2.1. Combinatorial principles from adding Cohen reals (This section is based on[12]and [1])
The last 40 years have seen a furious activity in proving results that are independent of the usual axioms of set theory, that is ZFC. As the methods of these independence proofs (e.g.
forcing or the fine structure theory of the constructible universe) are often rather sophisticated, while the results themselves are usually of interest to “ordinary” mathematicians (e.g. topologists or analysts), it has been natural to try to isolate a relatively small number of principles, i.e.
independent statements that a) aresimpleto formulate and b) areuseful in the sense that they have many interesting consequences. Most of these statements, we think by necessity, are of combinatorial nature, hence they have been called combinatorial principles.
In this section we present several new combinatorial principles that are all statements about P(ω), the power set of the natural numbers. In fact, they all concern matrices of the form hA(α, n) :hα, ni ∈κ×ωi, where A(α, n) ⊂ ω for each hα, ni ∈ κ×ω, and, in the interesting cases,κis a regular cardinal withc= 2ω≥κ > ω1.
We show that these statements are valid in the generic extensions obtained by adding any number of Cohen reals to any ground modelV, assuming that the parameter κis a regular and ω-inaccessible cardinal inV ( i.e. λ < κimpliesλω< κ).
Then we present a large number of consequences of these principles, some of them com-binatorial but most of them topological, mainly concerning separable and/or countably tight topological spaces. (This, of course, is not surprising because these are objects whose structure depends basically onP(ω).)
2.1.1. The combinatorial principles. The principles we formulate here are all statements onκ×ω matrices of subsets ofω claiming – roughly speaking – that all these matrices contain large “submatrices” satisfying certain homogeneity properties.
To simplify the formulation of our results we introduce the following pieces of notation. IfS is an arbitrary set andkis a natural number then let
(S)k={s∈Sk:|rans|=k}
and
(S)<ω= [
k<ω
(S)k. ForD0, . . . , Dk−1⊂S we let
(D0, . . . , Dk−1) ={s∈(S)k : ∀i∈k(s(i)∈Di)}.
Definition 2.1.1.If S is a set of ordinals denote by M(S) the family of allS×ω-matrices of subsets of ω, that is, A ∈ M(S) if and only if A =hA(α, i) :α∈S, i < ωi, where A(α, i)⊂ω for each α ∈ S and i < ω. IfA =hA(α, i) :α∈S, i < ωi ∈ M(S) and R ⊂ S we define the restriction of A to R, A |`R in the straightforward way: A |`R = hA(α, i) :α∈R, i < ωi. If A=hA(α, i) :α∈S, i < ωi ∈ M(S),t∈ω<ω ands∈(S)|t| then we let
A(s, t) = \
i<|t|
A(s(i), t(i)).
83
Now we formulate our first and probably most important principle that we call Cs(κ). We also specify a weaker version ofCs(κ)denoted byC(κ)because in most of the applications (2.1.23, 2.1.25, 2.1.30, 2.1.33, 2.1.37, 2.1.43 ) we don’t need the full power ofCs(κ).
Definition 2.1.2.ForT ⊂ω<ω a matrixA=hA(α, i) :α∈S, i∈ωi ∈ M(S)is called T-adic if for eacht∈T ands∈(S)|t|we have A(s, t)6=∅.
Definition 2.1.3.Forκ= cf(κ)> ω principleCs(κ)(C(κ)) is the following statement:
For everyT ⊂ω<ω andA ∈ M(κ)we have (1) or (2) below:
(1) there is a stationary (cofinal) setS⊂κsuch thatA |`S isT-adic,
(2) there aret∈T and stationary (cofinal) subsetsD0, D1,. . ., D|t|−1 ofκsuch that for every s∈(D0, . . . , D|t|−1)we have
A(s, t) =∅.
Next we formulate a dual version of principlesCs(κ)andC(κ). Let us remark that we don’t know whetherCs(κ)(C(κ)) impliesCˆs(κ)(C(κ)) or vice versa.ˆ
Definition 2.1.4.Ifκ= cf(κ)> ω, then principleCˆs(κ)(C(κ)) is the following statement:ˆ For everyT ⊂ω<ω andA ∈ M(κ)we have (1) or (2) below:
(1) there is a stationary (cofinal) setS⊂κsuch that for eacht∈T ands∈(S)|t|
|A(s, t)|< ω,
(2) there aret∈T and stationary (cofinal) subsetsD0, D1,. . ., D|t|−1 ofκsuch that for every s∈(D0, . . . , D|t|−1)we have
|A(s, t)|=ω.
