nLab family of supports

A family of supports in a topological space $X$ is a family $\phi$ of closed subsets $S\subset X$ such that

1. if $S_1,S_2\in \phi$ then $S_1\cup S_2\in \phi$;
2. if $S_1\in \phi$ and $S_2\subset S_1$ is closed, then $S_2\in \phi$.

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 $\phi$ 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 $F$ be a sheaf of abelian groups over a topological space $X$. Denote by $\Gamma_\phi(X,F)$ the subset of the space of all sections $f\in \Gamma(X,F) = F(X)$ for which $supp\,f\in \phi$. This gives rise to a covariant left exact functor $F\mapsto \Gamma_\phi(X,F)$. Its right-derived functors

$$H_\phi^k(X,F) := R^k\Gamma_\phi(X,F)$$

are called the cohomology groups of $X$ with coefficients in the sheaf $F$ and with supports in the family $\phi$ of supports. Or sometimes one simply says sheaf cohomology with supports.