# Definition:Jordan Curve

## Definition

Let $f$ be a Jordan arc from $\tuple {x_1, y_1}$ to $\tuple {x_2, y_2}$.

Then $f$ is a Jordan curve if and only if $\tuple {x_1, y_1} = \tuple {x_2, y_2}$.

### 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 sources call this a simple closed curve, or a simple loop.

## Source of Name

This entry was named for Marie Ennemond Camille Jordan.