Open cover

An open cover of a subset $A$ is a family of open sets whose union contains $A$.

An open cover is acyclic for a specified sheaf if its restriction to every nonempty finite intersection has zero higher sheaf cohomology. The acyclic cover theorem then identifies the cover's Čech cohomology with sheaf cohomology. On a separated scheme, an affine open cover is acyclic for every quasi-coherent sheaf.

A topological space is quasi-compact when every open cover has a finite subcover. This is the form of compactness used for spaces that need not be Hausdorff, especially spectra and schemes.