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I'm writing my thesis about Baire Classes. I want to prove that the 13 base Conway function is in the Baire two class. I need a clear proof of this fact, a proof in which is described how to "build" a sequence of Baire one functions that converges pointwise to our function, or every other way that DOESN'T involve Borel Spaces. Thank you!

Luca Cardarelli
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  • Please share first your thoughts about the proof. – KBS May 03 '22 at 20:18
  • Perhaps Andrés E. Caicedo's answer to Examples of Baire class 2 functions, specifically his verification of his 2nd example -- a verification that he says is similar to how one can show the Conway base $13$ is Baire $2.$ The references I give in my answer to About references for Baire class one function and Baire class two function should also be helpful more generally for your thesis. – Dave L. Renfro May 03 '22 at 20:25
  • @DaveL.Renfro I already read these articles, but there is not a "real" proof. Thank you by the way for the advice. – Luca Cardarelli May 04 '22 at 08:11
  • but there is not a "real" proof --- Of course you'll have to fill in the details, but the overall strategy (the hard part I would think) is probably a useful approach: Look for functions with finitely many discontinuities (thus, Baire one) that converge to functions which in turn converge to the Conway base $13$ function, where the last sequence of functions arises in some standard way from its construction. Ordinarily this would only show Baire $3,$ but more than likely the first convergence is uniform, which makes the first limits continue to be Baire one functions. – Dave L. Renfro May 04 '22 at 11:56
  • @DaveL.Renfro The problem of the strategy you propose is that i'm not sure that the last limit is uniform, not even in the second example of the link you gave me. By the way, i'm currently not able to "build" a sequence of Baire 1 functions that pointwise converges to Conway function, or a sequence of Baire 2 functions that uniformly converges to Conway function, and i'm pretty sure that there are no books in which i can find it. Can you find an explicit way to solve the problem? Thank you! – Luca Cardarelli May 12 '22 at 07:06

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