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  "body": "\n## Definition\n\nLet $n \\ge 1$ be an integer and let $\\overline{\\R}^n_+ = \\set{x \\in \\R^n : x_1 \\ge 0, \\ldots, x_n \\ge 0}$ be the closed orthant, with the subspace topology.\nLet $A \\subseteq \\R^n$ be any subset.\nA map $f \\colon A \\to \\R^n$ is *smooth* if each $a \\in A$ has an open neighbourhood $W$ in $\\R^n$ with a [[smooth|0V33BNKZ9S]] map $g \\colon W \\to \\R^n$ equal to $f$ on $A \\cap W$.\nLet $M$ be a [[topological space|0CQHFPGJ45]].\nA *chart with corners* on $M$ is a [[homeomorphism|0QT00PB5G2]] $\\varphi$ from an open subset $U \\subseteq M$ onto an open subset of $\\overline{\\R}^n_+$.\nA *smooth atlas with corners* is a family of charts with corners whose domains cover $M$.\nIts transition maps $\\varphi_j \\circ \\varphi_i^{-1}$ must be smooth in the above sense.\nA *smooth manifold with corners of dimension $n$* is a space $M$ such that:\n\n1. $M$ is [[Hausdorff|0S1FJ8VEX9]].\n2. $M$ is [[second-countable|03PPNCM9GG]].\n3. $M$ carries a maximal smooth atlas with corners of dimension $n$.\n\n## Remarks\n\nThe square $[0, 1]^2$ is a manifold with corners.\nNear the vertex $(0, 0)$ the identity is a chart onto an open subset of $\\overline{\\R}^2_+$.\nNear $(1, 1)$ the chart is $(x, y) \\mapsto (1 - x, 1 - y)$, and near the other points one uses similar affine maps.\nAll transition maps are affine, hence smooth, and the charts generate a maximal atlas.\n\nThe solid square pyramid in $\\R^3$ is not a manifold with corners for this structure, because clause 3 fails at its apex.\nThe tangent cone of the pyramid at the apex has four faces.\nThe derivative of a chart at the apex would map it linearly onto the cone $\\overline{\\R}^3_+$, which has only three faces.\n\nA chart with corners is in particular a chart in the sense of an [[atlas|J7P32JQTVE]] modelled on $\\overline{\\R}^n_+$.\nAs a topological space, every manifold with corners is a [[manifold with boundary|7DS6B2BFK8]], because $\\overline{\\R}^n_+$ is homeomorphic to the half-space $\\set{x_n \\ge 0}$.\nIn Mathlib, manifolds with corners are modelled through a `ModelWithCorners`.\n",
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