On two classes of sets containing all Baire sets and all co-analytic sets
type: Item
instance of: scholarly article
title: On two classes of sets containing all Baire sets and all co-analytic sets
zbMATH Open document ID: 0691.54023
DOI: 10.1016/0166-8641(89)90106-5
published in: Topology and its Applications
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 54H05
Mathematics Subject Classification ID: 54D15
Mathematics Subject Classification ID: 54E99
Mathematics Subject Classification ID: 54C50
zbMATH DE Number: 4131248
zbMATH Keywords: Baire sets, perfect maps
zbMATH Keywords: S-perfect spaces
zbMATH Keywords: R-perfect spaces
author: Z. Balogh
author: Harold Bennett
MaRDI profile type: Publication
full work available at URL: 0166-8641(89)90106-5
OpenAlex ID: W2009060894
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description: scientific article; zbMATH DE number 4131248