Cardinal Sequences and Combinatorial Principles Lajos Soukup

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Cardinal Sequences and Combinatorial Principles

Lajos Soukup

D. Sc. Thesis

Submitted to the Hungarian Academy of Sciences

Budapest, 2010

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Cardinal sequences

1.1. Introduction and Summary

Scattered spaces. Definitions and basic facts. A topological space is calledscatteredif its every non-empty subspace has an isolated point. Since

• the union of two scattered spaces is scattered, and

• the increasing union of scattered, open subspaces is scattered,

using Zorn lemma one can prove that every topological spaceX has a unique partition into two subspaces,

X =P∪S, (1.1)

such that S is an open, scattered subspace, and P is perfect, i.e., closed and dense-in-itself. (Of course,P orS can be empty.)

This observation explains why it is natural to investigate the structures of scattered spaces.

In the first chapter of the dissertation we study the structure of scattered spaces. To do so, we assign invariants to the scattered spaces, and we investigate the possible values of these invariants.

The Cantor-Bendixson Hierarchy. All topological spaces will be assumed to be infinite Hausdorff in this thesis.

Since a space is scattered iff it is right-separated, we have|X| ≤w(X)for scattered spaces.

Given a topological spaceX, for each ordinal numberα, theα-th derived setofX,X^(α), is defined as follows: X⁽⁰⁾ = X, X^(α+1) is the derived set of X^(α), i.e. the collection of all limit points of X^(α), and if αis limit thenX^(α)=∩ν<αX^(ν). Since X^(α)⊇X^(β) forα < β we have a minimal ordinal αsuch thatX^(α) =X^(α+1). This ordinalα, denoted byht(X), is called the Cantor-Bendixson height, or just theheightofX. Clearly the subspace X^(α)does not have any isolated points, it is dense-in-itself. The derived sets are all closed, soX^(α)is perfect. Moreover, Y =X\X^(α) is scattered and so it has cardinality ≤w(X). (So P =X^(α) and S =X \X^(α) in 1.1) This observation yields the Cantor-Bendixson theorem: every space of countable weight can be represented as the union of two disjoint subspaces, of which one is perfect and the other is countable. G. Cantor and I. Bendixson proved this fact independently in 1883 for subsets of the real line.

Historically, the investigation of scattered spaces was started by Cantor. He proved, in [32], that if the partial sums of a trigonometric series

a0/2 +

∞

X

n=1

(ancosnx+bnsinnx)

converge to zero except possibly on a set of points of finite scattered height, then all coefficients of the series must be zero.

Cardinal sequences of scattered spaces. We will assign invariants to scattered spaces.

Denote byI(Y)the isolated points of a topological space Y. For each ordinalα define the α-thCantor-Bendixson levelof a topological space X,Iα(X), as follows:

Iα(X) = I X\ ∪{Iβ(X) :β < α}

. ClearlyIα(X) =X^(α)\X^(α+1), and so ifX is scattered, then

ht(X) = min{α: Iα(X) =∅}. (1.2)

WriteI<λ(X) =S

α<λIα(X).

1

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Observe thatX is scattered iffX = I<ht(X)(X)iffX^(ht(X)) =∅.

Given a scattered spaceX, define thewidthofX, wd(X), as follows:

wd(X) = sup{|I_α(X)|:α <ht(X)}. (1.3) Thecardinal sequenceof a scattered space X, CS(X), is the sequence of the cardinalities of its Candor-Bendixson levels, i.e.

CS(X) =

|I_α(X)|:α <ht(X)

. (1.4)

We shall use the notation hκi_α to denote the constant κ-valued sequence of length α. As usual, the concatenation of a sequence f of lengthαand of a sequence g of lengthβ is denoted byf^_g. So the domain ofh=f^_g is α+β, h(ξ) =f(ξ)for ξ < α, and h(α+η) =g(η)for η < β.

By definition, ifCS(X) =hκi_αthenX has heightαand widthκ.

Cardinal sequences of Boolean algebras. A Boolean algebraBissuperatomic iff every homomorphic image ofBis atomic. We abbreviate "superatomic Boolean algebra" to "sBA".

The essential questions concerning sBA-s were asked by Tarski and Mostowski, [87], in 1939.

The main categories of results are: cardinal invariants, isomorphism types and automorphism groups. The major cardinal invariants of sBA-s were also introduced in [87] in the following way.

Definition 1.1.1.LetAbe a Boolean algebra.

(a) LetAt(A)be the atoms of A.

(b) IfJ is an ideal inA, then letJ^∗ be the ideal generated by J ∪ {x∈A:x/J ∈At(A/J)}.

(c) LetJ0(A) ={0A},Jα+1(A) =Jα(A)^∗, ifαis limit thenJα(A) =S

α⁰<αJα⁰(A).

(d) TheheightofA,ht(A), is the leastδwithJδ(A) =Jδ+1(A).

(e) wdα(A) =|At(A/Jα(A)|,

(f) Thecardinal sequenceofA,CS(A), is defined as follows:

CS(A) =hwdα(A) :α <ht(A)i. (1.5) By [68, Proposition 17.8],Ais superatomic iff there is an ordinalαsuch thatA=Jα(A).

ABoolean spaceis a compact, 0-dimensional, Hausdorff space. The Stone duality establishes a 1–1 correspondence between Boolean spaces and Boolean algebras.

Under Stone duality, homomorphic images of a Boolean algebraAcorrespond to closed subspaces of its Stone spaceS(A), and atoms ofAcorrespond to isolated points ofS(A). Moreover, ifB is a superatomic Boolean algebra, then the dual space of B^(α) is (S(B))^(α) (see [68, Con- struction 17.7]). So we have the following fact.

Fact* 1.1.2.A Boolean algebraB is superatomic iff its dual spaceS(B) is scattered. Moreover, ht(B) = ht(S(B)), andCS(B) = CS(S(B)).

Historically, the cardinal sequences of Boolean algebras were defined earlier: this notion was introduced by Day, [33], in 1967. The notion of cardinal sequences of topological spaces was introduced by LaGrange.

The beginning. By a classical result of S. Mazurkiewicz and J. Sierpiński, (see [86] or [68, Theorem 17.11]) a countable, superatomic Boolean algebra B is determined completely by its cardinal sequence!

Theorem* 1.1.3.If B is a countable, superatomic Boolean algebra, then CS(B) = hβi_ω^_hni, where β + 1 = ht(B), and n = |wdβ(X)| is a natural number. Moreover, in this case B is homeomorphic to Boolean algebra of the clopen subsets of the compact ordinal space ω^β·n+ 1, i.e. S(B) =ω^β·n+ 1.

What about uncountable sBA-s? An sBA is calledthin-talliff it has widthω and height at leastω1. The question of whether thin-tall sBA can exist was asked by Telgarsky in 1968. This question was the actual starting point of the modern investigation of sBA-s.

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1.1. INTRODUCTION AND SUMMARY 3

Cardinal sequences of LCS^∗spaces. As we remarked, the study of cardinal sequences was actually originated in the theory of superatomic Boolean algebras. Since superatomic Boolean algebras correspond to compact scattered spaces via Stone duality, the efforts were concentrated on the study of cardinal sequences of compact scattered spaces.

Reduction. We say that a spaceX is anLCS^∗-spaceiff it is locally compact, scattered, but not compact. ¹

A locally compact scattered space X is non-compact iff either ht(X) is a limit ordinal, or ht(X) =δ+ 1 and the top level ofX,Iδ(X), is infinite.

