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Proceedings of the American Mathematical Society

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Note on metric dimension


Author: Reese T. Prosser
Journal: Proc. Amer. Math. Soc. 25 (1970), 763-765
MSC: Primary 54.70
DOI: https://doi.org/10.1090/S0002-9939-1970-0259873-9
MathSciNet review: 0259873
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Abstract: The metric dimension of a compact metric space is defined here as the order of growth of the exponential metric entropy of the space. The metric dimension depends on the metric, but is always bounded below by the topological dimension. Moreover, there is always an equivalent metric in which the metric and topological dimensions agree. This result may be used to define the topological dimension in terms of the metric entropy as the infimum of the metric dimension taken over all metrics.


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DOI: https://doi.org/10.1090/S0002-9939-1970-0259873-9
Keywords: Metric dimension, topological dimension, metric entropy
Article copyright: © Copyright 1970 American Mathematical Society