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.

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:


 * the complex plane $\C$
 * a real Euclidean space $\R^n$
 * a $T_2$ (Hausdorff) topological space $\struct { S, \tau_S }$

This is what 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

 * Definition:Jordan Arc


 * Definition:Simple Loop (Topology)


 * Jordan Curve Image Equals Set Homeomorphic to Circle