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.
Original formulation The original formulation of the Schoenflies problem 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 is a simple closed curve, then there is a homeomorphism such that is the unit circle in the plane. Elementary proofs can be found in , , and . 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 .
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.See:
Such a theorem is valid only in two dimensions. In three dimensions there are counterexamples such as Alexander's horned sphere. Although they separate space into two regions, those regions are so twisted and knotted that they are not homeomorphic to the inside and outside of a normal sphere.
Proofs of the Jordan–Schoenflies theorem For smooth or polygonal curves, the Jordan curve theorem can be proved in a straightforward way. Indeed, the curve has a tubular neighbourhood, defined in the smooth case by the field of unit normal vectors to the curve or in the polygonal case by points at a distance of less than ε from the curve. In a neighbourhood of a differentiable point on the curve, there is a coordinate change in which the curve becomes the diameter of an open disk. Taking a point not on the curve, a straight line aimed at the curve starting at the point will eventually meet the tubular neighborhood; the path can be continued next to the curve until it meets the disk. It will meet it on one side or the other. This proves that the complement of the curve has at most two connected components. On the other hand, using the Cauchy integral formula for the winding number, it can be seen that the winding number is constant on connected components of the complement of the curve, is zero near infinity and increases by 1 when crossing the curve. Hence the curve separates the plane into exactly two components, its "interior" and its "exterior", the latter being unbounded. The same argument works for a piecewise differentiable Jordan curve.
Polygonal curve Given a simple closed polygonal curve in the plane, the piecewise linear Jordan–Schoenflies theorem states that there is a piecewise linear homeomorphism of the plane, with compact support, carrying the polygon onto a triangle and taking the interior and exterior of one onto the interior and exterior of the other.See:
The interior of the polygon can be triangulated by small triangles, so that the edges of the polygon form edges of some of the small triangles. Piecewise linear homeomorphisms can be made up from special homeomorphisms obtained by removing a diamond from the plane and taking a piecewise affine map, fixing the edges of the diamond, but moving one diagonal into a V shape. Compositions of homeomorphisms of this kind give rise to piecewise linear homeomorphisms of compact support; they fix the outside of a polygon and act in an affine way on a triangulation of the interior. A simple inductive argument shows that it is always possible to remove a free triangle—one for which the intersection with the boundary is a connected set made up of one or two edges—leaving a simple closed Jordan polygon. The special homeomorphisms described above or their inverses provide piecewise linear homeomorphisms which carry the interior of the larger polygon onto the polygon with the free triangle removed. Iterating this process it follows that there is a piecewise linear homeomorphism of compact support carrying the original polygon onto a triangle.
Because the homeomorphism is obtained by composing finite many homeomorphisms of the plane of compact support, it follows that the piecewise linear homeomorphism in the statement of the piecewise linear Jordan-Schoenflies theorem has compact support.
As a corollary, it follows that any homeomorphism between simple closed polygonal curves extends to a homeomorphism between their interiors. For each polygon there is a homeomorphism of a given triangle onto the closure of their interior. The three homeomorphisms yield a single homeomorphism of the boundary of the triangle. By the Alexander trick this homeomorphism can be extended to a homeomorphism of closure of interior of the triangle. Reversing this process this homeomorphism yields a homeomorphism between the closures of the interiors of the polygonal curves.
Continuous curve The Jordan-Schoenflies theorem for continuous curves can be proved using Carathéodory's theorem on conformal mapping. It states that the Riemann mapping between the interior of a simple Jordan curve and the open unit disk extends continuously to a homeomorphism between their closures, mapping the Jordan curve homeomorphically onto the unit circle.See: To prove the theorem, Carathéodory's theorem can be applied to the two regions on the Riemann sphere defined by the Jordan curve. This will result in homeomorphisms between their closures and the closed disks |z| ≤ 1 and |z| ≥ 1. The homeomorphisms from the Jordan curve to the circle will differ by a homeomorphism of the circle which can be extended to the unit disk (or its complement) by the Alexander trick. Composition with this homeomorphism will yield a pair of homeomorphisms which match on the Jordan curve and therefore define a homeomorphism of the Riemann sphere carrying the Jordan curve onto the unit circle.
The continuous case can also be deduced from the polygonal case by approximating the continuous curve by a polygon.See: The Jordan curve theorem is first deduced by this method. The Jordan curve is given by a continuous function on the unit circle. It and the inverse function from its image back to the unit circle are uniformly continuous. So dividing the circle up into small enough intervals, there are points on the curve such that the line segments joining adjacent points lie close to the curve, say by ε. Together these line segments form a polygonal curve. If it has self-intersections, these must also create polygonal loops. Erasing these loops, results in a polygonal curve without self-intersections which still lies close to the curve; some of its vertices might not lie on the curve, but they all lie within a neighbourhood of the curve. The polygonal curve divides the plane into two regions, one bounded region U and one unbounded region V. Both U and V ∪ ∞ are continuous images of the closed unit disk. Since the original curve is contained within a small neighbourhood of the polygonal curve, the union of the images of slightly smaller concentric open disks entirely misses the original curve and their union excludes a small neighbourhood of the curve. One of the images is a bounded open set consisting of points around which the curve has winding number one; the other is an unbounded open set consisting of points of winding number zero. Repeating for a sequence of values of ε tending to 0, leads to a union of open path-connected bounded sets of points of winding number one and a union of open path-connected unbounded sets of winding number zero. By construction these two disjoint open path-connected sets fill out the complement of the curve in the plane.See:
, Lecture 36
Given the Jordan curve theorem, the Jordan-Schoenflies theorem can be proved as follows.
The first step is to show that a dense set of points on the curve are accessible from the inside of the curve, i.e. they are at the end of a line segment lying entirely in the interior of the curve. In fact, a given point on the curve is arbitrarily close to some point in the interior and there is a smallest closed disk about that point which intersects the curve only on its boundary; those boundary points are close to the original point on the curve and by construction are accessible. The second step is to prove that given finitely many accessible points Ai on the curve connected to line segments AiBi in its interior, there are disjoint polygonal curves in the interior with vertices on each of the line segments such that their distance to the original curve is arbitrarily small. This requires tessellations of the plane by uniformly small tiles such that if two tiles meet they have a side or a segment of a side in common: examples are the standard hexagonal tessellation; or the standard brickwork tiling by rectangles or squares with common or stretch bonds. It suffices to construct a polygonal path so that its distance to the Jordan curve is arbitrarily small. Orient the tessellation such no side of a tiles is parallel to any AiBi. The size of the tiles can be taken arbitrarily small. Take the union of all the closed tiles containing at least one point of the Jordan curve. Its boundary is made up of disjoint polygonal curves. If the size of the tiles is sufficiently small, the endpoints Bi will lie in the interior of exactly one of the polygonal boundary curves. Its distance to the Jordan curve is less than twice the diameter of the tiles, so is arbitrarily small.