Alexander-Spanier Cohomology
A fundamental result of de Rham cohomology is that the kth de Rham cohomology vector space of a manifold M is canonically isomorphic to the Alexander-Spanier cohomology vector space H^k(M;R) (also called cohomology with compact support). In the case that M is compact, Alexander-Spanier cohomology is exactly "singular" cohomology.