Possible Duplicate:
Is there “essentially only 1” Jordan arc in the plane?
Let $\gamma : [0,1] \to \mathbb{R}^2$ be a simple curve with $\gamma(0)=(0,0)$ and $\gamma(1) =(1,0)$. Clearly, $\gamma$ is a homeomorphism onto $\gamma([0,1])$. Is it possible to extend $\gamma$ to a homeomorphism $\varphi : \mathbb{R}^2 \to \mathbb{R}^2$?
