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  "body": "\n## Definition\n\nLet $X$ be a [[locally compact|0FE2SB8WVJ]] [[Hausdorff space|0S1FJ8VEX9]] and let $A$ be an [[abelian group|082GHNPJ53]].\nA *locally finite singular $i$-chain* on $X$ is a formal sum $c = \\sum_\\sigma a_\\sigma \\sigma$, possibly infinite, over singular $i$-simplices $\\sigma \\colon \\Delta^i \\to X$ with $a_\\sigma \\in A$.\nIt is required that every point of $X$ has a neighbourhood that meets the image of only finitely many $\\sigma$ with $a_\\sigma \\neq 0$.\nThese chains form a group $C^{\\mathrm{lf}}_i(X; A)$, and the usual boundary formula of [[singular homology|M0Q5CFJ7ZH]] defines $\\partial \\colon C^{\\mathrm{lf}}_i(X; A) \\to C^{\\mathrm{lf}}_{i-1}(X; A)$ with $\\partial \\circ \\partial = 0$.\nThe *Borel–Moore homology* of $X$ is the homology of this chain complex:\n$$H^{\\mathrm{BM}}_i(X; A) = H_i(C^{\\mathrm{lf}}_\\bullet(X; A)).$$\n\n## Remarks\n\nThe boundary of a locally finite chain is locally finite, because each face of $\\sigma$ has image inside the image of $\\sigma$.\nIf $X$ is compact, a locally finite chain is a finite chain, so $H^{\\mathrm{BM}}_i(X; A) = H_i(X; A)$.\nSome authors define the theory with sheaves.\nIf $X = Y \\setminus Z$ for a finite CW complex $Y$ and a subcomplex $Z$, then $H^{\\mathrm{BM}}_i(X; A) \\cong H_i(Y, Z; A)$.\n\nExample: on $X = \\R$, the chain $c = \\sum_{k \\in \\Z} \\sigma_k$, where $\\sigma_k$ is the affine path from $k$ to $k+1$, is locally finite, and $\\partial c = \\sum_k (\\set{k+1} - \\set{k}) = 0$.\nIts class generates $H^{\\mathrm{BM}}_1(\\R; \\Z) \\cong \\Z$, while $H^{\\mathrm{BM}}_0(\\R; \\Z) = 0$.\n\nNon-example: the infinite sum $\\sum_{k \\ge 1} \\tau_k$, where $\\tau_k$ is the affine path from $0$ to $1/k$ in $\\R$, is not locally finite: every neighbourhood of $0$ meets all the $\\tau_k$.\n\nFor an oriented $n$-manifold $M$, Poincaré duality takes the form $H^{\\mathrm{BM}}_i(M; \\Z) \\cong H^{n-i}(M; \\Z)$.\n",
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