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The Helly number of a family of sets with empty intersection is the size ofits largest inclusion-wise minimal sub-family with empty intersection. Let F bea finite family of open subsets of an arbitrary locally arc-wise connectedtopological space Gamma. Assume that for every sub-family G of F theintersection of the elements of G has at most r connected components, each ofwhich is a Q-homology cell. We show that the Helly number of F is at mostr(d_Gamma+1), where d_Gamma is the smallest integer j such that every open setof Gamma has trivial Q-homology in dimension j and higher. (In particulard_{R^d} = d). This bound is best possible. We prove, in fact, a strongertheorem where small sub-families may have more than r connected components,each possibly with nontrivial homology in low dimension. As an application, weobtain several explicit bounds on Helly numbers in geometric transversal theoryfor which only ad hoc geometric proofs were previously known; in certain cases,the bound we obtain is better than what was previously known.

Helly numbers of acyclic families