Some sufficiency conditions for \(D\)-spaces, dually discrete and its applications
type: Item
instance of: scholarly article
title: Some sufficiency conditions for \(D\)-spaces, dually discrete and its applications
zbMATH Open document ID: 1362.54021
author: Liang-Xue Peng
published in: Topology and its Applications
publication date: 2017-01-09T00:00:00Z
review text: Let \(X\) be a topological space and for each \(x\in X\) let \(\mathcal{W}(x)\) be a family of subsets of \(X\) containing \(x\). In [\textit{P. J. Collins} and \textit{A. W. Roscoe}, Proc. Am. Math. Soc. 90, 631--640 (1984; Zbl 0541.54034)], the following definition was given: \(\mathcal{W}=\{\mathcal{W}(x):x\in X\}\) satisfies the property \((F)\) if for every \(x\in X\) and every open neighborhood \(U\) of \(x\) there exists an open set \(V\) containing \(x\) such that \(x\in W\subseteq U\) for some \(W\in \mathcal{W}(y)\) whenever \(y\in V\). If \(\mathcal{P}\) is a property, then \(\mathcal{W}\) satisfies \(\mathcal{P}(F)\) if \(\mathcal{W}\) satisfies \((F)\) and for each \(x\in X\), \(\mathcal{W}(x)\) has the property \(\mathcal{P}\). We recall that a \textit{neighborhood assignment} in \(X\) is a family of open sets \(\{O_x:x\in X\}\) such that \(x\in O_x\) for every \(x\in X\). Using special neighborhood assignments or families \(\mathcal{W}\) satisfying \(\mathcal{P}(F)\) for \(\mathcal{P}\in\) \{well-ordered, chain, chain neighborhood, closure-preserving chain, \(\omega_1\)-sheltering, \(\omega_1\)-sheltering chain, etc.\}, the author of this paper proves several sufficiency conditions for a topological space \(X\) to be a \(D\)-space (paracompact \(D\)-space, hereditary \(D\)-space), dually discrete space or dually scattered space of rank \(\leq 2\). As corollaries of these theorems, the author obtains some previously known results. Several examples are given.
reviewed by: Ivan S. Gotchev
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54E30
zbMATH DE Number: 6672884
zbMATH Keywords: \(D\)-space
zbMATH Keywords: chain neighborhood \((F)\)
zbMATH Keywords: closure-preserving chain \((F)\)
zbMATH Keywords: dually discrete \(\kappa\)-sheltering \((F)\) space
MaRDI profile type: Publication
full work available at URL: j.topol.2016.12.007
OpenAlex ID: W2565434776
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DOI: 10.1016/J.TOPOL.2016.12.007
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description: scientific article; zbMATH DE number 6672884