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Wadge hierarchy; function spaces; pointwise convergence
We show that if $X$ is a $\Sigma _1^1$ separable metrizable space which is not $\sigma $-compact then $C_p^* (X)$, the space of bounded real-valued continuous functions on $X$ with the topology of pointwise convergence, is Borel-$\Pi _1^1$-complete. Assuming projective determinacy we show that if $X$ is projective not $\sigma $-compact and $n$ is least such that $X$ is $\Sigma _n^1$ then $C_p (X)$, the space of real-valued continuous functions on $X$ with the topology of pointwise convergence, is Borel-$\Pi _n^1$-complete. We also prove a simultaneous improvement of theorems of Christensen and Kechris regarding the complexity of a subset of the hyperspace of the closed sets of a Polish space.
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