{"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":null,"pages":null},"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}
引用次数: 0
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.
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.