o-Boundedness of free topological groups

Assuming the absence of Q-points (which is consistent with ZFC) we prove that the free topological group F ( X ) over a Tychonov space X is o-bounded if and only if every continuous metrizable image T of X satisfies the selection principle ⋃ fin ( O , Ω ) (the latter means that for every sequence 〈 u n 〉 n ∈ ω of open covers of T there exists a sequence 〈 v n 〉 n ∈ ω such that v n ∈ [ u n ] < ω and for every F ∈ [ X ] < ω there exists n ∈ ω with F ⊂ ⋃ v n ). This characterization gives a consistent answer to a problem posed by C. Hernándes, D. Robbie, and M. Tkachenko in 2000.