A question about compact abelian groups

Question

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.

Answer 1

For the first part, $\Bbb R$ has many pairs of discrete subgroups $H$ and $K$ such that $H+K$ is dense in $\Bbb R$.

Answer 2

For the second part. Consider the map $+:G\times G \to G$ given by $+(g,h)=g+h$ and use the fact that the image of a compact set is compact.