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Math Help - Is a compact space closed?

  1. #1
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    Is a compact space closed?

    Please read the following:
    Every compact set is closed. So the union of all compact sets is closed? If that's so, then a compact space, which is basically the union of the compact sets, is closed, right?
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  2. #2
    Senior Member TheAbstractionist's Avatar
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    Hi Mohit.

    Quote Originally Posted by Mohit View Post
    So the union of all compact sets is closed?
    No. The union of finitely many compact sets only is closed.
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  3. #3
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    Quote Originally Posted by TheAbstractionist View Post
    Hi Mohit.



    No. The union of finitely many compact sets only is closed.
    Yes, this is correct. For an example consider the following. K_n=[-n,n], each of which is clearly compact, now \bigcup_{n=0}^\infty K_n=\mathbb{R}, which is not compact (it is not bounded).
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    Quote Originally Posted by Mohit View Post
    Please read the following:
    Every compact set is closed (not necessarily true). So the union of all compact sets is closed? If that's so, then a compact space, which is basically the union of the compact sets, is closed, right?
    If your topological space is a standard topology on \mathbb{R}, then a subset A of \mathbb{R} is compact if and only if it is closed and bounded.
    More generally, if a topological space X is a compact Hausdorff space, then a subset A of X is compact if and only if it is closed. In this case, as Abstractionist mentioned, finite unions of compact sets is compact.

    However, open sets can be compact sets. For example, in a cofinite topology on \mathbb{R}, compact sets are not necessarily closed sets.
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    Thank you all, thanks a lot.
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  6. #6
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    Quote Originally Posted by Mohit View Post
    Please read the following:
    Every compact set is closed. So the union of all compact sets is closed? If that's so, then a compact space, which is basically the union of the compact sets, is closed, right?
    As pointed out by others, an infinite union of closed set is not necessarily closed so the union of all compact sets in a space is not necessarily closed in that space.

    But you mention a "compact space". Every topological space is closed in itself.
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