For any topological space and for any group
Thanks
Alternatively:
Think about it like this. If then we have that and so is closed. But, and so is open. So, in a connected space , . But, and so must be dense in . So more examples would be be , , etc.
The above also shows (like Focus) pointed out that your subgroup cannot have empty interior.
Also, note the above applies equally well to non-subgroup subsets.