Intersection of sets with 𝑛-connected unions

Source abstract

We show that if n n sets in a topological space are given so that all the sets are closed or all are open, and for each k ≤ n k\le n every k k of the sets have a ( k − 2 ) (k-2) -connected union, then the n n sets have a point in common. As a consequence, we obtain the following starshaped version of Helly’s theorem: If every n + 1 n+1 or fewer members of a finite family of closed sets in R n \mathbb {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.