On index of total boundedness of (strictly) o-bounded groups

Let G be a metrizable topological group. Denote by itb(G) the smallest cardinality of a cover of G by totally bounded subsets of G. A group G is defined to be σ-bounded if itb(G)⩽ℵ0. The group G is called o-bounded if for every sequence (Un)n∈ω of neighborhoods of the identity in G there exists a sequence (Fn)n∈ω of finite subsets in G such that G=⋃n∈ωFn·Un; G is called strictly o-bounded (respectively OF-determined) if the second player (respectively one of the players) has a winning strategy in the following game OF: two players, I and II, choose at every step n an open neighborhood Un of the identity in G and a finite subset Fn of G, respectively. The player II wins if G=⋃n∈ωFn·Un. second countable group G the following results are proven. itb(G)∈{0,1,ℵ0}∪[b,d]. If G is strictly o-bounded, then itb(G)⩽ℵ1 and G is σ-bounded or meager. If the space G is analytic, then the group is OF-determined and satisfies itb(G)∈{0,1,ℵ0,d}. G is σ-bounded if it is strictly o-bounded and one of the following conditions holds: (i) G is analytic; (ii) ℵ1<b; (iii) (MA+¬CH) holds; (iv) analytic games are determined; (v) there exists a measurable cardinal. Also we show that under (MA) every non-locally compact Polish Abelian divisible group contains a Baire o-bounded OF-undetermined subgroup.