Unit Circle is a 1-manifold in $\mathbf{R}^2$ and a function is not a coordinate patch

Question

I am trying to show that the unit circle $S^{1}$ is a 1 -manifold in $\mathbf{R}^{2}$, but the function $\alpha:[0,1) \rightarrow S^{1}$ given by $$ \alpha(t)=(\cos 2 \pi t, \sin 2 \pi t) $$ is not a coordinate patch on $S^{1}$.

Answer 1

Will this help you? It's from Carl Öhrnell's manuscript "Lie Groups and PDE"

Answer 2

Your coordinate patch has to be a homeomorphism, but the map $\alpha$ has no continuous inverse.

You can see this visually by noting that $t=0$ and $t \in (1-\epsilon,1)$ have arbitrarily close image, but are bounded apart by some positive constant. Hence there is some $\epsilon'>0$ so that no matter how close $x,x'$ are on $S^1$, $\|\alpha^{-1}(x) - \alpha^{-1}(x')\| \geqslant \epsilon'$.

I will leave the details to you.