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  "body": "\n## Statement\n\nLet $X$ and $Y$ be [[CW complexes|5W95QFHXPR]] with $n$-skeleta $X^n$ and $Y^n$.\nA map $g \\colon X \\to Y$ is *cellular* if $g(X^n) \\subseteq Y^n$ for all $n \\ge 0$.\nLet $f \\colon X \\to Y$ be a continuous map, and let $A \\subseteq X$ be a subcomplex such that $f$ restricted to $A$ is cellular.\nThen $f$ is [[homotopic|F2F143KQ2M]] [[relative to|ADDS6HE5SV]] $A$ to a cellular map.\n\n## Remarks\n\nThe proof proceeds cell by cell, by induction on the dimension.\nThe key lemma concerns a map from a closed $k$-disk into a space obtained by attaching an $n$-cell with $n \u003e k$.\nSuch a map can be deformed so that it misses a point of the open cell, and then pushed off the cell by radial projection.\nTaking $A = \\emptyset$ shows that every map is homotopic to a cellular map.\n\nExample: give $S^n$ the CW structure with one $0$-cell and one $n$-cell, and let $k \u003c n$.\nEvery map $S^k \\to S^n$ is homotopic to a cellular map, which sends $S^k = (S^k)^k$ into the $k$-skeleton of $S^n$, a single point.\nSo $\\pi_k(S^n) = 0$ for $k \u003c n$.\n\nThe theorem also shows that the inclusion $X^n \\to X$ induces an isomorphism on $\\pi_k$ for $k \u003c n$ and a surjection for $k = n$.\n\n## Explanations\n\n### L1 human\n\nA continuous map can be very wild, for example a curve can fill a whole square.\nThe theorem says that between CW complexes one can always straighten a map, up to homotopy, so that it sends low-dimensional cells into low-dimensional parts of the target.\nThis is why a loop in a sphere of dimension $2$ or more can always be shrunk to a point.\n\n### L2 human\n\nCellular approximation reduces many homotopy questions to combinatorics of cells.\nIt gives the vanishing of $\\pi_k(S^n)$ for $k \u003c n$.\nIt shows that the fundamental group of a CW complex depends only on its $2$-skeleton.\nIt also makes the maps induced on cellular chain complexes well defined.\nIts simplicial ancestor is the simplicial approximation theorem.\n",
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