o-Boundedness of free topological groups

TL;DR

This paper characterizes when free topological groups over Tychonov spaces are o-bounded, linking it to a specific selection principle in all continuous metrizable images, providing a solution to a longstanding problem.

Contribution

It provides a new characterization of o-boundedness in free topological groups using a selection principle, resolving a problem posed in 2000.

Findings

o-boundedness of free topological groups is characterized by a selection principle in all continuous metrizable images.

The result depends on the absence of Q-points, consistent with ZFC.

Provides a complete answer to a problem posed by Hernandes, Robbie, and Tkachenko.

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.