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Thread: 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.
    $\displaystyle x\ge 0~\&~ y<0$
    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: $\displaystyle \beta(S)=\{(x,0,z):x\ge 0\}$

    The interior points: $\displaystyle S^o=\{(x,y,z):x> 0~\&~y<0\}$
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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 $\displaystyle \mathbb{R}^1$.
    $\displaystyle S=(0,1)=\{x:0<x<1\}$ do you see that $\displaystyle \beta(S)=\{0,1\}~?$.

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