A New Topological Helly Theorem and Some Transversal Results

We prove that for a topological space X with the property that H-*(U) = 0 for* >= 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 <= j <= 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.