Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance

Author:
Gerald Beer

Journal:
Proc. Amer. Math. Soc. **95** (1985), 653-658

MSC:
Primary 54B20; Secondary 54C35, 54E45

DOI:
https://doi.org/10.1090/S0002-9939-1985-0810180-3

MathSciNet review:
810180

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Abstract | References | Similar Articles | Additional Information

Abstract: Atsuji has internally characterized those metric spaces for which each real-valued continuous function on is uniformly continuous as follows: (1) the set of limit points of is compact, and (2) for each , the set of points in whose distance from exceeds is uniformly discrete. We obtain these new characterizations: (a) for each metric space , the Hausdorff metric on , induced by a metric on compatible with the product uniformity, yields the topology of uniform convergence; (b) there exists a metric space containing an arc for which the Hausdorff metric on yields the topology of uniform convergence; (c) the Hausdorff metric topology on is at least as strong as the Vietoris topology. We also characterize those metric spaces whose hyperspace is such a space.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1985-0810180-3

Keywords:
Hausdorff metric,
topology of uniform convergence,
uniformly continuous function,
function space,
UC space,
hyperspace,
Vietoris topology

Article copyright:
© Copyright 1985
American Mathematical Society