Definition:Jordan Arc

Definition
Let $f: \closedint 0 1 \to \R^2$ be an injective path in the Euclidean plane.

Then $f$ is called a Jordan arc.

Also known as
Some texts refer to a Jordan arc as merely an arc.

Also defined as
Some texts define a Jordan arc $f: \closedint 0 1 \to X$ as an injective path, 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 defines as an arc.

Some texts, especially those on complex analysis, drop the condition about injectivity and instead state that:


 * $\map f {t_1} \ne \map f {t_2}$ for all $t_1 ,t_2 \in \hointr 0 1$ with $t_1 \ne t_2$


 * $\map f t \ne \map f 1$ for all $t \in \openint 0 1$

That is, either $f$ is injective and a Jordan Arc, or $\map f 0 = \map f 1$, when $f$ is a Jordan curve by the definition.

Some texts, especially those on topology, define a Jordan arc as topological subspace $\struct{C, \tau_C}$ of $\R^2$ or $X$, where $\struct{C, \tau_C}$ is homeomorphic to the closed interval $\closedint 0 1$.

This means they consider a Jordan arc to be a topological space rather than a mapping.

Also see

 * Definition:Jordan Curve


 * Definition:Arc (Topology)