Non-reflective categories of some kinds of weakly sober spaces (Q2140634)

The authors show that none of the categories of directed closure spaces, Rudin spaces, well-filtered determined spaces, quasisober spaces, weakly sober spaces and cut spaces is reflective in the category of \(T_0\) spaces with continuous mappings. Definitions of these relaxed concepts of sobriety and relations between them are recalled from the literature. The main results are achieved by applying appropriate non-\textbf{K} spaces to theorems designed to exclude \textbf{K}-reflections for the categories in question.