A New Topological Helly Theorem and some Transversals Results

We prove that for a topological space X with the property that $H_p(U)=0$ for $p\ge d$ and every open subset $U$ of $X$, a finite family of open sets in $X$ has nonempty intersection if for any subfamily of size $j$, $1\le j\le d+1$, 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.