$p^i\mathbb {Z}_p$ is an open subgroup of $\mathbb {Z}_p$ of index $p^i$, where $\mathbb {Z}_p$ is the group of p-adic integers

Since $\Bbb Z/p^i\Bbb Z$ has the discrete topology, $\{0\} $ is an open subset of $\Bbb Z/p^i\Bbb Z$, consequently, $\operatorname {Ker}\pi_i=\pi_i^{-1}\{0\}$ is open being the preimage of an open.