Definition:Atlas

Definition
Let $M$ be a topological space.

An atlas of class $C^k$ and dimension $d$ on $M$ is a set of charts $\mathscr F = \left\{{\left({U_\alpha, \phi_\alpha}\right): \alpha \in A}\right\}$ indexed by some set $A$ such that:


 * $(1): \quad \displaystyle \bigcup_{\alpha \mathop \in A} U_\alpha = M$


 * $(2): \quad$ Every two charts $(U,\phi)$ and $(V,\psi)$ are $C^k$-compatible. That is, their transition mapping $\psi \circ \phi^{-1}$ is of class $C^k$ as a map $\phi \left({U \cap V}\right) \to \psi\left({U \cap V}\right)$

Also known as
Some sources refer to an atlas as a pre-differentiable structure.

Also see

 * Definition:Topological Manifold
 * Definition:Compatible Atlases