Apartness, compactness and nearness
type: Item
instance of: scholarly article
title: Apartness, compactness and nearness
zbMATH Open document ID: 1156.54013
author: Hajime Ishihara
author: Peter Schuster
author: Douglas Bridges
author: Luminiţa Viţă
published in: Theoretical Computer Science
publication date: 2008-10-21T00:00:00Z
review text: The paper investigates the possibility of introducing a notion of compactness for apartness spaces. The latter have been proposed as a suitable framework for doing topology constructively, and correspond to the classical theory of proximity spaces. The authors set out to find a definition of compactness that, for uniform spaces, is equivalent to totally boundedness together with completeness, and that is classically equivalent to the usual Heine-Borel definition. The chosen approach is to reconstruct a base of entourages from a given apartness relation. It is shown that this approach almost succeeds and that many interesting properties hold. Unfortunately it is not shown that a uniform space that is compact with respect to the reconstructed uniformity is totally bounded or complete. The paper is of relevance to anybody interested in constructive topology, compactness, and foundations of mathematics.
reviewed by: Hannes Diener
Mathematics Subject Classification ID: 54E17
Mathematics Subject Classification ID: 03F65
Mathematics Subject Classification ID: 54E05
Mathematics Subject Classification ID: 54E15
zbMATH DE Number: 5354938
zbMATH Keywords: apartness
zbMATH Keywords: constructive topology
zbMATH Keywords: compact
MaRDI profile type: Publication
full work available at URL: j.tcs.2008.06.019
OpenAlex ID: W2000254072
cites work: Q5573965
cites work: Q3754620
cites work: Apartness spaces as a framework for constructive topology
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cites work: Investigation of constructive functions by the method of fillings
cites work: Q5606492
cites work: Q5556988
cites work: Q5310897
DOI: 10.1016/J.TCS.2008.06.019
description: scientific article; zbMATH DE number 5354938