1. ## 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.

2. ## Re: Sets and 3-space

Originally Posted by quantoembryo
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: $S^o=\{(x,y,z):x> 0~\&~y<0\}$

3. ## 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?

4. ## Re: Sets and 3-space

Originally Posted by quantoembryo
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 do you see that $\beta(S)=\{0,1\}~?$.

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