Boundary of a bounded open set in $\mathbb{R}^2$ Does the boundary of a bounded open set in $\mathbb{R}^2$ necessarily have infinite points? How do we prove that, or is there a counterexample?
It seems true to me, but I haven't been able to find a way to justify it.
Can we generalize this to $\mathbb{R}^n$ for $n>1$?

The boundary of the open set $\mathbb R^2 \backslash \{ (0,0) \}$ has only one point.
