The simplicial complex K(A) is defined to be the collection of simplices, and their proper subsimplices, representing maximal lattice free bodies of the form {x : Ax < b}, with A a fixed (n + 1) × n matrix. The topological space associated with K(A) is shown to be homeomorphic to Rn, and the space obtained by identifying lattice translates of these simplices is homeomorphic to the n-torus.