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Forcing with a coherent Souslin tree and locally countable subspaces of countably tight compact spaces
Topology and its Applications 195 (2015) 284–296
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Forcing with a coherent Souslin tree and locally countable subspaces of countably tight compact spaces A.J. Fischer a , F.D. Tall b,∗,1 , S. Todorcevic b,2 a b
Kurt Gödel Research Centre for Mathematical Logic, University of Vienna, Vienna, Austria Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada
a r t i c l e
i n f o
Article history: Received 25 March 2014 Received in revised form 22 July 2015 Available online 1 October 2015
a b s t r a c t We apply a method of Todorcevic in showing that PFA(S) implies that the coherent Souslin tree S forces various strengthenings of the assertion that every locally countable space of size ℵ1 that has a countably tight compactification must be σ-discrete. © 2015 Elsevier B.V. All rights reserved.
MSC: 54A35 54D65 54D55 54DA25 54D18 54D20 03E35 03E57 Keywords: PFA(S)[S] Forcing with a coherent Souslin tree (Locally) compact Countably tight Locally countable σ-discrete
1. Introduction The third author in 2001 developed technology for analyzing the forcing by a fixed Souslin tree S over a model obtained by iterating proper posets that preserve S. Parts of this work were circulated in the form of notes such as [13] and have been presented at several conferences.3 Now this work is finally available * Corresponding author. E-mail addresses: arthur.j.fi[email protected] (A.J. Fischer), [email protected] (F.D. Tall), [email protected] (S. Todorcevic). 1 Research supported by NSERC grant A-7354. 2 Research supported by NSERC grant 455916. 3 Such as, for example, the three lectures at the Erice conference on set-theoretic topology in 2008 (see [15]). http://dx.doi.org/10.1016/j.topol.2015.09.035 0166-8641/© 2015 Elsevier B.V. All rights reserved.
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in printed form in [14]. We shall use this technology to deduce a particular topological principle about subspaces of compact countably tight spaces. The proof that PFA(S)[S] implies that we give here uses the same machinery and notation as the proof in [11] that PFA(S)[S] implies locally compact normal spaces are ℵ1 -collectionwise Hausdorff. That machinery is not so elegant as that in [14], but it has the advantage of already being published, so that the reader who wants more details can find them there. We just make one small improvement over [11]: improving coding so as to obtain “σ-discrete”, rather than “σ-discrete on a club”. The other differences are related to the fact that we face two entirely different topological situations. It is quite remarkable that the set-theoretic techniques are the same for both. The [11] machinery (based on [13]) is easier to understand for topologists not so expert at forcing with elementary submodels as side conditions. We will take advantage of [11] actually being published to omit some explanatory material and details available there, since our proof here closely resembles the proof there. (In fact, the proof there was inspired by the idea of the proof here.) We define coherent Souslin trees below. At this point, all the reader need know is that they may be obtained from ♦. Given a coherent Souslin tree S, PFA(S) is the restriction of PFA to those proper posets preserving (the Souslinity of) S. Since countable support iterations of such posets preserve S [9], PFA(S) may be obtained by the usual Laver-diamond [3] method from a supercompact cardinal. We use the notation PFA(S)[S] implies Φ to mean given any model of PFA(S), forcing over it with S yields a model of Φ. We say Φ holds in a model of form PFA(S)[S] if forcing with S over a particular model of PFA(S) yields a model of Φ. Definition 1. An S-space is a hereditarily separable, T3 , non-Lindelöf space. The double use of ‘S’ is unfortunate, but it will always be clear which S is intended. MAω1 implies there are no compact S-spaces [10]; PFA implies there are no S-spaces [16]. Setting out the context for what we will be doing, we recall that Todorcevic in 2001 proved the following result (see [14]). Proposition 1. ([14]) PFA(S)[S] implies there are no compact S-spaces. One of the crucial parts of the proof is contained in the following result. Proposition 2. ([14]) PFA(S)[S] implies compact countably tight spaces are sequential. A particular application of this result was given in [5]: Proposition 3. ([5]) There is a model of form PFA(S)[S] in which every locally compact perfectly normal space is paracompact. The use of Proposition 2 in the proof of Proposition 3 was the application of an equivalent version of the result stated in the Abstract: Proposition 4. PFA(S)[S] implies that in a compact countably tight space, locally countable subspaces of size ℵ1 are σ-discrete. This follows from Proposition 2 plus: Proposition 5. PFA(S)[S] implies that in a compact sequential space, locally countable subspaces of size ℵ1 are σ-discrete.
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The conclusion of Proposition 4 was proved from MAω1 by Z.T. Balogh in [1], extending Szentmiklóssy’s work [10]. Propositions 4 and 5 were proved by Todorcevic in 2001 and can now be seen to follow from the proof of Theorem 8.6 in [14]. 2. The method and the proof We shall assume that all spaces considered are T2 . Notice that Proposition 4 is an extension of Proposition 1. Proposition 4 is proved using a “machine” that has little to do with the particular problem we are applying it to. The mechanism – due to the third author – is expounded at length in [14]. Proposition 4 (plus [7], where the weaker “axiom” we now call “MAω1 (S)[S]” is deployed) suffices to justify all the applications of PFA(S)[S] in [5]. However for [6] and [12] we need the implication from PFA(S)[S] to the following stronger principle considered in [1]. : Let X be a compact countably tight space. Let Y ⊆ X, |Y | = ℵ1 . Suppose {Wα }α∈ω1 , {Vα }α∈ω1 are open subsets of X such that 1. W α ⊆ Vα , 2. |Vα ∩ Y | ≤ ℵ0 , 3. Y ⊆ {Wα : α < ω1 }. Then Y is σ-closed-discrete in {Wα : α < ω1 }.
and its consequences, including the one mentioned in the Abstract, are useful in proving certain locally compact normal ℵ1 -collectionwise Hausdorff spaces are paracompact (where a space is ℵ1 -collectionwise Hausdorff if every closed discrete subspace of size ℵ1 can be separated by a disjoint collection of open sets). In [12] there are several examples of such arguments, which originated in [1]. It is then clearly of interest to obtain in models of set theory in which locally compact normal spaces are ℵ1 -collectionwise Hausdorff. That is what we shall do here. That PFA(S)[S] implies locally compact normal spaces are ℵ1 -collectionwise Hausdorff was proved by the second author [11]. Getting these two statements holding in the same model is not so easy. This is to be expected, since the second one is an important consequence of V = L [18]. In fact, there is a model of PFA(S)[S] in which locally compact normal spaces are collectionwise Hausdorff [11]. We shall prove here that: Theorem 6. PFA(S)[S] implies
.
Forcing over a model of PFA(S), we shall assume we have a name Z˙ S-forced to be a compact countably tight space. For convenience, we shall assume the locally countable subspace Y is actually ω1 , of course not with the latter’s usual topology. It will also be convenient to assume the universe of Z is some cardinal Υ. Re-ordering the open sets if necessary, we may assume we have S-names Wα, Vα , α < ω1 such that S forces: a) Wα and Vα are open, b) α ∈ Wα ⊆ W α ⊆ Vα , c) |Vα ∩ ω1 | ≤ ℵ0 While proving , we shall avoid the necessity of assuming or proving Proposition 2, but we shall use a weaker version of it – see Lemma 12 below. In fact, we shall use a Souslin-tree-forcing version of a standard order for σ-discretizing a given locally countable space. It is a natural variation of the Souslin-tree-forcing version, appearing in Proposition 8.6 of [14], of the standard poset that forces a discrete subspace of a given locally countable space. The idea is to code the locally countable subspace by the generic branch B of the
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Souslin tree and generically partition it into countably many pieces, freezing intersections of the pieces with the witnesses to local countability, so that the intersections remain finite. Toward obtaining , there were three obstacles to overcome. The first was to get rid of the reliance on a closed unbounded set, which seemed to play a prominent role in the proof of Proposition 1 in [13], as well as in the machinery developed in [11]. This was already accomplished by the third author in his unpublished proof of Proposition 4 (see the proof of Proposition 8.6 in [14]). One way to achieve this is to realize that we don’t really need a closed unbounded set of α’s such that all the nodes on the α’th level decide what is happening below α. Details will be found below. The second obstacle was to get ω1 to be the union of countably many “nice” (in our case, discrete) subsets, rather than just having an uncountable nice subset. This complicates the proof, but again was already accomplished in the proof of Proposition 4, as well as in [11] (modulo the removal of the first obstacle). The third obstacle was to improve the σ-discrete of Proposition 4 to σ-closed-discrete within the union of the witnesses to local countability. The observation there is that in order to prove that the locally countable {α : α < ω1 } is σ-closed-discrete within W = {Wα : α < ω1 }, it suffices to partition {α : α < ω1 } into Yn ’s, n < ω, such that each Yn has finite intersection with each Wα . Since the space is T2 , that will ensure that each subset of Yn is closed in W , so that Yn is closed discrete. With these observations, we can now define the required partial order. First, some notation. Construct a strictly increasing sequence {να }α<ω1 of countable ordinals such that να is the least ordinal greater than νβ , β < α, such that να > α, and every node on the να ’th level of S decides Vγ ∩ ω1 , Wγ ∩ ω1 , and W γ ∩ ω1 for all γ ≤ α. We let C 0 = {να : α < ω1 } and for each ν ∈ C 0 denote by ν − the unique α < ω1 such that ν = να . C 0 can then play the same role as successor elements of the club C did in [13] and [11], but that now, γ − : γ ∈ C 0 = ω1 , rather than = C. Let P be the collection of all pairs p = fp , Np where 1) fp is a finite partial function from S | C 0 to ω. (We think of colouring the nodes with ω-many colours.) Let lp denote the maximum element of the range of fp . Let doml fp = {s : fp (s) = l}. We require that each non-empty dom fp consists of nodes of different heights. 2) Np is a finite ∈-chain of countable elementary submodels of Hκ , ∈, <κ where κ is regular and sufficiently large and <κ is a fixed well-ordering of Hκ , containing all relevant objects (in particular, C 0 ), such that Np separates each doml fp in the sense that if s, s ∈ doml fp with s = s , then there is an N ∈ Np such that s ∈ N and s ∈ / N. For p, q ∈ P, we let p ≤ q if and only if: 3) fp | dom fq = fq , 4) Np ⊇ Nq , ˙ ht(t)− ]. 5) (∀t ∈ dom fq )(∀s ∈ dom fp − dom fq )[(s lq . Then fq ∪ {s, m }, Nq is the required extension of q in Ds . 2
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Once we show P is proper and preserves S, we will be able to finish the proof of
:
Proof. Let G be P-generic for the Ds ’s. Let Yn = {ht(s)− : s ∈ B (the generic branch) and for some p ∈ G, fp (s) = n}. Claim ω1 = n<ω Yn . For given γ − ∈ ω1 , γ ∈ C 0 , there is an s ∈ B of height γ. There is a p ∈ G with s ∈ dom fp , say fp (s) = n. Then γ − ∈ Yn . By T2 and Lemma 7, each subset of Yn is closed in W , since once an s of height νβ is coloured n, Yn ’s intersection with Wβ is finite and frozen. By construction then, each Yn is closed discrete in W . 2 We shall now start the proof that P is a proper poset that preserves S. We follow the scheme detailed in [11], but with some improvements. Lemma 8. P is proper and preserves S if, for all sufficiently large regular θ, and for a closed cofinal family C (in [Hθ ]ℵ0 ) of countable elementary submodels M of Hθ with P, S ∈ M , letting δ = M ∩ ω1 , for every p ∈ P ∩ M , there is a q ≤ p such that for all s ∈ S of height δ, q, s is (P × S, M )-generic. Proof. This is due to Miyamoto [9]. There is also a proof in [11].
