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

Abstract

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 ${\mathcal{O}}_Z(Y)=\{f^{-1}(U)\in \mathcal{O}(Y):f\in C(Y,Z)\mathrm{and}U\in \mathcal{O}(Z)\}$, introducing new problems in the field.

This paper appeared in the journal Q and A in General Topology, Volume 32, Number 1 (2014)