We build monotone complete C∗-algebras from equivalence relations on topological spaces. This is applied to orbit equivalence relations associated with the action of a countable group G. In general, these algebras may be identified with monotone cross-product algebras arising from actions of G on commutative monotone complete C∗-algebras. Since different groups can give rise to the same orbit equivalence relation, this can be used to show that, apparently different monotone cross-product algebras, are in fact, isomorphic.