Path as Parameterization of Contour/Corollary 2

Theorem
If $\gamma$ is a Jordan arc, then $C$ is a simple contour, and if $\gamma$ is a Jordan curve, then $C$ is a simple closed contour.

Proof
Let $k_1, k_2 \in \set {1, \ldots, n}$, and $t_1 \in \hointr {a_{k_1 - 1} } {a_{k_1} }, t_2 \in \hointr {a_{k_2 - 1} } {a_{k_2} }$.

Then by the definition of Jordan arc, or Jordan curve:


 * $\map \gamma {t_1} \ne \map \gamma {t_2}$

so:


 * $\map {\gamma_{k_1} } {t_1} \ne \map {\gamma_{k_2} } {t_2}$

Let instead:


 * $k \in \set {1, \ldots, n}$

and:


 * $t \in \hointr {a_{k - 1} } {a_k}$ with $t \ne a_1$.

Then by the definition of Jordan arc, or Jordan curve:


 * $\map \gamma t \ne \map \gamma {a_n}$

so:


 * $\map {\gamma_k} t \ne \map {\gamma_n} {a_n}$

By definition, it follows that $C$ is a simple contour.

A Jordan curve is a closed path by its definition.

Path as Parameterization of Contour/Corollary 1 shows that if $\gamma$ is a Jordan curve, then $C$ will be closed, so $C$ is a simple closed contour.