Characterizable groups: Some results and open questions

Let X be an Abelian topological group and X ∧ its dual group. A subgroup H of X is called characterized if there is a sequence { u n } in X ∧ such that H = { x ∈ X : ( u n , x ) → 1 } . A Polish Abelian group G is called characterizable if there is a continuous monomorphism p from G into a compact metrizable Abelian group X with dense image such that p ( G ) is a characterized subgroup of X. Every characterizable group is locally quasi-convex. We prove that every second countable locally compact Abelian group X is characterizable. Thus, every second countable locally compact Abelian group is the dual group of a complete countable maximally almost periodic group. It is shown that each characterizable Abelian group of finite exponent is locally compact. Analogously to the Abelian case, we define characterized subgroups of non-Abelian compact metrizable groups and non-Abelian characterizable groups. Using the ℓ p -sum of metric groups with two-sided invariant metrics, it is proved that every characterized subgroup admits a Polish group topology.