Let $\gamma:[0;1]\to\mathbb{R}^2$ be regular Jordan curve. Consider $\epsilon\in\mathbb{R}_{>0}$, definition a continuous map
$$\Gamma:[0,1]\times (-\epsilon,\epsilon)\to\mathbb{R}^2, (t,s)\mapsto\gamma(t)+sN(t).$$
Proof where $\epsilon$ enough small, $\Gamma$ is injection and homeomorphic to the image.
Here, $T(t)=\gamma^{'} / \Vert\gamma^{'}\Vert$ and $N(t)=J.T(t)$ where $J=\begin{align*} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \end{align*}$
To proof $\Gamma$ is injection, I consider $(t_1,s_1),(t_2,s_2)$. Suppose $\Gamma(t_1,s_1)=\Gamma(t_2,s_2)$.
Is the way right? By the way, i don't prove $\Gamma$ is injection.
And why $\Gamma$ is homeomorphic to image??