It is well-known that a *connected* topological group can be generated by any neighborhood of the identity. There are non-connected topological groups for which this is still true, such as $\mathbb{Q}$. My question is: is there any characterization of those non-connected topological groups for which the result is true? (I am specifically thinking about p-adics $\mathbb{Q}\_p$, where
$\mathbb{Q}$ is dense. I think the result is not true in this case, but I do not know how to prove/disprove it).



system of generators for non-connected topological groups
Immediate restatements: let $G$ be a topological group. Then $G$ is generated by any of its neighborhoods of 1 $\Leftrightarrow$ the only open subgroup of $G$ is $G$ $\Leftrightarrow$ $G$ has no nontrivial continuous action on any discrete set. All these properties hold if $G$ is connected and you're asking about when the converse holds.


For a locally compact group $G$, $G$ has no proper open subgroup iff $G$ is connected.


On the other hand, there exists an abelian Polish group $G\neq\{1\}$ that is totally disconnected (hence not connected) but has no proper open subgroup. Reference: T. Christine Stevens, Connectedness of complete metric groups. Colloquium Mathematicae 50(2) (1986) 233-240 [(eudml link)](https://eudml.org/doc/264581).


I guess that even in the realm of Polish groups, you can't expect more than trivial restatements, except if looking at very restricted subclasses. 


