Topological types of convergence for nets of multifunctions

This article proposes a unified concept of topological types of convergence for nets of multifunctions from one topological space ( X , τ) to another ( Y , σ). Any kind of convergence is representable by a (2n+2)-tuple, n = 0, 1,..., of functions u and l , such that the compositions u ∘ l and l∘u mean Choquet's supremum and infimum operations, respectively, on filters considered in terms of the upper Vietoris topology on the hyperspace of ( Y , σ). Convergence operators are obtained by composing functions from such (2n+2)-tuples. An allocation of places for the two distinguished functions in a convergence operator reflects the structure of the arguments of the used composition. The structure expresses a relationship between the directed set (Σ, ⪯) of the studied nets and the topological space ( X , τ). The set of all possible structures is an infinite monoid of the special three-parameter functions F (p,q) ◦ τ n : X → P 2n+2 (Σ × X), where p ≤ q ≤ 2n. The set of all convergence operators forms a finite monoid whose neutral element determines the pointwise convergence and possesses the structure given by the neutral element of the monoid of structures. We demonstrate the construction process of every convergence operator and show that the notions of the presented concept can characterize many well-known classical types of convergence. Of particular importance are the types of convergence derived from the concept of continuous convergence introduced by O. Frink in 1942. We establish some general theorems about the necessary and sufficient conditions for the continuity of limit multifunctions without any assumptions about the type of continuity of the members of the nets.