Jordan-Brouwer separation theorem

Statement
The Jordan-Brouwer separation theorem states that any subset of $$S^n$$ that is homeomorphic to $$S^{n-1}$$ must separate $$S^n$$ into two components, and must be precisely the common boundary between those two components.

The special case of this in two dimensions is the famous Jordan curve theorem.

Related results

 * Schoenflies theorem