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Strongly Almost Disjoint Sets and Weakly Uniform Bases TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 352, Number 11, Pages 4971{4987 S 0002-9947(00)02599-X Article electronically published on June 20, 2000 STRONGLY ALMOST DISJOINT SETS AND WEAKLY UNIFORM BASES Z. T. BALOGH, S. W. DAVIS, W. JUST, S. SHELAH, AND P. J. SZEPTYCKI Abstract. 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 + @0 (2 > @2) is assumed (see [2]). On the other hand, if CH holds and the space has at most @! 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 fAα : g 0 α<κ can be refined to disjoint families by removing small sets Aα from each f g 2 τ Aα. Let us sayT that a family Aα : α<κ is point-<τ if for every I [κ] the intersection α2I 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 fAα : α<κg that are 0 \almost point-<τ"canberefinedtopoint-<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. Key words and phrases. GCH, 2, strongly almost disjoint families, weakly uniform base, point countable base. The first author's research was partially supported by NSF grant DMS-9623391. The third author's research was done during visits at Rutgers University and The Hebrew University, Jerusalem, which were supported by NSF grant DMS-9704477 and the Landau Center. The fourth author was partially supported by the Israel Basic Research Fund. This is publication number 674 in Shelah's publication list. c 2000 American Mathematical Society 4971 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4972 Z. T. BALOGH, S. W. DAVIS, W. JUST, S. SHELAH, AND P. J. SZEPTYCKI In Section 4 we show that the positive answer to the question of [2], and more, follows already from Theorem 2. 1. The closed continuous 2-chain axiom Definition 1. Let τ be a regular cardinal. A set M is τ-closed if [M]<τ ⊂ M. We say that a set M is weakly τ-closed if for every I 2 [M]τ there exists J 2 [I]τ such that [J]<τ ⊂ M.WesaythatM is weakly closed if M is weakly τ-closed for all regular τ (equivalently: for all regular τ ≤|Mj). An 2-chain is a sequence hMξ : ξ<αi such that Mξ ⊂ Mη and hMξ : ξ ≤ ηi2Mη+1 for all ξ<η<α. The Closed Continuous 2-Chain Axiom for λ and τ (abbreviated CECAτ (λ)) is the following statement: For every cardinal Θ >λ, and for every pair of sets A; B with jAj = λ, jBj < λ, there exists a continuous 2-chain hMξiξ<cf(λ) of weakly τ-closed elementary submodels of H(Θ) such that { jMξj <λfor every ξ<cf(λ); 2 { B MS 0;and ⊂ { A ξ<cf(λ) Mξ. The Closed Continuous 2-Chain Axiom for λ (abbreviated CECA(λ)) is ob- tained by requiring that the models Mξ in the definition of CECAτ (λ)areweakly closed. The Closed Continuous 2-Chain Axiom (abbreviated CECA) asserts that CECA(λ) holds for all uncountable cardinals λ. Lemma 1. Let λ be a regular uncountable cardinal. The following are equivalent: (a) λ<λ = λ. (b) For every set A with jA|≤λ there exists a weakly closed M such that A ⊂ M and jMj = λ. (c) There exists a weakly λ-closed M such that jMj = λ. Proof. (a) ) (b). Suppose λ<λ = λ,andjAj = λ. Build recursively an increasing h i ⊂ <λ ⊂ j j sequence Mξ : ξ<λS such that A M0,[Mξ] Mξ+1,and Mξ = λ for all ≤ 2 τ 2 <τ ξ<λ.LetM = ξ<λ Mξ.Ifτ λ is regular, I [Mξ] and K [I] ,then <τ <λ K 2 [Mξ] ⊆ [Mξ] for