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  "body": "\n## Definition\n\nLet $M$ be a [[topological space|0CQHFPGJ45]] and let $n \\ge 0$ be an integer.\nA *chart* of dimension $n$ on $M$ is a pair $(U, \\varphi)$ with $U \\subseteq M$ [[open|02BH3MCBQB]].\nThe map $\\varphi$ must be a [[homeomorphism|0QT00PB5G2]] from $U$ onto an open subset $\\varphi(U)$ of [[Euclidean space|0XPR1E85W0]] $\\R^n$.\nAn *atlas* of dimension $n$ on $M$ is a family $\\set{(U_i, \\varphi_i)}_{i \\in I}$ of charts of dimension $n$ on $M$ such that $\\bigcup_{i \\in I} U_i = M$.\n\nFor two charts $(U_i, \\varphi_i)$ and $(U_j, \\varphi_j)$ of an atlas, the *transition map* is the homeomorphism\n$$\\varphi_j \\circ \\varphi_i^{-1} \\colon \\varphi_i(U_i \\cap U_j) \\to \\varphi_j(U_i \\cap U_j)$$\nbetween open subsets of $\\R^n$.\n\n## Remarks\n\nOn the circle $S^1 \\subset \\R^2$, let $N = (0, 1)$ and $S = (0, -1)$.\nThe 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 $S^1$.\nSo they form an atlas of dimension $1$; the transition map is $t \\mapsto 1/t$ on $\\R \\setminus \\set{0}$.\n\nThe single chart $S^1 \\setminus \\set{N}$ is not an atlas of $S^1$, because its domain misses the point $N$.\n\nA Hausdorff, second-countable space has an atlas of dimension $n$ exactly when it is a [[topological manifold|QQQ9B1DK4P]] of dimension $n$.\nAn atlas is called smooth when all its transition maps are smooth, and piecewise linear when they are piecewise linear.\nTwo smooth atlases are equivalent when their union is again a smooth atlas, and a smooth structure is an equivalence class of smooth atlases.\nIn Mathlib, a space with an atlas modelled on a space $H$ is a `ChartedSpace H M`.\n",
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