After learning about the duality between compact Abelian groups and discrete Abelian groups, I decided to look at exercises from various sources.
One question that stood out was the following:
If $G$ is a locally compact Abelian group with $H$ and $K$ being two closed subgroups of $G$, does it follow that the subgroup $H + K$ is closed?
Furthermore, is this subgroup closed if $G$ is a compact Abelian group?
I'm fairly certain this has something to do with the duality mentioned above. I'm having trouble thinking of counterexamples.