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Math Help - Baire Category Theorem?

  1. #1
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    Baire Category Theorem?

    Alright this question is from Carothers Real Analysis.

    Let (r_n) be an enumeration of \mathbb{Q}. For each each n, let I_n be the open interval centered at r_n of radius 2^{-n}, and let U= \cup_{n=1}^{\infty}I_n. Prove that U is a proper, open subset, dense subset of \mathbb{R} and that U^{c} is nowhere dense in \mathbb{R}.

    Where I am stuck is try to show that U is a proper subset of \mathbb{R}. My first thought was a proof by contradiction, by assuming that \cup_{n=1}^{\infty}I_n=\mathbb{R}. Then by the Baire Categroy theorem one of the int\{ \overline{I}_n\} \ne \emptyset but this didn't seem to go anywhere.


    Thanks for any input
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  2. #2
    Moo
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    Hello,

    I don't know if it leads anywhere, but you could see if U is a closed set or not.
    If it's not, then it cannot be \mathbb{R} nor \emptyset, which are both open and closed.
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  3. #3
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    The total length of the intervals in U is 2, so it's a very long way from being the whole of \mathbb{R}.

    More precisely, the Lebesgue measure of U is at most 2, while the measure of \mathbb{R} is infinite.
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