A new topological Helly theorem and some transversal results (Q464742)

A topological Helly-type theorem is proved, in the framework provided by a topological space \(X\), in which \(H_{k}(U)=0\), \(k\geq d\), for every \(U\subset X\) open and any subfamily of size \(j\) of the given sets has a zero \((d-j)\)-dimensional homology group of its intersection. As consequence, transversal theorems for families of convex sets are derived in three cases: transversal lines in the plane, transversal lines in three dimensions space and transversal hyperplanes in \(d\) dimensions space.