Proposition 14

In a locally compact group $G$, the identity component $C$ is the intersection of the open subgroups of $G$.

$C$ is a closed normal subgroup of $G$ ($§ 2$, no. 2, Proposition 7), and hence $G/C$ is locally compact (Proposition 13) and totally disconnected (Chapter I, $§ 11$, no. 5, Proposition 9). Since the inverse image of an open subgroup of $G/C$, under the canonical mapping of $G$ onto $G/C$, is an open subgroup of $G$ containing $C$, we see that we can restrict ourselves to proving the proposition for the group $G/C$. In other words, we may assume that $G$ is totally disconnected. We know then (Chapter II, $§ 4$, no. 4, Corollary to Proposition 6) that every compact neighbourhood $V$ of $e$ contains a neighbourhood $U$ of $e$ which is both open and closed. Since $U$ is compact and $B = C U$ is closed, there is a symmetric open neighbourhood $W$ of $e$ such that $W \subset U$ and $UW \cap BW = \varnothing$ ($§ 3$, no. 1 and Chapter II, $§ 4$, no. 3, Proposition 4), and a fortiori $UW \subset U$. By induction on $n$ it follows that $W^n \subset U$ for every integer $n > 0$. Hence the subgroup $W^\infty = \bigcup_{n > 0} W^n$, generated by $W$, is contained in $U$; but $W^\infty$ is open in $G$ ($§ 2$, no. 1, Corollary to Proposition 4). This completes the proof.

We have also proved: