This is a question from Chapter III, section 7 of Silverman's Arithmetic of Elliptic Curves. The section introduces the Tate module and then the author makes the offhand remark:
Since each $E[\ell^n]$ is a $\mathbb{Z}/\ell^n\mathbb{Z}$-module, we see that the Tate module has a natural structure as a $\mathbb{Z}_\ell$-module. Further, since the multiplication-by-$\ell$ maps are surjective, the inverse limit topology on $T_\ell(E)$ is equivalent to the $\ell$-adic topology that it gains by being a $\mathbb{Z}_\ell$-module.
I'm having trouble understanding this comment. I see that the Tate module is a $\mathbb{Z}_\ell$-module, since $\mathbb{Z}_\ell$ can act component-wise on the Tate module. For the topology, I assume that the inverse limit topology means the subspace topology that $T_\ell(E)$ inherits as a subspace of $\prod_{n \in \mathbb{N}} E[\ell^n]$, with each $E[\ell^n]$ given the discrete topology. What is the topology that $\mathbb{Z}_\ell$ gains by being a $\mathbb{Z}_\ell$-module, and why does the surjectivity of the multiplication-by-$\ell$ maps imply that these two topologies are equivalent?