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 + ℵ0 (2 > 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 : } 0 α<κ can be refined to disjoint families by removing small sets Aα from each { } ∈ τ Aα. Let us sayT 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 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 ∈-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 ∈ [M]τ there exists J ∈ [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 ≤|M|). An ∈-chain is a sequence hMξ : ξ<αi such that Mξ ⊂ Mη and hMξ : ξ ≤ ηi∈Mη+1 for all ξ<η<α. The Closed Continuous ∈-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 |A| = λ, |B| < λ, there exists a continuous ∈-chain hMξiξ – |Mξ| <λfor every ξ – |Mξ| <λfor every ξ 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 | | + 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 | | + 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 |P |≤ + + ξ λ; moreover, if ξ<λ with σ = cf(ξ) and x Sis a cofinal subset of ξ of + { ∈ } cardinality σ ,thenx canbewrittenintheformS x = xν : ν σ ,wherefor ∈ 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 {δ ∈ A : δ = cf(δ)} 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 ∈ A such thatScf(α) <αthere is x ⊂ α such that otp(x) < ∀ ∩ ∈ 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 ∈ I[κ] iff there exists asequencehCβ : β<κi such that: (a) Cβ is a closed subset of β; (b) if α ∈ nacc(Cβ) then Cα = Cβ ∩ α; (c) for some club E ⊆ κ, for every δ ∈ 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 ∈ I[λ+]. ≤ λ+ { + } For τ λ,letSτ = α<λ : cf(α)=τ . Theorem 1. Let τ<λbe infinite cardinals with τ regular and λ singular strong limit. The following are equivalent: λ+ ∈ + (a) Sτ I[λ ]; (b) Pr(λ, τ); 1Actually, the principle introduced in [3] is more general and contains an extra parameter τ, but the case τ = σ+ is most relevant for the results in [3] and fits most neatly into the framework of the present paper. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4974 Z. T. BALOGH, S. W. DAVIS, W. JUST, S. SHELAH, AND P. J. SZEPTYCKI (c) Pr0(λ, τ); + (d) CECAτ (λ ). Moreover, if τ = σ+, then each of the above is also equivalent to: (e) Sp(σ, λ). Proof. The implications from (b) to (c) and from (d) to (b) are obvious + + 0 (c) ⇒ (d). Let θ>λ ,andletN = hNi : i<λ i exemplify Pr (λ, τ). Let + M = hMξ : ξ<λ i be an increasing continuous sequence such that N ∈ M0,and for all ξ<λ+ we have: ∗ ∗ (1) Mξ ≺hH(Θ),< i (where < is some wellorder relation on H(Θ)); (2) |Mξ| = λ;and (3) M|(ξ +1)∈ M . S ξ+1 ∈ Let M = ξ<λ+ Mξ. Note that (3) implies in particular that ξ Mξ+1,and + + hence λ ⊂ M. Fix a bijection h : M → λ such that h|Mξ ∈ Mξ+1 for all ξ. ∗ Such h can be found by conditions (1) and (3): Recursively, let h|Mξ+1 be the < - + smallest bijection from Mξ+1 onto an ordinal that extends h|Mξ. For every ξ<λ , + + let η(ξ) be the smallest ordinal η ≥ ξ such that h[Mη]=η = Mη ∩ λ = Nη ∩ λ . By (2), η(ξ) <λ+ for all ξ. Define E = {δ ∈ LIM ∩ λ+ : ∀ξ<δ(η(ξ) <δ)}. Then E is a closed unbounded subset of λ+ that consists of fixed points of the + function η.Let{δε : ε<λ } be the increasing continuous enumeration of E. h +i + We show that Mδε : ε<λ witnesses CECAτ (λ ). Let ε<λand let ∈ τ ∈ τ ∩ + ∩ + I [Mδε ] . Notethatwehaveh[I] [δε] and Mδε λ = Nδε λ .Sothereis ⊆ | | <τ ⊆ ∩ J h[I]with J = τ and [J] Nδε ON. We distinguish two cases: 0 τ 0 Case 1: There exists J ∈ [J] with sup J <δε. 0 + Let α = η(sup J ). Then α<δε, Mα ∩ λ = α = h[Mα], and Mα ∩ λ = + 0 0 <τ Nα ∩ λ . Thus, by shrinking J if necessary, we may assume that [J ] ⊂ Nα. Let I0 = h−1J 0.ThenI0 ∈ [I]τ .Moreover,ifK ∈ [I0]<τ ,thenK = h−1L 0 <τ for some L ∈ [J ] .SinceL, h|Mα ∈ Mα+1 and L ⊂ Mα+1, it follows that ∈ ⊂ K Mα+1 Mδε , as desired. Case 2: There is no J 0 as in Case 1. 