Every arrangement %' of a&e hyperplanes in
Rd determines a partition of Rd into open topological cells. The face lattice L(X) of this partition was the object of a smdy by Barnabei and Brini, wko de.;ermined the homotopy type of its intervals. We use g:am&ic con~huctions from the theory of convex polytopes to prove the shellabiity of L,(Z) and to determine the combinatorial topology of its intervals up to homeomorphism. Let % be a finite set of afhne hyperplanes in a euclidean vector space Rd. 'Ihis arrangement determines a partition II(X) of Rd into open topologi& cells, w&b can be described as follows: for ea,ch hyperplane H E X, let zH be the partition of Rd into three parts given by H and the two open halfspaces determined by H. Then n(z) is the meet of the partitions zH (H E i%') in the lattice of all partitions of Rd. The parts of II(X) are non-empty intersections of parts of the partitions q## hence open topological cells [12, p. 2141 embedded convexly into Rd. Let P(Z) be the poset of these cells (i.e. the parts of n(R)), ordered by inclusion of their closures. The minimal elements of P(%') are all cells of the same dimension. By reduction to a suitable quotient space we can assume (without loss of generality for the study of P(X)) that the minimal elements of P(X) are in fact points, i.e. that X is reg&r in the sense of [1, p. 1141. Now let I,(%') be the face la&e of the dissection of Rd by %', that is, the poset P(X) with a minimal element 6 = 0 and a maximal element f = L(X) is a finite graded lattice of length d + 2. The face lattice L(X) of an arrangement was studied by who determined the homotopy type of its interv However, adding the "point at infimty" to compactification of Rd), it is easy to see that L(gyP) is the cell decomposition [12, p. 2161 of point at infinity) has been deleted. about the topology of L(Z). se