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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Metric characterizations of dimension for separable metric spaces


Authors: Ludvik Janos and Harold Martin
Journal: Proc. Amer. Math. Soc. 70 (1978), 209-212
MSC: Primary 54F45
MathSciNet review: 0474229
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Abstract: A subset B of a metric space (X, d) is called a d-bisector set iff there are distinct points x and y in X with $ B = \{ z:d(x,z) = d(y,z)\} $. It is shown that if X is a separable metrizable space, then $ \dim (X) \leqslant n$ iff X has an admissible metric d for which $ \dim (B) \leqslant n - 1$ whenever B is a d-bisector set. For separable metrizable spaces, another characterization of n-dimensionality is given as well as a metric dependent characterization of zero dimensionality.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1978-0474229-9
PII: S 0002-9939(1978)0474229-9
Keywords: Bisector set, star rigid metric, strongly rigid metric
Article copyright: © Copyright 1978 American Mathematical Society