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 S 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] and for every F ∈ [X] there exists n ∈ ω with F ⊂ ∪vn). This characterization gives a consistent answer to a problem posed by C. Hernandes, D. Robbie, and M. Tkachenko in 2000.