Topology of $\Bbb{Q}_p$

Question

Let $a\in \Bbb{Q}_p$. Is $ a+p^x\Bbb{Z}_p$ an open set around $a$ in the topology of $\Bbb{Q}_p$. Here $x \in \Bbb{Z}$. Also I have another question. Is $\mathbb{Z}_p$ open in $\Bbb{Q}_p$?

Answer 1

Yes, $\mathbb{Z}_p$ is both open and closed because the valuation is discrete. The possible non-zero distances in $\mathbb{Q}_p$ are $\{p^k:k\in \mathbb{Z}\}$, so the next largest distance after 1 is $p$. That means $\mathbb{Z}_p$ is both the closed ball of radius $1$ around $0$ (by definition), and the open ball of radius $1+\epsilon$. In the same way, $p^k \mathbb{Z}_p$ is the open ball of radius $p^{-k}+\epsilon$ around 0, and $a+p^k \mathbb{Z}_p$ the same around $a$.

Answer 2

Hint: Try expressing those sets using the metric.