I'm trying to cover $\mathbb{R}^2$ without 0 with polar coordinates. So I have the charts $\phi_i: (0, \infty) \times (\alpha_0, \alpha_0+2 \pi) \to \mathbb{R}^2$ without 0.
This means $x,y \in \mathbb{R}^2$ are described by $\phi(r, \alpha)= (r \ cos(\alpha), r \ sin(\alpha))$
Now I'm not sure how to define the transition functions between different charts to be sure that they are homeomorph. And what is about intersecting charts?
Maybe anybody has an idea? Thanks in advance!
If $\alpha_0=0$, then your chart will cover everything except for the positive $x$ axis. If $\alpha=-\pi$, then your chart covers everything except for the negative $x$-axis. So those two charts are enough.
To see that the translation functions are homeomorphisms, try writing $r$ and $\theta$ as functions of $x$ and $y$. Then note these functions are continuous. For example, $r=\sqrt{x^2+y^2}$. You need something similar for $\theta$.
Alternatively, you could use that $\mathbb{R}^2$ is locally compact to get continuity of the inverse.