Borel–Moore homology

In topology, Borel−Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by Armand Borel and John Moore in 1960.

For reasonable compact spaces, Borel−Moore homology coincides with the usual singular homology. For non-compact spaces, each theory has its own advantages. In particular, a closed oriented submanifold defines a class in Borel–Moore homology, but not in ordinary homology unless the submanifold is compact.

Note: Borel equivariant cohomology is an invariant of spaces with an action of a group G; it is defined as ${\displaystyle H_{G}^{*}(X)=H^{*}((EG\times X)/G).}$ That is not related to the subject of this article.