2019/09/20 by Xiaoquan Xu, Chong Shen, Xu, Xiaoquan +5 · 1 citation
Mathematics · Computer Science · #Advanced Topology and Set Theory #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1909.09303
We first introduce and study two new classes of subsets in T0 spaces - Rudin sets and \wdd sets lying between the class of all closures of directed subsets and that of irreducible closed subsets. Using such subsets, we define three new types of topological spaces - DC spaces, Rudin spaces and \wdd spaces. The class of Rudin spaces lie between the class of \wdd spaces and that of \dc spaces, while the class of \dc spaces lies between the class of Rudin spaces and that of sober spaces. Using Rudin sets and \wdd sets, we formulate and prove a number of new characterizations of well-filtered spaces and sober spaces. For a T0 space X, it is proved that X is sober iff X is a well-filtered Rudin space iff X is a well-filtered WD space. We also prove that every locally compact T0 space is a Rudin space, and every core compact T0 space is a \wdd space. One immediate corollary is that every core compact well-filtered space is sober, giving a positive answer to Jia-Jung problem. Using \wdd sets, we present a more directed construction of the well-filtered reflections of T0 spaces, and prove that the products of any collection of well-filtered spaces are well-filtered. Our study also leads to a number of problems, whose answering will deepen our understanding of the related spaces and structures.