Atlas

Remarks

On the circle $S^1 \subset \R^2$, let $N = (0, 1)$ and $S = (0, -1)$. The stereographic projections $\varphi_N(x, y) = x / (1 - y)$ on $S^1 \setminus \set{N}$ and $\varphi_S(x, y) = x / (1 + y)$ on $S^1 \setminus \set{S}$ are homeomorphisms onto $\R$, and their domains cover $S^1$. So they form an atlas of dimension $1$; the transition map is $t \mapsto 1/t$ on $\R \setminus \set{0}$.

The single chart $S^1 \setminus \set{N}$ is not an atlas of $S^1$, because its domain misses the point $N$.

A Hausdorff, second-countable space has an atlas of dimension $n$ exactly when it is a topological manifold of dimension $n$. An atlas is called smooth when all its transition maps are smooth, and piecewise linear when they are piecewise linear. Two smooth atlases are equivalent when their union is again a smooth atlas, and a smooth structure is an equivalence class of smooth atlases. In Mathlib, a space with an atlas modelled on a space $H$ is a ChartedSpace H M.