atlas

Atlas

Set

context	$\langle M,T\rangle$ … second-countable Hausdorff space
context	$n\in \mathbb N$
definiendum	$A\in$ it
inclusion	$A\subseteq$ chart $\left(\langle M,T\rangle,n\right)$
forall	$x\in M$
exists	$\langle U,\varphi\rangle\in A$
postulate	$x\in U$

Discussion

Idea

An atlas is a set of charts, so that no point $x\in M$ is left out from being mapped to $\mathbb R^n$ .

Alternative definitions

One can equivalently postulate that $M$ is covered by the union of all the open subsets $U$ given with the charts $\langle U,\varphi\rangle$ of an atlas $A$ :

$\bigcup_{chart\in A}\pi_1(chart)=M$ .

Reference

Wikipedia: Atlas (topology)

Parents

Context*

Second-countable Hausdorff space, Natural number

Subset of

Chart