LAUSR

We combine the classical Gromov-Hausdorff metric [ 5 ] with the \begin{document}$C^0$\end{document} distance to obtain the \begin{document}$C^0$\end{document} - Gromov - Hausdorff distance between maps of possibly different metric spaces. The latter is then combined with Walters's topological stability [ 11 ] to obtain the notion of topologically GH-stable homeomorphism . We prove that there are topologically stable homeomorphism which are not topologically GH-stable. Also that every topological GH-stable circle homeomorphism is topologically stable. Afterwards, we prove that every expansive homeomorphism with the pseudo-orbit tracing property of a compact metric space is topologically GH-stable. This is related to Walters's stability theorem [ 11 ] . Finally, we extend the topological GH-stability to continuous maps and prove the constant maps on compact homogeneous manifolds are topologically GH-stable.