For $\mathbb{R}$, o-minimality gives that an open set is a finite union of open intervals and points, but is there anything similar for the $p$-adic numbers? I looked at $p$-minimality but this doesn't give any good way of decomposing the open sets in $\mathbb{Q}_{p}$ similar to what $o$-minimality gives us. Since $\mathbb{Q}_{p}$ is a metric space, open sets are at least infinite unions of open balls, but can we reduce this to finite unions (or at least countably unions)?
Thank you.