The product of two ordinals is hereditarily dually discrete
type: Item
instance of: scholarly article
title: The product of two ordinals is hereditarily dually discrete
zbMATH Open document ID: 1234.54034
author: Liang-Xue Peng
published in: Topology and its Applications
publication date: 2011-12-06T00:00:00Z
review text: A topological space \((X,\tau)\) is called dually discrete (respectively, a \(D\)-space) if every mapping \(U: X\to\tau\) with \(x\in U(x)\) for each \(x\in X\) has a discrete (respectively, closed discrete) kernel, where a subset \(A\) of \(X\) is a kernel of \(U\) if \(\bigcup\{U(x)\mid x\in A\}= X\). \(D\)-spaces were introduced by \textit{E. K. van Douwen} in a joint paper with \textit{W. F. Pfeffer} [Pac. J. Math. 81, 371--377 (1979; Zbl 0409.54011)]. A recent survey on \(D\)-spaces by \textit{G. Gruenhage} can be found in [A survey of \(D\)-spaces. Babinkostova, L. (ed.) et al., Set theory and its applications. Annual Boise extravaganza in set theory, Boise, ID, USA, 1995--2010. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 533, 13--28 (2011; Zbl 1217.54025)]. Dually discrete spaces were introduced by \textit{J. van Mill}, \textit{V. V. Tkachuk} and \textit{R. G. Wilson} in [Topology Appl. 154, No. 10, 2127--2134 (2007; Zbl 1131.54022)] and subsequently studied by \textit{O. T. Alas}, \textit{R. Z. Buzyakova}, \textit{L. R. Junqueira}, \textit{V. V. Tkachuk}, \textit{R. G. Wilson} and the present author. In [ibid. 155, No. 13, 1420--1425 (2008; Zbl 1169.54010)], \textit{O. T. Alas}, \textit{L. R. Junqueira} and \textit{R. G. Wilson} have shown that every ordinal with its natural order topology is hereditarily dually discrete. As an answer to one of their questions it is shown here that, more generally, every product of two ordinals is hereditarily dually discrete. It follows that, if \(\mu\), \(\nu\) are ordinals and \(X\) is a subspace of \(\mu\times\nu\), then the following are equivalent: (i) \(X\) has countable spread; (ii) \(X\) is perfectly normal and has countable extent; and (iii) \(X\) is hereditarily a Lindelöf \(D\)-space.
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54G99
Mathematics Subject Classification ID: 54D15
Mathematics Subject Classification ID: 54F05
zbMATH DE Number: 5986567
zbMATH Keywords: dually discrete space
zbMATH Keywords: \(D\)-space
zbMATH Keywords: spread
zbMATH Keywords: extent
zbMATH Keywords: perfectly normal space
zbMATH Keywords: Lindelöf space
reviewed by: Harald Brandenburg
MaRDI profile type: Publication
full work available at URL: j.topol.2011.09.036
OpenAlex ID: W2032438987
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DOI: 10.1016/J.TOPOL.2011.09.036
description: scientific article; zbMATH DE number 5986567