The height of a compact scattered spaceX is a successor ordinal,ht(X) =δ+ 1, andIδ(X) is finite. SoY =X\Iδ(X)is an LCS^∗space, moreoverCS(X) = CS(Y)^_hni, wheren=|Iδ(X)|.

On the other hand, ifY is an LCS^∗ space, andX is the disjoint union ofncopies of the one point compactificationαY ofY, thenX is compact andCS(X) = CS(Y)^_hni. So we have

{CS(X) :X is a compact space}={CS(Y)^_hni:Y is an LCS^∗ space,n∈ω}. (1.6) Hence, instead of compact, scattered spaces we can study the cardinal sequences of LCS^∗spaces.

So Telgarsky’s question can be reformulated as follows:

Telgarsky’s Question: Is there an LCS^∗ space with cardinal sequence hωi_ω

1? His question was answered by Rajagopalan.

Theorem* 1.1.4 (Rajagopalan, 1976, [91]).There is an LCS^∗ spaceX with CS(X) =hωi_ω

1. There are strong limitations on cardinal sequences of LCS^∗ spaces.

Fact* 1.1.5.The cardinality of a scatteredT3, in particular of an LCS^∗spaceXis at most2^|I(X)|, henceht(X)<(2^|^I(X)|)⁺. Hence ifs=hκα:α < δiis the cardinal sequence of a scattered regular space, then for eachα < β < δ we have κ_β≤2^κ^α and|δ\α| ≤2^s(α).

The next definition simplifies the formulation of certain results.

Definition 1.1.6.We let C(α) denote the class of all cardinal sequences of length α of LCS^∗- spaces. We also put, for any fixed infinite cardinalλ,

Cλ(α) ={s∈ C(α) :s(0) =λ∧ ∀β < α[s(β)≥λ]}. (1.7) In 1978, Juhász and Weiss improved the above mentioned result of Rajagopalan.

Theorem* 1.1.7 (Juhász, Weiss, [59]).hωi_α∈ C(α)for each α < ω2. By Fact 1.1.5, if CH holds thenhωi_ω

2∈ C(ω/ ₂). Juhász and Weiss, [59], asked if it is consistent thathωi_α∈ C(α). Just proved that the failure of CH is not enough to get this consistency.

Theorem* 1.1.8 (Just, [64]).In the Cohen model there is no LCS^∗-space X with CS(X) = hωi_ω

2.

In 1985, Baumgartner and Shelah gave an affirmative answer to the question of Juhász and Weiss.

Theorem* 1.1.9 (Baumgartner, Shelah, [25]).It is consistent that there is an LCS^∗-space X withCS(X) =hωi_ω

2.

Their proof was based on the construction of a∆-function.

Definition 1.1.10 ([25]).Let f : ω₂²

→ ω₂≤ω

be a function with f{α, β} ⊂ α∩β for {α, β} ∈

ω2

2

. (1) We say that two finite subsetsxandyofω2aregood forf provided that for α∈x∩y,β∈x\y andγ∈y\xwe always have

(a) α < β, γ=⇒α∈f{β, γ}, (b) α < β=⇒f{α, γ} ⊂f{β, γ},

(c) β < γ=⇒f{α, β} ⊂f{α, γ}.

1In the literature anLCS spaceis defined as a locally compact, scattered space.

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(2) The functionf is a∆-functionif every uncountable family of finite subsets ofω2contains two elementsxandywhich are are good forf.

(3) The function f is a strong ∆-functionif every uncountable family A of finite subsets of ω2

contains an uncountable subfamilyBsuch that any two setsxandy fromB are good forf. To get Theorem 1.1.9 actually the following statements were proved:

• It is consistent that there is a∆-function.

• If there is a∆-function, the in some c.c.c generic extension of the ground model there is an LCS^∗-spaceX withCS(X) =hωi_ω

2.

It is worth to mention that there is a strong∆-function inL.

Theorem* 1.1.11 (Velickovic).If _ω₁ holds then there is a strong ∆-function.

Two types of questions were considered in connection with cardinal sequences:

(1) Given a sequence s of cardinals of length α decide whether s ∈ C(α) in a certain/in some model of ZFC.

(2) CharacterizeC(α)orCλ(α)in certain models of ZFC.

There are many results concerning Type 1 problems in the literature. In Section 1.2 we also consider such a problem. By Fact 1.1.5,ht(X)<(2^|I(X)|)⁺ for any LCS^∗ spaceX. Especially, if I₀(X)is countable, thenht(X)<(2^ω)⁺. It is not hard to prove (see e.g. Theorem 1.2.3 below) that the estimate above is sharp for LCS^∗ spaces with countably many isolated points:

• for eachα <(2^ω)⁺ there is an LCS^∗ spaceX with ht(X) =αand|I0(X)|=ω.

Much less is known about LCS^∗ spaces withω₁isolated points, for example it is a long standing open problem whether there is, in ZFC, an LCS^∗space of heightω2and widthω1. In fact, as was noticed by Juhász in the mid eighties, even the much simpler question if there is a ZFC example of an LCS^∗space of heightω2with onlyω1isolated points, turned out to be surprisingly difficult.

In section 1.2 we prove the main result of [9], and in Theorem 1.2.1 we give an affirmative answer to the above question of Juhász: There is, in ZFC, an LCS^∗ space of heightω2 and having ω1

isolated points. Although it is a ZFC result, but the main ingredient of the proof, Theorem 1.2.2, is a construction under CH!

To prove this result we had to develop a new amalgamation method of construction of LCS^∗ spaces.

In Sections 1.3, 1.4, 1.5 and 1.7 we consider “Type 2” problems, i.e. we try to give full characterization of the elements of certain classesCλ(α).

For countable sequences LaGrange [79], for sequences of length ω₁ Juhász and Weiss, [61], give full characterization:

Theorem* 1.1.12 ([61]).hκξ :ξ < ω1i ∈ C(ω1)iff κη ≤κ^ω_ξ holds wheneverξ < η < ω1.

It follows that cardinal arithmetic alone decides whether a sequence of cardinals of lengthω₁ belongs toC(ω₁)or not. The situation changes dramatically for longer sequences, in fact already for sequences of length ω1+ 1. For example, the question if hωi_ω

1

_hω2i₁ ∈ C(ω1+ 1) is not decided by the following cardinal arithmetic: 2^ω =ω₂ and 2^κ =κ⁺ for all κ > ω (see Just[64]

and Roitman[93]).

However, as we showed in [14], the elements ofC(α)can be characterized for allα < ω₂if we assume GCH. Section 1.3 contains this result. In order to characterize those sequences of length

< ω₂ which are cardinal sequences of LCS spaces, it suffices to characterize the classesC_λ(α)for any ordinal α < ω₂ and any infinite cardinalλ. In fact, this follows from the general reduction Theorem 1.3.2 that is valid in ZFC.

The promised GCH characterization of the classes Cλ(α) is given in Theorem 1.3.4. The main ingredient of the proof is Theorem 1.3.5 which is generalization of the main construction of Theorem 1.2.2.

In Section 1.4 we consider the classes Cλ(α) for α ≥ ω2 under GCH. In 1.4.1 we define a familyDλ(α)of sequences of cardinals. Using elementary topological considerations, this family is a natural “upper bound” ofCλ(α), i.e. GCH implies thatCλ(α)⊆ Dλ(α).

