On $T_0$ spaces determined by wellfiltered spaces
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 wellfiltered spaces and sober spaces. For a $T_0$ space $X$, it is proved that $X$ is sober if{}f $X$ is a wellfiltered Rudin space if{}f $X$ is a wellfiltered $\mathsf{WD}$ 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 wellfiltered space is sober, giving a positive answer to JiaJung problem. Using $\wdd$ sets, we present a more directed construction of the wellfiltered reflections of $T_0$ spaces, and prove that the products of any collection of wellfiltered spaces are wellfiltered. Our study also leads to a number of problems, whose answering will deepen our understanding of the related spaces and structures.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.09303
 Bibcode:
 2019arXiv190909303X
 Keywords:

 Mathematics  General Topology;
 06B35;
 06F30;
 54B99;
 54D30