Urysohn universal space, its development and Hausdorff's approach
type: Item
instance of: scholarly article
title: Urysohn universal space, its development and Hausdorff's approach
zbMATH Open document ID: 1159.54006
author: M. Hušek
published in: Topology and its Applications
publication date: 2008-08-06T00:00:00Z
review text: Three approaches to a direct construction of Urysohn universal space are compared, namely those of Urysohn, Hausdorff and Katětov. The focus is on Hausdorff's unpublished approach that is shown to work in a more general situation, and the following questions related to those constructions are added. {\parindent=6mm \begin{itemize}\item[1.] Katětov constructed a non-complete strongly \(\omega\)-universal and \(\omega\)-homogeneous space \(\mathcal{W}\). Can one describe such a space in terms of the space \(\mathcal{U}_\omega\)? \item[2.] Katětov proved that there is no \(\kappa\)-universal and \(\kappa\)-homogeneous space having density \(\kappa\) if \(\kappa <\kappa^{<\kappa}\). We believe that this result can be proved using \(\mathcal{U}_{\kappa}\). How do we do that? The author gives one possibility. \item[3.] V. Uspenskii used Katětov's construction for \(\kappa=\omega\) to prove universality of the group of isometries of Urysohn universal space. Is it possible to use the space \(\mathcal{U}_\kappa\) for similar results? \end{itemize}}
reviewed by: Takemi Mizokami
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54G99
Mathematics Subject Classification ID: 54-03
zbMATH DE Number: 5306963
zbMATH Keywords: separable metric space
zbMATH Keywords: universal space
MaRDI profile type: Publication
full work available at URL: j.topol.2008.03.020
OpenAlex ID: W2072203700
cites work: Q5847716
cites work: Q3784822
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cites work: Q5824065
cites work: Q3415741
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cites work: The Urysohn universal metric space is homeomorphic to a Hilbert space
DOI: 10.1016/J.TOPOL.2008.03.020
description: scientific article; zbMATH DE number 5306963