Remarks on the space \(\aleph_{1}\) in \(\mathbf {ZF}\)
type: Item
instance of: scholarly article
title: Remarks on the space \(\aleph_{1}\) in \(\mathbf {ZF}\)
zbMATH Open document ID: 1211.54038
author: Horst Herrlich
author: Kyriakos Keremedis
published in: Topology and its Applications
publication date: 2011-01-13T00:00:00Z
review text: A space \(X\) is said to be effectively normal if there exists a function \(F\) associating to every pair \(A, B\) of disjoint closed sets in \(X\) a pair \(C, D\) of disjoint open sets including \(A, B\) respectively. The authors prove in this paper that \(\aleph_{1}\) with the order topology is effectively normal. It is also proved that for every countable ordinal \(\alpha\), the ordered space \(\alpha\) is metrizable and so every closed subset of \(\alpha\) is a zero set and therefore the Čech-Stone extension of \(\alpha\) coincides with its Wallman extension. The Wallman extension for a space \(X\) is defined as follows: Let \(C\) be a \(T_1\) base for \(X\). \(\mathcal W(X,\mathcal C)\) denotes the set of all maximal \(C\)-filters \(\mathcal F\). The topology on \(\mathcal W(X,\mathcal C)\) having as a base for the closed sets the family \(\{A^{*}|A\in \mathcal C\}\) where \(A^{*}=\{\mathcal F \in \mathcal W(X,C):A\in \mathcal F\}\) is called the Wallman topology on \(\mathcal W(X,\mathcal C)\). The set \(\mathcal W(X,\mathcal C)\) with the Wallman topology is called the Wallman extension of \(X\) corresponding to \(\mathcal C\). If \(\mathcal C\) is the set \(\mathcal Z\) of all zero sets of \(X\), the Wallman extension is the Čech-Stone extension \(\beta X\) of \(X\). The authors prove that in the Feferman-Levy model where \(\aleph_{1}\) is singular, the ordinal space \(\aleph_{1}\) is base-Lindelöf but not Lindelöf. It is proved that \(\beta \aleph_{1}\) is compact iff its Wallman extension \(\mathcal W\aleph_1\) is compact. It is a nice result proved by the authors in this paper that the set \(L\) of all limit ordinals of \(\aleph_{1}\) is closed but not a zero set.
reviewed by: T. Thrivikraman
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 54D15
zbMATH DE Number: 5835750
zbMATH Keywords: Axiom of Choice
zbMATH Keywords: effectively normal spaces
zbMATH Keywords: compact spaces
zbMATH Keywords: Lindelöf spaces
zbMATH Keywords: ordinal spaces
zbMATH Keywords: Čech-Stone extensions
zbMATH Keywords: Wallman extensions
zbMATH Keywords: compactifications
MaRDI profile type: Publication
full work available at URL: j.topol.2010.11.007
OpenAlex ID: W2037219220
cites work: Q3994935
cites work: Q3270120
cites work: Continuing horrors of topology without choice
cites work: Axiom of choice
cites work: Q4955904
cites work: Q4210469
cites work: Q3852172
cites work: Wallman compactifications and Tychonoff's compactness theorem in ZF
cites work: Q3732526
cites work: Q5605131
DOI: 10.1016/J.TOPOL.2010.11.007
description: scientific article; zbMATH DE number 5835750