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On the classification of Polish metric spaces up to isometry. (English) Zbl 1012.54038
Mem. Am. Math. Soc. 766, 78 p. (2003).
A complete classification of a set $$X$$ of objects up to an equivalence relation $$E$$ on $$X$$ consists of finding a set of invariants $$I$$ and a map $$c:X\to I$$ such that $$xEy\Leftrightarrow c(x)=c(y)$$. Both $$I$$ and $$c$$ should be as simple as possible to be of any interest. The simplest case is when $$E$$ is concretely classifiable, i.e., when $$I$$ is a standard Borel space and $$c$$ is Borel. A. M. Vershik [Russ. Math. Surv. 53, No. 5, 921-928 (1998; Zbl 1005.53036)] points out that the classification of Polish spaces up to isometry is not concretely classifiable, thus quite complicated in some sense. By means of Borel reducibility the authors determine the exact complexity of various classification problems concerning Polish spaces up to isometry. They prove that the class of Polish spaces is bireducible to the universal orbit equivalence relation induced by a Borel action of a Polish group. Next they deal with special classes of Polish metric spaces including locally compact, ultrametric, $$0$$-dimensional, homogeneous and ultrahomogeneous spaces. They obtain also characterizations for isometry groups of various classes of Polish metric spaces.

##### MSC:
 54E35 Metric spaces, metrizability 03E75 Applications of set theory 54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) 03E15 Descriptive set theory
##### Keywords:
Polish group; the Urysohn space; Borel (bi)reducible; isometry
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