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Math Help - Compact spaces - Union and Intersection

  1. #1
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    Compact spaces - Union and Intersection

    Hey guys, i hope you can help me.

    I have to prove that if K_1, K_2,..., K_n are all compact subsets of the metric space (X,d):

    a) K_1 \cap K_2\cap K_3 \cap... \cap K_n is compact.

    b) K_1 \cup K_2 \cup K_3 \cup ... \cup K_n is compact.


    Im really kind of stuck on this, and ive been working on it for quite some time. Any of you guys who can tell me what to do or point me in the right direction?

    Thanks a lot.

    Morten
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  2. #2
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    Quote Originally Posted by MortenDK View Post
    I have to prove that if K_1, K_2,..., K_n are all compact subsets of the metric space (X,d):

    a) K_1 \cap K_2\cap K_3 \cap... \cap K_n is compact.

    b) K_1 \cup K_2 \cup K_3 \cup ... \cup K_n is compact.
    Can you show this:
    if K_1, K_2 are both compact subsets of the metric space (X,d):

    a) K_1 \cap K_2 is compact.

    b) K_1 \cup K_2 is compact.

    Think finite subcover.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Plato's explanation is great for the first one (remember the sum of two finite quantities is finite).

    For the second one, note that the makeup of metric spaces makes compact subspaces closed. Thus, K_1\cap K_2 is a closed subspace of K_1. Know any theorems about closed subspaces of compact spaces?
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  4. #4
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    Quote Originally Posted by Drexel28 View Post
    Plato's explanation is great for the first one (remember the sum of two finite quantities is finite).
    Do you mean the second one (the union)?

    Also note that if \left\{O_\alpha\right\} is an open covering of K_1\cap K_2 then \left\{O_\alpha\right\}\cup \{(K_1)^c\} is an open cover of K_2.
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Plato View Post
    Do you mean the second one (the union)?
    Yeah, sorry.

    Also note that if \left\{O_\alpha\right\} is an open covering of K_1\cap K_2 then \left\{O_\alpha\right\}\cup \{(K_1)^c\} is an open cover of K_2.
    That is kind of the proof for the fact about closed subspaces of compact spaces I was referring too haha. But, since we are talking about K_2 being the ambient space, it might be less confusing to put K_2-K_1
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