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Math Help - 2 disjoint clsed subsets of (R^2)

  1. #1
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    2 disjoint clsed subsets of (R^2)

    Q:Find two disjoint clsed subsets of (R^2) which are zero distance a part?

    my answer is A & B where

    A={(1/n) : n belong to z+}X{0}
    B={[-1,0]XR}

    Is it right

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  2. #2
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    Re: 2 disjoint clsed subsets of (R^2)

    No, it is not correct. Notice that \{1/n:n\in\mathbb{Z}^+\}=\mathbb{R}^+. So A=\{(r,0):r\in\mathbb{R}^+\}, and this set is not closed since (0,0) is a boundary point not in A.

    However, consider the sets C=\{(x,1/x):x\in\mathbb{R}^+\} and D=\{(x,0):x\in\mathbb{R}\}.
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  3. #3
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    Re: 2 disjoint clsed subsets of (R^2)

    Quote Originally Posted by hatsoff View Post
    However, consider the sets C=\{(x,1/x):x\in\mathbb{R}^+\} and D=\{(x,0):x\in\mathbb{R}\}.
    Thank u for your respond
    but C=\{(x,1/x):x\in\mathbb{R}^+\} looks like is not closed since D=\{(x,0):x\in\mathbb{R}\} is boundry point for the set C which not conained in D. Am I right or wrong
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  4. #4
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    Re: 2 disjoint clsed subsets of (R^2)

    D is a set, not a point. So D could not possibly *be* a boundary point.

    Think of it graphically: C is just the curve y=1/x in the first quadrant of the plane. D is the line y=0.
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