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Let $\gamma\colon [0,1]\to \mathbb{R}^2$ be a continuous injective map.

Is it true that for any inner point $t\in (0,1)$ there exist an open neighborhood $U$ of $\gamma(t)$ and a homeomorphism $f\colon U\tilde\to \mathbb{R}^2$ such that $f(U\cap Im(\gamma))$ is equal to a line $\mathbb{R}\subset \mathbb{R}^2$?

A reference would be helpful.

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    $\begingroup$ Embed the image of $\gamma$ in a Jordan curve. Then the statement is obvious by Jordan-Schoenflies. $\endgroup$
    – Wojowu
    Commented Sep 9, 2019 at 8:15
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    $\begingroup$ Here is a nice discussion about why every such $\gamma$ can be embedded in a Jordan curve: mathoverflow.net/questions/57766/… $\endgroup$
    – Dan Ramras
    Commented Sep 13, 2019 at 0:46

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