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

  • Alex Thompson (a1)

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.

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[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.
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[3]Gao, S., On automorphism groups of countable structures, this Journal, vol. 63 (1998), no. 3, pp. 891896.
[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. 221261.
[5]Hjorth, G. and Kechris, A. S., Recent developments in the theory of Borel reducibility, Fundamenta Mathematicae, vol. 170 (2001), pp. 2152.
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[7]Kechris, A. S., Classical descriptive set theory, Graduate Texts in Mathematics, Springer, 1995.
[8]Moschovakis, Y. N., Descriptive set theory, North-Holland Press, Amsterdam, 1980.

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

  • Alex Thompson (a1)

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