A composition structure is a sequence of consistent probability distributions for compositions (ordered partitions) of $n = 1, 2, \dots$. Any composition structure can be associated with an exchangeable random composition of the set of natural numbers. Following Donnelly and Joyce, we study the problem of characterizing a generic composition structure as a convex mixture of the "extreme" ones. We topologize the family $\mathscr{U}$ of open subsets of [0, 1] so that $\mathscr{U}$ becomes compact and show that $\mathscr{U}$ is homeomorphic to the set of extreme composition structures. The general composition struc-ture is related to a random element of $\mathscr{U}$ via a construction introduced by J. Pitman.