2
Remark. Because we want our locally countable subspace labeled as ω1 to be σ-discrete rather than merely include an uncountable discrete subspace, we need to have those dense sets Ds meeting our generic filter. This was achieved by Lemma 7. The price we paid, however, was having to weaken the familiar requirement that points be separated by elementary submodels to only require that different points coloured the same be so separated. When we get to the usual method of proving properness by reflecting the piece of a forcing condition outside of an elementary submodel M into M , the fact that points coloured differently need not be separated from each other renders the usual argument more difficult, especially notationally. Remark. It is worth emphasizing that for the next few pages, i.e. until Definition 5, we are developing a general machine for proving a partial order is proper and preserves S. The only requirements we place on the partial order are that: a) the elements of the partial order are of form fp , Np , where fp is a finite set, and Np is a finite ∈-chain of countable elementary submodels, b) the restriction q|M of a condition q to an elementary submodel M is in the partial order and is extended by q, c) some (weakening of the) requirement that Np separates the members of (dom) fp , d) a necessary requirement for fp , Np to extend fq , Nq is that fp | dom fq = fq and Nq ⊆ Np . Actually, in the particular case we are interested in, fp is a finite partial function from S. This simplifies some of the arguments. We shall use some combinatorial lemmas about Souslin trees and elementary submodels. Lemma 9. Let S be a Souslin tree and N a countable elementary submodel of some Hθ containing S. Suppose A ⊆ S, A ∈ N , t ∈ A − N . Suppose there is an s ∈ S ∩ N , s below t. Then there is a u ∈ [s, t) ∩ N such that A is dense above u. Proof. Folklore. See [11]. 2
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Definition 2. A chain is a linearly ordered subset of S. Unless we are considering the generic branch B, we shall only deal with finite chains. For such a finite chain, we write it as a tuple in increasing order. It will be convenient to sometimes consider chains a1 , . . . , am , a0 , where a0 is the maximal element of the chain. We will also consider chains with possible repetitions, in which the ordering of the a1 , . . . , am need not be strict. Definition 3. Let A be a family of chains with possible repetitions of a Souslin tree S. A is dense above s ∈ S if for each s extending s, there is an A ∈ A such that min A extends s . We shall use “s above s” and “s extends s” synonymously, and admit the possibility that s = s. Corollary 10. Let S be a Souslin tree and N a countable elementary submodel of some Hθ containing S. Suppose A is a family of chains with possible repetitions of S, A ∈ N , and suppose there is an A0 ∈ A, min A0 ∈ / N . Suppose s ∈ S ∩ N , s below t = min A0 . Then there is a u ∈ S ∩ N , u ∈ [s, t), such that A is dense above u. Proof. Let A∗ = {min A : A ∈ A}. Apply Lemma 9.
2
Before proceeding further, let us say what “coherent” means, since we will be using it. We quote from [17], Chapter 5. Definition 4. A coherent tree is a downward closed subtree S of <ω1 ω with the property that {ξ ∈ dom s ∩ dom t : s(ξ) = t(ξ)} is finite for all s, t ∈ S. A coherent Souslin tree is a Souslin tree given by a coherent family of functions in <ω1 ω closed under finite modifications. Note that for S a coherent Souslin tree, and s, t on the same (ηth) level of S, there is a canonical S isomorphism σst between the cones above (we think of our trees as growing upwards) s and t, defined by S letting σst (s )(α) be t(α) if α < η and s (α) otherwise, for each s extending s. These isomorphisms are such S S S S S −1 that σsu = σtu ◦ σst and σst = (σts ) . Let θ be a sufficiently large regular cardinal bigger than κ, and let M be a countable elementary submodel of Hθ containing everything relevant, in particular, S, a fixed well ordering of Hθ , and a fixed non-principal ultrafilter U 0 on ω. We may assume without loss of generality that the well ordering extends the tree ordering of the nodes of S, and is such that each level precedes the next one. (There are closed cofinally many such M .) Let δ = M ∩ ω1 . Let p ∈ P ∩ M . Let p M = fp , Np ∪ {M ∩ Hκ } . Then, by a standard argument, p M ∈ P. Let tM be an arbitrary node at the δth level of S. We shall show p M, tM is generic. Let D ∈ M be a given dense open subset of P × S and let q, t0 be a given extension of p M, tM . We need to show q, t0 is compatible with some member of D ∩ M . Extending q, t0 , we may assume that q, t0 ∈ D. Moreover, by extending further (since D is open), we may assume that t0 is not in the largest model of Nq , and that this model contains (all the members of dom) fq and ht(t0 ) > N ∩ ω1 , for every N ∈ Nq . Let qM = q | M = fq ∩M, Nq ∩M . Note that qM ∈ P ∩M and that q ≤ qM . That qM is in M is clear. To see that it is in P, the only point at issue is clause 2) — could the N ’s separating s and s ∈ doml fq ∩ M all have been in Nq −M ? Consider an N ∈ Nq such that ht(s) ∈ N and ht(s ) ∈ / N . Since ht(s ) ∈ M ∩Hκ ∈ Nq , N must be a member of M ∩ Hκ , so N ∈ NqM = Nq ∩ M . That q ≤ qM is assured by the way we defined the forcing. We then may assume that the maximal model NqM of NqM contains all the members of dom fqM (= dom fq ∩ M ), else we could have extended Nq to ensure this. Let δM be the intersection of ω1 with NqM . By taking NqM large enough, we may ensure that the projection of (dom fq ∪ {t0 }) − M on the δth level of S has the same size as its projection on the δM th level.
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Let {u0 , . . . , un }, {v0 , . . . , vn } respectively enumerate these projections on the δM th and δth levels, such that ui = vi | δM , i ≤ n, and such that u0 = t0 | δM and v0 = t0 | δ. For 0 ≤ i, j ≤ n, let σij be the canonical −1 isomorphism which moves ui to uj . Note σij = σji , and σii is the identity isomorphism. In [11] we observed that, defining S η = {s ∈ S : ht(s) ≥ η} , where η is the level we’re shifting cones over, σij extends to an automorphism of S η , and then extends to S η -names by recursively defining σij (x) ˙ = {σij (y), ˙ σij (u) : y˙ is an S η -name and u ∈ S η and y, u ∈ x} ˙ . We shall omit the “sub ij” except where needed. We then noted that σ(ˇ x) = x ˇ, for any x ∈ V . We also observed that if the only parameters in φ are S η -names y˙ 1 , . . . , y˙ m , and if s ∈ S η , then s S φ(y˙ 1 , . . . , y˙ m ) if and only if σ(s) S η φ(σ(y˙1 ), . . . , σ(y˙ m )). Since S η is dense in S, we can conclude s S φ(y˙ 1 , . . . , y˙ m ) if and only if σ(s) S φ(σ(y˙ 1 ), . . . , σ(y˙m )).
