Definition:Jordan Curve

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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