A unified method for completions of posets and closure spaces

Abstract

In this paper, we introduce the notion of a subset system on the category CL$_{0}$ of all $T_{0}$ closure spaces. And for each subset system Z, the concept of a Z-convergence space arises as a generalization of sober spaces and monotone convergence spaces. We define a Z-completion of a $T_{0}$ closure space X to be a Z-convergence space $X_{Z}$ together with a continuous function from X to $X_{Z}$ satisfying the universal property. In the case that Z is a hereditary subset system, we prove that: (1) for each $T_{0}$ closure space X, the set of all Z-tapered closed subsets of X endowed with the corresponding closure system is a Z-completion of X; (2) the category CS$_{Z}$ of all Z-convergence spaces is reflective in the category CL$_{0}$. The Dedekind–MacNeille completion, the Alexandroff completion, the Frink ideal completion, the ideal completion, the Z-completion for posets, the D-completion, the sobrification, the directed completion, etc., are special cases of the Z-completion.