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  "body": "\n## Definition\n\nLet $X$ be a [[Hausdorff space|0MXB4CKHFW]].\nFor $n \\ge 0$ let $D^n$ be the closed unit [[ball|05DSV40EB9]] in $\\R^n$, with interior $\\mathring{D}^n$ and boundary $S^{n-1}$.\nA *CW structure* on $X$ is a partition of $X$ into subsets $e_\\alpha$, called *cells*, with extra data.\nEach cell has a dimension $n_\\alpha \\ge 0$ and a [[continuous map|0SY9ECQ19Q]] $\\Phi_\\alpha \\colon D^{n_\\alpha} \\to X$, its *characteristic map*.\nThese data satisfy:\n\n1. $\\Phi_\\alpha$ restricts to a [[homeomorphism|05GZ11PWRY]] from $\\mathring{D}^{n_\\alpha}$ onto $e_\\alpha$.\n2. $\\Phi_\\alpha(S^{n_\\alpha - 1})$ is contained in the union of finitely many cells $e_\\beta$ with $n_\\beta \u003c n_\\alpha$ (closure finiteness).\n3. A subset $A \\subseteq X$ is closed if and only if $\\Phi_\\alpha^{-1}(A)$ is closed in $D^{n_\\alpha}$ for every $\\alpha$ (weak topology).\n\nA *CW complex* is a Hausdorff space with a CW structure.\nThe cell $e_\\alpha$ is an $n_\\alpha$-cell, and the union $X^n$ of all cells of dimension at most $n$ is the $n$-skeleton.\n\n## Remarks\n\nEquivalently, a CW complex is built inductively.\nThe skeleton $X^0$ is a discrete space, and $X^n$ is obtained from $X^{n-1}$ by attaching $n$-cells along maps $S^{n-1} \\to X^{n-1}$.\nThe union $X = \\bigcup_n X^n$ has the topology in which a set is closed when its intersection with each $X^n$ is closed.\n\nExample: the sphere $S^n$ with $n \\ge 1$ has a CW structure with one $0$-cell $\\set{p}$ and one $n$-cell $S^n \\setminus \\set{p}$.\nThe characteristic map $D^n \\to S^n$ collapses $S^{n-1}$ to $p$ and is a homeomorphism on the interior, so clause 1 holds.\nClause 2 holds since $\\Phi(S^{n-1}) = \\set{p}$, and clause 3 holds since $S^n$ is a quotient of $D^n \\sqcup \\set{p}$.\n\nNon-example: the closed disk $D^2$ with one $0$-cell at each point of the boundary circle and one $2$-cell, the interior, fails clause 2.\nThe boundary of the $2$-cell meets infinitely many $0$-cells.\n\nA CW complex is finite if it has finitely many cells; a finite CW complex is compact.\n",
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