A family of supports in a topological space is a family of closed subsets such that
In other words, it is an ideal in the lattice of closed subsets.
Families of supports are used to introduce a variant of sheaf cohomology with supports in and also for developing certain homology theories using sheaves (see the book by Bredon, Sheaf theory). Especially useful are the so-called paracompactifying families of supports on non-paracompact spaces.
Let be a sheaf of abelian groups over a topological space . Denote by the subset of the space of all sections for which . This gives rise to a covariant left exact functor . Its right-derived functors
H_\phi^k(X,F) := R^k\Gamma_\phi(X,F)
are called the cohomology groups of with coefficients in the sheaf and with supports in the family of supports. Or sometimes one simply says sheaf cohomology with supports.