Definition:Differentiable Structure

Definition
Let $M$ be a topological space.

Let $d$ be a natural number.

Let $k\geq1$ be a natural number.

An $n$-dimensional differentiable structure of class $\mathcal C^k$ on $M$ is an equivalence class of the set of $d$-dimensional $\mathcal C^k$-atlases on $M$ under the equivalence relation of compatibility.

Also defined as
An $n$-dimensional differentiable structure of class $\mathcal C^k$ is sometimes defined as a maximal $C^k$-atlas. See Bijection between Maximal Atlases and Differentiable Structures.

Also see

 * Compatibility of Atlases is Equivalence Relation