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Math Help - Collections of Bounded, Closed Subsets

  1. #1
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    Collections of Bounded, Closed Subsets

    Hello,

    There is a theorem that states,

    If there is a collection of compact subsets \{K_i\} of a metric space X which statifies the Finite Intersection Property, then \cap_i\{K_i\} \neq \emptyset.

    I am trying to find examples of collections of subsets from the reals,
    where the words
    1) closed
    2) bounded
    replace the word "compact" in the above statement, and show that the theorem is not true for either closed or bounded subsets, so the theorem only holds for compact subsets.

    That would imply, there exists collections of (1) closed or (2) bounded subsets whose intersection IS the null set.

    Thanks! Hope the question is clear... I am seeking two counterexamples, one each for collections of closed subsets and collections of bounded subsets.

    Cheers!
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  2. #2
    Senior Member roninpro's Avatar
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    For (1), how about the collection [1,3], [2,4], \mathbb{R}?
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  3. #3
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    If I understand correctly this should do for both

    S_n=[2n-1,2n] for n \in \mathbb{Z}
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  4. #4
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    I see that S_n=[2n-1,2n] is closed,
    however, I am not sure that it is bounded?

    Thank you for the replies!
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  5. #5
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    Sorry! Of course it is bounded, for a finite number of n. Yikes. Thank you!
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by matt.qmar View Post
    Hello,

    There is a theorem that states,

    If there is a collection of compact subsets \{K_i\} of a metric space X which statifies the Finite Intersection Property, then \cap_i\{K_i\} \neq \emptyset.

    I am trying to find examples of collections of subsets from the reals,
    where the words
    1) closed
    2) bounded
    replace the word "compact" in the above statement, and show that the theorem is not true for either closed or bounded subsets, so the theorem only holds for compact subsets.

    That would imply, there exists collections of (1) closed or (2) bounded subsets whose intersection IS the null set.

    Thanks! Hope the question is clear... I am seeking two counterexamples, one each for collections of closed subsets and collections of bounded subsets.

    Cheers!
    I personally like to consider \mathbb{R}-B_{n}}(0). These are all closed and unbounded and their intersection is null.
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