{"title":"弱偏度量空间上的拓扑与不动点","authors":"Mengqiao Huang , Xiaodong Jia , Qingguo Li","doi":"10.1016/j.topol.2025.109601","DOIUrl":null,"url":null,"abstract":"<div><div>For a weak partial metric space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span>, there is a canonical metric <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> on <em>X</em>, defined as <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>max</mi><mo></mo><mo>{</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>−</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>−</mo><mi>p</mi><mo>(</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>}</mo></math></span> for all <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>X</mi></math></span>. We prove that the partial metric topology and the Scott topology on <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> coincide if and only if the metric topology on <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> and the Lawson topology on <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> agree, provided that the weak partial metric space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> is a domain in its specialization order and its associated metric space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> is compact. We also discussed fixpoints of self maps defined on weak partial metric spaces.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109601"},"PeriodicalIF":0.5000,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Topologies and fixpoints on weak partial metric spaces\",\"authors\":\"Mengqiao Huang , Xiaodong Jia , Qingguo Li\",\"doi\":\"10.1016/j.topol.2025.109601\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>For a weak partial metric space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span>, there is a canonical metric <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> on <em>X</em>, defined as <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>max</mi><mo></mo><mo>{</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>−</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>−</mo><mi>p</mi><mo>(</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>}</mo></math></span> for all <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>X</mi></math></span>. We prove that the partial metric topology and the Scott topology on <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> coincide if and only if the metric topology on <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> and the Lawson topology on <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> agree, provided that the weak partial metric space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> is a domain in its specialization order and its associated metric space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> is compact. We also discussed fixpoints of self maps defined on weak partial metric spaces.</div></div>\",\"PeriodicalId\":51201,\"journal\":{\"name\":\"Topology and its Applications\",\"volume\":\"377 \",\"pages\":\"Article 109601\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2025-10-02\",\"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/S0166864125003992\",\"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/S0166864125003992","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Topologies and fixpoints on weak partial metric spaces
For a weak partial metric space , there is a canonical metric on X, defined as for all . We prove that the partial metric topology and the Scott topology on coincide if and only if the metric topology on and the Lawson topology on agree, provided that the weak partial metric space is a domain in its specialization order and its associated metric space is compact. We also discussed fixpoints of self maps defined on weak partial metric spaces.
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.