Thin-very tall compact scattered spaces which are hereditarily separable

We strengthen the property $\Delta$ of a function $f:[\omega_2]^2\rightarrow [\omega_2]^{\leq \omega}$ considered by Baumgartner and Shelah. This allows us to consider new types of amalgamations in the forcing used by Rabus, Juh\'asz and Soukup to construct thin-very tall compact scattered spaces. We consistently obtain spaces $K$ as above where $K^n$ is hereditarily separable for each $n\in\N$. This serves as a counterexample concerning cardinal functions on compact spaces as well as having some applications in Banach spaces: the Banach space $C(K)$ is an Asplund space of density $\aleph_2$ which has no Fr\'echet smooth renorming, nor an uncountable biorthogonal system.

The purpose of this work is to show that the existence of compact hereditarily separable scattered spaces of height ω 2 is consistent with the usual axioms of set theory.For a given ordinal θ let us consider the following notation: • A cw(θ) space is a compact scattered space of countable width and height equal to θ. • A hs(θ) space is a compact scattered space which is hereditarily separable and of height equal to θ. cw(ω 1 ) spaces are usually called thin-tall spaces and cw(ω 2 ) spaces are the thinvery tall spaces.First we remark that any hs(θ) space is a cw(θ) space as the Cantor-Bendixson levels form discrete subspaces.Whether there is or not in ZFC a cw(ω 1 ) space was a question posed by Telgársky in 1968 (unpublished) and first (consistently) answered by Ostaszewski [21], using ♦.Rajagopalan constructed the first ZFC example of a cw(ω 1 ) in [23].Further, Juhász and Weiss generalized these 1991 Mathematics Subject Classification.Primary 54G12; Secondary 03E35, 46B26.The research has been a part of Thematic Project FAPESP (2006/02378-7).The first author was supported by scholarships from CAPES (3804/05-4) and CNPq (140426/2004-3 and 202532/2006-2).She would like to thank Stevo Todorcevic and the second author, her Ph.D. advisors at the University of São Paulo and at the University of Paris 7, under whose supervision the results of this paper were obtained.
The second author was partially supported by Polish Ministry of Science and Higher Education research grant N N201 386234.
results (and simplified their proofs) in [12] proving in ZFC that for any ordinal θ < ω 2 , there is a cw(θ) space.
For higher θ's the situation changes: in any model of CH there are no cw(ω 2 ) spaces and Just proved in [13] that neither are there such spaces in the Cohen model (where ¬ CH holds).On the other hand, Baumgartner and Shelah [2] constructed by forcing the first consistent example of a cw(ω 2 ) space.An interesting point of this forcing construction was the use of a new combinatorial device called a function with the property ∆.
The main purpose of this work is to prove the consistency of the existence of a hs(ω 2 ) space.In fact, our space has even stronger properties: each of its finite powers is hereditarily separable.Whether consistently there are hs(ω 3 ) or even cw(ω 3 ) spaces remains a well-known open question.On the other hand, Martínez in [17] adopted the method of [2] to obtain the consistency of the existence of cw(θ) spaces for each θ < ω 3 .
It follows from an old result of Shapirovskiȋ [25] that for any compact space K, hd(K) ≤ hL(K) + .Our construction shows that the dual inequality does not follow from ZFC, since for our compact space K, we have that hL(K) = ℵ 2 ≤ ℵ 1 = hd(K) + .Nevertheless, the dual inequality holds under GCH for regular spaces: since the weight w(K) of a regular space K is less or equal to 2 d(K) (see, for example, [8]), we trivially conclude that hL(K) ≤ w(K) ≤ 2 d(K) = d(K) + ≤ hd(K) + under GCH.
Turning to properties of Banach spaces, let us first recall some definitions and results: a Banach space X is an Asplund space if every continuous and convex real-valued function on X is Fréchet smooth at all points of a G δ dense subset of X.For separable Banach spaces, this is equivalent to admitting a Fréchet smooth renorming (see [4]).Namioka and Phelps proved in [19] that C(K) is Asplund if and only if K is scattered.Thus, our C(K) is an Asplund space.
Haydon constructed in [7] the first nonseparable Asplund space C(K) which does not admit a Fréchet smooth renorming, concluding that the situation changes for nonseparable Asplund spaces.Later, Jiménez Sevilla and Moreno analyzed in [10] the structural properties of the space C(K), where K is the well-known Kunen line constructed under CH (see [20]).They showed, for the Kunen line K, that C(K) is also a nonseparable Asplund space with no Fréchet smooth renorming.
The weight of our space K is ℵ 2 , so that C(K) is an Asplund space of density ℵ 2 .The fact that K is compact scattered and every finite power of K is hereditarily separable implies, in the same way as for the Kunen line, that C(K) does not admit any Fréchet smooth renorming, but as in the case of the Kunen line we do not know if it admits a Gâteaux smooth renorming, or a Fréchet smooth bump function.
A biorthogonal system on a Banach space X is a family [29] (Theorem 9 together with the results of [3]) the existence of uncountable semi-biorthogonal systems in Banach spaces C(K) of density strictly greater than ℵ 1 .On the other hand, the fact that our space K is compact scattered and every finite power of K is hereditarily separable implies, in the same way as for the Kunen line, that C(K) does not admit an uncountable biorthogonal system.It follows that Todorcevic's result cannot be improved in ZFC by replacing the existence of uncountable semi-biorthogonal systems by the existence of uncountable biorthogonal systems in spaces C(K) of large density.On the other hand it is proved in [29] that it is consistent that every nonseparable Banach space has an uncountable biorthogonal system, showing that the existence of a Banach space like ours or Kunen's cannot be proved in ZFC.
Our construction is based on the Juhász and Soukup [11] interpretation of Rabus' work [22], where he modified the Baumgartner-Shelah forcing from [2] to obtain a countably tight space which is initially ω 1 -compact and noncompact, answering a question of Dow and van Douwen.
This paper is organized as follows: we finish this section by reviewing the method of Juhász and Soukup and some related results and definitions which we will need afterwards.In Section 2 we prove the key lemma which enables us to prove our main result in a straightforward way.This lemma introduces a new way of amalgamating conditions in forcings which add thin-very tall spaces.One can apply these amalgamations in the generic construction if one strengthens the property ∆ of a function involved in the forcing.In Section 3, we introduce the strong property ∆ and assuming the existence of a function which satisfies it, we prove the main results and analyze their consequences in topological and functional analytic terms.Section 4 is devoted to establishing the consistency of the existence of a function with the strong property ∆.Section 4 is due to the second author and the remaining sections to the first author.
The notation and terminology used are those of [11].Given a set X, ℘(X) is the power set of X and, given a cardinal κ, [X] κ (resp.[X] ≤κ and [X] <κ ) denotes the family of subsets of X of cardinality equal to κ (resp.less or equal to κ and less than κ).
Let us start by recalling the definition of the property ∆: To simplify notation, it is convenient to define the following: Definition 1.3 (Juhász, Soukup [11]).Given finite nonempty sets of ordinals x and y such that max x < max y, we define We now rewrite conditions 3.(a) and (b) of the definition of the forcing as To define the space K f , fix the ground model V and a generic filter G.
From Theorem 1.5 of [11], it follows that for all ξ < ω 2 , h(ξ) is a compact subspace of (ω 2 , τ ) and it easy to check that Therefore L f is a locally compact scattered zero-dimensional space.
We are now ready to define K f : Definition 1.5.
In particular, we use the following results.
Proposition 1.7.V P f satisfies "K f is a compact scattered zero-dimensional space".

