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Math Help - Prove the following

  1. #1
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    Prove the following

    If S is a subset of the Reals Prove:

    a) bd(S) = bd(R\S)
    b) int(S) intersect bd(S) = empty set
    c) S is open IFF bd(S) is a subset of R\S
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by Jdg6057 View Post
    If S is a subset of the Reals Prove:

    a) bd(S) = bd(R\S)
    b) int(S) intersect bd(S) = empty set
    c) S is open IFF bd(S) is a subset of R\S
    a) A point x\in\mathbb{R} is a boundary point of S iff every neighborhood around x contains points in S and R\backslash S. A point x\in\mathbb{R} is a boundary point of \mathbb{R}\backslash S iff every neighborhood around x contains points in \mathbb{R}\backslash S and S. What can you conclude from this?

    b) A point x\in S is an interior point of S iff there exists a neighborhood around x that is a proper subset of S. Compare this definition with the above definition and deduce that int(S)\cap bd(S)=\emptyset.

    c) \Rightarrow If S is open, then every point in S is an interior point, so bd(S)=\emptyset. \emptyset\subset\mathbb{R}\backslash S is trivially true.

    \Leftarrow If bd(S)\subset\mathbb{R}\backslash S, then bd(S)\cap S=\emptyset, meaning that every point in S is an interior point, so S is open.
    Last edited by redsoxfan325; October 16th 2009 at 08:27 AM.
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