No abstract

Given a function f : N→(ω+1)\{0}, we say that a faithfully indexed sequence {a_n: n ∈ N} of elements of a topological group G is: (i) f -Cauchy productive (f-productive) provided that the sequence {\prod_{n=0}^m a_n^{z(n)} : m ∈ N} is left Cauchy (converges to some element of G, respectively) for each function z : N → Z such that |z(n)|\leq f (n) for every n ∈ N; (ii) unconditionally f -Cauchy productive (unconditionally f -productive) provided that the sequence {a^{ϕ(n)}: n ∈ N} is ( f ◦ ϕ)-Cauchy productive (respectively, ( f ◦ ϕ)-productive) for every bijection ϕ : N → N. (Bijections can be replaced by injections here.) We consider the question of existence of (unconditionally) f -productive sequences for a given “weight function” f. We prove that: (1) a Hausdorff group having an f -productive sequence for some f contains a homeomorphic copy of the Cantor set; (2) if a non-discrete group is either locally compact Hausdorff or Weil complete metric, then it contains an unconditionally f -productive sequence for every function f : N → N \ {0}; (3) a metric group is NSS if and only if it does not contain an fω-Cauchy productive sequence, where f_ω is the function taking the constant value ω. We give an example of an f_ω-productive sequence {an: n ∈ N} in a (necessarily non-abelian) separable metric group H with a linear topology and a bijection ϕ : N→N such that the sequence {\prod_{n=0}^ma^{ϕ(n)}: m ∈ N} diverges, thereby answering a question of Dominguez and Tarieladze. Furthermore, we show that H has no unconditionally f_ω-productive sequences. As an application of our results, we resolve negatively a question from C_p(−, G)-theory

We call a subset $A$ of an abelian topological group $G$: (i) $absolutely$ $Cauchy$ $summable$ provided that for every open neighbourhood $U$ of $0$ one can find a finite set $F\subseteq A$ such that the subgroup generated by $A\setminus F$ is contained in $U$; (ii) $absolutely$ $summable$ if, for every family $\{z_a:a\in A\}$ of integer numbers, there exists $g\in G$ such that the net $\left\{\sum_{a\in F} z_a a: F\subseteq A\mbox{ is finite}\right\}$ converges to $g$; (iii) $topologically$ $independent$ provided that $0\not \in A$ and for every neighbourhood $W$ of $0$ there exists a neighbourhood $V$ of $0$ such that, for every finite set $F\subseteq A$ and each set $\{z_a:a\in F\}$ of integers, $\sum_{a\in F}z_aa\in V$ implies that $z_aa\in W$ for all $a\in F$. We prove that: (1) an abelian topological group contains a direct product (direct sum) of $\kappa$-many non-trivial topological groups if and only if it contains a topologically independent, absolutely (Cauchy) summable subset of cardinality $\kappa$; (2) a topological vector space contains $\mathbb{R}^{(\mathbb{N})}$ as its subspace if and only if it has an infinite absolutely Cauchy summable set; (3) a topological vector space contains $\mathbb{R}^{\mathbb{N}}$ as its subspace if and only if it has an $\mathbb{R}^{(\mathbb{N})}$ multiplier convergent series of non-zero elements. We answer a question of Hu\v{s}ek and generalize results by Bessaga-Pelczynski-Rolewicz, Dominguez-Tarieladze and Lipecki

It is a simple fact that a subgroup generated by a subset A of an abelian group is the direct sum of the cyclic groups a , a ∈ A if and only if the set A is independent. In [2] the concept of an independent set in an abelian group was generalized to a topologically independent set in a topological abelian group (these two notions coincide in discrete abelian groups). It was proved that a topological subgroup generated by a subset A of an abelian topological group is the Tychonoff direct sum of the cyclic topological groups a , a ∈ A if and only if the set A is topologically independent and absolutely Cauchy summable. Further, it was shown, that the assumption of absolute Cauchy summability of A can not be removed in general in this result. In our paper we show that it can be removed in precompact groups.In other words, we prove that if A is a subset of a precompact abelian group, then the topological subgroup generated by A is the Tychonoff direct sum of the topological cyclic subgroups a , a ∈ A if and only if A is topologically independent. We show that precompactness can not be replaced by local compactness in this result.All groups in this paper are assumed to be abelian and all topological groups are assumed to be Hausdorff. A topological group is precompact if it is a topological subgroup of a compact group. As usually, the symbols N and Z stay for the sets of natural numbers and integers respectively.Given an abelian group G, by 0 G we denote the zero element of G, and the subscript is omitted when there is no danger of confusion. Given a subset A of G, the symbol A stays for the subgroup of G generated by A. For a ∈ G, we use the symbol a to denote {a} . Following [2], the symbol S A stays for the direct sumand by K A we denote the unique group homomorphism K A : S A → G which extends each natural inclusion map a → G for a ∈ A. As in [2], we call the map K A the Kalton map associated with A.We say that A is the direct sum of cyclic groups a , a ∈ A provided that the Kalton map K A is an isomorphic embedding. When G is a topological group, we always consider a with the subgroup topology inherited from G and S A with the subgroup topology inherited from the Tychonoff product a∈A a . Finally, we say that A is a Tychonoff direct sum of cyclic groups a , a ∈ A if the Kalton map K A is at the same time an isomorphic embedding and a homeomorphic embedding.

In [2] various natural generalizations of the classical concepts of "convergent series", "unconditionally convergent series" and "absolutely convergent series" were thoroughly studied in the realm of topological groups. Nevertheless, some natural questions posed in [2] that have answers for abelian topological groups remained open in the realm of non-abelian topological groups. In our paper we provide a general schema for constructing strong counterexamples that answers many of those questions in negative.