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Math Help - compact subsets

  1. #1
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    compact subsets

    If S is a compact subset of R and T is a closed subset of S, then T is compact.

    A) prove this ussing the definition of compactness
    b) prove this using the Heine-Borel theorem
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  2. #2
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    Quote Originally Posted by luckyc1423 View Post
    If S is a compact subset of R and T is a closed subset of S, then T is compact.
    b) prove this using the Heine-Borel theorem
    S is compact thus it is closed and bounded (Heine-Borel Theorem). We only need the fact that S is bounded.

    Thus, T subset of S is bounded as well.

    But we know T is closed.

    Thus, T is a closed bounded subset thus it is compact (Heine-Borel Theorem).
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