Intersection of all open subgroups vs. the intersection of all open normal subgroups 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?
 A: $S_\infty$, 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
A: There are extreme examples of such behaviour, namely nonarchimedean (meaning that they have a basis at the identity consisting of open subgroups) Polish groups that are simple as abstract groups (meaning that they have no normal subgroups at all rather than just no closed ones).
First of all nonarchimedean Polish groups are plenty: groups of the form $\mathrm{Aut}(M)$ for a countable homogeneous structure $M$ are Polish, being isomorphic to a closed subgroup of $S_\infty$ and nonarchimedean, having a basis at the identity consisting of automorphisms pointwise stabilizing finite sets (indeed as the other answer mentions this is an iff).
Many such groups are known to be simple, the first example historically is probably the automorphism group of the random graph, shown to be simple by Truss, but you can look at the 2011 paper by Macpherson-Tent for many examples, historical remarks and general results on simplicity of such groups.
