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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Metric rigidity in $ E\sp{n}$


Authors: Brian M. Scott and Ralph Jones
Journal: Proc. Amer. Math. Soc. 53 (1975), 219-222
MSC: Primary 54E40
DOI: https://doi.org/10.1090/S0002-9939-1975-0377824-8
MathSciNet review: 0377824
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Abstract: In answer to a question rasied by L. Janos, it is shown (i) that $ {E^n}$ is not a finite union of metrically rigid subsets; (ii) that if $ {2^\omega } > {\kappa ^ + }$, then $ {E^n}$ is not the union of $ \kappa $ metrically rigid subsets, where $ \kappa $ is an infinite cardinal; and (iii) that if $ {2^\omega } = {\kappa ^ + }$, then $ {E^1}$ is the union of $ \kappa $ metrically rigid subsets, and hence that $ {E^1}$ is a countable union of metrically rigid subsets iff the continuum hypothesis holds. ( $ A \subseteq {E^n}$ is metrically rigid iff no two distinct two-point subsets of $ A$ are isometric.) Open question: assuming the continuum hypothesis, can $ {E^n}$ be written as a countable union of metrically rigid subsets if $ n > 1$?


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

  • [1] L. Janos, On rigidity of metrics, Notices Amer. Math. Soc. 18 (1971), 436. Abstract #71T-G40.
  • [2] -, On rigidity of subsets in metric spaces. Preliminary report, Notices Amer. Math. Soc. 20 (1973), A343-A344. Abstract #73T-G58.
  • [3] R. Jones, Rigid sets of cardinal $ c$ in $ {E^n}$ (submitted).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0377824-8
Keywords: Metric rigidity, rigid sets, continuum hypothesis
Article copyright: © Copyright 1975 American Mathematical Society