Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance
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- by Gerald Beer PDF
- Proc. Amer. Math. Soc. 95 (1985), 653-658 Request permission
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
- Masahiko Atsuji, Uniform continuity of continuous functions of metric spaces, Pacific J. Math. 8 (1958), 11–16; erratum, 941. MR 99023
- Jean-Pierre Aubin, Applied abstract analysis, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. Exercises by Bernard Cornet and Hervé Moulin; Translated from the French by Carole Labrousse. MR 0470034
- C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310
- Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
- Hermann Hueber, On uniform continuity and compactness in metric spaces, Amer. Math. Monthly 88 (1981), no. 3, 204–205. MR 619571, DOI 10.2307/2320473
- B. T. Levšenko, On the concept of compactness and point-finite coverings, Mat. Sb. N.S. 42(84) (1957), 479–484 (Russian). MR 0096185
- Somashekhar Amrith Naimpally, Graph topology for function spaces, Trans. Amer. Math. Soc. 123 (1966), 267–272. MR 192466, DOI 10.1090/S0002-9947-1966-0192466-4
- Jun-iti Nagata, On the uniform topology of bicompactifications, J. Inst. Polytech. Osaka City Univ. Ser. A 1 (1950), 28–38. MR 37501
- B. Penkov and Bl. Sendov, Hausdorffsche Metrik und Approximationen, Numer. Math. 9 (1966), 214–226 (German). MR 204927, DOI 10.1007/BF02162085
- 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
- John Rainwater, Spaces whose finest uniformity is metric, Pacific J. Math. 9 (1959), 567–570. MR 106448
- 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
- Gh. Toader, On a problem of Nagata, Mathematica (Cluj) 20(43) (1978), no. 1, 77–79. MR 530953
- W. C. Waterhouse, On $UC$ spaces, Amer. Math. Monthly 72 (1965), 634–635. MR 184200, DOI 10.2307/2313854
- Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581
Additional Information
- © Copyright 1985 American Mathematical Society
- 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