Here is the question:
If $X$ is a compact metric space that is totally disconnected, then for each $r > 0$ and each $x \in X,$ there is a clopen set $U$ such that $x \in U$ and $U \subseteq B_{r}(x).$
Definition:
A topological space $X$ is totally disconnected if for any two distinct points $x,y \in X,$ there is a separation $X = U \cup V$ of $X$ with $x \in U $ and $y \in V.$
Definition:
A subset of a topological space is clopen if it is both closed and open.
**My questions are: **
1-I was given a hint to show that $X-B_{r}(x)$ is compact, which I do not know how to show, so any help in that direction will be appreciated.
2- How to show the existence of such clopen set $U$?
Best Answer
For 1, $X - B_r(x)$ is the complement of an open set $B_r(x)$, which makes it closed. Since it is a closed subspace of a compact space $X$, it is compact too.
For 2, for any point $y \in X - B_r(x)$, you can use total disconnectedness to find disjoint open sets $U_y$ and $V_y$ such that $X = U_y \cup V_y$, and $x \in U_y$, with $y \in V_y$. These sets are complementary, and since both are open, they are both closed as well. Note that the $V_y$s cover every point $y \in X - B_r(x)$, which is compact, so a finite subcover must exist...
Are you able to finish it from there?