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Each closed subgroup $i: H \to G$ (each regular mono, in category-speak) induces a quotient map of topological groups $\hat{i}: \hat{G} \to \hat{H}$ (a regular epi), whose kernel is the annihilator. There is a one-one correspondence between such regular monos and regular epis under the Pontryagin dual equivalence. Furthermore, if $K$ is any closed subgroup ...
It seems the following. In general the answer is no, because compactness does not imply sequential compactness. Let $\Bbb T=\{z\in\Bbb C:|z|=1\}$ be the unit circle endowed with the standard topology. Put $G={\Bbb T}^{\Bbb T}$. By Tychonov Theorem, $G$ is a compact space. Let $K=\{e\}$ be the trivial subgroup of $G$. Select an element \$g=(g_z)_{z\in\Bbb ...