Let $X$ be a topological space and assume $X$ has a base $\mathcal{B}$ of clopen sets. Show $X$ is completely regular and a $T_{0}$ space.
My try:
First it is not hard to show that if $B \subset X$ then $\chi_{B}$, the characteristic function of $B$ is cts iff $B$ is clopen.
So let $F \subset X$ be a closed set and let $x \in X \setminus F$. Then since $X \setminus $ is open we can find $B \in \mathcal{B}$ such that $x \in B \subseteq X \setminus F$. Now define $\phi: X \rightarrow [0,1]$ by $\phi(x)= \chi_{B}(x)$ then since $B$ is clopen $\phi$ is a continuous map, $\phi(F)=\{0\}$ and $\phi(x)=1$, therefore $X$ is completely regular.
EDIT:
Sorry, Brian Scott is right, I'm trying to prove the following, if $X$ is $T_{0}$ and has a base of clopen sets then $X$ is completely regular and $T_{1}$. So I think the above proof is correct (i.e showing it is completely regular), how to show it is $T_{1}$?