Definition:
Suppose $\mathcal{A}$ and $\mathcal{B}$ are two covers of $X.$ We say that $\mathcal{B}$ refines $\mathcal{A}$ if each member of $\mathcal{B}$ is contained in some member of $\mathcal{A}.$ We say that $\mathcal{B}$ strictly refines $\mathcal{A}$ if each member of $\mathcal{B}$ is a proper subset of some member of $\mathcal{A}.$ if $\mathcal{B}$ strictly refines $\mathcal{A},$ we write $\mathcal{A} < \mathcal{B}.$
Here is the problem:
Let $X$ be a compact metric space that is totally disconnected, and let $\epsilon > 0.$
(a) Show that $X$ has a finite cover $\mathcal{A}$ clopen sets with diameter at most $\epsilon.$
( I am working on writing its details now.)
(b) Show that there is a clopen cover $\mathcal{B}$ such that $\mathcal{B}$ refines $\mathcal{A}$ and distinct numbers of $\mathcal{B}$ are disjoint.
My questions are:
1-I received a hint for the first part of (b) to consider the minimal (with respect to inclusion ) nonempty intersections of members of $\mathcal{A},$ which I am unable to implement its details, Could anyone help me in proving this , please?
2-For the second part of $(b)$ which is showing that distinct numbers of $\mathcal{B}$ are disjoint, I am stuck on this. could anyone help me in proving this please?
There is a simpler way to do b):
Just let $\mathcal{A} = \{A_1, A_2, \ldots A_n\}$.
Define $B_i = A_i \setminus \bigcup_{j < i} A_i$ for $1 \le i \le n$.
These form a clopen partition that refines $\mathcal{A}$. This idea works for countable covers too. It's a partition, because if $x \in X$, we can define $i(x)=\min \{i \in \{1,\ldots,n\}: x \in A_i\}$. Then $x \in B_{i(x)}$ and $x \in B_i$ iff $i=i(x)$.