The \(D\)-property of some Lindelöf spaces and related conclusions
type: Item
instance of: scholarly article
title: The \(D\)-property of some Lindelöf spaces and related conclusions
zbMATH Open document ID: 1110.54014
author: Liang-Xue Peng
published in: Topology and its Applications
publication date: 2007-02-09T00:00:00Z
review text: All spaces are \(T_1\)-spaces. A space \(X\) is called a \(D\)-space if for any neighborhood assignment \(\Phi\) for \(X\) there exists a closed discrete subset \(D\) of \(X\) such that \(X=\bigcup \{\phi(d):d\in D\}.\) A generalization of this concept is the notion of an \(\alpha D\)-space due to Arhangel'skiĭ and Buzyakova. One of the central questions on \(D\)-spaces posed by van Douwen is whether every Lindelöf space is a \(D\)-space. In the present article the author shows that the Lindelöf space \({\mathcal C}_p(\tau_\omega)\) is a \(D\)-space for any ordinal number \(\tau,\) where \(\tau_\omega=\{\alpha\leq \tau: \text{cf}(\alpha)\leq \omega\}.\) Furthermore a Lindelöf space of countable pseudocharacter due to Alster, which does not admit a continuous one-to-one mapping onto a first-countable Hausdorff space, is shown to be a \(D\)-space. The author then generalizes known results for \(D\)-spaces (\(\alpha D\)-spaces, respectively). It is shown that a space with a point-countable weak base is a \(D\)-space. As a corollary it follows that a countably compact space with a point-countable weak base is a compact metrizable space. It is also shown that spaces that are a finite union of \(\theta\)-refinable spaces are \(\alpha D\)-spaces.
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 5123632
zbMATH Keywords: \(D\)-space, \({\mathcal C}_p(\tau_\omega)\)
zbMATH Keywords: \(\theta\)-refinable space
zbMATH Keywords: Lindelöf space
zbMATH Keywords: weak base
reviewed by: Hans Peter Künzi
MaRDI profile type: Publication
full work available at URL: j.topol.2006.06.003
OpenAlex ID: W2078302866
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DOI: 10.1016/J.TOPOL.2006.06.003
description: scientific article; zbMATH DE number 5123632