Jordan Curve and Jordan Arc form Two Jordan Curves

Theorem
Let $\gamma: \left[{a \,.\,.\, b}\right] \to \R^2$ be a Jordan curve, where $\left[{a \,.\,.\, b}\right]$ is a closed real interval.

Denote the interior of $\gamma$ as $\operatorname{Int} \left({\gamma}\right)$, and denote the image of $\gamma$ as $\operatorname{Im} \left({\gamma}\right)$.

Let $\sigma: \left[{c \,.\,.\, d}\right] \to \R^2$ be a Jordan arc such that $\sigma \left({c}\right) \ne \sigma \left({d}\right)$, $\sigma \left({c}\right), \sigma \left({d}\right) \in \operatorname{Im} \left({\gamma}\right)$, and $\sigma \left({ t }\right) \in \operatorname{Int} \left({\gamma}\right)$ for all $t \in \left({c \,.\,.\, d}\right)$.

Put $t_1 = \gamma^{-1} \left({ \sigma \left({c}\right) }\right)$, and $t_2 = \gamma^{-1} \left({ \sigma \left({d}\right) }\right)$.

Suppose that $t_1 < t_2$.

Define $-\sigma: \left[{c \,.\,.\, d}\right] \to \operatorname{Im} \left({\sigma}\right)$ by $-\sigma \left({t}\right) = \sigma \left({c + d - t}\right)$.

Let $*$ denote concatenation of paths, and let $\gamma \restriction_{ \left[{a \,.\,.\, t_1}\right] }$ denote the restriction of $\gamma$ to $\left[{a \,.\,.\, t_1}\right]$.

Define $\gamma_1 = \gamma {\restriction_{ \left[{a \,.\,.\, t_1}\right] } }* \sigma * \gamma{ \restriction_{ \left[{t_2 \,.\,.\, b}\right] } }$, and define $\gamma_2 = \gamma{ \restriction_{ \left[{t_1 \,.\,.\, t_2}\right] } }* \left({ -\sigma }\right)$.

Then $\gamma_1$ and $\gamma_2$ are Jordan curves with $\operatorname{Int} \left({\gamma_1}\right) \subseteq \operatorname{Int} \left({\gamma}\right)$, and $\operatorname{Int} \left({\gamma_2}\right) \subseteq \operatorname{Int} \left({\gamma}\right)$.

Proof
As $\operatorname{Int} \left({\gamma}\right)$ and $\operatorname{Im} \left({\gamma}\right)$ are disjoint by the Jordan Curve Theorem, and $\sigma \left({ \left({c \,.\,.\, d}\right) }\right) \subseteq \operatorname{Int} \left({\gamma}\right)$, we have $\operatorname{Im} \left({\gamma}\right) \cap \operatorname{Im} \left({\sigma}\right) = \left\{ { \sigma \left({c}\right), \sigma \left({d}\right) }\right\}$.

As $\gamma$ is a Jordan curve, it follows that $\gamma {\restriction_{ \left[{a \,.\,.\, t_1}\right] } }$ and $\gamma{ \restriction_{ \left[{t_2 \,.\,.\, b}\right] } }$ only intersects in $\gamma \left({a}\right)$.

It follows that $\gamma_1$ is a Jordan arc.

As the initial point of $\gamma_1$ is $\gamma \left({a}\right)$, and the final point of $\gamma_1$ is $\gamma \left({b}\right) = \gamma \left({a}\right)$, it follows that $\gamma_1$ is a Jordan curve.

As $\operatorname{Im} \left({ -\sigma}\right) = \operatorname{Im} \left({\sigma}\right)$, it follows that $\gamma_2$ is a Jordan arc.

As $\gamma \left({t_1}\right) = \sigma \left({c}\right) = -\sigma \left({d}\right)$, it follows that $\gamma_2$ is a Jordan curve.

Denote the exterior of $\gamma$ as $\operatorname{Ext} \left({\gamma}\right)$.

Let $q_0 \in \operatorname{Ext} \left({\gamma}\right)$ be fixed, and let $q \in \operatorname{Ext} \left({\gamma}\right)$.

As $\operatorname{Ext} \left({\gamma}\right)$ is unbounded by the Jordan Curve Theorem, we can for all $N \in \N$ choose $q \in \operatorname{Ext} \left({\gamma}\right)$ such that $d \left({\mathbf 0, q}\right) > N$, where $d$ denotes the Euclidean metric on $\R^2$.

The same theorem shows that $\operatorname{Ext} \left({\gamma}\right)$ is open and connected.

As Connected Open Subset of Euclidean Space is Path-Connected, there is a path $\rho: \left[{0 \,.\,.\, 1}\right] \to \operatorname{Ext} \left({\gamma}\right)$ joining $q$ and $q_0$.

As $\operatorname{Im} \left({\gamma_1}\right) \subseteq \operatorname{Int} \left({\gamma}\right) \cup \operatorname{Im} \left({\gamma}\right)$, which is disjoint with $\operatorname{Ext} \left({\gamma}\right)$, it follows that $\rho$ is a path in either $\operatorname{Ext} \left({\gamma_1}\right)$ or $\operatorname{Int} \left({\gamma_1}\right)$.

As $d \left({\mathbf 0, q}\right)$ can be arbitrary large, and $\operatorname{Int} \left({\gamma_1}\right)$ is bounded, it follows that $\rho$ is a path in $\operatorname{Ext} \left({\gamma_1}\right)$.

Particularly, $q \in \operatorname{Ext} \left({\gamma_1}\right)$, so $\operatorname{Ext} \left({\gamma}\right) \subseteq \operatorname{Ext} \left({\gamma_1}\right)$.

If $q_1 \in \operatorname{Int} \left({\gamma_1}\right)$, then $q_1 \notin \operatorname{Ext} \left({\gamma}\right)$, as $\operatorname{Int} \left({\gamma_1}\right)$ and $\operatorname{Ext} \left({\gamma_1}\right)$ are disjoint.

It follows that $q_1 \in \operatorname{Int} \left({\gamma_1}\right)$, so $\operatorname{Int} \left({\gamma_1}\right) \subseteq \operatorname{Int} \left({\gamma}\right)$.

Similarly, it follows that $\operatorname{Int} \left({\gamma_2}\right) \subseteq \operatorname{Int} \left({\gamma}\right)$.