Complex Analysis - Stereographic Projection

Quote:

Originally Posted by habsfan31

Let S = {(ξ; η; ζ ) ∈ Σ : ζ ≥ ζ₀g}, where 0 < ζ₀ < 1 and let T be the corresponding
set in C. Show that T is the exterior of a circle centered at 0.

Im really confused by this problem, can anyone please help me.

I suppose the " $g$ " appearing in $S$ is a typo. Using the well known equations of the Stereographic Projection we get $\zeta=\frac{x^2+y^2}{1+x^2+y^2}\geq \zeta_0 \Leftrightarrow \ldots \Leftrightarrow x^2+y^2\geq \frac{ \zeta_0}{1- \zeta_0}$ and $(0,0,1)\to \infty$ . That is, $T$ is the the complement in $\Sigma$ of an open disk centered at $(0,0)$ and radius $r=\sqrt{\zeta_0/(1-\zeta_0)}$ (the complement, not the topological exterior).