 Perhaps the most simple example of this is due to Lemon and  Pratt[6]  (see e.g.,  1997).  Consider Euler Circles—where convex regions of the plane represent sets, and overlap of the regions represents non-empty intersection of the corresponding sets. A result of convex topology known as Helly’s Theorem states (for the 2 dimensional case) that if every triple of 4 convex regions has a non-empty intersection then all four regions must have a non-empty intersection.