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

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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]**Masahiko Atsuji,*Uniform continuity of continuous functions of metric spaces*, Pacific J. Math.**8**(1958), 11–16; erratum, 941. MR**0099023****[2]**Jean-Pierre Aubin,*Applied abstract analysis*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. Exercises by Bernard Cornet and Hervé Moulin; Translated from the French by Carole Labrousse; Pure and Applied Mathematics. MR**470034****[3]**C. Castaing and M. Valadier,*Convex analysis and measurable multifunctions*, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR**0467310****[4]**Ryszard Engelking,*General topology*, PWN—Polish Scientific Publishers, Warsaw, 1977. Translated from the Polish by the author; Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60]. MR**0500780****[5]**Hermann Hueber,*On uniform continuity and compactness in metric spaces*, Amer. Math. Monthly**88**(1981), no. 3, 204–205. MR**619571**, 10.2307/2320473**[6]**B. T. Levšenko,*On the concept of compactness and point-finite coverings*, Mat. Sb. N.S.**42(84)**(1957), 479–484 (Russian). MR**0096185****[7]**Somashekhar Amrith Naimpally,*Graph topology for function spaces*, Trans. Amer. Math. Soc.**123**(1966), 267–272. MR**0192466**, 10.1090/S0002-9947-1966-0192466-4**[8]**Jun-iti Nagata,*On the uniform topology of bicompactifications*, J. Inst. Polytech. Osaka City Univ. Ser. A. Math.**1**(1950), 28–38. MR**0037501****[9]**B. Penkov and Bl. Sendov,*Hausdorffsche Metrik und Approximationen*, Numer. Math.**9**(1966), 214–226 (German). MR**0204927****[10]**Bl. Sendov and V. A. Popov,*Exact asymptotic behavior of the best approximation by algebraic and trigonometric polynomials in the Hausdorff metric*, Mat. Sb. (N.S.)**89(131)**(1972), 138–147, 167 (Russian). MR**0308665****[11]**John Rainwater,*Spaces whose finest uniformity is metric*, Pacific J. Math.**9**(1959), 567–570. MR**0106448****[12]**B. Sendov,*Certain questions in the theory of approximations of functions and sets in the Hausdorff metric*, Uspehi Mat. Nauk**24**(1969), no. 5 (149), 141–178 (Russian). MR**0276648****[13]**Gh. Toader,*On a problem of Nagata*, Mathematica (Cluj)**20(43)**(1978), no. 1, 77–79. MR**530953****[14]**W. C. Waterhouse,*On 𝑈𝐶 spaces*, Amer. Math. Monthly**72**(1965), 634–635. MR**0184200****[15]**Stephen Willard,*General topology*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR**0264581**

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Additional Information

DOI:
http://dx.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