Abstract: We considered the persistence problem of homeomorphism on noncompact metric spaces. By using the definitions of persistence, equicontinuity, strongly topological stability, and persistent shadowing property of homeomorphisms, we prove that homeomorphisms that are equicontinuity and topologically stable are persistent, homeomorphisms have persistent shadowing properties if and only if they are persistent and have pseudoorbital shadowing properties, and an expansive homeomorphism with persistent shadowing property is strongly topologically stable.

Key words: persistence, equicontinuity, strongly topological stability, persistent shadowing property, noncompact metric space