0 −1 Then let I = h J. Note that we must have otp(J)=τ and sup J = δε.Thus,if 0 <τ −1 + K ∈ [I ] ,thenK = h L,whereL ∈ Nα for some α<δε with Nα = Mα∩H(λ ). Thus, arguing as in the previous case, one can show that K ∈ Mα+1 and hence 0 <τ ⊂ [I ] Mδε . + 0 (c) ⇒ (a). Let N = hNα : α<λ i be a sequence that witnesses Pr (λ, τ). By + thinning out the chain if necessary, we may assume that α ⊂ Nα for each α<λ . + For each α,letPα = Nα ∩P(α). We claim that the sequence hPα : α<λ i λ+ ∈ + witnesses that Sτ I[λ ]. Condition (a) of Definition 2 is obvious. To verify that ∈ λ+ \{ } (b) also holds, let α Sτ τ . Pick a subset I of α of order type τ such that τ τ <τ sup I = α.ThenI ∈ [Nα ∩ ON] ,andthereexistsJ ∈ [I] such that [J] ⊂ Nα. Since α isS a limit ordinal and the sequence N is continuous, the latter implies that <τ ⊂ P [J] β<α β. On the other hand, regularity of τ implies that otp(J)=τ<α and for every β<αthe set J ∩ β has cardinality <τ. Thus condition (b) holds, λ+ ∈ + andwehaveshownthatSτ I[λ ]. (a) ⇒ (d). We will actually prove something slightly more general which will allow us to show that the M sequence works for several τ’s simultaneously. Assume 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 4975 λ+ ⊂ ∈ + h i ∈ + Sτ D I[λ ]. Let Cα : α<λ and E be witnesses that D I[λ ]asin Lemma 3. Let µ = cf(λ), and let (κi)i<µ be a sequence of cardinals with supremum κi λ andsuchthatmax{µ, τ} <κ0 and 2 ≤ κi+1 for all i<µ.LetA, B be as in + + the assumptions of CECAτ (λ ), and let Θ >λ . Recursively construct a double h i + i sequence Mα : α<λS ,i < µ such that conditions 1–8 below are satisfied. For + i α<λ ,letMα = i<µ Mα. 1. Mα ≺ H(Θ); ⊂ 2. B SM0; ⊂ 3. A α<λ+ Mα; + 4. the sequence hMα : α<λ i is continuous and increasing; 5. hMβ : β ≤ αi∈Mα+1; | i | i ⊂ j 6. Mα = κi and Mα Mα for i are already covered by conditions 1–6. If αξ = β +1 for some β,thenMαξ is weakly τ-closed by condition 7. So consider the case when αξ is a limit ordinal, and let ∈ τ I [Mαξ ] . We distinguish three cases: 6 Case 1: cf(αξ)=τ = µ. S ∈ τ ⊂ i Then there exist J [I] and i<µsuch that J β∈C Mβ. We will show S αξ <τ ⊂ ∈ <τ ⊂ i ∈ that [J] Mαξ .LetK [J] .ThenK ∈ ∩ Mβ for some γ Cαξ . β Cαξ γ ∈ ∩ Without loss of generality, we may assume that γ naccCαξ .ThenCγ = Cαξ γ. | | | | ∈ Since Cαξ = τ<λ,also Cγ <λ. By condition 8, K Mγ+1, and hence ∈ K Mαξ , as required. Case 2: cf(αξ)=τ = µ. <τ ⊂ ∈ <τ ⊂ i We will show that [I] Mαξ .IfK [I] ,thenK Mβ+1 for some β<αξ. ∈ i+1 ⊂ By conditions 6 and 7, K Mβ+1 Mαξ . Case 3: cf(αξ) =6 τ. Then there exists β<αξ such that |I ∩ Mβ+1| = τ.SinceMβ+1 is weakly τ- ∈ ∩ τ <τ ⊂ closed and contained in Mαξ ,thereexistsJ [I Mβ+1] such that [J] Mαξ . ⇒ + h i + i (a) (e). Assume that τ = σ ,let Mα : α<λ,i < µ be as in the previous part of the proof, i.e., such that conditions 1–8 hold, and let the sequence + + σ hαξ : ξ<λ i be defined as above. For each ξ<λ ,letPξ =[ξ] ∩ Mξ+1. Now suppose that cf(ξ)=τ,andletx ⊂ ξ be cofinal of cardinality τ.We distinguish two cases: Case 1: σ<µ. Then we can let x = x for all ν<σ.Eachy ∈ [x]σ is contained in M i for ν S αη+1 σ ⊂ P some i<µand η<ξ.Thus[x] η<ξ η, as required. Case 2: σ ≥ µ. S ∩ ν ∅ ≥ ∈ σ Then let xν = x β∈C Mβ for ν<µand xν = for ν µ.Ify [xν ] , S αξ ⊂ ∩ ∈ ∈P then y γ ∩ for some γ naccCαξ , and hence y η for some η<ξwith γ Cαξ γ<αη. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4976 Z. T. BALOGH, S. W. DAVIS, W. JUST, S. SHELAH, AND P. J. SZEPTYCKI + + (e) ⇒ (a). Assume τ = σ ,andlethPξ : ξ<λ i witness that Sp(σ, λ)holds. For each ξ let Rξ be the union of Pξ and the set of initial segments of elements of P ∈ λ+ \{ } ⊂ ξ.LetS ξ Sτ τ .Nowifx ξ is any cofinal subset of ξ of order typeSτ,then R x = ν∈σ xν , where the family of all initial segments of xν is contained in η<ξ η for each ν<σ. At least one of these xν ’s must be cofinal in ξ and of order type τ, and this xν is exactly as required in Definition 2. Note that ∀τ ∈ Reg CECAτ (κ) does not always imply CECA(κ). For example, ℵn if 2 = ℵn+2 