Editing Smooth structure (section)

=== Maximal smooth atlases ===

By taking the union of all [[Atlas (topology)|atlases]] belonging to a smooth structure, we obtain a '''maximal smooth atlas'''. This atlas contains every chart that is compatible with the smooth structure. There is a natural one-to-one correspondence between smooth structures and maximal smooth atlases.   
Thus, we may regard a smooth structure as a maximal smooth atlas and vice versa.

In general, computations with the maximal atlas of a manifold are rather unwieldy. For most applications, it suffices to choose a smaller atlas. 
For example, if the manifold is [[Compact space|compact]], then one can find an atlas with only finitely many charts.