o-Boundedness of free topological groups

AbstractAssuming 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 〈un〉n∈ω of open covers of T there exists a sequence 〈vn〉n∈ω such that vn∈[un]<ω and for every F∈[X]