Can every two points be connected by a simple curve?

Question

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?

Possible duplicate of A question about path-connected and arcwise-connected spaces Also https://math.stackexchange.com/questions/155462/every-path-has-a-simple-subpath?rq=1 and many more. — Moishe Kohan, Sep 11 '19 at 16:03

score 5 · Accepted Answer · answered Sep 11 '19 at 06:43

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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.

answered Sep 11 '19 at 06:43

quarague

6,505

2

This result is... really astonishing! Thanks a lot :) – Yuxiao Xie Sep 11 '19 at 14:57

Can every two points be connected by a simple curve?

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