In this note a uniform convergence in the collection C ( E ) C(E) of nonempty, compact subsets of a separated uniform convergence space E is defined. This convergence is compared with the hyperspace convergence on C ( E ) C(E) and it is shown that the two convergences agree on Richardson’s class Γ \Gamma . In the case of a regular T 1 {T_1} topological space ( E, t ) this means that there is a uniform convergence structure on E , which induces t , such that uniform convergence in C ( E ) C(E) is convergences in the Vietoris topology on C ( E ) C(E) .