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 | References | Similar Articles | Additional Information
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.
- Felix Hausdorff, Dimension und äußeres Maß, Math. Ann. 79 (1918), no. 1-2, 157–179 (German). MR 1511917, DOI https://doi.org/10.1007/BF01457179
- Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493 A. N. Kolmogorov, On the Shannon theory of information transmission, IRE Trans. Information Theory (1956), 102-108.
- A. C. Vosburg, On the relationship between Hausdorff dimension and metric dimension, Pacific J. Math. 23 (1967), 183–187. MR 217776
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Keywords:
Metric dimension,
topological dimension,
metric entropy
Article copyright:
© Copyright 1970
American Mathematical Society