TITLE: A zero-dimensional Hausdorff space is totally disconnected
QUESTION [12 upvotes]: The full question: A space is zero-dimensional if the clopen subsets form a basis for the topology. Show that a zero-dimensional Hausdorff space is totally disconnected. Recall a space is totally disconnected if the only connected subsets are singletons (one-point subsets).
Let X = {X1, X2, ...} be the set of clopen subsets of the space. We know that X is a basis for the topology T, so any open set in T can be written as a union or finite intersection of elements in X.
In a topological space, we know the union/finite intersection of open sets are also open (by definition) and the union/finite intersection of closed sets are closed, so any union/finite intersection of clopen sets is also clopen.
Since X is a basis, then any open set in T is also closed, since it will be the union/finite intersection of clopen sets. Does this mean our space is discrete? If it is discrete, then the only connected subsets are singletons, and then our space is totally disconnected.
I have a strong feeling I've gone in circles and my argument is incorrect (especially the discrete part...) Any help would be appreciated.

REPLY [6 votes]: Suppose $X$ is a zero dimensional Hausdorff space, and $\mathcal{B}$ a clopen basis for it.  
Suppose $A$ is a set containing points $x$ and $y$. If they are distinct we can find an open set containing $x$ not containing $y$, hence we can find a basis element $B \in \mathcal{B}$ containing $x$ but not $y$. The sets $A\cap B$ and $A\cap B^c$ are both open in $A$ and disjoint, and their union is $A$, hence $A$ is disconnected. 
Thus the space doesn't even need to be Hausdorff - the $T_1$ axiom is sufficient.