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On closed P-sets with ccc in the space ω∗

arXiv:math/9303207v1 [math.LO] 15 Mar 1993On closed P-sets with ccc in the space ω∗R.Frankiewicz(Warsaw) S.Shelah(Jerusalem) P.Zbierski(Warsaw)Abstract. It is proved that – consistently – there can be no ccc closed P-sets in theremainder space ω∗.In this paper we show how to construct a model of set theory in which there are noP-sets satisfying ccc (countable antichain condition) in the ultrafilter space ω∗= β ⋄ω|\ω.The problem of the existence of such sets (which are generalizations of P-points) wasknown since some time and occurred explicitly in œvM-R—.

In the proof we follow theconstruction from œS— of a model in which there are no P-points. A particular case ofP-sets, which are supports of approximative measures has been settled in œM—, wherethe author shows that there can be no such measures on P(ω)/fin.

(Under CH, e.g. theGleason space G(2ω) of the Cantor space is a ccc P-set in ω∗which carries no approximativemeasure).Sec.1.

Closed P-sets in the space ω∗can be identified with P-filters F on ω. Thus,the dual ideal I = {ω \ A : A ∈F} has the property:(1.1)If An ∈I, for n ∈ω, then there is an A ∈Isuch that An ⊆∗A, for each n ∈ω.Further, the countable chain condition imposed upon F implies that I is fat in thefollowing sense (see œF-Z—):(1.2)if An ∈I, for n ∈ω and limn min An = ∞,then there is an infinite Z ⊆ω such that Sn∈Z An ∈I.Indeed, let en = An \ A, where A ∈I is as in (1.1). Since min An are arbitrarilylarge, we can find an infinite set Y ⊆ω such that the family {en : n ∈Y } is disjoint.

If{Yα : α < c} is an almost disjoint family of subsets of Y , then the unionsSα =[{en : n ∈Yα},α < care almost disjoint and hence the closures S∗α in the space ω∗are disjoint. By ccc we haveS∗α ∩\{B∗: B ∈F} = ∅,for some α and consequently Sα ∈I.

It follows that th e union[n∈YαAn =[n∈Yα(An ∩A) ∪[n∈Yα(An \ A)1

is in I as a subset of Sα ∪A.Let us fix a given ccc P-filter F and its dual I. We shall define a forcing P = P(F).A partial ordering (T, ≤T ), where T ⊆ω, will be called a tree, if for each i ∈T theset of predecessors {j ∈T : j ≤T i} is linearly ordered andi ≤T j implies i ≤j, for all i, j ∈T.We define a partial ordering for treesT ≤t S iff(S, ≤S) is a subordering of (T, ≤T )and each branch of T contains cofinally a (unique) branch of S.There is a tree T0 such that T0 ∈I and T0 is order isomorphic to the full binary treeof height ω.Deleting the numbers ≤n from T0 we obtain a subtree denoted by T (n)0(we haveT (n)0≤t T (m)0, for n ≤m).

Let T consist of all the trees T ∈I such thatT ≤t T (n)0, for some n ∈ω.Note that each tree T ∈T has finitely many roots.Definition. Elements of the forcing P are of the form p =< Tp, fp >, where Tp ∈Tand fp : Tp −→{0, 1}.

The ordering of P is defined thusp ≤q iffTp ≤t Tq and fp ⊇fq.Let {bα : α < c} be a fixed enumeration of all the branches of T0 in V . For a genericG ⊆P let TG = Sp∈G Tp and fG = Sp∈G fp.Each branch B of TG contains cofinally a unique bα.

Let us write B = Bα and defineXα = {i ∈ω : i ∈Bα and fG(i) = 1}Since Tp ∈I, for any p ∈P, hence ω \ Tp ∩A is infinite, for each A ∈F. It followsthat the setsDAαnε = {p ∈P : ∃i > n ⋄i ∈bpα and fp(i) = ε|}are dense, for each A ∈F, n ∈ω, α < c and ε = 0, 1 (here bpα denotes the branch of Tpextending bα).Thus, P adds uncountably many almost disjoint Gregorieff-like sets.Sec.2.

Let Q = Q(F) be a countable product of P = P(F). Thus the elements q ∈Qcan be written in the formq =< f q0 , f q1, .

. .

