TITLE: Property between totally disconnected and zero dimensional
QUESTION [13 upvotes]: Assume all spaces are Hausdorff.  
Definitions:
$X$ is totally disconnected if the only nonempty connected subsets of $X$ are singletons.
$X$ is zero dimensional if it has a base of clopen sets.
Now consider the following:

A) $X$ is zero dimensional,
B) for any $p\neq q\in X$ there exists a clopen set containing $p$
  missing $q$ (the quasicomponents of $X$ are singletons),
C) $X$ is totally disconnected (the components of $X$ are singletons)

It is clear to me that $A\Rightarrow B\Rightarrow C$, and I know that $A$ and $C$ are not equivalent.
Question: Is $B$ equivalent to either $A$ or $C$ ?  I think that $B\Leftrightarrow C$ if $X$ is compact since in this case quasicomponents are connected, but I'm not sure in general.
Edit: I am really interested in knowing if $B$ is strictly weaker than $A$, even for normal spaces.

REPLY [5 votes]: No two of the three properties are equivalent for Hausdorff spaces in general. In this answer Martin Sleziak gives a proof that the Erdős space $\ell^2\cap\Bbb Q^\omega$, the space of square-summable sequence of rational numbers, has (B) but not (A). The space obtained by removing the dispersion point $p=\left\langle\frac12,\frac12\right\rangle$ from the Knaster-Kuratowski fan has (C) but not (B): its quasicomponents are the segments $L_c\setminus\{p\}$ for $c\in C$, the middle-thirds Cantor set.