  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 $U_{fin}(O,\Omega)$ (the latter means that for every
sequence $<u_n>_{n\in w}$ of open covers of $T$ there exists a sequence
$<v_n>_{n\in w}$ such that $v_n\in [u_n]^{<w}$ and for every $F\in [X]^{<w}$
there exists $n\in w$ with $F\subset\cup v_n$). This characterization gives a
consistent answer to a problem posed by C. Hernandes, D. Robbie, and M.
Tkachenko in 2000.
