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More about metric spaces on which continuous functions are uniformly continuous

Published online by Cambridge University Press:  17 April 2009

Gerald Beer
Gerald Beer
Affiliation:
Department of Mathematics, California State University, Los Angeles, CA 90032, U.S.A.
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Abstract

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An Atsuji space is a metric space X such that each continuous function form X to an arbitrary metric space Y is uniformly continuous. We here present (i) characterizations of metric spaces with Atsuji completions; (ii) Cantor-type theorems for Atsuji spaces; (iii) a fixed point theorem for self-maps of an Atsuji space.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 33 , Issue 3 , June 1986 , pp. 397 - 406
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Atsuji, M., “Uniform continuity of continuous functions of metric spaces”, Pacific J. Math. 8 (1958), 11–16.CrossRefGoogle Scholar
[2]Aubin, J. P., Applied abstract analysis, Wiley, New York, (1977).Google Scholar
[3]Beer, G., “Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance”, Proc. Amer. Math. Soc. 95 (1985), 653–658.CrossRefGoogle Scholar
[4]Choquet, G., Lectures on analysis, vol. 1, Benjamin, New York, (1969).Google Scholar
[5]Engleking, R., General topology, Polish Scientific Publishers, Warsaw, (1977).Google Scholar
[6]Goldberg, R., Methods of real analysis, Wiley, New York, (1976).Google Scholar
[7]Hadziivanov, N. G., “Metric spaces in which every continuous function is uniformly continuous”, Annuaire Univ. Sofia Fac. Math. 59 (1964/1965), 105–115. (Bulgarian Russian summary).Google Scholar
[8]Hueber, H., “On uniform continuity and compactness in metric spaces”, Amer. Math. Monthly, 88 (1981), 204–205.CrossRefGoogle Scholar
[9]Istratescu, V., Fixed point theory, Reidel, Dordrecht, Holland, (1981).CrossRefGoogle Scholar
[10]Kuratowski, K., “Sur les espaces complets”, Fund. Math. 15 (1930), 301–309.CrossRefGoogle Scholar
[11]Rainwater, J., “Spaces whose finest uniformity is metric”, Pacific J. Math. 9 (1959) 567–570.CrossRefGoogle Scholar
[12]Simmons, G., Introduction to topology and modern analysis, McGraw-Hill, (1963).Google Scholar
[13]Toader, Gh., “On a Problem of Nagata”, Mathematica (Cluj), 20 (43) (1978), 78–79.Google Scholar
[14]Waterhouse, W., “On UC spaces”, Amer. Math. Monthly, 72 (1965), 634–635.CrossRefGoogle Scholar