In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles at which curves meet. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.
The north pole should correspond to the point at infinity. The plane is infinite, so the North Pole should not be excluded?
Because you want to avoid situation in which 2 points from the sphere are mapped onto the same point on the plane. See the example image.
In this case $S$ and $N$ would map to the same point on the plane.