Jordan Curve Theorem
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 is 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
generalizations to higher-dimensional spaces.
The Jordan curve theorem is, perhaps, the oldest result in set-theoretic topology and 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.
Jordan Curve Theorem.