On a property of pseudometrics and uniformities near to convexity. (English) Zbl 0671.54016

A pseudometric d on a set X is preconvex for distances less than c (where c is a positive real number) if for any x and y in X with \(d(x,y)<c\) and positive numbers r and s with \(d(x,y)<r+s\) there exists z in X with \(d(x,z)<r\) and \(d(z,y)<s\). d is called preconvex if d is preconvex for distances less than c for at least one c, and d is globally preconvex if it is preconvex for distances less than c for all positive c. The authors investigate properties of these “convex-like” pseudometrics which reflect the nature of usual convexity in linear spaces. A main result of the paper concerns preconvex uniform spaces (those possessing bases consisting of preconvex pseudometrics). Such uniform spaces are characterized in terms of certain covering and entourage properties, and this class of uniform spaces includes the strongly essential spaces, locally fine spaces, and sub-metric fine spaces.
Reviewer: S.C.Carlson


54E15 Uniform structures and generalizations
52A01 Axiomatic and generalized convexity
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