Take the complex $p$-adic filled unit disk: for some prime $p$, this is the set D = {c $\in$ $\mathbb{C}_p$ : |c|$_p$ $\leq$ $1$}, where $\mathbb{C}_p$ is the algebraically closed and complete field under the p-adic absolute value.
From a non-$p$-adic intuition, I would think the boundary of this set is S = {c $\in$ $\mathbb{C}_p$ : |c|$_p$ = $1$}. Is this correct? Does closure in p-adic land work the same way regardless of topological space? How could I rigorously determine the boundary?
Hint: It's straightforward to see that $D$ is closed; use the ultrametric property to show that it is also open.
If a set is both closed and open, what is its boundary according to the topological definition?