Direct sums and products in topological groups and vector spaces

Abstract

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

Topologically independent sets in precompact groups

“…Example 3.1 demonstrates, that precompactness can not be replaced by local compactness in Theorem 1.1. In [2] a result closely related to Theorem 1.1 was obtained. In order to state it, recall that by [2, Definition 3.1] a subset A of a topological group is absolutely Cauchy summable provided that for every neighborhood…”

“…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.…”

“…For a ∈ G, we use the symbol a to denote {a} . Following [2], the symbol S A stays for the direct sum…”

“…One can readily verify that in (Hausdorff) topological groups every topologically independent set is independent (see [2,Lemma 4.2]) and that these two notions coincide in discrete groups. Thus topological independence can be viewed as a natural generalization of independence.…”

Module-valued functors preserving the covering dimension

“…Examples of compact abelian self-slender groups that are not NSS can be found in [2]. Therefore, conditions from items (i) and (ii) of Lemma 8.4, as well as the corresponding assumptions in Corollary 8.5 and Theorem 9.5, cannot be "merged" into a single general statement.…”