Let $H$ a topological subgroups of a topological group $G$, and $H'$ the closure of $H$ as topological subspace. Are classic results the if $H'$ is a topological subgroup, and that it is normal if $H$ is normal this for a general topological group, and that if $G$ is topologically $T_2$ then $H'$ is commutative if $H$ is commutative. All these results are for example from "Topology I" of Bourbaki. But in literature nothing about the closure of a commutative subgroup if the topology is (more) general, and is not even a counterexample.

I ask a for a such counterexample, or for a more general condition about the commutativity of the closure of a commutative subgroup.