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We conjecture that GCH implies thatCλ(α) =Dλ(α), but in [15] we could prove just a weaker statement. Namely, as we prove it in Section 1.4, for each regular cardinalλit is consistent that GCH holds andCλ(α) =Dλ(α), see Theorem 1.4.3.

Using this result, in Theorem 1.4.4 we characterize those sequences of regular cardinals which can be obtained as a cardinal sequence of an LCS^∗ space in some model of ZFC + GCH.

To prove our theorems we should introduce the notion of universal spaces in 1.4.2 which is the main tool of some proofs in the forthcoming sections.

In Sections 1.5 and 1.7 we investigate the classC_ω(ω₂).

Baumgartner and Shelah proves that it is consistent thathωi_ω

2 ∈ Cω(ω2). Building on their method, Bagaria, [21], proved that Cω(ω2) ⊇ {s ∈ ^ω²{ω, ω1} : s(0) = ω} is also consistent.

However, he usedM A_ℵ₂ in his argument, andM A_ℵ₂ implies2^ω⁰≥ω3, and if2^ω⁰=ωα, then the natural “upper bound” ofCω(ω2)is a much larger family of sequences:

Cω(ω2)⊆ {s∈^ω²{ων:ν≤α}:s(0) =ω}. (1.8) These results naturally raised the questions whether we may have equality in (1.8). In Theorem 1.5.10 we answer in the positive: it is consistent that2^ω=ω₂ and

Cω(ω2) ={s∈^ω²{ω, ω1, ω2}:s(0) =ω}.

In Section 1.7 we improve the above mention result: we prove Theorem 1.6.1 which claims that it is consistent that2^ωis as large as you wish and every sequences=hsα:α < ω2iof infinite cardinals withsα≤2^ω is the cardinal sequence of some LCS space.

For a long timeω2 was a mystique barrier in both height and width. In that section we can construct wider spaces. However we should pay a price: by forcing we should introduce a morass type structure which helps us to obtain a very strong ∆-like functions in a generic extension which makes possible to force our desired space.

In Section 1.6 we prove certain stepping up theorems.

In a classical theorem, Roitman proved that it is consistent thathωi_ω

1

_hω₂i ∈ C_ω(ω₁+ 1).

Combining Koszmider’s strongly functions and Martinez’s orbit methods in Section 1.8 we improve Roitman’s result. We show e.g. that for eachβ < ω3 withcf(β) =ω2it is consistent that 2^ω¹ is as large as you wish andhω1i_β^_h2^ω¹i ∈ Cω₁(β+ 1).

Since the classes of the regular, of the zero-dimensional, and of the locally compact scattered spaces are different, it was a natural question what is the relationship between their cardinal sequences. Actually, Juhász raised the question whether there is a regular space with cardinal sequencehωi₂ω in ZFC. In [25, Theorem 10.1] the authors answered his question positively. for eachα <(2^ω)⁺ there is a regular (even zero-dimensional) scattered space with cardinal sequence hωi_α.

In Section 1.9 we succeeded in giving a complete characterization of the cardinal sequences of bothT₃ and zero-dimensional T₂ scattered spaces. Although the classes of the regular and of the zero-dimensional scattered spaces are different, it will turn out that they yield the same class of cardinal sequences.

E. van Douwen and, independently, A. Dow [34] have observed that under CH an initially ω₁-compact T₃ space of countable tightness is compact. (A space X is initially κ-compact if any open cover of X of size ≤ κhas a finite subcover, or equivalently any subset of X of size

≤κ has a complete accumulation point). Naturally, the question arose whether CH is needed here, i.e. whether the same is provable just in ZFC. The question became even more intriguing when in [23] , D. Fremlin and P. Nyikos proved the same result from PFA. Quite recently, A.

V. Arhangel’ski˘ıhas devoted the paper [20] to this problem, in which he has raised many related problems as well.

In [90] M. Rabus has answered the question of van Douwen and Dow in the negative. He constructed by forcing a Boolean algebra B such that the Stone spaceSt(B)includes a counterexam- pleX of sizeω2to the van Douwen–Dow question, in factSt(B)is the one point compactification ofX, henceX is also locally compact.

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In Sections 1.10, we give an alternative forcing construction of counterexamples to the van Douwen–Dow question, We directly force a topologyτ onω2 such that in the generic extension V^P out space X =hω2, τiis normal, locally compact, 0-dimensional, Frechet-Urysohn, initially ω₁-compact and non-compact space X of sizeω₂ having the following property: for every open (or closed) setAin X we have |A| ≤ω₁or |X\A| ≤ω₁.

In [20, problem 12] Arhangel’skii asks if it is provable in ZFC that a normal, first countable initiallyω₁-compact space is necessarily compact. We could not completely answer this question, but we show that the Frechet-Urysohn property (which is sort of half-way between countable tightness and first countability) in not enough to get compactness.

To achieve that we constructed a further generic extension of the model V^P in which X becomes Frechet-Urysohn but its other properties are preserved, for example,X remains initially ω1-compact and normal.

Arhangel’skii raised the question, [20, problem 3], whether CH can be weakened to 2^ω <

2^ω¹ in the theorem of van Douwen and Dow? We shall answer this question in the negative:

theorem 1.10.34 implies that the existence of a counterexample to the van Douwen–Dow question is consistent with practically any cardinal arithmetic that violates CH.

Improving the results of section 1.10, in Section 1.11 we answer a question of Arhangel’skii,[20, problem 12]: we force a first countable, normal, locally compact, initiallyω1-compact and non- compact spaceX=hω2×2^ω, τi.

Actually, Alan Dow conjectured that applying the method of [74] (that "turns" a compact space into a first countable one) to the space of Rabus in [90] yields an ω₁-compact but non- compact first countable space. How one can carry out such a construction was outlined by the second author in the preprint [71]. However, [71] only sketches some arguments as the language adopted there, which follows that of [90], does not seem to allow direct combinatorial control over the space which is forced. This explains why the second author hesitated to publish [71].

One missing element of [71] was a language similar to that of [11] which allows working with the points of the forced space in a direct combinatorial way. In section 1.11 we combine the approach of [11] with the ideas of [71] to obtain directly anω1-compact but non-compact first countable space. Consequently, our proofs follow much more closely the arguments of [11] than those of [90] or their analogues in [71].

As before, we again use a∆-function to make our forcing CCC but we need both CH and a

∆-function with some extra properties to obtain first countability.

It is immediate from the countable compactness ofX that its one-point compactificationX^∗ is not first countable. In fact, one can show that the character of the point at infinity∗in X^∗ is ω₂. AsX is initiallyω₁-compact, this means that every (transfinite) sequence converging fromX to∗must be of type cofinal withω₂. SinceX is first countable, this trivially implies that there is no non-trivial converging sequence of typeω₁in X^∗. In other words: the convergence spectrum of the compactumX^∗omitsω₁. As far as we know, this is the first and only (consistent) example of this sort.

Combinatorial principles. In the second chapter of the dissertation we consider combinatorial principles.

