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Math Help - Is this set thick in C0

  1. #1
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    Is this set thick in C0

    Let C0 be the set of all continuous functions from [a,b] to R

    Let A = { f C0: f(q) is irrational for all q Q} where Q is the set of rational numbers.

    The question is: Is A a thick subset of C0.

    EDIT: The answer I had here was very wrong
    Last edited by southprkfan1; March 8th 2010 at 05:49 AM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by southprkfan1 View Post
    Let C0 be the set of all continuous functions from [a,b] to R

    Let A = { f C0: f(q) is irrational for all q Q} where Q is the set of rational numbers.

    The question is: Is A a thick subset of C0.


    I said it is because if we let:

    Gj = { f C0: f(q) Br(p) where r = 1/j and p is irrational for all q Q}

    Then each Gj is open and dense and

    A = {j=1 to ] Gj

    so A is thick.

    ...thoughts?
    What does thick mean?
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  3. #3
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    Quote Originally Posted by Drexel28 View Post
    What does thick mean?
    Right, I forgot some people use different terms. A set is thick in a metric space M if it the intersection of countably many open and dense sets in M. I believe it is also called a 2nd category set or something.
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