A Non-expensive Homological Helly Theorem

We will discuss the following result: for a topological space X with the property that H k ( 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.