general topology - bijective continuous function on $mathbb R^n$ not homeomorphism?

Suppose we have a bijective continuous map $\mathbb{R}^n\to\mathbb{R}^n$ (relative to the standard topology). Must this map be a homeomorphism?

I have little doubt about this. I think that if it happens, I guess it's true, I've heard it is true, but I can not prove it.

Answer

Every such map is open according to invariance of domain and is therefore a homeomorphism. However, invariance of domain is highly nontrivial to prove with elementary topological methods. The slickest way would be via algebraic topology.