{"title":"关于与拟度量空间I相关的函数空间","authors":"Zhenhua Jia, 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":"{\"title\":\"On function spaces related to quasi-metric spaces I\",\"authors\":\"Zhenhua Jia, 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}
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 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 and a certain topological property , we obtain that under some conditions, has property if and only if possesses , where is the set of α-Lipschitz maps from to and . Additionally, for standard quasi-metric spaces and , we consider the function space , where denotes the set of α-Lipschitz continuous maps from to . 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.
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.