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On two classes of sets containing all Baire sets and all co-analytic sets. (English) Zbl 0691.54023

A subset A of a topological space X is said to be an \(S_{\delta}\) (respectively an \(R_{\delta})\) set in X if there is a countable family \({\mathcal S}\) of open subsets of X separating (resp. strongly separating) the points of A from the points of \(X\setminus A\) i.e. for every \(x\in A\) and \(y\not\in A\) there is a \(G\in {\mathcal S}\) with \(x\in G\), \(y\not\in G\) (resp. \(x\in G\), \(y\not\in clG\)- the closure of G). The problem is raised to characterize all metrizable spaces each subset of which is an \(S_{\delta}\) set. Some necessary conditions are given. A topological space X is said to be S-perfect (resp. R-perfect) if every closed subset of X is an \(S_{\delta}\) (resp. \(R_{\delta}\)-set in X). The example of a locally comact, locally countable, quasi-developable \(T_ 2\) space of cardinality \(2^{\omega}\) which is not S-perfect is presented. The images of S-perfect and R-perfect spaces under various types of mappings are investigated.
Reviewer: B.Aniszczyk

MSC:

54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54E99 Topological spaces with richer structures
54C50 Topology of special sets defined by functions
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