Is the space $\psi$ (described in problem 5I of L. Gillman and M. Jerison, Rings of continuous functions, Springer Verlag, 1976) a F-Z-space (i.e, space with cl(X-Z(f)) is a zero set for every f in C(X))?

Is the space $\psi$ (described in problem 5I of L. Gillman and M. Jerison, Rings of continuous functions, Springer Verlag, 1976) a F-Z-space (i.e, space with $cl(X-Z(f))$ is a zero set for every $f$ in $C(X)$)?

$\textbf{Clarification}$ A collection $\mathcal{A}$ of infinite subsets of $\mathbb{N}$ is said to be an almost disjoint family if $R\cap S$ is finite whenever $R,S\in\mathcal{A}$ and $R\neq S$.

Let $\mathcal{E}$ be a maximal almost disjoint family of subsets of $\mathbb{N}$. Let $D=(w_{E})_{E\in\mathcal{E}}$ be a set of new distinct points. Let $\psi=\mathbb{N}\cup D$. We shall give $\psi$ the topology as follows. The points in $\mathbb{N}$ are isolated. The neighborhood filter around a point $w_{E}$ is generated by sets of the form $w_{E}\cup R$ where $R\subseteq E$ and $E\setminus R$ is a finite set. In this space, $w_{E}\cup E$ is the one-point-compactification of the space $E$.

The problem in the book states that the space $\psi$ is completely regular, pseudocompact, but not countably compact. In particular, $\psi$ is not realcompact and $\psi$ is not normal. Furthermore, every subspace of $\psi$ is a $G_{\delta}$-set.