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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CNQXVBHCX]] and let $A \\subseteq X$ be a subspace.\nFor $n \\ge 0$ let $D^n$ be the closed unit [[ball|05DSV40EB9]] in $\\R^n$ with boundary $S^{n-1}$, where $S^{-1} = \\emptyset$.\nA *relative CW structure* on $(X, A)$ is a sequence of subspaces $A = X^{-1} \\subseteq X^0 \\subseteq X^1 \\subseteq \\cdots$ of $X$ such that:\n\n1. For each $n \\ge 0$ there are a set $J_n$ and maps $\\varphi_j \\colon S^{n-1} \\to X^{n-1}$, $j \\in J_n$, with the following property.\n   The space $X^n$ is the pushout of $X^{n-1} \\leftarrow \\bigsqcup_{j \\in J_n} S^{n-1} \\to \\bigsqcup_{j \\in J_n} D^n$, where the left map is given by the $\\varphi_j$.\n   That is, $X^n$ is the [[quotient|0V8046R0P9]] of the [[disjoint union|A486VR787Z]] $X^{n-1} \\sqcup \\bigsqcup_{j} D^n$ that identifies each $x \\in S^{n-1}$ in the $j$-th disk with $\\varphi_j(x)$.\n2. $X = \\bigcup_{n \\ge -1} X^n$.\n3. A subset $C \\subseteq X$ is closed if and only if $C \\cap X^n$ is closed in $X^n$ for every $n$.\n\nA *relative CW complex* is a pair $(X, A)$ with a relative CW structure.\nThe images of the open disks $D^n \\setminus S^{n-1}$ are the $n$-cells of $X$ relative to $A$.\n\n## Remarks\n\nWhen $A = \\emptyset$, a relative CW complex is the same as a [[CW complex|5W95QFHXPR]].\nIf $A$ is a subcomplex of a CW complex $X$, then $(X, A)$ is a relative CW complex.\n\nExample: let $A = S^1$ and $X = D^2$.\nTake $X^0 = X^1 = S^1$ with $J_0 = J_1 = \\emptyset$, and $X^2 = D^2$ with one $2$-cell attached by the identity $S^1 \\to S^1$.\nClause 1 holds since $D^2$ is the pushout of $S^1 \\leftarrow S^1 \\to D^2$, and clauses 2 and 3 hold since the filtration stops at $X^2 = X$.\n\nNon-example: let $A = \\set{0}$ and $X = \\set{0} \\cup \\set{1/m : m \\ge 1} \\subseteq \\R$.\nThe only cells available are $0$-cells, so $X = X^0$.\nClause 1 then forces $X^0$ to be the disjoint union of $A$ and a discrete set, but in $X$ the points $1/m$ converge to $0$, so clause 1 fails.\n",
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