(*)
We shall make use of this later. For any r, t ∈ P × S, define: Fr,t = {x ∈ (dom fr ∪ {t}) − NqM : for some ix , x | δM = uix and σ0ix (t) extends x}. Then we have: Fq,t0 = {x ∈ (dom fq ∪ {t0 }) − M : x | δ = some vix and σ0ix (t0 ) extends x} . If vi and vj are projections of elements of Fq,t0 , then σij (vi ) = vj . To see this, first note that if x ∈ Fq,t0 extends vi , then σi0 (vi ) ≤ σi0 (x) ≤ t0 . Hence σi0 (vi ) = v0 , since both are of height δ below t0 . It follows that σij (vi ) = σ0j ◦ σi0 (vi ) = σ0j (v0 ) = vj . By taking the maximal model and hence δM sufficiently large, we have that for such vi and vj , vi | [δM , δ) = vj | [δM , δ). For s above some ui , it will be convenient to write “ˆ s” for σi0 (s). Note that the fixed well order we have introduced allows us to consider any finite subset of S as an ordered tuple. For an (m + 1)-tuple s = s1 , . . . , sm , s0 , if sˆj is defined for 0 ≤ j ≤ m, let sˆ = ˆ s1 , . . . , sˆm , sˆ0 . Let Fˆq,t0 = ˆ x : x ∈ Fq,t0 . Similarly define l Fq,t = {t0 } ∪ {x ∈ dom fq ∩ Fq,t0 : fq (x) = l}, 0 l and Fˆq,t for l ∈ Lq = {l : doml fq = 0}. Note that Fˆq,t0 is a chain with possible repetitions. We can make 0 l analogous definitions of Fˆr,t , etc. Let cl = |Fq,t |, and c = {cl : l ∈ Lq }. 0 Let D0 = {r, t ∈ D : r, t ≤ qM , u0 and:
i) qM is an initial part of r, i.e. for each l, doml fr = doml fqM ∪ Fl , where Fl ⊆ S and each x ∈ doml fqM precedes the members of Fl in the well order on S; Nr = NqM ∪ N, where N is a finite ∈-chain, the first element of which contains NqM , ii) the height of each node in Fr,t − FqM ,u0 is > δM ,
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l iii) Lr = Lq , each |Fr,t | = cl , l l iv) fr (the jth element of Fr,t ) = fq (the jth element of Fq,t ) (recall we have ordered the nodes, which 0 l l l induces an order of Fr,t and Fq,t0 ), in other words, if x is in the jth position of Fr,t , then fr (x) is the l same value as fq (y) for whichever y is in the jth position of Fq,t0 , v) the height of t is greater than the height of any of the nodes in dom fr , vi) |Nr | = |Nq |.}
D0 is thus the set of all r, t ∈ D below qM , u0 that “look like q, t0 ”. We won’t actually use all of these requirements; we list them for possible future applications. Notice that the ui ’s and hence the σij ’s are in M ; Lq and |Nq | are just some finite sets of natural numbers and so are in M ; iv) can be described in terms of the order and a finite set of natural numbers; and so D0 ∈ M by definability. F = Fˆ : F = Fr,t for some r, t ∈ D0 is also in M and in Hκ as well. Since M ∩ Hκ ∈ N for each N ∈ Nq − M , it follows that F ∈ N , for all such N . Note that Fˆq,t0 ∈ F. l Note also that the terms of Fˆq,t are separated by models of Nq . 0 l Let Nq be a minimal subchain of Nq containing M ∩ Hκ at its bottom and separating Fq,t for each l. 0 ˆ Let Nq = {Na }a≤m ordered by inclusion, with N0 = M ∩ Hκ . Fq,t0 is a chain with possible repetitions; let us write it as: x1 , . . . , xm , t0
where xa = xa,1 , . . . , xa,da enumerates in increasing order Fˆq,t0 ∩ (Na − Na−1 ), a ≥ 1. Thus the length of the vector xa is equal to the size of Fq,t0 ∩ (Na − Na−1 ). Since F ∈ Nm ,
ˆ1 , . . . , ˆm , x ∈ F F(x1 , . . . , xm ) = x ∈ S : x x ˆm , t0 ), such that F(x1 , . . . , xm ) is dense above ym . ∈ Nm . By Lemma 9, there is a ym ∈ Nm ∩ S, ym ∈ [max x Next, consider:
F(x1 , . . . , xm−1 ) = x, y ∈ S dm +1 : x, y is a chain and F(x1 , . . . , xm−1 , x) is dense above y . ˆm , ym ∈ F(x1 , . . . , xm−1 ). As before, this time by Corollary 10, Then F(x1 , . . . , xm−1 ) ∈ Nm−1 and x ˆm−1 , ym , Nm−1 playing the roles of A, A0 , N respectively, we can find a ym−1 ∈ with F(x1 , . . . , xm−1 ), x ˆm−1 , min ˆm ), such that F( Nm−1 ∩ S, ym−1 ∈ [max x x x1 , . . . , xm−1 ) is dense above ym−1 . Continuing, in m steps we find a y0 ∈ N0 , y0 ∈ [u0 , v0 ), such that:
F(∅) = x, y ∈ S d1 +1 : x, y is a chain and F(x) is dense above y is ∈ N0 and dense above y0 . Let X˙1 be a name for: ξ ∈ [C ◦ ]d1 : for some z, w ∈ F(∅), {z, w} ⊆ B and for each d, ht(πd (z))− = πd (ξ) . Then X˙1 ∈ M . Claim. y0 X˙1 is uncountable and indeed for each d ≤ d1 , πd (X˙1 ) is uncountable.
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Proof. Given any α < ω1 , and any y0 extending y0 , since F(∅) is dense above y0 , we can find a z, w ∈ F(∅) extending y0 with minimal element of height greater than α. Take y0 above z, w . Then − − − y0 ht(z1 ) , . . . , ht(zd1 ) ∈ X˙1 and each ht(zd ) is ≥ α. 2 To prove , we need to deal with an arbitrary finite number of colours in the range of fq . Call the difference between Nk and Nk+1 , both in Nq , a layer. Without loss of generality, we shall assume each layer has non-empty intersection with dom fq . That intersection may be coloured by several colours; we can’t separate them, so we must remove parameters layer-by-layer. We must also reflect the layers layer-by-layer for the same reason. The way we will do this is essentially by regarding m-tuples of points of Z as points in Z m . Now it is finally time to leave the general scheme and start working with the topology needed for our situation here. The following definitions and lemmas used by the third author were crucial for his proof that PFA(S)[S] implies there are no compact S-spaces, and also are crucial for the proof of . Definition 5. Let U be a non-principal ultrafilter on a countable set T . We say a sequence of points xτ : τ ∈ T in a space X U -converges to a point x ∈ X, and write limτ →U xτ = x, if for each open set V containing x, {τ ∈ T : xτ ∈ V } ∈ U . Lemma 11. Suppose W ⊆ X, limτ →U T xτ = x, and x ∈ / W . Then τ ∈ T : xτ ∈ / W ∈ U. Proof. If τ ∈ T : xτ ∈ /W ∈ / U , then τ ∈ T : xτ ∈ W ∈ U. But then x ∈ W . To see this, let V be open containing x. Then {τ : xτ ∈ V } ∈ U. Then τ : xτ ∈ W ∩ V ∈ U. Then V ∩ W = ∅. But then V ∩ W = ∅. 2 Definition 6. Y ⊆ X is U -sequentially closed if whenever xt : t ∈ T is a sequence from Y that U-converges, then its U-limit is a member of Y . X is U -sequential if every U-sequentially closed subset of X is closed. Lemma 12. ([14, 8.2]) PFA(S)[S] implies that if X is compact countably tight, then for each ground model (i.e. before forcing with S) ultrafilter U on ω, X is U-sequential. In order to prove , it is helpful to reformulate “U-sequentially closed” so that if p is in the U -sequential closure of a set A, p ∈ / A, then there will be a sequence in A converging to p in some sense. The following definitions and lemma, taken almost verbatim from [14], accomplish this. Definition 7. A well-founded U-tree is a collection T of finite subsets of ω such that: 1) ∅ ∈ T , and if τ ∈ T , all initial segments of τ are in T ; 2) if τ ∪ {m} ∈ T for some m > max(τ ), then {n < ω : n > max(τ ) and τ ∪ {n} ∈ T } ∈ U ; 3) there is no infinite U ⊆ ω such that all finite initial segments of U belong to T . Let ∂T be the collection of all terminal nodes of T , i.e. those nodes with no proper end-extensions in T . By 3), there are such terminal nodes. Then for every compact Hausdorff space K, any assignment {xτ : τ ∈ ∂T } ⊆ K extends uniquely to the global assignment {xτ : τ ∈ T } defined recursively on the rank of T (see e.g. Exercise III (12) of [4]) as follows. If T has rank 1, i.e. T = {∅} ∪ {{n} : n ∈ U } for some U ∈ U , let x∅ = limn→U x{n} . If rank T > 1, for each {n} ∈ T apply the inductive hypothesis and extend the
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assignment {τ ∈ ∂T : n = min(τ )} to the assignment {τ ∈ T : n = min(τ )} and again let x∅ = limn→U x{n} . We call x∅ the limit of the assignment {xτ : τ ∈ ∂T } and write x∅ = lim xτ . τ ∈∂T
Having defined this limit, we note that it is in fact the limit of an ultrafilter on ∂T defined as follows. To see this, first define a U -subtree to be any T0 ⊆ T such that: 4) ∅ ∈ T0 ; 5) if τ ∈ T0 − ∂T , then {n < ω : n > max(τ ) and τ ∪ {n} ∈ T0 } ∈ U . Inducting on rank, we see that {∂T0 : T0 a U -subtree of T } generates an ultrafilter U T on ∂T such that for every assignment {xτ : τ ∈ ∂T } of points in some Hausdorff space X, we have lim xτ = lim xτ ,
τ ∈∂T
τ →U T
where the right hand side is as defined above Lemma 11, whenever one of these limits exists. Then: Lemma 13. ([14, 8.1]) Let K be compact and Y ⊆ K. Let Y Y
U
U
be the U-sequential closure of Y in K. Then
=
lim xτ : T a well-founded U -tree and {xτ : τ ∈ ∂T } ⊆ Y
τ ∈∂T
We now return to the proof of
.
