Almost isometric embeddings of metric spaces

2004/06/25 by Menachem Kojman, Saharon Shelah, Kojman, Menachem +1
Mathematics · #FOS: Mathematics #General Topology (math.GN) #Logic (math.LO) #math.GN #math.LO

paper · pdf · doi:10.48550/arxiv.math/0406530

published as Israel J. Math. 155 (2006) 309--334

arxiv created 2004/06/25 · arxiv updated 2009/12/01

Abstract

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 aleph₁ 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 aleph₂ so that every ultra-metric space of cardinality aleph₂ is almost isometrically embedded into one of them if aleph₂<2^aleph₀. We also prove that various spaces X satisfy that if a space X is almost isometric to X than Y is isometric to X.

Related

• On isometric universality of spaces of metrics
• Universal metric spaces according to W. Holsztynski
• Indivisible ultrametric spaces
• Can we classify complete metric spaces up to isometry?
• On homogeneous ultrametric spaces