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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-(a, β, θ, δ, I)-continuous if α(δ(V))) \ β(F+(θ(V))) ϵ I for each open subset V of Y; 2) lower-(α, β, θ, δ, I)-continuous if α(F-(δ(ν))) \ β(Γ-(θ(ν))) ϵ 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 η 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 η 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 η B) = β(Α) η β(Β) for any subset A, B of X.