chooses any $x\in G_2$ and responds with $$\sigma(O_1,O_2) = V(x,{\mathcal{O}},2)\in O_2.$$ This procedure describes a strategy for TWO in the game ${{\sf G}_1}({\mathcal{O}},{\mathcal{O}})$ on $G$.

To see that $\sigma$ is a winning strategy for TWO, consider any $\sigma$-play $$O_1,\, \sigma(O_1),\, O_2,\,\sigma(O_1,O_2),\,\cdots$$ Let an $x\in G$ be given. Fix the least $m$ with $x\in G_m$. By the definition of $\sigma$ we have an associated sequence $$U_n = U({\mathcal{O}}_n,n)$$ of subgroups of $G$ that are neighborhoods for the identity element where for each $n$ we have $U_{n+1}\subset U_n$, and ${\mathcal{O}}_1 = O_1$ while ${\mathcal{O}}_{n+1} = \{U\cap V:\, U\in {\mathcal{O}}_n \mbox{ and }V\in O_{n+1}\}\setminus\{\emptyset\}$, and elements $x_n$ of $G$ such that $$x_1*U_1 = F({\mathcal{O}}(U_1)) \mbox{ and } x_{n+1}*U_{n+1} = F({\mathcal{O}}(U_1),\cdots,{\mathcal{O}}(U_n)).$$

But then this is an $F$-play of ${{\sf G}_1}({{\mathcal{O}}_{nbd}},{\mathcal{O}})$ and thus won by TWO, meaning there are infinitely many $n$ with $x\in x_n*U_n$. Thus, fix an $n>m$ with $x\in x_n*U_n$. But then $G_n\cap x_n*G_n\neq \emptyset$ and thus $\sigma(O_1,\cdots,O_n)\supseteq x*U_n$, meaning $x\in \sigma(O_1,\cdots,O_n)$ $\Box$

\[sigmacompactGN\] For each infinite cardinal number $\kappa$ there is a $\sigma$-compact ${\sf T}_0$ topological group of cardinality $\kappa$ such that TWO has a winning strategy in the game ${{\sf G}_1}(\Omega,\Gamma)$.

\[rothbdedtorothb\] Let $(G,*)$ be a $\sigma$-compact ${\sf T}_0$ topological group 