How do I prove that $S^n$ is homeomorphic to $S^m \Rightarrow m=n$? This is what I have so far:
Assume $S^n$ is homeomorphic to $S^m$. Also, assume $m≠n$. So, let $m>n$.
From here I am not sure what is implied. Of course in this problem $S^k$ is defined as:
$S^k=\lbrace (x_0,x_1,⋯,x_{k+1}):x_0^2+x_1^2+⋯+x_{k+1}^2=1 \rbrace$ with subspace topology.

Hint: look at the topic Invariance of Domain