Let us remark that in the “plain” dual of principleCs(κ)we should have|A(s, t)|=∅in 2.1.41 and|A(s, t)| 6=∅ in 2.1.42, but this “principle” is easily provable in ZFC.
The principlesD(κ)and Ds(κ) that we introduce next easily follow from C(κ) and Cs(κ), respectively, but as their formulation is much simpler, we thought it to be worth while to have them as separate principles. We first give two auxiliary definitions.
Definition 2.1.5.IfA=hA(α, i) :α < κ, i < ωi ∈ M(κ), then we set Aˆ={Y ⊂ω:|{α < κ:∃i < ω A(α, i)⊂Y}|=κ}
and
Aˆs ={Y ⊂ω:{α < κ:∃i < ω A(α, i)⊂Y}is stationary inκ}.
Now we can formulateDs(κ)(D(κ)) as follows.
Definition 2.1.6.Forκ= cf(κ)> ω principleDs(κ)(D(κ)) is the following statement:
IfA ∈ M(κ)andAˆs (A) is centered then there is a stationary (cofinal) setˆ S⊂κsuch thatA |`S isω<ω-adic.
Theorem 2.1.7.Cs(κ) (C(κ))impliesDs(κ) (D(κ)).
Proof. We give the proof only forDs(κ)because the same argument works forD(κ).
LetA ∈ M(κ)so thatAˆs is centered and putT =ω<ω. ByCs(κ)either 2.1.3(1) or 2.1.3(2) holds.
IfS⊂κwitnesses 2.1.3(1) for ourT thenA |`S is clearlyω<ω-adic. So it is enough to show that 2.1.3(2) can not hold.
Assume, on the contrary, that there are t ∈ T =ω<ω and stationary subsets D0, D1, . . ., D|t|−1 ofκsuch that for eachs∈(D0, . . . , D|t|−1)we have
A(s, t) =∅. (+)
We can obviously assume that the setsDi are pairwise disjoint. LetXi=S
{A(δ, t(i)) :δ∈Di} for everyi <|t|. Then clearlyXi∈Aˆsfor everyi <|t|, while (+) implies T
i<k
Xi=∅, contradicting
thatAˆs is centered.
2.1. COMBINATORIAL PRINCIPLES FROM ADDING COHEN REALS 85
Definition 2.1.8.Ifκ= cf(κ)> ω, then principleFs(κ)(F(κ)) is the following statement:
For everyT ⊂ω<ω andA ∈ M(κ)(1) or (2) below holds:
(1) there is a stationary (cofinal) setS⊂κsuch that
|{A(s, t) :t∈T ands∈(S)|t|}| ≤ω.
(2) there aret∈T and stationary (cofinal) subsetsD0,D1, . . ., D|t|−1 ofκsuch that ifs0, s1∈ (D0, . . . , D|t|−1)withs0(i)6=s1(i)for each i <|t| then we have
A(s0, t)6=A(s1, t).
Clearly, ift, D0,. . .,D|t|−1 satisfy (2) then
|{A(s, t) :s∈(D0, . . . , D|t|−1)}|=κ.
There is a surprising connection between these principles and the dual versionsCˆs(κ)(C(κ))ˆ ofCs(κ)(C(κ)), respectively.
Theorem 2.1.9.Fs(κ) (F(κ))impliesCˆs(κ) ( ˆC(κ)).
Proof. LetA ∈ M(κ)andT ⊂ω<ω and apply Fs(κ)to AandT. Assume first that there is a stationary setS ⊂κsuch that the family
I ={A(s, t) :t∈T ands∈(S)|t|} is countable.
Now fort∈T,i <|t|andI∈ I ∩ ωω
set
D(I, t, i) ={α∈S:A(α, t(i))⊃I}.
If for somet∈T andI∈ I ∩ ωω
the set D(I, t, i)is stationary for eachi <|t|then this tand the setsD(I, t,0),. . .,D(I, t,|t| −1) witness 2.1.42.
