The normal way I use to prove that $\Bbb{R}$ and $\Bbb{R}^2$ are not homeomorphic is by removing a point and then using path connectedness.
But this method doesn't seem to work for $\Bbb{R}^m$ and $\Bbb{R}^n$ and it ends up that its better if you use fundamental groups.

But what if I remove a line instead of a point from both the topological spaces. Does the argument still fail or will it lead to a proof?