istvan juhasz, saharon shelah, lajos soukup and zoltan szentmiklossy- cardinal sequences and cohen...

8/3/2019 Istvan Juhasz, Saharon Shelah, Lajos Soukup and Zoltan Szentmiklossy- Cardinal Sequences and Cohen Real Exten

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CARDINAL SEQUENCES AND COHEN REAL

EXTENSIONS

ISTVAN JUHASZ, SAHARON SHELAH, LAJOS SOUKUP, AND ZOLTANSZENTMIKLOSSY

Abstract. We show that if we add any number of Cohen reals tothe ground model then, in the generic extension, a locally compactscattered space has at most (20)V many levels of size .

We also give a complete ZFC characterization of the cardinalsequences of regular scattered spaces. Although the classes of the

regular and of the 0-dimensional scattered spaces are different, weprove that they have the same cardinal sequences.

1. Introduction

Let us start by recalling that a topological space X is called scatteredif every non-empty subspace of X has an isolated point. Via the well-known Cantor-Bendixson analysis then X decomposes into levels, theth Cantor-Bendixson level ofX will be denoted by I(X). The height

of X, ht(X), is the least ordinal with I(X) = . The width of X,wd(X), is defined by wd(X) = sup{ | I(X)| : < ht(X)}. Our mainobject of study is the cardinal sequence of X, denoted by CS(X), thatis the sequence of cardinalities of the non-empty Candor-Bendixsonlevels of X, i.e.

CS(X) =

|I(X)| : < ht(X)

.

The cardinality of a T3 , in particular of a locally compact, scat-tered T2 (in short: LCS) space X is at most 2

| I0(X)|, hence clearly

1991 Mathematics Subject Classification. 54A25, 06E05, 54G12, 03E35.Key words and phrases. locally compact scattered space, superatomic Boolean

algebra, Cohen reals, cardinal sequence, regular space, 0-dimensional.The first, third and fourth authors were supported by the Hungarian National

Foundation for Scientific Research grant no. 37758 .The second author was supported by The Israel Science Foundation founded by

the Israel Academy of Sciences and Humanities. Publication 765.The third author was partially supported by Grant-in-Aid for JSPS Fellows No.

98259 of the Ministry of Education, Science, Sports and Culture, Japan.1


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2 I. JUHASZ, S. SHELAH, L. SOUKUP, AND Z. SZENTMIKLOSSY

ht(X) < (2| I0(X)|)+ and |I(X)| 2| I0(X)| for each . (Locally com-pact scattered spaces are closely related to superatomic boolean alge-

bras via Stone duality and the study of their cardinal sequences wasactually originated in that subject.) Thus, in particular, under CHthere is no scattered T3 space of height 2 and having only countablymany isolated points. After I. Juhasz and W. Weiss, [5, theorem 4],had proved in ZFC that for every < 2 there is an LCS space Xwith ht(X) = and wd(X) = , it was a natural question if the ex-istence of an LCS space of height 2 and width follows from CH.This question was answered in the negative by W. Just who proved,[6, theorem 2.13], that if one blows up the continuum by adding Cohenreals to a model ofCH then in the resulting generic extension there isno LCS space of height

2and width . On the other hand, in their

ground breaking work [1], J. Baumgartner and S. Shelah produced amodel in which there is a LCS space of height 2 and width , more-over they proved in ZFC that for each < (2)+ there is a scattered0-dimensional T2 space X with ht(X) = and wd(X) = . Buildingon the idea of the proof of this latter result, in section 3 we succeededin giving a complete characterization of the cardinal sequences of bothT3 and zero-dimensional T2 scattered spaces. Although the classes ofthe regular and of the zero-dimensional scattered spaces are different,it will turn out that they yield the same class of cardinal sequences. Weshould add that, with quite a bit of extra effort, in [8], J.-C. Martinez

extended the former result of Baumgartner and Shelah by producing amodel in which for every ordinal < 3 there is a LCS space of height and width . The question if it is consistent to have a LCS space ofheight 3 and width remains a big mystery.

