Summary: Let , be metric spaces, , . As is well known, converges continuously to iff for each sequence in , in . A beautiful result of Kuratowski states that if is compact, the sequence converges uniformly to if and only if converges continuously to . In this paper, we generalize the above result to convergences in hyperspaces. As an example, let , be Hausdorff pseudo-metrics on the power sets of , , respectively. The convergence is called Hausdorf-continuous-convergence iff for any nonempty subsets , of , implies . The following are equivalent:
(a) is a UC space i.e. every continuous function on to a uniform space is uniformly continuous;
(b) uniform convergence Hausdorff-continuous-convergence in ;
(c) uniform convergence Hausdorff-continuous-convergence in .