We introduce and study the topological concepts of ergodic shadowing, chain
transitivity and topological ergodicity for dynamical systems on non-compact
non-metrizable spaces. These notions generalize the relevant concepts for
metric spaces. We prove that a dynamical system with topological ergodic
shadowing property is topologically chain transitive, and that topological
chain transitivity together with topological shadowing property implies
topological ergodicity.