Amalgamating conditions
In this section, we present the key lemma needed to prove our main result.Let us start with some preliminaries and auxiliary lemmas.Definition 2.1.Let p 1 = (D 1 , h 1 , i 1 ), p 2 = (D 2 , h 2 , i 2 ) ∈ P f be two conditions.We say that p 1 and p 2 are isomorphic conditions if there is an order-preserving bijective function e : D 1 → D 2 satisfying the following conditions: In this case, if the order-preserving bijection e is such that ξ ≤ e(ξ) for every ξ ∈ D 1 we say that p 1 is lower than p 2 .
For example we have the following: Lemma 2.2.Let p 1 = (D 1 , h 1 , i 1 ), p 2 = (D 2 , h 2 , i 2 ) ∈ P f be two isomorphic conditions and let e : D 1 → D 2 be the order-preserving bijection.Then for every Proof.Directly from Definition 2.1.
Proof.Directly from Definition 2.3.
We prove the next lemma, for the reader's convenience.
Lemma 2.5 (Juhász, Soukup [11]).Let Suppose that η ∈ dom(δ 2 ) and which contradicts the minimality of δ 2 (η) and concludes the proof of the inclu- In the proof of c.c.c., Rabus, Juhász and Soukup considered the minimal amalgamation which is constructed in a symmetric way with respect to both of the conditions being extended.We will consider an asymmetric amalgamation.The lack of symmetry in our amalgamation is the result of using two functions, δ 2 and e, in the definition of the amalgamation.The final auxiliary lemma below characterizes the sets given by the operation * for elements of the extended condition.The role of the function g will be played by δ 2 or by e. Lemma 2.6.Let p = (D p , h p , i p ) ∈ P f and let D q ∈ [ω 2 ] <ω , h q : D q → ℘(D q ) and g : dom(g) → D p be such that (i) Then, for all ξ, η ∈ D p , ξ < η, we have that and so by (iii) On the other hand, if ξ / ∈ h q (η), then ξ / ∈ h p (η) and so by (iii) concluding the proof of the lemma.
Now we go to our key lemma: a strong hypothesis about the behaviour of the function f allows us to amalgamate two isomorphic conditions, one lower than the other, into a common extension q in such a way that h(ξ) ∩ D q ⊆ h[e(ξ)] ∩ D q for ξ in the domain of the lower of the two conditions.Lemma 2.7.Let p 1 = (D 1 , h 1 , i 1 ), p 2 = (D 2 , h 2 , i 2 ) ∈ P f be two isomorphic conditions and suppose p 1 is lower than p 2 .Let e : D 1 → D 2 be the order-preserving bijective function and assume that Then there is q ∈ P f , q ≤ p 1 , p 2 , such that for all ξ ∈ D 1 and all η ∈ D 2 : Proof.We define q = (D q , h q , i q ) by: Note that (A) implies that the set i q ({ξ, η}) is well-defined for any ξ, η ∈ D 1 ∩ D 2 , ξ = η; clearly i q is well-defined for the other pairs.Also, if ξ ∈ D 1 ∩D 2 , then the set h q (ξ) is well-defined because both of the conditions reduce to h q (ξ) = h 1 (ξ) ∪ h 2 (ξ) by Lemmas 2.5 and 2.2.(c).
We have to show that q ∈ P f , i.e., that q satisfies conditions 1, 2 and 3 from Definition 1.2.The fact that q satisfies conditions 1.2.1 and 1.2.3.(c)follows directly from the definition of q and from the fact that p 1 , p 2 ∈ P f .Condition 1.2.2. is satisfied because p 1 , p 2 ∈ P f and the functions e and δ 2 are nondecreasing.In what follows we will be using Lemma 2.6 for p = p 1 , p 2 and g = δ 2 , e respectively.The hypothesis of the lemma is satisfied for these objects by 2.4.(b) and 2.1.(b).Now we check conditions 1.2.3.(a) and (b).Let ξ, η ∈ D q , ξ < η, and we consider the following cases: It follows from the definition of q and from Lemma 2.6 that In this subcase, ζ ∈ h 1 (ξ) * h 1 (η) and there is γ In this subcase, it follows by the definition of q that ζ ∈ h q (γ), as we wanted.
Again we fix ζ ∈ h q (ξ) * h q (η) and we consider the following subcases: The proof in this subcase follows identically to the proof of Subcase 3.1.

