The Roelcke compactification of groups of homeomorphisms

Abstract

Let X be a zero-dimensional compact space such that all non-empty clopen subsets of X are homeomorphic to each other, and let H(X) be the group of all self-homeomorphisms of X with the compact-open topology. We prove that the Roelcke compactification of H(X) 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 H(X) is topologically simple and minimal, in the sense that it does not admit a strictly coarser Hausdorff group topology.