Dualities via ComPactness, Openness, and their Symmetry

通过紧凑性、开放性及其对称性实现的二元性

• 批准号: EP/Y015029/1
• 负责人: Achim Jung
• 金额: $ 23.84万
• 依托单位: University of Birmingham
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FY015029%2F1
• 关键词: Dualities; via; ComPactness; Openness; their

This project is positioned inside the study of Stone-like dualities and touches on themes in algebraic logic, domain theory andpointfree topology.The interplay between compact sets and open sets is a key tool in topology, and various conditions and constructions (such as localcompactness, the compact-open topology on function spaces, well-filtered spaces) hint at a symmetrical role between these twonotions. Indeed, a precise symmetry occurs in the spaces used by M.H. Stone to represent Boolean algebras (1936) and distributivelattices (1938). However, in some important generalisations of Stone dualities (such as the duality between sober spaces and spatialframes), the symmetry between compactness and openness is only partial.This project aims at demonstrating that the aforementioned partial symmetry is in fact a slice of a formal, complete symmetry in alarger framework. This framework will provide new explanations for the occurrence of certain conditions and constructions (such aslocal compactness, the compact-open topology, sobriety), will suggest new results (such as new dualities), and will allow goodstructural properties (such as Cartesian closure).

该项目定位在Stone-like对偶的研究中，涉及代数逻辑、Domain理论和点自由拓扑的主题。紧集和开集之间的相互作用是拓扑学中的一个关键工具，各种条件和构造（如局部紧性、函数空间上的紧开拓扑、良好过滤空间）暗示了这两个概念之间的对称作用。事实上，在M. H.斯通表示布尔代数（1936年）和distributivelattices（1938年）。然而，在Stone对偶的一些重要推广中（如清醒空间和空间框架之间的对偶），紧性和开放性之间的对称只是部分的，本项目旨在证明上述部分对称实际上是一个更大框架中形式的完全对称的一部分。这个框架将为某些条件和结构的出现提供新的解释（如局部紧性，紧开拓扑，清醒），将提出新的结果（如新的对偶），并将允许良好的结构性质（如笛卡尔闭包）。