0000$k^*=1$ (resp. $k^*<1$), then an approach space is probabilistic metrizable\u0000with respect to $*$ if and only if it is probabilistic metrizable with respect\u0000to the minimum (resp. product) t-norm.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181054","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
I-convergence of sequences in metric-like spaces 类度量空间中序列的 I- 收敛性
arXiv - MATH - General Topology Pub Date : 2024-08-10 DOI: arxiv-2408.13264
Prasanta Malik, Saikat Das
{"title":"I-convergence of sequences in metric-like spaces","authors":"Prasanta Malik, Saikat Das","doi":"arxiv-2408.13264","DOIUrl":"https://doi.org/arxiv-2408.13264","url":null,"abstract":"In this paper we introduce and study the notion of I-convergence of sequences\u0000in a metric-like space, where I is an ideal of subsets of the set N of all\u0000natural numbers. Further introducing the notion of I*-convergence of sequences\u0000in a metric-like space we study its relationship with I-convergence.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181057","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
Topological structure of projective Hilbert spaces associated with phase retrieval vectors 与相位检索矢量相关的投影希尔伯特空间的拓扑结构
arXiv - MATH - General Topology Pub Date : 2024-08-09 DOI: arxiv-2408.05317
Fahimeh Arabyani Neyshaburi, Ali Akbar Arefijamaal, Ghadir Sadeghi
{"title":"Topological structure of projective Hilbert spaces associated with phase retrieval vectors","authors":"Fahimeh Arabyani Neyshaburi, Ali Akbar Arefijamaal, Ghadir Sadeghi","doi":"arxiv-2408.05317","DOIUrl":"https://doi.org/arxiv-2408.05317","url":null,"abstract":"Projective Hilbert spaces as the underlying spaces of this paper are obtained\u0000by identifying two vectors of a Hilbert space $mathcal{H}$ which have the same\u0000phase and denoted by $hat{mathcal{H}}$. For a family $Phi$ of vectors of\u0000$mathcal{H}$ we introduce a topology $tau_{Phi}$ on $hat{mathcal{H}}$ and\u0000provide a topology-based approach for analyzing $hat{mathcal{H}}$. This leads\u0000to a new classification of phase retrieval property. We prove that\u0000$(hat{mathcal{H}}, tau_{Phi})$ is $sigma$-compact, as well as it is\u0000Hausdorff if and only if $Phi$ does phase retrieval. In particular, if $Phi$\u0000is phase retrieval, then we prove that $(hat{mathcal{H}}, tau_{Phi})$ is\u0000metrizable and $hat{mathcal{H}}$ is paracompact by a direct limit topology.\u0000Also, we make a comparison between $tau_{Phi}$ and some known topologies\u0000including the quotient topology, the weak topology and the direct-limit\u0000topology. Furthermore, we establish a metric $d_{Phi}$ on $hat{mathcal{H}}$\u0000and show that $d_{Phi}$ is weaker than the Bures-Wasserstein distance on\u0000$hat{mathcal{H}}$. As a result, in the finite dimensional case, we prove that\u0000$tau_{Phi}$ coincides with the weak topology and $tau_{d_{Phi}}$ on\u0000$hat{mathcal{H}}$ if and only if $Phi$ is phase retrieval.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181056","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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