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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.

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    $\begingroup$ For your first question, look for counterexamples in $\mathbb{R}$. $\endgroup$ – Chris Eagle Mar 10 '12 at 19:46
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For the first part, $\Bbb R$ has many pairs of discrete subgroups $H$ and $K$ such that $H+K$ is dense in $\Bbb R$.

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  • $\begingroup$ After looking at $\mathbb{R}$, I realized that $H = \mathbb{Z}$ and $K = \sqrt{2}\mathbb{Z}$ make the desired counterexample. $\endgroup$ – josh Mar 11 '12 at 13:49
  • $\begingroup$ @josh: Yep. Or indeed $a\Bbb Z$ and $b\Bbb Z$ for any $a,b$ such that $a/b$ is irrational. $\endgroup$ – Brian M. Scott Mar 11 '12 at 19:27
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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.

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