Consider a space of any dimension. For convenience I will define a point in three dimensions. We take for granted that the concept of ball of positive radius is understood. Now consider a nest of balls. This is simply any nonempty collection of balls with the property that for any and , we have either or . We define a point-producing nest of balls to have the property that the nest contains balls of arbitrarily small radius. To say it in yet another way, given any , there exists some ball such that the radius of is less than . We say that two nests and are related if every ball in meets every ball in .
Definition: We define the distance between two disjoint balls as the diameter of the smallest ball that can meet both simultaneously. If two balls meet, we define their distance to be zero.
Lemma: Suppose , and are point-producing nests. Then if every ball in meets every ball in , and if every ball in meets every ball in , then every ball in meets every ball in .
Proof: Suppose for a contradiction that there exist balls , , and such that meets , meets, but does not meet . For each positive integer , we can find such that the radius of each is less than . By assumption, we have that meets every and every meets . Therefore the distance between and is less than for every positive integer . This means the distance between and is zero, i.e. they are in contact. But this contradicts our assumption that does not meet , hence the Lemma is proved .
This equivalence relation organizes the collection of point-producing nests into equivalence classes. Each such equivalence class of nests corresponds to a point, and thus defines it.
Now we define what it means for a point to be a member of any nonempty closed region . Any point may be represented by any (point-producing) nest in its corresponding equivalence class. Let be such a nest. We say that if and only if meets every .