Note on metric dimension
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- by Reese T. Prosser
- Proc. Amer. Math. Soc. 25 (1970), 763-765
- DOI: https://doi.org/10.1090/S0002-9939-1970-0259873-9
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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.References
- Felix Hausdorff, Dimension und äußeres Maß, Math. Ann. 79 (1918), no. 1-2, 157–179 (German). MR 1511917, DOI 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
Bibliographic Information
- © Copyright 1970 American Mathematical Society
- 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