Riemann Sphere

A 1-D Complex Manifold C*, which is the one-point compactification of the Complex numbers $\Bbb{C}\cup\{\infty\}$, together with two charts. For all points in the Complex Plane, the chart is the Identity Map from the Sphere (with infinity removed) to the Complex Plane. For the point at infinity, the chart neighborhood is the sphere (with the Origin removed), and the chart is given by sending infinity to 0 and all other points $z$ to $1/z$.