Jan Spěvák - ATTI 2026 | Algebra, Topology and Their Interactions

Topological independence and groups with invariant linear spans

Topologically independent sets were introduced in [DSS]: A subset $A$ of nonzero elements of an abelian topological group $G$ is topologically independent provided that for every neighborhood $W$ of $0_G$ there exists neighborhood $U$ of $0_G$ such that for every finite set $B\subset A$ and every family $(z_a)_{a\in B}$ of integers the inclusion $\sum_{a\in B}z_aa\in U$ implies $z_aa\in W$ for all $a\in B$ .
If $G$ is moreover a topological linear space over the field of reals, it is possible to obtain a stronger notion of a topologically linearly independent set by replacing integers by real numbers in the above definition. These two notions do not coincide in the realm of topological vector spaces in general.
A topological group $G$ that can be embedded into a topological vector space has invariant linear span if all linear spans of $G$ under arbitrary embeddings into topological vector spaces are isomorphic as topological vector spaces. This notion was introduced recently in [PS].
The goals of our talk are:
1) To prove that if $G$ is a topological subgroup of a topological vector space that carries its weak topology, then its subset $A\subset G$ is topologically independent if and only if the topological group $\hull{A}$ is topologically isomorphic to the direct sum $\bigoplus _{a\in A} \hull{a}$ equipped with the Tychonoff topology. This extends the result from [JS] where the same was proved for a precompact group $G$ .
2) Using the result from 1) and a recent theorem from [PS] which says that for an arbitrary non-empty set $A$ the topological group $\mathbb Z^{(A)}$ has invariant linear span, we will show that topological independence and topological linear independence coincide in topological vector spaces with weak topologies.
3) Finally, we also intend to discuss further generalizations of the result from 1).

[DSS] D. Dikranjan, D. Shakhmatov and J. Spěvák, Direct sums and products in topological groups and vector spaces, J. Math. Anal. Appl. 437 (2016) 1257-1282.
[PS] E. Pernecká, J. Spěvák, Topological groups with invariant linear spans, Rev. Mat. Complut. (2021) DOI 10.1007/s13163-020-00383-7.
[JS] J. Spěvák, Topologically independent sets in precompact groups, Topol. Appl. 235 (2018) 269-274.