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Congruence and dimension of nonseparable metric spaces


Author: Yasunao Hattori
Journal: Proc. Amer. Math. Soc. 108 (1990), 1103-1105
MSC: Primary 54F45; Secondary 54E35, 54E40
DOI: https://doi.org/10.1090/S0002-9939-1990-1000155-8
MathSciNet review: 1000155
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Abstract: In this paper, we prove that, if a metrizable space $ {\mathbf{X}}$ has an admissible metric such that $ {\mathbf{X}}$ has no two distinct congruent subsets of cardinality 3, then $ {\text{ind}}{\mathbf{X}} \leq 1$. We also show that if a non-empty metrizable space $ {\mathbf{X}}$ has an admissible star-rigid metric, then $ {\text{ind}}{\mathbf{X}} = 0$. The latter answers a question of L. Janos and H. Martin [3].


References [Enhancements On Off] (What's this?)

  • [1] L. Janos, A metric characterization of zero-dimensional spaces, Proc. Amer. Math. Soc. 31 (1972), 268-270. MR 0288739 (44:5935)
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  • [3] L. Janos and H. Martin, Metric characterizations of dimension for separable metric spaces, Proc. Amer. Math. Soc. 70 (1978), 209-212. MR 0474229 (57:13876)
  • [4] J. Nagata, Modern general topology, 2nd ed., North-Holland, Amsterdam, 1985. MR 831659 (87g:54003)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1000155-8
Keywords: Metric spaces, congruence, small inductive dimension, star-rigid metric
Article copyright: © Copyright 1990 American Mathematical Society

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