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How to construct a real Baire function of class $\alpha$ which is not in the class $\alpha-1$, where $\alpha\in \mathbb{N}$? I believe the solution to this problem is equivalent to constructing a $\Sigma_{\alpha+1}^0$ set which is not in $\Sigma_{\alpha}^0$ (using characteristic functions), any such construction in $\mathbb{R}$?

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    $\begingroup$ Kechris's book "Classical Descriptive Set Theory" is a good reference for this sort of thing $\endgroup$ – Trevor Wilson Mar 2 '13 at 17:23
  • $\begingroup$ One of the standard arguments for producing sets in $\Sigma_{\alpha+1}^0 \setminus \Sigma_\alpha^0$ is using universal sets and diagonalization. See Theorems 22.3 and 22.4 in Kechris's book. $\endgroup$ – Martin Mar 3 '13 at 8:54
  • $\begingroup$ A recent post on MathOverflow on the same topic: Examples of Baire Class $\xi+1$ but not $\xi$ functions for each countable ordinal $\xi.$. $\endgroup$ – Martin Sleziak May 31 '18 at 10:33

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