The last 40 years have seen a furious activity in proving results that are independent of the usual axioms of set theory, that is ZFC. As the methods of these independence proofs (e.g.

forcing or the fine structure theory of the constructible universe) are often rather sophisticated, while the results themselves are usually of interest to “ordinary” mathematicians (e.g. topologists or analysts), it has been natural to try to isolate a relatively small number of principles, i.e.

independent statements that a) aresimpleto formulate and b) areuseful in the sense that they have many interesting consequences. Most of these statements, we think by necessity, are of combinatorial nature, hence they have been called combinatorial principles. The best known example is the Martin’s Axiom: to prove the consistency of this axiom you need to know the iterated forcing, but to apply this axiom in algebra, topology or combinatorics it is enough to know elementary set-theory.

The above formulated criteria a) and b) as to what constitutes a combinatorial principle are often contrary to each other: for more usefulness one often has to sacrifice some simplicity. It is

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not clear whether an ideal balance exists between them. It is up to the reader to judge if we have come close to this balance.

As we mentioned, Just proved that in the Cohen model neither hωi_ω

2 nor hωi_ω

1

_hω2i is a cardinal sequence of an LCS^∗ space. The two proofs are based on similar arguments. Using the same ideas we could prove other statements in the Cohen model, so it was a natural idea to investigate if we can combine these arguments into a “combinatorial principle”. That was the starting point of the investigation of combinatorial principles in the Cohen model. As we will see, we could find principles which are (a) true in the Cohen model, and (b) which implies the above mentioned results of Just.

In section 2.1 we present several new combinatorial principles that are all statements about P(ω), the power set of the natural numbers. In fact, they all concern matrices of the form hA(α, n) :hα, ni ∈κ×ωi, where A(α, n) ⊂ ω for each hα, ni ∈ κ×ω, and, in the interesting cases,κis a regular cardinal withc= 2^ω≥κ > ω1.

We show that these statements are valid in the generic extensions obtained by adding any number of Cohen reals to any ground modelV, assuming that the parameter κis a regular and ω-inaccessible cardinal inV ( i.e. λ < κimpliesλ^ω< κ).

Then we present a large number of consequences of these principles, some of them combinatorial but most of them topological, mainly concerning separable and/or countably tight topological spaces. (This, of course, is not surprising because these are objects whose structure depends basically onP(ω).)

Recently we could find an application of our principles in the theory of Banach spaces, [1].

Using a much finer analysis of the Cohen model, in Section 2.2 we recall the main result of [10]: in the Cohen model an LCS^∗may have at mostω₁many countable Cantor-Bendixson level.

We conjecture that in ZFC an LCS^∗ may have at mostω2 many countable Cantor-Bendicty level. In his celebrated theorem, Shelah proves that max pcf{ℵn : n < ω} < ℵω₄. One of the central question of set theory is to improve this result. If this conjecture is true, then max pcf{ℵn:n < ω}<ℵω₃.

I jointed to the investigation of the weak Freeze-Nation (wFN) property independently from the research of the combinatorial principles. However, at some point we realized that the wFN property of the posethP,⊂iitself has numerous interesting combinatorial and topological conse- quence, so it can be considered as a combinatorial principle. This observation explains why we discuss the wFN property of certain posets in 3 sections.

Definition 1.1.13.A poset P =hP,≤i has the weak Freese-Nation (wFN)property iff there is a functionf :P →

Pω

such that

• ifp,q∈P andp≤qthen there isr∈f(p)∩f(q)withp≤r≤q.

Let Q⊆ P. Write Q≤σ P if for each p ∈P the set {q ∈Q : q ≤p} has a countable cofinal subset, and the set{q∈Q:q≥p}has a countable coinitial subset.

These notions were introduced by Heindorf and Shapiro [51].

By [46], the posetP has the wFN property iff the family{Q∈ Pω1

:Q≤σ P} contains a club subfamily. In [7], we prove that a weaker assumption on the class {Q∈

Pω1

:Q≤σP}is enough to derive the wFN property ofP. This result is included in section 2.3.

We introduce a very weak box principle^∗∗∗µ , and prove Theorem 2.3.14: assume thatλis a cardinal with

(i) cf([µ]^ω,⊆) =µifω₁< µ < λ andcf(µ)≥ω₁, (ii) ^∗∗∗µ holds for eachµ < λwith cofinalityω.

Then for each posetP of cardinality≤λthe following are equivalent:

(w1) P has the wFN property,

(w2) for each large enough regular cardinalχ, ifM ≺ H(χ),P,ω1∈M, andM is the union of anω1 chain of countable elementary submodels ofH(χ)thenP∩M ≤σP.

We proved that the assumption of^∗∗∗ can not be omitted: if GCH holds and the Chang conjecture (ℵω+1,ℵω) (ℵ1,ℵ0). holds then the poset ([ℵω]^ℵ⁰,⊆) does not have the wFN property.

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Let us remark that if 2^ω = ω1 then (w2) holds for ([ℵω]^ℵ⁰,⊆). So GCH is not enough to prove the equivalence of (w1) and (w2)

In section 2.4 we solve some questions concerning wFN properties of Boolean algebras. While in section 2.3 we proved thathP(ω),⊂ihas the wFN propertyif we add arbitrary number of Cohen reals to a “nice‘” model of ZFC, (e.g. to, L), in Section 2.5 we proved that L can not be replaced by an arbitrary model of ZFC which satisfies GCH.

In Section 2.5 we show that the assumptionhP(ω),⊂ihas the wFN propertycan be considered as a useful combinatorial principle. Indeed, we proved, that ifhP(ω),⊂ihas the wFN property, then

(a) there is noℵ2-Luzin gap, andω2is not embeddable intohP(ω),⊆^∗i;

(b) hωi_ω

1

_hω2i∈ C(ω/ 1+ 1)andhωi_ω

2 ∈ C(ω/ 2); (c) non(M) =ω1 andcov(M)> ω1;

(d) a=ω1,

(e) anyω1-fold cover of the real real by closed sets can be partitioned intoω1 disjoint subcover.

The last application is from [2].

So far we investigated principles which hold in the Cohen model. However, there are other principles which can not hold in the Cohen model, e.g. ♣or •|.

Our starting point in the last section was to find a generic extension which allows to blow up the continuum without collapsing cardinals in such a way that •| holds in the generic extension.

To handle this problem we developed a new type of product of posets.

As it turned out, this new product is suitable to obtain models in which seemingly inconsistent statements hold. The basic idea is to add Cohen reals in such a way that in the generic extension there is no generic filter for the posetF n(ω1,2;ω).

We formulate just our most interesting result. As usual, M A(countable)denotes the statement that Martin’s Axiom holds for countable posets. So we proved that it is consistent that you can obtain a model without collapsing cardinals in such a way that in the generic extension2^ωis as large as you wish, and bothM A(countable)and •| hold.

1.2. A tall space with small bottom (This section is based on[9]) By theorems 1.1.7, 1.1.8 and 1.1.9,

• for eachα < ω2 there is an LCS^∗ spaceX withCS(X) =hωi_α,

• ZFC+¬CH does not decide if there is an LCS^∗ spaceX withCS(X) =hωi_ω

2.

The estimate ht(X)<(2^|I(X)|)⁺ is sharp for LCS^∗ spaces with countably many isolated points by theorem 1.2.3 below:

• for eachα <(2^ω)⁺ there is an LCS^∗ spaceX with ht(X) =αand|I₀(X)|=ω.

Much less is known about LCS^∗ spaces withω₁isolated points, for example it is a long standing open problem whether there is, in ZFC, an LCS^∗space of heightω₂and widthω₁. In fact, as was noticed by Juhász in the mid eighties, even the much simpler question if there is a ZFC example of an LCS^∗space of heightω₂with onlyω₁isolated points, turned out to be surprisingly difficult.

