$\omega$-well-filtered spaces, revisited

TL;DR

This paper characterizes $$-well-filtered spaces in topology, providing a new topological criterion, refining the concept of $$-well-filterification, and linking sobriety to specific convergence properties in second countable spaces.

Contribution

It offers a novel topological characterization of $$-well-filtered spaces and their filterification, and addresses a problem posed by Xiaoquan Xu.

Findings

Characterization of $$-well-filtered spaces via exclusion of certain topologies.

Refined topological description of $$-well-filterification.

Sobriety characterized by convergence of $$-Cauchy subsets in second countable spaces.

Abstract

We prove that a $T_0$ topological space is $\omega$-well-filtered if and only if it does not admit either the natural numbers with the cofinite topology or with the Scott topology as its closed subsets in the strong topology. Based on this, we offer a refined topological characterization for the $\omega$-well-filterification of $T_0$-spaces and solve a problem posed by Xiaoquan Xu. In the setting of second countable spaces, we also characterise sobriety by convergences of certain $\Pi^0_2$-Cauchy subsets of the spaces.