If I understand correctly, you're asking what

is (called the Riemann sphere). As a

__set__ it's simply the set of complex numbers

together with an additional point you call "infinity" and denote by

. As a

__topological space__, it's the one-point compactification of the complex plane: a set is open in

iff it's open in

*or* is the complement of a compact set (in

).

This definition should be enough for you to answer the two other questions.

Intuitively, imagine you grab the real line and wrap it around a circle. It's "natural" to say the line won't cover the whole circle: there will be one point missing, so you call this point infinity. Same goes for the complex plane folded around a sphere: one point of the sphere is missing, so you just cover the hole with an extra "point at infinity". That's why it's called the Riemann

__sphere__.

Hope it helped