$\omega$-bounded group; $\sigma$-bounded group; $o$-bounded group; Weil complete group; locally minimal group; Lie group
It is proven that an infinite-dimensional Banach space (considered as an Abelian topological group) is not topologically isomorphic to a subgroup of a product of $\sigma $-compact (or more generally, $o$-bounded) topological groups. This answers a question of M. Tkachenko.
Guran I.: On topological groups close to being Lindelöf
. Soviet Math. Dokl. 23 (1981), 173-175. Zbl 0478.22002
Hernández C.: Topological groups close to being $\sigma$-compact
. Topology Appl. 102 (2000), 101-111. MR 1739266