Corollary 3

Let $G$ be a locally compact group, let $H$ be a closed subgroup of $G$, and let $\varphi$ be the canonical mapping of $G$ onto $G/H$. Then the components of $G/H$ are the closures of the images under $\varphi$ of the components of $G$.

Let $C$ be the identity component of $G$. The components of $G$ are the sets $sC$, where $s \in G$ ($§ 2$, no. 2, Proposition 7); $\varphi(sC)$ is clearly connected, hence so is $\varphi(sC)$ (Chapter I, $§ 11$, no. 1, Proposition 1). But $\varphi(sC) = \varphi(sCH)$, and since $sCH$ is saturated with respect to the equivalence relation defined by $H$, and since this equivalence relation is open (§ 2, no. 4, Lemma 2), we have $\overline{\varphi(sCH)} = \varphi(\overline{sCH}) = \varphi(s.\overline{CH})$ (Chapter I, § 5, no. 3, Proposition 7).

Put $L = CH$. $L$ is a closed subgroup of $G$ which contains $C$ and $H$; hence to prove that the sets $\varphi(s.L) = s.\varphi(L)$ are the components of $G/H$ it is enough to show that the quotient space of $G/H$ by the equivalence relation whose classes are the sets $s.\varphi(L)$ is totally disconnected. Now this quotient space is homeomorphic to the homogeneous space $G/L$ (Chapter I, § 3, no. 4, Proposition 7); we are thus reduced to proving that when $C \subset H$, $G/H$ is totally disconnected. Since $G/H$ may be identified with $(G/C)/(H/C)$ (§ 2, no. 7, Proposition 22), we may even assume that $G$ itself is totally disconnected. Every neighbourhood of $\varphi(e)$ in $G/H$ contains a neighbourhood of the form $\varphi(V)$, where $V$ is a neighbourhood of $e$ in $G$, and therefore (Corollary 1) contains a neighbourhood of the form $\varphi(K)$, where $K$ is a compact open subgroup of $G$. $\varphi(K)$ is therefore both open and closed in $G/H$, and this shows that the component of $\varphi(e)$ in $G/H$ consists of $\varphi(e)$ alone. By translation, the same is true for the component of every point of $G/H$, and the corollary is proved.