Q:

Continuous surjection from simple closed curve to unit circle

My goal is to proof that every simple closed curve is homeomorphic to unit circle. I also have nice theorem for it.
But, I cannot do it before I find continuous surjection.
Let $f([0,1])\subseteq\mathbb{R}^{2}$ be the simple closed curve ($f:[0,1]\rightarrow \mathbb{R}^{2}$).
How I can find continuous surjection $g:f([0,1])\rightarrow S^{1}$?

A:

Both are homeomorphic to the quotient space $[0,1] / R$ where $R$ is the equivalence relation that only identifies $0$ and $1$ and no other points. For the circle you can use the map induced by the standard parametrisation, and $f$ has a similar role. And two spaces homeomorphic to the same space are homeomorphic.

