Bulletin of the
Korean Mathematical Society BKMS

We discuss the notions of positive expansivity, chain transitivity, uniform rigidity, chain mixing, weak specification, and pseudo orbital specification in terms of finite open covers for Hausdorff topological spaces and entourages for uniform spaces. We show that the two definitions for each notion are equivalent in compact Hausdorff spaces and further they are equivalent to their standard definitions in compact metric spaces. We show that a homeomorphism on a Hausdorff uniform space has uniform $h$-shadowing if and only if it has uniform shadowing and its inverse is uniformly equicontinuous. We also show that a Hausdorff positively expansive system with a Hausdorff shadowing property has Hausdorff $h$-shadowing.

Keywords: Hausdorff pseudo orbital specification, Hausdorff positive expansive, Hausdorff uniformly rigid

MSC numbers: Primary 37B02, 37B05, 37B65, 54B05, 26A18