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:

d_{G}(ξ,η) = sup_{(i,j)} d(x_{i}, y_{j}), where i ∈ I , j ∈ J belong to the respective index sets.

Theorem (Metricity): The set of ontic points is a metric space with distance d_{G}.

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 x_{i}≀y_{j }and d(x_{i}, y_{j}) > 0. So we find

ξ ≠ η ⇒ d_{G}(ξ, η) > 0

d_{G}(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 E_{G}.