# 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:

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

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.

## Sources

- 2000: James R. Munkres:
*Topology*(2nd ed.): $\S 66$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Jordan curve**