On $T_0$ spaces determined by well-filtered spaces

TL;DR

This paper introduces new classes of subsets and spaces in $T_0$ topology, providing characterizations of well-filtered and sober spaces, and proving that products of well-filtered spaces remain well-filtered, thus advancing the understanding of these topological structures.

Contribution

The paper defines Rudin and $ ext{wdd}$ sets and spaces, establishing new characterizations of well-filtered and sober spaces, and proves that products of well-filtered spaces are well-filtered.

Findings

Every locally compact $T_0$ space is a Rudin space.

Every core compact $T_0$ space is a $ ext{wdd}$ space.

Every core compact well-filtered space is sober.

Abstract

We first introduce and study two new classes of subsets in $T_0$ 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 - $\mathsf{DC}$ 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 $T_0$ space $X$, it is proved that $X$ is sober if{}f $X$ is a well-filtered Rudin space if{}f $X$ is a well-filtered $\mathsf{WD}$ space. We also prove that every locally compact $T_0$ space is a Rudin space, and every core compact $T_0$ space is a $\wdd$ 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 $T_0$ 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 answering will deepen our understanding of the related spaces and structures.