A Non-expensive Homological Helly Theorem

We will discuss the following result: for a topological space X with the property that Hk(U)=0 for k⩾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 reduced homology group of its intersection is zero. We also use this theorem to discuss new results concerning transversal affine planes to families of convex sets.