. The following topological fact is crucial.
Lemma 14. ([8]) A finite product of compact countably tight spaces is compact and countably tight. This will allow us to apply U 0 -sequentiality to a finite product of copies of Z. Actually, we shall work with a finite product of σi0 (Z)’s. Note U 0 remains an ultrafilter after forcing with S, since no new subsets of ω are added. For x ∈ dom fq − M , let ix be that i such that x is above vi . Consider Z˙ 1 = (the name of)
0 ˙ 1≤d≤d1 σix1,d 0 (Z). By Lemmas 12 and 14, this space is forced to be compact and U -sequential. There is a name p˙ 1 ∈ M which is forced by y0 to be a complete accumulation point for X˙1 as a subspace of Z˙ 1 and to have its projections be complete accumulation points. By Lemma 12, Lemma 13, and elementarity, there is a name T˙1 ∈ M for a well-founded U 0 -tree T1 , and a name in M for an assignment {ξτ : τ ∈ ∂T1 } ˙ with components in δ such that y0 forces {ξτ : τ ∈ ∂ T˙1 } ⊆ X˙1 and UT0˙ -converges to p˙ 1 . There is a limit 1 ˙ level ζ, δM < ζ < δ, at which all nodes of that level decide T˙1 and the ξτ , τ ∈ ∂ T˙1 . Furthermore, since ˙ y0 {ξτ : τ ∈ ∂ T˙1 } ⊆ X˙1 , by elementarity, we could also take ζ sufficiently high below δ such that the nodes ˙ of that level which are above y0 decide a zτ , wτ (ξτ ) – say the least one – ∈ F(∅) ∩ M , for each of the ξτ , τ ∈ ∂ T˙1 that they decide. In particular, there is a y0 , y0 < y0 < v0 , such that y0 makes these decisions. ˙ Since πd (p˙ 1 ) is forced by y0 to be a complete accumulation point, y0 forces πd (p˙ 1 ) ∈ / σix1,d 0 (W ht(x)− ), ˙ ˙ for any x ∈ dom fq − M . Therefore there is a name for an open Od in Z about πd (p1 ) such that y forces 0
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˙ ˙ σix1,d (O˙ d ) is disjoint from all those σix1,d 0 (W ht(x)− )’s. Therefore, by (*), σ0ix1,d (y0 ) forces Od is disjoint ˙ − ’s. from those W ht(x)
˙ ˙ Now since y0 forces {ξτ : τ ∈ ∂ T˙1 } UˇT01 -converges to p˙ 1 , it also forces {πd (ξτ ) : τ ∈ ∂ T˙1 } UˇT01 -converges to πd (p˙ 1 ). Then
ˇ y0 (∃U ∈ Uˇ0 )(∀τ ∈ U)(πd (ξτ ) ∈ O˙ d ),
so ˇ σ0ix1,d (y0 ) (∃U ∈ Uˇ0 )(∀τ ∈ U)(πd (ξτ ) ∈ σ0ix1,d (O˙ d )).
x1,d is above vix1,d and hence above σ0ix1,d (y0 ). Thus, ˇ ˙ ht(x )− ). x1,d (∃U ∈ Uˇ0 )(∀τ ∈ U)(πd (ξτ ) ∈ /W 1,d
Then there is an x1,d extending x1,d and a Ud ∈ U 0 such that ˇ ˙ ht(x )− ). x1,d (∀τ ∈ Ud )(πd (ξτ ) ∈ /W 1,d
˙ ht(x )− ∩ ω1 , so But x1,d decides W 1,d ˇ ˙ ht(x )− ). x1,d (∀τ ∈ Ud )(πd (ξτ ) ∈ /W 1,d
Notice that the same argument will work for any other x ∈ dom fq −M such that ix = ix1,d so as to conclude that there is a Ux ∈ U 0 such that ˇ ˙ ht(x)− ). x (∀τ ∈ Ux )(πd (ξτ ) ∈ /W
Let U1 =
{Ud : 1 ≤ d ≤ d1 } ∩
{Ux : ix = ix1,d for some d, 1 ≤ d ≤ d1 , x ∈ dom fq − M }.
Let τ1† be the least τ ∈ U1 , according to our fixed well-order. Consider zτ † , wτ † (ξτ † ) (the one y0 1 1 1 decided). Let
z1 = σ0i1,1 (π1,1 (zτ † ), . . . , σ0i1,d1 (π1,d1 (zτ † )) .
1
1
ˆ1 = ˆ1 , w † ∈ F(∅). Then z zτ † , and so z τ 1
1
We now need to iteratively peel off the remaining layers of Fq,t0 − {t0 }. Let X˙2 be a name for: ξ ∈ [C ◦ ]d2 : for some z, w , z, w ∈ F(z1 ), {z, w} ⊆ B and for each d, ht(πd (z))− = πd (ξ)}. We now carry out the same argument as before, forcing X˙2 to be uncountable, with all its projections uncountable, etc. ˆ1 , . . . , ˆm = Continuing, after m steps we will get z1 , . . . , zm with z z z 1 , . . . , z m for some z 1 , . . . , z m , w ∈ F. But then there is an r, t ∈ D0 ∩ M , such that Fˆr,t = z 1 , . . . , z m , t . Then w = t. By
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construction, w is below some y < v0 and so t ≤ t0 . We claim that r, t is compatible with q, t0 , which will finish the proof. Since r ≤ qM , it follows that fr ∪ fq is a function. We next note that Nr ∪ Nq is an ∈-chain, for by construction, Nr ∈ M , so Nr ∪ {M ∩ Hκ } is an ∈-chain. Now Nq = NqM ∪ (Nq − NqM ); the elements N of Nq − NqM all have M ∩ Hκ in them, for if not, such an N would be in M . Nr ∪ Nq is thus the ∈-chain Nr ∪ {M ∩ Hκ } ∪ (Nq − NqM ). Let R = fr ∪ fq , Nr ∪ Nq , t0 . Nr ∪ Nq separates doml (fr ∪ fq ) since r ≤ qM . Thus fr ∪ fq , Nr ∪ Nq ∈ P. We need to check that it extends r and q. The only case of interest is when x extends s, x ∈ dom fq − M , and s ∈ dom fr − dom fqM . Thus s = zd , for some component zd of some component of r. But then ht(s)− is some πd (ξτ † ) for some k
ξτ † arising in the construction. Then s is below some xk,d ∈ Fq,t0 − {t0 }, so s is below vik,d . But x is k ˙ ht(x)− as required. Thus above s, so x is above vik,d . So ik,d = ix . But then, as before, x forces ht(s)− ∈ /W fr ∪ fq , Nr ∪ Nq ∈ P and is below both r and q. But then R ∈ P × S is below both r, t and q, t0 as required. 2 The σ-closed-discrete within an open set formulation of is somewhat awkward. As Balogh noticed in [2], the following more easily stated version is what one really uses: ∗
: Let X be a locally compact space of Lindelöf number ℵ1 which does not include any perfect pre-images of ω1 . Let Y ⊆ X of size ℵ1 be (globally) locally countable, in the sense that each x ∈ X has a neighbourhood meeting countably many members of Y . Then Y is σ-closed-discrete.
Since: Lemma 15. ([1]) The one-point compactification of a locally compact space X is countably tight if and only if X does not include a perfect pre-image of ω1 . we see that
implies
∗
and so we have:
Corollary 16. PFA(S)[S] implies
∗
.
Acknowledgements In conclusion, let us thank the very careful and preternaturally patient referee of this manuscript for finding a number of crucial errors (and a larger number of less crucial ones), and Paul Larson, Assaf Rinot, and Alan Dow for suggesting many improvements. References [1] Z.T. Balogh, Locally nice spaces under Martin’s axiom, Comment. Math. Univ. Carol. 24 (1) (1983) 63–87. [2] Z.T. Balogh, Locally nice spaces and Axiom R, Topol. Appl. 125 (2) (2002) 335–341. [3] K.J. Devlin, The Yorkshireman’s guide to proper forcing, in: A.R.D. Mathias (Ed.), Surveys in Set Theory, in: London Math. Soc. Lecture Note Ser., vol. 87, Cambridge Univ. Press, Cambridge, 1983, pp. 60–115. [4] K. Kunen, Set Theory, North-Holland, Amsterdam, 1980. [5] P. Larson, F.D. Tall, Locally compact perfectly normal spaces may all be paracompact, Fundam. Math. 210 (2010) 285–300. [6] P. Larson, F.D. Tall, On the hereditary paracompactness of locally compact hereditarily normal spaces, Can. Math. Bull. 57 (2014) 579–584. [7] P. Larson, S. Todorcevic, Katětov’s problem, Trans. Am. Math. Soc. 354 (5) (2002) 1783–1791. [8] V. Malyhin, On tightness and Souslin number in exp X and in a product of spaces, Sov. Math. Dokl. 13 (1972) 496–499. [9] T. Miyamoto, ω1 -Souslin trees under countable support iterations, Fundam. Math. 142 (1993) 257–261. [10] Z. Szentmiklóssy, S-spaces and L-spaces under Martin’s axiom, in: Topology, 1978, in: Coll. Math. Soc. Janós Bolyai, vol. 23, North-Holland, Amsterdam, 1980, pp. 1139–1145.
