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