Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance
Author:
Gerald Beer
Journal:
Proc. Amer. Math. Soc. 95 (1985), 653658
MSC:
Primary 54B20; Secondary 54C35, 54E45
MathSciNet review:
810180
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Abstract: Atsuji has internally characterized those metric spaces for which each realvalued 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:
http://dx.doi.org/10.1090/S00029939198508101803
PII:
S 00029939(1985)08101803
Keywords:
Hausdorff metric,
topology of uniform convergence,
uniformly continuous function,
function space,
UC space,
hyperspace,
Vietoris topology
Article copyright:
© Copyright 1985
American Mathematical Society
