HFD and HFC type spaces, with applications

Topology and its Applications 126 (2002) 217–262

www.elsevier.com/locate/topol

HFD and HFC type spaces, with applications✩

István Juhász

Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Sciences,P.O. Box 127, Budapest H-1364, Hungary

Received 24 June 1998; received in revised form 28 January 2002

Abstract

The aim of this paper is to give a survey of HFD and HFC type spaces. These are subspaces ofCantor cubes 2λ with very flexible combinatorial properties that yield many basic examples ofS andL spaces as well as numerous other interesting topological spaces. 2002 Elsevier Science B.V. All rights reserved.

MSC:54A25; 03E55

Keywords:HFD and HFC spaces;S andL spaces; Cardinal functions; Independence results

1. Introduction

Throughout this paper we use standard terminology and notation as, e.g., in [8,12,17].ThusFn(A,B) denotes the set of all finite functionsε whose domainD(ε) ⊂ A and rangeR(ε) ⊂ B. We also setH(S) = Fn(S,2) for any setS.

Since the structures we will study are mostly subspaces of products of the form 2I (here2 is the two-point discrete space), we list below some special notation that will be used instudying these.

If ε ∈ Fn(I,2) = H(I) then

[ε] = {f ∈ 2I : ε ⊂ f

}denotes the elementary basic clopen set determined byε. If I is a set of ordinals andb ∈ [I ]<ω, b = {βi : i ∈ n = |b|} whereβi is theith member ofb in its increasing order.Now, if ε ∈ 2n then we denote byε ∗ b that element ofH(I) which hasb as its domain andsatisfiesε ∗ b(βi) = ε(i) for all i ∈ n.

✩ Research supported by OTKA grant no. 25745.E-mail address:[email protected] (I. Juhász).

0166-8641/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0166-8641(02)00080-9

218 I. Juhász / Topology and its Applications 126 (2002) 217–262

For any (usually infinite) cardinalµ andr ∈ ω we denote byDrµ(I) the collection of all

setsB ∈ [[I ]r ]µ such that the members ofB are pairwise disjoint. We shall write

Dµ(I) =⋃{

Drµ(I): r ∈ ω

}.

If B ∈ Dµ(I) thenn(B) = |b| for anyb ∈ B. Now, if B ∈ Dµ(I) andε ∈ 2n(B) then

[ε,B] =⋃{[ε ∗ b]: b ∈ B

}is called aDµ-set in 2I . The most important case of the above is whenµ = ω, so weshall often omit the lower indexω of D in this case. So, e.g., aD-set in 2I is aDω-set.Clearly, anyD-set is dense open and has product (Lebesgue) measure 1 in 2I . We call anuncountable setH ⊂ 2ω aD-Luzin set if for anyD-set[ε,B] in 2ω we have∣∣H \ [ε,B]∣∣ � ω.

So then any ordinary Luzin set or Sierpinski set is aD-Luzin set.If X = {fα : α ∈ κ} ⊂ 2λ then a matrixF :κ × λ → 2 may be defined by the formula

F(α,β) = fα(β),

and vice versa, ifF :κ × λ → 2 is a matrix then from

fα = F(α,−)

we obtain a subspaceX = {fα : α ∈ κ} of 2λ. In this case we say thatF representsX. Therepresentation is one-one ifα �= α′ impliesfα �= fα′ for any{α,α′} ∈ [κ]2.

Hereditarily separable (in short: HS) and hereditarily Lindelöf (in short: HL) subspacesof 2λ will play a crucial role in what follows. So in the following preliminary resultswe formulate criteria to decide when a subspaceX of 2λ is HS or HL. These criteriacan also serve as motivation for the concepts of HFD and HFC type structures, although,historically, this was not the case.

1.1. LetX ⊂ 2λ. Then

(i) X is HS iff for every setY ∈ [X]ω1 there are aZ ∈ [Y ]ω and I ∈ [λ]�ω such thatZ � (λ \ I) is dense inY � (λ \ I).

(ii) X is HL iff for everyDω1-set[ε,B] in 2λ there is aC ∈ [B]ω such that

[ε,B] ∩ X ⊂ [ε,C].

Proof. (i) The implication from left to right is obvious, so assume now thatX is not HS.Then there isY = {fα : α ∈ ω1} ⊂ X left-separated in thisω1-type ordering. So for eachα ∈ ω1 we have abα ∈ [λ]<ω such that[fα � bα] �� fβ wheneverβ ∈ α ∈ ω1. Without lossof generality, the collection{bα: α ∈ ω1} is a∆-system with rootb so thatfα � b = fβ � bfor all α,β ∈ ω1. But then for anyZ ∈ [Y ]ω andI ∈ [λ]�ω there is anα ∈ ω1 with β ∈ α

for all fβ ∈ Z andbα \ b ⊂ λ \ I , hence clearly[fα � (bα \ b)

] ∩ Z = ∅,and soZ � (λ \ I) is not dense inY � (λ \ I).

I. Juhász / Topology and its Applications 126 (2002) 217–262 219

(ii) Again, it is obvious that ifX is HL then the required condition is satisfied.Conversely, ifX is not HL then we have a right-separatedY = {fα ∈ α ∈ ω1} ⊂ X, wherefor everyα ∈ ω1 we havebα ∈ [λ]<ω so thatfβ /∈ [fα � bα] wheneverα ∈ β ∈ ω1. Likeabove, we may assume thatB = {bα: α ∈ ω1} ∈ Dn

ω1(λ) and thatfα � bα = ε ∗ bα for some

ε ∈ 2n and for allα ∈ ω1. Clearly, then we have

[ε,B] ∩ Y �⊂ [ε,C]for anyC ∈ [B]ω. ✷

As is well known, neither HS nor HL is a productive property. So we say that a spaceX

is HSn (HLn) if Xn is HS (HL). It is easy to extend the criteria of 1.1 to those characterizingwhenX ⊂ 2λ is HSn (respectively HLn).

Also, X is called strongly HS (respectively HL) if it is HSn (respectively HLn for alln ∈ ω). Finally, anSn space (Ln space) is aT3 space that is HSn but not HL (respectivelyHLn but not HS). Also, we can talk about strongS and strongL spaces.

2. HFD-type spaces

The aim of this section is to introduce certain general classes of spaces from which onecan obtainS spaces with all sorts of interesting properties. As we shall see in Section 3,the natural “duals” of these classes will do the same forL spaces.

2.1. Definition.

(i) A map F :κ × λ → 2 with κ � ω, λ � ω1 is called an HFDmatrix if for everyA ∈ [κ]ω, B ∈ Dω1(λ) andε ∈ 2n(B) there areα ∈ A andb ∈ B such that

fα = F(α,−) ⊃ ε ∗ b.

(Recall that the last line simply says thatF(α,βi) = ε(i) holds for eachi < n(B) =|b|, whereβi is theith member ofb in its increasing order.)

(ii) X ⊂ 2λ is said to be an HFD space if there exists an HFD matrixF :κ × λ → 2 suchthatX = {fα : α ∈ κ}. In this caseX is said to be represented byF .

The HFD matrix representation of an HFD space is not necessarily one–one, howeverit almost is. This is made clear by the next very simple observation, which in particularimplies that ifF :κ × λ → 2 representsX then|X| = κ .

2.2. If F :κ × λ → 2 is anHFD matrix then∣∣{α ∈ κ : fα = F(α,−) = f}∣∣ <ω

for everyf ∈ 2λ.

Proof. Suppose, indirectly, thatA ∈ [κ]ω is such thatfα = f for eachα ∈ A. We canchoose aB ∈ [λ]ω1 (∼= D1

ω1(λ)) and ani ∈ 2 with f (β) = i for all β ∈ B. But then for

ε = {〈0,1 − i〉} ∈ 21 we haveε ∗ {β} �⊂ fα wheneverα ∈ A andβ ∈ B, contradicting thatF is HFD. ✷


The matrix approach to HFD spaces makes the unified treatment and dualization easier.That is why we chose to follow it here, but it can be avoided. This is shown by the nextinternal characterization of HFD spaces that was the original definition. It also explains theterminology: HFD= Hereditarily Finally Dense.

2.3. The following two conditions(i) and(ii) are equivalent forX ⊂ 2λ (λ > ω):

(i) X is HFD;(ii) X is infinite and for everyA ∈ [X]ω there is aB ∈ [λ]ω such thatA (i.e.,A � (λ \ B))

is dense in2λ\B (such anA is also called finally dense).

Proof. (i) ⇒ (ii). Using 2.1 we can choose an HFD matrixF :κ × λ → 2 representingXin a one–one manner (i.e., such thatfα �= fβ if α �= β). GivenA ∈ [κ]ω we have to showthat A = {fα : α ∈ A} is finally dense. To this end we define by transfinite induction anincreasing sequence of countable setsBν ⊂ λ as follows. Ifν ∈ ω1 andBµ has been definedfor all µ ∈ ν putB ′

ν = ⋃{Bµ: µ ∈ ν}. If A is dense in 2λ\B ′ν we are done. Otherwise we

can choose anεν ∈ H(λ \ B ′ν ) such thatA ∩ [εν] = ∅, then we putBν = B ′

ν ∪ D(εν). Weclaim that there is aν ∈ ω1 such thatA is dense in 2λ\B ′

ν . If this were not the case then wecould choose a setI ∈ [ω1]ω1, ann ∈ ω, andε ∈ 2n such that, puttingbν = D(εν), we had

εν = ε ∗ bν

for eachν ∈ I . But thenA ∈ [κ]ω, B = {bν : ν ∈ I } ∈ Dω1(λ) andε would not satisfy therequirement for the HFD-ness ofF , a contradiction.

(ii) ⇒ (i). LetX = {fα : α ∈ κ} be a one–one enumeration ofX and defineF :κ×λ → 2by F(α,β) = fα(β). We claim thatF is an HFD matrix. Indeed, givenA ∈ [κ]ω, B ∈Dω1(λ) andε ∈ 2n(B) choose first a setC ∈ [λ]�ω such thatA = {fα : α ∈ A} is densein 2λ\C . But then for everyb ∈ B with b ∩ C = ∅ we haveA ∩ [ε ∗ b] �= ∅, i.e.,fα ⊃ ε ∗ b

for someα ∈ A. ✷Remark. In what follows we shall consider a formally more general notion of HFD spacesthen the one given above in that, instead ofλ, we may take any uncountable set (e.g.,in 2.3(ii)).

The most interesting property of HFD spaces is of course that they are HS. Beforeproving this however we introduce a larger class of spaces which still has this property. Thereason why we think it worth while to deal with the particular class of HFD spaces will,we hope, be clear from the extra results we can prove for them but not for the larger class.

2.4. Definition.

(i) F :κ × λ → 2 with κ,λ > ω is said to be an HFDw matrix (the subscript w stands for“weak”) if for everyA ∈ [κ]ω1, B ∈ Dω1(λ) andε ∈ 2n(B) there is anα ∈ A such that∣∣{b ∈ B: fα = F(α,−) ⊃ ε ∗ b

}∣∣ = ω1.

(ii) X ⊂ 2λ is called an HFDw space if it is represented by an HFDw matrix.


It should be clear that every HFD matrix (withκ > ω) is also HFDw, hence anuncountable HFD space is HFDw. That an HFDw spaceX must be uncountable (in fact,we have|X| = κ) follows from the following easy result corresponding to 2.2.

2.5. If F :κ × λ → 2 is HFDw then∣∣{α ∈ κ : fα = f }∣∣ � ω

for eachf ∈ 2λ.

Corresponding to 2.3, we have the following characterization of HFDw spaces.

2.6. LetX ⊂ 2λ with λ > ω, then the following are equivalent:

(i) X is HFDw;(ii) X is uncountable and for everyY ∈ [X]ω1 there is anA ∈ [Y ]ω that is finally dense.

Proof. (i) ⇒ (ii). Let F :κ × λ → 2 representX and chooseY ∈ [κ]ω1. Write

Y =⋃

{Yν : ν ∈ ω1},where|Yν | = ω for eachν ∈ ω1 and ν < µ implies Yν ⊂ Yµ. We propose to show thatYν = {fα : α ∈ Yν} is finally dense for someν ∈ ω1. To this end we again inductivelydefine setsBν ∈ [λ]�ω as follows. Forν ∈ ω1 putB ′

ν = ⋃{Bµ: µ ∈ ν}. If Yν is not densein 2λ\B ′

ν then pickεν ∈ H(λ \ B ′ν ) such thatYν ∩ [εν] = ∅ and then putBν = B ′

ν ∪ D(εν).If no Yν is finally dense then we can chooseI ∈ [ω1]ω1, n ∈ ω andε ∈ 2n in such a waythat

εν = ε ∗ bν

holds for eachν ∈ I (of coursebν = D(εν)). But thenB = {bν : ν ∈ I } ∈ Dnω1(λ) and for

anyµ ∈ ω1 andα ∈ Yµ we have

{ν ∈ I : ε ∗ bν = εν ⊂ fα} ⊂ µ,

contradicting thatF is HFDw.(ii) ⇒ (i) Let {fα : α ∈ κ} be a one–one enumeration ofX and defineF by F(α,β) =

fα(β). GivenA ∈ [κ]ω1, B ∈ Dω1(λ) andε ∈ 2n(B) choosea ∈ [A]ω andC ∈ [λ]�ω suchthat a is dense in 2λ\C . Clearly, then there is anα ∈ a ⊂ A such that∣∣{b ∈ B: ε ∗ b ⊂ fα}∣∣ = ω1. ✷

Now, applying 2.6(ii) and the general criterion 1.1(i) for deciding when a subspace of 2λ

is HS we immediately get what we wanted.

2.7. EveryHFDw (hence everyHFD) space isHS.

Now we can show howS spaces can be obtained from HFDw spaces. This will be aneasy consequence of the fact that the HFD and HFDw properties are very “stable” in thesense made precise by the following proposition.


2.8. LetX ⊂ 2λ beHFD (or HFDw).

(i) Infinite (uncountable) subspaces ofX are alsoHFD (HFDw).(ii) If I ∈ [λ]�ω1 thenX � I is alsoHFD (HFDw).(iii) For eachf ∈ X let f ′ ∈ 2λ be given such that∣∣{α ∈ λ: f (α) �= f ′(α)

}∣∣ � ω,

and putX′ = {f ′: f ∈ X}. ThenX′ is alsoHFD (HFDw).

Proof. (i) and (ii) are obvious. To see (iii) considerY ′ ∈ [X′]ω (Y ′ ∈ [X′]ω1) and chooseY ⊂ X such thatY ′ = {f ′: f ∈ Y }. (Note that 2.2 (respectively 2.5) implies|X| = |X′|.)In the first caseY is finally dense and in the second someZ ∈ [Y ]ω is. It is easy to see,however, that then so isY ′, respectivelyZ′ = {f ′: f ∈ Z}, because everyf ′ differs fromf only at a countable number of places.✷

This immediately yields one of the main results of this section.

2.9. If there is anHFDw space then there is anS space.

Remark. 2.9 contains a very weak assumption for the existence of anS space. In fact it isso weak that we do not even know whether the converse of 2.9 is valid, i.e., whether theexistence of anS space is equivalent to that of an HFDw space.

Proof of 2.9. If there is an HFDw space then by 2.8(i) and (ii) there is also one of the form

X = {fα : α ∈ ω1} ⊂ 2ω1.

Let us now define forα ∈ ω1 the functionf ′α ∈ 2ω1 by the following stipulations:

f ′α(ν) =

{0, if ν ∈ α;1, if ν = α;fα(ν), if α ∈ ν.

ThenX′ = {f ′α : α ∈ ω1} is right-separated in typeω1, henceX′ is not Lindelöf, but by

2.8(iii) it is HFDw, hence HS. Thus, in fact,X′ is a canonicalS space. ✷Thus we see that HFDw spaces give usS spaces and our next aim is to show that the

stronger HFD spaces, having pleasant topological properties, yield a number of furtherinteresting applications.

