PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 8, August 1998, Pages 2459–2465 S 0002-9939(98)04884-9

A ZFC DOWKER SPACE IN : AN APPLICATION ℵω+1 OF PCF THEORY TO TOPOLOGY

MENACHEM KOJMAN AND SAHARON SHELAH

(Communicated by Franklin D. Tall)

Abstract. The existence of a Dowker space of ω+1 and weight ℵ ω+1 is proved in ZFC using pcf theory. ℵ

1. Introduction A Dowker space is a normal Hausdorff whose product with the unit is not normal. The problem of existence of such spaces was raised by C. H. Dowker in 1951. C. H. Dowker characterized Dowker spaces as normal Hausdorff and not countably paracompact [4]. Exactly two Dowker spaces were constructed in ZFC prior to the construction of the space below. The existence of a Dowker space in ZFC was first proved by M. E. Rudin in 1971 [6], and her space was the only known Dowker space in ZFC for

over two decades. Rudin’s space is a subspace of n 1( n + 1) and has cardinality ≥ ℵ 0 . The problem of finding a Dowker space of smaller cardinality in ZFC was ωℵ Q ℵreferred to as the “small Dowker space problem”. Z. T. Balogh constructed [1] a Dowker space in ZFC whose cardinality is 2ℵ0 . Another, screenable, Dowker space of size 2ℵ0 was constructed by Balogh recently [2]. While both Rudin’s and Balogh’s spaces are constructed in ZFC, their respective are not decided in ZFC, as is well known by the independence results 0 0 of P. Cohen: both 2ℵ and ωℵ have no bound in ZFC (and may be equal to each other). ℵ The problem of which is the first α in which ZFC proves the existence of a Dowker space remains thus unansweredℵ by Rudin’s and Balogh’s results. In this paper we prove that there is a Dowker space of cardinality ω+1.A non-exponential bound is thus provided for the cardinality of the smallestℵ ZFC Dowker space. We do this by exhibiting a Dowker subspace of Rudin’s space of that cardinality. Our construction avoids the exponent which appears in the cardinality 0 of Rudin’s space by working with only a fraction of ωℵ . It remains open whether is the first cardinal at which there is a ZFC Dowkerℵ space. ℵω+1

Received by the editors June 15, 1996. 1991 Mathematics Subject Classification. Primary 54B99, 54G99; Secondary 04A20, 04A30. Key words and phrases. ZFC, Dowker space, cardinality, pcf theory. The second author’s research was supported by “The Israel Science Foundation” administered by The Israel Academy of Sciences and Humanities. Publication 609.

c 1998 American Mathematical Society

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We shall describe shortly the cardinal arithmetic developments which enable this result. The next three paragraphs are not necessary for understanding the proofs in this paper. In the last decade there has been a considerable advance in the understanding of the infinite exponents of singular cardinals, in particular the exponent ℵ0 .This ℵω exponent is the product of two factors: 2ℵ0 cf [ ]ℵ0 , . The second factor, the × h ℵω ⊆i cofinality of the partial ordering of inclusion over all countable subsets of ω,isthe least number of countable subsets of needed to cover every countableℵ subset of ℵω ; the first factor is the number of subsets of a single countable set. Since ℵ0 is ℵω ℵω the number of countable subsets of , the equality =2ℵ0 cf [ ]ℵ0 is clear. ℵω ℵω × h ℵω i While for 2ℵ0 it is consistent with ZFC to equal any cardinal of uncountable cofinality, the second author’s work on Cardinal Arithmetic provides a ZFC bound 0 of ω4 on the factor cf [ ω]ℵ , . ℵ h ℵ ⊆i 0 This is done by approximating cf [ ω]ℵ , by an interval of regular cardinals, h ℵ ⊆i 0 whose first element is ω+1 and whose last element is cf [ ω]ℵ , ,andsothat every regular cardinal λℵ in this interval is the true cofinalityh ℵ of a reduced⊆i product Bλ/J<λ of a set Bλ n :n<ω modulo an ideal J<λ over ω. The theory of reduced products of small⊆{ℵ sets of regular} cardinals, known now as pcf theory1,is Q used to put a bound of ω4 on the length of this interval. 0 Back to topology now, it turns out that the pcf approximations to ωℵ are concrete enough to “commute” with Rudin’s construction of a Dowker space.ℵ Rudin

defines a topology on a subspace of the functions space n>1( n + 1). What is ℵ 0 gotten by restricting Rudin’s definition to the first approximation of ωℵ is a closed R Q ℵ and cofinal Dowker subspace X of the Rudin space X of cardinality ω+1.The fact that X is Dowker follows, actually, from its closure and cofinality inℵ XR. Hardly any background is needed to state the pcf theorem we are using here. However, an interested reader can find presentations of pcf theory in either [3], the second author’s [8] or the first author’s [5]. The pcf theorem used here is covered in detail in each of those three sources.

