Jordan curve theorem

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.