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## Generalized Jordan Curve Theorem

If $A \subset \Bbb{R}^n$ is homeomorphic to $S^{n-1}$, then $\Bbb{R}^n \setminus A$ has two components, and $A$ is the boundary of each.

Generalized Jordan Curve Theorem

For the case $n=2$ you can find the solution in the following article: Jordan Curve Theorem bidimensional case