{"title":"D-completion, well-filterification and sobrification","authors":"Hualin Miao , Longchun Wang , Qingguo Li","doi":"10.1016/j.topol.2024.109050","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we study the <em>D</em>-completion, well-filterification and sobrification of a <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> space. First, we present an example of a tapered closed set which is neither the closure of a directed set nor a closed <em>KF</em>-set. In 2020, Xu et al. asked whether closed <em>RD</em>-sets are exactly closed <em>WD</em>-sets for every <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> space. This example also gives a negative answer to the above problem, since each tapered closed set is a closed <em>WD</em>-set. Second, we provide a direct characterization for the <em>D</em>-completion of a poset by using the notion of pre-<em>C</em>-compact elements. Finally, for a given <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> space, we give some sufficient conditions which guarantee that each pair of its standard <em>D</em>-completion, standard well-filterification and standard sobrification agrees.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"356 ","pages":"Article 109050"},"PeriodicalIF":0.6000,"publicationDate":"2024-08-26","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/S0166864124002359","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
In this paper, we study the D-completion, well-filterification and sobrification of a space. First, we present an example of a tapered closed set which is neither the closure of a directed set nor a closed KF-set. In 2020, Xu et al. asked whether closed RD-sets are exactly closed WD-sets for every space. This example also gives a negative answer to the above problem, since each tapered closed set is a closed WD-set. Second, we provide a direct characterization for the D-completion of a poset by using the notion of pre-C-compact elements. Finally, for a given space, we give some sufficient conditions which guarantee that each pair of its standard D-completion, standard well-filterification and standard sobrification agrees.
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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