I was skimming through some of this paper Measurable Dynamics and Simple $p$-adic Polynomials out of curiosity.
A few pages in, the author claims that closed balls are both open and compact sets in the $p$-adic topology on $\mathbb{Q}_p$. I have not been able to verify this, and would like to understand it before proceeding further.
For clarity, let a closed ball $B(x,r)=\{y\in\mathbb{Q}_p:|x-y|_p\leq p^{-r}\}$. Then why is $B(x,r)$ both open and compact in the $p$-adic topology?
I have been able to show that since $\mathbb{Q}_p$ has a non-Archimedian absolute value, then any point inside the ball can be taken to be the center, and from that, that any two closed balls $B(x,r)$ and $B(y,s)$ are either disjoint or one is contained in the other.
Thanks!