Admissible topologies on $C(Y,Z)$ and ${\mathcal{O}}_Z(Y)$

Let $Y$ and $Z$ be two given topological spaces, $\mathcal{O}(Y)$ (respectively, $\mathcal{O}(Z)$) the set of all open subsets of $Y$ (respectively, $Z$), and $C(Y,Z)$ the set of all continuous maps from $Y$ to $Z$. We study Scott type topologies on $\mathcal{O}(Y)$ and we construct admissible topologies on $C(Y,Z)$ and ${\mathcal O}_Z(Y)={f^{-1}(U)\in {\mathcal O}(Y): f\in C(Y,Z)
{\rm and}\ U\in {\mathcal O}(Z)}$, introducing new problems in the field.