Defn: (φ, U) is a chart or coordinate neighborhood if φ : U → U′ is a homeomorphism, where U is open in M and U′ is open in R^{n}.

Defn: Two charts, (φ, U) and (ψ, V) are C^{∞} compatible if the function φψ^{−1} : ψ(U ∩ V) → φ(U ∩ V) is a diffeomorphism.

Defn: A (pre) atlas or differentiable or smooth structure on M is a collection of charts on M satisfying the following two conditions:

i.) the domains of the charts form an open cover of M

ii.) Each pair of charts in the atlas is compatible.

Defn: An atlas is a (maximal or complete) atlas if it is maximal with respect to properties i) and ii).

A differential (or smooth or C^{∞}) n-manifold M is a topological n-manifold together with an atlas.

Theorem: A (pre) atlas can be uniquely enlarged to a maximal atlas.