Atlases

A SET OF CHARTS WHOSE OPEN SETS COVER THE SPACE

An atlas is a set of charts whose open sets cover the space: every point lies in at least one of them. That is the entire condition. The atlas is the family, not any one of its members.

The atlas is one object: the set of the charts, not any one of them. { } M U1 R x1(U1) (U1, x1) M U2 R x2(U2) (U2, x2) M U3 R x3(U3) (U3, x3) Each panel holds one chart: an open set of the space, and its image on the line R. The atlas A is the whole set, and the one condition on it is that the open sets cover the space: U1 ∪ U2 ∪ U3 = M Nothing here asks how few charts there are, or how they agree where they overlap.
Each panel is one chart, and the braces collect the three into one set.

The definition

Definition · atlas

Let \((M,\mathcal{O})\) be a \(d\)-dimensional topological manifold. An atlas of \((M,\mathcal{O})\) is a set of charts \[ \mathcal{A} = \{(U_{(\alpha)}, x_{(\alpha)}) \;\vert\; \alpha \in A\} \] whose domains cover \(M\): \[ \bigcup_{\alpha \in A} U_{(\alpha)} = M. \]

The index set \(A\) is unrestricted: an atlas may hold two charts or infinitely many, and the union condition is a statement about the open sets alone. That some atlas exists for a \(d\)-dimensional topological manifold is immediate: a chart is available around every point, and the set of all of them covers \(M\).

Covering is the whole requirement

Nothing is required about how two members agree where their open sets meet, and nothing requires the family to be as small as it could be. Adding a further chart to an atlas leaves an atlas.

Two charts that cover the circle, checked

Let \(M\) be the unit circle. Take \(U\) to be the circle without \((1,0)\), with \(x\) the angle from \((1,0)\) taken in \((0,2\pi)\); take \(V\) to be the circle without \((-1,0)\), with \(y\) the angle taken in \((-\pi,\pi)\). Each pair is a chart, and the covering condition is checked point by point: \((1,0)\) lies in \(V\), \((-1,0)\) lies in \(U\), and every other point of the circle lies in both. So \(U \cup V = M\), and \(\{(U,x), (V,y)\}\) is an atlas of two charts.

Neither open set covers alone: \(U\) misses \((1,0)\) and \(V\) misses \((-1,0)\), so each one-member family fails the union condition, and the pair together is what passes it.