Metrizable groups and strict $o$-boundedness

Abstract

We show that for metrizable topological groups %Tkachenko's notion ofbeing a strictly o-bounded group is equivalent to being a Hurewiczgroup. In [5] Hernandez, Robbie and Tkachenko ask if there arestrictly $o$-bounded groups $G$ and $H$ for which $G\times H$ isnot strictly $o$-bounded. We show that for metrizable strictly$o$-bounded groups the answer is no. In the same paper the authorsalso ask if the product of an $o$-bounded group with a strictly$o$-bounded group is again an $o$-bounded group. We show that ifthe strictly $o$-bounded group is metrizable, then the answer is yes.