I'd like help for the problem:
Let the additive group $2\pi \mathbb{Z}$ act on $\mathbb{R}$ on the right by $x · 2\pi n = x+2\pi n$, where $n$ is an integer. Show that the orbit space $\frac{\mathbb{R}}{2\pi n\mathbb{Z}} $ is a smooth manifold.
I proved that $\frac{\mathbb{R}}{2\pi n\mathbb{Z}} $ is hausdorff and second countable, but I don't know how to find an atlas, I was thinking about $ \psi([x])=e^{ix}$ but I don't know how to show that this is a homeomorphism.