$x_n*U_{x_n}$ is also a left coset of $U_{x_n}$, we have $x*N\cap x_n*U_{x_n} = \emptyset$. But then the family $\{x_n*U_{x_n}:n<\omega\}$ does not cover the subset $x*N$ of $G$, contradicting the fact that $\mathcal{F}$ is a cover of $G$. $\Box$

\[rothbwonbytwo\] For each infinite cardinal $\kappa$ there is a ${\sf T}_0$ Lindelöf $P$-group of cardinality $\kappa$ such that TWO has a winning strategy in the game ${{\sf G}_1}({\mathcal{O}},{\mathcal{O}})$.

Let $(G,*)$ be a Lindelöf $P$-group for which TWO has a winning strategy in ${{\sf G}_1}({\mathcal{O}}_{nbd},{\mathcal{O}})$ (as for example in Proposition \[twowinsrothberger\]). Let $F$ be TWO’s winning strategy in that game.

Define a strategy $\sigma$ for TWO in the game ${{\sf G}_1}({\mathcal{O}},{\mathcal{O}})$ as follows: When ONE plays the open cover $\mathcal{U}_1$, choose a neighborhood $N_1$ of the identity of $G$ as in Proposition \[pgpLebesgue\], and then let $\sigma(\mathcal{U}_1)$ be an element $U$ of $\mathcal{U}_1$ such that $F(N_1)*N_1\subseteq U$. When ONE plays the next open cover $\mathcal{U}_2$ choose a neighborhoood $N_2$ of the identity of $G$ as in Proposition \[pgpLebesgue\], and then let $\sigma(\mathcal{U}_1,\mathcal{U}_2)$ be an element $U$ of $\mathcal{U}_2$ such that $F(N_1,N_2)*N_2\subseteq U$, and so forth.

Then $\sigma$ is a winning strategy for TWO. $\Box$

Call an open cover $\mathcal{U}$ of a topological space 