Almost isometric embedding between metric spaces

Abstract

We investigate the relations ofalmost isometric embedding and ofalmost isometry between metric spaces.

These relations have several appealing features. For example, all isomorphism types of countable dense subsets of ∝ form exactly one almostisometry class, and similarly with countable dense subsets of Uryson's universal separable metric spaceU.

We investigate geometric, set-theoretic and model-theoretic aspects of almost isometry and of almost isometric embedding.

The main results show that almost isometric embeddability behaves in the category ofseparable metric spaces differently than in the category of general metric spaces. While in the category of general metric spaces the behavior of universality resembles that in the category of linear orderings —namely, no universal structure can exist on a regular λ > ℵ₁ below the continuum—in the category of separable metric spaces universality behaves more like that in the category of graphs, that is, a small number of metric separable metric spaces on an uncountable regular λ<2^ℵ ₀ may consistently almost isometrically embed all separable metric spaces on λ.