So we can assume that for allt∈T andI∈ I ∩ ωω
the set
b(I, t) ={i <|t|:D(I, t, i)is non-stationary inκ}
is not empty. Then the set D=[
{D(I, t, i) :I∈ I ∩ ωω
, t∈T, i∈b(I, t)}
is not stationary and so S0 =S\D is stationary. We claim that S0 witnesses 2.1.41. Assume on the contrary that t ∈ T, s ∈ (S0)|t| and I = A(s, t) is infinite. Then I ∈ I ∩
ωω
and s(i) ∈ D(I, t, i) for eachi < |t|. Since s(i) ∈/ D it follows thatD(I, t, i)is stationary for each i <|t|, that is,b(I, t) =∅, which is a contradiction.
Assume now that there are t∈T and stationary subsetsD0, D1, . . .,D|t|−1 ofκsuch that ifs0, s1∈(D0, . . . , D|t|−1)withs0(i)6=s1(i)for eachi <|t|then
A(s0, t)6=A(s1, t).
We show that in this case again 2.1.42 holds. Indeed, for eachI∈ ω<ω
picksI ∈(D0, . . . , D|t|−1) such thatA(sI, t) =I provided that there is such ans. LetR=S
{sI(i) :I∈ ω<ω
, i <|t|}and Di0 =Di\R fori <|t|. Now ifs∈(D00, . . . , D0|t|−1)then for anyI∈
ω<ω
we havesI(i)6=s(i) for eachi <|t|, henceI=A(sI, t)6=A(s, t). AsIwas an arbitrary element of
ω<ω
we conclude
that|A(s, t)|=ω.
The following observation is almost trivial.
Proposition 2.1.10.Ifκ= cf(κ)> cthenCs(κ)andFs(κ)are valid.
As was mentioned, our principles are of interest only for κ > ω1. In fact, forκ=ω1, they are all false!
Theorem 2.1.11 ([12]).C(ωˆ 1)(and soF(ω1)too) andD(ω1)are both false.
2.1.2. Consistency of the principles in the Cohen model. A cardinalκisω-inaccessible ifλω < κ holds for eachλ < κ. Given any infinite setI we denote byCI the posetFn(I,2, ω), i.e. the standard one adding|I|-many Cohen reals.
In this subsection we prove that ifκis a regularω-inaccessible cardinal in some ground model V and we addλ-many Cohen reals toV, whereλis an arbitrary cardinal, then in the extension the principlesCs(κ),Cˆs(κ)andFs(κ)are all satisfied. As we remarked in subsection 2.1.1 above the caseκ > λ is trivial, while the caseκ < λcan be reduced to the caseκ=λ.
Since the proof of the latter is long and technical, we first sketch the main idea. So let us be given a matrix A ∈ M(κ)and a set T ⊂ω<ω inV[G], whereGisCκ-generic overV. In the first part of the proof we find a setI∈
κω
and a stationary setS⊂κsuch that inV[G|`I]the sequenceshA(α, i) :i < ωiforα∈S have also pairwise isomorphic names with disjoint supports (contained in κ\I). This reduction, carried out in lemma 2.1.17, will be the place where we use that κ is regular and ω-inaccessible in V. In the second part of the proof, using slightly different arguments for Cs(κ) and forFs(κ), we show that if some A ∈ M(S)has names with these properties then either S witnesses 2.1.3(1) (or 2.1.8(1), respectively) or some stationary setsDi⊂S witness 2.1.3(2) (or 2.1.8(2), respectively). In this second step we don’t use thatκis ω-inaccessible or regular.
In our forcing arguments we follow the notation of Kunen [77]. Let us first recall definition [77, 5.11].
Definition 2.1.12.A CI-name B˙ of a subset of some ordinalµ is called niceif for each ν < µ there is an antichainBν ⊂ CI such that
B˙ ={hp,νiˆ :ν∈µ∧p∈Bν}=[
{Bν× {ˆν}:ν∈µ}.
We letsupp( ˙B) =S{dom(p) :p∈ S
ν<µ
Bν}.
It is well-known (see e.g. lemma [77, 5.12]) that every set of ordinals inV[G]has a nice name inV.
