Given a torsion abelian group $A$, prove that $A$ has a unique $\widehat{\mathbf{Z}}$-module structure and that $\widehat{\mathbf{Z}}\times A\to A$ is continuous if $A$ has the discrete topology.
I proved the first part, the module structure is given by letting an element $(a_k)_{k\geq 1}\in \widehat{\mathbf{Z}}$ act on an element $x\in A$ of order $n$ by $x^{a_n}$ (writing $A$ multiplicatively).
In order to show this action is continuous, I have to prove that the preimage of an element $x\in A$ of order $n$ is open. I think that the preimage is $(1+n\widehat{\mathbf{Z}} )\times \{x\}$, but I am not sure. For example there could be relations inside the group $E$ like two elements $x$ and $y$ such that $y^2=x^3$ and then we could have something like $\cdots \times \{y\}$ in the preimage. Could someone help here?
Continuity is local, and for any $a\in A$, $\widehat{\mathbf Z}\times\{a\}$ is open in $\widehat{\mathbf Z}\times A$ ($A$ has the discrete topology), so you only need to prove that the map is continuous on each $\widehat{\mathbf Z}\times\{a\}$.
But now for this one, $a$ is torsion, so this map factors as $\widehat{\mathbf Z}\times\{a\}\to \mathbf Z/n\mathbf Z\times\{a\}\to A$ for some $n$.