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Math Help - Topological Compactness and Compactness of a Set

  1. #1
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    Topological Compactness and Compactness of a Set

    I need help in each of the problems. See attached picture.

    Thanks
    Attached Thumbnails Attached Thumbnails Topological Compactness and Compactness of a Set-imag00264.jpg  
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by meks08999 View Post
    I need help in each of the problems. See attached picture.

    Thanks
    9. Let \{U_i\}_{i\in I} be an open cover of F and let U_k\in\{U_i\}_{i\in I} be the set that contains x_0. Since x_0 is a cluster point, U_k also contains all but finitely many x_n. Assume that it covers all but N points in \{x_n\}.Thus there are at most N more open sets in \{U_i\}_{i\in I} (in addition to U_k) that completely cover \{x_n\}, and so you have proved that every open cover has a finite subcover, and F is compact.

    I'm not sure about 10. I know in a compact space every sequence has a convergent subsequence, but I don't know how to find two of them.
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  3. #3
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    For 10 you have to show that there are at least two cluster points in the sequence. Assume that there is \alpha only one cluster point

    1. First, show that there is \varepsilon such that there are infiniely many (x_n)'s outside of B(\alpha,\varepsilon). If such an \varepsilon does not exist we would have that the sequence converge to \alpha

    2. For the subsequence formed by the (x_n)'s outside of B(\alpha,\varepsilon), observe that it is in the compact set K. Hence there exists a cluster point which can't be \alpha.
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