I am interested to know examples of topological groups $G$ for which the intersection $\bigcap\{H\leq G\mid H\text{ open}\}$ of all open subgroups of $G$ is the trivial subgroup but for which the intersection $\bigcap\{N\trianglelefteq G\mid N\text{ open}\}$ of all open normal subgroups is not the trivial subgroup.
Clearly (1) must be totally disconnected (2) cannot inject into a pro-discrete group by a continuous homomorphism and (3) it can't contain a topological subgroup isomorphic to $\mathbb{Q}$. I imagine that topological groups fitting this description exist and perhaps some are even important in some area I am not familiar with.
Does such a topological group exist? If so, is there an abundance of "standard" examples?
Best Answer
$S_\infty$, the group of all permutations of $\mathbb{N}$, has a neighborhood base of the identity of open subgroups. (In fact a Polish group with that property is isomorphic to a closed subgroup of $S_\infty$).
But without thinking about exactly which ones are open, $S_\infty$ has a very limited supply of normal subgroups outlined here: https://math.stackexchange.com/a/166472/29633