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In this paper, we prove that, if a metrizable space X has an admissible metric such that X has no two distinct congruent subsets of cardinality 3, then indX < 1 . We also show that if a non-empty metrizable space X has an admissible star-rigid metric, then ind X = 0 . The latter answers a question of L.doi:10.1090/s0002-9939-1990-1000155-8 fatcat:d7acud2u4vfxxailcam7thgyqq