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  "body": "\n## Statement\n\nLet $K$ and $L$ be [[simplicial complexes|V8GZ0W1ZZ5]] with $K$ finite, and let $\\abs{K}$ and $\\abs{L}$ be their geometric realizations.\nFor a point $y \\in \\abs{L}$, let $\\operatorname{supp}(y)$ be the smallest closed simplex of $\\abs{L}$ that contains $y$.\nA simplicial map $g \\colon K \\to L$ is a *simplicial approximation* of a continuous map $f \\colon \\abs{K} \\to \\abs{L}$ if $\\abs{g}(x) \\in \\operatorname{supp}(f(x))$ for every $x \\in \\abs{K}$.\nLet $f \\colon \\abs{K} \\to \\abs{L}$ be continuous.\nLet $\\operatorname{sd}^m K$ be the $m$-fold [[barycentric subdivision|J5DZEYCEX2]] of $K$, so that $\\abs{\\operatorname{sd}^m K} = \\abs{K}$.\nThen there are an integer $m \\ge 0$ and a simplicial map $g \\colon \\operatorname{sd}^m K \\to L$ that is a simplicial approximation of $f$.\n\n## Remarks\n\nA vertex map $g$ is a simplicial approximation of $f$ as soon as $f(\\operatorname{st} v) \\subseteq \\operatorname{st} g(v)$ for every vertex $v$, where $\\operatorname{st}$ denotes the open star.\nThe open stars of the vertices of $L$ form an open cover of $\\abs{L}$, so their preimages under $f$ form an open cover of the compact metric space $\\abs{K}$.\nRepeated barycentric subdivision makes the simplices smaller than a Lebesgue number of this cover.\nThen each open star of $\\operatorname{sd}^m K$ maps into some open star of $L$, and choosing such a vertex for each $v$ gives $g$.\n\nA simplicial approximation $g$ of $f$ is [[homotopic|F2F143KQ2M]] to $f$ by the straight-line homotopy, which stays in $\\operatorname{supp}(f(x))$.\n\nSubdivision is necessary.\nLet $K = L$ be the boundary of a triangle, so $\\abs{K}$ is a circle, and let $f$ wrap this circle twice around itself.\nA simplicial map $K \\to L$ is either a bijection on vertices, of degree $\\pm 1$, or misses a vertex, of degree $0$.\nSince a simplicial approximation is homotopic to $f$, which has degree $2$, no simplicial map $K \\to L$ approximates $f$.\n\n## Explanations\n\n### L1 human\n\nA continuous map between triangulated spaces can be very irregular.\nThe theorem says that after cutting the source into small enough triangles, the map can be replaced by one that sends triangles linearly onto triangles.\nThe new map is close to the old one and homotopic to it.\n\n### L2 human\n\nSimplicial approximation shows that the set of homotopy classes of maps $\\abs{K} \\to \\abs{L}$ is countable when $K$ and $L$ are finite.\nIt is used to prove that simplicial homology is a homotopy invariant and agrees with singular homology, and to compute $\\pi_k(S^n) = 0$ for $k \u003c n$.\nThe analogue for CW complexes is the cellular approximation theorem.\n",
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