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Article Dans Une Revue Topology and its Applications Année : 2006

Left and right uniform structures on functionally balanced groups

Résumé

Let $G$ be a Hausdorff topological group. It is shown that there is a class $\cal C$ of subspaces of $G$, containing all (but not only) precompact subsets of $G$, for which the following result holds: Suppose that for every real-valued discontinuous function on $G$ there is a set $A$ in $\cal C$ such that the restriction mapping $f_{|A}$ has no continuous extension to $G$; then the following are equivalent: (i) the left and right uniform structures of $G$ are equivalent, (ii) every left uniformly continuous bounded real-valued function on $G$ is right uniformly continuous, (iii) for every countable subset $A$ of $G$ and every neighborhood $V$ of the unit $e$ of $G$, there is a neighborhood $U$ of $e$ in $G$ such that $AU\subset VA$. As a consequence, it is proved that items (i), (ii) and (iii) are equivalent for every inframetrizable group. These results generalize earlier ones established by Itzkowitz, Rothman, Strassberg and Wu, by Milnes and by Pestov for locally compact groups, by Protasov for almost metrizable groups, and by Troallic for groups that are quasi-k-spaces.
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Dates et versions

hal-00372934 , version 1 (02-04-2009)

Identifiants

  • HAL Id : hal-00372934 , version 1

Citer

Ahmed Bouziad, Jean-Pierre Troallic. Left and right uniform structures on functionally balanced groups. Topology and its Applications, 2006, 153 (Issue 13, 1), pp.Volume 153, Issue 13, 1 July 2006, Pages 2351-2361. ⟨hal-00372934⟩
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