2. Notation and pcf In this section we present a few simple definitions needed to state the pcf theorem used in proving the existence of an ω+1-Dowker space. Suppose B ω is a subset of theℵ natural numbers. ⊆ Definition 1. 1. n B n = f :domf=B f(n)< n for n B . ∈ ℵ { ∧ ℵ ∈ } 2. n B( n +1)= f :domf=B f(n) n for n B . ∈ ℵ Q{ ∧ ≤ℵ ∈ } 3. For f,g n B( n +1)let: Q(a) fg(n) is finite, ≤ { } (d) f<∗giff n : f(n) g(n) is finite, { ≥ } (e) f =∗ g iff n : f(n) = g(n) is finite. { 6 } 4. A sequence fα : α<λ of functions in n B n is increasing in < ( ,<∗, ∗) h i ∈ ℵ ≤ ≤ iff α<β<λ fα

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6. g n B( n +1)isaleast upper bound of fα : α<δ n B n if and ∈ ∈ ℵ { }⊆ ∈ ℵ only if g is an upper bound of fα : α<δ n B n,andifg0 is an upper Q { }⊆ ∈ ℵ Q bound of fα : α<δ ,theng ∗g. { } ≤ Q Theorem 1 (Shelah). There is a set B = B ω and a sequence f = fα : ℵω+1 ⊆ h α< ω+1 of functions in n B n such that: ℵ i ∈ ℵ f is increasing in <∗, • Q f is cofinal: for every f n B n there is an α< ω+1 so that f<∗fα. • ∈ ∈ ℵ ℵ A sequence as in the theoremQ above will be referred to as an “ ω+1-scale”. By Theorem 1 we can find B ω and an -scale g = g ℵ : α< in ⊆ ℵω+1 h α ℵω+1i n B n.ThesetBis clearly infinite. Restricting every gα g toafixedco-finite set∈ ofℵ coordinates does not matter, so we assume without loss∈ of generality that 0Q, 1 / B. For notational simplicity we pretend that B = ω 0,1 ;ifthisisnot ∈ −{ } the case, we need to replace n in what follows by the n-th element of B.Wesum up our assumptions in the following:ℵ

Claim 2. We can assume without loss of generality that there is an ω+1-scale g = g : α< in . ℵ h α ℵω+1i n>1 ℵn Claim 3. There is an ωQ+1-scale f = fα : α< ω+1 in n so that for every ℵ h ℵ i n>1 ℵ δ< ,ifcfδ> and a least upper bound of f δ exists, then f is a least upper ℵω+1 ℵ0  Q δ bound of fδ. Proof. Fix an -scale g = g : α< in as guaranteed by Claim ℵω+1 h α ℵω+1i n>1 ℵn 2. Define fα by induction on α< ω+1 as follows: If α is a successor or limit of ℵ Q countable cofinality, let f be g for the first β (α, ) for which g >∗ f for α β ∈ ℵω+1 β γ all γ<α.Ifcfα> ,thenletg be a least upper bound to f δ := f : β<α, ℵ0 α  h β i if such least upper bound exists; else, define fα as in the previous cases. The sequence f = fα : α< ω+1 is increasing cofinal in n>1 n and by its definition satisfies theh required condition.ℵ i ℵ Q Claim 4. Suppose 0 k ℵn ζ α(ζ) ℵm g := sup g : ζ< is a least upper bound of f δ, Q • { ζ ℵm}∈ n>k ℵn  cfg(n)= m for all n>k, • ℵ Q g=∗ f . • δ Proof. Let g := sup g : ζ< .Sinceg:ζ< is increasing in <, { ζ ℵm} hζ ℵmi necessarily cfg(n)= m for all n k, and since g(n) n, it follows that g(n) < n for n>kand thereforeℵ g ≥ . ≤ℵ ℵ ∈ n>k ℵn Suppose that γ<δis arbitrary. There exists ζ< m such that γ<α(ζ), hence Q ℵ f <∗ f =∗ g g.Thusgis an upper bound of f δ. γ α(ζ) ζ ≤  To show that g is a least upper bound suppose that g0 is an upper bound of f δ.LetX:= n>k:g0(n)g0(n). Such ζ(n) can be found because g =supg :ζ< . ζ(n) {ζ ℵm} Let ζ∗ := sup ζ(n):n>1.Since > ,ζ∗< .Since g :ζ< is { } ℵm ℵ0 ℵm hζ ℵmi increasing in <, it holds that fζ( ) fζ(n)(n) >g0(n) for every n X .Butg0is ∗ ≥ ∈ an upper bound of fδ,sofζ( ) ∗g0 and X is therefore finite. ∗ ≤ By the definition of f we conclude that fδ is a least upper bound of fδ.Since both g and fδ are least upper bounds of fδ, it follows that g =∗ fδ.

