$\mathbb R^3$ minus a line is connected. Let $S\subseteq \mathbb R^3$ be homeomorphic to $\mathbb R$. Prove that $\mathbb R^3 \setminus S$ is connected.
I haven't been able to solve this, although my topology skills are pretty weak. My friend told me he managed to prove this using results from his "dimension topology" class. Although I think this should be solvable using more mainstream stuff like homology or something. But I'm not sure.
Regards.

Here's a nice hint: The zero-th homology $H_0(\mathbb{R}^3 - S)$ must be a direct sum of $n$ copies of $\mathbb{Z}$, where $n$ denotes the number of connected components of $\mathbb{R}^3 - S$. 
