On Uniform Convergence of Continuous Functions and Topological Convergence of Sets
1983 ◽
Vol 26
(4)
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pp. 418-424
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Keyword(s):
AbstractLet 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.
1985 ◽
Vol 28
(1)
◽
pp. 52-59
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1986 ◽
Vol 29
(4)
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pp. 463-468
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2009 ◽
Vol 29
(4)
◽
pp. 1141-1161
1908 ◽
Vol 28
◽
pp. 249-258
1979 ◽
Vol 20
(3)
◽
pp. 367-375
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Keyword(s):