for all n ∈ ω, then there are no weakly closed models of cardinality ℵn for n ∈ (0,ω), and thus CECA(ℵω) fails. But if τ = ℵn,thenforeachm ≥ n+1 ℵ ℵ ℵ there are plenty of n-closed models of cardinality m, and thus CECAℵn ( ω) holds. This anomaly cannot happen if κ is a successor cardinal: If κ = λ+ for aregularλ,thenCECAλ(κ) implies the existence of weakly λ-closed models of cardinality λ, which in turn implies that λ<λ. Now the proof of Lemma 2 can be adapted to derive CECA(κ). If κ is the successor of a singular limit cardinal λ and τ τ<λ,thenCECAτ + (κ)impliesthat2 ≤ λ. Since for cofinally many τ we also will have τ>cf(λ), K¨onig’s Theorem implies that 2τ is strictly less than λ.In + other words, if CECAτ (λ ) holds and λ is singular, then λ must be a strong limit cardinal. For such λ we have the following corollary to the proof of Theorem 1. Corollary 2. Let λ be a singular strong limit cardinal. Then the following are equivalent: (a) λ+ ∈ I[λ+]; λ+ ∈ + (b) Sτ I[λ ] for all regular τ<λ; (c) CECA(λ+); + (d) CECAτ (λ ) holds for all regular τ ∈ λ. Proof. The equivalence between (a) and (b) follows from Shelah’s observation that I[λ+] is a normal, and hence λ+-complete, ideal (see [5], [6], or [7]). The equivalence between (b) and (d) was established in Theorem 2. The implication (c) ⇒ (d) is obvious. To see that (a) implies (c), note that the last part of the proof of Theorem 2 + ∈ + h +i shows that if λ I[λ ], then the sequence Mαξ : ξ<λ constructed from + + D = λ witnesses CECAτ (λ ) simultaneously for all τ ∈ λ ∩ Reg. 2. The main theorem Let τ be a cardinal. Recall that a sequence of sets hAαiα<κ is said to be point- <τat a point a if |{α : a ∈ Aα}| <τ. We will say that a sequence of sets is h i point-<τ if it is point-<τ at everyT point. Notice that Aα α<κ is point <τ if for ∈ τ every I [κ] the intersection α∈I Aα is empty. Theorem 2. Assume CECA. Suppose that σ, τ are regular infinite cardinals, and let hAαiα<κ be a sequence of (not necessarily distinct) sets such that one of the following conditions is satisfied: T ∈ τ ∈ <τ | | (1.1) For every I [κ] there is J [I] such that Tα∈J Aα <σ. ∈ τ ∈ <τ | |≤ (1.2) For everyS I [κ] there is a J [I] such that α∈J Aα σ and for each ∈ σ |{ ⊂ }| ≤ S [ α<κ Aα] we have α : S Aα σ. h 0 i | 0 |≤ Then there exist Aα α<κ such that Aα σ for each α<κand the sequence h \ 0 i Aα Aα α<κ is point-<τ. 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 4977 Proof. Suppose the theorem is false, let κ be the smallest cardinal for which the theorem fails, and fix a counterexample A = hAαiα<κ and σ, τ that witness this fact.S Throughout the proof, let Θ denote a “sufficiently large” cardinal, and let A = α<κ Aα. In the proof we will consider all possible ways in which κ, σ, τ can be related to each other, and we will derive a contradiction in each case. To begin with, note that we may without loss of generality assume that τ ≤ κ;otherwise the conclusion of the theorem is vacuously true. Now let us eliminate the case σ ≥ κ. Lemma 4. Suppose that M ≺ H(Θ) is weakly τ-closed with respect to sets of or- dinals, i.e., for every I ∈ [M ∩ κ]τ there exists J ∈ [I]τ with [J]<τ ⊂ M. Moreover, suppose A ∈ M and σ ⊂ M. Then the sequence hAα \ Miα∈κ∩M is point-<τ. Corollary 3. σ<κ. Proof. Let M ≺ H(Θ) be weakly closed and such that σ ⊂ M, |M| = σ,and 0 ≥ M contains everything relevant. For each α<κ,letAα = M.Ifσ κ,then κ ∩ M = κ, and Lemma 4 implies the conclusion of Theorem 2. Proof of Lemma 4. SupposeT towards a contradiction that there are a and I ∈ ∩ τ ∈ \ [κ M] such that a α∈I Aα M.SinceM is weakly τ-closed with respect to sets of ordinals,2 by passing to a subset of I if necessary, we may assume that [I]<τ ⊂ M. Since (1.1) or (1.2) holds, we can find J ∈ [I]<τ such that \ (2) B = | Aα|≤σ. α∈J Since A, J ∈ M, it follows that B ∈ M.Sinceσ ⊂ M and |B|⊂σ, we conclude that B ⊂ M, which contradicts the assumption