THE REFLECTIVITY OF SOME CATEGORIES OF T0 SPACES IN DOMAIN THEORY

Abstract

Keimel and Lawson (2009) proposed a set of conditions for proving the reflectivity of a category of topological spaces in the category of all $T_0$ spaces. Recently, these conditions were used to prove the reflectivity of the category of all well-filtered spaces. We prove that, in certain sense, these conditions are not only sufficient but also necessary for a category of $T_0$ spaces to be reflective. By applying this general result, we can easily deduce that several categories proposed in domain theory are not reflective, thereby answering a few open problems.