I try solving the following excercise:
Show by stereoscopic projection that the $S^2$ sphere is locally homeomorphic to $R^2$.
I tried to solve this by using the cotangens function on the two angles defining the surface of the ball by the projection:
$\phi : S^2 \rightarrow \mathbb{R}^2, \ (\alpha, \beta) \mapsto \left(\cot \alpha, \cot \frac{\beta}{2}\right)$ for $\alpha \in (0, \pi), \beta \in (0, 2\pi)$
However the poles and one of the connecting lines ($\beta = 0$) is not defined by this map. Is there a neat trick to get this out of the way?