Definition:Differentiable Structure

Definition
Let $M$ be a topological space.

Let $d$ be a natural number.

Let $k \ge 1$ be a natural number.

A $d$-dimensional differentiable structure of class $\mathcal C^k$ on $M$ is a non-empty equivalence class of the set of $d$-dimensional $\mathcal C^k$-atlases on $M$ under the equivalence relation of compatibility.

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

Also see

 * Compatibility of Atlases is Equivalence Relation