×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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’AcadĂ©mie des Sciences Paris 218 pp 774– (1944)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.