The SIN property in homeomorphism groups

Abstract

Characterizations are given for a group of homeomorphisms of a uniform space, with weakly micro-transitive action, to be SIN. If the phase space is compact and the group is given the compact-open topology, the group is SIN if and only if it is precompact. Under very general conditions, the group is shown to be SIN if and only if it is uniformly equicontinuous with respect to a particular uniformity on the phase space. The SIN property of any Hausdorff topological group is characterized by its actions. A new condition is discovered for the group topology to be admissible.