Ifϕis a bijection between two setsIandJ thenϕlifts to a natural isomorphism betweenCI andCJ, which will be also denoted byϕ, as follows: forp∈ CI letdom(ϕ(p)) =ϕ00dom(p)and ϕ(p)(ϕ(ξ)) =p(ξ). Moreoverϕalso generates a bijection between the niceCI-names and the nice CJ-names (see [77, 7.12]): if B˙ is a nice CI-name then let ϕ( ˙B) = {D
ϕ(p),ξˆE :D
p,ξˆE
∈B}.˙ If I and J are sets of ordinals with the same order type thenϕI,J is the natural order-preserving bijection fromI ontoJ.
Definition 2.1.13.Assume I, J ⊂ κ, moreover A˙i and B˙i are nice Cκ-names of subsets of ω for i < ω, such that supp( ˙Ai) ⊂ I and supp( ˙Bi) ⊂ J. We say that the structures of names D
I,A˙i:i < ωE and D
J,B˙i:i < ωE
are twins if I and J have the same order type and for the order preserving bijectionϕI,J we have
(1) ϕI,J is the identity onI∩J, (2) ϕI,J( ˙Ai) = ˙Bi for each i < ω.
Definition 2.1.14.Assume thatI⊂κ, Gis aCκ-generic filter overV andH =G|`I. IfB˙ is a niceCκ-name of a subset of some ordinalµwe define inV[H]theCκ\I nameπH( ˙B)as follows:
πH( ˙B) ={hp|`κ\I,νˆi:hp,νi ∈ˆ B˙ ∧p|`I∈H}.
Lemma 2.1.15.πH( ˙B)is a niceCκ\I-name in V[H] andsupp(πH( ˙B))⊂supp( ˙B)\I, moreover val(πH( ˙B), G|`(κ\I)) = val( ˙B, G).
Proof. Straightforward from the construction.
Definition 2.1.16.Assume thatS ⊂κ. A matrixB˙=D
B(α, i) :˙ α∈S, i < ωE
of niceCκ-names of subsets ofω is called anice S-matrixif conditions (i) and (ii) below hold:
(i) puttingJα=S
i<ωsupp( ˙B(α, i))the sets{Jα:α∈S} are pairwise disjoint,
2.1. COMBINATORIAL PRINCIPLES FROM ADDING COHEN REALS 87
(ii) the structures of names{D
Jα,B(α, i) :˙ i < ωE
:α∈S}are pairwise twins.
We denote byN(S)the family of niceS-matrices.
Lemma 2.1.17. (Reduction lemma) Assume that κ is a regular, ω-inaccessible cardinal, G is Cκ-generic over V and A ∈ M(κ) in V[G]. Then there are a countable set I ⊂ κ and a
We need a strong version of Erdős-Rado∆-system theorem saying that there is a stationary setT ⊂κsuch that{Iα:α∈T}forms a∆-system with some kernelI, moreoversupI <minIα\I for eachα∈T. Although this statement is well-known, in [12] we presented a proof because we could not find any reference to it.
Erdős-Rado Theorem .If κ is an ω-inaccessible regular cardinal and X = {Xα : α < κ} is a family of countable sets then there is a stationary set I ⊂κsuch that {Xα : α∈I} forms a
∆-system.
Since2ω< κ= cf(κ)and there are only2ωdifferent isomorphism types of structures of names there is a stationary setS⊂T such that the structures of names{D
Iα,A(α, i) :˙ i < ωE S-sequenceif conditions (i) and (ii) below hold:
(i) Jα∈
We denote byS(S)the family of niceS-sequences.
Lemma 2.1.19. (Homogeneity lemma)Assume thatS⊂κandB˙=DD
Jα,B˙αE
:α∈SE is a nice S-sequence. If ϕ(x0, x1, . . . , xn−1, z)is a formula with free variablesx0, x1, . . . , xn−1, z and Z is an element of the ground model, then(1)or(2) below holds:
(1) 1Cκ “ ϕ( ˙Bs(0)B˙s(1), . . . ,B˙s(n−1),Zˆ)for alls∈(S)n”,
where G is Cκ-generic over V. Since the supports of p(β, i) for β ∈ S are pairwise disjoint a standard density argument gives thatDi∩A6=∅wheneverA∈
Sω
∩V, hence (a) holds.
