The Gruenhage property, property *, fragmentability, and \(\sigma \)-isolated networks in generalized ordered spaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 1286.54026
published in: Fundamenta Mathematicae
publication date: 2013-12-16T00:00:00Z
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 54E52
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 46A99
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 46B26
Mathematics Subject Classification ID: 46B50
zbMATH DE Number: 6238604
zbMATH Keywords: Gruenhage space
zbMATH Keywords: property *
zbMATH Keywords: fragmentable space
zbMATH Keywords: \(\sigma \)-isolated network
zbMATH Keywords: LOTS
zbMATH Keywords: linearly ordered topological space
zbMATH Keywords: GO-space
zbMATH Keywords: generalized ordered space
zbMATH Keywords: paracompactness
zbMATH Keywords: stationary sets
zbMATH Keywords: dense metrizable subspace
zbMATH Keywords: monotone normality
zbMATH Keywords: metrizability
zbMATH Keywords: \(G_\delta \)-diagonal
zbMATH Keywords: \(\sigma\)-disjoint \(\pi \)-base
zbMATH Keywords: quasi-developable space
zbMATH Keywords: Sorgenfrey line
zbMATH Keywords: Michael line
zbMATH Keywords: Eberlein compact
zbMATH Keywords: Talagrand compact
zbMATH Keywords: Gulko compact
zbMATH Keywords: strictly convex dual norm
author: Harold Bennett
author: David Lutzer
MaRDI profile type: Publication
full work available at URL: fm223-3-4
OpenAlex ID: W2015245074
DOI: 10.4064/FM223-3-4
title: The Gruenhage property, property *, fragmentability, and \(\sigma \)-isolated networks in generalized ordered spaces
review text: The \textit{Gruenhage property} of a space \(X\) refers to the existence of a sequence \(\langle \mathcal{V}(n): n<\omega\rangle\) of collections of open sets such that given any points \(x\neq y\) in \(X\), there is some \(n\) and some \(V\in\mathcal{V}(n)\) such that \(V\cap\{x,y\}\) is a singleton, and at least one of the numbers \(\mathrm{ord}(x,\mathcal{V}(n))\) and \(\mathrm{ord}(y,\mathcal{V}(n))\) is finite. (The number \(\mathrm{ord}(x,\mathcal{V}(n))\) is the cardinality of the set \(\{V\in\mathcal{V}(n): x\in V\}\).) This property was introduced by \textit{G. Gruenhage} [Proc. Am. Math. Soc. 100, 371--376 (1987; Zbl 0622.54020)].NEWLINENEWLINEProperty * of a space (introduced by \textit{J. Orihuela} et al. in [Proc. Lond. Math. Soc. (3) 104, No. 1, 197--222 (2012; Zbl 1241.46005)]) is a generalization of the Gruenhage property. A space \(X\) is said to have \textit{property *} if there is a sequence \(\langle \mathcal{V}(n): n<\omega\rangle\) of collections of open sets such that if \(x\neq y\) are any two points of \(X\), there is some \(n\) and some \(V\in\mathcal{V}(n)\) such that \(V\cap\{x,y\}\) is a singleton and no member of \(\mathcal{V}(n)\) contains both \(x\) and \(y\).NEWLINENEWLINEThese properties, together with fragmentability (a space \(X\) is \textit{fragmentable} if there is a metric \(d\) on \(X\), not directly related to the topology \(\tau\) of \(X\), such that for each \(\varepsilon>0\) and each nonempty \(E\subseteq X\) there is some \(U\in\tau\) with \(U\cap E\neq \emptyset\) and the \(d\)-diameter of \(U\cap E\) is smaller than \(\varepsilon\)) and the existence of \(\sigma\)-isolated networks, are carefully investigated in this paper in the context of linearly ordered topological spaces (LOTS), generalized ordered spaces (GO-spaces), and monotonically normal spaces. To quote the abstract:NEWLINENEWLINE``We show that any monotonically normal space with property * or with a \(\sigma\)-isolated network must be hereditarily paracompac…
reviewed by: Jorge Picado
description: scientific article; zbMATH DE number 6238604