Non-reflective categories of some kinds of weakly sober spaces

TL;DR

This paper investigates various non-sober space categories, demonstrating that none of these categories are reflective within the broader category of all $T_0$ spaces with continuous maps.

Contribution

It proves that six specific categories of non-sober spaces are not reflective in the category of all $T_0$ spaces, extending the understanding of their categorical properties.

Findings

None of the six non-sober space categories are reflective.

The categories include $ extsf{DC}$, $ extsf{RD}$, $ extsf{WD}$, quasisober, weakly sober, and cut spaces.

This result clarifies the categorical structure of these non-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.