Stereographic projection identifies the Riemann sphere with the unit two-sphere

Direct algebra gives $\mathrm{\Sigma }(z)\in S^2$ for every finite $z$, and the displayed formulas satisfy $\mathrm{\Sigma }(\mathrm{\Pi }(x,y,t))=(x,y,t)$ for $(x,y,t)\mathrm{\ne }(0,0,1)$ and $\mathrm{\Pi }(\mathrm{\Sigma }(z))=z$ for finite $z$, with $\mathrm{\Sigma }(\mathrm{\infty })=(0,0,1)$ and $\mathrm{\Pi }(0,0,1)=\mathrm{\infty }$.

On $\mathbb{C}$ and on $S^2\setminus \{(0,0,1)\}$ the formulas are rational with nonzero denominator, so both restrictions are continuous; and for the cap $U_t=\{(x,y,s):s>t\}\cup \{(0,0,1)\}$ one has ${\mathrm{\Sigma }}^{-1}(U_t)=\{z:\mathrm{\mid }z\mathrm{\mid }>\sqrt{(1+t)\mathrm{/}(1-t)}\}\cup \{\mathrm{\infty }\}$, which is a neighbourhood of $\mathrm{\infty }$ by [L1].

If $V=\widehat{\mathbb{C}}\setminus K$ is a neighbourhood of $\mathrm{\infty }$, compactness of $K$ gives $R>0$ with $K\subseteq \bar{D}(0,R)$, so the cap $U_{(R^2-1)\mathrm{/}(R^2+1)}$ satisfies $\mathrm{\Pi }(U_{(R^2-1)\mathrm{/}(R^2+1)})\subseteq V$; therefore $\mathrm{\Pi }$ is continuous at the north pole.

The maps $\mathrm{\Sigma }$ and $\mathrm{\Pi }$ are continuous inverse bijections by the preceding three steps, so $\mathrm{\Sigma }$ is a homeomorphism.