Strongly Almost Disjoint Sets and Weakly Uniform Bases

A combinatorial principle CECA is formulated and its equivalence with GCH + certain weakenings of 2λ for singular λ is proved. CECA is used to show that certain “almost point-< τ” families can be refined to point-< τ families by removing a small set from each member of the family. This theorem in turn is used to show the consistency of “every first countable T1-space with a weakly uniform base has a point-countable base.” This research was originally inspired by the following question of Heath and Lindgren [4]: Does every first countable Hausdorff space X with a weakly uniform base have a point-countable base? The answer to this question is negative if MA + (2א0 > א2) is assumed (see [2]). On the other hand, if CH holds and the space has at most אω isolated points, then the answer is positive (see [1]). The starting point of this paper was the observation that if in addition to GCH also the combinatorial principle 2λ holds for every singular cardinal λ, then no bound on the number of isolated points is needed. An analysis of the proof led to the formulation of a combinatorial principle CECA. It turns out that CECA is equivalent to GCH + some previously known weakenings of 2λ, but CECA has a different flavor than 2λ-principles and may be easier to work with. The equivalence will be shown in Section 1. In [3] it is shown that under certain conditions almost disjoint families {Aα : α < κ} can be refined to disjoint families by removing small sets Aα from each Aα. Let us say that a family {Aα : α < κ} is point-< τ if for every I ∈ [κ] the intersection ⋂ α∈I Aα is empty. In particular, a family is disjoint iff it is point-< 2. In Section 2 the main theorem of this paper (Theorem 2) is derived from CECA. Roughly speaking, Theorem 2 asserts that certain families {Aα : α < κ} that are “almost point-< τ” can be refined to point-< τ families by removing small sets Aα from each Aα. In Section 3, some related results for almost disjoint families are proved, and we explore how much of Theorem 2 can be derived from GCH alone rather than from CECA. Received by the editors March 17, 1998. 2000 Mathematics Subject Classification. Primary 03E05, 03E35, 03E75, 54D70.