Jordan curve theorem extended to multiple dimensions

Question

What if you extended Jordan curves to multiple dimensions? It would follow the same rules, that it does not intersect, break, and is continuous. For all extended Jordan curves (any amount of dimensions) does the theorem of having an inside and an out have proof for these?

Answer 1

This is the Jordan–Brouwer separation theorem see http://en.wikipedia.org/wiki/Jordan_curve_theorem

Let $X$ be a topological sphere in the $(n+1)$-dimensional Euclidean space $\Bbb R^{n+1}$, $(n > 0)$, i.e. the image of an injective continuous mapping of the $n$-sphere $S^n$ into $\Bbb R^{n+1}$. Then the complement $Y$ of $X$ in $\Bbb R^{n+1}$ consists of exactly two connected components. One of these components is bounded (the interior) and the other is unbounded (the exterior). The set $X$ is their common boundary.