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Math Help - Prove that the boundary of S is compact

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    Prove that the boundary of S is compact

    Let S be a compact subset of a hausdorff space X. Prove that the boundary of S is compact.
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    Re: Prove that the boundary of S is compact

    Quote Originally Posted by Plato13 View Post
    Let S be a compact subset of a hausdorff space X. Prove that the boundary of S is compact.
    Have you proved that in a Hausdorff space boundaries of sets are closed?

    What is known about closed subsets of compact sets?
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    Re: Prove that the boundary of S is compact

    No I haven't proved that. That proof would be helpful.
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    Re: Prove that the boundary of S is compact

    Quote Originally Posted by Plato13 View Post
    No I haven't proved that. That proof would be helpful.
    Well then prove that the boundary is a closed set. Show it contains all its limit points.
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    Re: Prove that the boundary of S is compact

    Let S be a subset of a hausdorff space X and let B be it's boundary. Let x be a limit point of B. Then every neighbourhood of x contains an element y in B different to x. So there exists disjoint open sets U,V such that x is in U, y is in V.
    Now i'm stuck.
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    Re: Prove that the boundary of S is compact

    Quote Originally Posted by Plato13 View Post
    Let S be a subset of a hausdorff space X and let B be it's boundary. Let x be a limit point of B. Then every neighborhood of x contains an element y in B different to x.
    Let's use \beta(S) for the boundary of S.

    What is the exact meaning of t\in\beta(S). (i.e. what is a boundary point?)

    Note how I edited your reply. Say U is a neighborhood of x.
    Then there is a neighborhood of y, V, such that x\not\in V\subset U..

    Now apply the definition of a boundary point.
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    Re: Prove that the boundary of S is compact

    Why do you say V is a subset of U? Also I don't know much about x being a boundary point except the boundary is closure/interior.
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    Re: Prove that the boundary of S is compact

    Quote Originally Posted by Plato13 View Post
    Why do you say V is a subset of U? Also I don't know much about x being a boundary point except the boundary is closure/interior.

    That is your whole problem in a nutshell.
    Let this be a lesson to you. Always post what you know about the question. I had no way of knowing that you were working with what I consider a non-standard definition.

    Notation: \mathcal{I}(S) stands for the interior of S.

    By your definition \beta(S)=\overline{S}\setminus\mathcal{I}(S).

    But \mathcal{I}(S) is open set. Therefore its complement is closed.

    Thus \beta(S) is the intersection of two closed sets. SO?
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    Re: Prove that the boundary of S is compact

    I'm sorry about that, I will bear your advice in mind in future. Where have you used X is hausdorff?
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    Re: Prove that the boundary of S is compact

    Quote Originally Posted by Plato13 View Post
    Where have you used X is hausdorff?
    It was not use. Being Hausdorff is not necessary.

    BTW: It is Hausdorff. That is a proper name.
    Last edited by Plato; December 19th 2012 at 08:10 AM.
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    Re: Prove that the boundary of S is compact

    Are you sure you've proved the boundary is compact, and not closed? Something's wrong because it is a necessary hypothesis

    BTW: are you highlighting the capital or that it should be droff and not dorff?
    Last edited by Plato13; December 19th 2012 at 08:04 AM.
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    Re: Prove that the boundary of S is compact

    Quote Originally Posted by Plato13 View Post
    Are you sure you've proved the boundary is compact, and not closed? Something's wrong because it is a necessary hypothesis.
    What was shown is that \beta(S) is a closed set.
    Any closed subset of a compact set in a Hausdorff space is compact.
    Thus \beta(S)\subseteq\overline{S} is compact.
    Last edited by Plato; December 19th 2012 at 08:41 AM.
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    Re: Prove that the boundary of S is compact

    Thanks for your help. In future please don't respond to my questions as other (better) helpers will see you're dealing with it and I will be stuck with your minimalist answers. I know your ethos is let the student do it themselves but sometimes it is more instructive for me to see the proof.
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