Riemann sphere: Does it absolutely need two or more charts?

Question

I am trying to understand the Riemann sphere. In this Wikipedia article it is stated that

...[the Riemann sphere] is the one-point compactification of a plane into the sphere.

So it seems to me that it should be possible to describe to entire sphere using a stereographic projection (identify the northpole with $\infty$).

But the article also says that it can be described using the two charts with north- and southpole removed, respectively.

What's the ''smallest'' atlas for the Riemann sphere?

Can one stereographic projection be used to fully describe the Riemann sphere or does it need two?

As I understand it the definition of the Riemann sphere is the complex plane with one additional point ''$\infty$''.

Answer 1

Stereographic projection is not defined at the pole from which you project. A chart is supposed to be a diffeomorphism, in particular, a function! So stereographic projection is not a global chart. Furthermore, any manifold which isn't (homeomorphic, diffeomorphic, biholomorphic, as you prefer) to ($\mathbb{R}^n,\mathbb{C}^n$, as you prefer) requires more than one chart, since a chart gives a (homeomorphism, diffeomorphism, biholomorphism) between its domain of definition and ($\mathbb{R}^n,\mathbb{C}^n$).

Answer 2

As the Riemann sphere is compact, isn't homeomorphic to an open subset of the plane and is impossible an atlas with only a chart (homeomorpish!) .