Spaces of continuous functions defined on Mrówka spaces
type: Item
instance of: scholarly article
title: Spaces of continuous functions defined on Mrówka spaces
zbMATH Open document ID: 1068.54022
author: Paul J. Szeptycki
author: Michael Hrušák
author: Angel Tamariz-Mascarúa
published in: Topology and its Applications
publication date: 2005-03-08T00:00:00Z
review text: A collection \({\mathcal A}\) of subsets of the natural numbers \(\omega\) is an almost disjoint family if each \(A\) in \({\mathcal A}\) is infinite, and for each two different elements \(A\), \(B\) in \({\mathcal A}\), \(|A\cap B|<\aleph_0\). A maximal almost disjoint family is a maximal element in the family of all almost disjoint families with the containment order. A topological space \(\Psi({\mathcal A})\) is a Mrówka space if it has the form \(\omega\cup{\mathcal A}\), where \({\mathcal A}\) is an almost disjoint family, and its topology is generated by the following base: each \(\{n\}\) is open for every \(n\in\omega\), and an open canonical neighborhood of \(A\in{\mathcal A}\) is of the form \(\{{\mathcal A}\}\cup B\) where \(B\subset\omega\) and \(A\setminus B\) is finite. The authors discuss normality and the Lindelöf property in spaces of continuous functions over \(\Psi({\mathcal A})\) and obtain the following results. Theorem 2.7. Let \(E\in\{\mathbb{I},\mathbb{R},\mathbb{P},2^\omega\}\). For every uncountable almost disjoint family \({\mathcal A}\) and every compactification \(b\Psi(A)\) of \(\Psi({\mathcal A})\), the space \(C_p(b\Psi({\mathcal A}),E)\) is not normal, where \(\mathbb{R}\) and \(\mathbb{P}\) are the spaces of real and irrational numbers with the natural topology, respectively, and \(\mathbb{I}\) denotes the unit interval \([0,1]\subset\mathbb{R}\). Theorem 3.4. Let \({\mathcal A}\) be an infinite maximal almost disjoint family on \(\omega\). Then the spaces \(C_p(\Psi({\mathcal A}), 2^\omega)\), \(C_p(\Psi({\mathcal A}), \omega^\omega)\), \(C_p(\Psi({\mathcal A}),\mathbb{I})\) and \(C_p(\Psi({\mathcal A}),\mathbb{R})\) are not normal, and their extent and Lindelöf numbers are all equal to \(|{\mathcal A}|\), where the extent of \(X\) means the supremum of the cardinalities of all the closed and discrete subspaces of \(X\). An almost disjoint family \({\mathcal A}\) of subsets of \(\omega\) is quasi-maximal if there is a maximal almost…
reviewed by: Akihiro Okuyama
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54G20
Mathematics Subject Classification ID: 54D15
zbMATH DE Number: 2142293
zbMATH Keywords: almost disjoint family
zbMATH Keywords: Mrówka space
zbMATH Keywords: function space with the pointwise convergence topology
zbMATH Keywords: normality Lindelöf property
zbMATH Keywords: extent
MaRDI profile type: Publication
full work available at URL: j.topol.2004.09.009
OpenAlex ID: W2151146316
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DOI: 10.1016/J.TOPOL.2004.09.009
description: scientific article; zbMATH DE number 2142293