Mrowka maximal almost disjoint families for uncountable cardinals

摘要

We consider generalizations of a well-known class of spaces, called by S. Mrowka, N U R, where R is an infinite maximal almost disjoint family (MADF) of countable subsets of the natural numbers N. We denote these generalizations by ψ = ψ(k, R) for k ≥ ω. Mrowka proved the interesting theorem that there exists an TZ such that |βψ/(ω, R) ψ(ω, R)| = 1. In other words there is a unique free z-ultrafilter p_0 on the space ψ. We extend this result of Mrowka to uncountable cardinals. We show that for k ≤ c, Mrowka's MADF 7? can be used to produce a MADF M is contained in [k]~ω such that |βψ(k,M) ψ (k,M)| = 1. For k > c, and every M is contained in [k]~ω, it is always the case that |βψ(k, M) ψ(k, M)| ≠1,yet there exists a special free z-ultrafilter p on ψ(k,M) retaining some of the properties of p_0. In particular both p and p_0 have a clopen local base in βψ (although βψ(k,M) need not be zero-dimensional). A result for k > c, that does not apply to po, is that for certain k > c, p is a P-point in βψ.