Definition:Jordan Arc/Also defined as

Jordan 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 $\mathsf{Pr} \infty \mathsf{fWiki}$ 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 $\mathsf{Pr} \infty \mathsf{fWiki}$ 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.

Sources

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• 2017: Thierry Vialar: Handbook of Mathematics: $10$: Topology: $\S 14$: Theory of Curves