Retracts of the closure space of filters in the lattice of all subsets
The paper concerns some subcategories \((S_{\alpha,\delta}\) for \(\alpha\), \(\delta\) being some cardinals) of the category S of closure spaces and continuous maps (introduced and studied earlier by the author). A pair (X,\({\mathcal F})\) is a closure space whenever \({\mathcal F}\) is a family of subsets of X such that \(X\in {\mathcal F}\), \(\emptyset \in {\mathcal F}\), and \({\mathcal F}\) is closed under arbitrary intersections. Clearly, for closure spaces (X,\({\mathcal F})\) and (X',\({\mathcal F}')\) a function \(f: X\to X'\) is said to be continuous if \(f^{-1}(A)\in {\mathcal F}\) for every \(A\in {\mathcal F}'\) (the author does not recall this definition). The author studies the notions of absolute retract and of absolute extensor for arbitrary subcategory M of S (they are natural generalizations of the corresponding notions well known for some categories of topological spaces). One of the numerous results is contained in Theorem 2.1. Using the notation AR(M) and AE(M) respectively for the class of absolute retracts and the class of absolute extensors in M, one can formulate this result as follows: \(AR(S_{\alpha,\delta})=AE(S_{\alpha,\delta})\). Most of the remaining results concern the lattice theory. Clearly, in the category of topological spaces every absolute extensor is an injective object. In Remark 2.1 the author shows that the analogous statement is false for the category S. The proof is based on Lemma 2.1, thus this lemma should precede Remark 2.1. Using the same characters for a structure and its universe \((X=(X,{\mathcal F}))\) is confusing. So is the numbering of statements.
scientific article; zbMATH DE number 4009293