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The notion of even-outer-semicontinuity for set-valued maps is introduced and compared with related ones from [Bagh, Wets, Set-Valued Anal. 4: 333–360, 1996] and [Kowalczyk, Demonstratio Math. 27: 79–87, 1994]. 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, { F σ   :  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 { F σ   :  σ  ∈ Σ} is evenly-outer semicontinuous; (2) the net { F σ   :  σ  ∈ Σ} is graph convergent to  F ; (3) the net { F σ   :  σ  ∈ Σ} is pointwise convergent to  F . This theorem generalizes some results from [Bagh, Wets, Set-Valued Anal. 4: 333–360, 1996] and [Kowalczyk, Demonstratio Math. 27: 79–87, 1994].

Graph Convergence of Set-Valued Maps and its Relationship to Other Convergences