To show (b) assume thatr∈Gand
V[G]|=“u∈(D0, . . . , Dk−1).”
Sinceuis finite we have u∈V. LetJ∗ = S
i<k
Ju(i) andψ = S
i<k
ϕu(i),s(i). Thenψ is a bijection betweenJ∗andJand so it extends to isomorphisms betweenCJ∗andCJ, and between the families of niceCJ∗-names and of niceCJ-names. LetΨbe the natural extension of ψto a permutation ofκ:
Ψ(ν) =
ψ(ν) ifν∈J∗, ψ−1(ν) ifν∈J,
ν ifν∈κ\(J∪J∗).
Then Ψ extends to an automorphism of Cκ, and also to an automorphism of nice Cκ-names.
Clearly ifq∈ CJ∗ andB˙ is a niceCJ∗-name then ψ(q) = Ψ(q)and ψ( ˙B) = Ψ( ˙B). Observe that Ψ(r) =randΨ( ˆZ) = ˆZ.
Let G∗ = Ψ00G. Then G∗ is also a Cκ-generic filter over V and since Ψ( ˙Bu(i)) = ˙Bs(i) it follows that
val( ˙Bu(i), G) = val( ˙Bs(i), G∗). (•) Butp(u(i), i)∈G, sop|`Js(i)=ψ(p(u(i), i))∈G∗. Thusp=r∪ S
i<k
p|`Js(i)∈G∗ as well. Since p¬ϕ( ˙Bs(0), . . . ,B˙s(n−1),Z)ˆ and soV[G∗]|=“¬ϕ(Bs(0), . . . , Bs(n−1), Z)”, by(•)this implies
V[G]|=“¬ϕ(Bu(0), . . . , Bu(n−1), Z)”
which was to be proved.
Theorem 2.1.20.Ifκis a regular, ω-inaccessible cardinal then for each cardinal λwe have VCλ |=“Cs(κ)andCˆs(κ)hold.”
Proof. We deal only withCs(κ) because the same argument works forCˆs(κ). As we observed in section 2.1.1 we can assume thatκ≤λ. First we investigate the caseλ=κ.
Assume that
1Cκ “A˙=D
A(α, i) :˙ α < κ;i < ωE
∈ M(κ)andT ⊂ω<ω.”
Applying the reduction lemma 2.1.17 and thatT is countable we can find a countable set I⊂κ and a stationary setS⊂κin V and a niceS-matrixBinV[G|`I]such that
V[G]|=“val( ˙A(α, i), G) = val( ˙B(α, i), G|`(κ\I))”
forα∈S andi∈ω, moreoverT∈V[G|`I].
We show that for eachq∈ Cκ there is a conditionr≤qinCκ such thatr“2.1.31 or 2.1.32 holds”. LetI0 =I∪dom(q).
For eacht∈T letϕt(x0, . . . , x|t|−1)be the following formula:
ϕ(hB0,k:k < ωi, . . . ,
B|t|−1,k :k < ω
)⇐⇒ \
i<|t|
Bi,t(i)6=∅.
Applying the homogeneity lemma 2.1.19 toV[G|`I0]as our ground model and to everyϕtwe get that either q“2.1.31 holds” orq∪p“2.1.32 holds”. Let us remark that 2.1.19(2)(a) implies that asS is stationary, so is eachDi.
Thus we have proved the theorem in the case κ=λ. Ifλ > κ andA ∈(M(κ))V[G], where GisCλ-generic overV, then there isJ ∈
λκ
such thatA ∈V[G|`J]. The stationary sets that witness 2.1.3(1) or 2.1.3(2) inV[G|`J]remain stationary inV[G], and so we are done.
Theorem 2.1.21.Ifκis a regular, ω-inaccessible cardinal then for each cardinal λwe have VCλ |=“Fs(κ) holds.”
2.1. COMBINATORIAL PRINCIPLES FROM ADDING COHEN REALS 89
Proof. As in 2.1.20 the important case is whenλ=κ, because the caseλ < κis trivial and the caseκ < λcan be reduced to the caseκ=λ.
We need the following lemma that is probably well-known.
