I want to prove that if $n\neq m$ then $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$.
This deceptively simple topology question came up on an algebraic topology worksheet on which the rest of the questions centre around the Mayer-Vietoris sequence and degrees of maps. I have to admit I have little idea about how to proceed, except maybe to use homotopy types (and I'm not even sure how to start there). A hint or two would be very handy...