o-Boundedness of free topological groups

Abstract

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 $U_{fin}(O,Ω)$ (the latter means that for every sequence $ _{n\in w}$ of open covers of $T$ there exists a sequence $ _{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.