Generating Borel measurable mappings with continuous mappings

We prove that the relative rank r ( B ( X ) : C ( X ) ) of the semigroup of Borel mappings B ( X ) from X to X (with the composition of mappings as the semigroup operation) with respect to the semigroup of continuous functions C ( X ) from X to X is equal to ℵ 1 if X is an uncountable Polish space which either can be retracted to a Cantor subset of X, or contains an arc, or is homeomorphic to its Cartesian square X 2 .