The R-completion of closure spaces

Abstract

In this paper, we consider a full subcategory R of the category CL0 of T0 closure spaces satisfying certain conditions. We give a construction for the universal R-completion of T0 closure spaces, which is a categorical reflection of the category CL0 onto the full subcategory R. The category CSZ of Z-convergence spaces is such a category R, where Z is a subset system on the category CL0. Hence the Z-completion is a special case of the R-completion. Specifically, the subset system Z is no longer required to be hereditary. Conversely, for every such category R, there is a subset system Z such that R = CSZ. Thus the R-completion and the Z-completion are in fact of the same level. In the case that Z is coarser than the subset system I of irreducible sets, the Z-completion can be restricted to the setting of topological spaces. The sobrification, the bounded sobrification, the D-completion, the conditional D-completion, the well filterification, the K-completion of topological spaces are shown to be special cases of the Z-completion.