Jordan curve theorem

The Jordan curve theorem states that every simple closed curve in the plane separates it into exactly two connected regions, with the curve as the boundary of each.

Jordan curve: A non-self-intersecting closed curve in the plane. The theorem asserts separation for precisely these curves.
Connected component: A maximal connected subset of a topological space. The theorem says the curve's complement has exactly two such components.
Camille Jordan: A French mathematician whose work included the first published proof of the planar curve-separation theorem. His 1887 proof gave the result its name, though its rigor was later disputed.
Point-in-polygon problem: The computational problem of deciding whether a point lies inside, outside, or on a polygon. Polygon algorithms rely on the separation between interior and exterior.
Jordan–Schoenflies theorem: A theorem stating that every simple closed curve in the plane can be carried to a circle by a homeomorphism of the plane. It strengthens separation by showing the enclosed region has the standard disk shape.
Complement (topology): The part of a space remaining after a specified subset is removed. Removing the curve is the operation whose connected components the theorem counts.
Bernard Bolzano: A Czech mathematician and philosopher whose work anticipated several ideas in rigorous analysis. His writings contain an early formulation related to separation by closed curves.
Ray casting algorithm: A method for testing point inclusion by counting intersections between a ray and a boundary. Its odd-even crossing rule exploits the two-region structure for polygonal curves.
Alexander horned sphere: A wild embedding of a sphere in three-dimensional space whose complement has a non-simply-connected component. It shows how higher-dimensional analogues can fail to behave as simply as planar curves.
Topological boundary: The set of points every neighborhood of which meets both a set and its complement. The theorem identifies the curve as the boundary of both complementary regions.

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