In section 2 we strengthened the result of Just by proving, in partic-ular, that in the same Cohen real extension no LCS space may have 2many countable (non-empty) levels. It seems to be an intriguing (andnatural) problem if the non-existence of an LCS space of width andheight 2 implies in ZFC the above conclusion, or more generally: isany subsequence of the cardinal sequence of an LCS space again sucha cardinal sequence? In connection with this problem let us remark

that, (as is shown in [2] or [3]), in the side-by-side random real exten-sion of a model of CH the combinatorial principle Cs(2) introducedin [4, definition 2.3] holds, consequently in such an extension there isno LCS space X of height 2 and width . In fact, by [4, theorem4.12], Cs(2) implies that { 2 : |I(X)| = } is non-stationary in2. However, we do not know if our above mentioned result, namelytheorem 2.1, holds there.


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CARDINAL SEQUENCES AND COHEN REAL EXTENSIONS 3

The morale of our above discussion may be concisely formulated asfollows: The cardinal sequences of regular or zero-dimensional scattered

spaces are only subject to the trivial inequality |X| 2|I0(X)|, howeverthose of the LCS spaces are much harder to determine, in particular,they are sensitive to the model of set theory in which we look at them.

2. Countable levels in Cohen real extensions

Let us formulate then the promised strengthening of Justs result.We note that no assumption (like CH) is made on our ground model.

Theorem 2.1. Let us set = (2)+ and add any number of Cohenreals to our ground model. Then in the resulting extension no LCSspace contains a-sequence {E : < } of pairwise disjoint countable

subspaces such that E E holds for all < < . In particular,for any LCS space X we have

{ : | I(X)| = } < .In fact, we shall prove a more general statement, but to formulate

that we need a definition. A family of pairs (of sets) D =

D0 , D1 :

I

is said to be dyadic over a set T iffD0 D1 = for each I

andD[] =

{D() : dom }

intersects T for each Fn(I, 2). We simply say that D is dyadic iffit is dyadic for some T, i.e. D[] = for each Fn(I, 2).

Now, it is obvious that in a LCS space

the compact open sets form a base that is closed under finiteunions,

there is no infinite dyadic system of pairs of compact sets.

Consequently, theorem 2.2 below immediately yields theorem 2.1 above.

Theorem 2.2. Set = (2)+ and add any number of Cohen reals tothe ground model. Then in the resulting generic extension the follow-ing statement holds: If X is any T2 space containing pairwise disjointcountable subspaces {E : < } such that E E for < < and X = E0 (i. e. E0 is dense in X) , moreover, for each x X,

we have fixed a neighbourhood base B(x) of x in X that is closed underfinite unions then there is an infinite set a

, for each a thereare disjoint finite subsets L0 and L

1 of E, and for each x L

0 L

1

there is a basic neighbourhoodV(x) B(x) such that the infinite familyof pairs

xL0

V(x),xL1

V(x)

: a

is dyadic.


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This topological statement in the Cohen extension in turn will followfrom a purely combinatorial one concerning certain matrices, namely

theorem 2.7.To formulate this theorem we again need some notation and defini-

tions.For an ordinal the interval [, + ) will be denoted by I.

Given two sets A and B we write f : Ap

B to denote that f is apartial function from A to B, i. e. a function from a subset of A intoB. As usual, we let

Fn(A, B) = {f : |f| < and f : Ap

B}.

If A On then for any partial function f : Ap

B we set

(f) =

min dom f if dom f = ,sup A if dom f = .

We let

=

A, B


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for each dom s.

The following easy observation will be applied later, in the proof oflemma 2.9:

Observation 2.4. If g : Sp

S and s : Sp

satisfy dom s ran g, and the pair (g, s g) is A-dyadic overU thens is A-dyadic overU, as well.

Definition 2.5. Fix a cardinal and let D M(, ). For s : p

we say that s is D-min-dyadic (m.d.) iff s is D-dyadic over I(s).

Moreover, we say that the matrix D is m.d.-extendible iff for eachfinite D-min-dyadic partial function s :

p and for each < (s)

there is an such that s {, } is also D-min-dyadic, i. e.

D-dyadic over I .Since I0 = , we clearly have the following.

Observation 2.6. If D M(, ) is m.d-extendible and s : p

is a finite D-min-dyadic partial function then s is D-dyadic over .

Finally, a matrix D M(, ) will be called -determinediffD,n D,m = implies D,n D,m = whenever < and n < m < .

