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?



Generalized Jordan theorem and winding number
For your first question the credit goes to Brouwer. It's sometimes called the *Jordan-Brouwer Separation Theorem*. In this context you do not need a sphere, your $n-1$-manifold can be any compact connected boundaryless submanifold of $\mathbb R^n$. I believe both appear in the Guillemin and Pollack textbook, and Bredon's Topology and Geometry textbook. 


Your last question has the *yes* answer expected. The proof is very much similar. I'm not sure if I would call either proof easier than the other. This one tends to be called the *open mapping theorem* and at this level of generality you would need Bredon's book, but most intro algebraic topology texts that cover singular homology cover this. 


