I am studying multivariable calculus and would like to ask if I have answered the following exercise correctly:
- Let D be the union of all lines through $P=(0,0,2)$ and the open ball $B((0,0,0),1)$ (that is, all lines passing through both P and that ball), show D is not open nor closed
- Describe the closure of D, the boundary points of D, and the interior points of D (there's no need to prove the correctness of the description).
I will not write the whole proof, just the general details. Please tell me if I have the right idea.
For 1., I show P is a boundary point that is contained in D (thus D is not open), and that $(0,1,0)$ is a limit point that is not in D (thus D is not closed).
For 2., the closure is the union of all lines passing through the /closed/ ball $B((0,0,0),1)$ and P, the interior points are the points in $D - P$, and the boundary points are the union of lines passing through a boundary point of the open ball $B((0,0,0),1)$, minus $B((0,0,0),1)$ itself and $B((0,0,3),1)$.