Geodesic
Pointwise convergence and the Wadge hierarchy
Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 1, pp. 159-172
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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.
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.
Classification : 03E15, 28A05, 54C35, 54H05
Keywords: Wadge hierarchy; function spaces; pointwise convergence 
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     author = {Andretta, Alessandro and Marcone, Alberto},
     title = {Pointwise convergence and the {Wadge} hierarchy},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {159--172},
     year = {2001},
     volume = {42},
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     zbl = {1052.03023},
     language = {en},
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Andretta, Alessandro; Marcone, Alberto. Pointwise convergence and the Wadge hierarchy. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 1, pp. 159-172. http://geodesic.mathdoc.fr/item/CMUC_2001_42_1_a11/