endowed with the relative topology is by Theorem \[Comfortcountable\] a Lindelöf ${\sf P}$-group. For a countable set $B\subset\kappa$, let $\Pi_B$ denote the projection of $G$ onto $\prod_{\alpha\in B}G_{\alpha}$. Then the set $$U_B = G^* \bigcap \Pi^{\leftarrow}_B\lbrack\{{\sf id}_B\}\rbrack$$ is a basic neighborhood of the identity element of $G^*$. Also $\mathcal{D} = \{U_B:B\in\lbrack\kappa\rbrack^{\leq\aleph_0}\}$ is a neighborhood basis for the identity element of $G^*$, and each $U_B$ is a subgroup of the group $G^*$. Since $G^*$ is a Lindelöf ${\sf P}$-group, each open cover of the form $\mathcal{O}(U_B)$ has a countable subcover, and this means that the subgroup $U_B$ has countably many distinct left cosets in $G^*$.

Now we show that TWO has a winning strategy in the game ${{\sf G}_1}({{\mathcal{O}}_{nbd}},{\mathcal{O}})$ played on $G^*$. Since $\mathcal{D}$ is a neighborhood basis of the identity element of $G^*$, we may assume that for each $n<\omega$ ONE’s $n$-th move is of the form ${\mathcal{O}}(U_{B_n})$, $B_n$ a countable subset of $\kappa$. And since TWO may replace ONE’s move ${\mathcal{O}}(U_B)$ with a move ${\mathcal{O}}(U_C)\subset {\mathcal{O}}(U_B)$ and respond to the replacement move instead, we may further assume that ONE’s moves are such that for each $n$, $B_n\subseteq B_{n+1}$, that is, $U_{B_{n+1}}\subseteq U_{B_n}$.

Also, for each move ${\mathcal{O}}(U_{B_n})$ by ONE, 