The reflectivity of some categories of T0 spaces in domain theory

TL;DR

This paper investigates the conditions under which categories of T0 spaces are reflective in domain theory, establishing their necessity and sufficiency, and applies these results to clarify which categories are not reflective.

Contribution

It proves that Keimel and Lawson's conditions are both necessary and sufficient for reflectivity of T0 space categories, resolving open problems in domain theory.

Findings

Keimel and Lawson's conditions are necessary and sufficient for reflectivity.

Several proposed categories in domain theory are not reflective.

The paper clarifies the structure of reflective categories in T0 spaces.

Abstract

Keimel and Lawson proposed a set of conditions for proving a category of topological spaces to be reflective in the category of all T0 spaces. These conditions were recently used to prove the reflectivity of the category of all well-filtered spaces. In this paper, we prove that, in certain sense, these conditions are not just sufficient but also necessary for a category of T0 spaces to be reflective. Using this general result, we easily deduce that several categories proposed in domain theory are not reflective, thus answered a few open problems.