On a condition for the union of spherical caps to be connected

Let C be a finite family of spherical caps of various sizes on a sphere in 3-space. A cap C∈C is called extremal if there is a great circle g such that the centers of those caps that intersect C lie in the same side of g, allowing some of them lie on g. We prove that if C contains no extremal cap, then the intersection graph of the caps in C is connected, and if furthermore every cap is smaller than a hemisphere, then the intersection graph is 2-connected. We also show that analogous assertions are no longer true in higher dimensions.