The Katětov construction modified for a \(T_0\)-quasi-metric space
type: Item
instance of: scholarly article
title: The Katětov construction modified for a \(T_0\)-quasi-metric space
zbMATH Open document ID: 1242.54012
author: Hans Peter Künzi
author: Manuel Sanchis
published in: Topology and its Applications
publication date: 2012-04-13T00:00:00Z
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54D65
Mathematics Subject Classification ID: 54E55
zbMATH DE Number: 6023686
zbMATH Keywords: \(T_0\)-quasi-metric space
reviewed by: Harald Brandenburg
MaRDI profile type: Publication
full work available at URL: j.topol.2011.11.001
OpenAlex ID: W2061985015
cites work: Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: A note on combinatorial properties of metric spaces
cites work: C-complete quasi-uniform spaces
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cites work: The Isbell-hull of a di-space
cites work: On strongly quasi-metrizable spaces
cites work: Q5191281
cites work: A double completion for an arbitrary \(T_0\)-quasi-metric space
cites work: On the geometry of Urysohn's universal metric space
cites work: Q3415741
DOI: 10.1016/J.TOPOL.2011.11.001
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review text: Recall that a pair \((X, d)\) is called a \(T_0\)-quasi-metric space if \(X\) is a set and \(d: X\times X\to[0,\infty)\) is a mapping satisfyingNEWLINENEWLINE (i) \(d(x, x)= 0\) whenever \(x\in X\);NEWLINENEWLINE (ii) \(d(x, z)\leq d(x, y)+ d(y, z)\) whenever \(x,y,z\in X\); andNEWLINENEWLINE (iii) \(d(x, y)= 0= d(y, x)\) implies that \(x=y\).NEWLINENEWLINE If \((X, d)\) is a \(T_0\)-quasi-metric space, then \((X, d^S)\) is a metric space, where \(d^S= \sup\{d, d^{-1}\}\) and \(d^{-1}(x, y)= d(y, x)\) whenever \(x,y\in X\). A \(T_0\)-quasi-metric space \((X, d)\) is called bicomplete (respectively, supseparable), if \((X, d^S)\) is complete (respectively, separable). It is shown that there exists a unique (up to isometry) \(T_0\)-quasi-metric space \(q\mathbb{U}\) with the following properties:NEWLINENEWLINE (a) \(q\mathbb{U}\) is bicomplete and supseparable;NEWLINENEWLINE (b) every isometry between two finite subspaces of \(q\mathbb{U}\) extends to an isometry of \(q\mathbb{U}\) onto itself; andNEWLINENEWLINE (c) \(q\mathbb{U}\) contains an isometric copy of every supseparable \(T_0\)-quasi-metric space.NEWLINENEWLINE The construction of \(q\mathbb{U}\) is similar to \textit{M. Katĕtov's} construction of Urysohn's famous universal metric space \(\mathbb{U}\) in [General topology and its relations to modern analysis and algebra VI, Proc. 6th Symp., Prague/Czech. 1986, Res. Expo. Math. 16, 323--330 (1988; Zbl 0642.54021)].
description: scientific article; zbMATH DE number 6023686