Jordan curve theorem

Statement

Denote by $\mathbb {R} ^{2}$ the Euclidean plane (with the usual Euclidean topology). A Jordan curve in $\mathbb {R} ^{2}$ is the image in $\mathbb {R} ^{2}$ of an injective continuous mapping from a circle. In other words, it is a non-self-intersecting simple closed curve in $\mathbb {R} ^{2}$.

The Jordan curve theorem states the following:

1. Any Jordan curve divides its complement in the plane into two connected components.
2. One of these connected components, which we call the interior component, is bounded. The other component, which we call the exterior component, is unbounded.

Related facts

Similar facts

• Jordan-Schoenflies theorem says that the interior component is homeomorphic to the open unit disk (which is homeomorphic to the Euclidean plane) and the exterior component is homeomorphic to the complement of the closed unit disk in $\mathbb {R} ^{2}$.
• Jordan-Brouwer separation theorem generalizes the Jordan curve theorem to higher dimensions. However, the Jordan_Schoenflies theorem does not generalize to higher dimensions.

Opposite facts

• The lakes of Wada topological space is a counterexample to some plausible but false generalizations of the Jordan curve theorem.
• The Alexander horned sphere provides a counterexample to the simplest attempted generalization of the Jordan-Schoenflies theorem to higher dimensions.