Zero sets in Mrówka spaces For a maximal almost disjoint family $\mathcal A$ of subsets of $\omega$ we choose a set $\{x_A:A\in\mathcal A\}$ of distinct points not in $\omega$ and define $\Psi (\mathcal A)=\omega\cup \{x_A:A\in \mathcal A\}$.  
A basis for a topology on $\Psi(\mathcal{A}$) is given by 
$$\{ \{n \}:n\in\omega\}\cup \{\{x_A\}\cup (A\setminus F):A\in\mathcal A, F\subseteq \omega\text{ is finite}\}.$$
Then $\Psi (\mathcal A)$ is 


*

*locally compact Hausdorff

*completely regular

*first countable

*separable

*zero dimensional

*pseudocompact


It is not countably compact as $\{x_A:A\in\mathcal A\}$ is an infinite closed discrete subspace. As normal plus pseudocompact implies countably compact, $\Psi (\mathcal A)$ also fails to be normal.
There is a theorem which says any compact metric space without isolated points is homeomorphic to the Stone-Čech remainder of some $\Psi (\mathcal A)$.  In particular the Stone-Čech remainder of $\Psi (\mathcal A)$ can be metric.  This means in some cases $\{x_A:A\in\mathcal A\}$, if it is a zero set, is not C$^*$-embedded in $\Psi (\mathcal A)$ since otherwise the Stone-Čech remainder would contain a copy of $\omega ^*$.  
For an arbitrary m.a.d. family $\mathcal A$ is there a nice way to characterize the zero sets of $\Psi (\mathcal A)$?  
Can you give an example of a closed subset of $\Psi (\mathcal A)$ which is not a zero set?
Can you prove $\{x_A:A\in\mathcal A\}$ is a zero set?
 A: Actually, $\{ x_A : A \in \mathcal{A} \}$ is always a zero-set in $\Psi(\mathcal{A})$:  define the function $f : \Psi(\mathcal{A}) \to [0,1]$ by $$f(n) = \tfrac{1}{n+1} \\ f(x_A) = 0.$$ This is clearly a continuous function (any neighbourhood of $0$ contains all but finitely many of the $\frac{1}{n+1}$, and so it's inverse image under $f$ includes a cofinite subset of $A$ for each $A \in \mathcal{A}$).
I am really unaware of any characterisation of the zero-sets in $\Psi(\mathcal{A})$.  The following are either easy or can be found in the literature:


*

*All finite subsets of $\{ x_A : A \in \mathcal{A} \}$ are zero-sets in $\Psi(\mathcal{A})$. (For $A \in \mathcal{A}$ define $f_A : \Psi(A) \to [0,1]$ by $$
f_A(x) = \begin{cases}
\tfrac{1}{n+1}, &\text{if }x \in A; \\
0, &\text{if }x=x_A; \\
1, &\text{otherwise.}\end{cases}$$
The almost-disjointness of $\mathcal{A}$ helps showing that each $f_A$ is continuous.)

*No countably infinite subset of $\{ x_A : A \in \mathcal{A} \}$ can be a zero-set in $\Psi(\mathcal{A})$. (Follows from the pseudocompactness of $\Psi(\mathcal{A})$ and the fact that no countably infinite subset of $\{ x_A : A \in \mathcal{A} \}$ is compact.) (This is problem 5I(6) (p.79) in Gillman & Jerison, Rings of Continuous Functions.)

*There is a mad family $\mathcal{A}$ such that an infinite $E \subseteq \{ x_A : A \in \mathcal{A} \}$ is a zero-set in $\Psi(\mathcal{A})$ iff it is co-countable. (Theorem 3.11 in Mrówka, Some set-theoretic constructions in topology, Fund. Math., vol.94 (1977) pp.83-92, link.)