that a ∈ B \ M. Now let us reveal how CECA will be used in the remainder of this proof. Lemma 5. Suppose that there is a continuous increasing chain hMξiξ Proof. Let the Mξ’s and Aα’s be as in the assumption. By the choice of κ and (ii) ∈ ∩ \ 0 ∈ ≤σ we can pick, for every ξ<κand α κ Mξ+1 Mξ,anAα [Aα] in such a way ∩ ⊂ 0 h \ 0 i that Aα Mξ Aα and the sequence Aα Aα α∈κ∩(Mξ+1\Mξ) is point-<τ. h \ 0 i ∈ We claim thatS the sequence Aα Aα α<κ is point-<τ. To see this, let a A, ∗ and let M = ξ + 2If M has only the property that for every subset I ∈ [M ∩ κ]τ there exists J ∈ [I]τ with <τ [J] ⊂ M, then the argument presented here shows that the sequence hAα \ Miα∈κ∩M is point- <τ+. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4978 Z. T. BALOGH, S. W. DAVIS, W. JUST, S. SHELAH, AND P. J. SZEPTYCKI • By (i), |{α ∈ κ ∩ Mξ : a ∈ Aα}| <τ. • |{ ∈ ∩ \ ∈ \ 0 }| By construction, α κ (Mξ+1 Mξ): a Aα Aα <τ. • ∈ \ ∩ ⊂ 0 ∈ \ If α κ Mξ+1,thenAα Mξ+1 Aα, and hence no α κ Mξ+1 satisfies ∈ \ 0 a Aα Aα. h \ 0 i It follows that the sequence Aα Aα α<κ is point-<τ at a. Case 2: a/∈ M ∗. We show that M ∗ is weakly τ-closed with respect to sets of ordinals. Let I ∈ [M ∗ ∩ κ]τ .Ifτ = cf(κ), then [κ]<τ ⊂ M ∗ and hence [I]<τ ⊂ M ∗.Ifτ =6 cf(κ), then there exists ξ It follows from Lemma 6 that if τ<κ, then we can find a sequence hMξiξ 3Note that in this argument, as well as in the proof of the next lemma, only the following + consequence of weak closedness is used: If I ∈ [M]σ , then there is J ∈ [I]σ such that J ∈ M. 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 4979 Case 1: µ = ν +1forsomeν. σ Since Nν+1 is weakly closed, there exists S ∈ [Aα] ∩Nν+1.By(4),h(S) ⊂ Nν+1. Since α in h(S), we have α ∈ Nν+1 = Nµ, contradicting the choice of α. Case 2: µ is a limit ordinal. We will show that σ (5) there are ν<µand S ∈ [Aα] such that S ∈ Nν+1. | ∩ |≥ + ∈ σ ∩ Indeed, sinceSNν is weakly closed and Aα Nν σ ,wecanpickS [Aα] Nν . ∈ Since Nν = ν<µ Nν ,thereisν<µsuch that S Nν+1. Now we can derive a contradiction as in the previous case. 3. Related theorems and examples We first present a modification of Theorem 2 exhibiting when we can obtain point-<τ families where τ is finite. Note that a family of sets is point-< 2ifand only if it is pairwise disjoint. Theorem 3. Assume CECA,andletn ≥ 1. Suppose that σ is a regular cardinal, +n and let hAαiα<κ be a sequence of (not necessarily distinct) sets such that |Aα|≤σ and one of the following conditions is satisfied: | ∩ | (1.3) Aα Aβ <σ for all α<β<κ; S | ∩ |≤ ∈ σ (1.4) Aα Aβ σ for all α<β<κ,andforeachS [ α<κ Aα] |{α : S ⊂ Aα}| ≤ σ. h 0 i | 0 |≤ Then there exist Aα α<κ such that Aα σ for each α<κand the sequence h \ 0 i Aα Aα α<κ is point- License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4980 Z. T. BALOGH, S. W. DAVIS, W. JUST, S. SHELAH, AND P. J. SZEPTYCKI observe that, as above, if α 6∈ Mξ then Aα ∩ Mξ =6 ∅ and, in addition, if α, β ∈ Mξ then Aα ∩ Aβ ⊆ Mξ. Now suppose m>1 and the lemma is true for all n The following example shows that Lemma 8 is in a sense the best possible. Example 1. For each n ∈ ω there is a point- (a) A ∩ B is finite for each pair of distinct A, B ∈An,and 0 (b) An has no point- n S = {t ∈ ωn : t(n − 1) = α and t(i) ∈ α for each i 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 4981 h i Corollary 5. Assume CECA. Suppose that Aα α<κ isT a sequence of sets such that ℵ ∈ 0 ⊂ | | ℵ every I [κ] has a finite subset J I such that α∈J Aα < 0. Then there 0 ≤ℵ 0 ∈ 0 h \ i are Aα [Aα] for α<κsuch that the sequence Aα Aα α<κ is point-finite. Letting σ = ℵ0 and τ = ℵ1 in (1.1), we get: Corollary 6. Assume CECA. Suppose that hAαiα<κ is a sequence of sets such that for every countably infinite set S,theset{α<κ: S ⊂ Aα} is countable. 