Topological Stability and Pseudo-orbit Tracing Property for Homeomorphisms on Uniform Spaces

In this paper, we investigate the topological stability and pseudo-orbit tracing property for homeomorphisms on uniform spaces. We introduce the concept of topological stability for homeomorphisms on compact uniform spaces and prove that if a homeomorphism on a compact uniform space is expansive and has pseudo-orbit tracing property, then it is topologically stable. Moreover, we discuss the topological stability for homeomorphisms on uniform spaces from the view of localization. We introduce definitions of topologically stable point and shadowable point for homeomorphisms on uniform spaces and show that every shadowable point of an expansive homeomorphism on a compact uniform space is topologically stable.