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A Metamathematical Condition Equivalent to the Existence of a Complete Left Invariant Metric for a Polish Group

Published online by Cambridge University Press:  12 March 2014

Alex Thompson
Alex Thompson*
Affiliation:
Mathematics Department, UCLA, Los Angeles, CA 90095-1555, USA, E-mail: [email protected], URL: www.math.ucla.edu/~act2
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Abstract

Strengthening a theorem of Hjorth this paper gives a new characterization of which Polish groups admit compatible complete left invariant metrics. As a corollary it is proved that any Polish group without a complete left invariant metric has a continuous action on a Polish space whose associated orbit equivalence relation is not essentially countable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 4 , December 2006 , pp. 1108 - 1124
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1]Becker, H. and Kechris, A. S., The descriptive set theory of Polish group actions, London Mathematical Society Lecture Notes Series, Cambridge University Press, Cambridge, 1996.CrossRefGoogle Scholar
[2]Dougherty, R., Jackson, S., and Kechris, A. S., The structure of hyperfinite Borel equivalence relations, Transactions of the American Mathematical Society, vol. 341 (1994), pp. 193–225.CrossRefGoogle Scholar
[3]Gao, S., On automorphism groups of countable structures, this Journal, vol. 63 (1998), no. 3, pp. 891–896.Google Scholar
[4]Hjorth, G., Orbit cardinals: On the effective cardinalities arising as quotient spaces of the form X/G where G acts on a Polish space X, Israel Journal of Mathematics, vol. 111 (1999), pp. 221–261.CrossRefGoogle Scholar
[5]Hjorth, G. and Kechris, A. S., Recent developments in the theory of Borel reducibility, Fundamenta Mathematicae, vol. 170 (2001), pp. 21–52.CrossRefGoogle Scholar
[6]Jech, T., Set theory, 3rd millenium ed., Springer Monographs in Mathematics, Springer, 2002.Google Scholar
[7]Kechris, A. S., Classical descriptive set theory, Graduate Texts in Mathematics, Springer, 1995.CrossRefGoogle Scholar
[8]Moschovakis, Y. N., Descriptive set theory, North-Holland Press, Amsterdam, 1980.Google Scholar