Topologically-ordered matter is a novel quantum state of matter observed only in a small number of physical systems, notably two-dimensional electron systems exhibiting fractional quantum Hall effects. It was recently proposed that a simple form of topological matter may be created in interacting systems of rotating ultra-cold atoms. We describe ensemble measurements on small, rotating clusters of interacting bosonic atoms, demonstrating that they can be induced into quantum ground states closely analogous to topological states of electronic systems. We report measurements of inter-particle correlations and momentum distributions of Bose gases in the fractional quantum Hall limit, making comparison to a full numerical simulation. The novel experimental apparatus necessary to produce and measure properties of these deeply entangled quantum states is described.