0 ≤ℵ 0 ∈ 0 h \ i Then there are Aα [Aα] for α<κsuch that the sequence Aα Aα α<κ is point-countable. Let us restate the first part of Theorem 3 in a more conventional way: Corollary 7. Assume CECA.Letσ be a regular infinite cardinal. Suppose that + hAαiα<κ is a sequence of sets of cardinality σ , each such that Aα ∩ Aβ has cardi- 0 ∈ ≤σ nality less than σ for all α<β<κ. Then there are Aα [Aα] for α<κsuch h \ 0 i that the sequence Aα Aα α<κ consists of pairwise disjoint sets. The consistency of Corollary 7 was already derived from a statement similar to CECA as Theorem 2.6 in [3]. Condition (1.4) allows us in certain circumstances to relax the assumption that the intersection of each pair of sets Aα,Aβ has cardinality strictly less than σ.In particular, if σ = ℵ0,thenweget Corollary 8. Assume CECA. Suppose that hAαiα<κ isasequenceofsetsofcar- dinality ℵ1 such that for every countably infinite set S,theset{α<κ: S ⊂ Aα} 0 ≤ℵ | ∩ |≤ℵ ∈ 0 is countable and Aα Aβ 0 for α<β<κ. Then there are Aα [Aα] for h \ 0 i α<κsuch that the sequence Aα Aα α<κ consists of pairwise disjoint sets. If the existence of certain large cardinals is consistent, none of the above corollar- ies is a consequence of GCH alone. To see this, suppose that hAαiα<κ is a sequence 0 ℵ 0 ∈ 0 h \ i of sets for which there are Aα [Aα] such that the sequence Aα Aα α<κ is point-countable. Then one can recursively construct forS each ξ<κpairwise dis- ℵ 0 joint sets B ∈ [κ] 0 and pairwise disjoint sets C ⊆ A \ A such that ξ ξ α∈Bξ α α ∩ \ 0 ∈ Cξ SAα Aα is infinite for each α Bξ. Now it is easy to construct a function 2 →{ } | 2 | c :[ α<κ Aα] 0, 1 such that c([Aα] ) =2forallα<κ. But, if the existence of a supercompact cardinal is consistent, one can construct a model of ZFC where h i ℵ GCH holds and where there exists a sequence Aα α<ℵω+1 of sets of size 1 each suchS that every two of these sets have finite intersection, and for every function →{ } | | c : α<ℵω+1 Aα 0, 1 there exists α with c[Aα] = 1 (see Theorem 4.6 of [3]). However, assuming only GCH, we can get weaker versions of Theorem 2 that are also of interest. For example: Theorem 4. Suppose that σ is a regular infinite cardinal and assume that λσ = λ for each cardinal λ of cofinality greater than σ. Suppose that A = hAαiα<κ is a sequence of (not necessarily distinct) sets such that T ℵ ∗ ∈ 0 ⊆ | |≤ ( ) For every I [κ] there is a finite J I such that α∈J Aα σ. h 0 i | 0 |≤ + Then there exists Aα α<κ such that Aα σ for each α<κand the sequence h \ 0 i Aα Aα α<κ is point-finite. Proof. Assume by induction that κ is minimal such that there is a counterexample to the theorem. Case 1: σ+ ≥ κ. Fix an elementary submodel M of size σ+ ≥ κ such that ∪{ }∪ + ∪{ +}⊆ ∈ 0 ∩ A A σ σ M.Foreachα κ let Aα = Aα M.Toseethat License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4982 Z. T. BALOGH, S. W. DAVIS, W. JUST, S. SHELAH, AND P. J. SZEPTYCKI h \ 0 i 6∈ { ∈ } Aα Aα α<κ is point-finite, fix x TM such that I = α : Tx Aα is infinite. ∗ ⊆ | |≤ ∈ By ( ), fix a finite J T I such that α∈J Aα σ. Clearly α∈J Aα M.But + ⊆ ⊆ 6∈ σ M implies that α∈J Aα M, contradicting x M. Case 2: σ+ <κ.Ifκ = λ+ for λ a cardinal of cofinality ≤ σ, we fix a continuous ∈-chain {Mξ : ξ<κ} of elementary submodels of some H(Θ) such that | | ⊆ ∈ (a) Mξ = λ and λ SMξ for each ξ κ, ∈ ⊆ (b) A M0 and A ξ∈κ Mξ,and S i (c) for ξ a successor ordinal we have Mξ = i 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 4983 Proof. We modify the proof of Theorem 4. First note that we may assume that σ+,τ < κ and that κ is the successor of a limit cardinal λ of cofinality ≤ σ.Fix Mξ satisfying (a), (b) and (c) of the previous proof. In addition, assume 0 i ≤τ ⊆ i+1 (c )[Mξ] Mξ , for each successor ξ<κand each i + Lemma 10. I = {α ∈ Mξ : a ∈ Aα} has cardinality <τ . Proof. Suppose not. Case 1: cf(λ) =6 τ. Then there are a successor η<ξT and an i Taking σ = ℵ0 in Theorem 4 gives the following. Corollary 9. Assume λω = λ for each cardinal λ of uncountable cofinality. Sup- pose that hAαiα<κ is a sequence of sets such that |Aα ∩ Aβ|≤ℵ0 for each pair of 0 ≤ℵ ∈ ∈ 1 distinct α, β κ. Then there are Aα [Aα] for α<κsuch that the sequence h \ 0 i Aα Aα α<κ is point-finite. Finally, we remark that Theorem 2 cannot be improved by demanding that the 0 { +} sets Aα be of cardinality <σ. Indeed, if σ is regular, then let Aα : α<σ be + a family of subsets of σ such that |Aα ∩ Aβ| <σwhile |Aα| = σ for each α<σ . No such family can be point-<σ+. Sonopoint-<σ+ family can be obtained by deleting sets of size less than σ from each Aα. 4. Topological applications In this section, we present some topological applications of the results just proved. We are particularly interested in the question, raised in [4], of whether a first countable space with a weakly uniform base must have a point-countable base. Definition 3. If n ∈ ω and B is a base for a topological space X,thenwesay B is an n-weakly uniform base for X provided that if A ⊆ X with |A| = n,then {B ∈B: A ⊆ B} is finite. We say a base B for X is a <ω-weakly uniform base provided that if A ⊆ X with |A|≥ℵ0, then there is a finite subset F ⊆ A with {B ∈B: F ⊆ B} finite. The notion of weakly uniform base, introduced in [4], is just what we have called a 2-weakly uniform base. We shall see later in this section that the properties defined above are all distinct. To avoid trivialities we will be interested only in the case n ≥ 2. Clearly, for n License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4984 Z. T. BALOGH, S. W. DAVIS, W. JUST, S. SHELAH, AND P. J. SZEPTYCKI It was shown in [4] that if X is a space with a 2-weakly uniform base B and x ∈ X is in the closure of a countable subset of X \{x},thenB is point-countable at x. In particular, if X is a first countable space with a weakly uniform base B, then B is point-countable at all nonisolated points of X. Using that result, it was shown in [4] that a first countable space in which the boundary of the set of isolated points is separable has a point-countable base. In [2], it is shown that a first countable space with a weakly uniform base and no more than ℵ1 isolated points has a point-countable base. In [1] this is improved (consistently) to the result that, assuming CH, every first countable space with a weakly uniform base and no more than ℵω isolated points has a point-countable base. ℵ0 An example is constructed in [2], assuming MA and ℵ2 < 2 , of a normal Moore space with a weakly uniform base which has no point-countable base. Such a space also could not be meta-Lindel¨of. Our next result establishes the independence of the existence of first countable spaces with weakly uniform bases but without point-countable bases, and it removes the cardinality restriction on the set of isolated points. Theorem 6. Assume CECA.IfX is a T1-space in which each non-isolated point is a cluster point of a countable set and X has a <ω-weakly uniform base, then X has a point-countable base. Proof. Suppose B is a <ω-weakly uniform base for X. Suppose x is a non-isolated point of X. We will show that B is point-countable at x. Choose a countable set C ⊆ X \{x} with x ∈ C.ForeachB ∈Bwith x ∈ B,thesetC ∩ B is infinite, and <ℵ hence there exists FB ∈ [C ∩ B] 0 such that {A ∈B : FB ⊂ A} is finite. Since ℵ ℵ [C]< 0 is countable and for each F ∈ [C]< 0 there are only finitely many B ∈B with FB = F , we conclude that {B ∈B : x ∈ B} is countable. Let hxα : α<κi list the isolated points of X,andletAα = {B ∈B : xα ∈ B}. Notice that since B is a <ω-weakly uniform base, we have that if I is an infinite subset of κ, then there exists a finite J ⊆ I such that |{B ∈B : {xα : a ∈ J}⊆B}| < ℵ0, T 0 ≤ℵ | | ℵ ∈ 0 i.e. α∈J Aα < 0. By Corollary 5, for each α there is Aα [Aα] such that h \ 0 i ∈B ∈ \ 0 Aα Aα : α<κ is point-finite, i.e. for each B we have B Aα Aα for only finitely many α’s. Hence for each B ∈Bthere is a finite set I(B)ofisolatedpoints { ∈B ∈ \ }⊆ 0 | 0 |≤ℵ such that, for