Metrizable groups and strict o-boundedness

Liljana Babinkostova

We show that for metrizable topological groups being a strictly o-bounded group is equivalent to being a Hurewicz group. In [5] Hernandez, Robbie and Tkachenko ask if there are strictly $o$-bounded groups $G$ and $H$ for which $G\times H$ is not strictly $o$-bounded. We show that for metrizable strictly $o$-bounded groups the answer is no. In the same paper the authors also ask if the product of an $o$-bounded group with a strictly $o$-bounded group is again an $o$-bounded group. We show that if the strictly $o$-bounded group is metrizable, then the answer is yes.