Fibers of continuous real-valued functions on ψ-spaces

We consider continuous real-valued functions with domain either a psi-space (studied by S. Mrowka, J. Isbell, and others) or a generalized psi-space introduced by A. Dow and J. Vaughan. A cardinal kappa >= omega is called a rich cardinal provided for every infinite, maximal almost disjoint family M of countably infinite subsets of kappa (MADF) and for every continuous f : psi(kappa, M) -> R defined on the associated space psi = psi (kappa.M) there exists r is an element of R such that vertical bar f(-1)(r)vertical bar = vertical bar psi vertical bar. Dow and Vaughan proved that omega is a rich cardinal if and only if a = c, where a is the smallest cardinality of a MADF on omega. We prove that a = c if and only if, for all omega <= kappa <= c, kappa is a rich cardinal, if and only if for every n < omega, omega(n), is a rich cardinal. We prove every kappa > c is rich using a set-theoretic hypothesis weaker than GCH. (C) 2015 Published by Elsevier B.V.