Subcase 4.2.2. θ > ξ.
Note that ζ ∈ h q (ξ) * h q (η) ⊆ h q (ξ).Since ζ and ξ satisfying the hypothesis of the Case 4.2.are in D 2 \ D 1 , it follows from the definition of h q that ζ ∈ h 2 (ξ).To finish, let us show the following: Finally, since p 2 ∈ P f , there is γ ∈ i 2 ({ξ, θ}) such that ζ ∈ h 2 (γ) ⊆ h q (γ).By condition (B).(i), which can be used by Fact 1, we have that Hence, γ ∈ i q ({ξ, η}) and ζ ∈ h q (γ), concluding the proof of Subcase 4.2, Case 4 and thus concluding the proof of Claim 2. Now we know that q ∈ P f and let us check the other conclusions: it follows easily from the definition of q and Lemma 2.5 that q ≤ p 1 and analogously it follows from the definition of q and Lemma 2.4 that q ≤ p 2 .
Finally, we verify the condition we want q to satisfy, that is, ξ ∈ h 2 (η)∪e −1 [h 2 (η)] if and only if e(ξ) ∈ h 2 (η): let ξ ∈ D 1 and η ∈ D 2 and we consider again the following cases: It follows from the fact that this case e(ξ) = ξ.
] if and only if e(ξ) ∈ h 2 (η), concluding the proof of the lemma.

