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 $\mathscr F = \family {\struct {U_\alpha, \phi_\alpha}: \alpha \in A}$ 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.
Smooth Atlas
A smooth atlas on $M$ is a $C^\infty$-atlas on $M$.
Maximal Atlas
Let $A$ be a $d$-dimensional atlas of class $C^k$ of $M$.
Definition 1
$A$ is a maximal $C^k$-atlas of dimension $d$ if and only if $A$ is not strictly contained in another $C^k$-atlas.
Definition 2
$A$ is a maximal $C^k$-atlas if and only if $A$ contains all charts of $M$ that are $C^k$-compatible with $A$.
Definition 3
$A$ is a maximal $C^k$-atlas if and only if $A$ is a maximal element of some differentiable structure, partially ordered by inclusion. That is, a maximal element of some equivalence class of the set of atlases of class $\CC^k$ on $M$ under the equivalence relation of compatibility.
Also known as
Some sources refer to an atlas as a pre-differentiable structure.
Some sources call a pre-atlas what on $\mathsf{Pr} \infty \mathsf{fWiki}$ is known as an atlas, and simply refer to a maximal atlas as an atlas.
Also see
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): atlas
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): atlas
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): atlas