The injective hull of ultra-quasi-metric versus \(q\)-hyperconvex hull of quasi-metric space
type: Item
instance of: scholarly article
title: The injective hull of ultra-quasi-metric versus \(q\)-hyperconvex hull of quasi-metric space
zbMATH Open document ID: 1347.54039
author: Olivier Olela Otafudu
published in: Topology and its Applications
publication date: 2016-04-13T00:00:00Z
reviewed by: Harald Brandenburg
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54E55
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 06B23
zbMATH DE Number: 6568024
zbMATH Keywords: \(T_0\)-quasi-metric space
zbMATH Keywords: injective hull
zbMATH Keywords: \(T_0\)-ultra-quasi-metric space
zbMATH Keywords: Dedekind-MacNeille completion
MaRDI profile type: Publication
full work available at URL: j.topol.2015.12.085
OpenAlex ID: W2272786780
Wikidata QID: Q114128091
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cites work: Endpoints in \(T_0\)-quasimetric spaces. II.
cites work: Six theorems about injective metric spaces
cites work: The Isbell-hull of a di-space
cites work: \(q\)-hyperconvexity in quasipseudometric spaces and fixed point theorems
cites work: The ultra-quasi-metrically injective hull of a \(T_0\)-ultra-quasi-metric space
cites work: Extremal function pairs in asymmetric normed linear spaces
cites work: Q3126163
DOI: 10.1016/J.TOPOL.2015.12.085
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review text: Given a \(T_0\)-ultra-quasi-metric space \((X,d)\), one can construct \(\nu(X,d)\), its injective hull with respect to \(T_0\)-ultra-quasi-metric spaces, see \textit{H.-P. A. Künzi} et al. [Appl. Categ. Struct. 21, No 6, 651--670 (2013; Zbl 1284.54040)], and \(\varepsilon(X,d)\), its injective hull with respect to \(T_0\)-quasi-metric spaces, see \textit{E. Kemajou} et al. [Topology Appl. 159, No. 9, 2463--2475 (2012; Zbl 1245.54023)].NEWLINENEWLINE In this paper it is observed that, if \((X,\leq)\) is a partially ordered set and \(d\) is the natural \(T_0\)-ultra-quasi metric on \(X\) defined by \(d(x,y)= 0\) if \(x\leq y\), and \(d(x,y)= 1\) otherwise, then \(\nu(X,d)\) coincides with the Dedekind-MacNeille completion of \(X\) and \(\nu(X,d)\subset\varepsilon(X,d)\).
description: scientific article; zbMATH DE number 6568024