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On characterized subgroups of Abelian topological groups X and the group of all X -valued null sequences

S. S. Gabriyelyan — 2014

Commentationes Mathematicae Universitatis Carolinae

Let X be an Abelian topological group. A subgroup H of X is characterized if there is a sequence 𝐮 = { u n } in the dual group of X such that H = { x X : ( u n , x ) 1 } . We reduce the study of characterized subgroups of X to the study of characterized subgroups of compact metrizable Abelian groups. Let c 0 ( X ) be the group of all X -valued null sequences and 𝔲 0 be the uniform topology on c 0 ( X ) . If X is compact we prove that c 0 ( X ) is a characterized subgroup of X if and only if X 𝕋 n × F , where n 0 and F is a finite Abelian group. For every compact Abelian...

On extensions of bounded subgroups in Abelian groups

S. S. Gabriyelyan — 2014

Commentationes Mathematicae Universitatis Carolinae

It is well-known that every bounded Abelian group is a direct sum of finite cyclic subgroups. We characterize those non-trivial bounded subgroups H of an infinite Abelian group G , for which there is an infinite subgroup G 0 of G containing H such that G 0 has a special decomposition into a direct sum which takes into account the properties of G , and which induces a natural decomposition of H into a direct sum of finite subgroups.

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