in topology, a jordan curve, sometimes called a "plane simple closed 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 of the nearby and far away exterior points, so that every 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 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.