Uniform Convergence for a Hyperspace

Abstract

Abstract.In this note a uniform convergence in the collection 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) and it is shown that the two convergences agree on Richardson's class T. In the case of a regular 7\ topological space (E, t) this means that there is a uniform convergence structure on E, which induces 7, such that uniform convergence in C(E) is convergences in the Vietoris topology on C(E).1. Definition of uniform convergence. Let C(£) be the collection of nonempty, compact subsets of a Hausdorff topological space (£, t). Then C(£) may be equipped with the Vietoris topology. If, in addition, (£, t) is completely regular and *% is a uniform structure which induces t, then C(£) also carries the uniform topology. A classical result is that the Vietoris and uniform topologies agree on C(£) (see [6]). Now let C(£) be the collection of nonempty, compact subsets of a Hausdorff convergence space (£, i) (see [3]). There is a reasonable way to generalize the Vietoris and uniform topologies to this setting. In fact, in [4] a convergence h(t) for C(£) was defined. It was shown that h(t) is the Vietoris topology for topological t. Moreover h(t) is Hausdorff (regular) (compact) whenever / is Hausdorff (regular) (compact and regular). But no additional properties of t (such as complete regularity) are needed to make a reasonable definition of uniform convergence in C(£) and this is done below. Let (£, J') be a separated uniform convergence space (see [2]). In [1] Cochran defined a U* base for ß to be a base ß for # such that each member of ß is coarser than the diagonal filter, each member of ß is its own inverse, the composition of any two members of ß exists and is finer than a third member, and the infimum of two members of ß is again in ß. Each uniform convergence space has a U* base; put ß=(Jecf:J[ A], J=J~1). It should also be pointed out that each Hausdorff topological space (£, t), indeed each Hausdorff convergence space (£, t), has a uniform convergence structure ß, constructively defined, such that Received by the editors February 5,1973. AMS (MOS) subject classifications (1970). Primary 54A05, 54A20, 54B20, 54E15.