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Math Help - Topology of the Reals proofs

  1. #1
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    Topology of the Reals proofs

    I need help proving the following statements. I've been doing this homework for 6 hours and I am finally burnt out. Any help is greatly appreciated.

    Let R denote the set of all reals.

    1) Let S be a subset of R and let x be an element of R. Prove that one and only one of the following three conditions holds:
    a) x is an element of the set of all interior points S (int S)
    b) x is an element of int (R\S)
    c) x is an element of the set of all boundary points S (bd S) = bd (R\S)


    2) Let S be a bounded infinite set and let x = sup S. Prove: If x is NOT an element of S, then x is an element of S' (set of all accumulation points of S).


    3) Prove: bd S = (cl S) intersection [cl (R\S)].


    Thank you. Any help would be awesome.
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  2. #2
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    Quote Originally Posted by noles2188 View Post
    I need help proving the following statements. I've been doing this homework for 6 hours and I am finally burnt out. Any help is greatly appreciated.

    Let R denote the set of all reals.

    1) Let S be a subset of R and let x be an element of R. Prove that one and only one of the following three conditions holds:
    a) x is an element of the set of all interior points S (int S)
    b) x is an element of int (R\S)
    c) x is an element of the set of all boundary points S (bd S) = bd (R\S)

    3) Prove: bd S = (cl S) intersection [cl (R\S)].
    Problems 1 & 3 are really about using the definition of boundary points.
    p is a boundary point of S if and only if each open set containing p contains a point of and a point not in S.
    From that statement it should be clear that p \in \beta (S)\; \Rightarrow \;p \in \beta (S^c ) every point is the boundary of S is in the boundary of S complement.
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  3. #3
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    Quote Originally Posted by noles2188 View Post
    I need help proving the following statements. I've been doing this homework for 6 hours and I am finally burnt out. Any help is greatly appreciated.

    Let R denote the set of all reals.

    1) Let S be a subset of R and let x be an element of R. Prove that one and only one of the following three conditions holds:
    a) x is an element of the set of all interior points S (int S)
    b) x is an element of int (R\S)
    c) x is an element of the set of all boundary points S (bd S) = bd (R\S)


    2) Let S be a bounded infinite set and let x = sup S. Prove: If x is NOT an element of S, then x is an element of S' (set of all accumulation points of S).


    3) Prove: bd S = (cl S) intersection [cl (R\S)].


    Thank you. Any help would be awesome.
    Another way to show (1) and (3) is that
    "int S" is the largest open set contained in S and "int R\S" is the largest open set contained in R\S. Since a union of open sets is an open set, "int S" \cup "int R\S" is an open set.
    Now, there is a closed set which is R\("int S" \cup "int R\S"). You need to show that the closed set is bd(S) using R\("int S" \cup "int R\S").

    Thus, "int S" , "int R\S", and "bd(S)" are disjoint sets whose union is R.

    For (2), the Bolzano-Weierstrass theorem states that "every bounded infinite subset of R has a limit point". If S is a closed set, limit points are in S. If S is an open set, limit points are either in S or S'.
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