ベール空間
A subset of Baire space has a corresponding subset of Cantor space under the map that takes each function from formula_41 to formula_41 to the characteristic function of its graph. A subset of Baire space is given the classification formula_13, formula_14, or formula_32 if and only if the corresponding subset of Cantor space has the same classification. An equivalent definition of the analytical hierarchy on Baire space is given by defining the analytical hierarchy of formulas using a functional version of second-order arithmetic; then the analytical hierarchy on subsets of Cantor space can be defined from the hierarchy on Baire space. This alternate definition gives exactly the same classifications as the first definition.