Let $(G,+)$ be a $T2$ topological group endowed with a linear topology. It means that there is a local basis around $0$ made of subgroups.
Pick two closed subgroups $C,D$; is it true that the sum $C+D$ is also closed?
The statement is clearly false on generic topological groups, I'm just asking if the hypothesis of linear topology it is enough to make it true.
attempt of proof: Let $g\in (C+D)^c$ and let $U$ be a basic open subgroup contained in $C+D$, then $g+U$ is contained in $(C+D)^c$ proving that $(C+D)^c$ is open.