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Thread: Baire Category Theorem?

  1. #1
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    Baire Category Theorem?

    Alright this question is from Carothers Real Analysis.

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

    Where I am stuck is try to show that $\displaystyle U$ is a proper subset of $\displaystyle \mathbb{R}$. My first thought was a proof by contradiction, by assuming that $\displaystyle \cup_{n=1}^{\infty}I_n=\mathbb{R}$. Then by the Baire Categroy theorem one of the $\displaystyle 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 $\displaystyle U$ is a closed set or not.
    If it's not, then it cannot be $\displaystyle \mathbb{R}$ nor $\displaystyle \emptyset$, 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 $\displaystyle \mathbb{R}$.

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