Borel resolvability of compact spaces and their subspaces
type: Item
instance of: scholarly article
title: Borel resolvability of compact spaces and their subspaces
zbMATH Open document ID: 0936.54039
DOI: 10.1007/BF02316285
author: Viatcheslav I. Malykhin
published in: Mathematical Notes
publication date: 2000-05-21T00:00:00Z
review text: 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.
reviewed by: Miroslav Repický
Mathematics Subject Classification ID: 54H05
Mathematics Subject Classification ID: 54B05
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E50
zbMATH DE Number: 1407579
zbMATH Keywords: \(F_\sigma\)-sets in Tikhonov cubes
zbMATH Keywords: \(CPCA\)-sets in compact spaces
zbMATH Keywords: dyadic compactum
zbMATH Keywords: perfectly normal compact space
zbMATH Keywords: dispersion character
zbMATH Keywords: \(A\)-sets in Tikhonov cubes
zbMATH Keywords: \(G_\delta\)-sets in Tikhonov cubes
zbMATH Keywords: Baire space
MaRDI profile type: Publication
cites work: A problem of set-theoretic topology
cites work: Q5788022
cites work: Q3311274
cites work: Resolvability: A selective survey and some new results
cites work: Groups with Only Resolvable Group Topologies
cites work: Q5541852
cites work: Q4955961
cites work: Q5686040
cites work: Q5547266
cites work: Maximal resolvability of bounded groups
cites work: Resolution property in generalized k-spaces
description: scientific article; zbMATH DE number 1407579