  Let $\mathcal{H}$ be a hypergraph of rank $r$. We show that the simplicial
complex whose simplices are the hypergraphs $\mathcal{F}\subset\mathcal{H}$
with covering number at most $p$ is
$\left(\binom{r+p}{r}-1\right)$-collapsible, and the simplicial complex whose
simplices are the pairwise intersecting hypergraphs
$\mathcal{F}\subset\mathcal{H}$ is $\frac{1}{2}\binom{2r}{r}$-collapsible.
