Classification of the spaces [formula omitted] within the Borel–Wadge hierarchy for a projective space X

Abstract

We study the complexity of the space Cp⁎(X) of bounded continuous functions with the topology of pointwise convergence. We are allowed to use descriptive set theoretical methods, since for a separable metrizable space X, the measurable space of Borel sets in Cp⁎(X) (also in the space Cp(X) of all continuous functions) is known to be isomorphic to a subspace of a standard Borel space. It was proved by A. Andretta and A. Marcone that if X is a σ-compact metrizable space, then the measurable spaces Cp(X) and Cp⁎(X) are standard Borel and if X is a metrizable analytic space which is not σ-compact then the spaces of continuous functions are Borel–Π11-complete. They also determined under the assumption of projective determinacy (PD) the complexity of Cp(X) for any projective space X and asked whether a similar result holds for Cp⁎(X).We provide a positive answer, i.e. assuming PD we prove, that if n≥2 and if X is a separable metrizable space which is in Σn1 but not in Σn−11, then the measurable space Cp⁎(X) is Borel–Πn1-complete. This completes under the assumption of PD the classification of Borel–Wadge complexity of Cp⁎(X) for X projective.