I am trying to prove Pontryagin duality for torsion abelian groups. It might appear that this question is a duplicate of this, but I assure you its not. Rather if the linked question had all the answers, I would have been happier.

Given a torsion abelian group $M$, we want to show $\hom(M,\mathbb{Q}/\mathbb{Z})$ is a profinite group. It is easy to see that $\hom(M,\mathbb{Q}/\mathbb{Z})\cong \varprojlim \hom(H,\mathbb{Q}/\mathbb{Z})$ where the $H\leq G$ and $|H|<\infty$, as groups. However, I need to show this equivalence as topological spaces as well, which is why this question. I have been told to assume topology of pointwise convergence on $\hom(M,\mathbb{Q}/\mathbb{Z})$. I am unable to show that the above map is open map.

Also, I am unable to construct a torsion abelian group given a profinite group. I do not want to assume the general Pontryagin duality for locally compact abelian groups as is done in the answer of the linked question.


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