Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
MathSciNet review: 810180
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Atsuji has internally characterized those metric spaces $ X$ for which each real-valued continuous function on $ X$ is uniformly continuous as follows: (1) the set $ X'$ of limit points of $ X$ is compact, and (2) for each $ \varepsilon > 0$, the set of points in $ X$ whose distance from $ X'$ exceeds $ \varepsilon $ is uniformly discrete. We obtain these new characterizations: (a) for each metric space $ Y$, the Hausdorff metric on $ C\left( {X,Y} \right)$, induced by a metric on $ X \times Y$ compatible with the product uniformity, yields the topology of uniform convergence; (b) there exists a metric space $ Y$ containing an arc for which the Hausdorff metric on $ C\left( {X,Y} \right)$ yields the topology of uniform convergence; (c) the Hausdorff metric topology on $ {\text{CL}}\left( X \right)$ is at least as strong as the Vietoris topology. We also characterize those metric spaces whose hyperspace is such a space.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54B20, 54C35, 54E45

Retrieve articles in all journals with MSC: 54B20, 54C35, 54E45


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0810180-3
PII: S 0002-9939(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