Let $G$ be a topological group and $H$ a subgroup of $G$. If $H$ and $G/H$ are totally disconnected, then $G$ is also totally disconnected.
With 'totally disconnected' we mean the every connected component of a given topological group has only one element.
My proof:
Let $R \subseteq G$ be a connected subset of $G$. We want to show that $R$ has only one element. (Can we suppose that $e \in R$??). We suppose that $e \in R$. Let $q$ be the quotient map $q: G \rightarrow G/H$. Then $q(R) \subseteq G/H$ is connected, because $q$ is continuous. We know that $G/H$ is totally disconnected, so $q(R)$ has only one element in $G/H$ and that should be $H$, i.e. $q(R)=H$, hence $R \subseteq H$. $H$ is also totally disconnected, so $R$ contains only one element and we are done.
Is this correct? Can someone look at this? I am thankful for every kind of tips.