Faithful zero-dimensional principal extensions
We prove that every topological dynamical system $(X,T)$ has a faithful zero-dimensional principal extension, i.e. a zero-dimensional extension $(Y,S)$ such that for every $S$-invariant measure $\nu $ on $Y$ the conditional entropy $h(\nu\,|\, X)$ is zero, and, in addition, every invariant measure on $X$ has exactly one preimage on $Y$. This is a strengthening of the authors' result in Acta Appl. Math. [to appear] (where the extension was principal, but not necessarily faithful).