\begin{definition}[Definition:Atlas]
Let $M$ be a topological space.
An '''atlas of class $C^k$ and dimension $d$''' on $M$ is a set of $d$-dimensional charts $\mathscr F = \family {\struct {U_\alpha, \phi_\alpha}: \alpha \in A}$ indexed by some set $A$ such that:
:$(1): \quad \ds \bigcup_{\alpha \mathop \in A} U_\alpha = M$
:$(2): \quad$ Every two charts $\struct {U, \phi}$ and $\struct {V, \psi}$ are $C^k$-compatible.
\end{definition}