Manifold

Atlas

A 𝐶𝛼-atlas 𝒜 is a set of charts whose codomains form an open cover and whose transition maps are [[Differentiability|𝐶𝛼]]. geo

Properties

1. Every 𝐶𝛼-atlas 𝒜 has a unique maximal 𝐶𝛼-atlas 𝒜′ containing 𝒜, i.e. so that no atlas is a superset of 𝒜′.

Proof

Let 𝒜′ be the set of all charts sharing 𝐶𝛼-transition maps with those in 𝒜. Then all the charts in 𝒜 have 𝐶𝛼-transition maps (just transition to a chart in 𝒜 and then out again). This structure is clearly unique and maximal, proving ^P1.

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