On the other hand, Martínez in [84, theorem 1] proved that

• it is consistent that for eachα < ω₃ there is a LCS^∗ spaceX withCS(X) =hω₁i_α. As the main result of [9], we gave an affirmative answer to the above question of Juhász:

Theorem 1.2.1.There is, in ZFC, an LCS^∗ space of height ω2 and havingω1 isolated points.

The main ingredient of the proof of Theorem 1.2.1 is the following result.

Theorem 1.2.2.If κ^<κ =κthen there is an LCS^∗ spaceX of heightκ⁺ with|I₀(X)|=κ.

In particular, if2^ω=ω1 then the above result yields an LCS^∗ spaceX withht(X) =ω2and

|I0(X)|=ω1. That such a space also exist under¬CH, hence in ZFC, follows from the following result.

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1.2. A TALL SPACE WITH SMALL BOTTOM 9

Theorem 1.2.3.For each α <(2^ω)⁺ there is a locally compact, scattered spaceXα with|Xα|=

|α|+ω,ht(Xα) =αand|I0(Xα)|=ω.

Proof. We do induction on α. If α=β+ 1then we letXα be the 1-point compactification of the disjoint topological sum of countably many copies ofXβ.

If α is limit then we first fix an almost disjoint family {A_β : β < α} ⊂ ω^ω

, which is possible by|α| ≤2^ω. Applying the inductive hypothesis, for eachβ < αwe can also fix a locally compact scattered space Xβ of cardinality ω +|β| and height β such that I0(Xβ) = Aβ and Xβ ∩Xγ = Aβ∩Aγ for {β, γ} ∈

α2

. Now amalgamate the spaces Xβ as follows: consider the topological space X = h∪β<αX_β, τi where τ is the topology generated by∪β<ατ_X_β. Since A_β∩A_γ is a finite and open subspace of both X_β and X_γ it follows that each X_β is an open subspace of X. Consequently, X is LCS^∗ with countably many isolated points, and ht(X) =

sup_β<αhtX_β=α.

Proof of Theorem 1.2.1 using Theorem 1.2.2 . If 2^ω =ω1, then theorem 1.2.2 gives such a space.

If2^ω> ω1then(2^ω)⁺≥ω3and so according to theorem 1.2.3 for eachα < ω3there is locally compact, scattered space of heightαand with countably many isolated points.

The rest of this section is devoted to the proof of Theorem 1.2.2.

Definition 1.2.4.Given a family of setsA we define the topological space X(A) =hA, τ_Ai as follows: τ_A is the coarsest topology in which the sets U_A(A) = A ∩ P(A) are clopen for each A∈ A, in other words: {U_A(A),A \U_A(A) :A∈ A}is a subbase forτ_A.

We shall writeU(A)instead ofU_A(A)ifAis clear from the context.

Clearly X(A) is a 0-dimensional T₂-space. A family A is called well-founded iff hA,⊂i is well-founded. In this case we can define the rank-functionrk :A −→Onas usual:

rk(A) = sup{rk(B) + 1 :B∈ A ∧ B(A}, and writeR_α(A) ={A∈ A: rk(A) =α}.

The familyAis said to be ∩-closed iffA∩B∈ A ∪ {∅}wheneverA, B∈ A.

It is easy to see that ifAis∩-closed, then a neighbourhood base inX(A)ofA∈ Ais formed by the sets

W(A;B1, . . . , Bn) = U(A)\

n

[

1

U(Bi),

wheren∈ω andBi(A fori= 1, . . . , n. ( For n= 0we haveW(A) = U(A).)

The following simple result enables us to obtain LCS^∗ spaces from certain families of sets.

Let us point out, however, that not every LCS^∗ space is obtainable in this manner, but we do not dwell upon this because we will not need it.

Lemma 1.2.5 ([9]).Assume that A is both∩-closed and well-founded. Then X(A) is an LCS^∗

space.

To simplify notation, ifX(A)is scattered then we writeI_α(A) = I_α(X(A)).

Clearly each minimal element ofAis isolated inX(A); more generally we have α≤rk(A)if A∈Iα(A), as is shown by an easy induction onrk(A).

Example 1.2.6.Assume that hT,≺i is a well-ordering, tphT,≺i =α, and let A be the family of all initial segments of hT,≺i, i. e. A= {T} ∪ {Tx : x∈ T}, where T_x ={t ∈T : t ≺ x}.

ThenAis well-founded,∩-closed and it is easy to see thatX(A)∼=α+ 1, i.e. the spaceX(A)is homeomorphic to the space of ordinals up to and includingα.

Example 1.2.6 above shows that, in general, Rα(A) and Iα(A) may differ even for α = 0.

Indeed, ifxis the successor ofyinhT,≺ithenTxis isolated inX(A)because{Tx}=W(Tx;Ty) = UA(Tx)\UA(Ty)is open, butrk(Tx) = tp(Tx)>0. However, for a wide class of families, the two kinds of levels do agree. Let us call a well-founded familyArk-goodiff the following condition is satisfied:

∀A∈ A ∀α <rk(A)|{A⁰∈ A:A⁰ ⊂A∧rk(A⁰) =α}| ≥ω.

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Then we have the following result.

Lemma 1.2.7 ([9]).If A is a well-founded, ∩-closed and rk-good family, then Iα(A) = Rα(A)

for eachα.

Example 1.2.8.For a fixed cardinalκand any ordinal γ < κ⁺ we define the familyE^γ ⊂ P(κ^γ) as follows:

E^γ=n

κ^1+α·ξ, κ^1+α·(ξ+ 1)

: α≤γ, κ^1+α·(ξ+ 1)≤κ^1+γo . Of course, throughout this definition exponentiation means ordinal exponentiation.

E^γ is clearly well-founded, ∩-closed, moreoverrk

κ^1+α·ξ, κ^1+α·(ξ+ 1)

=α, henceE^γ is also rk-good. Consequently X(E^γ) is an LCS space of height γ+ 1 in which the α^th level is n

κ^1+α·ξ, κ^1+α·(ξ+ 1)

:κ^1+α·(ξ+ 1)≤κ^1+γo

, i. e. all levels except the top one are of size κ.

To get an LCS^∗ space of height κ⁺ with “few” isolated points, our plan is to amalgamate the spaces {X(E^γ) :γ < κ⁺} into one LCS^∗ space X in such a way that|I₀(X)| ≤ κ^<κ. The following definition describes a situation in which such an amalgamation can be done.

Definition 1.2.9.A system of families {Ai : i ∈ I} is called coherent iff A∩B ∈ Ai∪ {∅}

whenever{i, j} ∈ I2

, A∈ Ai andB∈ Aj.

To simplify notation, we introduce the following convention. Whenever the system of families {Ai :i∈I} is given, we will write Ui(A)forU_A_i(A), andτi for τ_A_i. If the family Ais defined then we will writeU(A)forU_A(A), and τ forτ_A.

Lemma 1.2.10 ([9]. ).Assume that {Ai :i∈I} is a coherent system of well-founded, ∩-closed families andA=∪{Ai :i∈I}. Then for each i∈I and A∈ Ai we haveU(A) = U_i(A), A is also well-founded and ∩-closed, moreover τ_i|` U(A) =τ|` U(A). Consequently each X(Ai) is an open subspace of X(A)and thus{X(A_i) :i∈I} forms an open cover ofX(A).

