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Math Help - Sets and 3-space

  1. #1
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    Sets and 3-space

    Hello all,
    I'm looking for some guidance for a relatively simple question. We are supposed to describe the boundary points, the interior points, and state whether or not it is open, closed, or neither.

    x≥0, y<0

    So I draw each of these individually and restrict my focus to the overlapping areas. For x≥0, I get all x values greater than and equal to 0 in the y-z plane. For y<0 I get all values of y less than and not equal to 0 in the x-z plane. The overlapping portion of 3 space ends up being 2 octants.
    bdry(S) is the set of points (x,0) such that x≥0, and the set of points (0,y) such that 0≥y.
    Wouldnt the interior of S be the overlapping octants such that y<0, and hence all the points not touching the x-axis?

    Solution says it would be points not touching the y-axis, but I am not seeing it.
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  2. #2
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    Re: Sets and 3-space

    Quote Originally Posted by quantoembryo View Post
    We are supposed to describe the boundary points, the interior points, and state whether or not it is open, closed, or neither.
    x\ge 0~\&~ y<0
    bdry(S) is the set of points (x,0) such that x≥0, and the set of points (0,y) such that 0≥y.
    Wouldnt the interior of S be the overlapping octants such that y<0, and hence all the points not touching the x-axis?
    I would just describe the sets.
    Boundary points: \beta(S)=\{(x,0,z):x\ge 0\}

    The interior points: S^o=\{(x,y,z):x> 0~\&~y<0\}
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  3. #3
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    Re: Sets and 3-space

    I'm not seeing how your B(S) takes into consideration that y<0. Wouldn't that set you described have some values in the +y direction?
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  4. #4
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    Re: Sets and 3-space

    Quote Originally Posted by quantoembryo View Post
    I'm not seeing how your B(S) takes into consideration that y<0. Wouldn't that set you described have some values in the +y direction?
    Take a simple example in \mathbb{R}^1.
    S=(0,1)=\{x:0<x<1\} do you see that \beta(S)=\{0,1\}~?.

    There is not requirement that a boundary point belong to the set.
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