Almost isometric embedding between metric spaces |
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Authors: | Menachem Kojman Saharon Shelah |
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Institution: | (1) Department of Mathematics, Ben-Gurion University of the Negev, 84105 Beer-Sheva, Israel;(2) Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel;(3) Department of Mathematics, Rutgers University, 08854 New Brunswick, NJ, USA |
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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<2ℵ
0 may consistently almost isometrically embed all separable metric spaces on λ.
Research of the first author was supported by an Israeli Science foundation grant no. 177/01.
Research of the second author was supported by the United States-Israel Binational Science Foundation. Publication 827. |
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