A unified method for completions of posets and closure spaces

In this paper, we introduce the notion of a subset system on the category CL$$_{0}$$0 of all $$T_{0}$$T0 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 $$T_{0}$$T0 closure space X to be a Z-convergence space $$X_{Z}$$XZ together with a continuous function from X to $$X_{Z}$$XZ satisfying the universal property. In the case that Z is a hereditary subset system, we prove that: (1) for each $$T_{0}$$T0 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$$_{Z}$$Z of all Z-convergence spaces is reflective in the category CL$$_{0}$$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.