Given a system of families {Ai : i ∈ I} we would like to construct a coherent system of families {cAi :i∈I} such that Ai andAci are isomorphic for alli∈I. A sufficient condition for when this can be done will be given in lemma 1.2.12 below.

First, however, we need a definition. While reading it, one should remember that an ordinal is identified with the family of its proper initial segments.

Definition 1.2.11.Given a limit ordinal ρ and a family A with ρ ⊂ A ⊂ P(ρ), let us define the familyAbas follows. Consider first the functionk_A onρdetermined by the formula k_A(η) = U_A(η+ 1)forη∈ρand put

Ab={k⁰⁰_AA:A∈ A}.

Since ρ ⊂ A, for each η ∈ ρ we clearly have ∪U_A(η) = η and so k_A(η) = U_A(η+ 1) 6=

U_A(ξ+ 1) = k_A(ξ) whenever {η, ξ} ∈ ρ2

. Consequently, k_A is a bijection that yields an isomorphism betweenAandAb(and so the spacesX(A)andX(A)b are homeomorphic).

If the system of families{Ai:i∈I}is given, then we writeki forkA_i for each i∈I.

IfA ⊂ P(ρ)andξ≤ρthen we let

A |`ξ={A∩ξ:A∈ A}.

ForA₀6=A₁⊂ P(ρ)we let

∆(A0,A1) = min{δ:A0|`δ6=A1|`δ}.

Clearly we always have∆(A0,A1)≤ρ. If, in addition,ρ+ 1⊂ A0∩ A1, moreover both A0

andA1 are∩-closed then we also have

∆(A0,A1) = min{δ: U0(δ)6= U1(δ)}, because thenAi|`δ= Ui(δ)wheneveri∈2andδ≤ρ.

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1.2. A TALL SPACE WITH SMALL BOTTOM 11

Lemma 1.2.12 ([9]).Assume that κis a cardinal, {Ai :i∈I} ⊂ PP(κ)are ∩-closed families, κ+ 1^. ⊂ Ai for eachi ∈I, and ∆(Ai,Aj) is a successor ordinal whenever {i, j} ∈

I2

. Then

the system {cAi:i∈I}is coherent.

More is needed still if we want the "amalgamated" family to provide us a space with a small bottom, i.e. having not too many isolated points. This will be made clear by the following lemma.

Lemma 1.2.13.Letκbe a cardinal and{Ai:i∈I} ⊂ PP(κ) be a system of families such that (i) κ+ 1^. ⊂ A_i andA_i is well-founded and∩-closed for eachi∈I,

(ii) ∆(Ai,Aj)is a successor ordinal for each {i, j} ∈ I²

. Then

(a) the system{cAi:i∈I}is coherent and thus A=S

{Aci:i∈I} is well-founded,∩-closed and X(A)is covered by its open subspaces {X(cAi) :i∈I}.

If, in addition, we also have (iii) I₀(A_i)⊂

κ^<κ

for each i∈I, and

(iv) |Ui(ξ)|< κ for eachi∈I andξ∈κ, then

(b) I0(A)⊂ h

[κ]^<κi<κ^<κ .

Proof of lemma 1.2.13. The system {cA_i :i∈I} is coherent by lemma 1.2.12, thus (a) holds by lemma 1.2.10.

Consequently we have

I0(A) =[

{I0(cAi) :i∈I}.

Now ifA∈I0(Ai)andη∈κthen|A|< κ by (iii) andUi(η)∈h

κ<κi^<κ

by (iv), hence k⁰⁰_i A={Ui(η+ 1) :η∈A} ∈

h

κ<κi^<κ<κ

.

This, byI0(cAi) ={k⁰⁰_i A:A∈I0(Ai)}, proves (b).

Definition 1.2.14.Ifρis an ordinal andA ⊂ P(ρ)let us put

A^∗={A∩ξ:A∈ A ∧ξ≤ρ}=A ∪ {A∩ξ:A∈ A ∧ξ < ρ}.

Definition 1.2.15.A familyAis calledchain-closedif for each non-emptyB ⊂ AifBis ordered by⊂(i.e. if Bis a chain) then∪B ∈ A.

Lemma 1.2.16 ([9]).If ρ is an ordinal and A0,A1⊂ P(ρ) are chain-closed,∩-closed and well- founded families such that A0∗6=A1∗ then∆(A0∗,A1∗)is a successor ordinal.

The last result shows us that the operation * is useful because its application yields us families that satisfy condition (ii) of lemma 2.10. On the other hand, the following result tells us that the LCS spaces associated with certain families modified by * do not differ significantly from the spaces given by the original families, moreover they also satisfy condition (iii) of lemma 2.10.

Lemma 1.2.17.Let κbe a cardinal andA ⊂ κ^κ

be well-founded and∩-closed. Then so is A^∗, moreover

(a) X(A)is a closed subspace ofX(A^∗), (b) I0(A)⊆Iκ(A^∗).

(c) ht(A^∗)≥κ+ ht(A),^. (d) I₀(A^∗)⊂

κ<κ

.

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Proof of lemma 1.2.17. We shall writeU(A)forUA(A), and U∗(A)forUA^∗(A).

First observe that because

U_∗(A)∩ A=

U(A) ifA∈ A,

∅ ifA∈ A^∗\ A, X(A)is a closed subspace ofX(A^∗), hence (a) holds.

Now letA∈I0(A). Then there areB1, . . . , Bn∈U(A)\ {A} such that {A}= W(A;B1, . . . , Bn) = U(A)\

n

[

i=1

U(Bi).

Since hereBi(Aand|A|=κ, we can fixη∈Asuch that(A∩η)6⊂Bi for everyi= 1, . . . , n.

Now consider the basic neighbourhood

Z= W_∗(A;A∩η, B₁, . . . , B_n) = U_∗(A)\U_∗(A∩η)\

n

[

i=1

U_∗(B_i)

of A in X(A^∗). We claim that Z = {A∩ξ : η < ξ ≤ κ}. The inclusion ⊃ is clear from the choice of η. On the other hand, if C∩ξ ∈ Z with C ∈ A and ξ ≤ κ, then C∩ξ ⊂A hence C∩ξ = A∩C∩ξ, so as A is ∩-closed we can assume that C ⊆ A. If we had C 6= A then {A}= W(A;B1, . . . , Bn)would implyC⊂Bifor somei, henceC∩ξ∈U_∗(Bi)and soC∩ξ /∈ Z, a contradiction, thus we must have C =A. Moreover, sinceU_∗(A∩η)⊃ {A∩ν : ν ≤ η}, we must also haveξ > η.

By example 1.2.6 we haveX(Z)∼=κ+ 1. Moreover, the topologies τ_Z andτ_A^∗|`Z coincide because the above argument also shows that for eachC∈ Aandζ≤κwe have

U_∗(C∩ζ)∩ Z= U_∗(A∩C∩ζ)∩ Z=

U_Z(A∩ζ) ifA⊂C andζ > η;

∅ otherwise.

HenceX(Z)∼=κ+ 1^. is a clopen subspace ofX(A^∗)and so{A}= I_κ(Z) = Iκ(A^∗)∩ Z, what proves (b).

(c) follows immediately from (a) and (b).

Finally, I0(A^∗) ⊂ I<κ(A^∗) ⊂ (A^∗\ A) ⊂ κ<κ

, as follows immediately from (b), proving

(d).

