Let $Y$ and $Z$ be two arbitrary fixed topological spaces, $C(Y, Z)$ the set of all continuous maps from $Y$ to $Z$, and $\mathcal{O}_{Z}(Y)$ the set consisting of all open subsets $V$ of $Y$ such that $V = f^{-1}(U)$, where $f \in C(Y, Z)$ and $U$ is an open subset of $Z$. In this paper we continue the study of the $\mathcal{A}$-proper and $\mathcal{A}$-admissible topologies on $\mathcal{O}_{Z}(Y)$, where $\mathcal{A}$ is an arbitrary family of spaces, initiated in [6] and we offer new results concerning the finest $X$-proper topology $\tau(\{X\})$ on $\mathcal{O}_{Z}(Y)$ for several metrizable spaces $X$.