\(D\)-spaces and thick covers
type: Item
instance of: scholarly article
title: \(D\)-spaces and thick covers
zbMATH Open document ID: 1244.54056
author: H. F. Guo
author: Heikki J. K. Junnila
published in: Topology and its Applications
publication date: 2011-10-10T00:00:00Z
review text: This paper gives over 15 results relating nearly a dozen subtly defined properties. For readability, this review mentions only a few of these results relating only a few of these properties. The results mentioned in this review all end in ``then \(X\) is a \(D\)-space,'' but many of the results in this paper do not have this format. Recall that \(X\) is a \(D\)-space iff, given an open cover \(\{U_x: x \in X\}\) where each \(x \in U_x\), there is a closed discrete \(D \subseteq X\) so that \(X = \bigcup_{x \in D}U_x\). To give the flavor of this paper, here are three results: \textit{If \(X\) has an almost thick cover by closed Lindelöf \(D\)-sets, then \(X\) is a \(D\)-space.} Here a cover \(\mathcal L\) is almost thick iff \(\forall H \in [X]^{< \omega}\) there is \(L_H\) a finite union of elements of \(\mathcal L\) so that if \(A \subset X\) and \(A\) is not closed then there is \(H \in [A]^{< \omega}\) with \(L_H \cap\) cl\((A \setminus A) \neq \emptyset\). \textit{If \(X\) is \(t\)-metrizable, then \(X\) is a \(D\)-space.} Here a space \((X, \tau)\) is \(t\)-metrizable iff it has a finer metrizable topology \(\pi\) and a function \(J: [X]^{< \omega} \rightarrow [X]^{< \omega}\) so that if \(A \subset X\) then cl\(_{\tau}A \subset\) cl\(_{\pi}\bigcup_{H \in [A]^{< \omega}}J(H)\). \textit{If \(X\) is a union of finitely many screenable \(\sigma\)-spaces, then \(X\) is a \(D\)-space.} Here a \(\sigma\)-space is one with a \(\sigma\)-discrete closed network; a space is screenable iff each open cover has a \(\sigma\)-pairwise disjoint open refinement.
reviewed by: Judith Roitman
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54A25
zbMATH DE Number: 5956057
zbMATH Keywords: \(D\)-space
zbMATH Keywords: neighbornet
zbMATH Keywords: thick cover
zbMATH Keywords: \(t\)-metrizable
MaRDI profile type: Publication
full work available at URL: j.topol.2011.06.053
OpenAlex ID: W2041493944
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DOI: 10.1016/J.TOPOL.2011.06.053
description: scientific article; zbMATH DE number 5956057