Universal spaces for finite-dimensional closed images of locally compact metric spaces
type: Item
instance of: scholarly article
title: Universal spaces for finite-dimensional closed images of locally compact metric spaces
zbMATH Open document ID: 0923.54019
DOI: 10.1016/S0166-8641(97)00149-1
author: Kazuhiro Kawamura
author: Koichi Tsuda
published in: Topology and its Applications
publication date: 1999-10-25T00:00:00Z
review text: The authors provide spaces as in the title of the paper. In the separable case (and \(\dim\leq n\)) the universal space \(L(n)\) is obtained as a special quotient of \(\mu^n\smallsetminus\{*\}\) (Menger's universal compactum minus a point). In the non-separable case -- weight \(\kappa\) -- one takes a quotient of the sum of \(\kappa\) copies of \(L(n)\), employing a family \(\mathcal K\) of countable subsets of \(\kappa\). The universality of the resulting space is equivalent to the following properties of \(\mathcal K\): its cardinality should be \(\kappa\) and given any other family \(\mathcal C\) of countable subsets of \(\kappa\) of size \(\kappa\) there is a permutation \(\pi\) of \(\kappa\) such that for every \(C\in\mathcal C\) there is \(K\in\mathcal K\) such that \(\pi[C]\subseteq K\). If the GCH holds then every cardinal carries such a family but the authors leave open the question whether ZFC suffices for this.
Mathematics Subject Classification ID: 54C99
Mathematics Subject Classification ID: 54D45
Mathematics Subject Classification ID: 54E40
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 54F65
zbMATH DE Number: 1307999
zbMATH Keywords: Lashnev space
zbMATH Keywords: universal space
zbMATH Keywords: closed map
reviewed by: K. P. Hart
MaRDI profile type: Publication
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full work available at URL: s0166-8641(97)00149-1
OpenAlex ID: W1972293911
description: scientific article; zbMATH DE number 1307999