{"title":"Quasiorders for a Characterization of Iso-dense Spaces","authors":"Tom Richmond, Eliza Wajch","doi":"10.1007/s40840-024-01758-5","DOIUrl":null,"url":null,"abstract":"<p>A (generalized) topological space is called an iso-dense space if the set of all its isolated points is dense in the space. The main aim of the article is to show in <span>\\(\\textbf{ZF}\\)</span> a new characterization of iso-dense spaces in terms of special quasiorders. For a non-empty family <span>\\(\\mathcal {A}\\)</span> of subsets of a set <i>X</i>, a quasiorder <span>\\({{\\,\\mathrm{\\lesssim }\\,}}_{\\mathcal {A}}\\)</span> on <i>X</i> determined by <span>\\(\\mathcal {A}\\)</span> is defined. Necessary and sufficient conditions for <span>\\(\\mathcal {A}\\)</span> are given to have the property that the topology consisting of all <span>\\({{\\,\\mathrm{\\lesssim }\\,}}_{\\mathcal {A}}\\)</span>-increasing sets coincides with the generalized topology on <i>X</i> consisting of the empty set and all supersets of non-empty members of <span>\\(\\mathcal {A}\\)</span>. The results obtained, applied to the quasiorder <span>\\({{\\,\\mathrm{\\lesssim }\\,}}_{\\mathcal {D}}\\)</span> determined by the family <span>\\(\\mathcal {D}\\)</span> of all dense sets of a given (generalized) topological space, lead to a new characterization of non-trivial iso-dense spaces. Independence results concerning resolvable spaces are also obtained.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40840-024-01758-5","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
A (generalized) topological space is called an iso-dense space if the set of all its isolated points is dense in the space. The main aim of the article is to show in \(\textbf{ZF}\) a new characterization of iso-dense spaces in terms of special quasiorders. For a non-empty family \(\mathcal {A}\) of subsets of a set X, a quasiorder \({{\,\mathrm{\lesssim }\,}}_{\mathcal {A}}\) on X determined by \(\mathcal {A}\) is defined. Necessary and sufficient conditions for \(\mathcal {A}\) are given to have the property that the topology consisting of all \({{\,\mathrm{\lesssim }\,}}_{\mathcal {A}}\)-increasing sets coincides with the generalized topology on X consisting of the empty set and all supersets of non-empty members of \(\mathcal {A}\). The results obtained, applied to the quasiorder \({{\,\mathrm{\lesssim }\,}}_{\mathcal {D}}\) determined by the family \(\mathcal {D}\) of all dense sets of a given (generalized) topological space, lead to a new characterization of non-trivial iso-dense spaces. Independence results concerning resolvable spaces are also obtained.
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.