Definition:Atlas

Definition
Let $M$ be a topological space.

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

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

Some sources call a pre-atlas what on is known as an atlas, and simply refer to a maximal atlas as an atlas.

Also see

 * Definition:Topological Manifold
 * Definition:Compatible Atlases