Convergence in probabilistic semimetric spaces
type: Item
instance of: scholarly article
title: Convergence in probabilistic semimetric spaces
zbMATH Open document ID: 0689.54017
DOI: 10.1216/RMJ-1988-18-3-617
author: Jianyong Qiao
author: Sumit K. Garg
published in: Rocky Mountain Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: A probabilistic semimetric space (S,F) is a set S together with a function F defined on \(S\times S\) with values in the space \(\Delta^+\), which is a space of real-valued functions, satisfying some weak assumptions resembling those for a metric except for the triangular inequality. Since \(\Delta^+\) carries a metric which turns it into a (complete) metric space, S carries a natural Cauchy structure \({\mathcal C}\), namely the one generated by the set of all filters \(\phi\) such that F(\(\phi\) \(\times \phi)\) is a Cauchy filter in \(\Delta^+.\) The author shows that the additional assumption that F is Cauchy- continuous has quite a few interesting consequences, e.g.: (S,\({\mathcal C})\) is regular and \({\mathcal C}\) induces a pretopology on S and if (S,\({\mathcal C})\) is totally bounded then \({\mathcal C}\) is uniformizable. Furthermore the questions about the existence of completions and compactifications have very satisfactory solutions. Various examples show that the assumption of the Cauchy continuity of F cannot be dropped.
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 60B99
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54E52
zbMATH DE Number: 4129604
zbMATH Keywords: Cauchy space
zbMATH Keywords: probabilistic semimetric space
zbMATH Keywords: Cauchy structure
zbMATH Keywords: Cauchy filter
zbMATH Keywords: completions
zbMATH Keywords: compactifications
zbMATH Keywords: Cauchy continuity
MaRDI profile type: Publication
full work available at URL: rmj-1988-18-3-617
OpenAlex ID: W1992394046
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description: scientific article; zbMATH DE number 4129604