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A Boolean-valued space approach to separation axioms and sobriety of bitopological spaces 用布尔值空间方法研究比拓空间的分离公理和清醒度
arXiv - MATH - General Topology Pub Date : 2024-07-25 DOI: arxiv-2407.17728
Jing He, Dexue Zhang
{"title":"A Boolean-valued space approach to separation axioms and sobriety of bitopological spaces","authors":"Jing He, Dexue Zhang","doi":"arxiv-2407.17728","DOIUrl":"https://doi.org/arxiv-2407.17728","url":null,"abstract":"This paper presents a study of separation axioms and sobriety of\u0000bitopological spaces from the point of view of fuzzy topology via identifying\u0000bitopological spaces with topological spaces valued in the Boolean algebra of\u0000four elements. A system of separation axioms is proposed making use of\u0000Boolean-valued specialization order of bitopological spaces; The relationship\u0000between d-sobriety of bitopological spaces proposed by Jung and Moshier and\u0000sobriety of fuzzy topological spaces is studied; A Hofmann-Mislove theorem for\u0000bitopological spaces is established.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141786026","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 functor of comonotonically maxitive functionals 论最大单调函数的函子
arXiv - MATH - General Topology Pub Date : 2024-07-25 DOI: arxiv-2407.18345
Taras Radul
{"title":"On the functor of comonotonically maxitive functionals","authors":"Taras Radul","doi":"arxiv-2407.18345","DOIUrl":"https://doi.org/arxiv-2407.18345","url":null,"abstract":"We introduce a functor of functionals which preserve maximum of comonotone\u0000functions and addition of constants. This functor is a subfunctor of the\u0000functor of order-preserving functionals and contains the idempotent measure\u0000functor as subfunctor. The main aim of this paper is to show that this functor\u0000is isomorphic to the capacity functor. We establish such isomorphism using the\u0000fuzzy max-plus integral. In fact, we can consider this result as an idempotent\u0000analogue of Riesz Theorem about a correspondence between the set of\u0000$sigma$-additive regular Borel measures and the set of linear positively\u0000defined functionals.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869735","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
Generalizing $β$- and $λ$-maps β$-和λ$-映射的一般化
arXiv - MATH - General Topology Pub Date : 2024-07-24 DOI: arxiv-2407.16941
Ana Belén Avilez
{"title":"Generalizing $β$- and $λ$-maps","authors":"Ana Belén Avilez","doi":"arxiv-2407.16941","DOIUrl":"https://doi.org/arxiv-2407.16941","url":null,"abstract":"We generalize the notions of $beta$- and $lambda$-maps to general\u0000selections of sublocales, obtaining different classes of localic maps. These\u0000new classes of maps are used to characterize almost normality, extremal\u0000disconnectedness, $F$-frames, $Oz$-frames, among others types of locales, in a\u0000manner akin to the characterization of normal locales via $beta$-maps. As a\u0000byproduct we obtain a characterization of localic maps that preserve the\u0000completely below relation (that is, the right adjoints of assertive frame\u0000homomorphisms).","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141784924","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
Topologies derived from the old one via ideals 通过理想从旧拓扑图衍生出拓扑图
arXiv - MATH - General Topology Pub Date : 2024-07-24 DOI: arxiv-2407.17612
Faical Yacine Issaka, Murad Özkoç
{"title":"Topologies derived from the old one via ideals","authors":"Faical Yacine Issaka, Murad Özkoç","doi":"arxiv-2407.17612","DOIUrl":"https://doi.org/arxiv-2407.17612","url":null,"abstract":"The main purpose of this paper is to introduce and study minimal and maximal\u0000ideals defined on ideal topological spaces. Also, we define and investigate the\u0000concepts of ideal quotient and annihilator of any subfamily of $2^X$, where\u0000$2^X$ is the power set of $X.$ We obtain some of their fundamental properties.\u0000In addition, several relationships among the above notions have been discussed.\u0000Moreover, we get a new topology, called sharp topology via the sharp operator\u0000defined in the scope of this study, finer than the old one. Furthermore, a\u0000decomposition of the notion of open set has been obtained. Finally, we conclude\u0000our work with some interesting applications.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772225","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
Set convergences and uniform convergence of distance functionals on a bornology 生理学上距离函数的集合收敛性和均匀收敛性
arXiv - MATH - General Topology Pub Date : 2024-07-23 DOI: arxiv-2407.16408
Yogesh Agarwal, Varun Jindal
{"title":"Set convergences and uniform convergence of distance functionals on a bornology","authors":"Yogesh Agarwal, Varun Jindal","doi":"arxiv-2407.16408","DOIUrl":"https://doi.org/arxiv-2407.16408","url":null,"abstract":"For a metric space $(X,d)$, Beer, Naimpally, and Rodriguez-Lopez in ([17])\u0000proposed a unified approach to explore set convergences via uniform convergence\u0000of distance functionals on members of an arbitrary family $mathcal{S}$ of\u0000subsets of $X$. The associated topology on the collection $CL(X)$ of all\u0000nonempty closed subsets of $(X,d)$ is denoted by $tau_{mathcal{S},d}$. As\u0000special cases, this unified approach includes classical Wijsman, Attouch-Wets,\u0000and Hausdorff distance topologies. In this article, we investigate various\u0000topological characteristics of the hyperspace $(CL(X), tau_{mathcal{S},d})$\u0000when $mathcal{S}$ is a bornology on $(X,d)$. In order to do this, a new class\u0000of bornologies and a new metric topology on $CL(X)$ have been introduced and\u0000studied.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772226","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
Accessibility of countable sets in plane embeddings of arc-like continua 弧状连续体平面嵌入中可数集的可达性
arXiv - MATH - General Topology Pub Date : 2024-07-23 DOI: arxiv-2407.16792
