# Thread: analytic pointset, Baire space

1. ## 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.

2. 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 &bull; 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.)