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  "body": "\n## Statement\n\nLet $X$ be a [[topological space|0CQHFPGJ45]] and let $A, B \\subseteq X$ be subspaces whose [[interiors|0RZH6W98HK]] cover $X$.\nLet $H_n$ denote [[singular homology|M0Q5CFJ7ZH]] with coefficients in an abelian group.\nThen there are homomorphisms $\\partial \\colon H_n(X) \\to H_{n-1}(A \\cap B)$ such that the sequence\n$$\\cdots \\to H_n(A \\cap B) \\xrightarrow{\\Phi} H_n(A) \\oplus H_n(B) \\xrightarrow{\\Psi} H_n(X) \\xrightarrow{\\partial} H_{n-1}(A \\cap B) \\to \\cdots \\to H_0(X) \\to 0$$\nis [[exact|JB7ZBFBA9C]], where $\\Phi(x) = (i_* x, -j_* x)$ and $\\Psi(y, z) = k_* y + l_* z$ for the inclusions $i \\colon A \\cap B \\to A$, $j \\colon A \\cap B \\to B$, $k \\colon A \\to X$ and $l \\colon B \\to X$.\n\n## Remarks\n\nLet $C_\\bullet^{A+B}(X)$ be the complex of singular chains that are sums of simplices lying in $A$ or in $B$.\nThere is a short exact sequence of chain complexes $0 \\to C_\\bullet(A \\cap B) \\to C_\\bullet(A) \\oplus C_\\bullet(B) \\to C_\\bullet^{A+B}(X) \\to 0$.\nIts long exact sequence in homology is the Mayer–Vietoris sequence, once one knows that the inclusion $C_\\bullet^{A+B}(X) \\to C_\\bullet(X)$ induces isomorphisms in homology.\nThis last fact uses barycentric subdivision and needs the interiors of $A$ and $B$ to cover $X$.\nWithout this hypothesis the sequence can fail.\nTake $X = [0, 1]$, $A = [0, 1/2]$ and $B = (1/2, 1]$, so that $A \\cap B = \\emptyset$.\nExactness at $H_0(A) \\oplus H_0(B)$ would make the map $\\Z^2 \\to \\Z = H_0(X)$ injective, which is false.\nExample: write the sphere $S^n$, $n \\ge 1$, as the union of two open caps, each contractible, whose intersection deformation retracts to $S^{n-1}$.\nThe sequence in reduced homology gives $\\tilde H_k(S^n) \\cong \\tilde H_{k-1}(S^{n-1})$, hence $H_n(S^n; \\Z) \\cong \\Z$.\nThere is a reduced version when $A \\cap B \\neq \\emptyset$, and a dual version for singular cohomology with the arrows reversed.\n\n## Explanations\n\n### L1 human\n\nIf a space is glued from two pieces, its homology can be computed from the homology of the pieces and of their overlap.\nThe Mayer–Vietoris sequence is the exact rule for this computation.\nHoles in the whole space come either from holes in the pieces or from the overlap.\nIf the overlap has several parts, or holes that get filled in the pieces, these create a new hole one dimension higher in the whole space.\nFor example, a circle is two arcs overlapping in two small arcs, and the two components of the overlap produce the loop of the circle.\n\n### L2 human\n\nIt is the main computational tool of basic algebraic topology.\nIt is the analogue for open covers of the long exact sequence of a pair, and it can also be derived from that sequence by excision.\nIt gives the homology of spheres, of wedge sums and of connected sums, and it is the inductive step in proofs about manifolds, such as Poincaré duality.\nThe same pattern appears for de Rham cohomology, for sheaf cohomology and in any cohomology theory satisfying the Eilenberg–Steenrod axioms.\n",
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