Borel resolvability of compact spaces and their subspaces
Let \((X,\tau)\) be a topological space and \(B\) a family of its subsets. \(X\) is said to be \(B\)-resolvable if it has two disjoint dense sets belonging to \(B\). \(X\) is maximally \(B\)-resolvable if it has \(\Delta(\tau)\) disjoint dense sets in \(B\) where \(\Delta(\tau)=\min\{|V|:\emptyset\neq V\in\tau\}\). This is a generalization of resolvability introduced by Hewitt in 1943. Some work on Borel resolvability was done by \textit{J. G. Ceder} [Rev. Roum. Math. Pures Appl. 11, 89-94 (1966; Zbl 0158.41501)] and by \textit{R. Jiménez} and \textit{V. I. Malykhin} [Commentat. Math. Univ. Carolin. 39, No. 2, 379-387 (1998; Zbl 0936.54016), see the review above)]. In the paper under review the author deals with Baire and Borel resolvability and proves that these two notions coincide for Baire spaces. Moreover, the statement ``all perfectly normal compact spaces are Borel resolvable'' is independent of ZFC. He continues in the study of Borel resolvability of dyadic compacta and \(G_\delta\) subsets of Tikhonov cubes. At last he obtains several results on resolvability of \(A\)-, \(CA\)-, and \(PCA\)-subsets of compact spaces.
scientific article; zbMATH DE number 1407579