Helly's theorem asserts that if W is a family of compact convex sets in a j-dimensional linear space, and if any j + 1 members of ℱ have a non-empty intersection, then there is a point common to all members of ℱ. If one attempts to generalise this result to the case when ℱ consists of sets which are expressible as the union of at most n disjoint compact convex sets then, in general, one finds that there is no number h(n, j) such that if any h(n, j) members of ℱ have a non-empty intersection, then there is a point common to all members of ℱ. The difficulty lies in the fact that, in general, the intersections of members of ℱ are more complicated in structure than are the members of ℱ.