A Definition of a Geometrical Point

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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 left{ S_i right} _{i in I} with the property that for any S_{i_1} and S_{i_2}, we have either S_{i_1}subseteq S_{i_2} or S_{i_2}subseteq S_{i_1}. 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 varepsilon >0, there exists some ball S_j  in left{ S_i right} _{i in I} such that the radius of S_j is less than varepsilon. We say that two nests left{ S_i right} _{i in I} and left{ S'_j right} _{j in J} are related if every ball in left{ S_i right} _{i in I} meets every ball in left{ S'_j right} _{j in J}

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 left{ S_i right} _{i in I}left{ S'_j right} _{j in J} and left{ S''_k right} _{k in K} are point-producing nests. Then if every ball in left{ S_i right} _{i in I} meets every ball in left{ S'_j right} _{j in J}, and if every ball in left{ S'_j right} _{j in J} meets every ball in left{ S''_k right} _{k in K}, then every ball in left{ S_i right} _{i in I} meets every ball in left{ S''_k right} _{k in K}.
Proof: Suppose for a contradiction that there exist balls S_iS'_j, and S''_k such that S_i meets S'_jS'_j meetsS''_k, but S_i does not meet S''_k. For each positive integer n, we can find B_n in left{ S'_j right} _{j in J} such that the radius of each B_n is less than frac{1}{n} . By assumption, we have that S_i meets every B_n and every B_n meets S''_k. Therefore the distance between S_i and S''_k is less than frac{2}{n}  for every positive integer n. This means the distance between S_i and S''_k is zero, i.e. they are in contact. But this contradicts our assumption that  S_i does not meet S''_k, 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 R. Any point p may be represented by any (point-producing) nest in its corresponding equivalence class. Let left{ S_i right} _{i in I} be such a nest. We say that p in R if and only if R meets every S in left{ S_i right} _{i in I}.