Does there exist a topological group which can be covered by a collection of open abelian subgroups?

Question

I recently asked about the existence of topological groups which can be covered by a collection of open subgroups. It turns out that there do exist topological groups with this property. Now I want to know if there exists a topological group which can be covered by a collection of open abelian subgroups. Obviously every abelian topological group has this property, but what about nonabelian groups? If such a group has this property, then we can think of it as being "locally abelian" (note that in group theory, this has a different meaning). This might seem like a useful property for a topological group. Have the topological groups with this property been classified.