Mrówka Maximal Almost Disjoint Families for Uncountable Cardinals

We consider generalizations of a well-known class of spaces, called by S. Mrówka, N ∪R, where R is an infinite maximal almost disjoint family (MADF) of countable subsets of the natural numbersN . We denote these generalizations by ψ = ψ(κ,R) for κ ≥ ω. Mrówka proved the interesting theorem that there exists an R such that |βψ(ω,R) \ ψ(ω,R)| = 1. In other words there is a unique free z-ultrafilter p0 on the space ψ. We extend this result of Mrówka to uncountable cardinals. We show that for κ ≤ c, Mrówka’s MADF R can be used to produce a MADF M ⊂ [κ] such that |βψ(κ,M) \ ψ(κ,M)| = 1. For κ > c, and every M ⊂ [κ], it is always the case that |βψ(κ,M) \ ψ(κ,M)| 6= 1, yet there exists a special free z-ultrafilter p on ψ(κ,M) retaining some of the properties of p0. In particular both p and p0 have a clopen local base in βψ (although βψ(κ,M) need not be zero dimensional). A result for κ > c, that does not apply to p0, is that for certain κ > c, p is a P-point in βψ.