{"title":"A Boolean-valued space approach to separation axioms and sobriety of bitopological spaces","authors":"Jing He, Dexue Zhang","doi":"10.1016/j.fss.2024.109241","DOIUrl":null,"url":null,"abstract":"<div><div>This paper presents a study of separation axioms and sobriety of bitopological spaces from the point of view of fuzzy topology via identifying bitopological spaces with topological spaces valued in the Boolean algebra of four elements. A system of separation axioms is proposed making use of Boolean-valued specialization order of bitopological spaces; The relationship between d-sobriety of bitopological spaces proposed by Jung and Moshier and sobriety of fuzzy topological spaces is studied; A Hofmann-Mislove theorem for bitopological spaces is established.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"503 ","pages":"Article 109241"},"PeriodicalIF":3.2000,"publicationDate":"2024-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fuzzy Sets and Systems","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0165011424003877","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
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Abstract
This paper presents a study of separation axioms and sobriety of bitopological spaces from the point of view of fuzzy topology via identifying bitopological spaces with topological spaces valued in the Boolean algebra of four elements. A system of separation axioms is proposed making use of Boolean-valued specialization order of bitopological spaces; The relationship between d-sobriety of bitopological spaces proposed by Jung and Moshier and sobriety of fuzzy topological spaces is studied; A Hofmann-Mislove theorem for bitopological spaces is established.
期刊介绍：
Since its launching in 1978, the journal Fuzzy Sets and Systems has been devoted to the international advancement of the theory and application of fuzzy sets and systems. The theory of fuzzy sets now encompasses a well organized corpus of basic notions including (and not restricted to) aggregation operations, a generalized theory of relations, specific measures of information content, a calculus of fuzzy numbers. Fuzzy sets are also the cornerstone of a non-additive uncertainty theory, namely possibility theory, and of a versatile tool for both linguistic and numerical modeling: fuzzy rule-based systems. Numerous works now combine fuzzy concepts with other scientific disciplines as well as modern technologies.
In mathematics fuzzy sets have triggered new research topics in connection with category theory, topology, algebra, analysis. Fuzzy sets are also part of a recent trend in the study of generalized measures and integrals, and are combined with statistical methods. Furthermore, fuzzy sets have strong logical underpinnings in the tradition of many-valued logics.
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