Intersection properties of Helly families

Abstract The Helly convex-set theorem is extended onto topological spaces. From our results it follows that if there are given m +2 convex subsets of an m -dimensional contractible Hausdorff space and the intersection of each collection of m +1 the subsets is a nonempty contractible set, then the intersection of the whole collection of m +2 subsets is a nonempty set. Our results are stated in terms of Helly families, the definition of which involves k -connectedness of intersections of m − k sets for k =−1,0,…, m −1.