Classification of the spaces Cp⁎(X) within the Borel–Wadge hierarchy for a projective space X
Topology and its Applications
We study the complexity of the space C^*_p(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 C^*_p(X) (and also in the space C_p(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
... le spaces C_p(X) and C^*_p(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-Π^1_1-complete. They also determined under the assumption of projective determinacy (PD) the complexity of C_p(X) for any projective space X and asked whether a similar result holds for C^*_p(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 Σ^1_n but not in Σ^1_n-1 then the measurable space C^*_p(X) is Borel-Π^1_n-complete. This completes under the assumption of PD the classification of Borel-Wadge complexity of C^*_p(X) for X projective.