### A metamathematical condition equivalent to the existence of a complete left invariant metric for a Polish group

Alex Thompson
Source: J. Symbolic Logic Volume 71, Issue 4 (2006), 1108-1124.

#### 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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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1164060447
Digital Object Identifier: doi:10.2178/jsl/1164060447
Mathematical Reviews number (MathSciNet): MR2275851
Zentralblatt MATH identifier: 1105.03041

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