Can you tell me if this is correct:
I define $f_{(0,1)}: S^{1+} \setminus \{(0,1)\} \to \mathbb R$ as $ f((x,y)) =\frac{x}{1-y} $ where $S^{1+} = \{ (x,y) \in S^1 \mid y \geq 0 \}$. Since I removed the north pole this map is continuous and it is easy to verify that $f$ is also differentiable in each variable.
Hence $S^1$ is a differentiable manifold with the atlas $\{ (S^{1+} \setminus \{(0,1)\}, f_{(0,1)}), (S^{1-} \setminus \{(0,-1)\}, f_{(0,-1)}) \}$.
Another question I have is: is this also called stereographic projection? I google-searched the term but all results seemed to be only about $S^2 \to \mathbb R^2$.