O-Boundedness of free topological groups

摘要

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 is contained in ∪ v_n). This characterization gives a consistent answer to a problem posed by C. Hernandes, D. Robbie, and M. Tkachenko in 2000.