Def: A Tychonoff space $X$ is said to be strongly zero-dimensional if its Stone-Čech compactification $\beta X$ is totally disconnected (that is if the only connected subspaces of $\beta X$ are singletons).
I have heard that strong zero-dimensionality of $X$ provides that disjoint zero-sets are separated by clopen sets. My question is: why is this equivalent? Could you provide a source for proof or anything?
Thank you.
(By $\beta X$, we mean the Stone-Čech compactification of $X$).