On the topological Helly theorem

Abstract

The main result of this paper is the following theorem, related to the missing link in the proof of the topological version of the classical result of Helly: Let { X i } i = 0 2 be any family of simply connected compact subsets of R 2 such that for every i , j ∈ { 0 , 1 , 2 } the intersections X i ∩ X j are path connected and ⋂ i = 0 2 X i is nonempty. Then for every two points in the intersection ⋂ i = 0 2 X i there exists a cell-like compactum connecting these two points, in particular the intersection ⋂ i = 0 2 X i is a connected set.