a $\sigma$-compact Rothberger bounded group of cardinality $\kappa$. $\Box$

Using the method of proof of Proposition \[twowinsrothberger\], one proves

\[twowinssigmacompactrothberger\] In the groups of Corollary \[corsoncor\] TWO has a winning strategy in the game ${{\sf G}_1}({{\mathcal{O}}_{nbd}},{\mathcal{O}})$.

Note that for each finite subset $B$ of $\kappa$ the set $U_B =\{f\in G:(\forall \alpha\in B)(f(\alpha) = id_{G_{\alpha}}\}$ is in fact a subgroup of $G$, is a neighborhood of the identity element of $G$, and the set of such $U_B$ form a neighborhood basis of the identity element of $G$. Now apply the argument of Proposition \[twowinsrothberger\]. $\Box$

In fact the groups of Corollary \[corsoncor\] satisfy the stronger selection principle ${{\sf S}_1}({\mathcal{O}},{\mathcal{O}})$: One can prove more, namely

\[twowinssigmacompact\] In the groups of Corollary \[corsoncor\] TWO has a winning strategy in the game ${{\sf G}_1}({\mathcal{O}},{\mathcal{O}})$ on $G$.

Let $(G,*)$ be such a group and $F$ be a winning strategy for TWO in the game ${{\sf G}_1}({{\mathcal{O}}_{nbd}},{\mathcal{O}})$ on this group. Since the group is $\sigma$-compact we write $G = \bigcup_{n=0}^{\infty}G_n$ where for each $n$ $G_n\subseteq G_{n+1}$ and $G_n$ is compact.

For each open cover $\mathcal{U}$ of $G$ and for each $n$ choose a neighborhood $U(\mathcal{U},n)$ of the identity element of $G$ such that $U(\mathcal{U},n)$ is a 