Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-08T08:21:47.659Z Has data issue: false hasContentIssue false

Hausdorff Distance and a Compactness Criterion for Continuous Functions

Published online by Cambridge University Press:  20 November 2018

Gerald Beer*
Affiliation:
Department of Mathematics California State University, Los Angeles Los Angeles, California 90032
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let 〈X, dx and 〈Y, dY be metric spaces and let hp denote Hausdorff distance in X x Y induced by the metric p on X x Y given by p[(x1, y1), (x2, y2)] = max ﹛dx(x1, x2),dY(y1, y2)﹜- Using the fact that hp when restricted to the uniformly continuous functions from X to Y induces the topology of uniform convergence, we exhibit a natural compactness criterion for C(X, Y) when X is compact and Y is complete.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Atsuji, M., Uniform continuity of continuous functions of metric spaces, Pacific J. Math. 8 (1958), pp. 1116.Google Scholar
2. Beer, G., On uniform convergence of continuous functions and topological convergence of sets, Canad. Math. Bull. 26 (1983), pp. 418–24.Google Scholar
3. Beer, G., More on convergence of continuous functions and topological convergence of sets, Canad. Math. Bull. 28 (1985), pp. 5259.Google Scholar
4. Beer, G., Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance, Proc. Amer. Math Soc. 95 (1985), pp. 653–58.Google Scholar
5. Castaing, C. and Valadier, M., Convex analysis and measurable multifunctions, Springer-Verlag, Berlin, 1977.Google Scholar
6. Kuratowski, K., Topology, Vol. 1., Academic Press, New York, 1966.Google Scholar
7. Naimpally, S., Graph topology for function spaces, Trans. Amer. Math. Soc. 123 (1966) pp. 267 —72.Google Scholar
8. Penkov, B. and Bl. Sendov, Hausdorffsche metrik und approximationen, Num. Math 9 (1966), pp. 214–26.Google Scholar
9. Bl., Sendov, Hausdorff approximations, Bolgar. Acad. Nauk., Sofia, 1979 (Russian).Google Scholar
10. Stromberg, K., An introduction to classical real analysis, Wadsworth, Belmont, CA, 1981.Google Scholar