The main results
To apply the key lemma proved in the previous section, the function f on which the forcing P f depends must satisfy a stronger version of the property ∆: ≤ω has the strong property ∆ if f ({ξ, η}) ⊆ min{ξ, η} for all {ξ, η} ∈ [ω 2 ] 2 and for any uncountable ∆-system A of finite subsets of ω 2 , there are distinct a, b ∈ A and an order-preserving bijection e : a → b which is the identity on a ∩ b and such that ξ ≤ e(ξ) for all ξ ∈ a and for any τ ∈ a ∩ b, any ξ ∈ a \ b and any η ∈ b \ a we have: Finally we arrive at the main result of this paper.
Proof.We prove this by induction on n ∈ N: in V P f , fix n ∈ N and suppose that for all 0 ≤ i < n, K i f is hereditarily separable (take K 0 f = { * }) and let us show that K n f is hereditarily separable.We will be using a well-known fact that a regular space is hereditarily separable if and only if it has no uncountable left-separated sequence (see Theorem 3.1 of [24]).
In V , suppose ( ẋα ) α<ω1 is a sequence of names such that P f forces that ( ẋα ) α<ω1 is a left-separated sequence in K n f of cardinality ℵ 1 and for each α < ω 1 , we have that ẋα = ( ẋα 1 , . . ., ẋα n ), where each ẋα i is a name for an element of K f .
By thinning out, we can assume without loss of generality that (D α ) α<ω1 forms a ∆-system with root D such that for every α < β < ω 1 : • if e αβ : D α → D β is the order-preserving bijective function, then e αβ (x α i ) = x β i , for all 1 ≤ i ≤ n.Finally, we may assume that for all 1 ≤ i ≤ n we have: either x α i = x β i for all α < β < ω 1 ; or x α i / ∈ D for all α < ω 1 and actually the second case holds by our initial assumption about the sequence.
Since f has the strong property ∆, there are α < β < ω 1 such that for all ζ ∈ D, all ξ ∈ D α \ D and all η ∈ D β \ D: Note that p α and p β satisfy the hypothesis of Lemma 2.7.Hence, there is q ≤ p α , p β in P f such that for all ξ ∈ D α and all η ∈ D β , ξ ∈ h q (η) if and only if e αβ (ξ) ∈ h p β (η).
Then, for all 1 ≤ i ≤ n and all ξ ∈ D β , we have that But q ≤ p α , p β and then contradicting the hypothesis about ẋα i , ẋβ i and Ḟ β i .Corollary 3.3.It is relatively consistent with ZFC that there is a hereditarily separable compact scattered space of height ω 2 .
Proof.Since each level of the Cantor-Bendixson decomposition of Corollary 3.4.It is relatively consistent with ZFC that there is a hereditarily separable compact space with hereditary Lindelöf degree equal to ℵ 2 .In particular, it is relatively consistent with ZFC that there is a compact space K such that hL(K) ≤ hd(K) + ".

Proof. It follows from the fact that hL(K
is an open covering of K f \ { * } which does not admit a subcovering of strictly smaller cardinality.Corollary 3.5.It is relatively consistent with ZFC that there is an Asplund space C(K) of density ℵ 2 which does not admit any Fréchet smooth renorming and which does not contain an uncountable biorthogonal system.
Proof.Since every finite power of K f is hereditarily separable, Lemma 4.37 and Theorem 4.38 of [6] imply that C(K f ) is hereditarily Lindelöf relative to its pointwise convergence topology.But for compact scattered spaces K, the pointwise convergence topology and the weak topology of C(K) coincide (see Theorem 7.4 of [20]), so that C(K f ) is hereditarily Lindelöf relative to its weak topology.Now, if C(K f ) admits a Fréchet smooth renorming, by Corollaries 8.34 (due to Mazur [18]) and 8.36 of [6] (due to Jiménez Sevilla and Moreno [10]) it contains an uncountable bounded subset A such that for every x 0 ∈ A, x 0 is not in the (norm-) closed convex hull of A \ {x 0 }, that is, x 0 / ∈ conv(A \ {x 0 }).Since the weak and norm convex closures coincide in Banach spaces, A turns out to be an uncountable discrete family of C(K f ) relative to its weak topology, which contradicts the fact that C(K f ) is hereditarily Lindelöf relative to its weak topology.Now, if C(K f ) admits an uncountable biorthogonal system (x α , ϕ α relative to its weak topology, contradicting the fact that C(K f ) is hereditarily Lindelöf relative to its weak topology.

