\begin{definition}[Definition:Differentiable Structure]
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 $\CC^k$''' on $M$ is a non-empty equivalence class of the set of $d$-dimensional $\CC^k$-atlases on $M$ under the equivalence relation of compatibility.
\end{definition}