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Thread: Cantor Intersection Theorem

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    Cantor Intersection Theorem

    Suppose that H is a closed bounded set of real numbers and that (U sub n) is an expanding sequence of open sets.

    (a) Explain why the sequence of sets H \ (U sub n) is a contracting sequence of closed bounded sets.

    (b) Use the Cantor intersection theorem to deduce that if H \ (U sub n) does not equal an empty set for every n, then the intersection from n=1 to infinity, of (H \ (U sub n)) does not equal an empty set.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Slazenger3 View Post
    Suppose that $\displaystyle H$ is a closed bounded set of real numbers and that $\displaystyle U_n$ is an expanding sequence of open sets.

    (a) Explain why the sequence of sets $\displaystyle H-U_n$is a contracting sequence of closed bounded sets.
    Note that $\displaystyle U_n\subseteq U_{n+1}\implies H-U_{n}\supseteq H- U_{n+1}$. So, they're clearly contracting. Also, We know that $\displaystyle H$ is closed and $\displaystyle U_n$ open, thus $\displaystyle H-U_n=H\cap \left(U_n\right)'$ is the intersection of two closed sets, thus closed. Lastly, note that $\displaystyle H\supseteq H-U_1\supseteq H-U_2\supseteq\cdots$ and so $\displaystyle \text{diam }H\geqslant \text{diam }H-U_1\geqslant \text{diam }H-U_2\geqslant\cdots$

    (b) Use the Cantor intersection theorem to deduce that if $\displaystyle H-U_n$ does not equal an empty set for every n, $\displaystyle \bigcap_{n=1}^{\infty}\left\{H-U_n\right\}\ne\varnothing$
    Which Cantor intersection theorem? The one that deals with complete spaces or the one that deals with compact spaces? Noting the insistence on bounded subspaces I assume the latter.

    Merely note from the above that given a finite subclass $\displaystyle H-U_{n_1},\cdots,H-U_{n_j}$ of $\displaystyle \left\{H-U_n\right\}_{n\in\mathbb{N}}$ that $\displaystyle \left(H-U_{n_1}\right)\cap\cdots\cap \left(H-U_{n_j}\right)=H-U_\alpha$ where $\displaystyle \alpha=\max\{n_1,\cdots,n_j\}$ and since by assumption this is non-empty we see that $\displaystyle \left\{H-U_n\right\}_{n\in\mathbb{N}}$ is a class of closed subsets of the compact space $\displaystyle H$ with the FIP. It follows that $\displaystyle \bigcap_{n=1}^{\infty}\left\{H-U_n\right\}\ne\varnothing$
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