The existence of the required function f
In this section we prove the consistency of the existence of a function with the strong property ∆.It turns out that we are even able to prove the consistency of the existence of such a function with its range included in the family of finite (rather than countable) subsets of ω 2 .The method is quite involved but, as shown at the end of this section, forcings preserving CH (as in [2]) cannot serve for this purpose even if we were interested in a function with its range included in countable subsets of ω 2 .

4.1.
Forcing with side conditions in Velleman's simplified morasses.To construct a forcing which adds the required auxiliary function on pairs of ω 2 we will need a family of countable subsets of ω 2 with some strong properties.The following proposition establishes a list of the most useful properties: Proposition 4.1.It is relatively consistent with ZFC+CH that there exists a family F ⊆ [ω 2 ] ω which satisfies the following properties: Proof.We will prove that a simplified Velleman's (ω 1 , 1)-morass (see [30]) which is a stationary coding set (see [31]) satisfies the above properties.The proof relies heavily on the properties of Velleman's morasses obtained in [15].We will often refer to this paper, in particular we adopt definitions of simplified morass and stationary coding set from this paper (section 2).The consistency of the existence of such morasses can be immediately obtained from the corresponding proof for semimorasses in [14], Theorem 3 Section 2. 1) follows from Definition 2.1 of [15] and 2) from the fact that F is assumed to be a stationary coding set.To prove 3) apply 2.5 of [15].Now 4) is Fact 2.7 of [15], 5) is Fact 2.6 of [15] To obtain 6) apply Fact 2.8 of [15] to each X i obtaining Z(X i ) such that Z(X i ) ∈ M ∩ F and X i ∩ X ⊆ Z(X i ).Now use the elementarity of M and the directedness of F (see Definition 2.1.of [15]) to obtain Z as in 6).Now we will adopt a few facts from [16] and [15] concerning forcing with side conditions in F .As explained in these papers, to use elements of F as side conditions means to use forcings P whose conditions are of the form (p, A) where p is a finite condition of a natural forcing adding the structure in question and A is a finite subset of F .This is like using models as side conditions in the method of forcing with models as side conditions developed by Todorcevic (see [27]).The order is given by the forcing order on the first coordinate and inverse inclusion on the second coordinate.In addition we require the existence of some natural projections of p onto the elements of A as a part of the definition of the forcing notion.The properties 1) -6) above allow us to perform many maneuvers with ease; also the definitions are simpler.This method appears to be equivalent to the variant of Todorcevic's method where one employs matrices of models (see [28] Section 4, for an example with detailed definitions).The price we need to pay for this convenience is that P is not proper (unlike Todorcevic's forcings,) but only F -proper, i.e., there is a club C ⊆ [ω 2 ] ω such that for models M ≺ H(ω 3 ) such that M ∈ F ∩ C and p ∈ P ∩ M , there are (P, M )-generic conditions stronger than p.As F may be assumed to be stationary, F -properness implies the preservation of ω 1 (proof as for proper forcings, see [1]).The preservation of bigger cardinals follows from the ω 2 -chain condition.Note that the fact that the forcing is not proper but preserves cardinals is no limitation in the applications that one seeks here, i.e., consistent existence of structures of sizes bigger than ω 1 .Let us describe basic notions related to forcing with side conditions in F that we will use.Definition 4.2.Suppose F ⊆ [ω 2 ] ω .We say that a forcing notion P is F -proper if there is θ > (2 |P | ) + and a club set C ⊆ [H(θ)] ω such that whenever p ∈ M ∈ C and M ∩ ω 2 ∈ F then there is a (P, M )-generic p 0 ≤ p, i.e., D ∩ M is predense below p 0 for every D ∈ M which is dense in P .Proof.The proof is a straightforward version of Shelah's paradigmatic proof of preservation of ω 1 by proper forcings (see [26] or [1]).
