Let K be a convex set in R’, int (K) # a. If we remove a linite family of affine hyperplanes from R’, we disconnect K into a family of convex parts, which is called a topological dissection of K by an arrangement of hyperplanes. After the influential paper of Zaslavsky [9], the theory of topological dissections has been the object to renewed interest; for a survey of recent results and applications, see the paper of Cartier [3]. Following along the lines of [ 1,2, 81, we study dissections from the point of view of the homological theory of posets and Mobius functions. We associate to a dissection a partially ordered set-the poset of regions and faces-and determine the homotopy types of the classifying spaces of this poset and of its intervals. It is worthwile noting that our study is entirely developed by repeated applications of Quillen’s homotopy equivalence criterion for posets [7]. By this technique, we succeed in computing the Mobius function of the poset of regions and faces, which turns out to take only values 0, 1, - 1; this gives, as an immediate consequence, generalizations of Euler’s relation. We submit that, in the case when the convex set is the whole space, the Euler relation for regions and the Euler relation for bounded regions are nothing but the upper and the lower recursion for the Mobius function, respectively. Furthermore, the poset of regions and faces is homotopy Cohen-Macaulay. In the final section, we describe connections between our approach and that based on the notion of cut-intersection poset due to Zaslavsky.