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Let ω1 be the first uncountable ordinal, α < ω1 an ordinal, and Y, Z two  topological spaces. By Bα(Y,Z) we denote the set of all Borel maps of class  α from Y into Z and by GZα(Y) the set consisting of all subsets f -1(U),  where f  Bα(Y,Z) and U is an open subset of Z. In this paper we introduce  and investigate topologies on the sets Bα(Y,Z) and GZα(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 α.

Topologies on the set of Borel maps of class α