>, where < dm(f qi ), f qi >∈P, for each i < ω.By q(n) we denote the condition < gi : i < ω > where2

gi =f qi | dm(f qi )(n),for i < nf qifor i ≥nHere T (n) is a tree obtained from T by deleting the numbers ≤n.Lemma 2.1.For each decreasing sequence p0 ≥p1 ≥. .

. there is a q and an infiniteZ ⊆ω such thatq ≤p(n)n , for each n ∈Z.Proof.

Let Tni = dm(f pni ), where pn =< f pni:i < ω >. Since min T (n)ni≥n,we may use (1.2) to define by induction a descending sequence Z0 ⊇Z1 ⊇.

. .

of infinitesubsets of ω such thatSn∈Zi T (n)niis in I, for each i < ω.There is an infinite Z ⊆ω, such that Z ⊆∗Zi, for each i < ω. DefineTi = Tii ∪[n∈ZT (n)niandf qi = f pii∪[n∈Zf pni| T (n)ni .Then, dm(f qi ) = Ti and q =< f qi : i < ω > is as required.QEDFor q ∈Q and n ∈ω let S(q, n) be the set of all sequences s =< s0, . .

., sn−1 >satisfying the following properties1. s0, .

. ., sn−1 are finite zero-one functions.2.

The domains t0 = dm(s0), . .

., tn−1 = dm(sn−1) are finite trees such thatt0 ∩T (n)0= . .

. = tn−1 ∩T (n)n−1 = ∅,where T0 = dm(f q0), .

. ., Tn−1 = dm(f qn−1)3.

Ordered sums t0 ⊕T (n)0, . .

. , tn−1 ⊕T (n)n−1 are trees belonging to T .Note that from the definition of T it follows that S(q, n) is always finite.

Let us denotes ∗q(n) =< s0 ∪f q0 , . .

., sn−1 ∪f qn−1, f qn, . .

. >for q, n, s as above.

Obviously, we have(2.2)the set {s ∗q(n) : s ∈S(q, n)} is predense below q(n)(i.e. the boolean sum Ps∈S(q,n) s ∗q(n) = q(n)).3

Now, we obtain easily an analogue of VI, 4.5 in œS—. (2.3)For arbitrary p ∈Q, n < ω and τ ∈V (Q)such that Q ⊩“τ is an ordinal” there isa q ≤p and ordinals {α(s) : s ∈S(p, n)}so thatq(n) ⊩“_sτ = α(s)”Indeed, if S(q, n) = {s0, .

. ., sm−1}, then we define inductively conditions p0, .

. ., pmso that p0 = p and pk+1 ≤sk ∗p(n)kis such thatpk+1 ⊩“τ = α”, for some ordinal α = α(sk)Now, q = s ∗p(n)m , where s is such that p = s ∗p(k) (we may assume s ∈S(p, n)),satisfies (2.3).2.4.

Theorem.Q is α-proper, for every α < ω1, and has the strong PP-property.Proof. Let countable N ≺H(κ) for sufficiently large κ, be such that Q ∈N andsuppose that p ∈Q ∩N.

To prove that Q is proper we have to find a q ≤p, which isN-generic. Let {τn : n < ω} be an enumeration of all the Q-names for ordinals, such thatτn ∈N, for n < ω.

Using (2.3) we define inductively a sequence p0 = p ≥p1 ≥. .

. andordinals α(n, s) so thatp(n)n⊩“^i≤n_sτi = α(n, s)” for each n < ω(i.e.

in the n-th step we apply (2.3) for all names τ0, . .

., τn). Note that the p′s and α′scan be found in N, since N ≺H(κ).

By Lemma 2.1 there is a q and an infinite Z ⊆ωsuch thatq ≤p(n)n , for each n ∈Z.Hence also q(m) ≤p(n)nholds for arbitrarily large n and all m < ω and thusq(m) ⊩“τn ∈N”,for all n, m < ω.By III, 2.6 of œS—, each q(m) is N-generic.To see that Q is α-proper let < Nξ : ξ ≤α > be a continuous sequence of elementarycountable submodels of H(κ) such that Q ∈N0 and< Nξ : ξ ≤η >∈Nη+1, for each η < α.4

Assume that Q is β-proper, for each β < α and let q0 ∈Q ∩N0. If α = β + 1, wehave a q ≤q0 which is Nξ-generic, for each ξ ≤β and we may assume that the q(n) havethe same property, for all n < ω. since Nα ≺H(κ) and all the parameters are in Nα, sucha q can be found in Nα and as above we construct a qα ≤q which is Nα-generic and so arethe q(n)α , for n < ω.

