Metric characterizations of dimension for separable metric spaces

Janos, Ludvik; Martin, Harold

doi:10.1090/S0002-9939-1978-0474229-9

Metric characterizations of dimension for separable metric spaces
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by Ludvik Janos and Harold Martin PDF

Proc. Amer. Math. Soc. 70 (1978), 209-212 Request permission

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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