To illustrate the geometry, we
begin with the simplest elements in *H*(**R*** ^{n}*), namely the one point sets. In the 2002 REU,
John Mayberry, Audrey Powers, and I examined the subspace of

**Theorem 2: ***Let a, b be
points in ***R*** ^{n}. Let
A* = {

*if and only if there exist
positive real numbers s and t so that *

1. *C
is a subset of *(*A*)_{s}* *∩ (*B*)_{t} *and *

2. *there is a point c*_{0} *in
C with c*_{0}* in the intersection of *∂(*A*)_{s}*
*∩ ∂(*B*)_{t} *with the Euclidean line through a and
b. *

This theorem is illustrated by
the next applet. Any compact set in the shaded region that contains the black
point *c*_{0} on the *ab* axis is an element on the Hausdorff
line defined by *A* and *B*. The points *a* and *b* are
directly moveable as is the point *c*_{0}, by moving the black
point on the slider.

One interesting consequence of
Theorem 2 is that there may be many elements at a particular location on a
Hausdorff line. For example, if the point *c*_{0} described in
Theorem 2 is on the Euclidean segment defined by *a* and *b*, note
that the intersection of (*A*)* _{s}* with (

**Definition: ***Let A and B be distinct elements of H*(**R*** ^{n}*)

*h*(*A*,
*C*_{1}) *= h*(*A*, *C*_{2}) *and h*(*B*,
*C*_{1}) *= h*(*B*, *C*_{2}).

Unlike Euclidean lines which have
unique points at each location, Theorem 2 tells us that Hausdorff lines may
contain an infinite number or a finite number of elements at a given location.
We will see more about this later.