Manifold

Atlas

A -atlas is a set of charts whose codomains form an open cover and whose transition maps are . #m/def/geo

Properties

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

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

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