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[11] F.D. Tall, PFA(S)[S]: more mutually consistent topological consequences of PFA and V = L, Can. J. Math. 64 (2012) 1182–1200. [12] F.D. Tall, PFA(S)[S] and locally compact normal spaces, Topol. Appl. 162 (2014) 100–115. [13] S. Todorcevic, Chain condition methods in topology, II, Note of 01/10/01. [14] S. Todorcevic, Forcing with a coherent Souslin tree, Can. J. Math. (2015), to appear. [15] S. Todorcevic, Handwritten slides from his lectures at Advances in Set-Theoretic Topology, a Conference in Honor of T. Nogura, Erice, Italy, 2008. [16] S. Todorcevic, Partition Problems in Topology, Contemporary Mathematics, vol. 84, American Mathematical Society, Providence, RI, 1989. [17] S. Todorcevic, Walks on Ordinals and Their Characteristics, Progress in Mathematics, vol. 263, Birkhäuser Verlag, Basel, 2007. [18] W.S. Watson, Locally compact normal spaces in the constructible universe, Can. J. Math. 34 (5) (1982) 1091–1096.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Forcing with a coherent Souslin tree and locally countable subspaces of countably tight compact spaces A.J. Fischer a , F.D. Tall b,∗,1 , S. Todorcevic b,2 a b
Kurt Gödel Research Centre for Mathematical Logic, University of Vienna, Vienna, Austria Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada
a r t i c l e
i n f o
Article history: Received 25 March 2014 Received in revised form 22 July 2015 Available online 1 October 2015
a b s t r a c t We apply a method of Todorcevic in showing that PFA(S) implies that the coherent Souslin tree S forces various strengthenings of the assertion that every locally countable space of size ℵ1 that has a countably tight compactification must be σ-discrete. © 2015 Elsevier B.V. All rights reserved.
MSC: 54A35 54D65 54D55 54DA25 54D18 54D20 03E35 03E57 Keywords: PFA(S)[S] Forcing with a coherent Souslin tree (Locally) compact Countably tight Locally countable σ-discrete
1. Introduction The third author in 2001 developed technology for analyzing the forcing by a fixed Souslin tree S over a model obtained by iterating proper posets that preserve S. Parts of this work were circulated in the form of notes such as [13] and have been presented at several conferences.3 Now this work is finally available * Corresponding author. E-mail addresses: arthur.j.fi[email protected] (A.J. Fischer), [email protected] (F.D. Tall), [email protected] (S. Todorcevic). 1 Research supported by NSERC grant A-7354. 2 Research supported by NSERC grant 455916. 3 Such as, for example, the three lectures at the Erice conference on set-theoretic topology in 2008 (see [15]). http://dx.doi.org/10.1016/j.topol.2015.09.035 0166-8641/© 2015 Elsevier B.V. All rights reserved.
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in printed form in [14]. We shall use this technology to deduce a particular topological principle about subspaces of compact countably tight spaces. The proof that PFA(S)[S] implies that we give here uses the same machinery and notation as the proof in [11] that PFA(S)[S] implies locally compact normal spaces are ℵ1 -collectionwise Hausdorff. That machinery is not so elegant as that in [14], but it has the advantage of already being published, so that the reader who wants more details can find them there. We just make one small improvement over [11]: improving coding so as to obtain “σ-discrete”, rather than “σ-discrete on a club”. The other differences are related to the fact that we face two entirely different topological situations. It is quite remarkable that the set-theoretic techniques are the same for both. The [11] machinery (based on [13]) is easier to understand for topologists not so expert at forcing with elementary submodels as side conditions. We will take advantage of [11] actually being published to omit some explanatory material and details available there, since our proof here closely resembles the proof there. (In fact, the proof there was inspired by the idea of the proof here.) We define coherent Souslin trees below. At this point, all the reader need know is that they may be obtained from ♦. Given a coherent Souslin tree S, PFA(S) is the restriction of PFA to those proper posets preserving (the Souslinity of) S. Since countable support iterations of such posets preserve S [9], PFA(S) may be obtained by the usual Laver-diamond [3] method from a supercompact cardinal. We use the notation PFA(S)[S] implies Φ to mean given any model of PFA(S), forcing over it with S yields a model of Φ. We say Φ holds in a model of form PFA(S)[S] if forcing with S over a particular model of PFA(S) yields a model of Φ. Definition 1. An S-space is a hereditarily separable, T3 , non-Lindelöf space. The double use of ‘S’ is unfortunate, but it will always be clear which S is intended. MAω1 implies there are no compact S-spaces [10]; PFA implies there are no S-spaces [16]. Setting out the context for what we will be doing, we recall that Todorcevic in 2001 proved the following result (see [14]). Proposition 1. ([14]) PFA(S)[S] implies there are no compact S-spaces. One of the crucial parts of the proof is contained in the following result. Proposition 2. ([14]) PFA(S)[S] implies compact countably tight spaces are sequential. A particular application of this result was given in [5]: Proposition 3. ([5]) There is a model of form PFA(S)[S] in which every locally compact perfectly normal space is paracompact. The use of Proposition 2 in the proof of Proposition 3 was the application of an equivalent version of the result stated in the Abstract: Proposition 4. PFA(S)[S] implies that in a compact countably tight space, locally countable subspaces of size ℵ1 are σ-discrete. This follows from Proposition 2 plus: Proposition 5. PFA(S)[S] implies that in a compact sequential space, locally countable subspaces of size ℵ1 are σ-discrete.
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The conclusion of Proposition 4 was proved from MAω1 by Z.T. Balogh in [1], extending Szentmiklóssy’s work [10]. Propositions 4 and 5 were proved by Todorcevic in 2001 and can now be seen to follow from the proof of Theorem 8.6 in [14]. 2. The method and the proof We shall assume that all spaces considered are T2 . Notice that Proposition 4 is an extension of Proposition 1. Proposition 4 is proved using a “machine” that has little to do with the particular problem we are applying it to. The mechanism – due to the third author – is expounded at length in [14]. Proposition 4 (plus [7], where the weaker “axiom” we now call “MAω1 (S)[S]” is deployed) suffices to justify all the applications of PFA(S)[S] in [5]. However for [6] and [12] we need the implication from PFA(S)[S] to the following stronger principle considered in [1]. : Let X be a compact countably tight space. Let Y ⊆ X, |Y | = ℵ1 . Suppose {Wα }α∈ω1 , {Vα }α∈ω1 are open subsets of X such that 1. W α ⊆ Vα , 2. |Vα ∩ Y | ≤ ℵ0 , 3. Y ⊆ {Wα : α < ω1 }. Then Y is σ-closed-discrete in {Wα : α < ω1 }.
and its consequences, including the one mentioned in the Abstract, are useful in proving certain locally compact normal ℵ1 -collectionwise Hausdorff spaces are paracompact (where a space is ℵ1 -collectionwise Hausdorff if every closed discrete subspace of size ℵ1 can be separated by a disjoint collection of open sets). In [12] there are several examples of such arguments, which originated in [1]. It is then clearly of interest to obtain in models of set theory in which locally compact normal spaces are ℵ1 -collectionwise Hausdorff. That is what we shall do here. That PFA(S)[S] implies locally compact normal spaces are ℵ1 -collectionwise Hausdorff was proved by the second author [11]. Getting these two statements holding in the same model is not so easy. This is to be expected, since the second one is an important consequence of V = L [18]. In fact, there is a model of PFA(S)[S] in which locally compact normal spaces are collectionwise Hausdorff [11]. We shall prove here that: Theorem 6. PFA(S)[S] implies
.
