Schoenflies problem

In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often referred to as the Jordan–Schoenflies theorem.

Contents

1 Original formulation
2 Proofs of the Jordan–Schoenflies theorem

2.1 Polygonal curve
2.2 Continuous curve
2.3 Smooth curve

3 Generalizations
4 Notes
5 References

Original formulation
It states that not only does every simple closed curve in the plane separate the plane into two regions, one (the “inside”) bounded and the other (the “outside”) unbounded; but also that these two regions are homeomorphic to the inside and outside of a standard circle in the plane.
An alternative statement is that if

C

R

2

{\displaystyle C\subset \mathbb {R} ^{2}}

is a simple closed curve, then there is a homeomorphism

f
:

R

2

R

2

{\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}

such that

f
(
C
)

{\displaystyle f(C)}

is the unit circle in the plane. Elementary proofs can be found in Newman (1939), Cairns (1951), Moise (1977) and Thomassen (1992). The result can first be proved for polygons when the homeomorphism can be taken to be piecewise linear and the identity map off some compact set; the case of a continuous curve is then deduced by approximating by polygons. The theorem is also an immediate consequence of Carathéodory’s extension theorem for conformal mappings, as discussed in Pommerenke (1992), p. 25.
If the curve is smooth then the homeomorphism can be chosen to be a diffeomorphism. Proofs in this case rely on techniques from differential topology. Although direct proofs are possible (starting for example from the polygonal case), existence of the diffeomorphism can also be deduced by using the smooth Riemann mapping theorem for the interior and exterior of the curve in combination with the Alexander trick for diffeomorphisms of the circle and a result on smooth isotopy from differential topology.[1]
Such a theorem is valid only in two dime
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