2.10. If X ⊂ 2λ is HFD (HFDw) and C ⊂ X is compact then|C| < ω (respectively|C| � ω).

Proof. Indeed otherwiseC would be finally dense hence there would exist a setA ∈ [λ]�ω

such thatC � (λ\A) = C1 is dense in 2λ\A. ButC1 is also compact as the continuous imageof the compactC, hence we hadC1 = 2λ\A. But then, by 2.8(ii), 2λ\A � 2λ would be HFD(HFDw), which is clearly not the case (e.g., 2λ is not HS). ✷


Corollary. An HFD spaceX does not contain a non-trivial convergent sequence(even ifits limit point is not inX).

Indeed, a non-trivial convergent sequence with its limit point forms an infinite compactspace. Note, moreover, that ifX ⊂ 2λ is HFD andF ⊂ 2λ is any finite set thenX ∪ F isalso HFD.

In order to describe some further properties of HFD spaces we first need a definition.

2.10. Definition. Let X ⊂ 2λ be HFD andA ∈ [X]ω. We put

J (A) = {I ∈ [λ]ω: ∀ε ∈ H(I)

(∣∣A ∩ [ε]∣∣ = ω implies that

A ∩ [ε] is dense in 2λ\I)}.2.12. If X ⊂ 2λ is HFD andA ∈ [X]ω thenJ (A) is closed and unbounded in[λ]ω.

Proof. Suppose that{In: n ∈ ω} ⊂ J (A) andIn ⊂ In+1 for everyn ∈ ω. We have to showthat I = ⋃{In: n ∈ ω} ∈ J (A). But if ε ∈ H(I) then there is ann ∈ ω with ε ∈ H(In),hence|[ε] ∩ A| = ω implies that[ε] ∩ A is dense in 2λ\In and thus in 2λ\I as well. Thisshows thatJ (A) is closed.

To show the cofinality ofJ (A) fix anyK ∈ [λ]ω and then by induction onn ∈ ω defineIn ∈ [λ]ω as follows. PutI0 = K, and ifIn has been defined let

Hn = {ε ∈ H(In):

∣∣A ∩ [ε]∣∣ = ω}.

Then for eachε ∈ Hn there is a setJε ∈ [λ]ω such thatA ∩ [ε] is dense in 2λ\Jε . Then weput

In+1 = In ∪⋃

{Jε: ε ∈ Hn}.We claim thatI = ⋃{In: n ∈ ω} ∈ J (A). Indeed, ifε ∈ H(I) and|A∩ [ε]| = ω then thereis n ∈ ω with ε ∈ H(In), henceε ∈ Hn, consequentlyA ∩ [ε] is dense in 2λ\Jε and thus in2λ\I as well. ✷

The following technical result will have a number of important consequences.

2.13. Let X ⊂ 2λ be HFD, A ∈ [X]ω, I ∈ J (A) andg ∈ 2λ. If g � I is an accumulationpoint ofA � I (in 2I ) theng is an accumulation point ofA (in 2λ).

Proof. Let ε ∈ H(λ) be such thatg ∈ [ε]. We can writeε = ε1 ∪ ε2, whereε1 ∈ H(I)

andε2 ∈ H(λ \ I). Sinceg � I is an accumulation point ofA � I we have|A ∩ [ε1]| = ω,henceI ∈ J (A) implies thatA ∩ [ε1] is dense in 2λ\I . But [ε] = [ε1] ∩ [ε2] then implies|A∩ [ε]| = |A ∩ [ε1] ∩ [ε2]| = ω showing thatg is an accumulation point ofA. ✷

The projection map on subspaces of 2λ is in general not closed. The next result yields asomewhat weaker assertion of this sort for HFD spaces.


2.14. If X ⊂ 2λ is HFD, A ∈ [X]ω andI ∈ J (A) thenA � I is closed inX � I . (The closureA is of course taken inX!)

Proof. SinceA is dense inA and the projection map is continuous,A � I is dense inA � I . Therefore it suffices to show thatA � I contains every accumulation point ofA � I(in X � I ). Thus letg ∈ X be such thatg � I is an accumulation point ofA � I . By 2.13theng is an accumulation point ofA, i.e.,g ∈ A, consequentlyg � I ∈ A � I . ✷

Since an HFD spaceX is HS, every infinite closed set inX is of the formA for someA ∈ [X]ω. This is what makes 2.14 well-applicable, as, e.g., in the proof of the followingresult.

2.15. EveryHFD space is hereditarily collectionwise normal.

Proof. Since a normal HS space is collectionwise normal (cf. [8]) and being HFD is ahereditary property, it suffices to show that every HFD space is normal. Thus consider twodisjoint closed subsets of the HFD spaceX ⊂ 2λ; in view of our above remark we maytake them to beA andB whereA,B ∈ [X]ω. SinceA∩B = ∅ there is a setJ ∈ [λ]ω suchthatA � J ∩ B � J = ∅ and becauseJ (A) ∩ J (B) is (closed and) unbounded in[λ]ω wemay pick anI ∈ J (A) ∩J (B) such thatI ⊃ J .

From 2.14 we know thatA � I and B � I are both closed inX � I , moreover weclaim that they are also disjoint. Indeed, if this were false then we could findf ∈ A andg ∈ B such thatf � I = g � I . Now we distinguish two cases: First, iff � I ∈ A � I thenf � I = g � I /∈ B � I (sinceA � I ∩ B � I = ∅), consequentlyg � I is an accumulationpoint of B � I becauseg � I ∈ B � I andB � I is dense inB � I . But then, using 2.13and the fact thatf � I = g � I , we get thatf is also an accumulation point ofB, hencef ∈ B, contradictingA ∩ B = ∅. If, on the other handf � I = g � I /∈ A, theng � I is anaccumulation point ofA � I , hence with a similar argument as above we getg ∈ A, againa contradiction.

Thus we have indeed thatA � I andB � I are disjoint closed sets in the metrizable hencenormal spaceX � I ⊂ 2I , consequently we may choose disjoint open neighbourhoods forthem, sayU andV . But then the inverse images ofU andV under the projection mapfrom X to X � I yield disjoint open neighbourhoods ofA andB in X, and the proof iscompleted. ✷

Our next results will concern HFD spaces with a special property expressed by thefollowing definition.

2.15. Definition. Let * be a cardinal number. The setX ⊂ 2λ with λ > ω is said to be*-complete if for everyh ∈ Hω1(λ) = Fn(λ,2;ω1) we have|[h] ∩X| � *, where, of course,

[h] = {f ∈ 2λ: h ⊂ f

}.

Let us remark that obviously every 1-completeX ⊂ 2λ is alsoλω-complete, hence 2ω-complete, consequently any 1-completeX has cardinality� 2ω.


Now we show that HFD spaces that are complete in this sense have interesting topol-ogical properties.

2.17. If X ⊂ 2λ is a1-completeHFD thenX is countably compact. In particular thenX isanS space.

Proof. To show thatX is countably compact we pick anA ∈ [X]ω and show thatA has anaccumulation point inX. Using again thatJ (A) is unbounded in[λ]ω we chooseI ∈ J (A)

such thatA � I is infinite. Since 2I is (countably) compact there is anh ∈ 2I ⊂ Hω1(λ) suchthath is an accumulation point ofA � I . The 1-completeness ofX implies the existence ofanf ∈ X with h ⊂ f , moreover 2.13 implies that thisf is an accumulation point ofA.

To see thatX is an S space observe that a countably compact HFD space is neverLindelöf, because a countably compact Lindelöf space is compact, but by 2.10 everycompact subset of an HFD space is finite.✷

Another interesting restriction that completeness puts on the topology of HFD spaces isthe following.

2.18. If an HFD spaceX ⊂ 2λ is *-complete then for every infinite closed setF ⊂ X wehave|F | � *.

Proof. SinceX is HS, there is a setA ∈ [F ]ω such thatF = A. We can again choose anI ∈ J (A) such thatA � I is infinite. Leth ∈ 2I be an accumulation point ofA � I in 2I ,then by the*-completeness ofX we have∣∣{f ∈ X: h ⊂ f }∣∣ � *.

But by 2.13 everyf ∈ X with h ⊂ f is an accumulation point ofA, hence a member ofA = F , consequently|F | � *. ✷

We close this topic of complete HFD spaces by showing that it is not more difficultto produce complete HFD spaces than ordinary HFD spaces of cardinality� 2ω. Moreprecisely, we have the following result.

2.19. SupposeX ⊂ 2λ is anHFD space with|X| = * � 2ω. Then there is anHFD spaceX′ ⊂ 2λ which is also*-complete.

Proof. As is shown in 4.4(i), we must haveλ � 2ω, consequently we also have|Hω1(λ)| =2ω � *. ThereforeHω1(λ) can be written in the form

Hω1(λ) = {hν : ν ∈ *},where for eachh ∈ Hω1(λ) we have∣∣{ν ∈ *: hν = h}∣∣ = *.

Let us define the equivalence relation∼ onX as follows: Forf,g ∈ X let

f ∼ g ←→ ∣∣{α ∈ λ: f (α) �= g(α)}∣∣ � ω.


Then every equivalence class of∼ in X must be finite because ifA ∈ [X]ω would consistof pairwise∼-equivalent elements thenA could not be finally dense, just on the contrary,its elements would all coincide outside a countable subset ofλ. We may thus assume thatX

actually consists of pairwise∼-inequivalent elements.We can now writeX = {fν : ν ∈ *}, in this case however a one–one enumeration is

chosen. Let us then define forν ∈ ω1 a functionf ′ν ∈ 2λ as follows:

f ′ν(α) =

{hν(α), if α ∈ D(hν);fν(α), otherwise.

By 2.8(iii) thenX′ = {f ′ν : ν ∈ *} is HFD andX′ is also*-complete becauseν �= µ implies

f ′ν �= f ′

µ sincefν �∼ fµ and bothfν andfµ were only modified at a countable number ofplaces. ✷

We now turn to give an application of HFD spaces to certain topological games. To playsuch a game we choose a spaceX a propertyP of subsets ofX (examples ofP belowwill be P = D = dense inX andP = SD= somewhere dense inX) and an ordinalα. Thegame obtained in this way is denoted byGP

α (X) and is played by two playersI andII inthe following way:

A play ofGPα (X) consists ofα rounds, played in orderα, where each round consists of

first I choosing a non-empty openU ⊂ X and thenII picking a pointp ∈ U . PlayerI winsif the set of all points picked byII has propertyP , otherwiseII wins. These games wereintroduced in [5] where they were named point-picking games. The following results werealso proved there: IfX is T3 thenI has a winning strategy forGD

ω (X) [GSDω (X)] if and

only if π(X) = ω[π0(X) = ω], whereπ0(X) = min{π(G): G ∈ τ (X) \ {∅}}. The resultconcerning the case ofSD can easily be strengthened by replacingω with any ordinalα < ω2, simply because the union of finitely many nowhere dense sets is nowhere dense.This in turn implies the same kind of strengthening for the case ofD. Now, HFD spacescan be used to show that these strengthenings are already sharp. Note that ifλ > ω then forany dense subspaceX of 2λ we haveπ(X) = π0(X) = λ > ω (cf. [16]).

2.20. If X ⊂ 2λ is HFD and dense in2λ thenI has a winning strategy in the gameGDω2(X)

(hence inGSDω2 (X) as well).

Proof. Let us begin by describingI ’s strategy. He starts by choosing an arbitrary setI0 ∈ [λ]ω and in the firstω rounds he plays all the open sets[ε] ∩X with ε running throughH(I0). This is possible because|H(I0)| = ω. Now supposen ∈ ω \ {0} and the firstω · nrounds have already been played, moreover a setIn−1 ∈ [λ]ω has been defined. LetAn bethe set of all points picked by playerII in the firstω · n rounds. Then clearlyAn ∈ [X]ω,hence by 2.12 we can choose a setIn ∈ J (An) such thatIn−1 ⊂ In. PlayerI ’s strategy forthe nextω rounds then is to play all the open sets[ε] ∩ X whereε runs throughH(In).

Now we have to show that playerI ’s strategy described above is winning, i.e., that thesetA = ⋃{An: n ∈ ω} is dense inX (i.e., in 2λ). Let us putI = ⋃{In: n ∈ ω} and consideranyε ∈ H(λ). Thenε = ε1 ∪ ε2, whereε1 ∈ H(I) andε2 ∈ H(λ \ I). Let n ∈ ω be suchthatε1 ∈ H(In). SinceI ’s strategy forcesAn+1 � In to be dense in 2In we have∣∣[ε1] ∩ An+1

∣∣ = ω,


moreoverIn+1 ∈ J (An+1), ε1 ∈ H(In) ⊂ H(In+1) and ε2 ∈ H(λ \ I) ⊂ H(λ \ In+1),consequently

[ε2] ∩ [ε1] ∩ An+1 = [ε] ∩ An+1 �= ∅.This indeed shows thatA is dense. ✷

To conclude this topic, let us note that given any HFD spaceX ⊂ 2λ a very slightmodification ofX will yield an HFD of the same size that is dense in 2λ.

We have seen now that HFD and HFDw spaces can be used to obtainS spaces of variouskinds. But can they be used to obtain strongS spaces? The problem of course is that wedo not know whether finite powers of HFD spaces are necessarily HS or not. In fact, as weshall see later, this is not necessarily so, however a natural strengthening of the notions ofHFD and HFDw spaces can be found with the help of which this problem can be avoided.The basic definition here is as follows.

2.21. Definition.

(i) Let κ � ω (κ > ω), λ > ω, k ∈ ω andF :κ ×λ → 2.F is said to be an HFDk (HFDkw)

matrix if for everyA ∈ Dkω(κ) (A ∈ Dk

ω1(κ)), B ∈ Dω1(λ) andε0, . . . , εk−1 ∈ 2n(B)

there is ana ∈ A such that∣∣{b ∈ B: ∀i ∈ k[fαi = F(αi,−) ⊃ εi ∗ b

]}∣∣ = ω1,

whereαi is theith element ofa in its increasing order.Note that in the case of HFDk an equivalent definition is obtained if the above subsetof B is only required to be non-empty instead of having cardinalityω1. Thus weimmediately see that HFD1(w) = HFD(w) and that, ifκ > ω, HFDk → HFDk

w.

(ii) X ⊂ 2λ is called an HFDk(w) space if there is an HFDk(w) matrix F :κ × λ → 2 thatrepresents it.

(iii) If F :κ ×λ → 2 is HFDk (HFDkw) for all k ∈ ω thenF is called a strong HFD (HFDw)

matrix. Similarly,X ⊂ 2λ is defined to be a strong HFD (HFDw) space if it can berepresented by a strong HFD (HFDw) matrix.

The reason why HFDk(w) spaces were introduced is made clear by the following resultgeneralizing 2.7.

2.22. If X ⊂ 2λ is a HFDkw space thenXk is HS. Consequently a strongHFDw space is

stronglyHS and anS space that is strongHFDw is also a strongS space.

Proof. We prove this by induction onk. Of course the casek = 1 is just 2.7. Now assumethat statement to be valid fork and show that then it is also true fork + 1. Thus considerX ⊂ 2λ that is HFDk+1

w represented byF :κ × λ → 2 in a one–one manner. Note thatX

inherits fromF a κ-type ordering which we shall denote by≺. Assume, indirectly, thatXk+1 is not HS. ThenXk+1 contains a subset{−⇀xα : α ∈ ω1} left separated in typeω1, where−⇀xα = 〈xα,0, . . . , xα,k〉. With the help of the usual∆-system and counting arguments we


may assume without loss of generality that there is aB ∈ Dω1(λ), whereB = {bα: α ∈ ω1},andε0, . . . , εk ∈ 2n(B) such that the left separating neighbourhood of−⇀xα is of the form([ε0 ∗ bα] × · · · × [εk ∗ bα]) ∩ X.

For anyp ∈ X andi < k + 1 let us put

Sp,i = {α: xα,i = p} and Sp =k⋃

i=0

Sp,i .

We claim that|Sp| � ω for eachp ∈ X. Indeed,|Sp| = ω1 would imply |Sp,i | = ω1 forsomei < k + 1 and if we denote by−⇀yα the element ofXk obtained by deleting from−⇀xαits ith coordinatexα,i , then{−⇀yα: α ∈ Sp,i} would clearly be an uncountable left separatedsubset ofXk , contradicting our inductive assumption. Using that|Sp| � ω for all p ∈ X

allows us to perform a further thinning out of our original left separated sequence with theresulting (and re-enumerated) sequence already satisfying|Sp| � 1 for eachp ∈ X.

