Almost Isometric Embedding between Metric Spaces

We investigate the relations of almost isometric embedding and of almost isometry between metric spaces. These relations have several appealing features. For example, all isomorphism types of countable dense subsets of R form exactly one almost-isometry class, and similarly with countable dense subsets of Uryson’s universal separable metric space U. 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 of separable 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 exists 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 and uncountable regular λ < 2א0 may consistently almost isometrically embed all separable metric spaces on λ.