On c-sober spaces and ω*-well-filtered spaces

Based on countably irreducible version of Topological Rudin?s Lemma, we give some characterizations of c-sober spaces and ?*-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 ?*-well-filtered space. We also show that a locally compact ?+-well-filtered P-space is c-sober and a T0 space X is c-sober iff the one-point compactification of X is c-sober.