The Bohr compactification, modulo a metrizable subgroup

Comfort, W. ; Trigos-Arrieta, F. ; Wu, S.

Fundamenta Mathematicae, Tome 142 (1993), p. 119-136 / Harvested from The Polish Digital Mathematics Library

Résumé

The authors prove the following result, which generalizes a well-known theorem of I. Glicksberg [G]: If G is a locally compact Abelian group with Bohr compactification bG, and if N is a closed metrizable subgroup of bG, then every A ⊆ G satisfies: A·(N ∩ G) is compact in G if and only if {aN:a ∈ A} is compact in bG/N. Examples are given to show: (a) the asserted equivalence can fail in the absence of the metrizability hypothesis, even when N ∩ G = {1}; (b) the asserted equivalence can hold for suitable G and N with N closed in bG but not metrizable; (c) an Abelian group may admit two topological group topologies U and T, with U totally bounded, T locally compact,U ⊆ T, with U and T sharing the same compact sets, and such that nevertheless U is not the topology inherited from the Bohr compactification of ⟨ G, T⟩. There are applications to topological groups of the form kG for G a totally bounded Abelian group.

Détail

Publié le : 1993-01-01

Zbl 0812.22001

EUDML-ID : urn:eudml:doc:211996

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