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  "body": "\n## Summary\n\nBorel–Moore homology is a homology theory for locally compact spaces in which cycles may be infinite, as long as they are locally finite.\nIt is also called homology with closed supports.\nFor an oriented $n$-manifold $M$ it is dual to ordinary cohomology, $H^{\\mathrm{BM}}_i(M) \\cong H^{n-i}(M)$, even when $M$ is not compact.\nIt is the natural home of fundamental classes of non-compact manifolds and of closed subvarieties of complex varieties.\n\n## Explanations\n\n### L1 human\n\nIn ordinary homology a cycle is a finite sum of small pieces.\nIn Borel–Moore homology a cycle may have infinitely many pieces, provided only finitely many of them crowd near any one point.\nThe whole real line, cut into the intervals $[k, k+1]$ for all integers $k$, is such a cycle.\nIt has no boundary, because each endpoint is shared by two intervals and cancels.\nSo the line has a fundamental class in Borel–Moore homology, though it has none in ordinary homology.\n",
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