\begin{definition}[Definition:Riemann Sphere]
Let $f_1: \C \to \R^2$ be defined as:
:$\forall z \in \C: \map {f_1} z = \tuple {\map \Re z, \map \Im z}$
Let $f_2: \R^2 \to \R^3$ be the inclusion map:
:$\forall \tuple {a, b} \in \C^2: \map {f_2} {a, b} = \tuple {a, b, 0}$
Let $f = f_2 \circ f_1$.
Let $F: \C \to \map \PP {\R^3}$ be defined as the mapping which takes $z$ to the closed line interval from $\tuple {0, 0, 1}$ to $\map f z$ for all $z \in \C$.
Let $G = \set {x, y, z: x^2 + y^2 + z^2 = 1}$.
Then the Riemann map $R: \C \to \mathbb S^2$ is defined as:
:$\map R x = \map F z \cap G$
The set $R \sqbrk \C \cup \set {\tuple {0, 0, 1} } $ is called the '''Riemann sphere''', with the understanding that $\map f \infty = \tuple {0, 0, 1}$.
{{expand|According to the definition in Clapham & Nicholson, include the definition as the extended complex plane under a stereographic projection}}
\end{definition}