Abstract. The notion of even-outer-semicontinuity for set-valued maps is introduced and compared with related ones from [4] and [11]. The coincidence of these notions provides a new characterization of compactness and of local compactness. The following result is proved: Let X be a topological space, Y a uniform space, {Fσ : σ ∈ Σ} be a net of set-valued maps from X to Y and F be a set valued map from X to Y . Then any two of the following conditions imply the third: (1) the net {Fσ : σ ∈ Σ} is evenly-outer semicontinuous; (2) the net {Fσ : σ ∈ Σ} is graph convergent to F ; (3) the net {Fσ : σ ∈ Σ} is pointwise convergent to F . This theorem generalizes some results from [4] and [11].Graph convergence (that is Painlevé-Kuratowski convergence of graphs) of set-valued maps was studied in many books and papers (see for example [1,2,4,9,11]). In this topic we can include also graph convergence of single-valued maps [5,12,13], epiconvergence of lower semicontinuous functions [4,6,7] as well as Painlevé-Kuratowski convergence of graphs of partial maps [8] . In the books of Attouch [1], Aubin-Frankowska [2] 1991 Mathematics Subject Classification. 54C60, 54B20.

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.