Corollary 2

Let $G$ be a group and let $(H_\alpha)$ be a family of normal subgroups of $G$, directed with respect to the relation $H_\alpha \supset H_\beta$. If we endow $G$ with the group topology for which the $H_\alpha$ form a fundamental system of neighbourhoods of the identity element $e$, then the Hausdorff group associated with $G$ is isomorphic to $G_1 = G / \left( \bigcap_\alpha H_\alpha \right)$; $G_1$ has a completion $\hat{G}_1 = \tilde{G}$; and the canonical mapping $G_1 \to \tilde{G} = \varprojlim G/H_\alpha$ extends to an isomorphism of $\hat{G} = \hat{G}_1$ onto $\tilde{G}$.

The subgroup $H_\alpha$ of $G$ is open and therefore also closed ($\S$ 2, no. 1, Corollary to Proposition 4), and $G/H_\alpha$ is discrete ($\S$ 2, no. 6, Proposition 18); hence the conditions of Corollary 1 are satisfied.

For the remainder of this sub-section we shall assume that $G$ is Hausdorff and that $(H_\alpha)$ is a family of compact normal subgroups of $G$, which is directed with respect to the relation $H_\alpha \supset H_\beta$ and which satisfies the condition (AP); by virtue of Proposition 2, the mapping

is then an isomorphism of topological groups which permits us to identify $G$ and $\tilde{G}$. We denote by $f_\alpha$ the canonical mapping $G \to G/H_\alpha$.