Some remarks on extraresolvable spaces
type: Item
instance of: scholarly article
title: Some remarks on extraresolvable spaces
zbMATH Open document ID: 1016.54001
DOI: 10.1016/S0166-8641(01)00142-0
author: R. A. González-Silva
author: Salvador Garcia-Ferreira
published in: Topology and its Applications
publication date: 2002-08-22T00:00:00Z
review text: A space \(X\) is said to be extraresolvable if there is a family \(\mathcal D\) of dense subsets of \(X\) such that \(|\mathcal D|>\Delta(X)\) and the intersection of any two distinct members of \(\mathcal D\) is nowhere dense, where \(\Delta(X)=\min\{|U|:U\) is a nonempty open subset of \(X\}\). A space \(X\) is said to be strongly extraresolvable if there is a family \(\mathcal D\) of dense subsets of \(X\) such that \(|\mathcal D|>\Delta(X)\) and the intersection of any two distinct members of \(\mathcal D\) has size \(<\text{nwd}(X)\), where \(\text{nwd}(X)=\min\{|A|:A\subseteq X\) and \(\text{int}_X(\text{cl}_XA)\neq \emptyset\}\). The authors give an example of a countable extraresolvable space that is not strongly extraresolvable. They find some sufficient conditions and some necessary conditions for spaces of the form \(X\times\alpha\) to be strongly extraresolvable. In particular, they show that \(\mathbb Q\times\omega_1\) is strongly extraresolvable and if \(X\) is strongly extraresolvable and \(\text{nwd}(X)=\Delta(X)=\omega\), then \(X\times\omega\) is strongly extraresolvable.
reviewed by: Miroslav Repický
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54G20
zbMATH DE Number: 1787556
zbMATH Keywords: extraresolvable space
zbMATH Keywords: strongly extraresolvable space
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1787556