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Coset spaces of metrizable groups. (English) Zbl 1489.54032

A coset space is a space homeomorphic to \(G/H\), where \(G\) is a topological group with subgroup \(H\). By a classical result of G. S. Ungar [Trans. Am. Math. Soc. 212, 393–400 (1975; Zbl 0318.54037)], if \(X\) is homogeneous, locally compact, separable and metrizable, then it is a coset space of some Polish group (in fact, the group of all homeomorphisms of \(X\) endowed with a natural topology inspired by the compact-open topology). It is known that the assumption on local compactness is essential: it was shown by the reviewer that there is a homogeneous Polish space which is not a coset space of any \(\aleph_0\)-bounded topological group [J. van Mill, Isr. J. Math. 165, 133–159 (2008; Zbl 1153.54021)]. In the present interesting paper, several natural questions are addressed and solved. For example, the following one: is a separable metrizable (Polish) coset space a coset space of some separable metrizable (Polish) topological group? The authors prove for example that a separable metrizable coset space of some topological group has a Polish extension which is a coset space of some Polish group.

MSC:

54H15 Transformation groups and semigroups (topological aspects)
22F30 Homogeneous spaces
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
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