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Congruence and one-dimensionality of metric spaces


Author: Ludvik Janos
Journal: Proc. Amer. Math. Soc. 103 (1988), 1268-1270
MSC: Primary 54F45; Secondary 54E35, 54E40, 54F50
DOI: https://doi.org/10.1090/S0002-9939-1988-0955020-X
MathSciNet review: 955020
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Abstract: Two subsets $ A$ and $ B$ of a metric space $ (X,d)$ are said to be congruent if there is a bijection between them which preserves the distance $ d$. We show that if a separable locally compact metric space is such that no distinct subsets of cardinality 3 are congruent then its dimension is $ \leq 1$. We also show that the real line $ \mathbb{R}$ can be given a compatible metric with this property.


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

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0955020-X
Article copyright: © Copyright 1988 American Mathematical Society

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