TITLE: Any two points in a Stone space can be disconnected by clopen sets
QUESTION [28 upvotes]: Let $B$ be a Stone space (compact, Hausdorff, and totally disconnected).  Then I am basically certain (because of Stone's representation theorem) that if $a, b \in B$ are two distinct points in $B$, then $B$ can be written as a disjoint union $U \cup V$ of open sets where $a \in U, b \in V$.  
However, I can't seem to prove this directly.  The proof should be fairly straightforward, so I am sure I'm missing something obvious.  (As an exercise for myself, I'm trying to prove Stone's representation theorem, and I need this as a lemma.)

REPLY [13 votes]: The point of this answer is mostly to provide some terminology with which to express the distinction here.  
A topological space is totally disconnected if the only nonempty connected subsets are the singleton sets.  Equivalently, the connected components are all singleton sets.
A topological space is zero-dimensional if it admits a base of clopen [i.e., both open and closed] sets.
A topological space is separated (more traditionally: $T_1$) if all the singleton sets are closed.
First easy observation: A totally disconnected space is separated.  
(Proof: Otherwise the closure of a point would give a larger connected component.)
Second easy observation: In a separated zero-dimensional space $X$, for any two distinct points $x_1$, $x_2$, there exists a separation $X = U_1 \coprod U_2$ with $x_i \in U_i$. 
(In the terminology of Nuno's answer -- which is relatively standard if not well-known -- the conclusion is that the quasi-components are singleton sets.) In particular, $X$ is totally disconnected. 
(Proof: Let $x_1$, $x_2$ be distinct points in $X$.  Since $X$ is separated, $N = X \setminus \{x_2\}$ is open.  By definition of a base, there exists a clopen set $U_1$ with $x_1 \in U_1 \subset N$ and then $U_1$, $U_2 = X \setminus U_1$ is the desired separation.)
By contrast, we have the following
Nontrivial result: A locally compact Hausdorff space is zero-dimensional iff it is totally disconnected.  For instance this wikipedia article gives a reference.
Note that Nuno's answer (which should be the accepted one, IMO) gives a proof of this in the compact case.  
Perhaps a cleaner approach is to "redefine" a Stone space to be a space which is compact, Hausdorff and zero-dimensional.  Then when you are given a space that someone says is a Stone space, the first thing you'll do is look for a base of clopen sets, which tends to be a good idea anyway.  For instance, given any profinite space (i.e., an inverse limit of finite discrete spaces, yet another term for the same class of spaces!) it is very easy to do this.