We may also assume that the sequences−⇀xα are all isomorphic with respect to the abovementionedκ-type ordering≺ of X inherited fromF , hence by suitably arranging the orderof factors in the productXk+1 = X × · · · × X we may actually assume that for eachα wehavexα,i ≺ xα,j if i < j < k + 1. Thus if we now put forα ∈ ω1

aα = {xα,i: i < k + 1}then the HFDk+1

w property ofX implies the existence ofα ∈ ω1 such that∣∣{β ∈ ω1: −⇀xα ∈ [ε0 ∗ bβ] × · · · × [εk ∗ bβ]}∣∣ = ω1.

But this contradicts that, by left separation,−⇀xα /∈ [ε0 ∗ bβ] × · · · × [εk ∗ bβ]

wheneverα < β . ✷It is straight–forward to check that the argument given in the proof of 2.8(iii) is valid

for HFDk and HFDkw spaces as well for any givenk, i.e., if X is HFDk (HFDkw) and every

element ofX is modified at a countable number of coordinates then the resulting spaceX′ is also HFDk (HFDk

w). Hence the existence of a strong HFDw space implies that of astrongS space.

To obtain strongS spaces was the main reason why HFDk(w) spaces were introduced,

but as we shall show below they have other uses as well. To start with, however, we have todescribe a property of certain HFDw subspaces of 2ω1 that is quite interesting in itself. Inthis we shall use the following piece of notation: iff ∈ 2ω1 andf (ν) = 1 for someν ∈ ω1then

γ (f ) = min{ν ∈ ω1: f (ν) = 1

}.

If f (ν) = 0 for all ν ∈ ω1 then we putγ (f ) = ω1.

2.23. Suppose thatX ⊂ 2ω1 is anHFDw with the following property:

for everyα ∈ ω1 we have∣∣{f ∈ X: γ (f ) < α

}∣∣ � ω. (∗)


Then there exists aν ∈ ω1 such thatf ∈ 2ω1 andν � γ (f ) imply f ∈ X. (Of course, theclosure here is taken in2ω1.)

Proof. Assume, indirectly, that for eachν ∈ ω1 there is agν ∈ 2ω1 such thatγ (gν) � ν

andgν /∈ X. Let εν ∈ H(ω1) be chosen in such a way thatgν ∈ [εν] ⊂ 2ω1 \ X. We mayassume without any loss of generality that{D(εν): ν ∈ ω1} forms a∆-system with rootDsuch thatD <Dν = D(εν) \ D for eachν. Let us put

Y = {f ∈ X: ∃δ ∈ D

(f (δ) = 1

)}.

It follows from (∗) that |Y | � ω, consequentlyX \ Y is finally dense, i.e., there existsan α ∈ ω1, such thatX \ Y is dense in 2ω1\α . We can now pick aν ∈ ω1 such thatD ⊂ ν � γ (gν) andDν ⊂ ω1 \ α. But then for allδ ∈ D we have

εν(δ) = gν(δ) = 0,

hence iff ∈ X \ Y satisfiesf ∈ [εν � Dν], and such anf ∈ X \ Y exists becauseX \ Y isdense in 2ω1\α , thenf ∈ [εν] as well, a contradiction. ✷

If we examine the above proof we see that the HFDw property ofX was not used in itsfull strength. Instead, it would have been sufficient to assume thatX \ Y is finally densefor eachY ∈ [X]�ω. In the following corollary of 2.23 however the assumption thatX isHFDw is really essential because 2.23 has to be applied to each uncountable subset ofX.

2.24. Suppose thatX ⊂ 2ω1 is HFDw and satisfies condition(∗) of 2.23. Then every closedsubspace ofX is either countable or co-countable.

It turns out that the conclusion of 2.24 is actually sufficient to yield anS space.

2.25. If X is an uncountableT3 space in which every closed(or equivalently: open) set iseither countable or co-countable thenX is anS space.

Proof. We may assume that every pointp ∈ X has a countable open neighbourhoodUp

because, asX is T2, this may fail for at most one point ofX, and that point can be simply“thrown away”. It is also clear from our assumption onX thatX is CCC, hence if we takea maximal disjoint familyU of countable open subsets ofX then

⋃U is a countable dense

set inX. This shows thatX is separable and since our assumption on (the open sets in)X isinherited by all its subspaces we actually get thatX is HS. Finally, sinceX is uncountable,the open cover{Up: p ∈ X} shows thatX is not Lindelöf, i.e.,X is anS space. ✷

After this little detour let us get back to our original aim, the example that makes useof HFDk

w, more precisely HFD2w spaces. This concerns the following problem: SupposeX

is T3, |X| = κ and for every subspaceY ⊂ X with |Y | = |X| = κ we know thatY isseparable; is it true then thatX is HS? In other words, if everylarge subspace ofX isseparable, isX necessarily HS? It was shown in [13] that the answer to this question isaffirmative if κ = ω1 or cf(κ) �= ω1, moreover a consistent example was also constructedthere (under the assumption of♣ + 2ω � κ) showing that the above restriction on cf(κ) is


indeed necessary. Below we shall give another example to the same effect that, as we shallshow in Section 4, is compatible with 2ω < κ , in fact with 2ω = ω1 as well.

2.26. Supposecf(κ) = ω1 < κ and there exists anHFD2w spaceX ⊂ 2ω1 with |X| = κ . Then

there also is a0-dimensionalT2 (henceT3) spaceR such that|R| = κ , for everyS ∈ [R]κthe subspaceS is separable butR is notHS, in factR contains a discrete subspaceD ofsizeω1.

Proof. Using the appropriate version of 2.8(iii) we may assume that there is a subsetZ ⊂ X such that|Z| = ω1 andZ satisfies condition (∗) of 2.23.

Forα ∈ ω1 let gα ∈ 2ω1 be defined by

gα(ν) ={

1, if α = ν,0, if α �= ν,

moreoverh ∈ 2ω1 be such thath(ν) = 0 for everyν ∈ ω1. Let us putD = {gα: α ∈ ω1}.We may clearly assume that

X ∩ (D ∪ {h}) = ∅,

thenD is a closed discrete set inX ∪ D taken as a subspace of 2ω1, becauseh is the onlyaccumulation point ofD in 2ω1.

It follows from our assumptions thatX can be written in the form

X =⋃

{Xf : f ∈ Z},where (i) |Xf | < κ for eachf ∈ Z. We shall use the following piece of notation in ourconstruction:

If Y is any subset ofZ then

S(Y ) =⋃{{f } × Xf : f ∈ Y

}.

Clearly, we have|S(Z)| = κ .The spaceR that we wish to produce may now be defined as

R = ω1 ∪ S(Z)

with the topology specified as follows. Firstly,S(Z) will be an open subspace ofR withthe subspace topology that it inherits fromZ ×X. Note thatS(Z) with this topology is HSbecauseZ × X is; this is where we use the factX is HFD2

w, together with 2.22.Next, for anyα ∈ ω1, a neighbourhood basis forα in R will be formed by the sets of

the form

W(α,U) = {α} ∪ S(U \ {gα}),

whereU is any clopen neighbourhood ofgα in D ∪ Z such thatU ∩ D = {gα}. It is easyto see that in this wayW(α,U) will be a clopen neighbourhood ofα in R and that

W(α,U) ∩ ω1 = {α}.It easily follows then thatR is 0-dimensionalT2, moreover thatω1 is a closed discrete setin R, henceR is not HS.


We still have to show that every subspace ofR of sizeκ is separable. SinceT ⊂ R and|T | = κ imply |T ∩ S(Z)| = κ andS(Z) is HS, this will be an immediate consequence ofthe following claim: IfT ⊂ S(Z) and|T | = κ then|ω1 \ T R| � ω, i.e., all but countablymany points inω1 are in theR-closure ofT . To establish this claim let us first note thatby (i) for

Y = {f ∈ Z: T ∩ ({f } × Xf

) �= ∅}we have|Y | = ω1. But then 2.23 can be applied toY , hence there is aν ∈ ω1 such thatg ∈ 2ω1 and γ (g) � ν imply g ∈ Y . It follows then thatgα ∈ Y wheneverα � ν. Butthen for any clopen neighbourhoodU of gα in D ∪ Z we haveU ∩ Y �= ∅, consequentlyW(α,U) ∩ T �= ∅, because iff ∈ U ∩ Y then{f } × Xf ⊂ W(α,U). This completes theproof of the claim and thus of 2.26.✷

Hodel proved in [10] that under GCH for any infiniteT2 spaceX the number of allclosed subsets ofX of size|X| equals o(X), i.e., the number of all closed subsets ofX. Healso asked if this is provable in ZFC. The spaceR constructed above in 2.26, with someadditional cardinal arithmetic, yields a counterexample to this question.

2.27. Assumecf(κ) = ω1 < κ < 2ω1 andκω = κ , moreover there is anHFD2w space of size

κ . Then there is a0-dimensionalT2 spaceR with |R| = κ in which the number of closedsets of sizeκ is κ , whileo(R) = 2ω1 > κ .

Proof. The spaceR constructed fromX in 2.26 has clearly (at most)κω = κ closedsubsets of sizeκ since such a subset is separable, and o(R) � 2ω1 is immediate fromthe existence of an uncountable discrete subspace inR. Finally, o(R) � κω1 = 2ω1 becausehd(R) = ω1. ✷

3. HFC-type spaces

We start again with the “dual” notion of HFC and HFCw matrices. Observe that the dualdefinitions can simply be obtained by switching the roles ofA andB in the definitions ofHFD (HFDw) matrices.

3.1. Definition.

(i) A map F :κ × λ → 2 with κ � ω1 andλ � ω is called an HFCmatrix if for everyA ∈ [κ]ω1, B ∈ Dω(λ) andε ∈ 2n(B) there areα ∈ A andb ∈ B such that

fα = F(α,−) ⊃ ε ∗ b.

Accordingly, a (necessarily uncountable) setX ⊂ 2λ is said to be an HFC space if it isrepresented by an HFC matrix.

(ii) F :κ × λ → 2 with κ,λ > ω is said to be an HFCw matrix if for everyA ∈ [κ]ω1,B ∈ Dω1(λ) andε ∈ 2n(B) there is ab ∈ B such that∣∣{α ∈ A: fα ⊃ ε ∗ b}∣∣ = ω1.


Again,X ⊂ 2λ is a HFCw space if it can be represented by an HFCw matrix. Clearly,every HFC is also HFCw if λ > ω.

We leave it to the reader to verify that ifF is an HFCw matrix onκ × λ (or HFC ifλ = ω) then for everyf ∈ 2λ we have∣∣{α ∈ κ : fα = f }∣∣ � ω,

hence the matrix representation of an HFC or HFCw space can always be assumed to beone–one. Similarly as in the case of the HFD(w) spaces a simple matrix-free character-ization of HFC(w) spaces can be given:

3.2. (i) X ⊂ 2λ with |X| >ω is HFC if and only if for everyB ∈ Dω(λ) andε ∈ 2n(B)∣∣X \ [ε,B]∣∣ � ω,

i.e., everyDω-set in2λ finally coversX.(ii) X ⊂ 2λ with |X| >ω andλ > ω is HFCw if and only if for everyB ∈ Dω1(λ) there

is aC ∈ [B]ω such that for anyε ∈ 2n(B)∣∣X \ [ε,C]∣∣ � ω.

Roughly speaking, this says that everyDω1-set in 2λ has a sub-Dω-set which finallycoversX.

Proof. (i) If X = {fα : α ∈ κ} is a one–one enumeration ofX andF :κ × λ → 2 is definedby

F(α,β) = fα(β),

then it is immediate from the definition thatF is an HFC matrix exactly ifX satisfies theabove condition.

(ii) Just as above, ifF 1–1 representsX that satisfies the above condition then it isimmediate thatF is an HFCw matrix. To see the converse, assume, indirectly, thatF

is an HFCw matrix, B ∈ Dω1(λ) but for everyC ∈ [B]ω there is anε ∈ 2n(B) with|X \ [ε,C]| > ω. Let us now writeB = {bν : ν ∈ ω1} andCν = {bµ: µ ∈ ν} for ν ∈ ω1.Since 2n(B) is finite, anε can be chosen that works simultaneously for allCν . We may thenpick by induction onν ∈ ω1 pointsxν ∈ X \ [ε,Cν] such thatxν �= xµ if ν �= µ. Letαν ∈ κ

be such thatxν = fαν andA = {αν : ν ∈ ω1} ∈ [κ]ω1. SinceF is HFCw there is abµ ∈ B

with ∣∣{ν: fαν = xν ∈ [ε ∗ bµ]}∣∣ = ω1,

contradicting thatxν /∈ [ε ∗ bµ] ⊂ [ε,Cν] if µ ∈ ν. ✷Remark. Note that what (i) says in the case ofλ = ω is simply thatX ⊂ 2ω is an HFC ifand only ifX is aD-Luzin set in the sense of Section 1. In particular, then every ordinaryLuzin set or Sierpinski set in 2ω is an HFC.

Also, as a curiosity, let us mention here the following easily provable fact:X ⊂ 2λ is anHFD exactly if |X| � ω and|X \ [ε,B]| <ω for anyDω1-set[ε,B] in 2λ.


From 3.2 and the criterion 1.1(ii) for the HL-ness of subspaces of 2λ we immediatelyobtain the following basic result.

3.3. EveryHFCw (hence everyHFC) space isHL.

Similarly again to the HFD case, the HFC(w) property has analogous stability featuresthat we make precize in the following proposition. Here we again consider a slightlygeneralized version of HFC(w) spaces in that we allow them as subspaces of arbitraryproducts 2I .

3.4. (i) An arbitrary uncountable subspace of anHFC(w) space is againHFC(w).(ii) If X ⊂ 2I is HFC (HFCw) andJ ⊂ I is infinite (uncountable) thenX � J is HFC

(HFCw).(iii) If F :κ ×λ → 2 is anHFC(w) matrix andF ′ :κ ×λ → 2 is such that for eachβ ∈ λ∣∣{α ∈ κ : F ′(α,β) �= F(α,β)

}∣∣ � ω,

thenF ′ is alsoHFC(w).

Proof. (i) and (ii) are obvious from the definitions. To see (iii), considerA ∈ [κ]ω1,B ∈ Dω(λ) (respectivelyB ∈ Dω1(λ) in the HFCw case) andε ∈ 2n(B) (in the HFCwcase chooseB ′ ∈ [B]ω accordingly). Since there are only countably manyα ∈ κ withF ′(α,β) �= F(α,β) for someβ ∈ B (respectivelyβ ∈ B ′) the conclusion now followsimmediately. ✷

If X ⊂ 2ω1 is HFCw and the HFCw matrixF :ω1 × ω1 → 2 one–one represents it thenby 3.4(iii) the following matrixF ′, obtained by “slightly” modifyingF , is also HFCw:

F ′(α,β) ={0, if α ∈ β,

1, if α = β,F(α,β), if β ∈ α.

But if we set

f ′α = F ′(α,−)

thenX′ = {f ′α : α ∈ ω1} is clearly left-separated, henceX′ is anL space. Thus we got the

following result:

3.5. The existence of anHFCw space implies the existence of anL space.

HFC(w) spaces have an interesting property: they have large, more precizely maximumpossible weight (or character). Actually, this property was the original motivation behindthe introduction of HFC spaces. As it turns out, however, this property is already possessedby all the HFCw spaces.

3.6. If X ⊂ 2λ is HFCw (note that thenλ > ω) andp ∈ X has no countable neighbourhoodin X then

χ(p,X) = λ.

Consequentlyw(X) = χ(X) = λ.


