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A relationship between packing and topological dimensions

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

H. Joyce
H. Joyce
Affiliation:
University of Jyväskylä, Department of Mathematics, P.O. Box 35, FIN-40351, Jyväskylä, Finland.
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Abstract

The relationship between the topological dimension of a separable metric space and the Hausdorff dimensions of its homeomorphic images has been known for some time. In this note we consider topological and packing dimensions, and show that if X is a separable metric space, then

where and denote the topological and packing dimensions of X, respectively.

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Mathematika , Volume 45 , Issue 1 , June 1998 , pp. 43 - 53
Copyright
Copyright © University College London 1998

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References

1.Eilenberg, S. and Harrold, O. G. Jr. Continua of finite linear measure I. Amer. J. Math., 65 (1943), 137–146.CrossRefGoogle Scholar
2.Fremlin, D.. Spaces of finite length. Proc. London Math. Soc. (3), 64 (1992), 449–486.CrossRefGoogle Scholar
3.Harrold, O. G.. Continua of finite length and certain product sets. Bull. Amer. Math. Soc, 46 (1940), 951–953.CrossRefGoogle Scholar
4.Hurewicz, W. and Wallman, H., Dimension Theory (Princeton University, Press, 1941).Google Scholar
5.Mattila, P.. Geometry of Sets and Measures in Euclidean Spaces (Cambridge University Press, 1995).CrossRefGoogle Scholar
6.Raymond, X. Saint and Tricot, C.. Packing regularity of sets in n-space. Math. Proc. Camb. Phil. Soc, 103 (1988), 133–145.CrossRefGoogle Scholar
7.Szpilrajn, E.. La dimension et la mesure. Fund. Math., 28 (1937), 81–89.CrossRefGoogle Scholar
8.Taylor, S. J. and Tricot, C.. Packing measure and its evaluation for a Brownian path. Trans. Amer. Math. Soc, 288 (1985), 679–699.CrossRefGoogle Scholar
9.Taylor, S. J. and Tricot, C.. The packing measure of rectifiable subsets of the plane. Math. Proc. Camb. Phil. Soc, 99 (1986), 285–296.CrossRefGoogle Scholar
10.Tricot, C.. Two definitions of fractional dimension. Math. Proc. Camb. Phil. Soc, 91 (1982), 57–74.CrossRefGoogle Scholar
11.Whyburn, G. T.. Concerning continua of finite degree and local separating points. Amer. J. Math., 57 (1935), 11–18.CrossRefGoogle Scholar