TWO. Towards this we need another generalization of the Lebesgue Covering Lemma, this time for Lindelöf P-groups:

\[pgpLebesgue\] Let $(G,*)$ be a Lindelöf P-group. Then there is for each open cover $\mathcal{U}$ of $G$ a neighborhood $N$ of the identity of $G$ such that for each $x\in G$ there is a $U\in\mathcal{U}$ such that $x*N\subseteq U$.

Let $\mathcal{U}$ be an open cover of $G$. For each $x\in G$ choose a neighborhood $U_x$ of $G$’s identity such that $U_x$ is an open subgroup of $G$, and there is a $U\in\mathcal{U}$ with $x*U_x\subseteq U$. Since $G$ is Lindelöf we find $x_n,\, n<\omega$ such that $\mathcal{F} = \{x_n*U_{x_n}:n<\omega\}$ is an open cover of $G$ and refines $\mathcal{U}$. Now since $G$ is a $P$-space, choose an open neighborhood $N$ of the identity such that $N$ is a subgroup of $G$, and $N\subseteq \bigcap_{n<\omega}U_{x_n}$.

We claim that $N$ is as required. For consider any $x\in G$. Then $x*N$ is a left coset of $N$ in $G$. We claim there is an $n$ with $x*N\subseteq x_n*U_{x_n}$. For if not, then for each $n$ we have $x*N\not\subseteq x_n*U_{x_n}$. But we have $x*N\subseteq x*U_{x_n}$, a left coset of $U_{x_n}$ in the group $G$. Thus $x*U_{x_n}\neq x_n*U_{x_n}$, and as 