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 \(*\ge d\) 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.