Proof. Suppose thatχ(p,X) < λ andV is a neighbourhood base ofp in X with |V| < λ.Let us choosei ∈ 2 in such a way that|{β ∈ λ: p(β) = i}| = λ. Since

|V| < λ� ω1,

we may pick a neighbourhoodV ∈ V in such a way that|B| � ω1 where

B = {β ∈ λ: V ⊂ [εβ ]}

andεβ = {〈β, i〉}. SinceX is HFCw it follows that⋃{2λ \ [εβ]: β ∈ B} finally coversX,

henceV ⊂ ⋂{[εβ]: β ∈ B} must be countable.To see the second statement recall that, by 3.3,X is HL hence it may contain at most

countably many points with a countable neighbourhood, moreoverX itself is uncount-able. ✷

An immediate corollary of 3.6 is that ifλ > 2ω then any HFCw spaceX in 2λ is anLspace because a separableT3 space has weight� 2ω.

Next we consider the “dual” notions corresponding to the HFDk(w)

spaces. These can beused to obtain strongL spaces from appropriate HFC(w) spaces.

3.7. Definition. Fix k ∈ ω. A mapF :κ × λ → 2 with κ � ω1 andλ � ω (λ � ω1) is calledan HFCk (HFCk

w) matrix if for everyA ∈ Dkω1(κ) andB ∈ Dω(λ) [B ∈ Dω1(λ)] and for

anyε0, . . . , εk−1 ∈ 2n(B) there existsb ∈ B such that∣∣{a ∈ A: ∀i ∈ k(fαi ⊃ εi ∗ b)}∣∣ = ω1,

where{αi : i ∈ k} is the increasing enumeration of the elements ofa. F is astrongHFC(w)

matrix if it is HFCk(w) for all k ∈ ω. Finally HFCk

(w) spaces are the ones represented by thesame kind of matrices; similarly for strong HFC(w) spaces.

It is easy to check that every HFCk(w) space is HLk, hence we immediately obtain thefollowing result.

3.8. If there is a strongHFCw space then there is a strongL space.

This result, however, is also a consequence of the following one, which shows that thefull duality that is known to exist between strongS andL spaces also occurs betweenstrong HFD and HFC spaces.

3.9. A matrixF :κ ×λ → 2 is strongHFD(w) if and only if its dualF ∗ :λ×κ → 2 is strongHFC(w).

Proof. We show that ifF is strong HFD thenF ∗ is strong HFC. The remaining threeimplications can be proved in exactly the same way.

To see that, givenk ∈ ω, F ∗ is HFCk we considerB ∈ Dkω1(λ), A ∈ Dω(κ) and

δ0, . . . , δk−1 ∈ 2n(A). Then for eachi ∈ n(A) we defineεi ∈ 2k = 2n(B) as follows:

εi(j) = δj (i).


SinceF is HFDn(A) we conclude that there existb ∈ B and (uncountably many)a ∈ A

such that

F(αi,−) ⊃ εi ∗ b

wheneveri < n(A). But this is equivalent to

F ∗(βj ,αi) = δj (i)

for all i andj ; of course, hereαi is theith member ofa andβj is thej th member ofb,which is exactly what we wanted to show, namely

F ∗(βj ,−) ⊃ δj ∗ a. ✷

4. Existence results for HFDs and HFCs

We start this section by results which show that in certain generic extensions ofV wecan obtain HFDs and HFCs in a rather straightforward way. Actually, in order to givea general treatment of these results which also emphasizes the really important steps inthe forcing arguments we formulate the results in terms of the properties of the resultingextensions of the universeV , without the direct use of forcing. To this end we first introducesome definitions.

4.1. Definition. Let W ⊃ V be an extension ofV andr ∈ 2ω ∩ W . We say thatr is aD-real overV if r belongs to everyD-set in the sense ofV , i.e., for everyA ∈ D(ω)∩V andε ∈ 2n(A) we have

W |= r ∈ [ε,A].

Note that sinceD-sets are both dense open and of Lebesgue-measure 1 in 2ω, bothCohen and random generic reals areD-reals.

4.2. Suppose thatW ⊃ V is an extension ofV such that

(i) ωW1 = ωV

1 ;(ii) there is inW a D-real r overV ;(iii) if, in W , A ⊂ V and|A| = ω1 then there is aB ∈ [A]ω such thatB ∈ V .

Then there exists a strong HFCw (hence by 3.9 also a strong HFDw) matrix inW .

Proof. Let us start by choosing, inV , one–one mapshα :α → ω for all α ∈ ω1 in such away that ifα �= β then∣∣R(hα) ∩ R(hβ)

∣∣ <ω.

We claim then that if the matrixF :ω1 × ω1 → 2 satisfies

F(α,β) = r(hα(β)

)


wheneverβ < α < ω1 thenF is a strong HFCw matrix. Since, by (iii), for everyA ∈Dω1(ω1) there is aB ∈ [A]ω ∩ V , this will clearly follow if we show that for everyB ∈ Dω(ω1) ∩ V , a ∈ [ω1 \ ∪⋃

B]<ω andε :a × n(B) → 2 there is a setb ∈ B satisfying

F(α,βj ) = ε(α, j)

for all α ∈ a andj ∈ n(B), where, of course,βj is thej th member ofb.Since by our choice of the mapshα∣∣∣⋃{

R(hα) ∩ R(hα′):{α,α′} ∈ [a]2}∣∣∣ <ω,

we may assume that nohα[b] for α ∈ a andb ∈ B intersects this finite set, consequentlyfor everyb ∈ B

hα[b] ∩ hα′ [b] = ∅if α andα′ are distinct elements ofa. Thus if we put forb ∈ B

b =⋃{

hα[b]: α ∈ a},

then|b| = |b|.|a| andεb : b → 2 may be defined without any conflict by putting

εb(hα(βj )

) = ε(α, j).

Another consequence of this assumption is that ifb1, b2 ∈ B andb1 �= b2 thenb1 ∩ b2 = ∅,henceB = {b: b ∈ B} ∈ D(ω) ∩ V . Clearly, there is aδ ∈ 2n(B) such that∣∣{b ∈ B: εb = δ ∗ b

}∣∣ = ω,

consequently, asr is aD-real overV , there is ab ∈ B such thatδ ∗ b = εb ⊂ r, i.e.,

F(α,βj ) = r(hα(βj )

) = εb(hα(βj )

) = ε(α, j)

holds wheneverα ∈ a andβj is thej th member ofb, and this was to be shown.✷Since a generic extensionW = V [r] of V obtained by adding a Cohen or random real

r to V is well known to satisfy conditions (i)–(iii) of 4.2 we immediately get the followingcorollary.

4.3. If r is Cohen or random generic overV then there is a strongHFCw and thus also astrongHFDw matrix inV [r].

This result yields us strong HFDw and HFCw spaces and thus strongS andL spaces, butonly of the minimal possible cardinality and weightω1. Before we turn to our next resultsin which it is shown that one can obtain HFDs and HFCs of larger size we first formulatea result that gives the natural upper bounds for these sizes.

4.4. (i) If X ⊂ 2λ is HFDw then

λ � 2ω and |X| � 2ω1.

(ii) If X ⊂ 2λ is HFCw then

λ � 2ω1 and |X| � 2ω.


Proof. (i) We may of course assume thatX is dense in 2λ, henced(2λ) = d(X) = ω,which is only possible ifλ� 2ω. To see the second inequality consider, e.g., the projectionmap ofX into 2ω1. SinceX is HFDw this map must be countable-to-one because everyuncountable subset ofX is finally dense. Thus we clearly have|X| � 2ω1.

(ii) Here |X| � 2ω is obvious becauseX is HL. Now, to showλ � 2ω1 let us fix a setY ∈ [X]ω1 and for anyα ∈ λ andi ∈ 2 put

Y iα = {

f ∈ Y : f (α) = i}.

Clearly, it suffices to show that the mapα $→ Y 0α is countable-to-one. Assume, on the

contrary thatA ∈ [λ]ω1 andY 0α = Y 0 for all α ∈ A. Because of the symmetry of 0 and 1,

we may also assume that|Y 0| = ω1. This, however, is not possible ifX is HFCw, for then⋃{[{〈α,1〉}]: α ∈ A} must finally coverX. ✷Of course, 4.4 is equivalent to the statement that ifF :κ × λ → 2 is an HFDw (HFCw)

matrix thenκ � 2ω1 andλ� 2ω (respectivelyκ � 2ω andλ � 2ω1).Before giving our next result we need a definition again.

4.5. Definition. If W is an extension of the universeV andI ∈ V then a mapF : I → 2in W is said to be aD-map overV if the following is satisfied inW : For everyA ∈ [V ]ωthere are an “intermediate” extensionV ′ of V (i.e., such thatV ⊂ V ′ ⊂ W ) and a setJ ∈ V ′ ∩ [I ]ω with the property thatA ∈ V ′ and for every mapB ∈ V ′ that satisfies

D(B) ∈ Dω(I \ J )

and

B(b) ∈ 2b

for all b ∈ D(B) we haveF ⊃ B(b) for someb ∈ D(B), i.e.,

F ∈⋃{[

B(b)]: b ∈ D(B)

}.

Of course, this definition is motivated by the well known fact that both Cohen andrandom generic maps overV are alsoD-maps overV (cf., e.g., [18]).

4.6. SupposeV ⊂ W andF :κ × κ → 2 in W is aD-map overV , whereκ is an uncount-able cardinal inW . ThenF is both a strongHFD and strongHFCmatrix.

Proof. We give the proof for the strong HFD case, the other case being completelyanalogous. Thus we consider anyA ∈ Dk

ω(κ) and chooseV ′ andJ with V ⊂ V ′ ⊂ W andJ ∈ [κ ×κ]ω as in 4.5. We may assume thatJ = J1×J1 for someJ1 ∈ V ′ ∩[κ]ω. ThatF isHFDk will immediately follow if we show that for anyb ∈ [κ \J1]<ω andε0, . . . , εk−1 ∈ 2b

there is a membera ∈ A such that

F(αi, β) = εi(β)

holds for everyβ ∈ b and for eachi < k, with αi being theith element ofa. To see thisfirst note that

C = {a × b: a ∈ A} ∈ Dω(κ × κ \ J )∩ V ′,


hence if the mapB with domainC is defined in such a way thatB(a × b) ∈ 2a×b with

B(a × b)(αi, β) = εi(β)

for eacha × b ∈ C then

F ∈ [B(a × b)

]for somea ∈ A, which was to be shown.✷

In view of our above result we immediately get from 4.6 the following corollary.

4.7. If F :κ × κ → 2 is a Cohen or random generic map overV then, inV [F ], F is botha strongHFD and strongHFCmatrix.

To see that the bounds forκ andλ given is 4.4 are sharp we need one more result whoseproof combines the ideas of those of 4.2 and 4.6.

4.8. Suppose that our ground modelV satisfies2ω = λ, 2ω1 = κ and there is inV a family{Sα : α ∈ κ} ⊂ [λ]λ such that|Sα ∩ Sβ | � ω if {α,β} ∈ [κ]2. If W is an extension ofV inwhich there is aD-mapF :λ × λ → 2 overV , then there exists a strongHFD matrix onκ × λ (hence also a strongHFC matrix onλ × κ) in W .

Proof. Let us fix inV for eachα ∈ κ the increasing bijectionσα :λ → Sα . Then we define,in W , the matrixG :κ × λ → 2 by putting

G(α,ν) = F(ν,σα(ν)

).

In order to show thatG is HFDk we pickA ∈ Dkω(κ) and then, sinceF is aD-map

overV , we chooseJ ∈ [λ]ω and an intermediate extensionV ′ with A, J ∈ V ′ such that 4.5is satisfied (withJ × J instead ofJ ).

It follows from our assumption about the setsSα that the set

S =⋃{

Sα ∩ Sβ : {α,β} ∈[⋃

A]2}

is countable and alsoS ∈ V ′. We claim that ifb ∈ [λ \ (J ∪ S)]<ω andε0, . . . , εk−1 ∈ 2b

then there is ana ∈ A for which

G(αi, ν) = εi(ν)

wheneverαi is theith member ofa andν ∈ b. Indeed, for anya ∈ A let us put

c(a) = {⟨ν,σαi (ν)

⟩: ν ∈ b, i ∈ k

},

then byb ⊂ λ \ S we havec(a)∩ c(a′) = ∅ if a, a′ ∈ A anda �= a′, hence

C = {c(a): a ∈ A

} ∈ D(λ × λ) ∩ V ′.Thus, ifB with domainC is defined by

B(c(a)

)(ν,σαi (ν)

) = εi(ν),

then the fact thatF is aD-map yields exactly our claim.✷


Baumgartner has shown in [3] that it is consistent with ZFC to have inV the following:2ω = λ, 2ω1 = κ and there is a family{Sα : α ∈ κ} ⊂ [λ]λ as required in 4.8, whileλ � κ

are arbitrarily large regular cardinals. Thus we get the following corollary.

4.9. It is consistent withZFC to have a strongHFD matrixF : 2ω1 × 2ω → 2 (hence alsoa strongHFC matrix F ∗ : 2ω × 2ω1 → 2) with 2ω and 2ω1 being, independently of eachother, as big as you wish.

If we compare the above results with those of Sections 2 and 3 we immediately get anumber of interesting topological consequences, let us mention a few of these here.

4. 10. It is consistent with2ω = λ and 2ω1 = κ being arbitrarily big that there exists ahereditarily collectionwise normal strongS spaceX ⊂ 2λ with |X| = κ such that

(i) X is countably compact;(ii) every compact subspace ofX is finite;(iii) if F ⊂ X is closed inX then eitherF is finite or |F | = |X| = κ .

Before we formulate our result on HFC spaces we first give a lemma.

4.11. SupposeX ⊂ 2λ is anHFC(w) space withλ� 2ω and|X| = κ . Then there also existsan HFC(w) spaceX′ ⊂ 2λ with |X′| = κ and such that every countable subspace ofX′ isclosed and discrete.(Note that such anX′ is automatically anL space.)

Proof. Let F :κ × λ → 2 be an HFC(w) matrix representingX. Since|X| = κ � 2ω wecan arrange all pairs〈E,α〉 with E ∈ [κ]ω andα ∈ κ \ E in a sequence{〈Eν,αν〉: ν ∈ λ}.We may then defineF ′ :κ × λ → 2 as follows:

F ′(α, ν) ={0, if α ∈ Eν ,

1, if α = αν ,F(α, ν), otherwise.

By 3.4(iii) it follows thatF ′ is also an HFC(w) matrix, hence ifX′ is the space representedby F ′ thenX′ is as required. Indeed, ifE ∈ [X′]ω then by our construction no point ofX′outside ofE can be a limit point ofE, hence every countable subset ofX′ is closed andtherefore closed discrete.✷

Comparing this with 3.6 we get the following result.

4.12. It is consistent with2ω = κ and2ω1 = λ being arbitrarily big that there is a strongLspaceX ⊂ 2λ with |X| = κ such that every countable set inX is closed discrete, hence forany subspaceY ⊂ X we have eitherw(Y ) = ω or w(Y ) = χ(Y ) = λ = 2ω1.

As a curiosity we mention here the following “converse” to 4.12.

4.13. If X is aT3 space withw(X) > 2ω and such that for anyY ⊂ X eitherw(Y ) = ω orw(Y ) > 2ω, thenX is anL space.


Proof. Clearly, it suffices to show thatX is HL. Indeed, otherwiseX would contain asubspaceY right separated in typeω1. Thenw(Y ) � ω1 since it is right separated, whileevery proper initial segment ofY (in its right separating well-ordering) has weight� 2ω

henceω1 � w(Y ) � ω1 · 2ω = 2ω, a contradiction. ✷Now we turn to a new manner of obtaining HFDs and HFCs via the combinatorial

principlesW(κ) andV (κ). Let us start by giving the definition ofW(κ).

4.14. Definition. We useW(κ) to denote the following statement:There is a treeT of heightω1 + 1 and a functionS with domainω1 such that

(i) |Tω1| = κ and|Tα| � ω if α ∈ ω1;(ii) for α ∈ ω1 we haveS(α) ⊂ [Tα]ω and|S(α)| � ω;(iii) for any a ∈ [Tω1]ω there is aγ ∈ ω1 such thatpω1

α [a] ∈ S(α) if α ∈ ω1 \ γ .

Herepβα for α � β denotes the canonical projection from theβ th levelTβ of T to the

αth levelTα , i.e., if t ∈ Tβ thenpβα (t) is the (unique)T -predecessor oft at levelα.

