Upper and lower $(α, β,θ,δ,\mathcal{I})$-continuous multifunctions - C. Carpintero; E. Rosas; J. Sanabria; J. Vielma (Математичні Студії, 2021)
Let $(X, τ)$ and $(Y, σ)$ be topological spaces in which no separation axioms are assumed, unless explicitly stated and if $\mathcal{I}$ is an ideal on $X$. Given a multifunction $F\colon (X, τ)→ (Y, σ)$, $α,β$ operators on $(X, τ)$, $θ,δ$ operators on $(Y, σ)$ and $\mathcal{I}$ a proper ideal on $X$. We introduce and study upper and lower $(α, β,θ,δ,\mathcal{I})$-continuous multifunctions. A multifunction $F\colon (X, τ)→ (Y, σ)$ is said to be: {1)} upper-$(α, β,θ,δ,\mathcal{I})$-continuous if $α(F^{+}(δ(V)))\setminus β(F^{+}(θ(V)))\in \mathcal{I}$ for each open subset $V$ of $Y$;\ {2)} lower-$(α, β,θ,δ,\mathcal{I})$-continuous if $α(F^{-}(δ(V)))\setminus β(F^{-}(θ(V)))\in \mathcal{I}$ for each open subset $V$ of $Y$;\ {3)} $(α, β,θ,δ,\mathcal{I})$-continuous if it is upper-\ %$(α, β,θ,δ,\mathcal{I})$-continuous and lower-$(α, β,θ,δ,\mathcal{I})$-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 $(Y, σ)$: