Topologically independent sets in precompact groups

Abstract

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.

Topological groups with invariant linear spans

“…Topologically independent sets were introduced in [1, Definition 4.1] in an arbitrary abelian topological group. For further properties of these sets in precompact groups we refer to [7].…”

“…Given n ∈ N arbitrarily, by (5) we can fix a finite set B ⊂ A \ {a} and an indexed set (r n b ) b∈B of reals such that (7) a…”

Homomorphic encoders of profinite abelian groups I

“…However, this result does not answer Problem 1.2, since its actual proof gives no clue on how to get a generating set for G. Thus, Question 1.2, remains still open so far. Incidently, Dikranjan, Shakhmatov and Spevak have investigated the continuity of mappings defined on weak direct sums (see [3,23]). However, their results go in a different direction and Question 1.2 is not touched there.…”