Dualities via ComPactness, Openness, and their Symmetry

这个项目定位于类Stone对偶的研究，涉及代数逻辑、整环理论和无点拓扑学的主题。紧集和开集之间的相互作用是拓扑学中的一个关键工具，各种条件和结构(如局部紧性、函数空间上的紧开拓扑、良好过滤空间)暗示了这两个集合之间的对称作用。事实上，M.H.斯通用来表示布尔代数(1936)和分配格(1938)的空间具有精确的对称性。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)，紧性和开放之间的对称性只是部分的。本项目旨在证明上述部分对称性实际上是更大框架下形式上的完全对称性的一部分。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(如新的对偶性)，并将允许良好的结构性质(如笛卡尔闭包)。

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).