Fibers of continuous real-valued functions on ψ-spaces

We consider continuous real-valued functions with domain either a ψ-space (studied by S. Mrowka, J. Isbell, and others) or a generalized ψ  -space introduced by A. Dow and J. Vaughan. A cardinal κ≥ωκ≥ω is called a rich cardinal provided for every infinite, maximal almost disjoint family MM of countably infinite subsets of κ   (MADF) and for every continuous f:ψ(κ,M)→Rf:ψ(κ,M)→R defined on the associated space ψ=ψ(κ.M)ψ=ψ(κ.M), there exists r∈Rr∈R such that |f−1(r)|=|ψ||f−1(r)|=|ψ|. Dow and Vaughan proved that ω   is a rich cardinal if and only if a=ca=c, where aa is the smallest cardinality of a MADF on ω  . We prove that a=ca=c if and only if, for all ω≤κ≤cω≤κ≤c, κ   is a rich cardinal, if and only if for every n<ωn<ω, ωnωn is a rich cardinal. We prove every κ>cκ>c is rich using a set-theoretic hypothesis weaker than GCH.