In the paper [Pac. J. Math. 1, 5--31 (1951; Zbl 0044.11801)], \textit{R. Arens} and \textit{J. Dugundji} introduced and studied several types of topologies on the spaces of continuous functions such as admissible, proper and splitting topologies. In the paper under review, the authors study analogously Borel splitting and admissible structures on the set of Borel maps. In particular, they prove that there exists at most one Borel structure on \(\mathcal{B}(Y, Z)\) (the set of all Borel maps from a Borel space \(Y\) to a Borel space \(Z\)) which is both Borel splitting and admissible. Moreover, the authors ``define and study some relation between the Borel structures on \(\mathcal{B}(Y,Z) \) and the Borel structures on the set \(\mathcal{B}_Z(Y)\) (consisting of all subsets \(f^{-1}(B)\) of \(Y\), where \(f \in \mathcal{B}(Y,Z)\) and \(B\) is an element of the Borel structure on \(Z\)) concerning the notion'' of coordinately Borel \(\mathcal{A}\)-splitting and coordinately Borel \(\mathcal{A}\)-admissible Borel structures, where \(\mathcal{A}\) is an arbitrary family of Borel spaces. Finally some open question for Borel structures on \(\mathcal{B}(Y,Z)\) and \(\mathcal{B}_Z(Y)\) are posed in this paper.