The principleW(κ) is related to other well known combinatorial principles. First of all,W(ω1) is easily seen to be equivalent to CH, whileW(ω2) is a consequence of the existenceof an (ω1,1)-morass, thus in particularW(ω2) is valid in the constructible universe,L(see [6]).

Another way of obtainingW(κ) for largeκ is by forcing. We present the details of thishere because this has not been published before.

4.15. Suppose thatCH holds inV andW is an extension ofV such that the following threeconditions hold inW :

(1) for every ordinalα we have

cf(α) = cfV (α);(2) [V ]ω ⊂ V ;(3) there is a functionf ∈ ω

ω11 eventually majorizing everyg ∈ V ∩ ω

ω11 , i.e., for every

g ∈ V ∩ ωω11 there is aγ ∈ ω1 with g(α) < f (α) if α ∈ ω1 \ γ .

ThenW(κ) holds inW , whereκ = (2ω1)V .

Proof. Using CH, let us commence with writing inV for α ∈ ω1

PV (α) = {t(α)σ : σ ∈ ω1

}and [

PV (α)]ω = {

a(α)σ : σ ∈ ω1},

wherePV stands for the power set operator inV and the enumerations are 1–1 ifα � ω.Then, still inV , for anyt ∈ PV (ω1) we definegt ∈ ω

ω11 as follows:

gt (α) = min{σ : t ∩ α = t(α)σ

}.


Since cfV (κ) > ω1 we also have, by (1), cf(κ) > ω1, hence from|PV (ω1)| = κ and (3)we obtain inW a setH ⊂ PV (ω1) with |H | = κ and an ordinalα0 ∈ ω1 such that for anyt ∈ H we havegt (α) < f (α) if α ∈ ω1 \ α0.

We now define the treeT required in 4.14 by specifying its levelsTα and the mapspβα .

We putTω1 = H and forα ∈ ω1

Tα = {t ∩ (α0 + α): t ∈ H

},

moreover forα � β � ω1 andt ∈ Tβ we set

pβα (t) = t ∩ (α0 + α).

This clearly yields a tree of heightω1 + 1 for which|Tω1| = κ and forα ∈ ω1

Tα ⊂ {t(α0+α)σ : σ < f (α0 + α)

},

hence|Tα| � ω.Next we define the mapS onω1 by putting

S(α) = {a(α0+α)σ : σ < f (α0 + α)

} ∩ [Tα]ω.To see thatS satisfies 4.14(iii) consider anya ∈ [Tω1]ω = [H ]ω. Thena is in V by (2),hence so is the functionh ∈ ω

ω11 defined below:

h(α) ={

0, if∣∣{t ∩ α: t ∈ a}∣∣<ω,

σ, if {t ∩ α: t ∈ a} = a(α)σ .

Thus we can chooseγ ∈ ω1 such thath(α) < f (α) if α ∈ ω1 \ γ , moreover we may alsoassume that|{t ∩ α: t ∈ a}| = ω for theseα. But thenγ � α0 + α < ω1 implies

pω1α [a] = a

(α0+α)h(α0+α)

∈ S(α). ✷Next we show how one can find modelsV andW that satisfy the assumptions of 4.15

and consequently we obtain the consistency ofW(2ω1) with 2ω1 being as large as you wish.

4.16. Suppose thatCH holds inV . ThenV has a generic extensionW = V [G] such that(2ω1)W = (2ω1)V andW(2ω1) holds inW .

Proof. We define a notion of forcing〈P,�〉 as follows:

P = ω<ω11 × [

ωω11

]�ω,

moreover if〈p,K〉, 〈q,L〉 ∈ P then

〈p,K〉 � 〈q,L〉 ←→p ⊃ q ∧ K ⊃ L ∧ (∀g ∈ L)

(∀α ∈ D(p) \ D(q))(p(α) > q(α)

).

Standard arguments show that〈P,�〉 is ω1-closed and, using CH, that it has theω2-CC.Thus ifG is P -generic overV then (1) and (2) of 4.15 are satisfied inV [G].

Next we put

f =⋃{

p: (∃K)(〈p,K〉 ∈ G

)}and claim thatf is as required by 4.15(3).


Indeed, for anyα ∈ ω1 the set

Dα = {〈p,K〉 ∈ P : α ∈ D(p)}

is clearly dense in〈P,�〉, hencef ∈ ωω11 , moreover for anyg ∈ ω

ω11 ∩ V

Eg = {〈p,K〉 ∈ P : g ∈ K}

is also dense in〈P,�〉, and thus we get thatf eventually majorizesg.Finally, since|P | = 2ω1 in V , by counting nice names for subsets ofω1 (see [17]) we

obtain that(2ω1)V [G] = (2ω1)V , hence putting all this together and using 4.15 our proof iscompleted. ✷

Let us note for those who know what Baumgartner’s axiom is (see, e.g., [25] and [27])that the above notion of forcing〈P,�〉 has the necessary conditions for the applicatibilityof this axiom, assuming of course that we also have CH. Consequently, this partial order,with suitably chosen dense sets, may be used to show that BACH impliesW(κ) wheneverκω < 2ω1.

Our next aim is to show howW(κ) can be used to construct HFDs and HFCs. In orderto do this we first introduce some notation and then give an alternative version ofW(κ).

First, to homogenize the notation, we shall useS(ω1) to denote[Tω1]ω. We shall alsoomit the superscriptω1 from p

ω1α , i.e., we writepα = p

ω1α . For any setA ∈ S(α) (with

α � ω1) we define the ordinalβ(A) < ω1 as follows:

β(A) = min{β � α: pα

β �A is one–one∧ (∀γ )(β � γ � α → pαγ [A] ∈ S(γ )

)}.

Let us emphasize that according to the definition ofW(κ) thisβ(A) is less thanω1 even ifα = ω1. We shall writepα

β(A)[A] = A∗. Sincepαβ(A) � A is one–one, its inverse that maps

A∗ ontoA exists; we shall denote it bypA. Note that ifα1 ∈ α \ β(A) andA1 = pαα1

[A]thenβ(A) = β(A1), A∗ = A∗

1 andpA1 = pαα1

◦ pA.Now we are ready to formulate and prove the above mentioned alternative version of

W(κ).

4.17. LetT andS be as in4.14. Then we can define a mapR onω1 such that

(i) R(α) is a countable collection of one–one maps ofω into Tα for eachα ∈ ω1;(ii) for any one–one mapr :ω → Tω1 there is aγ ∈ ω1 such that ifα ∈ ω1 \ γ then

pα ◦ r ∈ R(α).

Proof. SinceW(κ) implies CH, for anyA ∈ ⋃{S(α): α ∈ ω1} we may arrange all theone–one maps ofω into A in a sequence{

hAν : ν ∈ ω1}.

Then, for anyα ∈ ω1 we put

R(α) = {pA ◦ hA

∗ν : A ∈ S(α) ∧ ν � α

}.


ClearlyR(α) is a countable collection of one–one maps ofω intoTα . To show (ii), considerany one–one mapr ∈ (Tω1)

ω with ranger[ω] = A. Thenpβ(A) ◦ r is a one–one map ofωinto A∗, hence there is aν ∈ ω1 for which

pβ(A) ◦ r = hA∗

ν .

Now, if γ = max{β(A), ν} then forα ∈ ω1 \ γ andB = pα[A] we haveB ∈ S(α) andA∗ = B∗, hence

pα ◦ r = pB ◦ pβ(A) ◦ r = pB ◦ hA∗

ν = pB ◦ hB∗

ν ∈ R(α),

thus the proof is completed.✷In what follows we shall denote byR(ω1) the set of all one–one maps ofω into T (ω1).

Then, ifr ∈ R(α) for someα � ω1 we defineβ(r) similarly asβ(A) was defined above:

β(r) = min{β � α: (∀γ )(β � γ � α → pα

γ ◦ r ∈ R(γ ))}.

Clearlyβ(r) < ω1, even ifr ∈ R(ω1).

4.18. W(κ) implies the existence of a strongHFD matrixF onκ × ω1 (hence also that ofa strongHFC matrix onω1 × κ).

Proof. Consider the treeT of 4.15 and the mapR of 4.17. We may assume thatTω1 = κ .By transfinite induction onν ∈ ω1 we are going to define maps

Gν :Tν → 2,

and then our matrixF will be defined by

F(α,β) = Gβ

(pβ(α)

)whenever〈α,β〉 ∈ κ × ω1.

As a preparation for the induction let us denote forν ∈ ω1 by J (ν) the obviouslycountable set of all triples〈r, k, ε〉 such thatr ∈ R(ν), k ∈ ω and ε ∈ 2k×b whereb ∈ [ν + 1 \ β(r)]<ω. The induction will be carried out in such a way that for anyν ∈ ω1the following inductive hypothesisI (ν) be satisfied:

I (ν): For every〈r, k, ε〉 ∈ J (ν) the set

Z(r, k, ε) = {n ∈ ω: (∀i ∈ k)(∀β ∈ b)

[ε(i, β) = Gβ

(pνβ

(r(n · k + i)

))]}is infinite.

Now, assumeν ∈ ω1 andGµ has already been defined for eachµ ∈ ν in such a way thatI (µ) holds. Let us first consider the setJ ′(ν) of all those triples〈r, k, ε〉 ∈ J (ν) for whichthe setb that occurs in the domaink × b of ε satisfiesb ⊂ ν (and not onlyb ⊂ ν + 1). Letus note that ifb ⊂ µ + 1 � ν then we clearly have

Z(r, k, ε) = Z(pνµ ◦ r, k, ε

),

wherepνµ ◦ r ∈ R(µ) if b �= ∅ hence byI (µ) we know thatZ(r, k, ε) is infinite.

Next we consider anω-type enumeration{〈rj , kj , εj ,πj 〉: j ∈ ω}


of the set of all quadruples〈r, k, ε,π〉 with 〈r, k, ε〉 ∈ J ′(ν) andπ ∈ 2k that isω-abundant,i.e., for any such quadruple〈r, k, ε,π〉 we have∣∣{j : 〈rj , kj , εj ,πj 〉 = 〈r, k, ε,π〉}∣∣ = ω.

The functionGν :Tν → 2 will now be defined recursively, inω steps in such a way that,for each step,Gν(t) is defined only for finitely manyt ∈ Tν . Suppose that we are to dostepj of this construction. Since, as we have seen above,Z(rj , kj , εj ) is infinite, we maychoosenj ∈ Z(rj , kj , εj ) in such a way thatnl �= nj for l < j , moreover for eachi ∈ kj

Gν

(rj (nj · kj + i)

)is still undefined. Then, what we actually do in stepj is to put

Gν

(rj (nj · kj + i)

) = πj (i)

for everyi ∈ kj . Note that this will implynj ∈ Z(rj , kj , εj ) where

εj = εj ∪ {⟨〈i, ν〉,πj (i)⟩: i ∈ kj

}.

If, after all theω steps have been done, there remaint ∈ Tν for whichGν(t) is not definedthen for theset the valuesGν(t) can be chosen arbitrarily.

To check thatI (ν) will be valid, consider a triple〈r, k, ε〉 ∈ J (ν) \ J ′(ν), i.e., suchthat for k × b = D(ε) we haveν ∈ b. We put ε′ = ε � k × (b \ {ν}), then we have〈r, k, ε′〉 ∈ J ′(ν), moreover we letπ ∈ 2k be defined by the stipulations

π(i) = ε(i, ν)

for i ∈ k. But then for anyj ∈ ω with⟨r, k, ε′,π

⟩ = 〈rj , kj , εj ,πj 〉we made sure in our construction that

nj ∈ Z(r, k, ε),

hence clearly|Z(r, k, ε)| = ω.Having definedGν :Tν → 2 for eachν ∈ ω1 we put as promised

F(α, ν) = Gν

(pν(α)

)for α ∈ κ andν ∈ ω1. To show thatF is strong HFD we considerA ∈ Dk

ω(κ) for somek ∈ ω. Let A = {a(n): n ∈ ω} be a one–one enumeration ofA, then we definer ∈ R(ω1)

with the following stipulations:

r(n · k + i) = α(n)i ,

where, of course,α(n)i is the ith member ofa(n). Clearly, it will suffice to show that for

everyb ∈ [ω1 \ β(r)]<ω and for everyε ∈ 2k×b there is ana(n) ∈ A such that

F(α(n)i , β

) = ε(i, β)

for all i ∈ k andβ ∈ b.To see this, chooseγ ∈ ω1 \ β(r) such thatb ⊂ γ + 1. Thenpγ ◦ r ∈ R(γ ), hence

〈pγ ◦ r, k, ε〉 ∈ J (γ ),


consequently, byI (γ ) we have

Z(pγ ◦ r, k, ε) �= ∅.But for anyn ∈ Z(pγ ◦ r, k, ε) we have

F(α(n)i , β

) = F(r(n · k + i), β

) = Gβ

(pβ

(r(n · k + i)

))= Gβ

(pγβ

(pγ ◦ r(n · k + i)

)) = ε(i, β)

wheneveri ∈ k andβ ∈ b, and this was to be shown.✷Of course we could again list a number of immediate topological consequences ofW(κ)

in the vein of, e.g., 4.10 and 4.12. These in turn could be broken into consequences ofspecial cases ofW(κ), namely CH,V = L (or the (ω,1)-morass), BACH etc. We thinkthat the details of this can be safely left to the reader. We mention only one corollaryof 4.18 (and of 2.26 and 2.27) whose analogue was not mentioned after 4.9. (Recall thatW(κ) implies CH!)

4.19. If cf(κ) = ω1 < κ andW(κ) holds then there exists a0-dimensionalT2-spaceRwith |R| = κ such that for everyS ∈ [R]κ the subspaceS is separable butR is not HS.Moreover, ifκω = κ < 2ω1 is also valid theno(R) = 2ω1 is greater than the number ofclosed subsets ofR of sizeκ . This condition, together withW(κ), will be satisfied, e.g.,in the generic extensionV [G] of 4.16 if one starts with a ground modelV in whichκω = κ < 2ω1 holds(in addition toCH).

Below we formulate a weakening ofW(κ) that is a natural candidate for a principlethat yields the analogues of the HFD and HFC type consequences ofW(κ) to the case ofHFDws and HFCws.

4.20. Definition. Let us denote byV (κ) the following statement:There is a treeT of heightω1 + 1 and a functionS with domainω1 such that

(i) |Tω1| = κ andTα � ω if α ∈ ω1;(ii) for α ∈ ω1 we haveS(α) ⊂ [Tα]ω and|S(α)| � ω;(iii) for any A ∈ [Tω1]ω1 there is ana ∈ [A]ω and aγ ∈ ω1 such thatpα[a] ∈ S(α) if

α ∈ ω1 \ γ .

Actually, to obtain the full analogue of 4.18, i.e., thatV (κ) implies the existence of astrong HFDw matrix onκ × ω1, we seem to need a modified, stronger version ofV (κ)

corresponding to the version ofW(κ) given in 4.17.This modified version, sayV ′(κ), is obtained by replacing in 4.20 the functionS with

another functionR, moreover (ii) and (iii) with conditions (ii′) and (iii′) as follows:

(ii ′) for α ∈ ω1, R(α) is a countable set of one–oneω-sequences of elements of[Tα]<ω;(iii ′) every one–oneω1-sequenceA :ω1 → [Tω1]<ω has anω-type subsequencea such

that for someγ ∈ ω1 we have−→pα ◦ a ∈ R(α) wheneverα ∈ ω1 \ γ .


The reader can easily check that essentially the same proof as in 4.18 yields indeed thatV ′(κ) implies the existence of a strong HFDw matrix onκ × ω1. The problem is, howeverthat we do not know whetherV (κ) implies V ′(κ) or not. We do get a weaker, but stillinteresting consequence ofV (κ):

4.21. V (κ) implies the existence of anHFDw matrix onκ × ω1.

Proof. Let T andS be as in 4.20. For any setA ∈ S(α) we shall put again

β(A) = min{β � α: pα

β � A is one–one∧ (∀γ )(β � γ � α) → pαγ [A] ∈ S(γ )

}.

