Let $\omega_1$ be the first uncountable ordinal, $\alpha<\omega_1$ an ordinal, and $Y,Z$ two topological spaces. By $\mathbf B^\alpha(Y,Z)$ we denote the set of all Borel maps of class $\alpha$ from $Y$ into $Z$ and by $\mathbf G^Z_\alpha(Y)$ the set consisting of all subsets $f^{-1}(U)$, where $f\in\mathbf B^\alpha(Y,Z)$ and $U$ is an open subset of $Z$. In this paper we introduce and investigate topologies on the sets $\mathbf B^\alpha(Y,Z)$ and $\mathbf G^Z_\alpha(Y)$. More precisely, we generalize the results presented by Arens, Dugundji, Aumann, and Rao (see [1], [2], [3], and [10]) for Borel maps of class $\alpha$.