de Rham complex is a resolution of the constant sheaf RX: where ΩXj is the sheaf of smooth j-forms and the map ΩXj → ΩXj+1 is the exterior derivative d. By the results above, the sheaves ΩXj are soft and therefore acyclic. It follows that the sheaf cohomology of X with real coefficients is isomorphic to the de Rham cohomology of X, defined as the cohomology of the complex of real vector spaces: The other part of de Rham's theorem is to identify sheaf cohomology and singular cohomology of X with real coefficients; that holds in greater generality, as discussed above. Čech cohomology Čech cohomology is an approximation to sheaf cohomology that is often useful for computations. Namely, let be an open cover of a topological space X, and let E be a sheaf of abelian groups on X. Write the open sets in the cover as Ui for elements i of a set I, and fix an ordering of I. Then Čech cohomology is defined as the cohomology of an explicit complex of abelian groups with jth group There is a natural homomorphism . Thus Čech cohomology is an approximation to sheaf cohomology using only the sections of E on finite intersections of the open sets Ui. If every finite intersection V of the open sets in has no higher cohomology with coefficients in E, meaning that Hj(V,E) = 0 for all j > 0, then the homomorphism from Čech cohomology to sheaf cohomology is an isomorphism. Another approach to relating Čech cohomology to sheaf cohomology is as follows. The Čech cohomology groups are defined as the direct limit of over all open covers of X (where open covers are ordered by refinement). There is a homomorphism from Čech cohomology to sheaf cohomology, which is an isomorphism for j ≤ 1. For arbitrary topological spaces, Čech cohomology can differ from sheaf cohomology in higher degrees. Conveniently, however, Čech cohomology is isomorphic to sheaf cohomology for any sheaf on a paracompact Hausdorff space. The isomorphism implies a description of H1(X,E) for any sheaf E of abelian groups on a topological space X: this group classifies the E-torsors (also called principal E-bundles) over X, up to isomorphism. (This statement generalizes to any sheaf of groups G, not necessarily abelian, using the non-abelian cohomology set H1(X,G).) By definition, an E-torsor over X is a sheaf S of sets together with an action