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Math Help - Show a set is compact

  1. #1
    Member billa's Avatar
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    Show a set is compact

    If A and B are compact subsets of some metric space M with metric d, then I think that the cross product A\times B is a compact subset of the metric space M^2 where the metric in this space is defined by

    p((x,y),(z,w))=\sqrt{(d(x,z)^2+d(y,w)^2}

    What is the best way to show that AxB is indeed compact? Is there a particularly good way to do this? I think I have a proof showing that every sequence in AxB has a convergent subsequence.

    Thanks.
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  2. #2
    Senior Member roninpro's Avatar
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    Showing that every sequence has a convergent subsequence is probably the easiest way to do it. You would need to use the fact that a sequence \{(x_n,y_n)\} in A\times B converges if and only if \{x_n\} converges in A and \{y_n\} converges in B.

    On the other hand, it might be possible to use the definition directly: take an open cover of A\times B and try to show that it has a finite subcover. You can relate this to the compactness of A and B by using the projection maps: \pi_1: A\times B\to A given by \pi_1(x,y)=x and \pi_2: A\times B\to B given by \pi_2(x,y)=y.
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  3. #3
    Senior Member
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    By the way, I think you mean "cartesian product" and not "cross product"

    I think that both methods suggested by roninpro should work.
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  4. #4
    Senior Member Tinyboss's Avatar
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    Show that the given metric induces the product topology, and invoke the theorem that finite (easy proof) or arbitrary (Tychonoff) products of compact spaces are compact.
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