A Zfc Dowker Space in ℵω+1: an Application of Pcf Theory
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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 cardinality !+1 and weight @ !+1 is proved in ZFC using pcf theory. @ 1. Introduction A Dowker space is a normal Hausdorff topological space whose product with the unit interval 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 ω@ Q ℵreferred to as the \small Dowker space problem". Z. T. Balogh constructed [1] a Dowker space in ZFC whose cardinality is 2@0 . Another, screenable, Dowker space of size 2@0 was constructed by Balogh recently [2]. While both Rudin's and Balogh's spaces are constructed in ZFC, their respective cardinalities are not decided in ZFC, as is well known by the independence results 0 0 of P. Cohen: both 2@ and ω@ 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 ω@ . 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 2459 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2460 MENACHEM KOJMAN AND SAHARON SHELAH 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 @0 .This ℵω exponent is the product of two factors: 2@0 cf [ ]@0 , . 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 @0 is ℵω ℵω the number of countable subsets of , the equality =2@0 cf [ ]@0 is clear. ℵω ℵω × h ℵω i While for 2@0 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 h ℵ ⊆i 0 This is done by approximating cf [ ω]@ , by an interval of regular cardinals, h ℵ ⊆i 0 whose first element is ω+1 and whose last element is cf [ ω]@ , ,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 ω@ 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 ω@ 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 . 2 ℵ { ∧ ℵ ∈ } 2. n B( n +1)= f :domf=B f(n) n for n B . 2 ℵ Q{ ∧ ≤ℵ ∈ } 3. For f,g n B( n +1)let: Q(a) f<g∈ iff 2n ℵB [f(n) <g(n)], (b) f g Qiff ∀n ∈ B [f(n) g(n)], ≤ ∀ ∈ ≤ (c) f ∗ g iff n : f(n) >g(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 2 ℵ ≤ ≤ iff α<β<λ fα <fβ (fα fβ,fα <∗ fβ,fα ∗ fβ). ⇒ ≤ Q ≤ 5. g n B( n +1)isanupper bound of fα : α<δ n B n if and only ∈ 2 ℵ { }⊆ 2 ℵ if fα ∗ g for all α<δ. Q≤ Q 1pcf means possible cofinalities. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A ZFC DOWKER SPACE IN 2461 @!+1 6. g n B( n +1)isaleast upper bound of fα : α<δ n B n if and ∈ 2 ℵ { }⊆ 2 ℵ only if g is an upper bound of fα : α<δ n B n,andifg0 is an upper Q { }⊆ 2 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 : @!+1 ⊆ h α< ω+1 of functions in n B n such that: ℵ i 2 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. • ∈ 2 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 set2 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 <m k<ω.Let α(ζ):ζ< m be strictly increasing with sup α(ζ):ζ< =δ<≤ .If gh :ζ< ℵisi a sequence of functions in { ℵm} ℵω+1 hζ ℵmi which is increasing in <,andg =∗f for every ζ< , then: n>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 .