×

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

MSC:

54E15 Uniform structures and generalizations
52A01 Axiomatic and generalized convexity
PDF BibTeX XML Cite
Full Text: EuDML

References:

[1] Aronszajn N., Panitchpakdi P.: Extensions of uniformly continuous transformations and hyperconvex metric spaces. Pacif J. Math. 6 (1956), 405-439. · Zbl 0074.17802
[2] Corson H., Klee V.: Topological classification of convex sets. Proc. Symp. Pure Math. VII (Convexity), AMS (1963), 37-51. · Zbl 0207.42901
[3] Hejcman J.: Metrization lemmas and equivalence of Lipschitz structures. General topology and its relations to modern analysis and algebra V (Proc. Fifth Prague Topological Symp. 1981), Heldermann Verlag, Berlin, 1983, 261-264.
[4] Isbell J. R.: Uniform spaces. Amer. Math. Soc., Providence, 1964. · Zbl 0124.15601
[5] Kuratowski K.: Topology I. Academic Press, New York and London, PWN, Warszawa, 1966. · Zbl 0158.40901
[6] Menger K.: Untersuchungen über allgemeine Metrik. Math. Ann. 100 (1928), 75-163. · JFM 54.0622.02
[7] Pelant J.: Locally fine uniformities and normal covers. Czechoslovak Math. J. 37(112) (1987), 181-187. · Zbl 0656.54020
[8] Rice M. D.: Metric-fine uniform spaces. J. London Math. Soc. 11 (1975), 53-64. · Zbl 0324.54018
[9] Rolfsen D.: Geometric methods in topological spaces. Topology conference ASU 1967, Tempe, Arizona 1968, 250-257.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.