  We introduce topological definitions of expansivity, shadowing, and chain
recurrence for homeomorphisms. They generalize the usual definitions for metric
spaces. We prove various theorems about topologically Anosov homeomorphisms
(maps that are expansive and have the shadowing property) on noncompact and
non-metrizable spaces that generalize theorems for such homeomorphisms on
compact metric spaces. The main result is a generalization of Smale's spectral
decomposition theorem to topologically Anosov homeomorphisms on first countable
locally compact paracompact Hausdorff spaces.
