The \(R\)-completion of closure spaces
Given a full subcategory \(\mathbf{R}\) of the category \(\mathbf{CL_0}\) of \(T_0\)-closure spaces and continuous maps, satisfying four necessary conditions, the authors construct an \(R\)-completion for closure spaces. This completion is a reflection into the category \(\mathbf{R}\). This general construction is applicable to several known situations. It encompasses previous work of some of the authors about \(Z\)-completions, where \(Z\) is a system of subsets on the category \(\mathbf{CL_0},\) [\textit{Z. Zhang} et al., Soft Comput. 23, No. 21, 10699--10708 (2019; Zbl 1436.54004)]. The authors show that every category of \(Z\)-convergence spaces is a category \(\mathbf{R}\) satisfying the four conditions. In this way the subset system \(Z\) of the \(Z\)-completion is not restricted to hereditary cases. Conversely, given a category \(\mathbf{R}\), there exists a subset system such that the category of \(Z\)-convergence spaces is exactly the category \(\mathbf{R}\) and the \(R\)-completion coincides with the \(Z\)-completion. Furthermore the authors discuss hereditary \(Z\)-subset systems in their new context and apply their framework to the \(K\)-completion of topological spaces in the sense of [\textit{K. Keimel} and \textit{J. D. Lawson}, Ann. Pure Appl. Logic 159, No. 3, 292--306 (2009; Zbl 1172.54016)]. Particular cases such as the (bounded) sobrification also fit into the new construction.
scientific article; zbMATH DE number 7434149
scientific article