Jordan curve theorem

Jordan curve theorem
Illustration of the Jordan curve theorem. The Jordan curve (drawn in black) divides the plane into an "inside" region (light blue) and an "outside" region (pink).

In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane. The Jordan curve theorem asserts that every Jordan curve divides the plane into an "interior" region bounded by the curve and an "exterior" region containing all far away points, so that any continuous path connecting a point of one region to a point of the other intersects that loop somewhere. While the statement seems intuitively obvious, it takes quite a bit of ingenuity to prove it by elementary means. More transparent proofs rely on the machinery of algebraic topology and lead to generalisations to higher-dimensional spaces.

The Jordan curve theorem is named after Camille Jordan, who found the first proof. For a long time, it was generally thought that this proof had been flawed and that the first rigorous proof was due to Oswald Veblen; however, this view has been challenged by Thomas Hales.

Contents

Definitions and the statement of the Jordan theorem

A Jordan curve or a simple closed curve in the plane R2 is the image C of an injective continuous map of a circle into the plane, φ: S1R2. A Jordan arc in the plane is the image of an injective continuous map of a closed interval into the plane.

Alternatively, a Jordan curve is the image of a continuous map φ: [0,1] → R2 such that φ(0) = φ(1) and the restriction of φ to [0,1) is injective. The first two conditions say that C is a continuous loop, whereas the last condition stipulates that C has no self-intersection points.

Let C be a Jordan curve in the plane R2. Then its complement, R2 \ C, consists of exactly two connected components. One of these components is bounded (the interior) and the other is unbounded (the exterior), and the curve C is the boundary of each component.

Furthermore, the complement of a Jordan arc in the plane is connected.

Proof and generalisations

The Jordan curve theorem was independently generalised to higher dimensions by H. Lebesgue and L.E.J. Brouwer in 1911, resulting in the Jordan–Brouwer separation theorem.

Let X be a topological sphere in the (n+1)-dimensional Euclidean space Rn+1, i.e. the image of an injective continuous mapping of the n-sphere Sn into Rn+1. Then the complement Y of X in Rn+1 consists of exactly two connected components. One of these components is bounded (the interior) and the other is unbounded (the exterior). The set X is their common boundary.

The proof uses homology theory. It is first established that, more generally, if X is homeomorphic to the k-sphere, then the reduced integral homology groups of Y = Rn+1 \ X are as follows:

\tilde{H}_{q}(Y)= \begin{cases}\mathbb{Z},\quad q=n-k \\ 0,\quad \text{otherwise}\end{cases}

This is proved by induction in k using the Mayer–Vietoris sequence. When n = k, the zeroth reduced homology of Y has rank 1, which means that Y has 2 connected components (which are, moreover, path connected), and with a bit of extra work, one shows that their common boundary is X. A further generalisation was found by J.W. Alexander, who established the Alexander duality between the reduced homology of a compact subset X of Rn+1 and the reduced cohomology of its complement. If X is an n-dimensional compact connected submanifold of Rn+1 (or Sn+1) without boundary, its complement has 2 connected components.

There is a strengthening of the Jordan curve theorem, called the Jordan–Schönflies theorem, which states that the interior and the exterior planar regions determined by a Jordan curve in R2 are homeomorphic to the interior and exterior of the unit disk. In particular, for any point P in the interior region and a point A on the Jordan curve, there exists a Jordan arc connecting P with A and, with the exception of the endpoint A, completely lying in the interior region. An alternative and equivalent formulation of the Jordan–Schönflies theorem asserts that any Jordan curve φ: S1R2, where S1 is viewed as the unit circle in the plane, can be extended to a homeomorphism ψ: R2R2 of the plane. Unlike Lebesgues' and Brouwer's generalisation of the Jordan curve theorem, this statement becomes false in higher dimensions: while the exterior of the unit ball in R3 is simply connected, because it retracts onto the unit sphere, the Alexander horned sphere is a subset of R3 homeomorphic to a sphere, but so twisted in space that the unbounded component of its complement in R3 is not simply connected, hence not homeomorphic to the exterior of the unit ball.

History and further proofs

The statement of the Jordan curve theorem may seem obvious at first, but it is a rather difficult theorem to prove. Bernard Bolzano was the first one to formulate a precise conjecture, observing that it was not a self-evident statement, but required a proof. It is easy to establish the result for polygonal lines, but the problem came in generalizing it to all kinds of badly behaved curves, which include nowhere differentiable curves, such as the Koch snowflake and other fractal curves, or even a Jordan curve of positive area constructed by Osgood (1903).

The first proof was given by Camille Jordan in his lectures on real analysis and published in his book Cours d'analyse de l'École Polytechnique.[1] There is some controversy about whether Jordan's proof was complete: the majority of authors have claimed that the first complete proof was given by Oswald Veblen, who said the following about Jordan's proof:

His proof, however, is unsatisfactory to many mathematicians. It assumes the theorem without proof in the important special case of a simple polygon, and of the argument from that point on, one must admit at least that all details are not given.[2]

However, Thomas Hales wrote:

Nearly every modern citation that I have found agrees that the first correct proof is due to Veblen... In view of the heavy criticism of Jordan’s proof, I was surprised when I sat down to read his proof to find nothing objectionable about it. Since then, I have contacted a number of the authors who have criticized Jordan, and each case the author has admitted to having no direct knowledge of an error in Jordan’s proof.[3]

Hales also pointed out that the special case of simple polygons is not only an easy exercise, but is not really used by Jordan anyway, and quoted Reeken as saying:

Jordan’s proof is essentially correct... Jordan’s proof does not present the details in a satisfactory way. But the idea is right and with some polishing the proof would be impeccable.[4]

Jordan's proof and another early proof due to de la Vallée-Poussin were later critically analyzed and completed by Shoenflies (1924).

Due to the importance of the Jordan curve theorem in low-dimensional topology and complex analysis, it received a lot of attention from prominent mathematicians of the first half of the twentieth century. Various proofs of the theorem and its generalizations were found by J. W. Alexander, Antoine, Bieberbach, Brouwer, Denjoy, Hartogs, Kerékjártó, Pringsheim, Schoenflies.

New elementary proofs of the Jordan curve theorem, as well as simplifications of earlier proofs, continue to be designed.

The first formal proof of the Jordan curve theorem was created by Hales (2007a) in the HOL Light system, in January 2005 and contained about 60,000 lines. Another rigorous 6,500-line formal proof was produced in 2005 by an international team of mathematicians using the Mizar system. Both the Mizar and the HOL Light proof rely on libraries of previously proved theorems, so these two sizes are not comparable. Nobuyuki Sakamoto and Keita Yokoyama (2007) showed that the Jordan curve theorem is equivalent in proof-theoretic strength to the weak König's lemma.

See also

  • Lakes of Wada
  • Quasi-Fuchsian group, a group preserving a Jordan curve

Notes

  1. ^ Camille Jordan (1887)
  2. ^ Oswald Veblen (1905)
  3. ^ Hales (2007b)
  4. ^ Ibid
  5. ^ Czes Kosniowski, A First Course in Algebraic Topology

References

External links


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