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Metric Characterizations of Dimension for Separable Metric Spaces

Ludvik Janos and Harold Martin
Proceedings of the American Mathematical Society
Vol. 70, No. 2 (Jul., 1978), pp. 209-212
DOI: 10.2307/2042090
Stable URL: http://www.jstor.org/stable/2042090
Page Count: 4
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Metric Characterizations of Dimension for Separable Metric Spaces
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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 $\dim(X) \leqslant n \operatorname{iff} X$ 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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