Topological groups close to being σ-compact

Two new classes of topological groups are introduced: o-bounded and strictly o-bounded groups. We give an example of a topological group which is strictly o-bounded but is not a subgroup of a σ-compact group. A characterization of o-bounded groups in terms of their second countable continuous homomorphic images is presented. Also, we construct a topological group which iso-bounded but not strictly o-bounded.