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3. The space Definition 5. Let XR = h ( +1): m n[ 1 ℵn ∃ ∀ ℵ0 ℵm } The space XR is the Rudin spaceQ from [6] with the Hausdorff topology defined by letting, for every f1 ℵn (1) (f,gQ]:= h XR :f1,then U is not empty. n⊆ n ⊆ n n>1 n These two facts establish by [4] that XR is Dowker. T Let f = fα : α< ω+1 be as provided by Claim 3. We use this scale to extract a closed Dowkerh subspaceℵ i of cardinality from Rudin’s space. ℵω+1 R Definition 7. X = h X : α< [h =∗ f ] . { ∈ ∃ ℵω+1 α } R Since h X : h =∗ fα = ω for every α< ω+1,itisobviousthat X= |{ . ∈ }| ℵ ℵ | | ℵω+1 Since f is totally ordered by <∗, for every h X there exists a unique α< ∈ ℵω+1 such that h =∗ fα. Consequently, the space X is totally quasi-ordered by <∗, namely the following trichotomy holds:

(2) h, k X h<∗ k k<∗ h h=∗ k . ∀ ∈ ∨ ∨ Claim 4 translates to a propertyh of X: i

Claim 8. Suppose that 0 1( n + 1) be defined by h(n)= n for n k and h(n)=g(n)for ∈ ℵ R ℵ ≤ n>k.Thenh X and h =∗ f .Thush Xand h =∗ g as required. Q ∈ δ ∈ Claim 9. X is a closed subspace of XR. Proof. Suppose t clX and t XR. For every h X let E(h, t):= n>1: h(n)=t(n) . ∈ ∈ ∈ { } Claim 10. If h t and h X,thenE(h, t) is either finite or co-finite. ≤ ∈

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Proof. Suppose to the contrary that h t, h X and E(h, t) = ω E(h, t) = 0. Let, for n>1, ≤ ∈ | | | − | ℵ 0ifnE(h, t), ∈ f(n)= h(n)ifn(ωE(h, t)). ∈ − Clearly fh(n) for all n (w E(h, t)), then k <∗ h k =∗ h ∈ ∈ − 6 ∧ 6 because w E(h, t) is infinite and so h<∗ kby the trichotomy (2). Since E(h, t) is infinite and− n>1:k(n) h(n) is finite, there is an n E(h, t) such that k(n) >h(n)=t{(n) and therefore≤ k/}(f,t]. ∈ ∈ We need a definition: Definition 11. W := w ω : f for all n>1 it follows that⊆ f1:cft(n)= . Likewise, M = M . m { ℵm} f(k, ω). Without loss of generality we can choose hζ so that h (n) >γn for all n k in M . ζ ζ ≥ m By Claim 8 there is some h X with h(n)=∗ sup h (n):ζ< .In ∈ { ζ ℵm} particular, h(n)=t(n) for all but finitely many n k in Mm.SinceMmis infinite, E(h, t) is infinite, and we are done. ≥

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Claim 15. X is collectionwise normal. Proof. Clear from Claim 9 and Theorem 2.

