I have difficulties testing this statement: If $(X,d)$ is a compact metric space totally disconnected, then for $\varepsilon> 0$ there is a finite open cover $ (A_i)_{i=1}^{i=m}$ of $X$ such that $A_i\cap A_j=\emptyset$ and $diam(A_k)< \varepsilon$ for all $k$.

I appreciate any suggestion.

  • 3
    $\begingroup$ Use that $X$ is zero-dimensional, i.e. it has a clopen base. Then cover $X$, given an $\varepsilon >0$, by clopen sets of diameter smaller than that. Compactness will reduce to finitely many. If not yet disjoint then they’re easy to make disjoint. $\endgroup$ – Henno Brandsma Jan 1 '18 at 23:23

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