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Hyper-continuous convergence in function spaces. (English) Zbl 1066.54019
Summary: Let $(X, d)$, $(Y, e)$ be metric spaces, $f_n$, $f\in C(X, Y)$. As is well known, $f_n$ converges continuously to $f$ iff for each sequence $x_n\to x$ in $X$, $f_n(x_n)\to f(x)$ 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 $f_n\to f$ is called Hausdorff-continuous-convergence iff for any nonempty subsets $A_n$, $A$ of $X$, $d_H(A_n, A)\to 0$ implies $e_H(f_n(A_n),f(A))\to 0$. 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 $\Rightarrow$ Hausdorff-continuous-convergence in $C(X,Y)$; (c) uniform convergence $\Leftrightarrow$ Hausdorff-continuous-convergence in $C(X,Y)$.

MSC:
 54C35 Function spaces (general topology) 54B20 Hyperspaces (general topology) 54E05 Proximity structures and generalizations 54E15 Uniform structures and generalizations 54A20 Convergence in general topology