Let $G$ be a LCA group. It is well-known that to every closed subgroup $H$ of $G$ correspond a closed subgroup in the dual group $\widehat{G}$, namely the annihilator of $H$.
My question is this : is there a one-to-one correspondance between the closed subgroups of $G$ and the closed subgroups of the dual group $\widehat{G}$ ? Is every closed subgroup of $\widehat{G}$ an annihilator for some closed subgroup $H$ of $G$ ?
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 of the LCA group $\hat{G}$, then the quotient $\hat{G}/K$ is also LCA. So we have one-one correspondences between (1) closed subgroups of $G$, (2) LCA quotient groups of $\hat{G}$, and (3) closed subgroups of $\hat{G}$.
