Proposition 4

Let $B_G$ be the set of elements of G which generate a relatively compact subgroup of G. Then $B_G$ is a closed subgroup of G and $B^{\bot}_G$ is the neutral component of $\widehat{G}$.

The set $B_G$ is a subgroup of G since the product of two compact subsets of G is a compact subset of G.

Let H be an open subgroup of G generated by a compact neighbourhood of $e$. There exist positive integers $p$ and $q$ and a compact group K such that the subgroup H is isomorphic to $\mathbf{R}^p\times \mathbf{Z}^q\times K$ (Prop. 1 of II, p. 246). If these groups are identified, we see that $B_G\cap H = K$ is closed in H. Since the family of open subgroups H generated by compact neighbourhoods is an open covering of G (for example, $x$ belongs to the subgroup generated by $U\cap \{x\}$ for every fixed compact neighbourhood U of $e$), it follows that $B_G$ is closed.

We now calculate $B^{\bot}_G$. By Prop. 3, a), there exist a positive integer $n\geqslant 0$ and a group L admitting a compact open subgroup such that G can be identified with $\mathbf{R}^n\times L$. Then $B_G$ is identified with $\{0\} \times B_L$, and $B^{\bot}_G$ with $\mathbf{R}^n\times B^{\bot}_L$. We are therefore reduced to the case where G = L admits a compact open subgroup K.

On then has $K\subset B_G$; if one identifies $G/\widehat{K}$ with $K^{\bot}$ (Theorem 4 of II, p. 226), the orthogonal $(B_G/K)^{\bot}$ of $B_G/K$ in $\widehat{G}/K$ is identified with $B^{\bot}_G$. But on the other hand $B_G/K = B_{G/K}$, and since $K^{\bot}$ is an open subgroup of $\widehat{G}$ (Cor. 2 of II, p. 233), the connected component $\widehat{G}_0$ of $\widehat{G}$ is also the connected component of $K^{\bot}$ (TG, III, p. 35, Prop. 14). Thus the assertion for the discrete group $G/K$ is equivalent to that for G.

Finally, suppose that G is discrete. The group $\widehat{G}$ is then compact (Prop. 18 of II, p. 233). The connected component $(\widehat{G})_0$ is the intersection of the open subgroups of $\widehat{G}$ (TG, III, p. 35, Prop. 14) and a subgroup of $\widehat{G}$ is open if and only if it is closed and of finite index, or equivalently if its orthogonal is finite (Corollary 2 of II, p. 233); Corollary 4 of II, p. 228 shows that $(\widehat{G})^{\bot}_0$ is the union of the finite subgroups of G, which is none other than $B_G$ since G is discrete. We conclude by duality.