Characterizations Of Upper And Lower (Α, Β, Θ, Δ, I)-Continuous Multifunctions

Abstract

Let (X, τ ) and (Y, σ) be topological spaces in which no separation axioms are assumed, unless explicitly stated and if I is an ideal on X. Given a multifunction F : (X, τ ) → (Y, σ), α, β operators on (X, τ ), θ, δ operators on (Y, σ) and I a proper ideal on X. We introduce and study upper and lower (α, β, θ, δ, I)-continuous multifunctions. A multifunction F : (X, τ ) → (Y, σ) is said to be: 1) upper-(α, β, θ, δ, I)-continuous if α(F +(δ(V ))) \ β(F +(θ(V ))) ∈ I for each open subset V of Y ; 2) lower-(α, β, θ, δ, I)-continuous if α(F −(δ(V ))) \ β(F −(θ(V ))) ∈ I for each open subset V of Y ; 3) (α, β, θ, δ, I)-continuous if it is upper- and lower-(α, β, θ, δ, I)- continuous. In particular, the following statements are proved in the article (Theorem 2): Let α, β be operators on (X, τ ) and θ, θ∗ , δ operators on (Y, σ): 1. The multifunction F : (X, τ ) → (Y, σ) is upper (α, β, θ ∩ θ ∗ , δ, I)-continuous if and only if it is both upper (α, β, θ, δ, I)-continuous and upper (α, β, θ∗ , δ, I)-continuous. 2. The multifunction F : (X, τ ) → (Y, σ) is lower (α, β, θ ∩ θ ∗ , δ, I)-continuous if and only if it is both lower (α, β, θ, δ, I)-continuous and lower (α, β, θ∗ , δ, I)-continuous, provided that β(A ∩ B) = β(A) ∩ β(B) for any subset A, B of X.