Definition:Jordan Curve

Definition

Let $f : \closedint 0 1 \to \R^2$ be a path in the Euclidean plane such that:

$\map f {t_1} \ne \map f {t_2}$ for all $t_1 ,t_2 \in \hointr 0 1$ with $t_1 \ne t_2$
$\map f 0 = \map f 1$

Then $f$ is called a Jordan curve.

Interior

It follows from the Jordan Curve Theorem that $\R^2 \setminus \Img f$ is a union of two disjoint connected components, one of which is bounded.

This bounded component is called the interior of $f$, and is denoted as $\Int f$.

Exterior

It follows from the Jordan Curve Theorem that $\R^2 \setminus \Img f$ is a union of two disjoint connected components, one of which is unbounded.

This unbounded component is called the exterior of $f$, and is denoted as $\Ext f$.

Also known as

Some texts refer to a Jordan curve as a simple closed curve, or a simple loop.

Also defined as

Some texts change the definition of the codomain of a Jordan curve from $\R^2$ to $X$, where $X$ is alternatively defined as:

This is what $\mathsf{Pr} \infty \mathsf{fWiki}$ defines as an simple loop.

Some texts drop the condition that $\map f 0 = \map f 1$ and replace it with the condition:

$\map f t \ne \map f 1$ for all $t \in \openint 0 1$

which means they consider a Jordan arc to be a Jordan curve.

Some texts, especially those on topology, define a Jordan curve as a topological subspace $\struct{C, \tau_C}$ of $\R^2$ or $X$, where $\struct{C, \tau_C}$ is homeomorphic to the unit circle $\mathbb S^1$.

Jordan Curve Image Equals Set Homeomorphic to Circle shows the connection between the definition of Jordan curve as a path, and the definition as a topological space.

Also see

• Results about Jordan curves can be found here.

Source of Name

This entry was named for Marie Ennemond Camille Jordan.