I have a problem trying to prove the following theorem.
If $X$ is a $T_{0}$ space with a base $C$ of clopen sets, then X is totally disconnected.
By definition, a space is totally disconnected if and only if $C(x) = {x} \quad \forall x \in X$. I try to prove it by contradiction, suposing that $\exists y \in C(x) , x \neq y $ and trying to use the $T_{0}$ property for finding a contradiction, but I am stucked.
I haven't found any solution to the problem in the web, so I would aprecciate if somebody could help me.
Regards.
HINT: If $x\ne y$, then either there is an open $U$ such that $x\in U$ and $y\notin U$, or there is an open $U$ such that $y\in U$ and $x\notin U$. Without loss of generality assume that there is an open $U$ such that $x\in U$ and $y\notin U$. Since $X$ has a clopen base, there is a clopen $B$ such that $x\in B\subseteq U$. Use $B$ to show that $y\notin C(x)$.