Let $p$ be any prime number and $G=\prod\limits_{n=1}^{\infty}\mathbb Z/(p^n)$ be the direct product of the finite cyclic groups $\mathbb Z/(p^n)$. Let $H=\mathbb Z_p$ be the group of $p$-adic integers with its usual norm $|.|_p$. Then the function $f_n:G\to H$ sending a sequence $(x_k)_{k=1}^{\infty} \in G$ to its $n$-th term $x_n \in \{0, 1, \dots , p^n − 1\} \subset \mathbb Z_p$ is a continuous $p^{−n}$–homomorphism.
As there is a term "continuity" of the map $f_n:G\to H$, there should be topologies on both $G$ and $H$, with respect to which $f_n$ is continuous.
Here, I've two questions.
Question 1 : It seems that the topology on $H=\mathbb Z_p$ is generated by the metric induced by the norm $|.|_p$. Is it true?
Question 2 : What is the topology on $G=\prod\limits_{n=1}^{\infty}\mathbb Z/(p^n)$?