Schönflies Theorem

If $J$ is a simple closed curve in $\Bbb{R}^2$, the closure of one of the components of $\Bbb{R}^2-J$ is Homeomorphic with the unit 2-Ball. This theorem may be proved using the Riemann Mapping Theorem, but the easiest proof is via Morse Theory.

The generalization to $n$-D is called Mazur's Theorem. It follows from the Schönflies theorem that any two Knots of $\Bbb{S}^1$ in $\Bbb{S}^2$ or $\Bbb{R}^2$ are equivalent.

See also Jordan Curve Theorem, Mazur's Theorem, Riemann Mapping Theorem

References

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 9, 1976.

Thomassen, C. ``The Jordan-Schönflies Theorem and the Classification of Surfaces.'' Amer. Math. Monthly 99, 116-130, 1992.