  For any topological space there is a sheaf cohomology. A Grothendieck
topology is a generalization of the classical topology such that it also
possesses a sheaf cohomology. On the other hand any noncommutative
$C^*$-algebra is a generalization of a locally compact Hausdorff space. Here we
define a Grothendieck topology arising from $C^*$-algebras which is a
generalization of the topology of the spectra of commutative $C^*$-algebras.
This construction yields a noncommutative generalization of the sheaf
cohomology of topological spaces. The presented here theory gives a unified
approach to the Gelfand duality and the duality between the commutative von
Neumann algebras and measure locales. The generalization of the Dixmier-Douady
theory concerning $C^*$-algebras of foliations is also discussed.
