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

TL;DR

This paper proves that the topological space N^N^N does not satisfy Normann's condition, which is significant for understanding the hierarchy of functionals in Computable Analysis.

Contribution

It establishes that N^N^N fails Normann's condition, providing insight into the structure of function spaces relevant to Computable Analysis.

Findings

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

Implication for the non-coincidence of functional hierarchies

Advances understanding of topological properties in Computable Analysis

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.