The Roelcke compactification of groups of homeomorphisms

Let X be a zero-dimensional compact space such that all non-empty clopen subsets of X are homeomorphic to each other, and let Aut X be the group of all self-homeomorphisms of X, equipped with the compact-open topology. We prove that the Roelcke compactification of AutX can be identified with the semigroup of all closed relations on X whose domain and range are equal to X. We use this to prove that the group AutX is topologically simple and minimal.