Geodesic
On Uniform Convergence of Continuous Functions and Topological Convergence of Sets
Canadian mathematical bulletin, Tome 26 (1983) no. 4, pp. 418-424

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-1983-069-6
Mots-clés : 40A30, 54C35, 54B20, 54A20, uniform convergence, topological convergence of sets, Hausdorff metric 
Beer, Gerald. On Uniform Convergence of Continuous Functions and Topological Convergence of Sets. Canadian mathematical bulletin, Tome 26 (1983) no. 4, pp. 418-424. doi: 10.4153/CMB-1983-069-6
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