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Math Help - Need Help with Cluster Points

  1. #1
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    Exclamation Need Help with Cluster Points

    Can someone give me hints on how to solve this problem?
    This is from Pugh's Real Mathematical Analysis
    Problem 89.
    Recall that p is a cluster point of S if each M_r_p contains infinitely many points of S. The set of cluster points of S is denoted as S'
    Prove:
    a) If S is a subset of T then S' is a subset of T'
    b) (The union of S and T)'=Union of S' and T'
    c) S' = (cl(S))'
    d) S' is closed in M; that is, S" is a subset of S' where S"=(S')'
    e) Calculate N', Q', R', (R\Q)', Q"
    f) Let T be the set of points {1/n:n in N}. Calculate T' and T"
    g) Give an example showing that S" can be a proper subset of S'
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  2. #2
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    If x is a point such that every neighborhood of x contains infinitely points of S then x is a cluster point of S and x is in S.

    If S is a subset of T and every neighborhood of x contains infinitely points of S, then does every neighborhood of x contains infinitely points of T? Does that prove that S is a subset of T?

    We know that S \subseteq \left( {S \cup T} \right) does that mean S' \subseteq \left( {S \cup T} \right)^\prime? Now you can finish the second one.

    All of the others have similar proofs.
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