Thus, qα and all the q(n)αare Nξ-generic for all ξ ≤α. If α is a limitordinal, we fix an increasing sequence < ξn : n < ω > such that α = supn<ω ξn and by theinductive hypothesis there is a sequence q0 ≥qξ0 ≥qξ1 ≥.

. .

such that, for each n < ω,qξn is Nξ-generic, for each ξ ≤ξn and qξn ∈Nξn+1 and that q(m)ξnhave the same propertyfor each m < ω. By Lemma 2.1 there is a q ∈Q such that q ≤q(n)ξn , for infinitely manyn < ω.

Thus, q ≤q0 and q is Nξ-generic for each ξ < α and hence also for each ξ ≤α.Finally, to prove the PP-property let h : ω −→ω diverge to infinity and suppose thatp ⊩“f : ω −→ω”. Definekn = min{i : h(i) > 2n · cardS(p, n)}, for n < ωand, using (2.3), define inductively the sequence p = p0 ≥p1 ≥.

. .

such thatp(n)n⊩“^i

Let Xnα be the α-th set added by n-th factor of the product Q = Pω. Supposethat for some r ∈R and a ccc P-filter E ∈V (R) we haver ⊩“F ⊆E”Note that for each n < ω, the relation Xnα ∈E hold for at most countably many α’s,since E is ccc.

Hence, there is an α such that for all n < ω we have ω \ Xnα ∈E and, sinceE is a P-filter, there is an A ∈E and a function g, so that A ⊆Tn<ω(ω \ Xnα) ∪⋄0, g(n))i.e. for some r1 ≤r we have(2.6)r1 ⊩“\n<ω(ω \ Xnα) ∪⋄0, g(n)) ∈E”5

Since R is ωω-bounding we may assume that g ∈V . By the assumption, Q is acomplete subforcing of R and hence there is a q ∈Q such that r is compatible with eachq′ ≤q.On the other hand, since Tn = dm(f qn) ∈T , there is a set B ∈I and an increasingsequence a0 < a1 < .

. .

such that Tn \ ⋄0, an) ⊆B, g(n) < an and ⋄an, an+1) \ B ̸= ∅,for each n < ω. Define q′ ≤q as follows. For a given n extend Tn by adding elementsof ⋄an, an+1) \ B on the α-th branch bqα and put f q′n (i) = 1, for each i ∈⋄an, an+1) \ B.Obviously, we haveq′ ⊩“(ω \ Xnα) ∪⋄0, g(n)) ∩⋄an, an+1) \ B = ∅”, for each nand henceq′ ⊩“\n<ω(ω \ Xnα) ∪⋄0, g(n)) \ B ∩[n<ω⋄an, an+1) = ∅”Consequently q′ ⊩“ Tn<ω(ω \ Xnα) ∪⋄0, g(n)) ⊆∗B”, which contradicts (2.6).QEDThe rest of the proof is routine.

Beginning with a model V of 2ω = ω1 and 2ω1 = ω2we iterate with countable supports the forcings Q(F), for all ccc P-filters F booked ateach stage α < ω2 of the iteration. From œS—, V.4 we know that the resulting forcing R(obtained after ω2 stages) is proper and ωω-bounding.

Hence, in V ⋄G| there are no cccP-sets.ReferencesœF-Z— R.Frankiewicz, P.Zbierski “Strongly discrete subsets in ω∗” Fund. Math.

129(1988) pp 173-180œM—A.H.Mekler “Finitely additive measures on N and the additive property”Proc.AMS Vol.92 No3 Nov.1984 pp 439-444œvM-R— J.vanMill, G.M.Reed editors “Open problems in topology”, North HollandElsevier Science Publishers B.V 1990œS— S.Shelah “Proper forcing” Lecture Notes in Mathematics 940 Springer VerlagRyszard Frankiewicz Institute of Mathematics, Polish Academy of Sciences, War-saw, PolandSaharon Shelah Institute of Mathematics, The Hebrew University, Jerusalem, IsraelPawe lZbierski Institute of Mathematics, Warsaw University, Warsaw, Poland6

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