Topological groups with invariant linear spans

TL;DR

This paper investigates the property of invariant linear span in topological groups, showing that certain groups like ^{(A)} have it, while others like free abelian groups over non-discrete spaces do not, answering a key open question.

Contribution

It proves that ^{(A)} groups have invariant linear span and provides examples of groups that embed into topological vector spaces without this property.

Findings

^{(A)} groups have invariant linear span.

Free abelian topological groups over non-discrete spaces lack invariant linear span.

Answers a question of D. Dikranjan et al. in positive.

Abstract

Given a topological group $G$ that can be embedded as a topological subgroup into some topological vector space (over the field of reals) we say that $G$ has invariant linear span if all linear spans of $G$ under arbitrary embeddings into topological vector spaces are isomorphic as topological vector spaces. For an arbitrary set $A$ let $\mathbb{Z}^{(A)}$ be the direct sum of $|A|$-many copies of the discrete group of integers endowed with the Tychonoff product topology. We show that the topological group $\mathbb{Z}^{(A)}$ has invariant linear span. This answers a question of D. Dikranjan et al. in positive. We prove that given a non-discrete sequential space $X$, the free abelian topological group $A(X)$ over $X$ is an example of a topological group that embeds into a topological vector space but does not have invariant linear span.