Three-dimensional Schoenflies theorem
= Three-dimensional Schoenflies theorem
{title2=$\overline{C_+}\cong B^3,\quad\overline{C_-}\cong B^3$}
A smooth, piecewise-linear or locally flat embedded two-sphere in the <three-sphere> has two complementary components, each with closure homeomorphic to a three-ball. Equivalently a tame embedded two-sphere in $\mathbb R^3$ bounds a ball on its compact side. Local flatness is essential; arbitrary wild sphere embeddings are excluded.