INTERSECTION OF SETS WITH N-CONNECTED UNIONS

We show that if n sets in a topological space are given so that all th e sets are closed or all are open, and for each k less than or equal t o n every k of the sets have a (k - 2)- connected union, then the n se ts have a point in common. As a consequence, we obtain the following s tarshaped version of Helly's theorem: If every n fl or fewer members o f a finite family of closed sets in R-n have a starshaped union, then all the members of the family have a point in common. The proof relies on a topological KKM-type intersection theorem.