Suppose that $(X, d)$ is compact zero-dimensional metric space. That it, $X$ has basis of clopen sets.
Fix $\varepsilon > 0$. Can we find open cover consisting of disjoint balls of radius at most $\varepsilon$?
This statement is true for Cantor set, as standard basis of clopen sets consists of open balls. We know that any such space $X$ can be embedded in Cantor set.
Initially, I thought that those sets can be transferred via inverse image. That is if $f:X \to C$ is said embedding, and $U_i$ are open disjoint balls covering $C$, then $f^{-1}(U_i)$ could do the job if we choose diameters of $U_i$ to be smaller than some appropriate $\delta > 0$. We can bound diameter of those inverse images using uniform continuity on compact space. But the issue is, we can't guarantee that they will be balls in original space $(X, d)$.
I don't know if this is true, but for me seems likely to be. Any ideas?