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Polish Metric Spaces: Their Classification and Isometry Groups

Clemens, John D. and Gao, Su and Kechris, Alexander S. (2001) Polish Metric Spaces: Their Classification and Isometry Groups. Bulletin of Symbolic Logic, 7 (3). pp. 361-375. ISSN 1079-8986. doi:10.2307/2687754.

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In this communication we present some recent results on the classification of Polish metric spaces up to isometry and on the isometry groups of Polish metric spaces. A Polish metric space is a complete separable metric space (X,d). Our first goal is to determine the exact complexity of the classification problem of general Polish metric spaces up to isometry. This work was motivated by a paper of Vershik [1998], where he remarks (in the beginning of Section 2): "The classification of Polish spaces up to isometry is an enormous task. More precisely, this classification is not 'smooth' in the modern terminology." Our Theorem 2.1 below quantifies precisely the enormity of this task. After doing this, we turn to special classes of Polish metric spaces and investigate the classification problems associated with them. Note that these classification problems are in principle no more complicated than the general one above. However, the determination of their exact complexity is not necessarily easier. The investigation of the classification problems naturally leads to some interesting results on the groups of isometries of Polish metric spaces. We shall also present these results below.

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Additional Information:© 2001 Association for Symbolic Logic. Received November 15, 2000; revised March 27, 2001. We would like to thank G. Hjorth, V. Kanovei, A. Louveau and S. Solecki for many useful comments and for allowing us to include their results in this announcement.
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Official Citation:Polish Metric Spaces: Their Classification and Isometry Groups John D. Clemens, Su Gao and Alexander S. Kechris The Bulletin of Symbolic Logic , Vol. 7, No. 3 (Sep., 2001), pp. 361-375 Published by: Association for Symbolic Logic Article Stable URL:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:38565
Deposited By: Jason Perez
Deposited On:22 May 2013 17:07
Last Modified:09 Nov 2021 23:38

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