Proposition 5

Let $(G_\alpha, f_{\alpha\beta})$ be an inverse system of Hausdorff topological groups such that the $f_{\alpha\beta}$ are surjective strict morphisms with compact kernels. Then for each $\alpha \in I$ the canonical mapping $f_\alpha$ of $G = \varprojlim G_\alpha$ into $G_\alpha$ is a surjective strict morphism whose kernel is compact.

The facts that $f_\alpha$ is surjective and that its kernel is compact are consequences of Chapter I, § 9, no. 6, Corollary 1 to Proposition 8. It remains to show that $f_\alpha$ is a strict morphism. Let $e$ (resp. $e_\alpha$) denote the identity element of $G$ (resp. $G_\alpha$). Each neighbourhood $V$ of $e$ in $G$ contains a set of the form $f_\beta^{-1}(V_\beta)$, where $V_\beta$ is a neighbourhood of $e_\beta$ in $G_\beta$, and we may suppose that $\beta \geq \alpha$; since $f_{\alpha \beta}$ is a surjective strict morphism, $f_{\alpha \beta}(V_\beta)$ is a neighbourhood of $e_\alpha$ in $G_\alpha$, and since $f_\beta$ is surjective, we have $V_\beta \subset f_\beta(V)$, whence

this shows that $f_\alpha(V)$ is a neighbourhood of $e_\alpha$ in $G_\alpha$.

If $H_\alpha = f_\alpha^{-1}(e_\alpha)$, then each $H_\alpha$ is a compact normal subgroup of $G$; the $H_\alpha$ clearly satisfy the condition (AP) of no. 3; and $G_\alpha$ can be identified with $G/H_\alpha$. In particular Propositions 3 and 4 of no. 3 apply to $G$ and the $H_\alpha$.