every α<κ, B : xα B I(B) Aα,and Aα 0. Hence B1 = {B \ I(B):B ∈B}is a point-countable open family of subsets of X which ∗ forms a base at every non-isolated point of X.ThusB = B1 ∪{{xα} : α<κ} is a point-countable base for X. Corollary 10. It is consistent with ZFC that every first countable T1-space with a weakly uniform base has a point-countable base. Combining that corollary with the example of [2] completes the independence result. Assuming only λω = λ for each cardinal λ of uncountable cofinality, we can obtain a weaker version of Theorem 6. The proof is identical, using Corollary 9 in place of Corollary 5. 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 4985 Theorem 7. Assume λω = λ for each cardinal λ of uncountable cofinality. If X is a T1-space in which each non-isolated point is a cluster point of a countable set and X has a <ω-weakly uniform base, then X has a point-< ℵ2 base. Corollary 11. Assume λω = λ for each cardinal λ of uncountable cofinality. Every first countable T1-space with a weakly uniform base has a point-< ℵ2 base. Note that if we weaken the conclusion of <ωweakly uniform base to “{B ∈ B : F ⊆ B} is countable”, then we obtain a notion that implies point-countable at nonisolated points and gives both Theorem 7 and its corollary. It would be interesting to know when this weaker notion sufficed to conclude that the space has a point-countable base. Definition 4. If n ∈ ω, then we say that a topological space X is n-metacompact provided that for every open cover U of X there is an open refinement V≺U such that if A ⊆ X with |A| = n,then{V ∈V : A ⊆ V } is finite. We say X is <ω-metacompact provided that, for every open cover U of X there is an open refinement V≺Usuch that if A ⊆ X with |A|≥ℵ0, then there is a finite subset F ⊆ A with {V ∈V : F ⊆ V } finite. It is clear that 1-metacompact is just metacompact and that, for n Theorem 8. Assume CECA.IfX is a <ω-metacompact T1-space, then X is meta-Lindel¨of. Proof. Suppose X is <ω-metacompact and U is an open cover of X.LetV be an open, <ω-weakly uniform refinement of U. List the points of X as hxα : α<κi. For each α<κ,letAα = {V ∈V : xα ∈ V }. By Corollary 5, for each α<κthere 0 ≤ℵ 0 ∈ 0 h \ ∈ i V is Aα [Aα] such that Aα Aα : α κ is point-finite on the set .Foreach ∈V { ∈ ∈ \ 0 } | | ℵ V ,weletI(V )= xα V : V Aα Aα .NotethatI(V ) < 0.Further, { ∈V ∈ \ }⊆ 0 | 0 |≤ℵ for each α<κwe have V : xα V I(V ) Aα and Aα 0.Foreach α<κ,chooseVα ∈Vsuch that xα ∈ Vα.ForV ∈V,let ∗ V =(V \ I(V )) ∪{xα ∈ I(V ):V = Vα}. Note that V ∗ ⊆ V and V \ V ∗ is finite. Let V∗ = {V ∗ : V ∈V}.ThenV∗ U ∈ ∗ ∈ 0 ∪{ } V∗ is an open refinement of ,andifxα V ,thenV Aα Vα .Thus is point-countable. Corollary 12. It is consistent with ZFC that every T1-space with a weakly uniform base is meta-Lindel¨of. Again, combining this corollary with the example in [2] completes the indepen- dence result. Example 2. For each natural number n ≥ 1, there is a Moore space Xn of scattered height 2 which has an (n+1)-weakly uniform base, but Xn does not have an n-weakly uniform base. Proof. Let L be a subset of R ×{0} with |L| = ℵ1,andletD = {pn : n ∈ ω} be a countable subset of R × (0, ∞) which is dense in the Euclidean topology. It is shown in [2] that there is a collection H of countably infinite subsets of L and a partition {Hn : n ∈ ω} of H such that License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4986 Z. T. BALOGH, S. W. DAVIS, W. JUST, S. SHELAH, AND P. J. SZEPTYCKI (1) if H1, H2 ∈Hand H1 =6 H2,then|H1 ∩ H2| < ℵ0,and (2) if Y ⊆ L and |Y | = ℵ1,thenforeachn ∈ ω,thereexistsH ∈Hn such that | ∩ | ℵ Y H = 0. S ∈ { ∈H} For each n ω,letYn = (pn,H):H n ,andletY = n∈ω Yn.Let X = L ∪ Y .Foreachn ∈ ω and each x ∈ L,letBn(x)={(pi,H):(pi,H) ∈ Yi, −n x ∈ H,andpi is an element of the Euclidean open ball in R × (0, ∞)ofradius2 which is tangent to the axis R ×{0} at the point x}.Ifx ∈ L and n ∈ ω,thenwe define