Find the boundary, the closure and the interior of the following sets

Question

Find the boundary, the closure and the interior of the following sets:

$$F = \{ (x,y) \in R^2 : (x^2+y^2-1)(4-x^2-y^2) >0\}$$ $$G = \{ (x,y,z) \in R^3 : x + y + z < 1\}$$ $$H = \{ (x,y,z) \in R^3 : x + y + z < 1, x > 0, y > 0, z > 0\}$$

As there's no possible graphical representation of some of them, I'm completely lost. Could you tell me how to start or any reference to consult?

Thanks in advance.

Answer 1

Hints:

1. Show that $F = \{ (x,y) \in R^2 : (x^2+y^2-1)(4-x^2-y^2) >0\}= \{(x,y) \in \mathbb R^2: 1<x^2+y^2<4\}.$ (annulus, draw a picture !).

2. For $F = \{ (x,y,z) \in R^3 : x + y + z < 1\}$ think about $\{ (x,y,z) \in R^3 : x + y + z = 1\} $ a plane !

3. For $F = \{ (x,y,z) \in R^3 : x + y + z < 1, x > 0, y > 0, z > 0\}$ consult 2. and the first octant.