If J is a simple closed curve in R^2, the closure of one of the components of 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 dimensions is called Mazur's theorem. It follows from the Schönflies theorem that any two knots of S^1 in S^2 or R^2 are equivalent.

Jordan curve theorem | Mazur's theorem | Riemann mapping theorem