Is there a non-compact open subset of the ring of $p$-adic intergers $\Bbb{Z}_p, p$ a prime?

Question

Can anyone give me some idea? I cant find it at all.

Answer 1

Hint: try $\mathbf Z_p - \{0\}$.

Answer 2

The ring of $p$-adic integers is compact Hausdorff. Suppose every open subset is compact, hence closed. Then every closed subset would be open, so any singleton would be open and the topology would be discrete.