Forcing over a model of PFA(S), we shall assume we have a name Z˙ S-forced to be a compact countably tight space. For convenience, we shall assume the locally countable subspace Y is actually ω1 , of course not with the latter’s usual topology. It will also be convenient to assume the universe of Z is some cardinal Υ. Re-ordering the open sets if necessary, we may assume we have S-names Wα, Vα , α < ω1 such that S forces: a) Wα and Vα are open, b) α ∈ Wα ⊆ W α ⊆ Vα , c) |Vα ∩ ω1 | ≤ ℵ0 While proving , we shall avoid the necessity of assuming or proving Proposition 2, but we shall use a weaker version of it – see Lemma 12 below. In fact, we shall use a Souslin-tree-forcing version of a standard order for σ-discretizing a given locally countable space. It is a natural variation of the Souslin-tree-forcing version, appearing in Proposition 8.6 of [14], of the standard poset that forces a discrete subspace of a given locally countable space. The idea is to code the locally countable subspace by the generic branch B of the
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Souslin tree and generically partition it into countably many pieces, freezing intersections of the pieces with the witnesses to local countability, so that the intersections remain finite. Toward obtaining , there were three obstacles to overcome. The first was to get rid of the reliance on a closed unbounded set, which seemed to play a prominent role in the proof of Proposition 1 in [13], as well as in the machinery developed in [11]. This was already accomplished by the third author in his unpublished proof of Proposition 4 (see the proof of Proposition 8.6 in [14]). One way to achieve this is to realize that we don’t really need a closed unbounded set of α’s such that all the nodes on the α’th level decide what is happening below α. Details will be found below. The second obstacle was to get ω1 to be the union of countably many “nice” (in our case, discrete) subsets, rather than just having an uncountable nice subset. This complicates the proof, but again was already accomplished in the proof of Proposition 4, as well as in [11] (modulo the removal of the first obstacle). The third obstacle was to improve the σ-discrete of Proposition 4 to σ-closed-discrete within the union of the witnesses to local countability. The observation there is that in order to prove that the locally countable {α : α < ω1 } is σ-closed-discrete within W = {Wα : α < ω1 }, it suffices to partition {α : α < ω1 } into Yn ’s, n < ω, such that each Yn has finite intersection with each Wα . Since the space is T2 , that will ensure that each subset of Yn is closed in W , so that Yn is closed discrete. With these observations, we can now define the required partial order. First, some notation. Construct a strictly increasing sequence {να }α<ω1 of countable ordinals such that να is the least ordinal greater than νβ , β < α, such that να > α, and every node on the να ’th level of S decides Vγ ∩ ω1 , Wγ ∩ ω1 , and W γ ∩ ω1 for all γ ≤ α. We let C 0 = {να : α < ω1 } and for each ν ∈ C 0 denote by ν − the unique α < ω1 such that ν = να . C 0 can then play the same role as successor elements of the club C did in [13] and [11], but that now, γ − : γ ∈ C 0 = ω1 , rather than = C. Let P be the collection of all pairs p = fp , Np where 1) fp is a finite partial function from S | C 0 to ω. (We think of colouring the nodes with ω-many colours.) Let lp denote the maximum element of the range of fp . Let doml fp = {s : fp (s) = l}. We require that each non-empty dom fp consists of nodes of different heights. 2) Np is a finite ∈-chain of countable elementary submodels of Hκ , ∈, <κ where κ is regular and sufficiently large and <κ is a fixed well-ordering of Hκ , containing all relevant objects (in particular, C 0 ), such that Np separates each doml fp in the sense that if s, s ∈ doml fp with s = s , then there is an N ∈ Np such that s ∈ N and s ∈ / N. For p, q ∈ P, we let p ≤ q if and only if: 3) fp | dom fq = fq , 4) Np ⊇ Nq , ˙ ht(t)− ]. 5) (∀t ∈ dom fq )(∀s ∈ dom fp − dom fq )[(s
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Once we show P is proper and preserves S, we will be able to finish the proof of
:
Proof. Let G be P-generic for the Ds ’s. Let Yn = {ht(s)− : s ∈ B (the generic branch) and for some p ∈ G, fp (s) = n}. Claim ω1 = n<ω Yn . For given γ − ∈ ω1 , γ ∈ C 0 , there is an s ∈ B of height γ. There is a p ∈ G with s ∈ dom fp , say fp (s) = n. Then γ − ∈ Yn . By T2 and Lemma 7, each subset of Yn is closed in W , since once an s of height νβ is coloured n, Yn ’s intersection with Wβ is finite and frozen. By construction then, each Yn is closed discrete in W . 2 We shall now start the proof that P is a proper poset that preserves S. We follow the scheme detailed in [11], but with some improvements. Lemma 8. P is proper and preserves S if, for all sufficiently large regular θ, and for a closed cofinal family C (in [Hθ ]ℵ0 ) of countable elementary submodels M of Hθ with P, S ∈ M , letting δ = M ∩ ω1 , for every p ∈ P ∩ M , there is a q ≤ p such that for all s ∈ S of height δ, q, s is (P × S, M )-generic. Proof. This is due to Miyamoto [9]. There is also a proof in [11].
2
Remark. Because we want our locally countable subspace labeled as ω1 to be σ-discrete rather than merely include an uncountable discrete subspace, we need to have those dense sets Ds meeting our generic filter. This was achieved by Lemma 7. The price we paid, however, was having to weaken the familiar requirement that points be separated by elementary submodels to only require that different points coloured the same be so separated. When we get to the usual method of proving properness by reflecting the piece of a forcing condition outside of an elementary submodel M into M , the fact that points coloured differently need not be separated from each other renders the usual argument more difficult, especially notationally. Remark. It is worth emphasizing that for the next few pages, i.e. until Definition 5, we are developing a general machine for proving a partial order is proper and preserves S. The only requirements we place on the partial order are that: a) the elements of the partial order are of form fp , Np , where fp is a finite set, and Np is a finite ∈-chain of countable elementary submodels, b) the restriction q|M of a condition q to an elementary submodel M is in the partial order and is extended by q, c) some (weakening of the) requirement that Np separates the members of (dom) fp , d) a necessary requirement for fp , Np to extend fq , Nq is that fp | dom fq = fq and Nq ⊆ Np . Actually, in the particular case we are interested in, fp is a finite partial function from S. This simplifies some of the arguments. We shall use some combinatorial lemmas about Souslin trees and elementary submodels. Lemma 9. Let S be a Souslin tree and N a countable elementary submodel of some Hθ containing S. Suppose A ⊆ S, A ∈ N , t ∈ A − N . Suppose there is an s ∈ S ∩ N , s below t. Then there is a u ∈ [s, t) ∩ N such that A is dense above u. Proof. Folklore. See [11]. 2
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Definition 2. A chain is a linearly ordered subset of S. Unless we are considering the generic branch B, we shall only deal with finite chains. For such a finite chain, we write it as a tuple in increasing order. It will be convenient to sometimes consider chains a1 , . . . , am , a0 , where a0 is the maximal element of the chain. We will also consider chains with possible repetitions, in which the ordering of the a1 , . . . , am need not be strict. Definition 3. Let A be a family of chains with possible repetitions of a Souslin tree S. A is dense above s ∈ S if for each s extending s, there is an A ∈ A such that min A extends s . We shall use “s above s” and “s extends s” synonymously, and admit the possibility that s = s. Corollary 10. Let S be a Souslin tree and N a countable elementary submodel of some Hθ containing S. Suppose A is a family of chains with possible repetitions of S, A ∈ N , and suppose there is an A0 ∈ A, min A0 ∈ / N . Suppose s ∈ S ∩ N , s below t = min A0 . Then there is a u ∈ S ∩ N , u ∈ [s, t), such that A is dense above u. Proof. Let A∗ = {min A : A ∈ A}. Apply Lemma 9.
2
Before proceeding further, let us say what “coherent” means, since we will be using it. We quote from [17], Chapter 5. Definition 4. A coherent tree is a downward closed subtree S of <ω1 ω with the property that {ξ ∈ dom s ∩ dom t : s(ξ) = t(ξ)} is finite for all s, t ∈ S. A coherent Souslin tree is a Souslin tree given by a coherent family of functions in <ω1 ω closed under finite modifications. Note that for S a coherent Souslin tree, and s, t on the same (ηth) level of S, there is a canonical S isomorphism σst between the cones above (we think of our trees as growing upwards) s and t, defined by S letting σst (s )(α) be t(α) if α < η and s (α) otherwise, for each s extending s. These isomorphisms are such S S S S S −1 that σsu = σtu ◦ σst and σst = (σts ) . Let θ be a sufficiently large regular cardinal bigger than κ, and let M be a countable elementary submodel of Hθ containing everything relevant, in particular, S, a fixed well ordering of Hθ , and a fixed non-principal ultrafilter U 0 on ω. We may assume without loss of generality that the well ordering extends the tree ordering of the nodes of S, and is such that each level precedes the next one. (There are closed cofinally many such M .) Let δ = M ∩ ω1 . Let p ∈ P ∩ M . Let p M = fp , Np ∪ {M ∩ Hκ } . Then, by a standard argument, p M ∈ P. Let tM be an arbitrary node at the δth level of S. We shall show p M, tM is generic. Let D ∈ M be a given dense open subset of P × S and let q, t0 be a given extension of p M, tM . We need to show q, t0 is compatible with some member of D ∩ M . Extending q, t0 , we may assume that q, t0 ∈ D. Moreover, by extending further (since D is open), we may assume that t0 is not in the largest model of Nq , and that this model contains (all the members of dom) fq and ht(t0 ) > N ∩ ω1 , for every N ∈ Nq . Let qM = q | M = fq ∩M, Nq ∩M . Note that qM ∈ P ∩M and that q ≤ qM . That qM is in M is clear. To see that it is in P, the only point at issue is clause 2) — could the N ’s separating s and s ∈ doml fq ∩ M all have been in Nq −M ? Consider an N ∈ Nq such that ht(s) ∈ N and ht(s ) ∈ / N . Since ht(s ) ∈ M ∩Hκ ∈ Nq , N must be a member of M ∩ Hκ , so N ∈ NqM = Nq ∩ M . That q ≤ qM is assured by the way we defined the forcing. We then may assume that the maximal model NqM of NqM contains all the members of dom fqM (= dom fq ∩ M ), else we could have extended Nq to ensure this. Let δM be the intersection of ω1 with NqM . By taking NqM large enough, we may ensure that the projection of (dom fq ∪ {t0 }) − M on the δth level of S has the same size as its projection on the δM th level.
