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On Uniform Convergence of Continuous Functions and Topological Convergence of Sets

Published online by Cambridge University Press:  20 November 2018

Gerald Beer
Gerald Beer*
Affiliation:
Department of Mathematics, California State University, Los AngelesLos Angeles, California 90032
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Abstract

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Let X and Y be metric spaces. This paper considers the relationship between uniform convergence in C(X, Y) and topological convergence of functions in C(X, Y), viewed as subsets of X×Y. In general, uniform convergence in C(X, Y) implies Hausdorff metric convergence which, in turn, implies topological convergence, but if X and Y are compact, then all three notions are equivalent. If C([0, 1], Y) is nontrivial arid topological convergence in C(X, Y) implies uniform converger ce then X is compact. Theorem: Let X be compact and Y be loyally compact but noncompact. Then topological convergence in C(X, Y) implies uniform convergence if and only if X has finitely many components. We also sharpen a related result of Naimpally.

Keywords

40A3054C3554B2054A20uniform convergencetopological convergence of setsHausdorff metric
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 26 , Issue 4 , 01 December 1983 , pp. 418 - 424
Copyright
Copyright © Canadian Mathematical Society 1983

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