Ordinal remainders of \(\psi \)-spaces
type: Item
instance of: scholarly article
title: Ordinal remainders of \(\psi \)-spaces
zbMATH Open document ID: 1227.54025
author: Alan Dow
author: Jerry E. Vaughan
published in: Topology and its Applications
publication date: 2011-09-22T00:00:00Z
review text: For an infinite cardinal \(\kappa\) and any almost disjoint family (ADF) \(\mathcal{A}\subset [\kappa]^{\omega}\), let \(\psi(\kappa,\mathcal{A})\) denote the space with underlying set \(\kappa\cup \mathcal{A}\) and with the topology having as a base all singletons \(\{\alpha\}\) for \(\alpha<\kappa\) and all sets of the form \(\{ A\}\cup (A\setminus F)\) where \(A\in\mathcal{A}\) and \(F\) is finite. When \(\kappa\) and the MADF \(\mathcal{M}\subset [\kappa]^{\omega}\) are understood, the space \(\psi(\kappa,\mathcal{M})\) is called a \(\psi\)-space. It is known that all \(\psi\)-spaces are locally compact, so the remainder \(\beta\psi\setminus\psi\) is closed in \(\beta\psi\), hence compact. Thus if the remainder \(\beta\psi\setminus\psi\) is homeomorphic to an ordinal, it must be a successor ordinal. In the main result of the paper under review the authors show that the cardinal \(\mathfrak{c}^+\) is the largest cardinal satisfying the following property: for every cardinal \(\omega\leq\kappa\leq\mathfrak{c}^+\), there exists a MADF \(\mathcal{M}\subset [\kappa]^{\omega}\) such that the Stone-Čech remainder of \(\psi(\kappa,\mathcal{M})\) is homeomorphic to \(\kappa+1\) with the order topology. The article is a continuation of the authors' paper [Topology Appl. 157, 1379--1394 (2010; Zbl 1196.54060)].
reviewed by: Tomasz Natkaniec
Mathematics Subject Classification ID: 54D40
Mathematics Subject Classification ID: 03E25
Mathematics Subject Classification ID: 03E10
zbMATH DE Number: 5949219
zbMATH Keywords: maximal almost disjoint families
zbMATH Keywords: \(\psi\)-spaces
zbMATH Keywords: cardinal numbers
zbMATH Keywords: ordinal numbers
zbMATH Keywords: Stone-Čech compactification
zbMATH Keywords: Stone-Čech remainders
MaRDI profile type: Publication
full work available at URL: j.topol.2011.06.021
OpenAlex ID: W2049332386
cites work: Partition Algebras for Almost-Disjoint Families
cites work: Spaces of continuous functions over a \(\Psi\)-space
cites work: Mrówka maximal almost disjoint families for uncountable cardinals
cites work: Q3994935
cites work: Q3270120
cites work: Q5630317
cites work: Set theory. An introduction to independence proofs
cites work: Separation in \(\Psi\)-spaces
cites work: Ordering MAD families a la Katětov
cites work: On completely regular spaces
cites work: Q5609113
cites work: Some set-theoretic constructions in topology
cites work: Spaces \(N\cup\mathcal R\) and their dimensions
cites work: Q4820394
DOI: 10.1016/J.TOPOL.2011.06.021
Recommended article: Ordinal remainders of classical ψ-spaces
Recommended article: Almost disjoint families and topology
Recommended article: Small cardinals and the pseudocompactness of hyperspaces of subspaces of \(\beta\omega\)
Recommended article: Mrówka maximal almost disjoint families for uncountable cardinals
Recommended article: On two problems of Erdos and Hechler: New methods in singular madness
Recommended article: Extending the class of known Stone-Čech remainders for \(\psi\)-spaces
Recommended article: Q4090330
Recommended article: Maximally almost disjoint families of representing sets
Recommended article: MAD SPECTRA
Recommended article: Fibers of continuous real-valued functions on \(\psi\)-spaces
description: scientific article; zbMATH DE number 5949219