Smooth manifolds and their smooth charts

A smooth $n$-manifold is a pair $(M,\mathcal{S})$ in which $M$ is a topological $n$-manifold (Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces) and $\mathcal{S}$ is a smooth structure on $M$: a maximal smooth atlas (Each smooth atlas is contained in a unique maximal smooth atlas). Because Each smooth atlas is contained in a unique maximal smooth atlas sends every smooth atlas to the unique maximal atlas containing it, a smooth manifold is equivalently specified by a topological manifold $M$ together with any one smooth atlas $\mathcal{A}$, the structure being the generated $[\mathcal{A}]$. A chart $(U,\phi )\in \mathcal{S}$ is called a smooth chart (or a chart of the smooth structure); its domain is a coordinate domain and its coordinate functions are smooth coordinates on $U$. When the structure is clear from context, the manifold itself is written $M$ in place of $(M,\mathcal{S})$.

• The maximal atlas is the structure. Two smooth atlases present the same smooth manifold exactly when they generate the same maximal atlas, which by Each smooth atlas is contained in a unique maximal smooth atlas holds exactly when their union is again a smooth atlas.

• A smooth chart is a chart of the structure, nothing more. Membership in $\mathcal{S}$ is what licenses calling a coordinate map smooth; a chart that is merely a homeomorphism onto an open set need not be smooth relative to $\mathcal{S}$.