Abstract. The aim of this paper is to study uniform and topological structures on spaces of multifunctions. Uniform structures on hyperspaces compatible with the Fell, the Wijsman and the HausdorfF metric topology respectively are studied and the links between them are explored. Topologies induced by the above uniformities on spaces of multifunctions are considered and compared. Also connections between uniform convergence of multifunctions and their equi-semicontinuity are investigated.Continuing the investigation of [Mcl], [Mc2] of uniform topologies on compacta on spaces of multifunctions, we realized that the study of uniform structures on hyperspaces allows us to find relationships between uniform topologies on compacta on spaces of multifunctions and also sheds more light on definitions of equi-semicontinuity (for multifunctions) scattered in the literature [Pa2], [Ko], [BW], [DDH]. For this we first deal with uniform structures on hyperspaces.We concentrate upon three important uniformities: a uniformity compatible with the Fell, the Wijsman and the Hausdorff metric topologies respectively. In the literature [Be] we can find complete results concerning relations between the Fell, Wijsman and Hausdorff metric topology, however necessary and sufficient conditions for the coincidence of uniformities are not known. In our paper we clarify also the relationships between the uniformities.Then we utilize the results concerning uniformities on hyperspaces in the study of uniform topologies on compacta on spaces of multifunctions.

The space C(X, Y) of continuous functions from a topological space X to a Hausdorff space Y can be thought of as a subset of the hyperspace of closed subsets of X × Y by identifying each element of C(X, Y) with its graph. A study is made of C(X, Y) with the topology inherited by the Fell topology on hyperspaces. The emphasis is on real‐valued functions where Y=ℝ, in which case the function space is denoted by C(X). A number of characterizations are given of topological properties of C(X) and C(X, Y) in terms of properties on X (and Y). These properties often turn out to involve both local compactness and connectedness type of conditions.

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