Lemma 2.1.22 ([12]).If H is aCκ-generic filter overV andI,J are disjoint subsets ofκthen
Applying the homogeneity lemma 2.1.19 toV[G|`I0]as our ground model we get that (A) or (B) below holds: so it lifts up to an isomorphism betweenCJ∗ andCJ and between the families of niceCJ∗-names and niceCJ-names.
From this it is obvious that we have
1Cκ {B(t, s) :˙ t∈T ∧s∈(S)|t|} ⊂ I=[
{It:t∈T}.
whereI is countable asT is.
Assume now that (A) fails and so (B) holds.
LetGbeCκ-generic withp∈Gandhγ0, . . . , γk−1i,hδ0, . . . , δk−1i ∈(D0, . . . , Dk−1)such that V[G]|=“{γi, δi} ∈
Di2
fori < kare pairs of distict ordinals”.
Let J∗ = S
The theorem is proved.
2.1.3. Applications. We start with presenting some combinatorial applications because they are quite simple and so they nicely illustrate the use of our principles.
Kunen [76] proved that if one adds Cohen reals to a model of CH, then in the generic extension there is no strictly⊂∗-increasing chain of subsets ofω of lengthω2. The first theorem we prove easily yields Kunen’s above result.
Theorem 2.1.24.IfD(κ)orC(κ)ˆ holds, thenκis not embeddable into P(ω)/f in.
Remark .In the original version of the paper the statement of theorem 2.1.24 above was derived only from principleC(κ). This strengthening of our result was pointed out by the referee.
Proof. Assume that{Aα:α < κ}is a strictly⊂∗-increasing chain in
The next theorem can be considered as a kind of dual to 2.1.23.
Theorem 2.1.25 ([12]).If C(κ) holds then for each A ⊂
Next we prove a consequence of theorem 2.1.25, but first we give a definition.
Definition 2.1.26.Letκbe a regular cardinal andA ⊂ ωω
2.1. COMBINATORIAL PRINCIPLES FROM ADDING COHEN REALS 91
Anω1-Luzin gap can be constructed in ZFC and simple forcings give models in which there are2ω-Luzin gaps while2ω is as large as you wish. The next corollary of theorem 2.1.25 implies that one can not constructω2-Luzin gaps from the assumption2ω≥ω2alone .
Corollary 2.1.27.If C(κ)holds then there is noκ-Luzin gap.
Proof. Assume thatA ⊂ ωω
is an almost disjoint family of sizeκ. Then we can not get a even a two element subfamily B ⊂ Asatisfying 2.1.25(a). So applying theorem 2.1.25 for this Aand for k = 2 there are subfamilies B ⊂ A and D ⊂ A of size κ such that (SB)∩(SD) is finite.
HenceX =SBwitnesses thatAis not aκ-Luzin gap.
Now we turn to applying our principles to topology. We start with an application of the relatively weak principleD(κ).
A. Dow [36] proved that if we add ω2 Cohen reals to a model of GCH then in the generic extensionβωcan be embedded into every separable, compactT2space of size> c=ω2. Here we show thatc=ω2= 2ω1 together withD(ω2)suffice to imply this statement.
First we need a lemma based on the observation that large separable spaces contain many
“similar” points.
Given a topological space X and a point x ∈ X we denote by VX(x) the neighbourhood filter of x in X, that is, VX(x) = {U ⊂ X : x ∈ intX(U)}. If D is a dense subset of X let VX(x)|`D={U ∩D:D∈ VX(x)}. We omit the subscript X if it may not cause any confusion.
In section 2.1.1 we defined the operationAˆforA ∈ M(κ). By an abuse of notation we define Aˆfor every familyAof subsets ofω as follow:
Aˆ={X⊂ω:|A ∩ P(X)|=|A|}.
Lemma 2.1.28.Assume that X is a separable regular topological space of size > c<c, where c= 2ω,D∈
Xω
,D=X. Then there are a pointx∈X and a family A={Aα, Bα:α < c} ⊂ P(D)such that
(1) Aα∩Bα=∅ for each α < c, (2) A ⊂ Vˆ (x)|`D.
