First countability, $\omega$wellfiltered spaces and reflections
Abstract
We first introduce and study two new classes of subsets in $T_0$ spaces  $\omega$Rudin sets and $\omega$wellfiltered determined sets lying between the class of all closures of countable directed subsets and that of irreducible closed subsets, and two new types of spaces  $\omega$$d$ spaces and $\omega$wellfiltered spaces. We prove that an $\omega$wellfiltered $T_0$ space is locally compact iff it is core compact. One immediate corollary is that every core compact wellfiltered space is sober, answering JiaJung problem with a new method. We also prove that all irreducible closed subsets in a first countable $\omega$wellfiltered $T_0$ space are directed. Therefore, a first countable $T_0$ space $X$ is sober iff $X$ is wellfiltered iff $X$ is an $\omega$wellfiltered $d$space. Using $\omega$wellfiltered determined sets, we present a direct construction of the $\omega$wellfiltered reflections of $T_0$ spaces, and show that products of $\omega$wellfiltered spaces are $\omega$wellfiltered.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.13201
 Bibcode:
 2019arXiv191113201X
 Keywords:

 Mathematics  General Topology
 EPrint:
 17 pages. arXiv admin note: substantial text overlap with arXiv:1909.09303 and arXiv:1911.11617