On two classes of sets containing all Baire sets and all co-analytic sets
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.
scientific article; zbMATH DE number 4131248
scientific article