Ana Anušić, Logan C. Hoehn
{"title":"Accessibility of countable sets in plane embeddings of arc-like continua","authors":"Ana Anušić, Logan C. Hoehn","doi":"arxiv-2407.16792","DOIUrl":"https://doi.org/arxiv-2407.16792","url":null,"abstract":"We consider the problem of finding embeddings of arc-like continua in the\u0000plane for which each point in a given subset is accessible. We establish that,\u0000under certain conditions on an inverse system of arcs, there exists a plane\u0000embedding of the inverse limit for which each point of a given countable set is\u0000accessible. As an application, we show that for any Knaster continuum $K$, and\u0000any countable collection $mathcal{C}$ of composants of $K$, there exists a\u0000plane embedding of $K$ in which every point in the union of the composants in\u0000$mathcal{C}$ is accessible. We also exhibit new embeddings of the Knaster\u0000buckethandle continuum $K$ in the plane which are attractors of plane\u0000homeomorphisms, and for which the restriction of the plane homeomorphism to the\u0000attractor is conjugate to a power of the standard shift map on $K$.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772179","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
Assouad type dimensions of homogeneous Moran sets and Cantor-like sets 同质莫兰集和类康托集的阿苏阿德类型维数
arXiv - MATH - General Topology Pub Date : 2024-07-20 DOI: arxiv-2407.14837
Yanzhe Li, Jun Li, Shuang Liang, Manli Lou
{"title":"Assouad type dimensions of homogeneous Moran sets and Cantor-like sets","authors":"Yanzhe Li, Jun Li, Shuang Liang, Manli Lou","doi":"arxiv-2407.14837","DOIUrl":"https://doi.org/arxiv-2407.14837","url":null,"abstract":"In this paper, we give the Assouad dimension formula and the upper bound of\u0000the lower dimension for homogeneous Moran sets under the condition $sup_{kge\u00001}{n_{k}}<+infty$. We also give the Assouad spectrum and the lower spectrum\u0000formulas for Cantor-like sets.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772227","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
Remoteness in the category of bilocales 二元对立范畴中的远程性
arXiv - MATH - General Topology Pub Date : 2024-07-20 DOI: arxiv-2407.14755
Mbekezeli Nxumalo
{"title":"Remoteness in the category of bilocales","authors":"Mbekezeli Nxumalo","doi":"arxiv-2407.14755","DOIUrl":"https://doi.org/arxiv-2407.14755","url":null,"abstract":"In locale theory, a sublocale is said to be remote in case it misses every\u0000nowhere dense sublocale. In this paper, we introduce and study a new class of\u0000sublocales in the category of bilocales, namely (i,j)-remote sublocales. These\u0000are bilocalic counterparts of remote sublocales and are the sublocales missing\u0000every (i,j)-nowhere dense sublocale, with (i,j)-nowhere dense sublocales being\u0000bilocalic counterparts of (tau_{i},tau_{j})-nowhere dense subsets in\u0000bitopological spaces. A comprehensive study of (i,j)-nowhere dense sublocales\u0000is given and we show that in the class of balanced bilocales, a sublocale is\u0000(i,j)-nowhere dense if and only if its bilocale closure is nowhere dense. We\u0000also consider weakly (i,j)-remote sublocales which are those sublocales missing\u0000every clopen (i,j)-nowhere dense sublocale. Furthermore, we extend\u0000(i,j)-remoteness to the categories of bitopological spaces as well as normed\u0000lattices. In the class of sup-T_{D} bitopological spaces, a subset A of a\u0000bitopological space (X,tau_{1},tau_{2}) is (tau_{i},tau_{j})-remote if and\u0000only if the induced sublocale widetilde{A} of tau_{1}veetau_{2} is\u0000(i,j)-remote.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772230","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
Priestley duality and representations of recurrent dynamics 普利斯特里对偶性与循环动力学表征
arXiv - MATH - General Topology Pub Date : 2024-07-19 DOI: arxiv-2407.14359
William Kalies, Robert Vandervorst
{"title":"Priestley duality and representations of recurrent dynamics","authors":"William Kalies, Robert Vandervorst","doi":"arxiv-2407.14359","DOIUrl":"https://doi.org/arxiv-2407.14359","url":null,"abstract":"For an arbitrary dynamical system there is a strong relationship between\u0000global dynamics and the order structure of an appropriately constructed\u0000Priestley space. This connection provides an order-theoretic framework for\u0000studying global dynamics. In the classical setting, the chain recurrent set,\u0000introduced by C. Conley, is an example of an ordered Stone space or Priestley\u0000space. Priestley duality can be applied in the setting of dynamics on arbitrary\u0000topological spaces and yields a notion of Hausdorff compactification of the\u0000(chain) recurrent set.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737310","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
Some variants of remote sublocales 远程子域的一些变体
arXiv - MATH - General Topology Pub Date : 2024-07-18 DOI: arxiv-2407.13400
Mbekezeli Nxumalo
{"title":"Some variants of remote sublocales","authors":"Mbekezeli Nxumalo","doi":"arxiv-2407.13400","DOIUrl":"https://doi.org/arxiv-2407.13400","url":null,"abstract":"We introduce and study some variants of remote sublocales, namely sublocales\u0000that are remote from dense sublocales and those that are *remote from dense\u0000sublocales. We show that the coframe of sublocales coincides with the\u0000collection of all sublocales remote from the Booleanization. Furthermore, the\u0000supplement of the Booleanization of any locale is the largest sublocale *remote\u0000from the Booleanization. We give conditions on localic maps such that their\u0000induced localic image and pre-image functions preserve sublocales that are\u0000remote (resp. *remote) from dense sublocales. We introduce new types of localic\u0000maps called f-remote preserving maps.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737201","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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