T0-spaces and the lower topology

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS Mathematical Structures in Computer Science Pub Date : 2024-09-19 DOI:10.1017/s0960129524000240
Jimmie Lawson, Xiaoquan Xu
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Using the lower topology, one can define and study new properties of the original space that provide deeper insight into its structure. One focus of study is the property R, which asserts that if the intersection of a family of finitely generated sets <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000240_inline4.png\"/> <jats:tex-math> $\\mathord{\\uparrow } F$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000240_inline5.png\"/> <jats:tex-math> $F$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> finite, is contained in an open set <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000240_inline6.png\"/> <jats:tex-math> $U$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, then the same is true for finitely many of the family. We first show that property R is equivalent to several other interesting properties, for example, the property that all closed subsets of the original space are compact in the lower topology. We then find conditions under which these spaces are compact, well-filtered, and coherent, a weaker variant of stably compact spaces. We also investigate what have been called strong <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000240_inline7.png\"/> <jats:tex-math> $d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-spaces, develop some of their basic properties, and make connections with the earlier considerations involving spaces satisfying property R. Two key results we obtain are that if a dcpo <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000240_inline8.png\"/> <jats:tex-math> $P$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with the Scott topology is a strong <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000240_inline9.png\"/> <jats:tex-math> $d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-space, then it is well-filtered, and if additionally the Scott topology of the product <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000240_inline10.png\"/> <jats:tex-math> $P\\times P$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the product of the Scott topologies of the factors, then the Scott space of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000240_inline11.png\"/> <jats:tex-math> $P$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is sober. We also exhibit connections of this work with de Groot duality.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2024-09-19","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/s0960129524000240","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

The authors’ primary goal in this paper is to enhance the study of $T_0$ topological spaces by using the order of specialization of a $T_0$ -space to introduce the lower topology (with a subbasis of closed sets $\mathord{\uparrow } x$ ) and studying the interaction of the original topology and the lower topology. Using the lower topology, one can define and study new properties of the original space that provide deeper insight into its structure. One focus of study is the property R, which asserts that if the intersection of a family of finitely generated sets $\mathord{\uparrow } F$ , $F$ finite, is contained in an open set $U$ , then the same is true for finitely many of the family. We first show that property R is equivalent to several other interesting properties, for example, the property that all closed subsets of the original space are compact in the lower topology. We then find conditions under which these spaces are compact, well-filtered, and coherent, a weaker variant of stably compact spaces. We also investigate what have been called strong $d$ -spaces, develop some of their basic properties, and make connections with the earlier considerations involving spaces satisfying property R. Two key results we obtain are that if a dcpo $P$ with the Scott topology is a strong $d$ -space, then it is well-filtered, and if additionally the Scott topology of the product $P\times P$ is the product of the Scott topologies of the factors, then the Scott space of $P$ is sober. We also exhibit connections of this work with de Groot duality.
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T0 空间和下拓扑
作者在本文中的主要目标是通过使用 $T_0$ -space 的特化阶引入下拓扑(具有闭集的子基础 $\mathord{\uparrow } x$ ),并研究原始拓扑与下拓扑的相互作用,从而加强对 $T_0$ 拓扑空间的研究。利用低级拓扑,我们可以定义和研究原始空间的新属性,从而更深入地了解其结构。研究的一个重点是属性 R,它断言如果有限生成集合 $\mathord{\uparrow } 的交集为F$ ,$F$ 有限,包含在一个开集 $U$ 中,那么这个族中的有限个集也是如此。我们首先证明性质 R 等价于其他几个有趣的性质,例如,原始空间的所有封闭子集在下拓扑中都是紧凑的性质。然后,我们找到了这些空间紧凑、过滤良好且连贯的条件,这是稳定紧凑空间的较弱变体。我们还研究了所谓的强 $d$ -空间,发展了它们的一些基本性质,并把它们与前面涉及满足性质 R 的空间的考虑联系起来。我们得到的两个关键结果是:如果具有斯科特拓扑的 dcpo $P$ 是一个强 $d$ -空间,那么它就是过滤良好的;如果另外的乘积 $P\times P$ 的斯科特拓扑是各因子的斯科特拓扑的乘积,那么 $P$ 的斯科特空间就是清醒的。我们还展示了这项工作与德格鲁特对偶性的联系。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science 工程技术-计算机:理论方法
CiteScore
1.50
自引率
0.00%
发文量
30
审稿时长
12 months
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.
期刊最新文献
On Hofmann–Streicher universes T0-spaces and the lower topology GADTs are not (Even partial) functors A linear linear lambda-calculus Countability constraints in order-theoretic approaches to computability
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