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 .