On the topological Helly theorem

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 is an element of {0, 1, 2} the intersections X-i boolean AND X-j are path connected and boolean AND(2)(i=0) X-i is nonempty. Then for every two points in the intersection boolean AND(2)(i=0) X-i there exists a cell-like compactum connecting these two points, in particular the intersection boolean AND(2)(i=0)=0 X-i is a connected set. (C) 2005 Elsevier B.V. All rights reserved.