{"title":"Hofmann-Mislove type definitions of non-Hausdorff spaces","authors":"Chong Shen, Xiaoyong Xi, Xiaoquan Xu, Dongsheng Zhao","doi":"10.1017/S0960129522000196","DOIUrl":null,"url":null,"abstract":"Abstract One of the most important results in domain theory is the Hofmann-Mislove Theorem, which reveals a very distinct characterization for the sober spaces via open filters. In this paper, we extend this result to the d-spaces and well-filtered spaces. We do this by introducing the notions of Hofmann-Mislove-system (HM-system for short) and \n$\\Psi$\n -well-filtered space, which provide a new unified approach to sober spaces, well-filtered spaces, and d-spaces. In addition, a characterization for \n$\\Psi$\n -well-filtered spaces is provided via \n$\\Psi$\n -sets. We also discuss the relationship between \n$\\Psi$\n -well-filtered spaces and H-sober spaces considered by Xu. We show that the category of complete \n$\\Psi$\n -well-filtered spaces is a full reflective subcategory of the category of \n$T_0$\n spaces with continuous mappings. For each HM-system \n$\\Psi$\n that has a designated property, we show that a \n$T_0$\n space X is \n$\\Psi$\n -well-filtered if and only if its Smyth power space \n$P_s(X)$\n is \n$\\Psi$\n -well-filtered.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Structures in Computer Science","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1017/S0960129522000196","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract One of the most important results in domain theory is the Hofmann-Mislove Theorem, which reveals a very distinct characterization for the sober spaces via open filters. In this paper, we extend this result to the d-spaces and well-filtered spaces. We do this by introducing the notions of Hofmann-Mislove-system (HM-system for short) and
$\Psi$
-well-filtered space, which provide a new unified approach to sober spaces, well-filtered spaces, and d-spaces. In addition, a characterization for
$\Psi$
-well-filtered spaces is provided via
$\Psi$
-sets. We also discuss the relationship between
$\Psi$
-well-filtered spaces and H-sober spaces considered by Xu. We show that the category of complete
$\Psi$
-well-filtered spaces is a full reflective subcategory of the category of
$T_0$
spaces with continuous mappings. For each HM-system
$\Psi$
that has a designated property, we show that a
$T_0$
space X is
$\Psi$
-well-filtered if and only if its Smyth power space
$P_s(X)$
is
$\Psi$
-well-filtered.
期刊介绍:
Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.