Quasi-shadowing property on compact Hausdorff spaces

In this paper, two topological versions of quasi-shadowing property defined in terms of finite open covers and uniformities are introduced, which coincide on compact Hausdorff spaces and are equivalent to the standard quasi-shadowing property stated by means of a metric on compact metric spaces. At the same time, some general properties about uniform quasi-shadowing are investigated. Also, a map has uniform quasi-shadowing if and only if the induced map on the hyperspace of compact subsets of a compact Hausdorff space does.