Let $G$ be a Hausdorff group. Let $A$ be a closed subgroup of $G$ and $B$ be a discrete subgroup of $G$ (note $B$ is closed since $G$ is Hausdorff). Is the product set $AB$ necessarily closed in $G$?
2026-03-27 16:27:51.1774628871
Product of a closed subgroup and a discrete subgroup in a Hausdorff group
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The answer is no, assuming you meant for $B$ to be the discrete subgroup, which seems likely. In the additive group $\mathbb R$ we have two discrete, closed subgroups $\mathbb Z$ and $\mathbb Z\sqrt 2$, but the sum of these subgroups is dense and not all of $\mathbb R$ and hence is in particular not closed.