Question

specify the boundary and the interior of the sets S in 3-space whose points (x, y, z) satisfy the given conditions. Is S open, closed, or neither? x > 0, y > 1, . >1. ​

Answer 1

$\bold{ANSWER≈}$

I need to find the interior and boundary of this set:

A={(x,y,z)∈R3:0≤x≤1,0≤y≤2,0≤z<3}∖{(0,0,0)}.

We defined the interior as the set of all interior points, where we defined an interior point as:

point a∈Rn is interior for A⊆Rn if ∃r>0 so that K(a,r)⊆A. (K being an open ball with a centre in a and a radius of r).

I understand the definitions in a logical sense but don't know how to apply the "ball condition" to a real example. I also don't understand how the different boundaries (<,≤) impact it.

Answer 2

Answer:

A={(x,y,z)∈R3:0≤x≤1,0≤y≤2,0≤z<3}∖{(0,0,0)}.

We defined the interior as the set of all interior points, where we defined an interior point as:

point a∈Rn is interior for A⊆Rn if ∃r>0 so that K(a,r)⊆A. (K being an open ball with a centre in a and a radius of r).

I understand the definitions in a logical sense but don't know how to apply the "ball condition" to a real example. I also don't understand how the different boundaries (<,≤) impact it.

$thanks$