The Hausdorff metric measures the distance between nonempty compact sets in
, the collection of
which is denoted
.
Betweenness in
can be defined in the same manner as betweenness in Euclidean geometry. But unlike betweenness
in
, for some
elements
and
of
there can be many
elements between
and
at a fixed
distance from
.
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
such that there
exist elements
and
having
exactly
different
elements between
and
at each
distance from
while proving the surprising result that no such
and
exist
for
.
In this vein, we prove that there do not exist elements
and
with
exactly a countably infinite number of elements at any location between
and
.
