{"title":"THE BAIRE CLOSURE AND ITS LOGIC","authors":"G. BEZHANISHVILI, D. FERNÁNDEZ-DUQUE","doi":"10.1017/jsl.2024.1","DOIUrl":null,"url":null,"abstract":"<p>The Baire algebra of a topological space <span>X</span> is the quotient of the algebra of all subsets of <span>X</span> modulo the meager sets. We show that this Boolean algebra can be endowed with a natural closure operator, resulting in a closure algebra which we denote <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240127131651504-0594:S002248122400001X:S002248122400001X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbf {Baire}(X)$</span></span></img></span></span>. We identify the modal logic of such algebras to be the well-known system <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240127131651504-0594:S002248122400001X:S002248122400001X_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathsf {S5}$</span></span></img></span></span>, and prove soundness and strong completeness for the cases where <span>X</span> is crowded and either completely metrizable and continuum-sized or locally compact Hausdorff. We also show that every extension of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240127131651504-0594:S002248122400001X:S002248122400001X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathsf {S5}$</span></span></img></span></span> is the modal logic of a subalgebra of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240127131651504-0594:S002248122400001X:S002248122400001X_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbf {Baire}(X)$</span></span></img></span></span>, and that soundness and strong completeness also holds in the language with the universal modality.</p>","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":"58 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Journal of Symbolic Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/jsl.2024.1","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
The Baire algebra of a topological space X is the quotient of the algebra of all subsets of X modulo the meager sets. We show that this Boolean algebra can be endowed with a natural closure operator, resulting in a closure algebra which we denote $\mathbf {Baire}(X)$. We identify the modal logic of such algebras to be the well-known system $\mathsf {S5}$, and prove soundness and strong completeness for the cases where X is crowded and either completely metrizable and continuum-sized or locally compact Hausdorff. We also show that every extension of $\mathsf {S5}$ is the modal logic of a subalgebra of $\mathbf {Baire}(X)$, and that soundness and strong completeness also holds in the language with the universal modality.