#### Vol. 7, No. 5, 2014

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Infinite cardinalities in the Hausdorff metric geometry

### Alexander Zupan

Vol. 7 (2014), No. 5, 585–593
##### Abstract

The Hausdorff metric measures the distance between nonempty compact sets in ${ℝ}^{n}$, the collection of which is denoted $\mathsc{ℋ}\left({ℝ}^{n}\right)$. Betweenness in $\mathsc{ℋ}\left({ℝ}^{n}\right)$ can be defined in the same manner as betweenness in Euclidean geometry. But unlike betweenness in ${ℝ}^{n}$, for some elements $A$ and $B$ of $\mathsc{ℋ}\left({ℝ}^{n}\right)$ there can be many elements between $A$ and $B$ at a fixed distance from $A$. Blackburn et al. (“A missing prime configuration in the Hausdorff metric geometry”, J. Geom., 92:1–2 (2009), pp. 28–59) demonstrate that there are infinitely many positive integers $k$ such that there exist elements $A$ and $B$ having exactly $k$ different elements between $A$ and $B$ at each distance from $A$ while proving the surprising result that no such $A$ and $B$ exist for $k=19$. In this vein, we prove that there do not exist elements $A$ and $B$ with exactly a countably infinite number of elements at any location between $A$ and $B$.

##### Keywords
Hausdorff metric, betweenness, metric geometry
Primary: 51F99
Secondary: 54B20
##### Milestones
Received: 22 September 2010
Revised: 23 April 2014
Accepted: 11 May 2014
Published: 1 August 2014

Communicated by Józef H. Przytycki
##### Authors
 Alexander Zupan Department of Mathematics University of Texas at Austin 1 University Station C1200 Austin, TX 78712 United States