Exercise 5, § 7

Let $G$ be a Hausdorff topological group and let $X$ be a Hausdorff topological space on which $G$ operates continuously; 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 $X_\alpha$ denote $X/H_\alpha$. Show that the canonical mapping $X \to \varprojlim X_\alpha$ is a homeomorphism.