By induction onν ∈ ω1 we define mapsGν :Tν → 2 as follows. Let, forν ∈ ω1, J (ν) bethe set of all pairs〈A,ε〉 with A ∈ S(ν) andε ∈ H(ν + 1\ β(A)), moreover

J ′(ν) = {〈A,ε〉 ∈ J (ν): ε ∈ H(ν \ β(A)

)}.

Throughout the construction of the mapsGν we keep valid the following inductivehypothesisI (ν): for every 〈A,ε〉 ∈ J (ν) we have|Z(A,ε)| = ω, whereZ(A,ε) ={t ∈ A: (∀β ∈ D(ε))(ε(β) = Gβ(p

νβ(t)))}. The actual construction ofGν requiresω-

abundantly enumerating in anω-sequence all triples〈A,ε, i〉 with 〈A,ε〉 ∈ I ′(ν) andi ∈ 2,and then in stepn puttingGν(t) = i for somet ∈ Z(A,ε) where〈A,ε, i〉 is thenth termof the above enumeration. It is easy to check then that, if we identifyTκ with κ , the matrix

F(α,β) = Gβ

(pβ(α)

)is indeed HFDw. ✷

We just briefly mention here the combinatorial principle “stick” that claims the existenceof a familyA ⊂ [ω1]ω such that|A| = ω1 and for every uncountableB ⊂ ω1 there isA ∈ Awith A ⊂ B. Thus “stick” is a weakening of both CH and the principle “club”. We leave itto the reader to verify that, e.g., the principle “stick” impliesV (ω1) and hence the existenceof a HFDw.

Let us now return to the definitions of HFDs and HFCs, cf. 2.1 and 3.1. The reader mighthave wondered why we allowed the case of countable HFD spaces (i.e., HFD matrices onω × λ) or HFC spaces of countable weight (i.e., HFC matrices onκ × ω). Certainly, theseobjects do not seem to yield directly neitherS or L spaces, even though their definitionsdo make sense. We show below, however, that these defects are only apparent, becausethe existence of a countable HFD implies that of an uncountable HFD and similarly theexistence of an HFC of countable weight implies that of an HFC of uncountable weight.

Before we formulate the exact statements of these results we recall the following wellknown result (see [17], Theorem 5.9).

4.22. There exists a sequence〈fα : α ∈ ω1〉 of one–one mapsfα :α → ω such that ifα < β < ω1 thenfα =∗ fβ � α (i.e.,fα(ν) = fβ(ν) for all but finitely manyν ∈ α).

We may now formulate and prove the above mentioned results about transformingcountable HFDs (respectively HFCs of countable weight) into uncountable HFDs (respec-tively HFCs of weightω1).


4.23. (i) For every matrixF :ω × ω1 → 2 there exists a matrixG :ω1 × ω1 → 2 such thatif, for anyk ∈ ω, F is HFDk then so isG. In particular, if F is strongHFD then so isG.

(ii) For every matrixF :ω1 ×ω → 2 there is a matrixG :ω1 ×ω1 → 2 such that ifF isHFCk then so isG. In particular, if F is strongHFC then so isG.

Proof. (i) Let us consider the sequence〈fα : α ∈ ω1〉 given in 4.22 and, givenF , let usdefine the matrixG :ω1 × ω1 → 2 by setting

G(α,β) = F(fβ(α),β

)wheneverα ∈ β ∈ ω1.

Now, if F is HFDk we have to show thatG is also HFDk . To see this we letA ∈ Dkω(ω1),

B ∈ Dω1(ω1) andε0, . . . , εk−1 ∈ 2n(B). We may assume, without any loss of generality, thatfor someγ ∈ ω1⋃

A ⊂ γ < min⋃

B

and that there is a finite sets ∈ [γ ]<ω such that for everyβ ∈ ⋃B we have

fβ � γ \ s = fγ � γ \ s.

We might also assume that⋃

A ∩ s = ∅, hence for anyα ∈ ⋃A andβ ∈ ⋃

B we have

fβ(α) = fγ (α).

Now {fγ [a]: a ∈ A

} ∈ Dkω(ω),

and asF is HFDk onω × ω1 there exista ∈ A andb ∈ B such that

F(fγ (αi), βj

) = εi(j)

for all i ∈ k andj ∈ n(B). But, by the above,

F(fγ (αi), βj

) = F(fβj (αi), βj

) = G(αi,βj ),

and the proof is completed.(ii) In this caseG :ω1 × ω1 → 2 is chosen in such a way that

G(α,β) = F(α,fα(β)

)holds wheneverβ < α < ω1. The verification of the fact that thisG will be HFCk if F is,is completely analogous to the above argument for the HFD case and therefore we omit thedetails. ✷

According to our remark made after 3.2 we know thatX ⊂ 2ω is HFC if and only ifit is a D-Luzin set. Thus we immediately obtain that the existence of aD-Luzin set (inparticular of a Luzin or a Sierpinski set) implies the existence of an HFC in 2ω1 and thusthe existence of anL space.

We shall now deal with a couple of more specialized existence results concerning HFDwand HFCw spaces (or matrices). They will make use of a common weakening of the notionsof HFDw and HFCw that we formulate below. This will be given only for the case ofmatrices onω1 × ω1 because this suffices for the applications.


4.24. Definition. F :ω1 × ω1 → 2 is said to be an HFw matrix if for everyA ∈ [ω1]ω1,B ∈ Dω1(ω1) andε ∈ 2n(B) there areα ∈ A andb ∈ B such that

F(α,−) = fα ⊃ ε ∗ b.

It is clear that both HFDw and HFCw matrices are HFw. It is more surprising howeverthat, in a sense made precise by our next result, we can also say that an HFw matrix iseither HFDw or yields an HFCw matrix.

4.25. LetF :ω1 × ω1 → 2 be anHFw matrix.

(i) If F is notHFDw then there is anA ∈ [ω1]ω1 such thatF �A × ω1 is HFCw.(ii) If F is notHFCw then there is anA ∈ [ω1]ω1 such thatF � A × ω1 is HFDw.

Proof. If F is not HFDw then there areA ∈ [ω1]ω1, B ∈ Dω1(ω1) andε ∈ 2n(B) such thatfor everyα ∈ A we have∣∣{b ∈ B: fα ⊃ ε ∗ b}∣∣ � ω.

In other words, we can define a mapg :A → ω1 such that for anyα ∈ A if b ∈ B andb ⊂ ω1 \ g(α) thenfα �⊃ ε ∗ b. We may of course assume that ifα,β ∈ A andα < β theng(α) < g(β).

We now show thatF � A × ω1 is HFCw. Suppose, indirectly, that this is not the case,i.e., we can findA′ ∈ [A]ω1, C ∈ Dω1(ω1) andη ∈ 2n(C) such that∣∣{α ∈ A′: fα ⊃ η ∗ c

}∣∣ � ω

wheneverc ∈ C. Of course, we may again find a maph :C → ω1 such that ifc ∈ C andα ∈ A′ \ h(c) thenfα �⊃ η ∗ c. We may also assume thatc ⊂ h(c) holds wheneverc ∈ C.

Now, by induction onν ∈ ω1 we pickαν ∈ A′, bν ∈ B andcν ∈ C satisfying

αν < bν < cν

in the following way. Supposeµ ∈ ω1 andαν , bν , cν have already been defined forν ∈ µ.Then firstαµ ∈ A′ is picked in such a way thath(cν) < αµ for everyν < µ. Givenαµ wechoosebµ ∈ B such thatbµ ⊂ ω1 \ g(αµ) and thencµ ∈ C satisfyingbµ < cµ.

Having completed the induction, we putdµ = bµ ∪ cµ and consider the sets

{αν : ν ∈ ω1} ∈ [ω1]ω1

and

{dµ: µ ∈ ω1} ∈ Dω1[ω1].SinceF is HFw we must haveν,µ ∈ ω1 such thatfαν ⊃ δ ∗ dµ, whereδ = εDη. This,however, is impossible because ifν � µ then bµ ⊂ ω1 \ g(αµ) ⊂ ω1 \ g(αν) impliesfαν �⊃ ε ∗ bµ, and if µ < ν thenαν > h(cµ) implies fαν �⊃ η ∗ cµ. (Note thatδ ∗ dµ =ε ∗ bµ ∪ η ∗ cµ.) This contradiction shows us thatF � A × ω1 is indeed HFCw.

The completely analogous proof of (ii) is left to the reader.✷


If, as usual, we say that a subspaceX ⊂ 2ω1 is HFw if it can be represented by an HFwmatrix, then 4.24 may also be phrased as follows: IfX ⊂ 2ω1 is HFw and not HFDw (notHFCw), then it has an HFCw (HFDw) subspace. Moreover, ifX ⊂ 2ω1 satisfies|X| = ω1then it is easy to check (and is left to the reader) that the following statements (i)–(iii) areequivalent.

(i) X is an HFw;(ii) everyY ∈ [X]ω1 is finally dense;(iii) everyDω1-set in 2ω1 finally coversX.

In view of the above we see that the existence of an HFw implies the existence of anSspace or anL space. On the other hand, we do not know a model of set theory in whichthere is no HFw. Compare this with the following “empirical” facts: there is a model withnoS spaces but none is known with noL spaces.

It follows from 2.24 that the existence of an HFDw space implies also the existenceof an uncountable subspaceX ⊂ 2ω1 with the property that every open set inX is eithercountable or co-countable. Next we present certain partial converses of this fact; in theproofs HFws will play a crucial role.

4.26. Let τ be a topology onω1 satisfying the following two properties:

(∗) everyU ∈ τ is either countable or co-countable;(∗∗) for everyα ∈ ω1 there exist two disjoint open setsU0,U1 ∈ τ such that

ω1 \ β ⊂ U0 \ α ∩ U1 \ α

for someβ ∈ ω1. (Clearly, the latter assumption onU0 andU1 is by (∗) equivalentto requiring that bothU0 \ α andU1 \ α be uncountable.)

Then there is an HFDw space.

Proof. First, using (∗∗), we inductively pick elementsαν ∈ ω1 and setsU(ν)0 ,U

(ν)1 ∈ τ

such that for eachν ∈ ω1 we haveU(ν)0 ∩ U

(ν)1 = ∅ and

ω1 \ αν+1 ⊂ U(ν)0 \ αν ∩ U

(ν)1 \ αν.

This can also be done in such a way thatµ< ν impliesαµ < αν .For anyε ∈ H(ω1) we define

Uε =⋂{

U(ν)ε(ν): ν ∈ D(ε)

}.

Let us put for anyε ∈ H(ω1) \ {∅}ε = minD(ε).

We claim thatUε \αε �= ∅ for eachε ∈ H(ω1)\{∅}. This is proved by induction onn = |ε|.Forn = 1 the claim is obvious. Now, if|ε| = n + 1> 1 let us putε′ = ε � (D(ε) \ {ε}). Bythe inductive hypothesis we haveUε′ \ αε′ �= ∅, moreover

αε < αε+1 � αε′ ,


consequently, becauseU(ε)

ε(ε)\ αε is dense inω1 \ αε+1, we have

Uε′ ∩ U(ε)

ε(ε)\ αε = Uε \ αε �= ∅.

By (∗), for everyν eitherU(ν)0 or U(ν)

1 is countable, thus we may assume without any loss

of generality thatU(ν)0 is countable for allν. Now, for everyα ∈ ω1 we definefα ∈ 2ω1 by

the stipulation

fα(β) = 0 ←→ α ∈ U(β)0 .

We shall show thatX = {fα : α ∈ ω1} ⊂ 2ω1 is HFDw. In fact, first we show thatX is HFw.To see this we letA ∈ [ω1]ω1 andB ∈ Dω1(ω1), moreoverε ∈ 2n(B). It follows from ourabove considerations that the set

G(B,ε) =⋃

{Uε∗b: b ∈ B} ∈ τ

is uncountable, hence by (∗) it is in fact co-countable. In particular we haveA∩G(B,ε) �=∅, hence there areα ∈ A andb ∈ B such that

α ∈ Uε∗b.But it is clear from the definition offα that then

fα ⊃ ε ∗ b

holds as well. This shows thatX is indeed HFw. Moreover, for eachβ ∈ ω1 we have∣∣{α: fα(β) = 0}∣∣ = ∣∣U(β)

0

∣∣ � ω,

hence the familyB = {{β}: β ∈ ω1} ∈ Dω1(ω1) and the function{〈0,0〉} show that noY ∈ [X]ω1 can be HFCw, consequently, by 4.25,X is HFDw. ✷

We do not know whether, for aT2 or T3 topologyτ , condition (∗∗) is actually impliedby (∗), but our next result shows that this is at least consistently so.

4.27. AssumeMAω1 for countable partial orders(or equivalently that the real line is notthe union ofω1 nowhere dense sets). Then the following three statements are equivalent:

(i) There exists aHFDw space.(ii) There exists an uncountable0-dimensionalT2 (henceT3) space in which every open

set is countable or co-countable.(iii) There exists an uncountableT2 space in which every open set is countable or co-

countable.

Proof. (i) ⇒ (ii) is shown in 2.24 and (ii)⇒ (iii) is trivial. Moreover, the proofs of theseimplications do not make use of MAω1(countable). To prove (iii)⇒ (i) assume thatτ is aT2 topology onω1 satisfying (∗). Clearly, we may assume that the weight ofτ is ω1, sinceω1 many open sets will show that it isT2.

Now we distinguish two cases. If for everyα ∈ ω1 there are distinct elementsβ0, β1of ω1 such that wheneverβi ∈ Ui ∈ τ (i ∈ 2) then|Ui \ α| = ω1 then, sinceτ is T2, (∗∗)of 4.26 is clearly satisfied.


Otherwise there is anα ∈ ω1 such that for all but oneβ ∈ ω1 \ α there is aτ -neighbourhoodU of β such that|U \ α| � ω. Thus, by possibly throwing away countablymany points, we may assume that every point has a neighbourhood with countable closure.Then, possibly with the use of a further shrinking, we may assume thatτ has the following,even stronger, property: every pointα ∈ ω1 has a neighbourhoodU such thatU ⊂ α + 1.We show that thenτ again satisfies (∗∗).

To see this, fixα ∈ ω1 and then let us put

ν = min{µ ∈ ω1 \ α: |µ \ α| = ω1

}.

That thisν ∈ ω1 exists follows from the fact thatτ is HS, established in the proof of 2.25.Obviouslyν is a limit ordinal, thus there is anω-type increasing sequence{νi : i ∈ ω} ⊂ ν

such that⋃{νi : i ∈ ω} = ν. For everyi ∈ ω then|νi \ α| � ω, hence there is aγ ∈ ω1 \ α

such thatω1 \ γ ⊂ ν \ α and⋃{νi \ α: i ∈ ω

} ⊂ γ.

Note that then for everyG ∈ τ if G \ γ �= ∅ thenG ∩ ν is cofinal inν.Let B be a basis of the topologyτ of cardinalityω1 and put

B′ = {B ∈ B: B \ γ �= ∅}.We define a partial order〈P,�〉 as follows:P consists of all functionsh :n → ν \ α,wheren ∈ ω andh(i) � νi for all i ∈ n. If h,h′ ∈ P then we seth � h′ if and only ifh ⊃ h′. Obviously,P is countable. It is also easy to see that for alln ∈ ω and for allB ∈ B′the sets

EB,n = {h ∈ P : ∃k � n

[h(k) ∈ B

]}are dense in〈P,�〉, becauseB ∩ ν is cofinal inν. Thus by MAω1(countable) there exists ageneric setG ⊂ P over the family of dense sets{EB,n: n ∈ ω,B ∈ B′}. Let us putH = ⋃

GthenH :ω → ν \ α andH(i) � νi for all i ∈ ω, henceS = R(H) is anω-type (cofinal)subset ofν \ α such thatS ∩ B �= ∅, in fact |S ∩ B| = ω, for everyB ∈ B′.

Since MAω1(countable) also implies thatω1 many infinite subsets ofω always havepropertyB (or equivalently: can be reaped, cf. [9]),S can be written in the formS =S0 ∪ S1, S0 ∩ S1 = ∅, |S0| = |S1| = ω, where|Si ∩ B| = ω holds wheneveri ∈ 2 andB ∈ B′. Thus we also haveω1 \ γ ⊂ Si for i ∈ 2.

