A direct approach to K-reflections of T0 spaces

Abstract

In this paper, we provide a direct approach to K-reflections of T0 spaces. For a full subcategory K of the category of all T0 spaces and a T0 space X, let K(X)={A⊆X:A is closed and for any continuous mapping f:X⟶Y to a K-space Y, there exists a unique yA∈Y such that f(A)‾={yA}‾} and PH(K(X)) the space of K(X) endowed with the lower Vietoris topology. It is proved that if PH(K(X)) is a K-space, then the pair 〈Xk=PH(K(X)),ηX〉, where ηX:X⟶Xk, x↦{x}‾, is the K-reflection of X. We call K an adequate category if for any T0 space X, PH(K(X)) is a K-space. Therefore, if K is adequate, then K is reflective in Top0. It is shown that the category of all sober spaces, that of all d-spaces, that of all well-filtered spaces and the Keimel and Lawson's category are all adequate, and hence are all reflective in Top0. Some major properties of K-spaces and K-reflections of T0 spaces are investigated. In particular, it is proved that if K is adequate, then the K-reflection preserves finite products of T0 spaces. Our study also leads to a number of problems, whose answering will deepen our understanding of the related spaces and their categorical structures.