Before we could apply the amalgamation result to the familiesE^γ, however, we need some further preparation that will be useful in ensuring the fulfillment of condition (iv) in 1.2.13.

Definition 1.2.18.A family Ais calledtree-likeiff A∩A⁰ 6=∅ implies that A⊂A⁰ orA⁰ ⊂A, wheneverA, A⁰∈ A.

It is easy to see that the familiesE^γ given in example 1.2.8 are both tree-like and chain-closed.

Also, tree-like families are clearly∩-closed.

Lemma 1.2.19([9]).Ifδ is an ordinal andA ⊂ P(δ)is tree-like, well-founded and chain-closed

then so isA |`ξfor each ξ≤δ.

Definition 1.2.20.Given a familyA ⊂ P(δ)andα, β∈δ let us put S^A(α, β) =∪{A∈ A:α∈Aandβ /∈A}.

Lemma 1.2.21.Assume that δ is an ordinal and A ⊂ P(δ) is a tree-like, well-founded and chain-closed family withδ∈ A. Then

A \ {∅}={δ} ∪ {S^A(α, β) :α, β∈δ} \ {∅}.

Consequently,|A| ≤ |δ|².

Proof of lemma 1.2.21. Given α, β ∈δ, the familyS ={A∈ A :α ∈A, β /∈ A} is ordered by ⊂ because A is tree-like. Thus either S =∅ and so S^A(α, β) =∪S =∅, or if S 6= ∅ then S^A(α, β) =∪S ∈ A, forAis chain-closed.

Assume now that A ∈ A \ {∅, δ} and let D = {D ∈ A : A (D}. Clearly δ ∈ D. Since A is tree-like, D is ordered by ⊂ , so it has a ⊂-least element, say D, because hA,⊂i is also

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1.3. CARDINAL SEQUENCES OF LENGTH < ω2 UNDER GCH 13

well-founded. Pickβ∈D\Aand letα∈A. We claim thatA= S^A(α, β). ClearlyA⊂S^A(α, β) becauseα∈Aandβ /∈A. On the other hand, ifA⁰∈ A, α∈A⁰ andβ /∈A⁰ then eitherA⁰⊂A or A ⊂A⁰ because A is tree-like. Butβ /∈ A⁰ implies that A⁰ ∈ D, i.e./ A( A⁰ can not hold.

ThusA⁰⊂Aand so S^A(α, β) =A is proved.

Now we are ready to collect the fruits of all the preparatory work.

Proof of theorem 1.2.2. For each γ < κ⁺ consider the well-founded,∩-closed,rk-good family E^γ constructed in example 1.2.8:

E^γ =n

κ^1+α·ξ, κ^1+α·(ξ+ 1)

: α≤γ, κ^1+α·ξ < κ^γo .

Fix a bijection fγ : κ^γ −→ κ, and letFγ ={fγ00E : E ∈ E^γ}, i.e. Fγ is simply an isomorphic copy ofE^γ on the underlying setκ. AsE^γ is also chain-closed and tree-like, hence so isFγ.

We shall now show that the∗-modified families{Fγ∗:γ < κ⁺}satisfy conditions (i)-(iv) of lemma 1.2.13. Sinceκ∈ Fγ it follows thatκ+ 1^. ⊂ Fγ∗

and so (i) is true. For{γ, δ} ∈ κ⁺2

, the height ofX(E^γ)isγ+ 1and the height of X(E^δ)isδ+ 1, hence E^γ andE^δ are not isomorphic.

ThusFγ 6=Fδ and soFγ∗ 6=Fδ∗ as well because Fγ =Fγ∗∩ κκ

andFδ =Fδ∗∩ κκ

. Hence

∆(Fγ∗,Fδ∗)is a successor ordinal by lemma 1.2.16, i.e. (ii) is satisfied.

(iii) holds by 1.2.17.(d.)

To show (iv), let us fixξ < κ. ThenUF_γ^∗(ξ) =Fγ∗|`ξ=∪{Fγ|`ζ:ζ ≤ξ} where|Fγ|`ζ| ≤

|ζ|² for allζ≤ξby lemmas 1.2.19 and 1.2.21, consequently|Fγ∗

|

`ξ| ≤ |ξ|³< κ.

Thus we may apply lemma 1.2.13 to the family F =∪{Fdγ∗ : γ < κ⁺} and conclude that the space X = X(F) is LCS, moreover |I₀(X)| ≤ (κ^<κ)^<κ^<κ

=κ. Since for every γ ∈ κ⁺ the space X(F_γ^∗)is an open subspace of X, we have ht(X) ≥ ht(X(F_γ^∗)) > γ, consequently

ht(X)≥κ⁺.

1.3. Cardinal sequences of length < ω₂ under GCH (This section is based on[14])

The cardinal sequences of LCS^∗spaces with heightω1 has the following characterization in ZFC:

Theorem 1.3.1 (Juhasz, Weiss, [61]).hκξ : ξ < ω₁i ∈ C(ω1) iff κ_η ≤ κ^ω_ξ holds whenever ξ < η < ω1.

It follows that cardinal arithmetic alone decides whether a sequence of cardinals of lengthω1

belongs toC(ω1)or not. The situation changes dramatically for longer sequences, in fact already for sequences of length ω1+ 1. For example, the question if hωi_ω

1

_ hω2i₁ ∈ C(ω1+ 1) is not decided by the following cardinal arithmetic: 2^ω =ω₂ and 2^κ =κ⁺ for all κ > ω (see Just[64]

and Roitman[93]).

However, as we showed in [14], the elements of C(α)can be characterized for all α < ω2 if we assume GCH. This section contains that result.

In order to characterize those sequences of length< ω2which are cardinal sequences of LCS^∗ spaces, it suffices to characterize the classesCλ(α)for any ordinalα < ω2and any infinite cardinal λ. In fact, this follows from the following general reduction theorem that is valid in ZFC.

Theorem 1.3.2([14]).For any ordinalαand any sequencef of cardinals of lengthαthe following are equivalent:

(1) f ∈ C(α)

(2) for some natural number n there is a decreasing sequence λ0 > λ1>· · · > λn−1 of infinite cardinals and there are ordinals α₀, . . . α_n−1 such that α=α₀+· · ·+α_n−1 and f =f₀ ^_ f₁^_· · · ^_f_n−1 with f_i∈ Cλi(α_i)for each i < n.

We prove Theorem 1.3.2 at the end of this section.

From here on let us assume GCH. Our aim is to characterize the classesCλ(α)withα < ω2. (For an ordinal αand a set B, as usual, we let ^αB denote the set of all sequences of length α

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taking values inB.) Now, for anys∈^α{λ, λ⁺} we write

A_λ(s) ={β∈α:s(β) =λ}=s⁻¹{λ}.

Definition 1.3.3.Ifαis any ordinal, a subsetL⊂αis calledκ-closed inα, whereκis an infinite cardinal, iffsuphαi:i < κi ∈ L∪ {α} for each increasing sequencehαi:i < κi ∈^κL. The setL is< λ-closed inαprovided it isκ-closed inαfor each cardinalκ < λ. We say thatLissuccessor closed inαifβ+ 1∈L∪ {α} for allβ ∈L.

We are now ready to present the promised GCH characterization of the classes Cλ(α) and consequently, in view of 1.3.2, the characterization ofC(α)for allα < ω2.

