N^N^N does not satisfy Normann's condition

Abstract

We prove that the Kleene-Kreisel space $N^N^N$ does not satisfy Normann's condition. A topological space $X$ is said to fulfil Normann's condition, if every functionally closed subset of $X$ is an intersection of clopen sets. The investigation of this property is motivated by its strong relationship to a problem in Computable Analysis. D. Normann has proved that in order to establish non-coincidence of the extensional hierarchy and the intensional hierarchy of functionals over the reals it is enough to show that $N^N^N$ fails the above condition.