X We show next that X is not countably paracompact. Let Dn = f X : X { ∈ m n [f(m)= m] for n>1. It is straightforward that Dn is closed and that ∃ D≥X = . ℵ } n n ∅ X TClaim 16. If Un X is open, and D Un for all n>1, then Un is not empty. ⊆ n ⊆ The truth of the matter is that this follows trivially from the analogousT property R X R in X and the closedness of X: Suppose that Dn Un X and Un is open ⊆ ⊆ R for all n. The definition of Dn above is absolute between X and X ,soVn := R Un (X X) is open and contains Dn. Rudin’s proof in [6] shows that if Dn Vn and∪V is− open for all n, then there is some f such that h V⊆ for n ∈ n>1 ℵn ∈ n n all h>f in XR.Sinceforeverysuchfthere is an h X with h>f,weseethat Q ∈ T Un Xis not empty. For∩ the sake of completeness, though (but not less, for the reader’s amusement) Twe shall prove this property directly for X using elementary submodels. Proof of Claim 16. Suppose that U DX is open for n>1. We need to prove n ⊇ n that n Un is not empty. We shall prove that there is some f n>1 n such that every h>fin X belongsT to this intersection. ∈ ℵ It suffices to show that for each n>1thereissomeQ f such that n ∈ n>1 ℵn h X h>f h U , because then f =sup f :11 is fixed and for every function f n ∈ n>1 ℵ there is some function hf >fin X Um.Sincehf /Dm, it follows that hf (n) < n for all n m. − ∈ Q ℵ ≥ For a given f ,letg =suph :f0 (m, ω) E(f 0,f) .Since f { f0 ∈ n>1 ℵn ∧ ⊆ } this supremum is taken over m many functions hf 0 , it follows from the above that g (n) < for all n>m. Also,ℵ clearly g (i)=Q for 1

f, X and the functions f hf and f gf belong to M0, • M has cardinality and7→M : ξ<ζ7→ M for all ζ<ω. • ζ ℵ1 h ξ i∈ ζ+1 1 For every ζ let χζ (n) := sup(Mζ n) for all n>1. Since Mζ = 1, it follows that χ (n) < for all n and hence∩ℵχ . | | ℵ ζ ℵn ζ ∈ n>1 ℵn Since χξ Mζ for ξ<ζ<ω1, by elementarity also hχ and gχ belong to Mζ ∈ Q ξ ξ and consequently hχξ ,gχξ <χζ.

Therefore, if ξ<ζ<ω1,thenχξ mand χ0(i)= i for 1

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A ZFC DOWKER SPACE IN 2465 ℵω+1 Theorem 3. There is a ZFC Dowker space of cardinality . ℵω+1 It is straightforward to verify that the space X constructed above has weight and character . ℵω+1 ℵω Problem 17. Is ω+1 the first cardinal in which one can prove the existence of a Dowker space in ZFC?ℵ Acknowledgements The first author wishes to thank J. Baumgartner, for inviting him to the 11th Summer Conference on and Applications in Maine in the sum- mer of 95, Z. T. Balogh, for both presenting the small Dowker space problem in that meeting and for very pleasant and illuminating conversations that followed, P. Szeptycki, for suggesting that the subspaces we construct may actually be closed, and Mirna Djamonja, for detecting an inaccuracy in an earlier version of this paper. References

[1] Zoltan T. Balogh, A small Dowker space in ZFC, Proc. Amer. Math. Soc. 124 (1996), no. 8, 2555–2560. MR 96j:54026 [2] Zoltan T. Balogh, A normal screenable nonparacompact space in ZFC, Proc. Amer. Math. Soc. 126 (1998), 1835–1844. CMP 97:15 [3] M. .R. Burke and M. Magidor, Shelah’s pcf theory and its applications, Ann. Pure Appl. Logic 50 (1990) 207–254. MR 92f:03053 [4] C. H. Dowker, On countably paracompact spaces, Canad. J. Math. 3 (1951), 219–224. MR 13:264c [5] M. Kojman, The A,B,C of pcf: a pcf companion,preprint. [6] M. E. Rudin, AnormalspaceXfor which X I is not normal, Fund. Math. 73 (1971) 189–186. MR 45:2660 × [7] M. E. Rudin, Dowker Spaces,inHandbook of set-theoretic topology, K. Kunen and J. E. Vaughan eds. Elsevier Science Publishers, 1984. MR 86c:54018 [8] S. Shelah, Cardinal Arithmetic, Oxford University Press, 1994. MR 96e:03001

Department of Mathematics, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213 Current address: Department of Mathematics, Ben Gurion University of the Negev, POB 653, 84105 Beer-Sheva, Israel E-mail address: [email protected] Institute of Mathematics, The Hebrew University of Jerusalem, and Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903 E-mail address: [email protected]

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