Recently, i have read the assertion that in $Q_p$, the p-adics, every open set is a disjoint union of open balls. This is not true for a general metric space, see for example How to make open covers disjoint in $\mathbb{R}$?. My question is:
How to prove this assertion / What properties (in general) does the topology need to have in order for this to hold?
For example: I have the feeling that the striking property in $Q_p$ is that two open balls are either disjoint or one is contained in the other. Is it true that, if a metric space (with/without the demand that the metric is an ultrametric?) has this property, then the assertion that every open set has a disjoint covering of open balls follows?
The property above indicates how an algorithm could look like: If $$U = \bigcup_{i \in \mathcal{I}} B_{\epsilon_i}(x_i)$$ then one goes through the $x_i$ and if there is some $x_j$ s.t. $B_{\epsilon_i}(x_i) \cap B_{\epsilon_j}(x_j) \neq \emptyset$ then if $B_{\epsilon_i}(x_i) \subset B_{\epsilon_j}(x_j)$, one removes $x_i$ from the list and otherwise one removes $x_j$. Of course, one needs some set theoretic argument that assures that this process keeps manageable. I played around by finding equivalence relations (like $x_i \sim x_j$ iff. their balls are equal) and with the lemma of Zorn directly by sorting all the "sublists" $(x_j)_{j \in \mathcal{J}}$ s.t. their union of balls is disjoint and so on but nothing really gave me a disjoint union that covered whole $U$.
Thanks in advance,
Fabian Werner