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Hausdorff quasi-uniformities inducing the same hypertopologies

The question is investigated when two quasi-uniformities on a set X give rise to Hausdorff quasi-uniformities inducing the same topologies on the set P0(X) of nonempty subsets of X . Some conditions are also given under which such hypertopologies are induced by a unique Hausdorff quasi-uniformity. Our results should be compared to investigations on H-equivalence of uniformities due to Smith, Ward and others.