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

Journal of Symbolic Logic 71 (4):1108 - 1124 (2006)
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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