example: The underlying set of the group $G$ is $$G:=\{f\in\,^{\omega_1}2:\vert\{\alpha: f(\alpha)\neq 0\}\vert<\aleph_0\};$$ Endow $G$ with the [G]{}$_{\delta}$ topology. [@CR] shows that $(G,\oplus)$ is a Lindelöf $P$-group (and thus ${\sf T}_4$). Thus $(G,\oplus)$ is an uncountable Rothberger bounded group that is not contained in a $\sigma$-compact group. Theorem 2.3 of Comfort in [@Comfort] implies the following generalization of the above example:

\[Comfortcountable\] Let $(G_i,*_i)$, $i\in I$, be a family of countable topological groups. Endow the product $\prod_{i\in I}G_i$ with the countable box topology. Then the subgroup $$G:=\{f\in\prod_{i\in I}G_i: \vert\{j\in I:f(j)\neq {\sf id}_j\}\vert<\aleph_0\}$$ is a Lindelöf $P$-group.

In particular, for each uncountable cardinal number $\kappa$ there is a ${\sf T}_0$ Lindelöf $P$ group of cardinality $\kappa$. One can prove an analogue of Theorem \[rboundedinsigmacompact\] also for Lindelöf $P$-groups, because: Galvin proved a result that implies that if a space is a Lindelöf P-space then it is a Rothberger space - see the Lemma in Section 2 of [@GN]. Thus, Lindelöf $P$ groups satisfy the stronger selection principle ${{\sf S}_1}({\mathcal{O}},{\mathcal{O}})$. But the following are equivalent:

1.  [Topological space $X$ satisfies ${{\sf S}_1}({\mathcal{O}},{\mathcal{O}})$;]{}

2.  [ONE has no winning strategy in the game ${{\sf G}_1}({\mathcal{O}},{\mathcal{O}})$ played on $X$;]{}

3.  [For each positive integer $k$, $\Omega\rightarrow({\mathcal{O}})^2_k$ 