Path as Parameterization of Contour

Theorem
Let $\closedint a b$ be a closed real interval.

Let $\gamma: \closedint a b \to \C$ be a path.

Let there exist $n \in \N$ and a subdivision $\set {a_0, a_1, \ldots, a_n}$ of $\closedint a b$ such that:
 * $\gamma {\restriction_{\closedint {a_{k - 1} } {a_k} } }$ is a smooth path for all $k \in \set {1, \ldots, n}$

where $\gamma {\restriction_{\closedint {a_{k - 1} } {a_k} } }$ denotes the restriction of $\gamma$ to $\closedint {a_{k - 1} } {a_k}$.

Then there exists a contour $C$ with parameterization $\gamma$ and these properties:


 * $(1): \quad$ If $\gamma$ is a closed path, then $C$ is a closed contour.


 * $(2): \quad$ If $\gamma$ is a Jordan arc or a Jordan curve, then $C$ is a simple contour.


 * $(3): \quad$ If $\gamma$ is a Jordan curve, then $C$ is a simple closed contour.

Proof
Put $\gamma_k = \gamma {\restriction_{\closedint {a_{k - 1} } {a_k} } } : \closedint {a_{k - 1} } {a_k} \to \C$.

By definition, it follows that there exists a directed smooth curve $C_k$ with parameterization $\gamma_k$.

For all $k \in \set {1, \ldots, n - 1}$, we have:


 * $\map {\gamma_k} {a_k} = \map {\gamma_{k + 1} } {a_k}$

Define the contour $C$ as the concatenation $C_1 \cup C_2 \cup \ldots \cup C_n$.

Then, it follows by definition that $\gamma$ is a possible parameterization of $C$.

Suppose that $\gamma$ is a closed path.

Then:


 * $\map \gamma a = \map {\gamma_1} {a_0} = \map {\gamma_n} {a_n}$

so:


 * $C_1$ has start point $\map \gamma a$

and:


 * $C_n$ has end point $\map \gamma a$.

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

Suppose that $\gamma$ is a Jordan arc or a Jordan curve.

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.

Suppose that $\gamma$ is a Jordan curve.

By its definition, a Jordan curve is a closed path.

It follows from what is shown above that $C$ is a simple closed contour.