  We prove that the class of numerable open covers of topological spaces is the
smallest class that contains covers with pairwise disjoint elements and
numerable covers with two elements, closed under composition and coarsening of
covers. We apply this result to establish an analogue of the Brown--Gersten
property for numerable open covers of topological spaces: a simplicial presheaf
on the site of topological spaces satisfies the homotopy descent property for
all numerable open covers if and only if it satisfies it for numerable covers
with two elements and covers with pairwise disjoint elements. We also prove a
strengthening of these results for manifolds, ensuring that covers with two
elements can be taken to have a specific simple form. We apply these results to
deduce a representability criterion for stacks on topological spaces similar to
arXiv:1912.10544. We also use these results to establish new simple criteria
for chain complexes of sheaves of abelian groups to satisfy the homotopy
descent property.
