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  "body": "\n## Definition\n\nLet $M$ be a [[topological manifold|QQQ9B1DK4P]] of dimension $n$.\nA chart $(U, \\varphi)$ on $M$ is an open set $U \\subseteq M$ with a [[homeomorphism|0QT00PB5G2]] $\\varphi$ from $U$ onto an open subset of $\\R^n$.\nTwo charts $(U, \\varphi)$ and $(V, \\psi)$ are *smoothly compatible* if $U \\cap V = \\emptyset$, or if the transition map $\\psi \\circ \\varphi^{-1} \\colon \\varphi(U \\cap V) \\to \\psi(U \\cap V)$ is [[smooth|0V33BNKZ9S]] with smooth inverse.\nA *smooth [[atlas|J7P32JQTVE]]* on $M$ is a family of pairwise smoothly compatible charts whose domains cover $M$.\nA *smooth structure* on $M$ is a smooth atlas $\\mathcal{A}$ that is maximal.\nThis means that every chart that is smoothly compatible with all charts of $\\mathcal{A}$ already belongs to $\\mathcal{A}$.\n\n## Remarks\n\nEvery smooth atlas is contained in a unique maximal smooth atlas, so a smooth structure is usually given by any one atlas.\nExample: on $\\R^n$ the single chart $(\\R^n, \\id)$ is a smooth atlas, and the maximal atlas containing it is the standard smooth structure.\nNon-example: the atlas $\\set{(\\R, \\id), (\\R, t \\mapsto t^3)}$ on $\\R$ is not a smooth atlas.\nThe transition map $t \\mapsto t^{1/3}$ is not differentiable at $0$, so the two charts are not smoothly compatible.\nEach of the two charts alone defines a smooth structure; the two structures are different but diffeomorphic.\nTwo smooth structures on $M$ are usually counted as the same when they are diffeomorphic, not only when they are equal.\n",
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