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Math Help - Analysis, Borel sets

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    Analysis, Borel sets

    Show that if X and Y are topological spaces, and f : X --> Y is a continuous mapping, then f^-1(B) is a Borel set in X whenever B is a Borel set in Y.
    see attached image for hints.
    Attached Thumbnails Attached Thumbnails Analysis, Borel sets-hints.png  
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  2. #2
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    Quote Originally Posted by JJMC89 View Post
    Show that if X and Y are topological spaces, and f : X --> Y is a continuous mapping, then f^-1(B) is a Borel set in X whenever B is a Borel set in Y.
    see attached image for hints.
    As the hint suggests, it is easy to show that for all  E, E_n \in B :

    A.  f^{-1}(\bigcup E_n) = \bigcup (f^{-1}(E_n))
    B.  f^{-1}(E^c) = (f^{-1}(E))^c

    Recall that a sigma algebra is a set closed under compliments and countable unions; and a Borel sigma algebra is the "smallest" sigma algebra containing every open set; thus, you have to show 3 things.

    1. If B is open or closed then so is  f^{-1}(B) . This follows from continuity of f.

    2. If B = union of open and/or closed sets, then  f^{-1}(B) is Borel. This follows from 1 and A.

    3. If B is a compliment of a set from 1 or 2, then  f^{-1}(B^c) is borel. This follows from 1/2 and B.
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    Quote Originally Posted by southprkfan1 View Post
    As the hint suggests, it is easy to show that for all  E, E_n \in B :

    A.  f^{-1}(\bigcup E_n) = \bigcup (f^{-1}(E_n))
    B.  f^{-1}(E^c) = (f^{-1}(E))^c


    I don't know how to show A.
    Last edited by JJMC89; April 22nd 2010 at 09:01 PM.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by JJMC89 View Post
    I don't know how to show A and B.
    Come on. Why don't you try?
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  5. #5
    Moo
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    Quote Originally Posted by Drexel28 View Post
    Come on. Why don't you try?
    Come on. Why don't you help a little ?

    To the OP : consider an element in the set that is LHS of the equation, and prove it belongs to the set that is RHS of the equation.
    And vice-versa
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Moo View Post
    Come on. Why don't you help a little ?
    Because someone doing measure theory and algebraic topology should be expected to just know let alone be able to prove a common fact about mappings.
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