THE BAIRE CLOSURE AND ITS LOGIC

G. BEZHANISHVILI, D. FERNÁNDEZ-DUQUE
{"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 Abstract Image$\mathbf {Baire}(X)$. We identify the modal logic of such algebras to be the well-known system Abstract Image$\mathsf {S5}$, 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 Abstract Image$\mathsf {S5}$ is the modal logic of a subalgebra of Abstract Image$\mathbf {Baire}(X)$, and that soundness and strong completeness also holds in the language with the universal modality.

贝叶封闭及其逻辑
拓扑空间 X 的贝叶尔代数是 X 的所有子集调制集代数的商。我们证明,这个布尔代数可以被赋予一个自然闭包算子,从而得到一个闭包代数,我们将其命名为 $\mathbf {Baire}(X)$ 。我们确定这种代数的模态逻辑是著名的 $\mathsf {S5}$ 系统,并证明了 X 是拥挤的、完全可元化的和连续体大小的或局部紧凑的 Hausdorff 的情况下的健全性和强完备性。我们还证明$edmathsf {S5}$的每一个扩展都是$\mathbf {Baire}(X)$ 的一个子代数的模态逻辑,并且在具有普遍模态的语言中健全性和强完备性也成立。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信