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 = {x ∈ X : 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.
Highlights
Inspired by the notion of T -characterized subgroup, we introduce the stronger one of
We study the basic properties of K- and of N -characterized subgroups respectively in Sections 7 and 8
For a topological abelian group X and a subgroup L of X, the weak topology σ(X, b generated by the elements of L considered as characters of is the totally bounded group topology of X
Summary
The symbol c is used to denote the cardinality of continuum. The symbols Z, P, N and N+ are used for the set of integers, the set of primes, the set of non-negative integers and the set of positive integers, respectively. For p ∈ P, we denote by Z(p∞ ) and Jp , respectively, the Prüfer group and the p-adic integers. We denote by Hom(G, T) the group of the homomorphisms G → T. For a subset A of X, we denote by A the closure of A in (X, τ ) (we write only A when there is no possibility of confusion). A topological abelian group X is totally bounded if for every open subset U of 0 in X, there exists a finite subset F of X, such that U + F = X. We say that a sequence v ∈ X n ≥ n0 , and we say that v is non-trivial if it is not trivially null
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