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 $U_{fin}(O,\Omega )$ (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.

24 pages, submitted

References in corpus (2)

• A semifilter approach to selection principles
• o-Boundedness of free objects over a Tychonoff space