Graph Convergence of Set-Valued Maps and its Relationship to Other Convergences
Ida Del Prete, M. Di Iorio, Ľubica Holá · Journal of Applied Analysis · 2000
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].