Let $\mathbb{R}^2$ and $\mathbb{R}^3$ each be a D816: Euclidean real Cartesian product.
The Riemann sphere projection is the D4363: Euclidean real function \begin{equation} \mathbb{R}^2 \to \mathbb{R}^3, \quad (x, y) \mapsto \left( \frac{x}{1 + x^2 + y^2}, \frac{y}{1 + x^2 + y^2}, \frac{x^2 + y^2}{1 + x^2 + y^2} \right) \end{equation}