This is a naive question as I am new to topological group theory.
Let $G$ be a Locally Compact Abelian group (LCA group).
I know that to every closed subgroup $H$ of $G$ correspond a closed subgroup in the dual group $\widehat{G}$, namely the annihilator of $H$ - that is the set of all characters $\gamma$ over $G$ such that $\gamma(h) = 1$ for all $h \in H$.
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 $G$ ?
Yes, by Theorem 53 from [Pon] each closed subgroup of LCA group is the annihilator of its annihilator.
[Pon] Lev S. Pontrjagin, Continuous groups, 2nd ed., M., (1954) (in Russian).