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Hyper-continuous convergence in function spaces. (English) Zbl 1066.54019

Summary: Let (X,d), (Y,e) be metric spaces, f n , fC(X,Y). As is well known, f n converges continuously to f iff for each sequence x n x in X, f n (x n )f(x) in Y. A beautiful result of Kuratowski states that if X is compact, the sequence f n converges uniformly to f if and only if f n converges continuously to f. In this paper, we generalize the above result to convergences in hyperspaces. As an example, let d H , e H be Hausdorff pseudo-metrics on the power sets of X, Y, respectively. The convergence f n f is called Hausdorf-continuous-convergence iff for any nonempty subsets A n , A of X, d H (A n ,A)0 implies e H (f n (A n ),f(A))0. The following are equivalent:

(a) X is a UC space i.e. every continuous function on X to a uniform space is uniformly continuous;

(b) uniform convergence Hausdorff-continuous-convergence in C(X,Y);

(c) uniform convergence Hausdorff-continuous-convergence in C(X,Y).

MSC:
54C35Function spaces (general topology)
54B20Hyperspaces (general topology)
54E05Proximity structures and generalizations
54E15Uniform structures and generalizations
54A20Convergence in general topology