Proposition 3

a) There exists an integer $n\geqslant 0$ and a subgroup L such that G is direct product of L and a subgroup isomorphic to $\mathbf{R}^n$, and moreover L admits a compact open subgroup K such that $L/K$ is discrete;

b) The group G is union of an increasing filtrant family of open subgroups, each of which is a projective limit of groups isomorphic to groups of the form $\mathbf{R}^p\times \mathbf{T}^q\times \mathbf{Z}^r\times A$, where A is a finite group and $p,q,r$ are positive integers.

Prove b). For every compact neighbourhood V of $e$, denote by $G_V$ the subgroup of G generated by V. It is open, and by the corollary of Prop. 2, the group $G_V$ is the projective limit of groups of the form $\mathbf{R}^p\times \mathbf{T}^q\times \mathbf{Z}^r\times A$, where A is a compact group and $p,q$ and $r$ are in $\mathbf{N}$. When V runs through the compact neighbourhoods of $e$, these subgroups $G_V$ form a filtered family (since $G_V$ and $G_W$ are contained in $G_{V\cup W}$ for all compact neighbourhoods V and W of $e$). Finally, the group G is the union of the subgroups $G_V$.

Let H be an open subgroup of G generated by a compact neighbourhood of $e$. There exists a compact group K and positive integers $p$ and $q$ such that H is isomorphic to $\mathbf{R}^p\times \mathbf{Z}^q\times K$ (Prop. 1 of II, p. 246); identify H with this product. The canonical surjection of H onto the divisible group $\mathbf{R}^p$ extends to a morphism $\pi$ of G onto $\mathbf{R}^p$ (Lemma 3 of II, p. 245). It is a projector $\pi$ of G onto $\mathbf{R}^p$, which is continuous since its restriction to the open subgroup H is continuous. Thus G is the direct product of $\mathbf{R}^p$ and the kernel L of $\pi$ (TG, III, p. 47, cor.). We have $\mathbf{Z}^q\times K = H\cap L$, hence $\mathbf{Z}^q\times K$ is an open subgroup of L. Thus K is a compact open subgroup of L, and consequently $L/K$ is discrete.