Write S = {σn: n ∈ ω} in its increasing,ω-type order and for eachn ∈ ω pick Un ∈ τ

such that

σn ∈ Un ⊂ Un ⊂ σn + 1.

Then for eachn ∈ ω

Vn = Un \⋃{

Um: m< n}

is a neighbourhood ofσn such thatVn ∩ Vm = ∅ if n �= m. Thus if we put

Gi =⋃

{Vn: σn ∈ Si}for i ∈ 2, thenG0 andG1 are disjoint open sets withSi ⊂ Gi \ α, hence clearly|Gi \ α| =ω1, showing that (∗∗) is indeed satisfied. ✷


In our next application of 4.25 we show that the existence of a Suslin tree implies theexistence of an HFCw space. This yields anL space quite different from a Suslin line,because the latter is first countable while the former is very much not so (cf. 3.6).

4.28. The existence of a Suslin tree implies the existence of anHFCw space.

Proof. Let T be a normal Suslin tree, we may assume thatT ⊂ 2<ω1 with extension as thetree ordering. Theαth levelTα of T is then obtainable as

Tα = {t ∈ T : D(t) = α

} = T ∩ 2α.

Let us note that the normality ofT implies that ift ∈ Tα andε ∈ H(ω1 \ α) thent has anextensions ∈ T such thatε ⊂ s.

Now, for anyt ∈ T with t ∈ Tα we defineft ∈ 2ω1 as follows:

ft (ν) ={t (ν), if ν ∈ α,0, if ν ∈ ω1 \ α.

We claim thatX = {ft : t ∈ T } is HFCw. Since no countable subset ofX can be finallydense it is trivial thatX has no HFDw subspace, thus by 2.25 it will suffice to show thatX

is HFw.To see this we consider anyA ∈ [T ]ω1, then by a well known property of Suslin trees

there is ans ∈ T such thatA is dense aboves, i.e., every extension ofs in T has an exten-sion inA. An immediate consequence of this and our above remark is that for everyε ∈H(ω1 \ D(s)) there is at ∈ A such thatε ⊂ ft , which clearly implies thatX is HFw. ✷

5. CCC-destructibility

In this section we examine the question if some of the HFD and HFC type propertiescan or cannot be destroyed by CCC forcings. Of course, this is very closely related to thequestion if they contradict MAω1 or not.

In [19] this question had already been touched upon, for example Silver’s result is giventhat MAω1(σ -centered) (or equivalentlyp > ω1) destroys both HFDs and HFCs. On theother hand, Szentmiklóssy’s following celebrated result from [24] is also treated in detail:the existence of anS-space is consistent with MAω1. This result is relevant to what we dohere because its proof proceeds by first constructing from CH a special kind of HFD space(called tight HFD) and then showing that the HS-ness of a tight HFD is CCC-indestructible(while, of course, HFD-ness will be lost by theσ -centered forcing that introduces MAω1

(σ -centered)).What about the “dual” question of obtaining aCCC-indestructibleL-space from an

HFC? Surprizingly, this cannot be done, at least not from CH! Indeed, Soukup provedin [22] that if CH holds then anyX ⊂ 2ω1 that is an HFC contains a strong HFC subspace,and so its HL-ness is CCC-destructible. The latter statement, as well as its dual, followfrom the next result.

5.1. (i) If X ⊂ 2ω1 is a strongHFDw then theHS-ness ofX is CCC-destructible.(ii) If X ⊂ 2ω1 is a strongHFCw then theHL-ness ofX is CCC-destructible.


Proof. We only give the proof of (ii), since that is what we need next; the proof of (i)is completely analogous. So letX = {fα : α ∈ ω1} be a 1–1 enumeration ofX (clearly,|X| = ω1 may be assumed). We can also assume thatfα(α) = 1 holds for everyα ∈ ω1.

Now define the poset〈P,�〉 by P = {p ∈ [ω1]<ω: ∀β,α ∈ p(β < α ⇒ fα(β) = 0)}andp � q iff p ⊃ q . Then〈P,�〉 is CCC: By a simple∆-system and counting argumentfor this it suffices to show that if{pν : ν ∈ ω1} ⊂ P with {pν : ν ∈ ω1} ∈ Dn

ω1(ω1) then there

areµ< ν < ω1 such that, maxpµ < minpν , andfα(β) = 0 for all α ∈ pν andβ ∈ pµ, asthenpν ∪ pµ ∈ P extends bothpν andpµ. This in turn is clear from the HFCnw propertyof X.

Since〈P,�〉 is CCC, there is somep ∈ P such thatp � |Γ | = ω1, hencep � |⋃Γ | =ω1, whereΓ is the name for aP -generic set. It is also clear that

⋃Γ is forced to be a right

separated subspace, hencep forces thatX is not HL. ✷Now we present Soukup’s result.

5.2. If CH holds then anyHFCspaceX ⊂ 2ω1 contains a strongHFCsubspace.

Proof. For any x ∈X, B ∈Dnω(ω1) and ε∈ 2n let W(B,ε, x)= {b∈B: ε ∗ b ⊂ x},

moreover letU(B) = {x ∈ X: ∃ε ∈ 2n(|W(B,ε, x)| < ω)}. Then U(B) is alwayscountable: Indeed,|U(B)| = ω1 would imply that for some fixedε ∈ 2n andB0 ∈ [B]<ω

we had{x ∈ X: W(B,ε, x) = B0} = ω1, contradicting 3.2(i) withB \ B0 instead ofB.Now let ϑ be a large enough regular cardinal and〈Nα : α ∈ ω1〉 be an∈-chain (i.e.,

Nα ∈ Nα+1 for everyα ∈ ω1) of countable elementary submodels ofHϑ with X ∈ N0.By CH we haveHω1 ⊂ N = ⋃{Nα : ∈ ω1}, in particular we haveDω(ω1) ⊂ N . For anyα ∈ ω1 we may also pick agα ∈ X ∩ (Nα+1 \ Nα). We claim thatY = {gα : α ∈ ω1} is astrong HFC.

To see this letB ∈ Dnω(ω1) and α ∈ ω1 be chosen so thatB ∈ Nα . Fix k ∈ ω and

−⇀ε = 〈εi : i ∈ k〉 ∈ (2n)k ; if we can show that for anyα � α0 < · · · < αk−1 there is ab ∈ B

with εi ∗ b ⊂ gαi for all i ∈ k, thenY is shown to be HFCk , and so we will be done.Now B ∈ Nα impliesU(B) ∈ Nα and so, by|U(B)| � ω, U(B) ⊂ Nα as well. Thus

gα0 /∈ U(B), henceB1 = W(B,εo, gα0) is infinite. But clearly,B1 ∈ Nα0+1 ⊂ Nα1, sogα1 /∈ U(B1) ⊂ Nα1 and, as before,B2 = W(B1, ε1, gα1) is infinite. Continuing this ink steps we arrive at an infiniteBk ⊂ B such that for everyb ∈ Bk we haveεi ∗ b ⊂ gαi forall i ∈ k, hence the proof is completed.✷

We now turn to another result of Soukup (see [23]) which yields a very powerfulmethod for constructing CCC-indestructible structures, in particular HFDn

ws and HFCnwsfor any fixedn. The result has gone through several “stages of development” before it gotits present succinct and easily applicable form. Unfortunately, the proof of it is still tootechnical and lengthy to be given here. We start with a few definitions.

5.3. Definition.

(i) Let K be any set andn ∈ ω. Then we writeSn(K) for the family of all setsS such thatS ⊂ Fn(ω1,K), |S| = ω1, |s| = n for eachs ∈ S, moreoverD(s) ∩ D(t) = ∅ for any


{s, t} ∈ [S]2. (In other words,S is a set of functions whose domains form an elementof Dn

ω1(ω1) and whose ranges are contained inK.) Note that anys ∈ S is ann-element

subset ofω1 × K.(ii) Now let E ⊂ [ω1 × K]2, i.e.,E is a graph onω1 × K. ThenE is calledn-solid if for

anyS ∈ Sn(K) there is a pair{s, t} ∈ [S]2 such that[s, t] ⊂ E, i.e., for anyv ∈ s andw ∈ t we have{v,w} ∈ E.

The graphE onω1 × K is called strongly solid if it isn-solid for everyn ∈ ω.

Soukup’s above mentioned result now reads as follows.

5.4. Assume2ω1 = ω2 andE be a strongly solid graph onω1 ×K. Then for any fixedn ∈ ω

there is aCCC notion of forcingP such that, inV P , the graphE is CCC-indestructiblyn-solid.

For us, perhaps the most interesting application of 5.4 is that, starting from a strongHFDw (respectively HFCw), for any fixedn ∈ ω both CCC-indestructible HFDnw spacesand CCC-indestructible HFCnw spaces may be obtained consistently. The followingdefinition tells us how to chooseK and the graphE ⊂ [ω1 × K]2 to a givenX ⊂ 2ω1

that is a strong HFDw (respectively HFCw).

5.4. Definition. Let X = {fα : α ∈ ω1} ⊂ 2ω1, where thefα ’s form a 1–1 enumerationof X. Also, letD be a countable dense subset of 2ω1. SetK = [ω1]<ω × D and define thegraphE onω1 × K as follows. (For simplicity, we shall write the elements ofω1 × K inthe form〈ν, b, d〉 instead of〈ν, 〈b, d〉〉.)

First, letJ = {〈ν, b, d〉 ∈ ω1 × K: ν ∩ b = ∅} and defineE ∩ [J ]2: If both 〈ν, b, d〉 ∈ J

and 〈ν′, b′, d ′〉 ∈ J then {〈ν, b, d〉, 〈ν′, b′, d ′, 〉} ∈ E if either d �= d ′ or ν < ν′ andfν ⊃d � b′ = d ′ � b′. Any other edge, i.e., one having an end point outside ofJ , is put inE, orformally:E ⊃ [ω1 × K]2 \ [J ]2.

The connection between the HFDnw property ofX and then-solidity of E is now given:

5.6. For anyn ∈ ω, X is HFDnw exactly ifE is n-solid.

Proof. AssumeX is HFDnw and letS = {sα : α ∈ ω1} be chosen fromSn(K). Clearly, we

may assume thatsα ⊂ J for everyα ∈ ω1. For anyα we havesα = {〈να,i , bα,i, dα,i〉: i ∈ n}with να,i < να,j if i < j . By thinning out if necessary, we may assume that

(i) dα,i = di is the same for allα ∈ ω1;(ii) for eachbα = ⋃{bα,i : i ∈ n} we have|bα| = m, bα < να+1,0, henceB = {bα: α ∈

ω1} ∈ Dmω1(ω1), anddi � bα = εi ∗ bα with a fixedεi ∈ 2m for everyi ∈ n.

Setting aα = {να,i : i ∈ n} we may then apply the HFDnw property of X to A ={aα: α ∈ ω1} ∈ Dn

ω1(ω1), B ∈ Dm

ω1(ω1) and 〈εi : i ∈ n〉 and so obtainα < β < ω1


such thatfνα,i ⊃ εi ∗ bβ holds for everyi ∈ n. Now, let σi = 〈να,i, bα,i, di〉 ∈ sα and*j = 〈νβ,j , bβ,j , dj 〉 ∈ sβ . Then eitherdi �= dj , hence{σi, *j } ∈ E, or di = dj and so

fνα,i ⊃ εi ∗ bβ = dj � bβ = dj � bβ,j ,hence again{σi, *j } ∈ E, proving[sα, sβ ] ⊂ E.

Now, to see the converse, assume thatE is n-solid and to check thatX is HFDnw

considerA = {aα = {να,i : i ∈ n}: α ∈ ω1} ∈ Dnω1(ω1), B = {bα: α ∈ ω1} ∈ Dm

ω1(ω1) and

−⇀ε = 〈εi : i ∈ n〉 ∈ (2m)n. Without loss of generality we may assume thataα < bα < aα+1hold for all α ∈ ω1. Also, asD is dense in 2ω1, we may pick forα ∈ ω1 andi ∈ n a pointdα,i in D such thatdα,i ⊃ εi ∗ bα . Thus we can definesα ∈ Fn(ω1,K) with domainaα foranyα by

sα = {〈να,i , bα, dα,i〉: i ∈ ω}.

Then να,i ∈ aα < bα imply sα ∈ J for everyα. Moreover, asD is countable, we mayassume thatdα,i = di is the same for everyα ∈ ω1 andi ∈ n.

By n-solidity, there areα < β < ω1 such that[sα, sβ ] ⊂ E, in particular{〈να,i, bα, di〉, 〈νβ,i, bβ, di〉} ∈ E ∩ [J ]2

for every i ∈ n. But να,i < νβ,i then impliesfνα,i ⊃ di � bβ = di,β � bβ = εi ∗ bβ for alli ∈ n proving the HFDnw property ofX. ✷

It should be clear that 5.6 has a dual for HFCnws: In the definition of the graphE one

has to replace the conditionν < ν′ with ν > ν′ to obtain the graphE onω1 × K, and thenthe above proof dualizes to the following:

5.7. X = {fα : α ∈ ω1} ⊂ 2ω1 is HFCnw exactly ifE is n-solid.

Now, from 5.4, 5.6 and 5.7 we immediately obtain that if in our ground modelV wehave 2ω1 = ω2 and there is a strong HFDw (HFCw) spaceX then for any fixedn ∈ ω thereis a CCC extensionV P of V in whichX is CCC indestructibly HFDnw (HFCn

w).So by a further CCC forcing that introduces MAω1 we obtain a model which shows

the compatibility of MAω1 with the existence of a HFDnw (HFCnw). In fact, as was shown

by Soukup, these procedures can be dovetailed to obtain a model in which all these holdsimultaneously:

5.8. Assume2ω1 = ω2 and there is a strongHFDw space(hence a strongHFCw as well).Then in a suitableCCCextension we haveMAω1 and for everyn ∈ ω there areHFDn

w andHFCn

w spaces, henceSn andLn spaces as well.

Of course, this is the most we may expect because by Kunen’s classical result there areno strongS or L spaces under MAω1.

The following application of 5.4 will start again from a strong HFDw (respectivelyHFCw) but will yield a CCC-indestructibleS-group (respectivelyL-group) in 2ω1. Aswas shown by Roitman in [20], ifX ⊂ 2ω1 is strongly HS (respectively HL) thenA(X),the subgroup of 2ω1 generated byX is HS (respectively HL). Thus if, e.g.,X is strongly


HFDw thenA(X) is HS, however the problem is how to makeA(X) CCC-indestructiblyHS? This question is answered by the following result of Soukup.

If X = {fα : α ∈ ω1} ⊂ 2ω1 then for anya ∈ [ω1]<ω we writefa = ∑{fα : α ∈ a}, then

A(X) = {fa : a ∈ [ω1]<ω

}.

Now, with an application of 5.4 in mind, letK = [ω1]<ω ×[ω1]<ω ×D, whereD is a fixedcountable dense set in 2ω1. Similarly as above, we shall write the elements ofω1 × K inthe form〈ν, a, b, d〉, and single out a subsetJ ⊂ ω1 × K by

J = {〈ν, a, b, d〉 ∈ ω1 × K: ν = mina � minb andfa ⊃ d � b}.

Then we define the graphE ⊂ [ω1 × K]2 by

[ω1 × K]2 \ [J ]2 ⊂ E

and

E ∩ [J ]2

= {{〈ν, a, b, d〉, ⟨ν′, a′, b′, d ′⟩}:(d �= d ′) or

(ν < ν′ andfa ⊃ d � b′ = d ′ � b′)}.

The following lemma can be proved then by similar arguments as 5.6 was.

5.9. LetX andE be as above.

(i) If X is stronglyHFDw thenE is strongly solid.(ii) If E is 1-solid thenA(X) is HS.

Again, if X is strongly HFCw and in the second part of the definition ofE we replaceν < ν′ by ν > ν′, then a dual version is obtained. So with applying 5.4 we get the followingresult of Soukup.

5.10. If 2ω1 = ω2 and there is a strongHFDw (henceHFCw, as well), then in a suitableCCC extension of our ground model there areCCC-indestructibleS and L subgroupsof 2ω1. In particular, the existence of bothS andL groups is compatible withMAω1 .

