Someone argues: "Longitude-latitude coordinates

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

Invariance, Ch Manifolds, charts and tangent spaces

Problem

Someone argues: "Longitude-latitude coordinates (θ,ϕ)(\theta,\phi), with θ∈(0,π)\theta\in(0,\pi) the colatitude and ϕ∈[0,2π)\phi\in[0,2\pi) the longitude, assign a unique pair of numbers to every point of the sphere except the north and south poles. We can just declare θ=0\theta=0 and θ=π\theta=\pi to be the poles by continuity. So (θ,ϕ)(\theta,\phi) is a single global chart covering the whole sphere smoothly." Find the error(s) in this reasoning, identifying every point at which the definition of a coordinate chart fails, and explain why no amount of patching up the domain of (θ,ϕ)(\theta,\phi) can turn it into a single global chart for the sphere.

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