A fundamental result of de Rham cohomology is that the th de Rham cohomology vector space of a manifold is canonically isomorphic to the Alexander-Spanier cohomology vector space (also called cohomology with compact support). In the case that is compact, Alexander-Spanier cohomology is exactly "singular" cohomology.
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Weisstein, Eric W. "Alexander-Spanier Cohomology." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Alexander-SpanierCohomology.html