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Axioms 2015, 4(4), 459-491; doi:10.3390/axioms4040459

Characterized Subgroups of Topological Abelian Groups

Dipartimento di Matematica e Informatica, Università degli Studi di Udine, Via delle Scienze 206, Udine 33100, Italy
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Academic Editor: Sidney A. Morris
Received: 2 September 2015 / Revised: 27 September 2015 / Accepted: 8 October 2015 / Published: 16 October 2015
(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)
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Abstract

A subgroup H of a topological abelian group X is said to be characterized by a sequence v = (vn) of characters of X if H = {xX : vn(x) → 0 in T}. We study the basic properties of characterized subgroups in the general setting, extending results known in the compact case. For a better description, we isolate various types of characterized subgroups. Moreover, we introduce the relevant class of auto-characterized groups (namely, the groups that are characterized subgroups of themselves by means of a sequence of non-null characters); in the case of locally compact abelian groups, these are proven to be exactly the non-compact ones. As a by-product of our results, we find a complete description of the characterized subgroups of discrete abelian groups. View Full-Text
Keywords: characterized subgroup; T-characterized subgroup; T-sequence; TB-sequence; von Neumann radical; convergent sequence; null sequence; abelian group characterized subgroup; T-characterized subgroup; T-sequence; TB-sequence; von Neumann radical; convergent sequence; null sequence; abelian group
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Dikranjan, D.; Giordano Bruno, A.; Impieri, D. Characterized Subgroups of Topological Abelian Groups. Axioms 2015, 4, 459-491.

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