  Let $Y$ and $Z$ be two given topological spaces, ${\cal O}(Y)$ (respectively,
${\cal O}(Z)$) 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.
