$\omega$-Rudin spaces, well-filtered determined spaces and first-countable spaces

TL;DR

This paper explores the relationships between various countability properties and classes of $T_0$ spaces, establishing conditions under which spaces are sober, well-filtered, or Rudin, with a focus on first-countability and sobrification.

Contribution

It provides new characterizations of $ ext{ω}$-Rudin and well-filtered spaces based on countability and sobrification properties, advancing the understanding of their structure.

Findings

First-countable sobrification implies the space is $ ext{ω}$-Rudin.

$ ext{ω}$-well-filtered spaces are sober if their sobrification is first-countable.

Second-countable or countable first-countable spaces are $ ext{ω}$-Rudin.

Abstract

We investigate some versions of $d$-space, well-filtered space and Rudin space concerning various countability properties. The main results include: (i) if the sobrification of a $T_0$ space $X$ is first-countable, then $X$ is an $\omega$-Rudin space; (ii) every $\omega$-well-filtered space is sober if its sobrification is first-countable; (iii) if a $T_0$ space is second-countable or first-countable and with a countable underlying set, then it is a $\omega$-Rudin space; (iv) every first-countable $T_0$ space is well-filtered determined; (v) every irreducible closed subset in a first-countable $\omega$-well-filtered space is countably-directed; (vi) every first-countable $\omega$-well-filtered $\omega^\ast$-$d$-space is sober.