Jerry E. Vaughan and Catherine Payne

Abstract: We consider $\psi$-spaces constructed from an infinite cardinal $\kappa$ and a maximal almost disjoint family $\mathcal M\subset [\kappa]^\omega$ of countably infinite subsets of $\kappa$. A cardinal $\kappa$ is called a rich cardinal provided for every $\mathcal M\subset [\kappa]^\omega$ and for every continuous $f:\psi(\kappa,\mathcal M)\rightarrow \mathbb R$, there exists $r\in \mathbb R$ such that $|f^{-1}(r)|=|\psi|$. A. Dow and J. Vaughan proved that $\omega$ is a rich cardinal if and only if $\mathfrak a=\mathfrak c$, where $\mathfrak a$ is the smallest cardinality of a MADF on $\omega$. We extend this result for cardinals greater than $\omega$. We prove $\mathfrak a=\mathfrak c$ iff ``every $\kappa\leq\mathcal \mathfrak c$ is rich" iff ``$(\forall n<\omega$) $\omega_n$ is rich." Under assumptions weaker than GCH, we prove every $\kappa>\mathfrak c$ is a rich cardinal.