I have the following problem:
On the intervall $[0,2\pi]$ endowed with the subspace topology one declare two points to be equivalent if they are equal ore one is zero and the other one is $2\pi$. Denote by $M$ the quotient space with the quotient topology and by $p:[0,2\pi]\rightarrow M$ the canonical projection. Consider the map $$f:[0,2\pi]\rightarrow S^1$$ We first need to show that there exists exaclty one $g:M\rightarrow S^1$ such that $f=g\circ p$. And then that $g$ is bijective.
I found $g([\Theta])=(cos(\Theta),sin(\Theta))$ But I don't think that this function is injective isn't it? Could you maybe help me or give me a hint?
Thank you.