Intersection properties of Helly families

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.