Let $G$ be a group and $\omega$ be set of all subgroup of $G$.
Since $\omega$ is closed under intersection, it is trivial to check that $\omega$ satisfies to conditions to be a base.
Thus,Let $T$ be topology on $G$ induced by $\omega$.
One trivial observation is that every subgroup of $G$ is open under this topology and every automorphism of $G$ is also continious under this topology since inverse of "subgroups" are "subgroups"and so are their unions which are really interesting for me.
I wonder whether this topology has importance in terms of "Topology" or "Group Theory", any observation or comment is welcome.