Jordan Curve and Jordan Arc form Two Jordan Curves/Corollary

From ProofWiki
Jump to: navigation, search

Corollary to Jordan Curve and Jordan Arc form Two Jordan Curves

Let $\closedint a b$ denote the closed real interval between $a \in \R, b \in \R: a \le b$.

Let $\gamma: \closedint a b \to \R^2$ be a Jordan curve.

Let the interior of $\gamma$ be denoted $\map {\operatorname {Int} } \gamma$.

Let the image of $\gamma$ be denoted $\Img \gamma$.


Let $\sigma: \closedint c d \to \R^2$ be a Jordan arc such that:

$\map \sigma c \ne \map \sigma d$
$\map \sigma c, \map \sigma d \in \Img \gamma$

and:

$\forall t \in \openint c d: \map \sigma t \in \map {\operatorname {Int} } \gamma$

Let $t_1 = \map {\gamma^{-1} } {\map \sigma c}$.

Let $t_2 = \map {\gamma^{-1} } {\map \sigma d}$.

Let $t_1 < t_2$.


Define:

$-\sigma: \closedint c d \to \Img \sigma$ by $-\map \sigma t = \map \sigma {c + d - t}$

Let $*$ denote concatenation of paths.

Let $\gamma \restriction_{\closedint a {t_1} }$ denote the restriction of $\gamma$ to $\closedint a {t_1}$.


Define:

$\gamma_1 = \gamma {\restriction_{\closedint a {t_2} } } * \paren {-\sigma} * \gamma {\restriction_{\closedint {t_1} b} }$

Define:

$\gamma_2 = \gamma {\restriction_{\closedint {t_2} {t_1} } } * \sigma$


Then $\gamma_1$ and $\gamma_2$ are Jordan curves such that:

$\map {\operatorname {Int} } {\gamma_1} \subseteq \map {\operatorname {Int} } \gamma$

and:

$\map {\operatorname {Int} } {\gamma_2} \subseteq \map {\operatorname {Int} } \gamma$


Proof

This is proved in the same way as Jordan Curve and Jordan Arc form Two Jordan Curves.

$\blacksquare$