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Math Help - Compact Sets.

  1. #1
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    Compact Sets.

    Let (X,d_{x}) be a continuous function. Suppose that K\subset X is compact and that A\subset\ \mathbb{R}^k is compact.
    Prove that f(K)\bigcap A\ =\ \emptyset implies dist (f(K),A)>0

    It's analysis and there isn't an extra class for that unlike the other modules (even though it's considered the hardest).
    Thnx in advance
    Last edited by mr fantastic; December 10th 2011 at 08:08 PM. Reason: Title.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Compact Sets.

    Quote Originally Posted by maxgunn555 View Post
    Let (X,d_{x}) be a continuous function. Suppose that K\subset X is compact and that A\subset\ \mathbb{R}^k is compact.
    Prove that f(K)\bigcap A\ =\ \emptyset implies dist (f(K),A)>0

    It's analysis and there isn't an extra class for that unlike the other modules (even though it's considered the hardest).
    Thnx in advance

    You know that f(K) is compact, and so f(K)\times A is compact, so the mapping d:f(K)\times A\to\mathbb{R} obtains a minimum value d(f(k_0),a_0) for some (f(k_0),a_0)\in f(K)\times A. Clearly though d(f(k_0),a_0)>0 otherwise f(k_0)=a_0 which contradicts f(K)\cap A=\varnothing.
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