Three-dimensional Schoenflies theorem ($\overline{C_{+}}\cong B^3,\overline{C_{-}}\cong B^3$)

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.