# A topological property of curves on the plane $\mathbb{R}^2$

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 $$\mathbb{R}\subset \mathbb{R}^2$$?

• Embed the image of $\gamma$ in a Jordan curve. Then the statement is obvious by Jordan-Schoenflies. – Wojowu Sep 9 '19 at 8:15
• Here is a nice discussion about why every such $\gamma$ can be embedded in a Jordan curve: mathoverflow.net/questions/57766/… – Dan Ramras Sep 13 '19 at 0:46