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.

  • 6
    $\begingroup$ Embed the image of $\gamma$ in a Jordan curve. Then the statement is obvious by Jordan-Schoenflies. $\endgroup$ – Wojowu Sep 9 '19 at 8:15
  • 2
    $\begingroup$ Here is a nice discussion about why every such $\gamma$ can be embedded in a Jordan curve: mathoverflow.net/questions/57766/… $\endgroup$ – Dan Ramras Sep 13 '19 at 0:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.