A Property of Totally Disconnected Space On pg. 4 of this article on profinite groups is defined the following:

A topological space $X$ is said to be totally disconnected if each connected component of $X$ is a singleton.

Then the author remarks that a space $X$ is totally disconnected if and only if for any two distinct points in $X$, there is a closed and open subset of $X$ which contains one of the points but not the other.
I am unable to show this. One direction is trivial. But I am not able to conclude from the definition of total disconnectedness that one can "separate any two points by a disconnection."

The statement is false in general. Two points of $X$ are in the same quasicomponent if there is no clopen set in $X$ containing exactly one of them. Thus, the quasicomponents are singletons if for any two points there is a clopen set containing exactly one of them. The component of a point is always a subset of the quasicomponent of the point, but the two need not be equal. They are always equal in compact Hausdorff spaces; this answer and this one contain proofs. In this answer I gave an example of a space in which they are not equal.
