Scott topology on Smyth power posets

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS Mathematical Structures in Computer Science Pub Date : 2023-07-27 DOI:10.1017/s0960129523000257
Xiaoquan Xu, Xinpeng Wen, Xiaoyong Xi
{"title":"Scott topology on Smyth power posets","authors":"Xiaoquan Xu, Xinpeng Wen, Xiaoyong Xi","doi":"10.1017/s0960129523000257","DOIUrl":null,"url":null,"abstract":"\n\t <jats:p>For a <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline1.png\" />\n\t\t<jats:tex-math>\n$T_0$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> space <jats:italic>X</jats:italic>, let <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline2.png\" />\n\t\t<jats:tex-math>\n$\\mathsf{K}(X)$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> be the poset of all nonempty compact saturated subsets of <jats:italic>X</jats:italic> endowed with the Smyth order <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline3.png\" />\n\t\t<jats:tex-math>\n$\\sqsubseteq$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula>. <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline4.png\" />\n\t\t<jats:tex-math>\n$(\\mathsf{K}(X), \\sqsubseteq)$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> (shortly <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline5.png\" />\n\t\t<jats:tex-math>\n$\\mathsf{K}(X)$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula>) is called the Smyth power poset of <jats:italic>X</jats:italic>. In this paper, we mainly discuss some basic properties of the Scott topology on Smyth power posets. It is proved that for a well-filtered space <jats:italic>X</jats:italic>, its Smyth power poset <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline6.png\" />\n\t\t<jats:tex-math>\n$\\mathsf{K}(X)$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> with the Scott topology is still well-filtered, and a <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline7.png\" />\n\t\t<jats:tex-math>\n$T_0$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> space <jats:italic>Y</jats:italic> is well-filtered iff the Smyth power poset <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline8.png\" />\n\t\t<jats:tex-math>\n$\\mathsf{K}(Y)$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> with the Scott topology is well-filtered and the upper Vietoris topology is coarser than the Scott topology on <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline9.png\" />\n\t\t<jats:tex-math>\n$\\mathsf{K}(Y)$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula>. A sober space <jats:italic>Z</jats:italic> is constructed for which the Smyth power poset <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline10.png\" />\n\t\t<jats:tex-math>\n$\\mathsf{K}(Z)$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> with the Scott topology is not sober. A few sufficient conditions are given for a <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline11.png\" />\n\t\t<jats:tex-math>\n$T_0$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> space <jats:italic>X</jats:italic> under which its Smyth power poset <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline12.png\" />\n\t\t<jats:tex-math>\n$\\mathsf{K}(X)$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> with the Scott topology is sober. Some other properties, such as local compactness, first-countability, Rudin property and well-filtered determinedness, of Smyth power spaces, and the Scott topology on Smyth power posets, are also investigated.</jats:p>","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":" ","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2023-07-27","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/s0960129523000257","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}
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Abstract

For a $T_0$ space X, let $\mathsf{K}(X)$ be the poset of all nonempty compact saturated subsets of X endowed with the Smyth order $\sqsubseteq$ . $(\mathsf{K}(X), \sqsubseteq)$ (shortly $\mathsf{K}(X)$ ) is called the Smyth power poset of X. In this paper, we mainly discuss some basic properties of the Scott topology on Smyth power posets. It is proved that for a well-filtered space X, its Smyth power poset $\mathsf{K}(X)$ with the Scott topology is still well-filtered, and a $T_0$ space Y is well-filtered iff the Smyth power poset $\mathsf{K}(Y)$ with the Scott topology is well-filtered and the upper Vietoris topology is coarser than the Scott topology on $\mathsf{K}(Y)$ . A sober space Z is constructed for which the Smyth power poset $\mathsf{K}(Z)$ with the Scott topology is not sober. A few sufficient conditions are given for a $T_0$ space X under which its Smyth power poset $\mathsf{K}(X)$ with the Scott topology is sober. Some other properties, such as local compactness, first-countability, Rudin property and well-filtered determinedness, of Smyth power spaces, and the Scott topology on Smyth power posets, are also investigated.
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Smyth幂偏序集上的Scott拓扑
对于$T_0$空间X,设$\mathsf{K}(X)$是赋予Smyth阶$\sqsubsteq$的X的所有非空紧致饱和子集的偏序集$(\mathsf{K}(X),\sqsubsteq)$(shortly$\mathsf{K}(X)$)称为X的Smyth幂偏序集。本文主要讨论了Smyth功率偏序集上Scott拓扑的一些基本性质。证明了对于一个良好滤波的空间X,其具有Scott拓扑的Smyth幂偏序集$\mathsf{K}(X)$仍然是良好滤波的,并且$T_ 0$空间Y被很好地过滤,如果具有Scott拓扑的Smyth幂偏序集$\mathsf{K}(Y)$被很好的过滤。构造了具有Scott拓扑的Smyth幂偏序集$\mathsf{K}(Z)$不清醒的清醒空间Z。给出了$T_0$空间X的几个充分条件,在此条件下,具有Scott拓扑的Smyth幂偏序集$\mathsf{K}(X)$是清醒的。还研究了Smyth幂空间的一些其他性质,如局部紧性、第一可数性、Rudin性质和良好滤波的确定性,以及Smyth功率偏序集上的Scott拓扑。
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.
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