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Math Help - Discrete metrics (basic topology)

  1. #1
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    Question Discrete metrics (basic topology)

    Let X be a set donated by the discrete metrics
    d(x,y) = 0 if x=y,
    1 if x̸=y.

    Show that a subset Y of X is compact iff this set is closed
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by hebby View Post
    Let X be a set donated by the discrete metrics
    d(x,y) = 0 if x=y,
    1 if x̸=y.

    Show that a subset Y of X is compact iff this set is closed
    I don't understand. Maybe I am speaking out of turn here, but isn't every subset of X closed? For, to claim that it wasn't closed would be to say that the subset doesn't contain one of it's limit points but no point in X has a limit point (as can be seen by taking a neighborhood of radius one-half). Did you mean finite?
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    Exclamation

    Yes finite, im sorry I was reading another question at the same time
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by hebby View Post
    Yes finite, im sorry I was reading another question at the same time
    A finite point set is always compact. Conversely, suppose that Y was compact, but infinite. Then Y must have a limit point....so


    THIS IS ROUGH ON PURPOSE. Clean it up.
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    Another way to see this is to consider an open cover by open balls with radius \frac 12.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by putnam120 View Post
    Another way to see this is to consider an open cover by open balls with radius \frac 12.
    Another way to do what? This surely isn't compact, but you have given one example, not a proof.
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    Member Abu-Khalil's Avatar
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    First, notice \left(x_n\right)_{n\in\mathbb{N}}\subset X converges iff it becomes constant after a certain N\in\mathbb{N}. With that, every subset is closed.

    Now, im not sure about this excercise. I think Y must be finite... if not we may choose X=\mathbb{N}, Y=X-\{1\} and the sequence x_n=n+1. that doesn't have any convergent sub-sequence 'cause it never become constant.
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    Y is finite
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    Question

    HI

    What do mean by constant?...could you explain this? thanks
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    Quote Originally Posted by Drexel28 View Post
    Another way to do what? This surely isn't compact, but you have given one example, not a proof.
    I was just giving a place to start from. Obviously finite sets are compact. Then using this you can show that any compact set must be finite.
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  11. #11
    Member Abu-Khalil's Avatar
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    Quote Originally Posted by hebby View Post
    HI

    What do mean by constant?...could you explain this? thanks
    \exists A,N:x_n=A,\forall n\geq N.

    If Y is finite, then (x_n)\subset Y have at least one term infinitly repeated and here u have your sub-sequence for compactness.
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