Non-reflective categories of some kinds of weakly sober spaces

Abstract

Ern\'e weakened the concept of sobriety in order to extend the theory of sober spaces and locally hypercompact spaces to situations where directed joins were missing, and introduced and discussed three kinds of non-sober spaces: cut spaces, weakly sober spaces, and quasisober spaces. Three other kinds of non-sober spaces, namely $\mathsf{DC}$ space, $\mathsf{RD}$ space and $\mathsf{WD}$ space, were introduced and investigated by Xu, Shen, Xi and Zhao. All these six kinds of spaces are strictly weaker than sober spaces. In this paper, it is shown that none of the category of all $\mathsf{DC}$ spaces, that of all $\mathsf{RD}$ spaces, that of all $\mathsf{WD}$ spaces, that of all quasisober spaces, that of all weakly spaces and that of all cut spaces is reflective in the category of all $T_0$ spaces with continuous mappings.