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Lemma 2.9 if $(X,d)$ is an ultrametric space, then any two open balls of radius $r>0$ are eithier equal or disjiont. In particular, if X is compact, then every open ball of radius r is clopen and the collection of all open balls of radius r forms a partition. let $B_r(x) $ and $B'_r(y)$ be open ...
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