On $c$-sober spaces and $\omega^*$-well-filtered spaces

Abstract

Based on countably irreducible version of Topological Rudin's Lemma, we give some characterizations of $c$-sober spaces and $\omega^*$-well-filtered spaces. In particular, we prove that a topological space is $c$-sober iff its Smyth power space is $c$-sober and a $c$-sober space is an $\omega^*$-well-filtered space. We also show that a locally compact $\omega^*$-well-filtered $P$-space is $c$-sober and a $T_0$ space $X$ is $c$-sober iff the one-point compactification of $X$ is $c$-sober.