Space of Continuous Set-Valued Mappings with Closed Unbounded Values

Abstract

We consider a space of continuous set-valued mappings defined on a locally compact space T with a countable base. The values of these mappings are closed (not necessarily bounded) sets in a metric space (X, d(·)) whose closed balls are compact. The space (X, d(·)) is locally compact and separable. Let Y be a countable dense set in X. The distance ρ(A,B) between sets A and B belonging to the family CL(X) of all nonempty closed subsets of X is defined as follows:

$$\rho (A, B) = \sum_{i=1}^\infty\frac{1}{2^i}\frac{|d(y_i, A) - d(y_i, B)|}{1+|d(y_i, A) - d(y_i, B)|},$$

where d(y_i,A) is the distance from the point y_i ∈ Y to the set A. This distance is independent of the choice of the set Y, and the function ρ(A,B) is a metric on the space CL(X). The convergence of a sequence of sets A_n, n ≥ 1, in the metric space (CL(X), ρ(·)) is equivalent to the Kuratowski convergence of this sequence. We prove the completeness and separability of the space (CL(X), ρ(·)) and give necessary and sufficient conditions for the compactness of sets in this space. The space C(T,CL(X)) of all continuous mappings from T to (CL(X), ρ(·)) is endowed with the topology of uniform convergence on compact sets in T. We prove the completeness and separability of the space C(T,CL(X)) and give necessary and sufficient conditions for the compactness of sets in this space. These results are reformulated for the space C(T,CCL(X)), where T = [0, 1], X is a finite-dimensional Euclidean space, and CCL(X) is the space of all nonempty closed convex sets in X with metric ρ(·). This space plays a crucial role in the study of sweeping processes. We give a counterexample showing the significance of the assumption of compactness of closed balls in X.