I'm interested in the proof that space of endpoints of the Lelek fan is almost zero-dimensional. As much is claimed in On homogeneous totally disconnected 1-dimensional spaces by Kawamura, Oversteegen, Tymchatyn, where they say that the proof is easy and reference A Topological Characterization of R-Trees by Mayer, Oversteegen in which no such thing is proved.
Some context: A space $X$ is almost zero-dimensional if there exists a basis $\mathcal B$ of open sets such that for all $U \in \mathcal B$ the complement of the closure of $B$ is a union of clopen sets. The Lelek fan $L$ is the unique smooth fan with a dense set $E(L)$ of endpoints. Consider $E(L)$ with the induced topology.
I'd like a description of a basis which witnesses the almost zero-dimensionality of $E(L)$.