Acyclic intersection cover homology bound
= Acyclic intersection cover homology bound
If a space is a union of $n\geq2$ subcomplexes and every nonempty intersection is acyclic, its homology vanishes in degrees at least $n-1$. Induction with the <Mayer–Vietoris sequence> proves the bound; the boundary of an $(n-1)$-simplex covered by its facets shows sharpness.