Metric Space of Ontic Points has a Completion

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.

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