Proof. Fix an enumeration {Dξ : ξ < c} ofP(D)and let Dα = {Dξ :ξ < α} for α < c. For x∈X and α < cletV(x, α) = (V(x)|`D)∩ Dα. A point x∈X is calledspecial if there is an α < c such that V(x, α)6=V(y, α) for each y ∈X \ {x}. Clearly there are at most c<c special points inX. Since|X|> c<cwe can pick a pointx∈X which is not special. Then for eachα < c we can find a point xα 6=xin X such thatV(xα, α) =V(x, α). Since X is regular the pointsx andxαhave neighbourhoodsUαandWα, respectively, withUα∩Wα=∅. LetAα=Uα∩Dand Bα=Wα∩D.
Now assume thatE ∈Aˆand pick ξ < c withE =Dξ. We can find α < csuch thatξ < α and eitherAα⊂E orBα⊂E. HenceE∈ V(x, α)∪ V(xα, α) =V(x, α). ThereforeE∈ V(x)|`D
which was to be proved.
Let us now recall the definition of aµ-dyadic system from [62].
Definition 2.1.29.If X is a topological space a family {hA(α,0), A(α,1)i: α∈ µ} of pairs of closed subsets ofX is a µ-dyadic systemsuch that
(1) A(α,0)∩A(α,1) =∅for eachα < µ, (2) for each∈Fn(µ,2, ω)we have T
α∈dom()
A(α, (α))6=∅.
Theorem 2.1.30.IfD(c)holds,X is a separable compactT2space of size> c<cthenX contains a c-dyadic system, consequently X maps continuously onto [0,1]c ( and so βω can be embedded intoX ).
Proof. Fix a countable dense subsetD ofX. By lemma 2.1.28 there is a familyA={Aα, Bα: α < c} ⊂ P(D) such that Aα∩Bα = ∅ for α < c and Aˆ is centered. Let D(α,0) = Aα, D(α,1) = Aα and D(α, n) = D for α < κ and n ≥ 2 and consider the κ×ω-matrix D = hD(α, i) :α < κ, i < ωi. Since Aˆ = ˆD we can apply D(c) to get a cofinal S ⊂ c such that
the family
Aα, Bα:α < c
is c-dyadic. Now we can apply theorem [62, 3.18] to get the other
consequences.
Since the cardinality of a locally compact, scattered separable space is at most2ω by [88], the height of such a space is less then (2ω)+. So underCH there is no such a space of height ω2. M. Weese asked whether the existence of such a space of heightω2follows from ¬CH. This question was answered in the negative by W. Just, who proved, [64, theorem 2.13 ], that if one adds Cohen reals to a model of CH then in the generic extension there are no locally compact scattered thin spaces of heightω2.
The next theorem is a generalization of the above mentioned result of Just.
Theorem 2.1.31.If Cs(κ)holds then there is no locally compact, thin scattered space of height κ.
Proof. Assume on the contrary that there is such a space X. We can assume that Iα(X) = {α} ×ω forα <ht(X). For eachα <ht(X)fix compact open neighbourhoodsU(α, n)ofhα, ni forn∈ω such thatU(α, n)⊂ {hα, ni} ∪S{Iβ(X) :β < α} and the sets U(α, n)forn < ω are pairwise disjoint.
PutA(α,2n) =U(α, n)∩I0(X)andA(α,2n+ 1) = I0(X)\S{U(α, m) :m≤n}. Let T ={t∈ω<ω: t(0)is even andt(i)is odd for i >0}.
Now applyCs(κ)to the matrixhA(α, n) :α < κ, n < ωi ∈ M(κ)andT. Observe that A(β,2n)∩ T
i<k
A(αi,2ni+ 1) =∅ iff U(β, n)∩I0(X)⊂ S
i<k
U(αi, ni)∩I0(X) iff U(β, n)⊂ S
i<k
U(αi, ni).
Thus ift=h2n,2n0+ 1, . . . ,2nk−1+ 1i ∈T andhβ, α0, . . . , αk−1i ∈(κ)k+1 then A(β,2n)∩ T
i<k
A(αi,2ni + 1) = ∅ implies β ≤ max
i<k αi. This excludes 2.1.3(2). So 2.1.3(1) holds, that is we have a stationary set S ⊂ κ such that if t = h2n,2n0+ 1, . . . ,2nk−1+ 1i ∈ T and hβ, α0, . . . , αk−1i ∈(S)k+1 then
A(β, n)∩\
i<k
A(αi,2ni+ 1)6=∅, that is
U(β, n)\ [
i<k
[
j≤ni
U(αi, j)6=∅.
