A study on monotonically metacompact and property \((A)((B))\)
type: Item
instance of: scholarly article
title: A study on monotonically metacompact and property \((A)((B))\)
zbMATH Open document ID: 1398.54043
author: Liang-Xue Peng
author: Lijun Wang
published in: Topology and its Applications
publication date: 2018-07-25T00:00:00Z
review text: A topological space \(X\) is \textit{(countably) metacompact} if each (countable) open cover of \(X\) has a point-finite open refinement which also covers \(X\), and \(X\) is said to be \textit{monotonically (countably) metacompact} if there is a function \(r\) that associates with each (countable) open cover \(\mathcal U\) of \(X\) a point-finite open refinement \(r(\mathcal U)\) that also covers \(X\) in such a way that the following statement holds: if \(\mathcal U\) and \(\mathcal{V}\) are open covers of \(X\), then \(r(\mathcal U)\) refines \(r(\mathcal V)\) whenever \(\mathcal U\) refines \(\mathcal V\). A generalized ordered topological space (GO-space) is a triple \((X,\tau,<)\), where \(<\) is a linear ordering on \(X\) and \(\tau\) is a Hausdorff topology on \(X\) which is generated by a base of order-convex sets. It \(\tau\) is the open interval topology of the given ordering, then \((X,\tau,<)\) is a linearly ordered topological space (LOTS). Properties (A) and (B) where introduced by \textit{T. Chase} and \textit{G. Gruenhage} [Topology Appl. 213, 135--144 (2016; Zbl 1356.54031)]. Let \(X\) be a topological space and \(P(X)\) be the collection of all pairs \((x,U)\) where \(x \in X\) and \(U\) is an open neighbourhood of \(x\). A space has property (A) (resp. property (B)) if to each \((x,U) \in P(X)\) one can assign an open set \(V(x,U)\) such that \(x \in V(x,U) \subseteq U\) and for any collection \(\{(x_\alpha,U_\alpha): \alpha \in A\} \subseteq P(X)\), either \(\bigcap\limits_{\alpha \in A} V(x_\alpha,U_\alpha) = \emptyset\) or there is a finite \(A' \subseteq A\) such that for each \(\alpha \in A\) there is some \(\beta \in A'\) with \(V(x_\alpha,U_\alpha) \subseteq U_\beta\) (resp. \(\{x_\alpha: \alpha \in A\} \subseteq \bigcup\limits_{\beta \in A'} U_\beta\)). Chase and Gruenhage have shown, in the paper referred to, that compact spaces which satisfy property (A) are metrizable, and a separable space which satisfies property (A) is metrizable. I…
reviewed by: Samuel Gomes da Silva
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54E30
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54F05
zbMATH DE Number: 6910242
zbMATH Keywords: monotonically (countably) metacompact
zbMATH Keywords: property \((A)\)
zbMATH Keywords: property \((B)\)
zbMATH Keywords: GO-space
MaRDI profile type: Publication
full work available at URL: j.topol.2018.06.003
OpenAlex ID: W2809241422
cites work: Monotone normality
cites work: A note on monotonically metacompact spaces
cites work: Compact monotonically metacompact spaces are metrizable
cites work: Monotone covering properties and properties they imply
cites work: Some properties of the Sorgenfrey line and related spaces
cites work: Q3994935
cites work: Monotone meta-Lindelöf spaces
cites work: Monotonically Normal Spaces
cites work: Q5636033
cites work: Q2938883
cites work: Linearly ordered extensions of GO spaces
cites work: Q3404188
cites work: The D-property of monotone covering properties and related conclusions
cites work: A note on monotone covering properties
cites work: Q3842327
cites work: C-scattered and paracompact spaces
cites work: Classes defined by stars and neighbourhood assignments
DOI: 10.1016/J.TOPOL.2018.06.003
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description: scientific article; zbMATH DE number 6910242