Strongly countable dimensional compacta form the Hurewicz set
type: Item
instance of: scholarly article
title: Strongly countable dimensional compacta form the Hurewicz set
zbMATH Open document ID: 1113.54004
author: Paweł Krupski
author: Alicja Samulewicz
published in: Topology and its Applications
publication date: 2007-03-08T00:00:00Z
review text: \textit{R.~Cauty} [Fundam. Math. 138, 35--58 (1991; Zbl 0770.54015)] proved that the metric space of all differentiable functions \(f:[0,1]\to\mathbb R\) with the supremum norm is homeomorphic to the Hurewicz set~\(\mathcal H\) of countable compact subsets of \([0,1]\). \textit{H.~Gladdines} and \textit{J. van Mill} [ibid. 142, 173--188 (1993; Zbl 0811.54028)] proved that the hyperspace of locally connected continua in the cube \([0,1]^3\) is homeomorphic to the infinite Cartesian power of the pseudoboundary of the Hilbert cube~\(\ell^\infty\). Let us recall that a~space is strongly countable dimensional if it is a~countable union of closed finite dimensional subsets. The authors of the paper under review apply some of the ideas of the above mentioned papers (in particular, the powerful method of absorbers) and prove that the hyperspace of all strongly countable dimensional compacta in~\(\ell^\infty\) of dimension at least~\(k\) with \(k\geq1\) and the hyperspace of all strongly countable dimensional continua in~\(\ell^\infty\) of dimension at least~\(k\) with \(k\geq2\) are homeomorphic to~\(\mathcal H\). These results hold for any true dimension in the sense of Dobrowolski and Rubin which includes the covering dimension dim and the cohomological dimension dim\(_G\) with respect to any Abelian group.
reviewed by: Miroslav Repický
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 57N20
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 54H05
zbMATH DE Number: 5131628
zbMATH Keywords: analytic set
zbMATH Keywords: absorber
zbMATH Keywords: cohomological dimension
zbMATH Keywords: countable dimensional space
zbMATH Keywords: Hilbert cube
zbMATH Keywords: Hurewicz set
zbMATH Keywords: hyperspace
MaRDI profile type: Publication
full work available at URL: j.topol.2005.02.015
OpenAlex ID: W2050369891
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DOI: 10.1016/J.TOPOL.2005.02.015
description: scientific article; zbMATH DE number 5131628