I'm working on a problem that I think has a very intuitive result, but I'm having a hard time coming up with a rigorous proof. The problem reads
If $B$ is a bounded subset of $\mathbb{R}^{n}$ where $n\geq2$, then the complement of $B$ in $\mathbb{R}^{n}$ has exactly one unbounded component.
I naively intuit that because $B$ is bounded, the complement is of course unbounded and the complement of $B$ must be connected. I guess I think of this as making a hole in $\mathbb{R}^{n}$ which of course leaves one connected set and therefore exactly one unbounded component.