By the generalized Jordan theorem any continuous injective map $S^{n-1} \hookrightarrow R^n$ splits $R^n$ into two regions, one being bounded (interior) and the other one unbounded (exterior). It must exist a stronger statement that the winding number for the points of the interior with respect to this sphere is always $\pm 1$. Anybody can tell me a reference how this theorem is called and who must be given credit for?

In fact a friend of mine asked me a different question. Is it true that any continuous injective map $R^n\hookrightarrow R^n$ is always open? I expect the answer is yes. It can be easily reduced to the previous one, but perhaps there is a simpler proof?