ButU(β, n)is compact and eachU(α, n)is open so it follows that for everyβ∈Sandn∈ωthe set
D(β, n) =U(β, n)\[
{U((α, m) :α∈S\ {β} ∧m∈ω}
is not empty. For every suchβ andnlethγ(β, n), m(β, n)i ∈D(β, n).
SinceIβ(X) is dense in X\S{Iα(X) :α < β} for every β ∈κthere is k(β)∈ω such that hβ, k(β)i ∈U(β∗,0), whereβ∗= minS\β+ 1. Thushβ, k(β)i∈/D(β, k(β))and soγ(β, k(β))< β for eachβ∈S. The setS is stationary so there are a stationary setS0⊂S, and ordinals γ < κ and k, m < ω such that k(β) = k, γ(β, k) = γ and m(β, k) = m whenever β ∈ S0. Thus hγ, mi ∈ D(β, k)for each β ∈ S0, whileD(β, k)∩D(β0, k) = ∅ for any {β, β0} ∈
S02 by the construction. This is a contradiction, hence the theorem is proved.
Remark .It is easy to see that the proof of theorem 2.1.31 goes through if, instead of assuming that all levels of the space are countable, we only require that
(i) (i) all levels are of size< κ,
(ii) (ii) there are stationary many countable levels.
In [64] W. Just also proved that if one adds at leastω2 Cohen reals to a model ofCH then in the generic extension there is no locally compact, scattered topological space X such that
2.1. COMBINATORIAL PRINCIPLES FROM ADDING COHEN REALS 93
ht(X) = ω1+ 1, I0(X) is countable, |Iα(X)| ≤ ω1 for α < ω1 and |Iω1(X)| = ω2. The next theorem shows how to get a generalization of this result from our principles.
Theorem 2.1.32.If cf(λ) ≥ ω1 and F(λ+) holds then there is no locally compact, scattered topological space X such that ht(X) =λ+ 1,I0(X)is countable, |Iα(X)| ≤λfor all α < λ and
|Iλ(X)|=λ+.
Proof. See in [12].
Following the terminology of [49] a Hausdorff space is called P2 if it does not contain two uncountable disjoint open sets. Hajnal and Juhász in [49] constructed a ZFC example of a first countable,P2 space of size ω1 as well as consistent examples of size2ω with2ω as large as you wish . On the other hand, using a result of Z. Szentmiklóssy they proved that it is consistent with ZFC that2ωis as large as you wish and there are no first countableP2 spaces of size≥ω3. However their method was unable to replace here ω3 with ω2. Our next result does just this because, as is shown in [49], everyP2 space is separable.
Theorem 2.1.33.If C(κ) holds then every first countable, separable T2 topological space X of sizeκcontains two disjoint open setsU andV of cardinalityκ.
Proof. Let D be a countable dense subset of X. For each x ∈ X fix a neighborhood base {U(x, n) :n∈ω}ofxinX. ApplyC(κ)to the matrixhU(x, n)∩D:x∈X, n < ωiandT =ω2.
Definition 2.1.34.Let X be a topological space and D ⊂ X. We say that D is sequentially dense inX iff for eachx∈X there is a sequence Sxfrom Dwhich converges tox. A spaceY is said to besequentially separableif it contains a countable sequentially dense subset.
Definition 2.1.35.Given a topological spacehX, τiand a subspaceY ⊂X a functionf is called aneighbourhood assignment on Y in X ifff :Y −→τ andy∈f(y)for eachy∈Y.
Our next result says that underC(κ)if a sequentially separable spaceX does not contain a discrete subspace of sizeκ, (i.e. ˆs(X)≤κusing the notation of [62]) thenX does not contain left or right separated subspaces of sizeκeither. This can be written asˆh(X) ˆz(X)≤κ. Since in [11]
a normal, Frechet-Urysohn, separable (hence sequentially separable) spaceX is forced such that z(X)≤ω1 buth(X) =ω2, this result is not provable in ZFC. First, however, we need a lemma.
a normal, Frechet-Urysohn, separable (hence sequentially separable) spaceX is forced such that z(X)≤ω1 buth(X) =ω2, this result is not provable in ZFC. First, however, we need a lemma.