# Can every two points be connected by a simple curve? [duplicate]

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

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.