Finally, as a third application of 5.4 we mention the following result of Soukup: Wecan have a CCC-indestructible first countable O-space. (An O-space is an uncountableT3-space in which every open set is either countable or co-countable.) Since any O-spaceis anS-space, this also yields a CCC-indestructible first countableS-space. It should bementioned that this latter result was first obtained by Avraham and Todorcevic [1], andtheir method served as the starting point for Soukup’s results.

Let us now sketch the construction. The first step of this, similarly as in [1], is to forcea generic first countable, 0-dimensionalT2, right separated topologyτ on ω1. This isdone by the following notion of forcingQ: The elements ofQ are triplesq = 〈a,n,u〉where a ∈ [ω1]<ω, n ∈ ω and u :a × n → P(a) such thatα ∈ u(α, k) ⊂ α + 1 forα ∈ a and k ∈ n. For q = 〈a,n,u〉 and q ′ = 〈a′, n′, u′〉 we haveq � q ′ iff a ⊃ a′,n � n′, u′(α, k) = u(α, k) ∩ a′ for 〈α, k〉 ∈ a′ × n′, moreoveru′(α, i) ∩ u′(β, j) = ∅


impliesu(α, i) ∩ u(β, j) = ∅ andu′(α, i) ⊂ u′(β, j) impliesu(α, i) ⊂ u(β, j) for suitable〈α, i〉 and 〈β, j 〉. It is easy to check that ifG is Q-generic then, inV [G], the setsU(α, k) = ⋃{u(α, k): 〈a,n,u〉 ∈ G, α ∈ a, k < n} form a clopen base for a first countableT2 topologyτ onω1 which is right separated and such thatω, in fact every infinite groundmodel subset ofω, is τ -dense.

We also setK = ω × ω and single out a subsetJ ⊂ ω1 × K by J = {〈α, k, d〉: d ∈U(α, k)}, and then define a graphE ⊂ [ω1 ×K]2 by [ω1 ×K]2 \ [J ]2 ⊂ E andE ∩[J ]2 ={{〈α, k, d〉, 〈α′, k′, d ′〉}: d �= d ′ or α ∈ U(α′, k′) or α′ ∈ U(α, k)}.

The following lemma, with a standard amalgamation proof, yields both thatQ is CCCand thatE will be strongly solid inV [G].

5. 11. Assumen ∈ ω and {qα: α ∈ ω1} ⊂ Q, moreover{sα: α ∈ ω1} ∈ Sn(K) are givenin V . Then there areα < β < ω1 and aq ∈ Q such thatq � qα, qβ andq � [sα, sβ ] ⊂ E.

Finally, the following statement insures that 5.4 be applicable to yield a CCC extensionin which 〈ω1, τ 〉 is a CCC-indestructible O-space.

5. 12. If E is 2-solid, then every element ofτ (that is the topology generated by the setsU(α, k)) is either countable or co-countable.

Proof. AssumeH ∈ τ andω1 \ H are both uncountable. Pick{να : α ∈ ω1} ∈ [H ]ω1,{µα: α ∈ ω1} ∈ [ω1 \ H ]ω1 with µα < νβ for α < β < ω1 andkα, dα ∈ ω for all α ∈ ω1such thatU(να, kα) ⊂ H anddα ∈ ω ∩U(να, kα). We may assume thatdα = d is the samefor all α ∈ ω1. Since, by genericity, we clearly haveU(µ,0) = µ + 1 for eachµ ∈ ω1, wecan also haved ∈ U(µα,0), hence〈να, kα, d〉 ∈ J and〈µα,0, d〉 ∈ J for all α. Now, settingsα = {〈να, kα, d〉, 〈µα,0, d〉} we have{sα : α ∈ ω1} ∈ S2(K) and so by the 2-solidity ofEthere areα < β < ω1 with [sα, sβ ] ⊂ E∩[J ]2. In particular, then{〈µα,0, d〉, 〈νβ, kβ, d〉} ∈E ∩ [J ]2 which byµα /∈ H ⊃ U(νβ, kβ) implies νβ ∈ U(µα,0) = µα + 1, contradictingµα < νβ . ✷

Putting these together we obtain the following.

5. 13. If 2ω1 = ω2 then in a suitableCCC extension there is aCCC-indestructible firstcountableO-space. So the existence of a first countableO-space is compatible withMAω1.

Note that, by 4.27, this also yields an alternative proof of the compatibility of an HFDwwith MAω1.

6. Higher cardinal versions

There are many ways in which HFDs and the other concepts introduced in Sections 2and 3 may be extended to higher cardinals. In this section we are going to consider severalapplications of these, without any attempt to give a complete treatment.

Perhaps the most fruitful higher cardinal version is the following:


6.1. Definition. Fix an uncountable cardinalκ and a natural numberk. A matrix F :κ ×κ → 2 is called aκ-HFDk

w (respectivelyκ-HFCkw) matrix if for anyA = {aα: α ∈ κ} ∈

Dkκ(κ), B = {bα: α ∈ κ} ∈ Dκ (κ) and 〈εi : i ∈ k〉 ∈ (2n(B))k there areα < β < κ

(respectivelyβ < α < κ) such thatF(ζα,i,−) = fζα,i ⊃ εi ∗ bβ for every i ∈ k, whereζα,i is theith member ofaα in its increasing order.

F is a strongκ-HFDw if it is κ-HFDkw for all k ∈ ω. Also, the dual definition with

β < α instead ofα < β gives theκ-HFCkw (respectively strongκ-HFCw) matrices. Finally,

X = {fα : α ∈ κ} ⊂ 2κ is aκ-HFDkw (etc.) space if it can be represented by aκ-HFDk

w (etc.)matrix.

Of course, the stronger concepts ofκ-HFD, etc. can be also be defined, as was done,e.g., in [11], and their existence can be shown to be consistent, e.g., from 2<κ = κ or byforcing. The amazing thing about the strongκ-HFDws (orκ-HFCws) is that their existenceis provable in ZFC, e.g., for cardinals of the formκ = λ+ whereλ is any uncountableregular cardinal (and also for many singularλ).

This is a highly non-trivial fact that follows from Shelah’s celebrated coloring theoremto be given below. Let us emphasize that the perhaps most significant case ofλ = ω1 (i.e.,κ = ω2) had been missing until quite recently and was only proven in [21]. We start with adefinition.

6.2. Definition. Let κ � ω andσ be any cardinals. Then Col(κ, σ ) denotes the followingstatement: There is a coloringc : [κ]2 → σ such that for anyA = {aα: α ∈ κ} ∈ Dκ (κ) andfor anyh :n(A) × n(A) → σ there areα < β < κ such thatc({ζα,i, ζβ,j }) = h(i, j) holdsfor every〈i, j 〉 ∈ n(A) × n(A), whereζα,i is theith element ofaα in its natural ordering.

Shelah’s main coloring theorem now can be formulated as follows:

6.3. If λ is an uncountable regular cardinal andκ = λ+ thenCol(κ, κ) holds.

It is now straightforward to check that the next statement is valid.

6.4. If Col(κ,2) holds andc : [κ]2 → 2 establishes this, then letF :κ × κ → 2 be definedso thatF(α,β) = c({α,β}) for any{α,β} ∈ [κ]2. ThenF is both a strongκ-HFDw and astrongκ-HFCw matrix.

Many of the results of Sections 2–4 that were formulated for (strong) HFDw or HFCwspaces may now be “lifted” to (strong)κ-HFDw and κ-HFCw spaces and yield ZFCtheorems that are higher cardinal versions of the consistency results given forω1. Thisis so easy that we do not give any details here.

We do give however several other applications that show the usefulness of theseconcepts in quite different areas as well.

Our first application taken from [15] solves a problem of Scott Williams that askedif any scattered Tychonov space has a weaker topology that is compactT2 (see [26],Problem 2.1.23).


6.5. Let κ be a regular cardinal andX = {fα : α ∈ κ} ⊂ 2κ be aκ-HFDw such that

fα(α) = 1 and fα(β) = 0 wheneverβ < α < κ. (∗)

ThenX is scattered and no finer topology onX is compactT2.

Proof. Let us note first of all that theκ-version of 2.8(iii) implies that if there is aκ-HFDwthen there is also one having property(∗). Clearly,X is right-separated (i.e., scattered),moreover theκ-version of 2.24 implies that every open (or closed) setS in X satisfieseither|S| < κ or |X \ S| < κ .

Now, assume indirectly thatτ is a weaker compactT2 topology onX. Then by theabove,X has a unique completeτ -accumulation point, sayg.

For everyfα ∈ X\{g} thenfα has aτ -neighbourhoodUα with g /∈ U τα and so|U τ

α | < κ .Let us pick for any suchα a finite setDα ∈ [κ]<ω such thatEα = X ∩ [εα] ⊂ Uα , whereεα = fα � Dα . There is a setS ∈ [κ]κ such that{Dα : α ∈ S} forms a∆-system with rootD.Let δ ∈ κ be chosen so thatD ⊂ δ. Note that for everyα ∈ κ \ δ we havefα � D ≡ 0.

Let us now set for eachα ∈ κ

Fα =⋂{

E τβ : β ∈ κ \ α

}.

Then〈Fα : α ∈ κ〉 is an increasing sequence ofτ -closed sets, hence, as every subset ofX

in its original topology, and so inτ as well, has a dense subset of size< κ , it must beeventually constant, i.e., there is anα0 ∈ κ so thatFα = Fα0 wheneverα ∈ κ \ α0.

Now, letV be aτ -neighbourhood ofg such thatFα0 ∩V τ = ∅. Sinceg is a completeτ -accumulation point of{fα : α ∈ S} we have|S′| = κ , whereS′ = {α ∈ S: fα ∈ V }. SinceXis κ-HFDw, its subsetY = {fα : α ∈ S′ \ δ} is finally dense in 2κ , i.e., there is anα1 ∈ κ \ δ

so thatY is dense in 2κ\α1. Clearly there is anα2 ∈ κ \ α0, with Dα \ D ⊂ κ \ α1 forα � α2, hence for any finite seta ∈ [κ \ α2]<ω we haveY ∩ ⋂{Eα : α ∈ a} �= ∅, henceV ∩ ⋂{Eα : α ∈ a} �= ∅. But then, by compactness we also had∅ �= V τ ∩ ⋂{Eτ

α : α ∈κ \ α2} = V τ ∩ Fα2 = V τ ∩ Fα0, contradicting the choice ofV . ✷

Note that, as Col(ω2,2) is provable in ZFC, 6.5 yields us a ZFC counterexample ofsizeω2 to S. Williams’ question and that the existence of an ordinary HFDw gives one ofsizeω1, the minimum possibility. However, it remains an open question if the existence ofa counterexample of sizeω1 is provable in ZFC?

Before turning to the next application, let us note that if we have a coloringc : [κ]2 → σ

that establishes Col(κ, σ ) then the same way as in 6.4 we can define a subspaceX ={fα : α ∈ κ} ⊂ D(σ)κ (whereD(σ) is the discrete space onσ ) that hasκ-HFDw-likeproperties, in particular ifκ = λ+ thenhd(X), hL(X) � λ. This observation is the basisfor the following result which yields ZFC examples of 0-dimensionalT2-spaces showingthat the well known inequality|X| � 2s(X)ψ(X) cannot be improved to|X| � s(X)ψ(X). Sofar (see, e.g., [14, 1.1]), only consistent examples of this sort have been known.

6.6. Let c : [κ]2 → ω establishCol(κ,ω), whereκ = λ+. We define a subspaceX ={fα : α ∈ κ} of (ω + 1)κ as follows:

fα(β) ={ω, if β = α,c({α,β}), if β �= α.

Thenψ(X) = ω, whilehd(X), hL(X) � λ.


Proof. It is clear thatψ(X) = ω, moreover theκ-HFDw (respectivelyκ-HFCw) likeproperty ofX easily implieshd(X) � λ (respectivelyhL(X) � λ) and sos(X) � λ aswell. It suffices for this to note that a basic neighbourhoodBε of fα in X is still determinedby a finite functionε ∈ Fn(κ,ω), where

Bε = {g ∈ X: g(ν) = ε(ν) if ν �= α andg(α) � ε(α)

}.

Hence the non-existence of a left (right) separated subset ofX of sizeκ follows from thatof the appropriate object inD(ω)κ .

If λ = 2ω is regular thenκ = λ+, by 6.3, satisfies Col(κ,ω) and so we get a spaceXfrom 6.6 withs(X) � hd(X), hL(X) � 2ω, ψ(X) = ω but |X| = (2ω)+. This is still not aZFC example, however the choiceλ = (2ω)+ yields one! ✷

Our final example involves still another modification of the concept of an HFD. We saythat an infinite subspaceX ⊂ 2κ is a (κ,µ)-HFD if for every infinite setY ⊂ X there isa set of co-ordinatesJ ∈ [κ]<µ so thatY is dense in 2κ\J . Also, X ⊂ 2κ is said to be a(κ,→)-HFD if for everyY ∈ [X]ω there is anα in κ such thatY is dense in 2κ\α. Clearly,if κ is regular then(κ,→)-HFD is the same as a(κ, κ)-HFD. Concerning this concept wehave the following interesting open problem that was raised in [7]:

6.7. Is the existence of a(c,→)-HFD provable inZFC?

We think that the answer to this question is quite likely affirmative, however we onlyhave the following partial result. In order to formulate it, we recall that the reaping numberr is defined as the smallest cardinal of a familyA ⊂ [ω]ω that cannot be reaped by somesetb ⊂ ω (or equivalently,A does not have propertyB). In [2] it has been shown thatr = min{πχ(p,ω∗): p ∈ ω∗}, and so asπχ(p,ω∗) � cf(c) for all p ∈ ω∗ by [4], r � cf(c),hencer = c if c is regular. Also, it is routine to check that MA(countable) impliesr = c,hencer = c is consistent with any cardinal arithmetic.

6.8. If r = c then there is a(c,→)-HFD.

Proof. Let us set[ω]ω = {Aα: α ∈ c}. By induction onα ∈ c we are going to definefunctionsfn,α ∈ 2α for n ∈ ω such that the following two inductive hypotheses be valid:

I (α): if β < α then fn,β ⊂ fn,α for any n ∈ ω; J (α): for every β < α the set{fn,α : n ∈ Aβ} is dense in 2α\β .

Now assume thatα ∈ c and thefn,β have been defined for allβ < α so thatI (β) andJ (β) hold.

If α is limit then we simply set

fn,α =⋃

{fn,β : β < α},clearlyI (α) andJ (α) will be satisfied. Next, ifα = β + 1, then for everyε ∈ H(β) set

Zβε = {n: fn,β ⊃ ε}

and

Zβ = {Aβ} ∪ ({Zβε : ε ∈ H(β)

} ∩ [ω]ω).


Then|Zβ | < c = r, hence there is a setB ⊂ ω that reapsZβ , i.e., |B ∩ Z| = |Z \ B| = ω

for all Z ∈ Zβ . Let us then definefn,α by

fn,α ⊃ fn,β and fn,α(β) = 1 iff n ∈ B.

It is obvious thatI (α) andJ (α) will be again satisfied.Having completed the induction we simply put

fn =⋃

{fn,α : α ∈ c}for all n ∈ ω and observe that, from the inductive hypothesesJ (α), {fn: n ∈ Aβ} will bedense in 2c\β for all β ∈ c, hence the set{fn: n ∈ ω} is indeed a(c,→)-HFD. ✷

6.8 is taken from [7] where it was also noticed that ifS ⊂ 2κ is a countable(κ,→)-HFDthenS ∪ (2κ)0 is a separable non-compact but initially cf(κ)-compact space (here(2κ)0 isthe set of allf ∈ 2κ that are eventually equal to 0). Thus, in particular, if cf(c) > ω1 andthere is(c,→)-HFD then there is a separable initiallyω1-compact and non-compact 0-dimensionalT2-space. To get this conclusion, however, we do not need the full force of a(c,→)-HFD, but only that

(i) every infinite subset ofS has a limit point in(2c)0;(ii) S has a limit point outside of(2c)0.

Such anS ⊂ 2c one can actually get in ZFC.

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HFD and HFC type spaces, with applications

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