# real analysis: finite cover of clopen sets with maximum diameter \$ epsilon. \$

Here is the problem:

Leave $$X$$ be a compact metric space that is totally disconnected and leave $$epsilon> 0.$$

(a) Show that $$X$$ it has a finite cover $$mathcal {A}$$ clopen sets with a maximum diameter $$epsilon.$$

My judgment

With the help of many people here on this site, I was able to demonstrate that:

Yes $$X$$ is a compact metric space that is totally disconnected, then for each $$r> 0$$ and every $$x in X,$$ there is a clopen set $$U$$ such that $$x in U$$ Y $$U subseteq B_ {r} (x).$$

I feel this can help me in testing the first question, but I don't know how, could someone please clarify this for me?

Also, I got hints here from a finite deck of clopen sets. but still, I can't write the solution thoroughly. Any help would be appreciated.