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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nA *triangulation* of $X$ is a pair $(K, h)$ with $K$ a [[simplicial complex|V8GZ0W1ZZ5]].\nThe map $h \\colon \\abs{K} \\to X$ must be a [[homeomorphism|0QT00PB5G2]] from the geometric realization $\\abs{K}$ of $K$ onto $X$.\n$X$ is *triangulable* if it has a triangulation.\n\n## Remarks\n\nLet $K$ be the simplicial complex of all proper faces of a $3$-simplex in $\\R^3$, with $4$ vertices, $6$ edges and $4$ triangles.\nRadial projection from the barycentre onto the sphere centred there is a homeomorphism $\\abs{K} \\to S^2$, so $K$ with this map is a triangulation of $S^2$.\n\nThe CW structure of $S^2$ with one $0$-cell and one $2$-cell is not a triangulation.\nIts $2$-cell is glued to the point along its whole boundary circle, while in a simplicial complex each triangle has three distinct vertices and three edges.\n\nSome authors allow only finite or locally finite complexes $K$, and some require $h$ to be piecewise linear in a given structure.\nEvery smooth manifold is triangulable, while some topological $4$-manifolds, such as the $E_8$ manifold, are not.\n",
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