Construction of certain maximal almost disjoint families of countable subsets of an arbitrary cardinal.
by
Jerry E Vaughan
University of North Carolina at Greensboro
Coauthors: Alan Dow

Let k ≥ w be a cardinal number and A a maximal almost disjoint family (MADF) of countably infinite subsets of k. Let y(k, A) denote the space whose underlying set is k∪A with that topology which has as a base all singletons {a} for a < k and all sets of the form {A}∪(A\F) where A ∈ A and F is finite. For the case k = w, y(w, A) is the well known space of S. Mrówka which he denoted N∪R. Mrówka constructed a R ⊂ [w]^w MADF such that |b(y(w, R))\y(w, R)|=1. We discuss our method of constructing M ⊂ [k]^w MADF of countable subsets of an arbitrary cardinal k so that M plays the analogous role for k as that played by Mrówka's MADF R for the countable case k = w. One major difference betwee the countable and uncountable case is that for k > c, the Stone-Cech compactification of y(k, M) (for any M) will have infinitely many points, but nevertheless contains a special free z-ultrafiler which retains some of the properties of the unique free z-ultrafilter on Mrówka's space y(w, R).

Date received: May 4, 2008