On continuous convergence of nets of multifunctions

Abstract

Abstract. 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. Introduction and preliminariesFor a subset A of a topological space (X, π) we denote by Cl(A) and Int(A) the closure and the interior of A, respectively. By a multifunction F : X → Y we mean a correspondence which assigns to each element x of X a nonempty subset F (x) of Y. The upper and lower inverse images of a set B ⊂ Y under F are defined by F + (B) = {x ∈ X : F (x) ⊂ B} and F − (B) = {x ∈ X : F (x) ∩ B = ∅}, respectively. For any A ⊂ X its image under F is the set F (A) = {F (x) ⊂ Y : x ∈ A}. Definition 1.1. A multifunction F : (X, π) → (Y, τ ) is said to be (a) upper semi continuous (briefly u.s.c.) (resp. lower semi continuous (briefly l.s.c.)) at point x ∈ X if for each subsetThe set of all points at which F is u.s.c. (resp. l.s.c.) is denoted by [28]; (b) upper α-continuous (briefly u.α.c.) (resp. lower α-continuous (briefly l.α.c.)) at point x ∈ X if for each subset V ∈ τ such that x ∈ F + (V ) (resp. x ∈ F − (V )), x ∈ Int(Cl(Int(F + (V )))) (resp. x ∈ Int(Cl(Int(F − (V )))) ). The set of all points at which F is u.α.c. (resp. l.s.c.) is denoted by αC u (F ) (resp. αC l (F )), [26];2000 Mathematics Subject Classification: 54A20, 54B20, 54C08, 54C60, 54E15.

A study of function space topologies for multifunctions

“…Our study here is purely topological, unlike [11], where metric spaces and normed spaces are considered for similar results. Similarly, the continuous convergence introduced in our paper is different from that of [2] and [18]. In [2] and [18], upper and lower topologies, defined on the second space, are used for defining continuous convergence.…”

“…Similarly, the continuous convergence introduced in our paper is different from that of [2] and [18]. In [2] and [18], upper and lower topologies, defined on the second space, are used for defining continuous convergence. However our definition is more straight forward and appears similar to its counterpart of single-valued functions.…”