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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.