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We combine the classical Gromov-Hausdorff metric [*Gromov*-*Hausdorff* *distance* between maps of possibly different metric spaces. The latter is then combined with Walters's topological stability [*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 [

We propose a definition of topological stability for set-valued maps. We prove that a single-valued map which is topologically stable in the set-valued sense is topologically stable in the classical sense [

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