Lemma 4.6.If P is simply F -proper, then P is F -proper.
4.2.Adding a function with the strong property ∆.Fix a family F ⊆ [ω 2 ] ω satisfying 1) -6) of Proposition 4.1.We will assume familiarity of the reader with elementary submodels of structures H(θ).In particular we will make use of facts such as that countable elements of such models are their subsets or that such models contain ω.See [5] for more on this subject.We consider the following forcing P whose conditions p are of the form: p = (a p , f p , A p ) where a) The order is just the inverse inclusion, i.e., p ≤ q if and only if a p ⊇ a q , f p ⊇ f q , A p ⊇ A q .Fact 4.7.P is simply F -proper.
Define q|M = (a q ∩ M, f q |M, A q ∩ M ).Introduce notation δ = M ∩ ω 1 = rank(M ), where the second equality follows from 4) of Proposition 4.1.Note that A q ∩ M = A q|M = {X ∈ A q : X ⊂ X 0 }.This follows from 5) of Proposition 4.1.The fact that [M ] <ω ⊆ M implies that a q|M , A q|M ∈ M .Also as d) of the definition of the forcing is satisfied for q and α, β ∈ a q , we have that f q (α, β) ⊆ X 0 = M ∩ ω 2 for α, β ∈ a q ∩ X 0 .So, we may conclude that f q|M ∈ M , in other words we have q|M ∈ M ∩ P .It is clear that q|M ≤ p.By 6) of Proposition 4.1 and the fact that ) be the formula which says that x 0 is a condition of the partial order x 4 which extends in x 4 the condition x 3 and such that the difference between the first coordinate of x 0 and x 2 is disjoint from x 1 .
Proof of the Claim.It is clear that φ(q, Z, a q|M , q|M, P ) holds in H(ω 3 ).Now let s ∈ M be a condition satisfying φ(s, Z, a q|M , q|M, P ) i.e., s extends in P the condition q|M and a s \ a q|M is disjoint from Z. Define the common extension r of q and s as follows: . Such an f r is a function on [a r ] 2 since q|M ≥ q, s.Clearly all clauses of the definition of the forcing P but d) are trivially satisfied by r.So let us prove d).Let α, β ∈ a r and X ∈ A r , we will consider a few cases.
It is trivial because s ∈ P .
Case 2. α, β ∈ a q , X ∈ A q It is trivial because q ∈ P .
Case 3. α, β ∈ a s , X ∈ A q .Since φ(s, Z, a q|M , q|M, P ) holds in M we have that either rank(X) d) for s and 3) of Proposition 4.1 or rank(X) < δ and then by the definition of φ and Z we get that α, β ∈ a s ∩ a q , so we are again in Case 2.
This means that α, β ∈ M , because s ∈ M , i.e., α, β ∈ a s ∩ a q so we are again in Case 1.
Case 5. α ∈ a s \ a q and β ∈ a q \ a s .
The proof of the claim completes the proof of Fact 4.7.Definition 4.9.We say that two conditions p, q of P are isomorphic (via π : supp(p) → supp(q)) if π : supp(p) → supp(q) is an order preserving bijection constant on supp(p) ∩ supp(q) and i Lemma 4.10.Suppose p, q ∈ P are isomorphic via π : supp(p) → supp(q).Then they are compatible.
Proof.Define the common extension r of p and q as follows: a r = a p ∪ a q , f r = f p ∪f q ∪h, A r = A p ∪A q , where h({α, β}) = ∅ for {α, β} ∈ [a p ∪a q ] 2 −([a p ] 2 ∪[a q ] 2 ).The only non-automatic condition of the definition of P which needs to be checked is d).
Similar to the previous case.
Case 3. α ∈ a r \ a q , β ∈ a q \ a r .In this case h is empty.

4.3.
CH and the strong property ∆.In this section we prove that CH implies that there is no function f such as in the previous section, even if we allow f to take countable sets as values.This also proves that the strong property ∆ cannot be obtained as in Baumgartner and Shelah [2], that is, by a forcing which preserves CH.Let M ≺ H(ω 3 ) be closed under its countable subsets (here we use CH) |M | = ω 1 , ω 1 ⊆ M ; ω 1 , ω 2 , f ∈ M and such that sup(M ∩ ω 2 ) = γ has an uncountable cofinality.

Fact 4 . 3 .
Suppose F ⊆ [ω 2] ω is a stationary set and P is an F -proper forcing notion, then P preserves ω 1 .

Definition 4 . 8 .
For p ∈ P , call the set a p ∪ f [[a p ] 2 ] ∪ A p the support of p and denote it by supp(p).