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  "body": "\n## Definition\n\nLet $n \\ge 0$ be an integer and let $M$ be a [[topological space|0CQHFPGJ45]].\n$M$ is a *topological manifold of dimension $n$*, or a topological $n$-manifold, if:\n\n1. $M$ is [[Hausdorff|0S1FJ8VEX9]].\n2. $M$ is [[second-countable|03PPNCM9GG]].\n3. $M$ is locally Euclidean of dimension $n$: each point $x \\in M$ has an [[open|02BH3MCBQB]] neighbourhood $U$ [[homeomorphic|0QT00PB5G2]] to an open subset of $\\R^n$, the [[Euclidean space|0XPR1E85W0]].\n\n## Remarks\n\nThe unit circle $S^1 = \\set{(x, y) \\in \\R^2 : x^2 + y^2 = 1}$ is a $1$-manifold.\nIt is Hausdorff and second-countable as a subspace of $\\R^2$.\nThe map $(x, y) \\mapsto x$ sends the open upper half $\\set{y \u003e 0}$ of $S^1$ homeomorphically onto the interval $(-1, 1)$.\nThe three other open half circles are treated in the same way, and together the four cover $S^1$.\n\nThe line with two origins is the quotient of $\\R \\times \\set{0, 1}$ obtained by identifying $(t, 0)$ with $(t, 1)$ for all $t \\ne 0$.\nIt satisfies clauses 2 and 3, but it fails clause 1: the two origins have no disjoint neighbourhoods.\n\nA nonempty manifold has only one dimension, by [[invariance of domain|6X2D0JDFQW]]: a nonempty open subset of $\\R^m$ is not homeomorphic to an open subset of $\\R^n$ when $m \\ne n$.\nA pair $(U, \\varphi)$ with $\\varphi$ a homeomorphism from $U$ onto an open subset of $\\R^n$ is called a chart.\nA compact manifold without boundary is called a closed manifold.\n",
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