Exercise 4, § 7

Let $G$ be a Hausdorff topological group and let $(H_\alpha)$ be a family of compact normal subgroups of $G$, directed with respect to the relation $\supseteq$, and satisfying condition (AP) of no. 3. Let $(L_\alpha)$ be a family of closed subgroups of $G$ such that $H_\alpha \subset L_\alpha$ for each $\alpha \in I$ and $L_\alpha = H_\alpha L_\beta$ for $\alpha \leq \beta$, and let $L = \bigcap_\alpha L_\alpha$. Show that $L_\alpha = H_\alpha L$ for each $\alpha$ (use Proposition 3).