关于与拟度量空间I相关的函数空间

IF 0.5 4区 数学 Q3 MATHEMATICS
Zhenhua Jia, Qingguo Li
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Then, for quasi-metric spaces <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo><mo>,</mo><mo>(</mo><mi>Y</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>Y</mi></mrow></msub><mo>)</mo></math></span> and a certain topological property <span><math><mi>R</mi></math></span>, we obtain that under some conditions, <span><math><mo>(</mo><mi>Y</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>Y</mi></mrow></msub><mo>)</mo></math></span> has property <span><math><mi>R</mi></math></span> if and only if <span><math><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo><mo>,</mo><mover><mrow><mi>d</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span> possesses <span><math><mi>R</mi></math></span>, where <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span> is the set of <em>α</em>-Lipschitz maps from <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span> to <span><math><mo>(</mo><mi>Y</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>Y</mi></mrow></msub><mo>)</mo></math></span> and <span><math><mover><mrow><mi>d</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><msub><mrow><mi>sup</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow></msub><mo>⁡</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>Y</mi></mrow></msub><mo>(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span>. Additionally, for standard quasi-metric spaces <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>Y</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>Y</mi></mrow></msub><mo>)</mo></math></span>, we consider the function space <span><math><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo><mo>,</mo><mover><mrow><mi>d</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span> denotes the set of <em>α</em>-Lipschitz continuous maps from <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span> to <span><math><mo>(</mo><mi>Y</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>Y</mi></mrow></msub><mo>)</mo></math></span>. These results will provide useful tools for identifying Cartesian closed subcategories within the category of quasi-metric spaces, thereby providing denotational semantics for certain higher-order functional programming languages.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109498"},"PeriodicalIF":0.5000,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On function spaces related to quasi-metric spaces I\",\"authors\":\"Zhenhua Jia,&nbsp;Qingguo Li\",\"doi\":\"10.1016/j.topol.2025.109498\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>This paper investigates a series of issues concerning the preservation of certain properties of topological spaces in passing to function spaces. First, we introduce two quasi-metrics on <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> spaces: one is a KF-quasi-metric and the other is an IR-quasi-metric, and we prove that <em>X</em> is a quasi-metrizable sober space if and only if <em>X</em> is a quasi-metrizable well-filtered space, if and only if <em>X</em> is KF-quasi-metrizable, if and only if <em>X</em> is IR-quasi-metrizable. Then, for quasi-metric spaces <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo><mo>,</mo><mo>(</mo><mi>Y</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>Y</mi></mrow></msub><mo>)</mo></math></span> and a certain topological property <span><math><mi>R</mi></math></span>, we obtain that under some conditions, <span><math><mo>(</mo><mi>Y</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>Y</mi></mrow></msub><mo>)</mo></math></span> has property <span><math><mi>R</mi></math></span> if and only if <span><math><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo><mo>,</mo><mover><mrow><mi>d</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span> possesses <span><math><mi>R</mi></math></span>, where <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span> is the set of <em>α</em>-Lipschitz maps from <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span> to <span><math><mo>(</mo><mi>Y</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>Y</mi></mrow></msub><mo>)</mo></math></span> and <span><math><mover><mrow><mi>d</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><msub><mrow><mi>sup</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow></msub><mo>⁡</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>Y</mi></mrow></msub><mo>(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span>. Additionally, for standard quasi-metric spaces <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>Y</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>Y</mi></mrow></msub><mo>)</mo></math></span>, we consider the function space <span><math><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo><mo>,</mo><mover><mrow><mi>d</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span> denotes the set of <em>α</em>-Lipschitz continuous maps from <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span> to <span><math><mo>(</mo><mi>Y</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>Y</mi></mrow></msub><mo>)</mo></math></span>. These results will provide useful tools for identifying Cartesian closed subcategories within the category of quasi-metric spaces, thereby providing denotational semantics for certain higher-order functional programming languages.</div></div>\",\"PeriodicalId\":51201,\"journal\":{\"name\":\"Topology and its Applications\",\"volume\":\"373 \",\"pages\":\"Article 109498\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2025-07-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Topology and its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0166864125002962\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Topology and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166864125002962","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

本文研究了拓扑空间在传递给函数空间时某些性质的保留问题。首先,我们在T0空间上引入了两个拟度量:一个是kf -拟度量,一个是ir -拟度量,并证明了X是拟可度量的清晰空间,当且仅当X是拟可度量的良滤空间,当且仅当X是kf -拟可度量的,当且仅当X是ir -拟可度量的。然后,对于准度量空间(X,dX),(Y,dY)和一定的拓扑性质R,我们得到了在某些条件下,(Y,dY)有性质R当且仅当(Lα(X,Y),d})具有R,其中,Lα(X,Y)是由(X,dX)到(Y,dY)的α-Lipschitz映射集,且d} (f1,f2)=supx∈X∈dY(f1(X), f2(X))。另外,对于标准拟度量空间(X,dX)和(Y,dY),我们考虑函数空间(Lα(X,Y),d -,其中,Lα(X,Y)表示从(X,dX)到(Y,dY)的α-Lipschitz连续映射集。这些结果将为识别准度量空间范畴内的笛卡尔闭子范畴提供有用的工具,从而为某些高阶函数式编程语言提供指称语义。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On function spaces related to quasi-metric spaces I
This paper investigates a series of issues concerning the preservation of certain properties of topological spaces in passing to function spaces. First, we introduce two quasi-metrics on T0 spaces: one is a KF-quasi-metric and the other is an IR-quasi-metric, and we prove that X is a quasi-metrizable sober space if and only if X is a quasi-metrizable well-filtered space, if and only if X is KF-quasi-metrizable, if and only if X is IR-quasi-metrizable. Then, for quasi-metric spaces (X,dX),(Y,dY) and a certain topological property R, we obtain that under some conditions, (Y,dY) has property R if and only if (Lα(X,Y),dˆ) possesses R, where Lα(X,Y) is the set of α-Lipschitz maps from (X,dX) to (Y,dY) and dˆ(f1,f2)=supxXdY(f1(x),f2(x)). Additionally, for standard quasi-metric spaces (X,dX) and (Y,dY), we consider the function space (Lα(X,Y),dˆ), where Lα(X,Y) denotes the set of α-Lipschitz continuous maps from (X,dX) to (Y,dY). These results will provide useful tools for identifying Cartesian closed subcategories within the category of quasi-metric spaces, thereby providing denotational semantics for certain higher-order functional programming languages.
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来源期刊
CiteScore
1.20
自引率
33.30%
发文量
251
审稿时长
6 months
期刊介绍: Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.
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