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An example of a Polish group. (English) Zbl 1160.03031
A topological group is called Polish if its group topology is separable and completely metrizable. A Polish group is called CLI if it admits a compatible complete left-invariant metric, i.e., a metric $$d$$ inducing the group’s topology and such that $$d(zx, zy) = d(x, y)$$ for all $$x, y, z \in G$$. The classical theorem of Birkhoff–Kakutani implies that every metrizable topological group $$G$$ admits a left-invariant compatible metric. The first known Polish group that was proved not to admit such a metric was $$\text{Hom}([0, 1])$$, the group of homeomorphisms of unit interval with supremum metric.
In this paper, the author gives a characterization of CLI subgroups of $$S_\infty$$, the group of all permutations of natural numbers, considered with pointwise convergence topology. Then he uses this characterization to construct a non-discrete Polish group which is not CLI and none of whose non-discrete subgroups is CLI.

##### MSC:
 03E15 Descriptive set theory 54H11 Topological groups (topological aspects) 22A05 Structure of general topological groups
##### Keywords:
Polish group; CLI group
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##### References:
 [1] DOI: 10.1006/jabr.1997.7243 · Zbl 0896.22001 · doi:10.1006/jabr.1997.7243 [2] Topology and its Applications (2006) · Zbl 1113.54001 [3] DOI: 10.1090/S0894-0347-98-00258-6 · Zbl 0894.03027 · doi:10.1090/S0894-0347-98-00258-6 [4] Oligomorphic Permutation Group 152 (1990) · Zbl 0813.20002 [5] Comptes Rendus de l’Académie des Sciences Paris 218 pp 774– (1944)
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