Жордан қисығының теоремасы және оның жалғасы

A closed curve divides the plane into two regions

In topology, the Jordan curve theorem asserts that every Jordan curve (a plane simple closed 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. Every continuous path connecting a point of one region to a point of the other intersects with the curve somewhere. While the theorem seems intuitively obvious, it takes some ingenuity to prove it by elementary means. "Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it." More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher dimensional spaces. The Jordan curve theorem is named after the mathematician Camille Jordan (1838–1922), who published its first claimed proof in 1887. 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 overturned by Thomas C. Hales and others.

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. 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:

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 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: 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 Lebesgue's 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.

The Jordan curve theorem can be proved from the Brouwer fixed point theorem (in 2 dimensions), and the Brouwer fixed point theorem can be proved from the Hex theorem: "every game of Hex has at least one winner", from which we obtain a logical implication: Hex theorem implies Brouwer fixed point theorem, which implies Jordan curve theorem. It is clear that Jordan curve theorem implies the "strong Hex theorem": "every game of Hex ends with exactly one winner, with no possibility of both sides losing or both sides winning", thus the Jordan curve theorem is equivalent to the strong Hex theorem, which is a purely discrete theorem. The Brouwer fixed point theorem, by being sandwiched between the two equivalent theorems, is also equivalent to both. In reverse mathematics, and computer formalized mathematics, the Jordan curve theorem is commonly proved by first converting it to an equivalent discrete version similar to the strong Hex theorem, then proving the discrete version.

The Steinhaus chessboard theorem in some sense shows that the 4 neighbor grid and the 8 neighbor grid "together" implies the Jordan curve theorem, and the 6 neighbor grid is a precise interpolation between them. The theorem states that: suppose you put bombs on some squares on a chessboard, so that a king cannot move from the bottom side to the top side without stepping on a bomb, then a rook can move from the left side to the right side stepping only on bombs.

In computational geometry, the Jordan curve theorem can be used for testing whether a point lies inside or outside a simple polygon. From a given point, trace a ray that does not pass through any vertex of the polygon (all rays but a finite number are convenient). Then, compute the number n of intersections of the ray with an edge of the polygon. Jordan curve theorem proof implies that the point is inside the polygon if and only if n is odd.

Adler, Daskalakis and Demaine prove that a computational version of Jordan's theorem is PPAD complete. As a corollary, they show that Jordan's theorem implies the Brouwer fixed point theorem. This complements the earlier result by Maehara, that Brouwer's fixed point theorem implies Jordan's theorem.