In topology, the Jordan curve theorem states that every non-self-intersecting loop in the plane divides the plane into an "inside" and an "outside". It was proved by Oswald Veblen in 1905. The precise mathematical statement is as follows. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... Oswald Veblen (24 June 1880 - 10 August 1960) was an American mathematician. ...

Let c be a simple closed curve (i.e. a Jordan curve) in the plane R². Then the complement of the image of c consists of two distinct connected components. One of these components is bounded (the interior) and the other is unbounded (the exterior). Also, c is the boundary of each component. In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. ...

The statement of the Jordan curve theorem seems obvious, but it was a very difficult theorem to prove. The first to attempt a proof was Bernard Bolzano, followed by a number of other mathematicians including Camille Jordan, after whom the theorem is named. None could provide a correct proof, until Oswald Veblen finally did so in 1905. Several alternative proofs were found since then. Bernard Bolzano Bernard Placidus Johann Nepomuk Bolzano (October 5, 1781 – December 18, 1848) was a Czech mathematician, theologian, philosopher and logician. ... Marie Ennemond Camille Jordan (January 5, 1838 – January 22, 1922) was a French mathematician, known both for his foundational work in group theory and for his influential Cours danalyse. ... Oswald Veblen (24 June 1880 - 10 August 1960) was an American mathematician. ...

A rigorous 200,000-line formal proof of the Jordan curve theorem was produced in 2005 by an international team of mathematicians using the Mizar system. The Mizar system consists of a language for writing strictly formalized mathematical definitions and proofs, a computer program which is able to check proofs written in this language, and a library of definitions and proved theorems which can be referred to and used in new articles. ...

Generalizations

There is a generalisation of the Jordan curve theorem to higher dimensions.

Let X be a continuous, injective mapping of the sphere Sⁿ into Rⁿ⁺¹. Then the complement of the image of X consists of two distinct connected components. One of these components is bounded (the interior) and the other is unbounded (the exterior). The image of X is their common boundary. In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ... For images in Wikipedia, see Wikipedia:Images. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. ...

There is a generalisation of the Jordan curve theorem called the Jordan-Schönflies theorem which states that any Jordan curve in the plane can be extended to a homeomorphism of the plane. This is a much stronger statement than the Jordan curve theorem. This generalisation is false in higher dimensions, and a famous counterexample is Alexander's horned sphere. The unbounded component of the complement of Alexander's horned sphere is not simply connected, and so the mapping of Alexander's horned sphere cannot be extended to all of R³. In mathematics, the Jordan-Schönflies theorem in geometric topology is a sharpening of the Jordan curve theorem in two dimensions. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... The Alexander horned sphere is one of the most famous pathological examples in mathematics. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...

References

• Oswald Veblen, Theory on plane curves in non-metrical analysis situs, Transactions of the American Mathematical Society 6 (1905), pp. 83–98.
• Ryuji Maehara, The Jordan curve theorem via the Brouwer fixed point theorem, American Mathematical Monthly 91 (1984), no. 10, pp. 641–643.

External links

• The full 200,000 line formal proof of Jordan's curve theorem
• Historical material relating to the Jordan curve theorem
• A simple proof of Jordan curve theorem (PDF)