Gao, Su and Kechris, Alexander S. (2003) On the classification of Polish metric spaces up to isometry. Memoirs of the American Mathematical Society, 161 (766). pp. 1-78. ISSN 0065-9266. https://resolver.caltech.edu/CaltechAUTHORS:20130516-152757565
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Abstract
We study the classification problem of Polish metric spaces up to isometry and the isometry groups of Polish metric spaces. In the framework of the descriptive set theory of definable equivalence relations, we determine the exact complexity of various classification problems concerning Polish metric spaces. We start with the class of all Polish metric spaces and prove that it is Borel bireducible to the universal orbit equivalence relation induced by Borel actions of Polish groups. We then turn to special classes of Polish metric spaces, including locally compact, ultrametric, zero-dimensional, homogeneous, and ultrahomogeneous spaces. In the investigation of the classification problems we also obtain characterizations for isometry groups of various classes of Polish metric spaces.
| Item Type: | Article | |||||||||
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| Additional Information: | © 2003 American Mathematical Society. Received by the editor August, 14, 2000. Alexander S. Kechris' research was partially supported by NSF Grant DMS-9987437. | |||||||||
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| Subject Keywords: | Polish group, the Urysohn space, Borel (bi)reducible, isometry | |||||||||
| Issue or Number: | 766 | |||||||||
| Classification Code: | 2000 MSC: Primary 03E15, 54E35; Secondary 54H05, 03E75 | |||||||||
| Record Number: | CaltechAUTHORS:20130516-152757565 | |||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20130516-152757565 | |||||||||
| Official Citation: | Gao, Su and Kechris, Alexander S. (2003) On the classification of Polish metric spaces up to isometry. Memoirs of the American Mathematical Society, 161 (766). pp. 1-78. | |||||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||
| ID Code: | 38548 | |||||||||
| Collection: | CaltechAUTHORS | |||||||||
| Deposited By: | Tony Diaz | |||||||||
| Deposited On: | 30 May 2013 19:31 | |||||||||
| Last Modified: | 03 Oct 2019 04:58 |
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