For a locally compact space, replace finite singular chains by chains for which every compact subset meets the images of only finitely many simplices with nonzero coefficients. The boundary operator preserves this local finiteness, giving the Borel-Moore chain complex and its homology. On a compact space it reduces to ordinary singular homology. A noncompact manifold has a locally finite fundamental class, with its orientation local system when necessary. The chain groups and homology are zero in negative degrees.
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Borel–Moore homology is a homological algebraic concept that arises in the study of algebraic varieties, particularly in the context of algebraic geometry and algebraic topology. It is a form of homology theory designed to handle locally compact topological spaces, with particular application to appropriate classes of varieties, such as quasi-projective varieties or complex algebraic varieties.