THE INFIMAL VALUE FUNCTIONAL AND THE UNIFORMIZATION OF HIT-AND-MISS HYPERSPACE TOPOLOGIES

We give necessary and sufficient conditions for the uniformizability o f hit-and-miss and proximal hit-and-miss hyperspace topologies defined on the nonempty closed subsets CL(X) of a Hausdorff uniform space (X, U). In the case of uniformizability, one can always,find a family F o f continuous functions on X into [0, 1] so that the hyperspace topolog y is the weak topology induced by {m(f): f is an element of F}, where for each f, m(f) :CL(X) --> [0, 1] is the infimal value functional def ined by m(f)(A) = inf{f(x) : x is an element of A}.