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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.

03E15 Descriptive set theory
54H11 Topological groups (topological aspects)
22A05 Structure of general topological groups
Full Text: DOI
[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’AcadĂ©mie des Sciences Paris 218 pp 774– (1944)
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