Dowker spaces revisited
type: Item
instance of: scholarly article
title: Dowker spaces revisited
zbMATH Open document ID: 1203.54015
author: Lew Ludwig
author: Peter Nyikos
author: John E. Porter
published in: Tsukuba Journal of Mathematics
publication date: 2010-09-28T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tkbjm/1283967404
review text: It is a classical result of \textit{C. H. Dowker} [Can. J. Math. 3, 219--224 (1951; Zbl 0042.41007)] that a \(T_1\)-space \(X\) is countably paracompact and normal if and only if \(X\times[0,1]\) (or, equivalently, \(X\times(\omega+ 1)\)) is normal. The main results of this paper are (somewhat technical) analogs of Dowker's result for \(\beta\)-normality and for \(\alpha\)-normality, where a topological space \(X\) is called \(\beta\)-normal (\(\alpha\)-normal) if for every pair \(A\), \(B\) of disjoint closed subsets of \(X\) there exist open subsets \(U\) and \(V\) of \(X\) such that \(A\cap U\) is dense in \(A,B\cap V\) is dense in \(B\), and \(\text{cl\,}U\cap\text{cl\,}V= \emptyset\), (\(U\cap V=\emptyset\), respectively) (see \textit{A. V. Arhangel'skii} and \textit{L. Ludwig} [Commentat. Math. Univ. Carol. 42, No.~3, 507--519 (2001; Zbl 1053.54030)]). The authors call a topological space \(X\) \(\alpha\)-countably paracompact if for every decreasing sequence \((A_n)_{n<\omega}\) of closed subsets of \(X\) satisfying \(\bigcap_{n<\omega} A_n=\emptyset\) there exists a sequence \((U_n)_{n<\omega}\) of open subsets of \(X\) such that \(U_n\cap A_n\) is dense in \(A_n\) for each \(n<\omega\) and \(\bigcap_{n< \omega}\text{cl\,} U_n=\emptyset\). It follows from their main results that if \(X\times(\omega+ 1)\) is \(\beta\)-normal, then \(X\) is \(\beta\)-normal and \(\alpha\)-countably paracompact. Whether the converse is true is an open problem. Additionally, it is shown that there exists a completely regular \(T_1\)-space that is first countable, collectionwise Hausdorff, \(\alpha\)-countably paracompact, and \(\alpha\)-normal but not \(\beta\)-normal. Moreover, there exists a Moore space that is collectionwise Hausdorff and \(\alpha\)-normal but not \(\beta\)-normal.
Mathematics Subject Classification ID: 54D15
zbMATH DE Number: 5792241
zbMATH Keywords: \(\alpha\)-normal space
zbMATH Keywords: \(\beta\)-normal space
zbMATH Keywords: \(\alpha\)-countably paracompact space
zbMATH Keywords: collectionwise Hausdorff space
zbMATH Keywords: Moore space
reviewed by: Harald Brandenburg
MaRDI profile type: Publication
OpenAlex ID: W1596620261
DOI: 10.21099/TKBJM/1283967404
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description: scientific article; zbMATH DE number 5792241