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

Summary: Let $\left(X,d\right)$, $\left(Y,e\right)$ be metric spaces, ${f}_{n}$, $f\in C\left(X,Y\right)$. As is well known, ${f}_{n}$ converges continuously to $f$ iff for each sequence ${x}_{n}\to x$ in $X$, ${f}_{n}\left({x}_{n}\right)\to f\left(x\right)$ 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 Hausdorf-continuous-convergence iff for any nonempty subsets ${A}_{n}$, $A$ of $X$, ${d}_{H}\left({A}_{n},A\right)\to 0$ implies ${e}_{H}\left({f}_{n}\left({A}_{n}\right),f\left(A\right)\right)\to 0$. 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 $⇒$ Hausdorff-continuous-convergence in $C\left(X,Y\right)$;

(c) uniform convergence $⇔$ Hausdorff-continuous-convergence in $C\left(X,Y\right)$.

##### 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