I am always passing through this example while reading about manifolds that I don't quite get.
It is when describing the unit 2-sphere $S^2$ as an example of a manifold.
They say, initially it may be defined as the surface $x^2+y^2+z^2=1$ embedded in $R^3$. They add, it is common to use the usual spherical polar coordinates $\theta$, $\phi$ with $$ x=\sin\theta \cos\phi, \qquad y=\sin\theta\sin\phi , \qquad z=\cos\theta.$$ Then, they say, that this is fine for some purposes, but it does not define a good coordinate chart at the poles $\theta=0, \pi$, since these points have no unique values of $\phi$.
Then there is the discussion of stereographic projection, a method to introduce coordinate charts to define a manifold structure. Authors say, there are two patches whose union covers the sphere, namely $M_1$, consisting of the sphere with south pole deleted, and $M_2$, which is the sphere with north pole deleted.
I don't get there pole deleted argument as well as the line in bold in the first paragraph, how does this come about?