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In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a simple closed curve. 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 of the nearby and far away exterior points, so that any continuous path connecting a point of one region to a point of the other intersects with that loop somewhere. While the statement of this theorem seems to be intuitively obvious, it takes quite a bit of ingenuity to prove it by elementary means. More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces.

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).

The Jordan curve theorem is named after the mathematician Camille Jordan, who found its first proof. For decades, mathematicians generally thought that this proof was flawed and that the first rigorous proof was carried out by Oswald Veblen. However, this notion has been challenged by Thomas C. Hales and others.

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.

With these definitions, the Jordan curve theorem can be stated as follows:

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 generalizations

The Jordan curve theorem was independently generalized 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 (n > 0), 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:

$$\tilde{H}_{q}(Y)= \begin{cases}\mathbb{Z},\quad q=n-k \\ 0,\quad \text{otherwise}.\end{cases}$$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: S1 → R2, where S1 is viewed as the unit circle in the plane, can be extended to a homeomorphism ψ: R2 → R2 of the plane. Unlike Lebesgues' and Brouwer's generalization 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, and hence not homeomorphic to the exterior of the unit ball.

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 generalization was found by J. W. Alexander, who established the Alexander duality between the reduced homology of a compact subset X of R n+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 generalization 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, and 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 to formulate a precise conjecture, observing that it was not a self-evident statement, but that it required a proof. It is easy to establish this 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 of this theorem was given by Camille Jordan in his lectures on real analysis, and was 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 commenters on it have claimed that the first complete proof was given later 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 C. 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 was not really used by Jordan anyway, and quoted Michael 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 by de la Vallée-Poussin were later critically analyzed and completed by Schoenflies (1924).

Due to the importance of the Jordan curve theorem in low-dimensional topology and complex analysis, it received much attention from prominent mathematicians of the first half of the 20th century. Various proofs of the theorem and its generalizations were constructed by J. W. Alexander, Louis Antoine, Bieberbach, Luitzen Brouwer, Denjoy, Hartogs, Béla Kerékjártó, Alfred Pringsheim, and Schoenflies.

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

A short elementary proof of the Jordan curve theorem was presented by A. F. Filippov in 1950.[5]
A proof using non-standard analysis by Narens (1971).
A proof using constructive mathematics by Gordon O. Berg, W. Julian, and R. Mines et al. (1975).
A proof using the Brouwer fixed point theorem by Maehara (1984).
A proof using non-planarity of the complete bipartite graph K3,3 was given by Thomassen (1992).
A simplification of the (?whose) proof by Helge Tverberg.[6]

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.

Quasi-Fuchsian group, a mathematical group that preserves a Jordan curve
Complex analysis

Notes

Camille Jordan (1887)
Oswald Veblen (1905)
Hales (2007b)
Hales (2007b)
A. F. Filippov, An elementary proof of Jordan's theorem, Uspekhi Mat. Nauk, 5:5(39) (1950), 173–176

Czes Kosniowski, A First Course in Algebraic Topology

References

Berg, Gordon O.; Julian, W.; Mines, R.; Richman, Fred (1975), "The constructive Jordan curve theorem", Rocky Mountain Journal of Mathematics 5 (2): 225–236, doi:10.1216/RMJ-1975-5-2-225, ISSN 0035-7596, MR 0410701
Hales, Thomas C. (2007a), "The Jordan curve theorem, formally and informally", The American Mathematical Monthly 114 (10): 882–894, ISSN 0002-9890, MR 2363054
Hales, Thomas (2007b), "Jordan's proof of the Jordan Curve theorem" (PDF), Studies in Logic, Grammar and Rhetoric 10 (23)
Jordan, Camille (1887), Cours d'analyse (PDF), pp. 587–594
Maehara, Ryuji (1984), "The Jordan Curve Theorem Via the Brouwer Fixed Point Theorem", The American Mathematical Monthly (Mathematical Association of America) 91 (10): 641–643, doi:10.2307/2323369, ISSN 0002-9890, JSTOR 2323369, MR 0769530
Narens, Louis (1971), "A nonstandard proof of the Jordan curve theorem", Pacific Journal of Mathematics 36: 219–229, doi:10.2140/pjm.1971.36.219, ISSN 0030-8730, MR 0276940
Osgood, William F. (1903), "A Jordan Curve of Positive Area", Transactions of the American Mathematical Society (Providence, R.I.: American Mathematical Society) 4 (1): 107–112, ISSN 0002-9947, JFM 34.0533.02, JSTOR 1986455
Ross, Fiona; Ross, William T. (2011), "The Jordan curve theorem is non-trivial", Journal of Mathematics and the Arts (Taylor & Francis) 5 (4): 213–219, doi:10.1080/17513472.2011.634320. author's site
Sakamoto, Nobuyuki; Yokoyama, Keita (2007), "The Jordan curve theorem and the Schönflies theorem in weak second-order arithmetic", Archive for Mathematical Logic 46 (5): 465–480, doi:10.1007/s00153-007-0050-6, ISSN 0933-5846, MR 2321588
Thomassen, Carsten (1992), "The Jordan–Schönflies theorem and the classification of surfaces", American Mathematical Monthly 99 (2): 116–130, doi:10.2307/2324180, JSTOR 2324180
Veblen, Oswald (1905), "Theory on Plane Curves in Non-Metrical Analysis Situs", Transactions of the American Mathematical Society (Providence, R.I.: American Mathematical Society) 6 (1): 83–98, ISSN 0002-9947, JSTOR 1986378