Generalizations of almost disjointness, c-sets, and the baire number of βN-N

AbstractWe consider certain combinatorial problems and their consequences in βN-N. Let F be any family of infinite subsets of N. Then we are interested in conditions under which it is possible to find an almost disjoint family A = {AF:FϵF} such that Aϵ⊆F. If such a family A exists, we call F separable. We note that any family of cardinality less than c is separable, but that it is independent of the negation of the Continuum Hypothesis as to whether or not a union of ℵ1 separable families is separable. We consider the separability of unions of ultrafilters and use our results to show that it is consistent that if S is any nowhere dense subset of βN-N, then there exists a family of c pairwise disjoint open sets each of which is disjoint from S but contains S in its closure.We also consider rectangular arrays of subsets of N subject to the condition that each row and each column is an almost disjoint family, and we consider various maximality questions concerning these. We then note that if there exists such an array with K rows and K+ columns and the columns are all maximal, then βN-N may be covered by K+ nowhere dense sets