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  "body": "\n## Definition\n\nA *$3$-manifold* is a topological space $M$ such that:\n\n1. $M$ is [[Hausdorff|0S1FJ8VEX9]];\n2. $M$ is second countable;\n3. every point of $M$ has an open neighbourhood [[homeomorphic|0QT00PB5G2]] to an open subset of $\\R^3$.\n\nEquivalently, it is a [[topological manifold|QQQ9B1DK4P]] of dimension $3$.\nA *$3$-manifold with boundary* allows open subsets of the closed half-space $\\set{x \\in \\R^3 : x_3 \\ge 0}$ in clause 3, and a *closed* $3$-manifold is a compact one without boundary.\n\n## Remarks\n\nExample: the $3$-torus $T^3 = \\R^3 / \\Z^3$ is a closed $3$-manifold.\nThe quotient map $\\R^3 \\to T^3$ restricted to any open cube of side $1/2$ is a homeomorphism onto an open subset, and $T^3$ is Hausdorff, second countable and compact.\n\nNon-example: the cone $C = (T^2 \\times [0, 1]) / (T^2 \\times \\set{0})$ on the torus fails clause 3 at the cone point $v$.\nThe local homology group $H_3(C, C \\setminus \\set{v})$ is $\\tilde H_2(T^2) \\cong \\Z$, which matches $\\R^3$, but $H_2(C, C \\setminus \\set{v}) \\cong \\tilde H_1(T^2) \\cong \\Z^2$, while at a point of $\\R^3$ this group is $0$.\n\nOther standard examples are the $3$-sphere $S^3$, the [[lens spaces|AZBTMF196E]] and the [[Poincaré homology sphere|NHERCK7C43]].\n\n## Explanations\n\n### L1 human\n\nThe definition asks that every point have a small region around it shaped like a piece of ordinary space.\nThe first two clauses exclude pathological spaces.\nGluing opposite faces of a cube gives an example with no edge or corner anywhere.\nA cone on a torus is not an example, because the tip of the cone does not look like a point of space.\n",
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