The Gromov-Hausdorff distance measures how closely two metric space can be embedded into a third metric space. For compact metric spaces, Gromov-Hausdorff distance \(0\) implies isometry, but this is false for arbitrary Polish spaces. We consider the equivalence relation \(E_{GH}\) between Polish spaces defined by having Gromov-Hausdorff distance 0 from the viewpoint of Borel reducibility, and show that it is at least as complicated as isometry of Polish spaces. In order to compare \(E_{GH}\) with isometry, we study the complexity of isometry on single \(E_{GH}\) classes and show how to realize various equivalence relations such as the iterated Friedman-Stanley jumps of the equality relation.
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