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Math Help - analytic pointset, Baire space

  1. #1
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    analytic pointset, Baire space

    Prove that the inverse image g^{-1}[A] of an analytic pointset A by a continuous function g : \mathcal{N} \rightarrow \mathcal{N} is analytic.

    Hint. Aim for an equivalence of the form y \in g^{-1}[A] \text{ } \Leftrightarrow \text{ }(\exists x) \text{ } [y=f(\rho_1(x))=g(\rho_2(x))] where f is continuous and \rho_n are defined by \rho_n(z) = (i \mapsto z(\rho(n, i))), and then use the result that says:
    If f, g : \mathcal{N} \rightarrow \mathcal{N} are continuous functions, then the set E=\{ x | f(x)=g(x) \} of points on which they agree is closed.

    I am not sure how to prove this. I would appreciate a few hints or suggestions. In our book, \mathcal{N} denotes the Baire space. Also, subsets of the Baire space are called pointsets. Thanks in advance.
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  2. #2
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    This is problem from Moschovakis' book, problem x10.5., p.153.
    In the hint the author refers to the proof of another theorem on p. 153.

    The same problem was posted here:
    Re: analytic pointset, Baire space (In this thread Henno Brandsma gave a pointer to a proof of this fact in Jech's set theory.)
    S.O.S. Mathematics CyberBoard :: View topic - analytic pointset, Baire space
    http://www.mymathforum.com/viewtopic...c453858c85d8f6
    http://www.mathhelpforum.com/math-he...ire-space.html
    Art of Problem Solving • View topic - analytic pointset, Baire space In this point I tried to explain the hint in the way it is given in the book (since I do not find the context given in the post sufficient.)
    (I'm posting these links in order to save the time of the helpers - in case the problem will be solved in one of the forums.)
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