I saw people wrote that a zero-dimensional space $X$ is Tychonoff since the characteristic function of a clopen set is continuous.
The confusion is that to show that X is Tychonoff we need to show that for every closed set $C$ and for every point $d$ outside $C$, there exists a continuous function that separates $C$ and $d$. Of course if $C$ is clopen, then it is very easy. But not every closed set in a zero-dimensional space is clopen.
So how does showing that the characteristic function of a clopen set is continuous prove that $X$ is Tychonoff? Thanks.