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On continuous convergence of nets of multifunctions

Przemski M.

Demonstratio Mathematica

2011

Vol. 44, nr 1

181--200

This paper expands the classical concept of the continuous convergence of nets of multifunctions introduced by Cao, Reilly and Vamanamurthy in [7]. We introduce some new types of properties of convergence of such nets which guarantee the upper or lower semicontinuity of the limit multifunction. Furthermore, we obtain some analogous results concerning generalized continuity properties of multifunctions.

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].