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.

**[1]**M. Atsuji,*Uniform continuity of continuous functions of metric spaces*, Pacific J. Math.**8**(1958), 11-16. MR**0099023 (20:5468)****[2]**J. P. Aubin,*Applied abstract analysis*, Wiley, New York, 1977. MR**470034 (81e:54001)****[3]**C. Castaing and M. Valadier,*Convex analysis and measurable multifunctions*, Springer-Verlag, Berlin, 1977. MR**0467310 (57:7169)****[4]**R. Engelking,*General topology*, Polish Scientific Publishers, Warsaw, 1977. MR**0500780 (58:18316b)****[5]**H. Hueber,*On uniform continuity and compactness in metric spaces*, Amer. Math. Monthly**88**(1981), 204-205. MR**619571 (82h:54045)****[6]**B. Levshenko,*On the concept of compactness and point-finite coverings*, Mat. Sb.**42**(1957), 479-484. MR**0096185 (20:2679)****[7]**S. Naimpally,*Graph topology for function spaces*, Trans. Amer. Math. Soc.**123**(1966), 267-272. MR**0192466 (33:691)****[8]**J. Nagata,*On the uniform topology of bicompactifications*, J. Inst. Polytech. Osaka City University**1**(1950), 28-38. MR**0037501 (12:272a)****[9]**B. Penkov and Bl. Sendov,*Hausdorffsche metrik und approximationen*, Numer. Math.**9**(1966), 214-226. MR**0204927 (34:4762)****[10]**V. Popov and Bl. Sendov,*The exact asymptotic behavior of the best approximation by algebraic and trigonometric polynomials in the Hausdorff metric*, Math. USSR-Sb.**18**(1972), 139-149. MR**0308665 (46:7779)****[11]**J. Rainwater,*Spaces whose finest uniformity is metric*, Pacific J. Math.**9**(1959), 567-570. MR**0106448 (21:5180)****[12]**Bl. Sendov,*Certain questions in the theory of approximation of functions and sets in the Hausdorff metric*, Russian Math. Surveys**24**(1969), 143-183. MR**0276648 (43:2390)****[13]**Gh. Toader,*On a problem of Nagata*, Mathematica (Cluj)**20**(1978), 78-79. MR**530953 (82k:54024)****[14]**W. Waterhouse,*On UC spaces*, Amer. Math. Monthly**72**(1965), 634-635. MR**0184200 (32:1673)****[15]**S. Willard,*General topology*, Addison-Wesley, Reading, Mass., 1968. MR**0264581 (41:9173)**

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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