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  "body": "\n## Statement\n\nLet $X$ be a topological space and let $A \\subseteq X$ be a subspace.\nLet $Z \\subseteq A$ be a subset whose [[closure|0QEAJB91GZ]] in $X$ is contained in the [[interior|0RZH6W98HK]] of $A$.\nThen the inclusion of pairs $(X \\setminus Z, A \\setminus Z) \\to (X, A)$ induces isomorphisms of [[relative homology|4BVWH0BN28]] groups\n$$H_n(X \\setminus Z, A \\setminus Z) \\to H_n(X, A)$$\nfor every $n \\ge 0$.\n\n## Remarks\n\nAn equivalent form: if $A, B \\subseteq X$ are subspaces whose interiors cover $X$, then the inclusion $(B, A \\cap B) \\to (X, A)$ induces isomorphisms on all $H_n$.\nThe proof uses repeated [[barycentric subdivision|J5DZEYCEX2]] to show that every singular chain is homologous to a sum of small simplices, each lying in the interior of $A$ or in $X \\setminus Z$.\nThe chains made of such small simplices compute the same homology as all chains, and on them the excision map is an isomorphism at chain level.\n\nExample: if $A$ is a nonempty closed subspace that is a deformation retract of some open neighbourhood in $X$, then excision gives $H_n(X, A) \\cong \\tilde H_n(X / A)$, where $\\tilde H$ is [[reduced homology|NJQMWVK3QK]].\nFor $(D^n, S^{n-1})$ this gives $H_n(D^n, S^{n-1}) \\cong \\tilde H_n(S^n)$.\n\nThe closure hypothesis is needed.\nTake $X = [-1, 1]$, $A = [0, 1]$ and $Z = A$; the closure of $Z$ is not inside the interior $(0, 1]$ of $A$.\nThen $H_0(X \\setminus Z, A \\setminus Z) = H_0([-1, 0)) \\cong \\Z$, while $H_0(X, A) = 0$ because $X$ is path-connected and $A$ is nonempty.\n\n## Explanations\n\n### L1 human\n\nRelative homology of $X$ modulo $A$ only depends on what happens near and outside $A$.\nSo one can cut away a piece deep inside $A$ without changing it.\nThis turns a hard global computation into a local one, for example the homology near a single point.\n\n### L2 human\n\nExcision is one of the Eilenberg–Steenrod axioms, and it is the axiom that fails for homotopy groups.\nWith the long exact sequence of a pair it gives the Mayer–Vietoris sequence and the suspension isomorphism $\\tilde H_n(X) \\cong \\tilde H_{n+1}(\\Sigma X)$.\nFor manifolds it gives $H_n(M, M \\setminus \\set{x}) \\cong H_n(\\R^n, \\R^n \\setminus \\set{0}) \\cong \\Z$, the starting point of orientations.\n",
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