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  "body": "\n## Definition\n\nLet $n \\ge 0$ be an integer.\n\n1. Let $X$ be a [[CW complex|5W95QFHXPR]]. The *$n$-skeleton* $X^{(n)}$ of $X$ is the union of all closed cells of $X$ of dimension at most $n$.\n2. Let $K$ be a [[simplicial complex|V8GZ0W1ZZ5]]. The *$n$-skeleton* $K^{(n)}$ of $K$ is the subcomplex of all simplices of $K$ of dimension at most $n$.\n\nThe notation $X^n$ is also common.\n\n## Remarks\n\nThe torus $T^2$ has a CW structure with one $0$-cell, two $1$-cells $a$ and $b$, and one $2$-cell attached along $a b a^{-1} b^{-1}$.\nIts $0$-skeleton is a point, its $1$-skeleton is a wedge of two circles, and its $2$-skeleton is $T^2$.\n\nThe skeleton depends on the cell structure and not only on the space.\nThe sphere $S^2$ with one $0$-cell and one $2$-cell has a point as $1$-skeleton.\nThe sphere $S^2$ as the boundary of a tetrahedron has as $1$-skeleton a graph with $4$ vertices and $6$ edges.\nSo \"the $1$-skeleton of $S^2$\" has no meaning before a cell structure is chosen.\n\nFor a CW complex, $X^{(n)}$ is obtained from $X^{(n-1)}$ by attaching $n$-cells along maps from their boundary spheres, and $X$ is the union of its skeleta with the weak topology.\nThe inclusion $X^{(n)} \\to X$ induces an isomorphism on [[homotopy groups|P976WYEFMB]] $\\pi_k$ for $k \u003c n$ and a surjection for $k = n$.\n",
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