Lemma 1

Let $(G_{\alpha}, f_{\alpha \beta})$ be a projective system of topological groups relative to a filtered index set I, and G its limit. Assume that the canonical maps $f_{\alpha}: G\rightarrow G_{\alpha}$ are surjective.

a) The subgroups $D(G_{\alpha})$ (resp. $C(G_{\alpha})$, resp. $C(G_{\alpha})_0$) form a projective system of subsets of $G_{\alpha}$.

b) We have D(G) = lim$_{_{\leftarrow -}\alpha}\overline{D(G_{\alpha})}$ and C(G) = lim$_{_{\leftarrow -}\alpha}C(G_{\alpha})$.

c) If $G_{\alpha}$ is compact for all $\alpha \in I$, then $C(G)_0=$ lim$_{_{\leftarrow -}\alpha}C(G_{\alpha})_0$.

Let $\alpha , \beta$ be two elements of I, with $\alpha \leq \beta$. Then $f_{\alpha \beta}(D(G_{\beta}))\subset D(G_{\alpha})$, and $f_{\alpha \beta}(C(G_{\beta}))\subset C(G_{\alpha})$ since $f_{\alpha \beta}$ is surjective; since $f_{\alpha \beta}$ is continuous, it follows that $f_{\alpha \beta}(\overline{D(G_{\beta})})\subset \overline{D(G_{\alpha})}$ and $f_{\alpha \beta}(C(G_{\beta})_0)\subset C(G_{\alpha})_0$, hence a). Since $f_{\alpha}$ is surjective, $f_{\alpha}(D(G)) = D(G_{\alpha})$ (Algebra, Chap. I, §6, no. 2, Prop. 6), so $\overline{D(G)}=$ lim$_{\leftarrow -}\overline{D(G_{\alpha})}$ (General Topology, Chap. I, §4, no. 4, Cor. of Prop. 9). The surjectivity of $f_{\alpha}$ also implies the inclusion $f_{\alpha}(C(G))\subset$ $C(G_{\alpha})$ and hence $C(G)\subset$ lim$_{\leftarrow -}C(G_{\alpha})$; the opposite inclusion is immediate. Finally, assertion c) follows from b) and General Topology, Chap. III, §7, no. 2, Prop. 4).