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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 in general topology 54B20 Hyperspaces in general topology 54E05 Proximity structures and generalizations 54E15 Uniform structures and generalizations 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)