Leray numbers of projections and a topological Helly-type theorem

Let X be a simplicial complex on the vertex set V. The rational Leray number L(X) of X is the minimal d, such that (H) over tilde (i) (Y; Q) = 0 for all induced subcomplexes Y subset of X and i >= d. Suppose that V = U(i=1)(m) V(i) is a partition of V such that the induced subcomplexes X[V(i)] are all 0-dimensional. Let pi denote the projection of X into the (m - 1)-simplex on the vertex set {1, ... ,m} given by pi(v) = i if v is an element of V(i). Let r = max{vertical bar pi(-1)(pi(x))vertical bar : x is an element of vertical bar X vertical bar}. It is shown that L(pi(X)) <= rL(X) + r - 1. One consequence is a topological extension of a Helly-type result of Amenta. Let F be a family of compact sets in R(d) such that for any F' subset of F, the intersection boolean AND F' is either empty or contractible. It is shown that if G is a family of sets such that for any finite G' subset of G, the intersection boolean AND G' is a union of at most r disjoint sets in F, then the Helly number of G is at most r(d + 1).