Congruence and one-dimensionality of metric spaces

Janos, Ludvik

doi:10.1090/S0002-9939-1988-0955020-X

Congruence and one-dimensionality of metric spaces
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Proc. Amer. Math. Soc. 103 (1988), 1268-1270 Request permission

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

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Modern dimension theory