If is a simple closed curve in , the closure of one of the components of 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 -D is called Mazur's Theorem. It follows from the Schönflies theorem that any two Knots of in or 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.