On T_0 spaces determined by well-filtered spaces

摘要

We first introduce and study two new classes of subsets in T-0 spaces - Rudin sets and WD 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 WD spaces. Rudin spaces lie between WD spaces and DC spaces, while DC spaces lie between Rudin spaces and sober spaces. Using Rudin sets and WD sets, we formulate and prove a number of new characterizations of well-filtered spaces and sober spaces. For a T-0 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 T-0 space is a Rudin space, and every core compact T-0 space is a WD space. One immediate corollary is that every core compact well-filtered space is sober, giving a positive answer to Jia-Jung problem. Using WD sets, we present a more direct construction of the well-filtered reflections of T-0 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 answers will deepen our understanding of the related spaces and structures. (C) 2020 The Author(s). Published by Elsevier B.V.