Strongly almost disjoint sets and weakly uniform bases

Abstract

Abstract. A combinatorial principle CECA is formulated and its equivalence with GCH + certain weakenings of 2 λ 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 T 1 -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 + (2 ℵ0 > ℵ 2 ) is assumed (see [2]). On the other hand, if CH holds and the space has at most ℵ ω 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 2 λ 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 2 λ , but CECA has a different flavor than 2 λ -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 {A α : α < κ} can be refined to disjoint families by removing small sets A α from each A α . Let us say that a family {A α : α < κ} is point-< τ if for every I ∈ [κ] τ the intersection α∈I A α 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 {A α : α < κ} that are "almost point-< τ" can be refined to point-< τ families by removing small sets A α from each A α .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.

The mathematics of Zoltán T. Balogh

“…He retained an interest in such "base-multiplicity" topics throughout his professional life. Probably his most interesting work in this area are the results (with S. Davis, W. Just, S. Shelah, and P. Szeptycki) in [BDJSS00]. The primary motivation for the results in this paper is an old (circa 1976) question of R. Heath and W. Lindgren: Does every first-countable space with a weakly uniform base have a (possibly different) point-countable base?…”

“…In [BDJSS00], the authors finish off the problem, obtaining a consistent positive answer to the Heath-Lindgren question, with no restriction on the number of isolated points. The deep set-theoretic combinatorial results they develop to prove this are sure to have many other applications.…”