Gn(x)={x}∪Bn(x)andlet{Gn(x):n ∈ ω} be a neighborhood base at x. If y ∈ Y ,then{y} is open. It is shown in [2] that with this topology X is a Moore space with a weakly uniform base, and if U is any open cover of X which refines the open cover {G0(x):x ∈ L}∪{{y} : y ∈ Y }, then there exists y ∈ Y such that {U ∈U : y ∈ U} is infinite. Suppose k is a natural number and k ≥ 1. We shall use the space X to construct aMoorespaceXk of scattered height 2 with a (k + 1)-weakly uniform base but no k-weakly uniform base. Let Xk = L ∪ (Y × k). Points of Y × k will be isolated. If ∈ ∈ k { }∪ × h k ∈ i x L and n ω,letGn(x)= x (Bn(x) k). Notice that Gn(x):n ω is a decreasing sequence, and thus this is a valid assignment of neighborhoods. Letting G { k ∈ }∪{{ } ∈ × } hG ∈ i n = Gn(x):x L y : y Y k ,weseethat n : n ω is a k development for Xk, and since each Gn(x)isclopen,Xk is a 0-dimensional space of scattered height 2. S B G We now show that = n∈ω n is a (k + 1)-weakly uniform base for Xk. Suppose A ⊆ Xk with |A| = k +1.If|A ∩ L|≥2, then |{B ∈B: A ⊆ B}| =0. | ∩ | { } ∩ ∈ k ∩ { } If A L =1,thenlet x = A L.Choosen ω such that Gn(x) A = x . Then |{B ∈B: A ⊆ B}| ≤ n,sinceforeachm ∈ ω there is only one element of Gm which contains x. Finally, if |A ∩ L| =0,thenletA = {(y1,n1),...,(yk+1,nk+1)}, where yi ∈ Y and ni ∈ k for 1 ≤ i ≤ k +1.Choosei, j ≤ k +1suchthatni = nj. { ∈B ⊆ }⊆{ ∈B { }⊆ } k Then B : A B B : (yi,ni), (yj,nj) B .NowforB = Gn(x), ∈ k ∈ we have that (yj,nj) Gn(x) implies yi Gn(x). So |{B ∈B : {(yi,ni), (yj,nj)}⊆B}| = |{Gn(x):{yi,yj}⊆Gn(x)}| < ℵ0, since {Gn(x):n ∈ ω,x ∈ L} is shown in [2] to be a weakly uniform collection. We now show that Xk has no k-weakly uniform base. Suppose V is any open cover of Xk which refines G1.Foreachx ∈ L,chooseVx ∈Vand n(x) ∈ ω with k ⊆ U { ∈ }∪{{ } ∈ } Gn(x)(x) Vx.NowinthespaceX, = Gn(x)(x):x L y : y Y refines {G1(x):x ∈ L}∪{{y} : y ∈ Y }. Hence there is a point y ∈ Y so that {U ∈U : y ∈ U} is infinite, and so {x ∈ L : y ∈ Gn(x)(x)} is infinite. Let { }× | | ∈ ∈ k ⊆ A = y k.Then A = k,andforeachi k we have (y, i) Gn(x)(x) Vx.So {x ∈ L : A ⊆ Vx} is infinite, and thus V is not a k-weakly uniform base for Xk. Example 3. There is a space Y which has <ω-weakly uniform base, but does not have an n-weakly uniform base for any n ∈ ω. Proof. For each natural number n ≥ 1, let Xn be as constructed in Example 2. Let Y be the disjoint union of the spaces Xn. The natural base is easily seen to be <ω-weakly uniform, since if A ⊆ Y and |A| = ℵ0, then either A ⊆ Xn for some n, in which case any subset of A of size n + 1 would be contained in only finitely many elements of the base, or A contains two points from distinct Xn’s, in which case no element of the base contains that two-point subset. To see that Y cannot have an n-weakly uniform base, it is enough to observe that Xn is an open subspace of Y . 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 4987 Thus we have shown that these properties are all distinct. References [1] A. V. Arhangel’skii, W. Just, E. A. Reznichenko, and P. J. Szeptycki; Sharp bases and weakly uniform bases versus point-countable bases, to appear in Topology and its Applications. [2]S.W.Davis,G.M.Reed,andM.L.Wage;Further results on weakly uniform bases, Houston Journal of Mathematics, 2 (1976) 57–63. 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MR 95h:03112 Department of Mathematics, Miami University, Oxford, Ohio 45056 E-mail address: [email protected] Department of Mathematics, Miami University, Oxford, Ohio 45056 E-mail address: [email protected] Department of Mathematics, Ohio University, Athens, Ohio 45701 E-mail address: [email protected] Institute of Mathematics, The Hebrew University, Givat Ram, 91904 Jerusalem, Israel E-mail address: [email protected] Department of Mathematics, Ohio University, Athens, Ohio 45701 E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use