of X to the integers Z, whereas sheaf cohomology H0(X,ZX) is the group of locally constant functions from X to Z. These are different, for example, when X is the Cantor set. Indeed, the sheaf cohomology H0(X,ZX) is a countable abelian group in that case, whereas the singular cohomology H0(X,Z) is the group of all functions from X to Z, which has cardinality For a paracompact Hausdorff space X and any sheaf E of abelian groups on X, the cohomology groups Hj(X,E) are zero for j greater than the covering dimension of X. (This does not hold in the same generality for singular cohomology: for example, there is a compact subset of Euclidean space R3 that has nonzero singular cohomology in infinitely many degrees.) The covering dimension agrees with the usual notion of dimension for a topological manifold or a CW complex. Flabby and soft sheaves A sheaf E of abelian groups on a topological space X is called acyclic if Hj(X,E) = 0 for all j > 0. By the long exact sequence of sheaf cohomology, the cohomology of any sheaf can be computed from any acyclic resolution of E (rather than an injective resolution). Injective sheaves are acyclic, but for computations it is useful to have other examples of acyclic sheaves. A sheaf E on X is called flabby (French: flasque) if every section of E on an open subset of X extends to a section of E on all of X. Flabby sheaves are acyclic. Godement defined sheaf cohomology via a canonical flabby resolution of any sheaf; since flabby sheaves are acyclic, Godement's definition agrees with the definition of sheaf cohomology above. A sheaf E on a paracompact Hausdorff space X is called soft if every section of the restriction of E to a closed subset of X extends to a section of E on all of X. Every soft sheaf is acyclic. Some examples of soft sheaves are the sheaf of real-valued continuous functions on any paracompact Hausdorff space, or the sheaf of smooth (C∞) functions on any smooth manifold. More generally, any sheaf of modules over a soft sheaf of commutative rings is soft; for example, the sheaf of smooth sections of a vector bundle over a smooth manifold is soft. For example, these results form part of the proof of de Rham's theorem. For a smooth manifold X, the Poincaré lemma says that the