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  "body": "\n## Definition\n\nLet $n \\ge 1$ be an integer and let $\\mathbb{H}^n = \\set{x \\in \\R^n : x_n \\ge 0}$ be the closed upper half-space of [[Euclidean space|0XPR1E85W0]], with the subspace topology.\nA [[topological space|0CQHFPGJ45]] $M$ is a *topological manifold with boundary of dimension $n$* if:\n\n1. $M$ is [[Hausdorff|0S1FJ8VEX9]].\n2. $M$ is [[second-countable|03PPNCM9GG]].\n3. Each point $x \\in M$ has an [[open|02BH3MCBQB]] neighbourhood that is [[homeomorphic|0QT00PB5G2]] to an open subset of $\\mathbb{H}^n$.\n\nA point $x \\in M$ is a *boundary point* if some homeomorphism $\\varphi$ from an open neighbourhood of $x$ onto an open subset of $\\mathbb{H}^n$ has $\\varphi(x) \\in \\set{x_n = 0}$.\nThe set of boundary points is the *boundary* $\\partial M$, and $M \\setminus \\partial M$ is the *interior* of $M$.\n\n## Remarks\n\nThe closed disk $D^2 = \\set{x \\in \\R^2 : \\abs{x} \\le 1}$ is a manifold with boundary of dimension $2$, with $\\partial D^2 = S^1$.\nNear a point of the open disk it is an open subset of $\\R^2$, which is homeomorphic to an open subset of the interior of $\\mathbb{H}^2$.\nNear a point of $S^1$, polar coordinates $(r, \\theta) \\mapsto (\\theta, 1 - r)$ map a neighbourhood onto an open subset of $\\mathbb{H}^2$, sending $S^1$ into the line $\\set{x_2 = 0}$.\n\nThe cross $\\set{(x, y) \\in \\R^2 : xy = 0}$ satisfies clauses 1 and 2 but fails clause 3 at the origin, where four arcs meet.\n\nBy [[invariance of domain|6X2D0JDFQW]], no point is a boundary point for one chart and an interior point for another, so the boundary is well defined.\nThe interior is a [[topological manifold|QQQ9B1DK4P]] of dimension $n$, and $\\partial M$ is a topological manifold of dimension $n - 1$ with empty boundary.\nA topological manifold is the same as a manifold with boundary whose boundary is empty.\nSome authors use $\\set{x_1 \\ge 0}$ or $\\set{x_1 \\le 0}$ as the model half-space; the notion does not change.\n",
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