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

The Hausdorff metric measures the distance between nonempty compact sets in n, the collection of which is denoted (n). Betweenness in (n) can be defined in the same manner as betweenness in Euclidean geometry. But unlike betweenness in n, for some elements A and B of (n) 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.

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

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