some ξ<λ, and hence K 2 Mξ+1 ⊆ M, as required. The implications (b) ) (c) and (c) ) (a) are obvious. Corollary 1. CECA ) GCH. Lemma 2. Assume GCH.Ifλ is an uncountable limit cardinal or if λ = κ+ for some regular infinite cardinal κ,thenCECA(λ) holds. Proof. Let λ be as above, let Θ >λ,andletA; B be such that jAj = λ, jBj <λ.In both of these cases we can construct a continuous 2-chain hMξiξ<cf(λ) of elementary submodels of H(Θ) such that { jMξj <λfor every ξ<cf(λ); <jM j { jMξ+1j is regular and [Mξ+1] ξ+1 ⊂ Mξ+1 for every ξ<cf(λ); 2 { B MS 0;and ⊂ { A ξ<cf(λ) Mξ. It remains to show that if δ<cf(λ) is limit, then Mδ is weakly closed. So τ assume δ<cf(λ) is limit, let τ be a regular cardinal, and let I 2 [Mδ] . If cf(δ) <τ, then there is ξ<τwith jMξ \ Ij = τ.Fixsuchξ,andlet <τ <τ <jMξ+1j J = I \ Mξ+1.Then[J] ⊂ [Mξ+1] ⊆ [Mξ+1] ⊂ Mξ+1 ⊂ Mδ. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use STRONGLY ALMOST DISJOINT SETS AND WEAKLY UNIFORM BASES 4973 S ≥ <τ ⊂ If cf(δ) τ,then[I] ξ<δ Mξ = Mδ,andwecantakeJ = I. Thus GCH implies that CECA(κ)holdsforallκ, except perhaps if κ = λ+ for a singular strong limit cardinal λ. Fortunately, it turns out that in this case CECA(κ)isequivalenttoaweakeningof2λ that has been extensively studied by the fourth author. To prove the equivalence, let us introduce some notation. We say that Pr(λ, τ) holds if there exists an increasing continuous chain N = h +i j j + Ni : i<λS such that Ni = λ for each i<λ ,eachNi is weakly τ-closed, and + ⊂ λ i<λ+ Ni. We say that Pr0(λ, τ) holds if there exists an increasing continuous chain N = h +i j j + Ni : i<λ such that Ni = λ for eachS i<λ ,eachNi is weakly τ-closed with + ⊂ respect to sets of ordinals, and λ i<λ+ Ni. λ + 0 S Note that if 2 = λ ,theninPr(λ, τ)andPr (λ, τ) we can demand that + i<λ+ Ni = H(λ ). In Definition 1.4 of [3], the following principle Sp(σ, λ) was introduced:1 There + + σ exists a sequence hPξ : ξ<λ i such that for all ξ<λ we have Pξ ⊂ [ξ] and jP |≤ + + ξ λ; moreover, if ξ<λ with σ = cf(ξ) and x Sis a cofinal subset of ξ of + f 2 g cardinality σ ,thenx canbewrittenintheformS x = xν : ν σ ,wherefor 2 σ ⊂ P each ν σ we have [xν ] η<ξ η. The ideal I[κ] was defined in [5] and [6]. We use here two equivalent definitions of I[κ] given in [7]. Definition 2. For a regular uncountable cardinal κ,letI[κ] be the family of all sets A ⊆ κ such that the set fδ 2 A : δ = cf(δ)g is not stationary in κ and for some hPα : α<κi we have: (a) Pα is a family of <κsubsets of α;and (b) for every limit α 2 A such thatScf(α) <αthere is x ⊂ α such that otp(x) < 8 \ 2 P α =supx and β<α(x β γ<α γ). The following characterization of I[κ] appears as Claim 1.2 in [7]. The abbrevi- ation nacc(C) stands for \nonaccumulation points of C" (in the order topology). Lemma 3. Let κ be a regular uncountable cardinal. Then D 2 I[κ] iff there exists asequencehCβ : β<κi such that: (a) Cβ is a closed subset of β; (b) if α 2 nacc(Cβ) then Cα = Cβ \ α; (c) for some club E ⊆ κ, for every δ 2 D \ E we have cf(δ) <δand δ =supCδ and otp(Cδ)=cf(δ);and (d) nacc(Cδ) is a set of successor ordinals. + It is clear from the above lemma that Jensen's principle 2λ implies that λ 2 I[λ+]. ≤ λ+ f + g For τ λ,letSτ = α<λ : cf(α)=τ .
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