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Let {u0 , . . . , un }, {v0 , . . . , vn } respectively enumerate these projections on the δM th and δth levels, such that ui = vi | δM , i ≤ n, and such that u0 = t0 | δM and v0 = t0 | δ. For 0 ≤ i, j ≤ n, let σij be the canonical −1 isomorphism which moves ui to uj . Note σij = σji , and σii is the identity isomorphism. In [11] we observed that, defining S η = {s ∈ S : ht(s) ≥ η} , where η is the level we’re shifting cones over, σij extends to an automorphism of S η , and then extends to S η -names by recursively defining σij (x) ˙ = {σij (y), ˙ σij (u) : y˙ is an S η -name and u ∈ S η and y, u ∈ x} ˙ . We shall omit the “sub ij” except where needed. We then noted that σ(ˇ x) = x ˇ, for any x ∈ V . We also observed that if the only parameters in φ are S η -names y˙ 1 , . . . , y˙ m , and if s ∈ S η , then s S φ(y˙ 1 , . . . , y˙ m ) if and only if σ(s) S η φ(σ(y˙1 ), . . . , σ(y˙ m )). Since S η is dense in S, we can conclude s S φ(y˙ 1 , . . . , y˙ m ) if and only if σ(s) S φ(σ(y˙ 1 ), . . . , σ(y˙m )).
(*)
We shall make use of this later. For any r, t ∈ P × S, define: Fr,t = {x ∈ (dom fr ∪ {t}) − NqM : for some ix , x | δM = uix and σ0ix (t) extends x}. Then we have: Fq,t0 = {x ∈ (dom fq ∪ {t0 }) − M : x | δ = some vix and σ0ix (t0 ) extends x} . If vi and vj are projections of elements of Fq,t0 , then σij (vi ) = vj . To see this, first note that if x ∈ Fq,t0 extends vi , then σi0 (vi ) ≤ σi0 (x) ≤ t0 . Hence σi0 (vi ) = v0 , since both are of height δ below t0 . It follows that σij (vi ) = σ0j ◦ σi0 (vi ) = σ0j (v0 ) = vj . By taking the maximal model and hence δM sufficiently large, we have that for such vi and vj , vi | [δM , δ) = vj | [δM , δ). For s above some ui , it will be convenient to write “ˆ s” for σi0 (s). Note that the fixed well order we have introduced allows us to consider any finite subset of S as an ordered tuple. For an (m + 1)-tuple s = s1 , . . . , sm , s0 , if sˆj is defined for 0 ≤ j ≤ m, let sˆ = ˆ s1 , . . . , sˆm , sˆ0 . Let Fˆq,t0 = ˆ x : x ∈ Fq,t0 . Similarly define l Fq,t = {t0 } ∪ {x ∈ dom fq ∩ Fq,t0 : fq (x) = l}, 0 l and Fˆq,t for l ∈ Lq = {l : doml fq = 0}. Note that Fˆq,t0 is a chain with possible repetitions. We can make 0 l analogous definitions of Fˆr,t , etc. Let cl = |Fq,t |, and c = {cl : l ∈ Lq }. 0 Let D0 = {r, t ∈ D : r, t ≤ qM , u0 and:
i) qM is an initial part of r, i.e. for each l, doml fr = doml fqM ∪ Fl , where Fl ⊆ S and each x ∈ doml fqM precedes the members of Fl in the well order on S; Nr = NqM ∪ N, where N is a finite ∈-chain, the first element of which contains NqM , ii) the height of each node in Fr,t − FqM ,u0 is > δM ,
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l iii) Lr = Lq , each |Fr,t | = cl , l l iv) fr (the jth element of Fr,t ) = fq (the jth element of Fq,t ) (recall we have ordered the nodes, which 0 l l l induces an order of Fr,t and Fq,t0 ), in other words, if x is in the jth position of Fr,t , then fr (x) is the l same value as fq (y) for whichever y is in the jth position of Fq,t0 , v) the height of t is greater than the height of any of the nodes in dom fr , vi) |Nr | = |Nq |.}
D0 is thus the set of all r, t ∈ D below qM , u0 that “look like q, t0 ”. We won’t actually use all of these requirements; we list them for possible future applications. Notice that the ui ’s and hence the σij ’s are in M ; Lq and |Nq | are just some finite sets of natural numbers and so are in M ; iv) can be described in terms of the order and a finite set of natural numbers; and so D0 ∈ M by definability. F = Fˆ : F = Fr,t for some r, t ∈ D0 is also in M and in Hκ as well. Since M ∩ Hκ ∈ N for each N ∈ Nq − M , it follows that F ∈ N , for all such N . Note that Fˆq,t0 ∈ F. l Note also that the terms of Fˆq,t are separated by models of Nq . 0 l Let Nq be a minimal subchain of Nq containing M ∩ Hκ at its bottom and separating Fq,t for each l. 0 ˆ Let Nq = {Na }a≤m ordered by inclusion, with N0 = M ∩ Hκ . Fq,t0 is a chain with possible repetitions; let us write it as: x1 , . . . , xm , t0
where xa = xa,1 , . . . , xa,da enumerates in increasing order Fˆq,t0 ∩ (Na − Na−1 ), a ≥ 1. Thus the length of the vector xa is equal to the size of Fq,t0 ∩ (Na − Na−1 ). Since F ∈ Nm ,
ˆ1 , . . . , ˆm , x ∈ F F(x1 , . . . , xm ) = x ∈ S : x x ˆm , t0 ), such that F(x1 , . . . , xm ) is dense above ym . ∈ Nm . By Lemma 9, there is a ym ∈ Nm ∩ S, ym ∈ [max x Next, consider:
F(x1 , . . . , xm−1 ) = x, y ∈ S dm +1 : x, y is a chain and F(x1 , . . . , xm−1 , x) is dense above y . ˆm , ym ∈ F(x1 , . . . , xm−1 ). As before, this time by Corollary 10, Then F(x1 , . . . , xm−1 ) ∈ Nm−1 and x ˆm−1 , ym , Nm−1 playing the roles of A, A0 , N respectively, we can find a ym−1 ∈ with F(x1 , . . . , xm−1 ), x ˆm−1 , min ˆm ), such that F( Nm−1 ∩ S, ym−1 ∈ [max x x x1 , . . . , xm−1 ) is dense above ym−1 . Continuing, in m steps we find a y0 ∈ N0 , y0 ∈ [u0 , v0 ), such that:
F(∅) = x, y ∈ S d1 +1 : x, y is a chain and F(x) is dense above y is ∈ N0 and dense above y0 . Let X˙1 be a name for: ξ ∈ [C ◦ ]d1 : for some z, w ∈ F(∅), {z, w} ⊆ B and for each d, ht(πd (z))− = πd (ξ) . Then X˙1 ∈ M . Claim. y0 X˙1 is uncountable and indeed for each d ≤ d1 , πd (X˙1 ) is uncountable.
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Proof. Given any α < ω1 , and any y0 extending y0 , since F(∅) is dense above y0 , we can find a z, w ∈ F(∅) extending y0 with minimal element of height greater than α. Take y0 above z, w . Then − − − y0 ht(z1 ) , . . . , ht(zd1 ) ∈ X˙1 and each ht(zd ) is ≥ α. 2 To prove , we need to deal with an arbitrary finite number of colours in the range of fq . Call the difference between Nk and Nk+1 , both in Nq , a layer. Without loss of generality, we shall assume each layer has non-empty intersection with dom fq . That intersection may be coloured by several colours; we can’t separate them, so we must remove parameters layer-by-layer. We must also reflect the layers layer-by-layer for the same reason. The way we will do this is essentially by regarding m-tuples of points of Z as points in Z m . Now it is finally time to leave the general scheme and start working with the topology needed for our situation here. The following definitions and lemmas used by the third author were crucial for his proof that PFA(S)[S] implies there are no compact S-spaces, and also are crucial for the proof of . Definition 5. Let U be a non-principal ultrafilter on a countable set T . We say a sequence of points xτ : τ ∈ T in a space X U -converges to a point x ∈ X, and write limτ →U xτ = x, if for each open set V containing x, {τ ∈ T : xτ ∈ V } ∈ U . Lemma 11. Suppose W ⊆ X, limτ →U T xτ = x, and x ∈ / W . Then τ ∈ T : xτ ∈ / W ∈ U. Proof. If τ ∈ T : xτ ∈ /W ∈ / U , then τ ∈ T : xτ ∈ W ∈ U. But then x ∈ W . To see this, let V be open containing x. Then {τ : xτ ∈ V } ∈ U. Then τ : xτ ∈ W ∩ V ∈ U. Then V ∩ W = ∅. But then V ∩ W = ∅. 2 Definition 6. Y ⊆ X is U -sequentially closed if whenever xt : t ∈ T is a sequence from Y that U-converges, then its U-limit is a member of Y . X is U -sequential if every U-sequentially closed subset of X is closed. Lemma 12. ([14, 8.2]) PFA(S)[S] implies that if X is compact countably tight, then for each ground model (i.e. before forcing with S) ultrafilter U on ω, X is U-sequential. In order to prove , it is helpful to reformulate “U-sequentially closed” so that if p is in the U -sequential closure of a set A, p ∈ / A, then there will be a sequence in A converging to p in some sense. The following definitions and lemma, taken almost verbatim from [14], accomplish this. Definition 7. A well-founded U-tree is a collection T of finite subsets of ω such that: 1) ∅ ∈ T , and if τ ∈ T , all initial segments of τ are in T ; 2) if τ ∪ {m} ∈ T for some m > max(τ ), then {n < ω : n > max(τ ) and τ ∪ {n} ∈ T } ∈ U ; 3) there is no infinite U ⊆ ω such that all finite initial segments of U belong to T . Let ∂T be the collection of all terminal nodes of T , i.e. those nodes with no proper end-extensions in T . By 3), there are such terminal nodes. Then for every compact Hausdorff space K, any assignment {xτ : τ ∈ ∂T } ⊆ K extends uniquely to the global assignment {xτ : τ ∈ T } defined recursively on the rank of T (see e.g. Exercise III (12) of [4]) as follows. If T has rank 1, i.e. T = {∅} ∪ {{n} : n ∈ U } for some U ∈ U , let x∅ = limn→U x{n} . If rank T > 1, for each {n} ∈ T apply the inductive hypothesis and extend the
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assignment {τ ∈ ∂T : n = min(τ )} to the assignment {τ ∈ T : n = min(τ )} and again let x∅ = limn→U x{n} . We call x∅ the limit of the assignment {xτ : τ ∈ ∂T } and write x∅ = lim xτ . τ ∈∂T
Having defined this limit, we note that it is in fact the limit of an ultrafilter on ∂T defined as follows. To see this, first define a U -subtree to be any T0 ⊆ T such that: 4) ∅ ∈ T0 ; 5) if τ ∈ T0 − ∂T , then {n < ω : n > max(τ ) and τ ∪ {n} ∈ T0 } ∈ U . Inducting on rank, we see that {∂T0 : T0 a U -subtree of T } generates an ultrafilter U T on ∂T such that for every assignment {xτ : τ ∈ ∂T } of points in some Hausdorff space X, we have lim xτ = lim xτ ,
τ ∈∂T
τ →U T
where the right hand side is as defined above Lemma 11, whenever one of these limits exists. Then: Lemma 13. ([14, 8.1]) Let K be compact and Y ⊆ K. Let Y Y
U
U
be the U-sequential closure of Y in K. Then
=
lim xτ : T a well-founded U -tree and {xτ : τ ∈ ∂T } ⊆ Y
τ ∈∂T
We now return to the proof of
.
