G}_1}({\mathcal{O}},{\mathcal{O}}_H)$.]{}

4.  [On $H$ ONE has no winning strategy in the game ${{\sf G}_1}({\mathcal{O}}_{nbd},{\mathcal{O}})$.]{}

5.  [For each positive integer $k$, $(G,*)$ satisfies $\Omega\rightarrow ({\mathcal{O}}_H)^2_k$]{}

\(1) implies that $(G,*)$ satisfies ${{\sf S}_1}({\mathcal{O}}_{nbd},{\mathcal{O}}_H)$. But then by Theorem 13 of [@coc11], $(H,*)$ satisfies ${{\sf S}_1}({\mathcal{O}}_{nbd},{\mathcal{O}})$ in the relative topology, proving (2). Since (2) states that $H$ is a Rothberger bounded subset of $G$, Theorem \[rboundedinsigmacompact\] gives the implication from (2) to (3) and from (3) to (5). Also, (3) implies (4) which implies (2). The proof that (5) implies (1) uses the fact that ${{\sf S}_1}({\mathcal{O}},{\mathcal{O}}_H)$ is equivalent to ${{\sf S}_1}(\Omega,{\mathcal{O}}_H)$. $\Box$\
Corollary \[rothbbdedgp\] improves Theorem 22 of [@coc11] in that it does not require the group $(G,*)$ to be metrizable. One might wonder how widely applicable Corollary \[rothbbdedgp\] really is. We shall address this in the next two sections by showing that:

-   [For each infinite cardinal number $\kappa$ there is a ${\sf T}_0$ Rothberger bounded group $(G,*)$ of cardinality $\kappa$ which is not a subgroup of any $\sigma$-compact group, and TWO has a winning strategy in the game ${{\sf G}_1}({\mathcal{O}}_{nbd},{\mathcal{O}})$ on $(G,*)$.]{}

-   [For each infinite cardinal number $\kappa$ there is a ${\sf T}_0$ $\sigma$-compact Rothberger bounded group of cardinality $\kappa$ 