If radial projection is bijective then is it a homeomorphism?

Question

Suppose $S$ is a regular surface in $\mathbb{R}^3 $ and $0\not\in S$. Now consider the radial projection $f: S\to\mathbb{S}^2$ given by $$f(x)=\frac{x}{||x||} \hspace{5mm}\mbox{ for all $x\in S$}$$ Now suppose $f$ is bijective, then does this imply that $f$ is a homeomorphism ? If not in general, is it true when $S$ is compact ?

Answer 1

If $f$ is a homeomorphism $S$ must be compact, as $\mathbb{S}^2$ is. On the other hand, every bijective continuous map from a compact space onto a Hausdorff space is a homeomorphism, as Selim Ghazouani already said in his comment.

So we conclude: the bijective map $f$ is a homeomorphism if and only if $S$ is compact.