. The following topological fact is crucial.
Lemma 14. ([8]) A finite product of compact countably tight spaces is compact and countably tight. This will allow us to apply U 0 -sequentiality to a finite product of copies of Z. Actually, we shall work with a finite product of σi0 (Z)’s. Note U 0 remains an ultrafilter after forcing with S, since no new subsets of ω are added. For x ∈ dom fq − M , let ix be that i such that x is above vi . Consider Z˙ 1 = (the name of)
0 ˙ 1≤d≤d1 σix1,d 0 (Z). By Lemmas 12 and 14, this space is forced to be compact and U -sequential. There is a name p˙ 1 ∈ M which is forced by y0 to be a complete accumulation point for X˙1 as a subspace of Z˙ 1 and to have its projections be complete accumulation points. By Lemma 12, Lemma 13, and elementarity, there is a name T˙1 ∈ M for a well-founded U 0 -tree T1 , and a name in M for an assignment {ξτ : τ ∈ ∂T1 } ˙ with components in δ such that y0 forces {ξτ : τ ∈ ∂ T˙1 } ⊆ X˙1 and UT0˙ -converges to p˙ 1 . There is a limit 1 ˙ level ζ, δM < ζ < δ, at which all nodes of that level decide T˙1 and the ξτ , τ ∈ ∂ T˙1 . Furthermore, since ˙ y0 {ξτ : τ ∈ ∂ T˙1 } ⊆ X˙1 , by elementarity, we could also take ζ sufficiently high below δ such that the nodes ˙ of that level which are above y0 decide a zτ , wτ (ξτ ) – say the least one – ∈ F(∅) ∩ M , for each of the ξτ , τ ∈ ∂ T˙1 that they decide. In particular, there is a y0 , y0 < y0 < v0 , such that y0 makes these decisions. ˙ Since πd (p˙ 1 ) is forced by y0 to be a complete accumulation point, y0 forces πd (p˙ 1 ) ∈ / σix1,d 0 (W ht(x)− ), ˙ ˙ for any x ∈ dom fq − M . Therefore there is a name for an open Od in Z about πd (p1 ) such that y forces 0
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˙ ˙ σix1,d (O˙ d ) is disjoint from all those σix1,d 0 (W ht(x)− )’s. Therefore, by (*), σ0ix1,d (y0 ) forces Od is disjoint ˙ − ’s. from those W ht(x)
˙ ˙ Now since y0 forces {ξτ : τ ∈ ∂ T˙1 } UˇT01 -converges to p˙ 1 , it also forces {πd (ξτ ) : τ ∈ ∂ T˙1 } UˇT01 -converges to πd (p˙ 1 ). Then
ˇ y0 (∃U ∈ Uˇ0 )(∀τ ∈ U)(πd (ξτ ) ∈ O˙ d ),
so ˇ σ0ix1,d (y0 ) (∃U ∈ Uˇ0 )(∀τ ∈ U)(πd (ξτ ) ∈ σ0ix1,d (O˙ d )).
x1,d is above vix1,d and hence above σ0ix1,d (y0 ). Thus, ˇ ˙ ht(x )− ). x1,d (∃U ∈ Uˇ0 )(∀τ ∈ U)(πd (ξτ ) ∈ /W 1,d
Then there is an x1,d extending x1,d and a Ud ∈ U 0 such that ˇ ˙ ht(x )− ). x1,d (∀τ ∈ Ud )(πd (ξτ ) ∈ /W 1,d
˙ ht(x )− ∩ ω1 , so But x1,d decides W 1,d ˇ ˙ ht(x )− ). x1,d (∀τ ∈ Ud )(πd (ξτ ) ∈ /W 1,d
Notice that the same argument will work for any other x ∈ dom fq −M such that ix = ix1,d so as to conclude that there is a Ux ∈ U 0 such that ˇ ˙ ht(x)− ). x (∀τ ∈ Ux )(πd (ξτ ) ∈ /W
Let U1 =
{Ud : 1 ≤ d ≤ d1 } ∩
{Ux : ix = ix1,d for some d, 1 ≤ d ≤ d1 , x ∈ dom fq − M }.
Let τ1† be the least τ ∈ U1 , according to our fixed well-order. Consider zτ † , wτ † (ξτ † ) (the one y0 1 1 1 decided). Let
z1 = σ0i1,1 (π1,1 (zτ † ), . . . , σ0i1,d1 (π1,d1 (zτ † )) .
1
1
ˆ1 = ˆ1 , w † ∈ F(∅). Then z zτ † , and so z τ 1
1
We now need to iteratively peel off the remaining layers of Fq,t0 − {t0 }. Let X˙2 be a name for: ξ ∈ [C ◦ ]d2 : for some z, w , z, w ∈ F(z1 ), {z, w} ⊆ B and for each d, ht(πd (z))− = πd (ξ)}. We now carry out the same argument as before, forcing X˙2 to be uncountable, with all its projections uncountable, etc. ˆ1 , . . . , ˆm = Continuing, after m steps we will get z1 , . . . , zm with z z z 1 , . . . , z m for some z 1 , . . . , z m , w ∈ F. But then there is an r, t ∈ D0 ∩ M , such that Fˆr,t = z 1 , . . . , z m , t . Then w = t. By
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construction, w is below some y < v0 and so t ≤ t0 . We claim that r, t is compatible with q, t0 , which will finish the proof. Since r ≤ qM , it follows that fr ∪ fq is a function. We next note that Nr ∪ Nq is an ∈-chain, for by construction, Nr ∈ M , so Nr ∪ {M ∩ Hκ } is an ∈-chain. Now Nq = NqM ∪ (Nq − NqM ); the elements N of Nq − NqM all have M ∩ Hκ in them, for if not, such an N would be in M . Nr ∪ Nq is thus the ∈-chain Nr ∪ {M ∩ Hκ } ∪ (Nq − NqM ). Let R = fr ∪ fq , Nr ∪ Nq , t0 . Nr ∪ Nq separates doml (fr ∪ fq ) since r ≤ qM . Thus fr ∪ fq , Nr ∪ Nq ∈ P. We need to check that it extends r and q. The only case of interest is when x extends s, x ∈ dom fq − M , and s ∈ dom fr − dom fqM . Thus s = zd , for some component zd of some component of r. But then ht(s)− is some πd (ξτ † ) for some k
ξτ † arising in the construction. Then s is below some xk,d ∈ Fq,t0 − {t0 }, so s is below vik,d . But x is k ˙ ht(x)− as required. Thus above s, so x is above vik,d . So ik,d = ix . But then, as before, x forces ht(s)− ∈ /W fr ∪ fq , Nr ∪ Nq ∈ P and is below both r and q. But then R ∈ P × S is below both r, t and q, t0 as required. 2 The σ-closed-discrete within an open set formulation of is somewhat awkward. As Balogh noticed in [2], the following more easily stated version is what one really uses: ∗
: Let X be a locally compact space of Lindelöf number ℵ1 which does not include any perfect pre-images of ω1 . Let Y ⊆ X of size ℵ1 be (globally) locally countable, in the sense that each x ∈ X has a neighbourhood meeting countably many members of Y . Then Y is σ-closed-discrete.
Since: Lemma 15. ([1]) The one-point compactification of a locally compact space X is countably tight if and only if X does not include a perfect pre-image of ω1 . we see that
implies
∗
and so we have:
Corollary 16. PFA(S)[S] implies
∗
.
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