Someone argues: "Stereographic projection from the north pole is given by

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Manifolds, charts and tangent spaces

Problem

Someone argues: "Stereographic projection from the north pole

(0,0,1)

is given by

(u,v) = (x,y)/(1-z)

. At the north pole itself,

z=1

, so

u = x/0

and

v = y/0

are infinite. But that just means the north pole maps to a single 'point at infinity'. So the map

(u,v) = (x,y)/(1-z)

is a perfectly good one-to-one map of the *whole* sphere onto

\mathbb{R}^2 \cup \{\infty\}

, and it is therefore a single chart covering the entire sphere." Find the error in this argument.

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