Pointwise convergence and the Wadge hierarchy

Alessandro Andretta; Alberto Marcone

Commentationes Mathematicae Universitatis Carolinae (2001)

  • Volume: 42, Issue: 1, page 159-172
  • ISSN: 0010-2628

Abstract

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We show that if X is a Σ 1 1 separable metrizable space which is not σ -compact then C p * ( X ) , the space of bounded real-valued continuous functions on X with the topology of pointwise convergence, is Borel- Π 1 1 -complete. Assuming projective determinacy we show that if X is projective not σ -compact and n is least such that X is Σ n 1 then C p ( X ) , the space of real-valued continuous functions on X with the topology of pointwise convergence, is Borel- Π 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.

How to cite

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Andretta, Alessandro, and Marcone, Alberto. "Pointwise convergence and the Wadge hierarchy." Commentationes Mathematicae Universitatis Carolinae 42.1 (2001): 159-172. <http://eudml.org/doc/248817>.

@article{Andretta2001,
abstract = {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.},
author = {Andretta, Alessandro, Marcone, Alberto},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Wadge hierarchy; function spaces; pointwise convergence; Wadge hierarchy; function spaces; pointwise convergence; analytic sets},
language = {eng},
number = {1},
pages = {159-172},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Pointwise convergence and the Wadge hierarchy},
url = {http://eudml.org/doc/248817},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Andretta, Alessandro
AU - Marcone, Alberto
TI - Pointwise convergence and the Wadge hierarchy
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 1
SP - 159
EP - 172
AB - 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.
LA - eng
KW - Wadge hierarchy; function spaces; pointwise convergence; Wadge hierarchy; function spaces; pointwise convergence; analytic sets
UR - http://eudml.org/doc/248817
ER -

References

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  11. Kuratowski K., Topology, vol. 1, Academic Press, 1966. MR0217751
  12. Lutzer D., van Mill J., Pol R., Descriptive complexity of function spaces, Trans. Amer. Math. Soc. 291 (1985), 121-128. (1985) Zbl0574.54042MR0797049
  13. Marciszewski W., On analytic and coanalytic function spaces C p ( X ) , Topology Appl. 50 (1993), 241-248. (1993) MR1227552
  14. van Mill J., Infinite-Dimensional Topology, North-Holland, 1989. Zbl1027.57022MR0977744
  15. Okunev O., On analyticity in cosmic spaces, Comment. Math. Univ. Carolinae 34 (1993), 185-190. (1993) Zbl0837.54009MR1240216
  16. Wadge W., Reducibility and Determinateness on the Baire Space, Ph.D. thesis, University of California at Berkeley, 1983. 

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