Results 1 to 2 of 2

Math Help - analytic pointset, Baire space

  1. #1
    Junior Member
    Joined
    Mar 2009
    Posts
    29

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Mar 2010
    From
    Bratislava
    Posts
    115
    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.)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. example, closed pointset
    Posted in the Discrete Math Forum
    Replies: 7
    Last Post: April 18th 2010, 03:21 PM
  2. Baire space homeomorphism question - again
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: January 25th 2010, 11:57 AM
  3. powers of Baire space are homeomorphic to Baire space
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: December 15th 2009, 03:46 PM
  4. base for the topology of baire space N^N
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: December 12th 2009, 07:49 AM
  5. [SOLVED] How do I prove R is a baire space?
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: June 24th 2008, 06:33 AM

Search Tags


/mathhelpforum @mathhelpforum