Schönflies Theorem
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.
See also
Jordan Curve Theorem, Mazur's Theorem, Riemann Mapping TheoremExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Schönflies TheoremCite this as:
Weisstein, Eric W. "Schönflies Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SchoenfliesTheorem.html