### Almost Isometric Embeddings of Metric Spaces

by Kojman and Shelah. [KjSh:827]

Israel J Math, 2006

We investigate a relations of almost isometric embedding
and almost isometry between metric spaces and prove that with
respect to these relations:
(1) There is a countable universal metric space.
(2) There may exist fewer than continuum separable metric spaces on
aleph_1 so that every separable metric space is almost
isometrically embedded into one of them when the continuum
hypothesis fails.
(3) There is no collection of fewer than continuum metric spaces of
cardinality aleph_2 so that every ultra-metric space of
cardinality aleph_2 is almost isometrically embedded into one of
them if aleph_2<2^{aleph_0} .
We also prove that various spaces X satisfy that if a space X is
almost isometric to X than Y is isometric to X .

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