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

  • I. Del Prete , M. Di Iorio and L̆. Holá

Abstract

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, {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 [Bagh, Wets, Set-Valued Anal. 4: 333–360, 1996] and [Kowalczyk, Demonstratio Math. 27: 79–87, 1994].

Received: 1999-06-01
Revised: 2000-02-28
Published Online: 2010-06-07
Published in Print: 2000-December

© Heldermann Verlag

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