Well-quasi-ordering and the Hausdorff quasi-uniformity
type: Item
instance of: scholarly article
title: Well-quasi-ordering and the Hausdorff quasi-uniformity
zbMATH Open document ID: 0922.54024
DOI: 10.1016/S0166-8641(97)00151-X
published in: Topology and its Applications
publication date: 1999-06-24T00:00:00Z
review text: It is a thesis of the first author that hereditary precompactness can often be substituted in the study of quasi-uniform spaces for the classical symmetric property, total boundedness. This thesis is supported in the paper by a surprising characterization of hypercompactness: The quasi-uniform space \(({\mathcal P}_0(X),{\mathcal U}^*)\) is compact if and only if \((X,{\mathcal U})\) is compact and \((X_m,{\mathcal U}^{-1}| X_m)\) is hereditarily precompact, where \(X_m= \{y\in X:y\) is a minimal element in the (specialization) quasi-order of the space \((X,{\mathcal U})\}\). The following corollaries are of independent interest. Let \((X,{\mathcal U})\) be a \(T_1\) quasi-uniform space. If \({\mathcal U}\) is stable, or \({\mathcal U}\) is the well-monotone quasi-uniformity, then \(({\mathcal P}_0(X),{\mathcal U}^*)\) is compact if and only if \((X,{\mathcal U})\) is compact. Reviewer's remarks: Anyone reading this paper will be struck by its connections to combinatorics.
reviewed by: P. Fletcher
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 54B20
zbMATH DE Number: 1307990
zbMATH Keywords: hypercompactness
zbMATH Keywords: quasi-uniform space
author: Hans Peter Künzi
author: Salvador Romaguera
MaRDI profile type: Publication
cites work: On Quasi-Uniformities in Hyperspaces
cites work: Q4834996
cites work: C-complete quasi-uniform spaces
cites work: Ordering by Divisibility in Abstract Algebras
cites work: Q3852172
cites work: Stability in quasi-uniform spaces and the inverse problem
cites work: Q4326507
cites work: Q4371050
cites work: Q4033051
cites work: Q4834599
cites work: Q3123097
cites work: Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture
cites work: The theory of well-quasi-ordering: a frequently discovered concept
cites work: Q5658811
cites work: Foundations of BQO Theory
cites work: The Set of Better Quasi Orderings is ∏
cites work: Q3691785
cites work: Q5520569
cites work: Q3715155
cites work: The metrisability of precompact sets
cites work: On hereditary precompactness and completeness in quasi-uniform spaces
cites work: Constructing a quasi-uniform function space
cites work: Q4871753
full work available at URL: s0166-8641(97)00151-x
OpenAlex ID: W1989615129
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description: scientific article; zbMATH DE number 1307990