I came across this problem that given a subgroup $L$ of a compact abelian group $A$, how can we tell if it is closed?
I think Pontryagin duality might be helpful. Identify $A$ with its double dual $\hat{\hat{A}}$. Since $A$ is compact, $\hat{A}$ is discrete and the compact-open topology on $\hat{\hat{A}}$ coincides with the topology induced by pointwise convergence, i.e., a sequence $\{x_n\}$ in $A$ converges to $x\in A$ iff $\forall \chi \in \hat{A}$, $\chi(x_n) \to \chi(x)$. Thus $L$ is closed if one can show that $L$ has property D, which is a term I made up and is defined as below.
Property D: A subgroup $L$ of $A$ is said to have property D if for any $a\notin L$, there exists $f\ \in \hat{A}$ satisfying $$\inf_{l\in L} |f(la^{-1})-1| > 0$$
I wonder if my reasoning above is correct and if this reduction has made things simpler. It seems quite manageable using Fourier analysis, but I haven't figured out how to do this.
And any comment on the original problem using other approaches is highly appreciated as well!