Topological groups with invariant linear spans

Given a topological group G that can be embedded as subgroup into some vector space (over the field of reals) we say has invariant linear span if all spans under arbitrary embeddings spaces are isomorphic spaces. For an set A let $${{\mathbb {Z}}}^{(A)}$$ direct sum |A|-many copies discrete integers endowed with Tychonoff product topology. We show span. This answers question from paper Dikranjan et al. (J Math Anal Appl 437:1257–1282, 2016) in positive. prove given non-discrete sequential X, free abelian A(X) over X is example embeds but does not have