Clopen subsets of a compact metric space

Solution 1:

It seems that we can show the claim as follows. It is well known and easy to prove that each metrizable compact space $X$ has a countable base. Fix such a base $\mathcal B$. Let $U$ be a clopen (that is, closed and open) subset of $X$. Each point $x\in U$ has a neighborhood $U_x\in\mathcal B$ such that $U_x\subset U$. Since $U$ is compact, there exists a finite subset $Y$ of $U$ such that $U=\bigcup\{U_x:x\in Y\}$. Hence the cardinality of the family of all clopen subsets of $X$ is not greater than the cardinality of the family of all finite subsets of $\mathcal B$, which is countable.