# Definition:Jordan Curve

## Contents

## 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 $\R^2 \setminus \operatorname{Im} \left({f}\right)$ 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 $\operatorname{Int} \left({f}\right)$.

### Exterior

It follows from the Jordan Curve Theorem that $\R^2 \setminus \operatorname{Im} \left({f}\right)$ 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 $\operatorname{Ext} \left({f}\right)$.

## Also known as

Some sources call this a **simple closed curve**, or a **simple loop**.

## Also see

## 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): Entry:**Jordan curve**