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Uniform structures on hyperspaces and uniform topologies on spaces of multifunctions

Del Prete I., Di Iorio M., Holá Ľ.

Demonstratio Mathematica

2003

Vol. 36, nr 4

985--1000

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.

Graph convergence of set-valued maps and its relationship to other convergences

Del Prete I., Di Iorio M., Holá L.

Journal of Applied Analysis

2000

Vol. 6, nr 2

213-226

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 ∈ ∑} 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 ∈ ∑} is evenly-outer semicontinuous; (2) the net {{Fσ : σ ∈ ∑} is graph convergent to F; (3) the net {Fσ : σ ∈ ∑} is pointwise convergent to F. This theorem generalizes some results from [4] and [11].