Exercise 19, § 4

Let $G$ be a locally compact group, let $G_0$ be the identity component of $G$, and let $H$ be a closed subgroup of $G$. Show that if the homogeneous space $G/H$ is locally connected, then $G_0H$ is open in $G$. Let $H_0$ be the identity component of $H$; then there is an open subgroup $L$ of $H$ such that $L/H_0$ is compact (no. 6, Corollary 1 to Proposition 14); show that on the one hand $LG_0$ is closed in $G$, by using Corollary 1 of Proposition 1, and on the other hand that $LG_0$ is open in G, by considering the canonical image of $G_0$ in $G/L$ and using Corollary 3 of Proposition 14 and the fact that a quotient space of a locally connected space is locally connected [Chapter I, § 11, no. 6, Proposition 12)].