nLab
family of supports

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

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

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 $\varphi$ 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 ${\Gamma }_{\varphi }(X,F)$ the subset of the space of all sections $f\in \Gamma (X,F)=F(X)$ for which $\mathrm{supp}f\in \varphi$. This gives rise to a covariant left exact functor $F\mapsto {\Gamma }_{\varphi }(X,F)$. Its right-derived functors

$$H_{\varphi }^k(X,F):=R^k{\Gamma }_{\varphi }(X,F)$$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 $\varphi$ of supports. Or sometimes one simply says sheaf cohomology with supports.