To build geometric space let us introduce point-like constructs.
Ontic point: Let ξ ⊂ Θ be a family of things. We say that ξ is a complete family of united things if it satisfies:
1. Any two things of ξ are united.
2. For anything x̸ ∉ ξ there is a thing y ∈ ξ separated of x.
Now we define a distance between ontic points
Distance between ontic points: The distance between ontic points is:
dG(ξ,η) = sup(i,j) d(xi, yj), where i ∈ I , j ∈ J belong to the respective index sets.
Theorem (Metricity): The set of ontic points is a metric space with distance dG.
Proof: The first three distance conditions are satisfied because d is a pseudo-metric. To show that the fourth is satisfied observe that if ξ ≠ η, there are xi≀yj and d(xi, yj) > 0. So we find
ξ ≠ η ⇒ dG(ξ, η) > 0
dG(ξ, η) > 0 ⇒ ξ = η
The isometric completion theorem guarantees that the metric space of ontic points has a completion. This justifies the definition of Geometric space: The completion of ontic space is the geometric space EG.