Classification of the spaces $C_p^*(X)$ within the Borel-Wadge hierarchy for a projective space $X$

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Abstract

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 $\sigma$-compact metrizable space, then the measurable spaces $C_p(X)$ and $C^*_p(X)$ are standard Borel and if $X$ is a metrizable analytic space which is not $\sigma$-compact then the spaces of continuous functions are Borel-$\Pi^1_1$-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 $n \geq 2$ and if $X$ is a separable metrizable space which is in $\Sigma^1_n$ but not in $\Sigma^1_{n-1}$ then the measurable space $C^*_p(X)$ is Borel-$\Pi^1_n$-complete. This completes under the assumption of PD the classification of Borel-Wadge complexity of $C^*_p(X)$ for $X$ projective.