Definition:Riemann Sphere

Let $$f_1:\C \to \R^2 \ $$ be defined $$f_1(z) = (\Re(z), \Im(z)) \ $$, let $$f_2:\R^2 \to \R^3 \ $$ be the inclusion map $$f_2(a,b) = (a,b,0) \ $$, and let $$f= f_2 \circ f_1 \ $$.

Let $$F:\C \to \mathcal{P} \left({\R^3}\right)$$ be defined as the function taking $$z \ $$ to the closed line interval from $$(0,0,1) \ $$ to $$f(z), z \in \C \ $$.

Let $$G = \left\{{x,y,z|x^2+y^2+z^2 =1}\right\} \ $$.

Then the Riemann map $$R:\C \to \mathbb{S}^2 \ $$ is defined $$R(z) = F(z) \cap G \ $$.

The set $$R(\C) \cup \left\{{(0,0,1)}\right\} \ $$ is called the Riemann sphere, with the understanding that $$f(\infty) = (0,0,1) \ $$.