The reflectivity of some categories of T0 spaces in domain theory
Chong Shen, Xiaoyong Xi, Dongsheng Zhao
TL;DR
The paper addresses when a subcategory of $Top_0$ is reflective by proving a necessary-and-sufficient criterion: if $K\subseteq Top_0$ with $K\nsubseteq Top_1$ satisfies (K2), then $K$ is reflective in $Top_0$ iff it satisfies (K1)--(K4); equivalently, $K$ is productive and $b$-closed-hereditary iff $K$ is productive and has equalizers. This framework also ensures that every reflection is a $b$-dense embedding. Applying the result to domain-theory subcategories, the authors show that Co-Sob, strong $d$-spaces (SD), $k$-bounded sober spaces (KSob), and open well-filtered spaces (OWF) are not reflective in $Top_0$, resolving several open problems. The work provides a practical criterion to disprove reflectivity and clarifies the role of the $b$-topology in understanding sober completions and categorical reflectivity within ${f Top_0}$. These insights have potential implications for identifying and analyzing other non-reflective subcategories in domain theory and related areas.
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.