Intersection of sets with 𝑛-connected unions

Horvath, Charles; Lassonde, Marc

doi:10.1090/S0002-9939-97-03622-8

Intersection of sets with $n$-connected unions
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by Charles D. Horvath and Marc Lassonde PDF

Proc. Amer. Math. Soc. 125 (1997), 1209-1214 Request permission

Abstract:

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

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