Theorem 1.3.4.Assume GCH and fixα < ω2. (i) C_ω(α) ={s∈^α{ω, ω₁}:s(0) =ω}.

(ii) Ifλ >cf(λ) =ω,

Cλ(α) ={s∈^α{λ, λ⁺}:s(0) =λand A_λ(s)isω₁-closed in α}.

(iii) Ifcf(λ) =ω1,

Cλ(α) ={s∈^α{λ, λ⁺}:s(0) =λand

Aλ(s)is both ω-closed and successor-closed inα}.

(iv) Ifcf(λ)> ω1,

Cλ(α) ={hλiα}.

The main ingredient of the proof is the following statement:

Theorem 1.3.5.Let λbe a cardinal with µ= cf(λ)> ω and satisfying λ=λ^<µ. Then for any cardinalκwithλ < κ≤λ^µ and for every ordinalα < µ⁺ withcf(α) =µwe havehλi_α^_hκi_µ+∈ C(µ⁺).

To prove theorems above we need some preparation.

IfX is a scattered space andx∈X then we writeht(x, X) =αiffx∈Iα(X). Trivially, then ht(X) = min{β :∀x∈X[ht(x, X)< β]}.

It is obvious that ifY ⊂X thenht(x, X)≥ht(x, Y)wheneverx∈Y, and ifY is alsoopenin X then actually ht(x, X) = ht(x, Y). On the other hand, for the points of X outside of Y one can get the following upper bound.

Fact 1.3.6.IfY is an open subspace of the scattered spaceX then for every pointx∈X\Y we haveht(x, X)≤ht(Y) + ht(x, X\Y). Consequently, ht(X)≤ht(Y) + ht(X\Y).

Indeed, this can be proved by a straight-forward transfinite induction on ht(x, X), using Y ⊂I_<ht(Y₎(X).

It is well-known that any ordinal, as an ordered topological space, is locally compact and scattered. It is easy to see that ifα < β are ordinals then ht(α, β) =γ iffα can be written in the formω^γ·(2δ+ 1), or equivalently,γ is minimal such that αcan be written asα=ε+ω^γ. Note that in the notationht(α, β)the ordinals play a double role: αis considered as a "point" in the setβ. Using the above characterization of the Cantor-Bendixson levels of ordinal spaces, it is easy to show that for any infinite cardinalλand for any ordinalα < λ⁺ we havehλiα∈ C(α).

This section is based on [14] which is is a natural sequel to [61], so now we recall a few general statements concerning cardinal sequences from [61] that will be needed later.

Fact* 1.3.7([61, lemma 1]).If s∈ C(β)then|β| ≤2^s(0) ands(α)≤2^s(0) for eachα < β.

Fact* 1.3.8([61, lemma 2]).If s∈ C(β)andα+ 1< β thens(α+ 1)≤s(α)^ω.

Fact* 1.3.9([61, lemma 3]).If s∈ C(β),δ < βis a limit ordinal and C is any cofinal subset of δ, then

s(δ)≤Y

{s(α) :α∈C} .

We shall also need the following general construction from [61] that is used to obtain an LCS^∗ space by gluing together certain others.

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1.3. CARDINAL SEQUENCES OF LENGTH < ω2 UNDER GCH 15

Lemma 1.3.10 ([61, lemma 7]). Let X be an LCS^∗ space with a closed discrete subset S such that for each s ∈ S there is given a sequence hUs,n : n ∈ωi of pairwise disjoint compact open subsets ofX\S, converging to the points. Also, for eachs∈S letYsbe a separable LCS^∗ space such that the collection of spaces {X} ∪ {Ys:s∈S} is disjoint. Then there is an LCS^∗ spaceZ with the following three properties (i) - (iii):

(i) Z = (X \S)∪S{Ys :s∈S} with X\S as an open subspace and eachYs as a closed subspace. Moreover,{Ys:s∈S} forms a discrete collection in Z.

(ii) ht(x, Z) = ht(x, X)forx∈X\S.

(iii) ht(y, Z) = δs+ ht(y, Ys) for y ∈ Ys, where δs is the least ordinal δ such that the set {n < ω:Us,n∩Iδ(X)6=∅}is finite. (Clearly, δs≤ht(s, X).)

Definition 1.3.11.For any family of setsA we define the topological spaceX(A) =hA, τ_Aias follows: τ_A is the coarsest topology onA such that the setsU_A(A) =A ∩ P(A)are clopen for eachA∈ A. In other words: {U_A(A),A \U_A(A) :A∈ A}is a subbase for τ_A.

ClearlyX(A)is a 0-dimensionalT₂-space.

A familyAis calledwell-founded iff the partial orderhA,⊂iis well-founded. Ais said to be

∩-closed iffA∩B ∈ A ∪ {∅}wheneverA, B∈ A.

It is easy to see that ifAis∩-closed, then a neighbourhood base ofA∈ Ain the spaceX(A) is formed by the clopen sets

W_A(A;B₁, . . . , B_n) = U_A(A)\

n

[

1

U_A(B_i),

wheren∈ωandB_i (Afori= 1, . . . , n. ( Forn= 0we haveW_A(A) = U_A(A).) We shall write U(A)instead ofU_A(A)ifAis clear from the context, and similarly for the W’s.

The following statement, that was proved in [9, Lemma 2.2], shows the relevance of the above concepts to the subject matter of this section.

Fact 1.3.12.Assume that Ais both ∩-closed and well-founded. ThenX(A)is an LCS^∗ space.

To simplify notation, if X(A) is scattered then we write Iα(A) instead of Iα(X(A)), and I<α(A)instead of∪{Iζ(A) :ζ < α}. In the same spirit, forA∈ A we sometimes writeht(A,A) instead ofht(A, X(A)).

We shall say that A is an ordinal family if all members of A are sets of ordinal numbers, moreover A is both ∩-closed and well-founded. (As usual, we shall denote by On the class of all ordinals.) The following definition makes sense for any ordinal family A and will play an important role in our construction.

IfAis an ordinal family andξis any ordinal then we let Aξ={A∩ξ:A∈ A}

and

A^∗=[

{Aξ:ξ∈On}.

SoA^∗is simply the family consisting of all initial segments of all members ofA. Clearly, A^∗=A ∪ {A∩ξ:A∈ A ∧ξ∈A}.

It is easy to see that ifA is an ordinal family then so isA^∗, hence both X(A) andX(A^∗) are LCS^∗ spaces. A key ingredient in our construction is, just like in [9], the clarification of the relationship between these two spaces. The following technical lemma will play a significant role in this. As indicated above, we shall write U(A) instead ofU_A(A), U_∗(A) instead ofU_A∗(A), and similarly for the W’s.

Lemma 1.3.13 ([14]).Let Abe an ordinal family. Then for any A∈ A we have ht(U_∗(A))≤sup{ht(U_∗(A⁰)) :A⁰ ∈U(A)\ {A})}+ ht(tpA+ 1)

What we shall really need in our construction is the following corollary of lemma 1.3.13.

Lemma 1.3.14([14]).Let Abe an ordinal family such that, for a fixed indecomposableα∈On, we have|A|<cf(α) andht(tpA)< α for allA∈ A. Then ht(X(A^∗))< α.

Cardinal Sequences and Combinatorial Principles Lajos Soukup

Cardinal Sequences and Combinatorial Principles

Lajos Soukup

D. Sc. Thesis

Submitted to the Hungarian Academy of Sciences

Budapest, 2010

Contents

Cardinal sequences