Let $f$ be the canonical quotient map of $\mathbb R^{n+1} - \{0\}$ onto $\mathbb R P^n$. Restrict $f$ to the hyperplane $H$ away from zero, so let $g = f|H$. Then I want to show that $g$ is a homeomorphism. It is easy to check that $g$ is bijective and continuous. Also easy to show that $f$ is an open mapping.
But how to show that $g$ is an open mapping?
There are infinitely many such hyperplanes. Okay, let us pick one. If $g$ were a homeomorphism, we would get a homeomorphism $\mathbb R^n \to \mathbb RP^n$. But this cannot exist.