Atlas

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

Properties

Every $𝐶 𝛼$ -atlas $𝒜$ has a unique maximal $𝐶 𝛼$ -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 $𝐶 𝛼$ -transition maps with those in $𝒜$ . Then all the charts in $𝒜$ have $𝐶 𝛼$ -transition maps (just transition to a chart in $𝒜$ and then out again). This structure is clearly unique and maximal, proving ^P1.

#state/tidy | #lang/en | #SemBr