## Real Analysis Exchange

### The Hausdorff dimension of the hyperspace of compact sets

Mark McClure

#### Abstract

Let $(X,\rho)$ be a separable metric space and let $(\mathcal{K}(X),\widetilde{\rho})$ denote the space of non-empty compact subsets of $X$ with the Hausdorff metric. The purpose of this paper is to investigate the relationship of the Hausdorff dimension of a set $E \subset X$ to that of $\mathcal{K}(E) \subset \mathcal{K}(X)$.

#### Article information

Source
Real Anal. Exchange, Volume 22, Number 2 (1996), 611-625.

Dates
First available in Project Euclid: 22 May 2012

https://projecteuclid.org/euclid.rae/1337713144

Mathematical Reviews number (MathSciNet)
MR1460975

Zentralblatt MATH identifier
0943.28013

#### Citation

McClure, Mark. The Hausdorff dimension of the hyperspace of compact sets. Real Anal. Exchange 22 (1996), no. 2, 611--625. https://projecteuclid.org/euclid.rae/1337713144

#### References

• Elaine Boardman, Some Hausdorff measure properties of the space of compact subsets of $[0,1]$. Quart. Journal Math. Oxford Ser. (2), 24 (1973), 333–341.
• G. A. Edgar, Measure, Topology, and Fractal Geometry. Springer-Verlag, 1990.
• K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications. John Wiley & and Sons, 1990.
• P. R. Goodey, Some Results Concerning the Dimension of the Space of Compact Subsets of the Unit Interval. Quart. Journal Math. Oxford Ser. (2), 27 (1976), 467–473.
• P. R. Goodey, Hausdorff Measure Functions in the Space of Compact Subsets of the Unit Interval. Transactions of the American Mathematical Society, 226 (1977), 203–208.
• Mark McClure, Entropy Dimensions of the Hyperspace of Compact Sets. Real Analysis Exchange 21 (1996), 294–202.
• St. Raymond and C. Tricot, Packing regularity of sets in $n$-space. Math. Proc. Camb. Phil. Soc. 103 (1988), 133–145.
• C. A. Rogers, Hausdorff Measures. Cambridge University Press, 1970.