  We prove that for a topological space X with the property that $H_p(U)=0$ for
$p\geq d$ and every open subset $U$ of $X$, a finite family of open sets in $X$
has nonempty intersection if for any subfamily of size $j$, $1\leq j \leq d+1$,
the $(d-j)$-dimensional homology group of its intersection is zero. We use this
theorem to prove new results concerning transversal affine planes to families
of convex sets.
