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Let $X$ be a path-connected subset of $\mathbb{R}^2$. For $x,y\in X$, $x\neq y$, does there necessarily exist a simple curve connecting $x,y$? In other words, is there an injective continuous map $\gamma:[0,1]\to X$ such that $\gamma(0)=x$, $\gamma(1)=y$?

I could neither prove this nor find any counterexample. It is easy when $\gamma$ has finitely many self-intersections. But is there any result for the general case?


marked as duplicate by Moishe Kohan, YuiTo Cheng, ThorWittich, Community Sep 12 at 14:51

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Wikipedia has the answer if you connect the dots. You are asking whether a subset of $\mathbb{R}^2$ that is path-connected is als arc-connected. If you use the induced topology from $\mathbb{R}^2$ any subspace will be Hausdorff and for Hausdorff spaces path-connected and arc-connected are equivalent. So the answer is yes.

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    $\begingroup$ This result is... really astonishing! Thanks a lot :) $\endgroup$ – Colescu Sep 11 at 14:57

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