First countability, $\omega$-well-filtered spaces and reflections

We first introduce and study two new classes of subsets in $T_0$ spaces - $\omega$-Rudin sets and $\omega$-well-filtered 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 $\omega$-well-filtered spaces. We prove that an $\omega$-well-filtered $T_0$ space is locally compact iff it is core compact. One immediate corollary is that every core compact well-filtered space is sober, answering Jia-Jung problem with a new method. We also prove that all irreducible closed subsets in a first countable $\omega$-well-filtered $T_0$ space are directed. Therefore, a first countable $T_0$ space $X$ is sober iff $X$ is well-filtered iff $X$ is an $\omega$-well-filtered $d$-space. Using $\omega$-well-filtered determined sets, we present a direct construction of the $\omega$-well-filtered reflections of $T_0$ spaces, and show that products of $\omega$-well-filtered spaces are $\omega$-well-filtered.