of Theorem \[smzpowTh9\] for $\sigma$-compact topological groups:

\[rboundedinsigmacompact\] Let $(G,*)$ be a $\sigma$-compact topological group and let $X$ be a subset of $G$. The following are equivalent:

1.  [$(G,*)$ satisfies ${{\sf S}_1}({\mathcal{O}},{\mathcal{O}}_X)$.]{}

2.  [$(G,*)$ satisfies ${{\sf S}_1}({\mathcal{O}}_{nbd},{\mathcal{O}}_X)$.]{}

3.  [$X$ is a Rothberger bounded subset of $(G,*)$.]{}

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

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

The implications $(2)\Rightarrow (3)$, $(4)\Rightarrow (5)$ and $(5)\Rightarrow(1)$ follow the ideas of the proof of Theorem 9 of [@smzpow], with some standard modifications. We prove $(3)\Rightarrow(4)$ here since its proof uses the perhaps new Lemma \[lebesguecov\].

[ ]{} Let $F$ be a strategy for player ONE in the game ${{\sf G}_1}({\mathcal{O}},{\mathcal{O}}_X)$ on $G$. Since $G$ is $\sigma$-compact, write $G=\bigcup_{n<\omega}G_n$ where for each $n$ we have $id_G\in G_n\subseteq G_{n+1}$ and $G_n$ is compact. For each $n$, $X_n = X\cap G_n$ is a Rothberger bounded subset of $G$. To defeat ONE’s strategy TWO will concentrate attention on specific $X_n$’s in specific innings. To this end, partition $\omega$ into infinitely many infinite subsets $S_n$. For innings numbered by members of $S_n$ TWO will focus on $X_n$.

We now use ONE’s strategy $F$ to recursively define a sequence $(N_k:k<\omega)$ 