Proposition 2

Suppose G compact. There exists a decreasing filtrant family $(H_i)_{i\in I}$ of closed subgroups of G such that

a) the group G identifies with the projective limit of the $G/H_i$;

b) for every $i$, there exists an integer $q\geqslant 0$ and a finite group A such that $G/H_i$ is isomorphic to $\mathbf{T}^q\times A$.

Indeed, $\widehat{G}$ is discrete (prop. 18 of II, p. 233), hence union of an increasing filtrant family $(D_i)_{i\in I}$ of subgroups of finite type. Put $H_i= D^{\bot}_i$ ; the group G identifies with the projective limit of the $G/H_i($II, p. 234, cor. 3), and $(H_i)$ is a decreasing filtrant family.

Let $i\in I$. There exists an integer $q\geqslant 0$ and a finite group A such that the group $D_i$ is isomorphic to $\mathbf{Z}^q\times A$ (A, VII, p. 22, th. 3), hence $G/H_i$ is isomorphic to $\mathbf{T}^q\times \widehat{A}($cf. corollary 1 of II, p. 232).