With this we now have all the necessary ingredients to formulate andprove the promised combinatorial statement that will be valid in anyCohen real extension.

Theorem 2.7. Set = (2)+ and add any number of Cohen reals tothe ground model. Then in the resulting generic extension for every -determined and m.d.-extendible matrixD M(, ) there is an infinite

D-dyadic partial function h : p

.

Before proving theorem 2.7, however, we show how theorem 2.2 canbe deduced from it.

Proof of theorem 2.2 using theorem 2.7. We can assume without anyloss of generality that E = I for each < and then will define anappropriate matrix D M(, ).

To this end, for coding purposes, we first fix a bijection :

2

and let : and : be the co-ordinate functions ofits inverse, i. e. k = ({(k), (k)}) and (k) < (k) for each k < .

Since X is T2, for each n < we can simultaneously pick basicneighbourhoods Bn (m) B( + m) of the points + m E = Ifor all m < n such that the sets {Bn(m): m < n} are pairwise disjoint.

Now we define D = D,k : , k M(, ) as follows:

D,k = B(k)((k)) .


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This matrix D is clearly -determined because E0 = I0 = is densein X. It is a bit less easy to establish the following

Claim . D is also m.d.-extendible.

Proof of the claim. Let s : p

be a finite D-min-dyadic partialfunction and let < (s).

Since the sets {D[s, ] : dom s2} are all open in the subspace and they all intersect I(s), moreover every element of I(s) is anaccumulation point of I, it follows that D[s, ] I must be infinitefor each dom s2. Thus we can easily pick two disjoint finite subsetsA0 and A1 ofI such that every D[s, ] intersects both A0 and A1. Letn < be chosen in such a way that A0 A1 { + m : m < n},and set Ki = {{m, n} : m < n + m Ai} for i < 2. Since isone-to-one we have K0 K1 = , hence = K0, K1 , moreover

() mK0

D,m

mK1

D,m

=

because the elements of the family {Bn(m) : m < n} are pairwisedisjoint.

Now put t = s {, }. Then for each dom t2 we clearly have

() A() D[t, ] = ,

hence () and () together yield that the extension t of s is D-dyadicover I = I(t).

Thus we may apply theorem 2.7 to the matrix D to obtain an infiniteD-dyadic partial function h :

p . Set a = dom h and for each

a and i < 2 put Li = { + (k) : k i(h())}. For x Li put

V(x) = {B(k)((k)) : x = + (k) and k i(h())}.

Then V(x) B(x) because B(x) is closed under finite unions. Sincefor i < 2

{V(x) : x Li}

= {D,k : k i(h())}

and

{D,k : k 0(h())} {D,k : k 1(h())} = ,we have

{V(x) : x Li}

{V(x) : x Li}

=

because the latter intersection is an open set which does not intersectthe dense set I0 . Hence the infinite family

xL0

V(x),xL1

V(x)

: a


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is indeed dyadic. 2.2

Proof of theorem 2.7. The proof will be based on the following two lem-mas, 2.9 and 2.10. For these we need some more notation and a newand rather technical notion of extendibility for set matrices.

Given a set A we set

F(A) = {f Fn(A, A) : f is injective and dom(f) ran(f) = }.

Each function f F(A) can be extended in natural way to a bijectionf : A A as follows:

f(a) =

f(a) if a dom f,f1(a) if a ran f,

a otherwise.Definition 2.8. If S and T are sets of ordinals then the matrix A M(S, T) is called nicely extendible iff for each f F(S) there are a

family N(f) Fn(S, ) and a function Kf : N(f)

S

suchthat

(1) the pair (f, s) is A-dyadic whenever f F(S) and s N(f),(2) N(f) for each f F(S),(3) for f, g F(S) and s N(f) if f | Kf(s) = g | Kf(s) then

s N(g).(4) for any f F(S), s N(f) and S (s) there is such

that s {, } N(f).

Clearly, this last condition (4) is what explains our terminology.

Lemma 2.9. If > 1 is regular and A M(, ) is a nicely ex-

tendible matrix then there is an infinite partial function h : p

that is A-dyadic .

Proof. By induction on n we will define functions h0 h1 . . . hn . . . from Fn(, ) such that |hn| = n and for each

()n there is g F() such that (g) > , ran g = dom hn and hn g N(g).

First observe that h0 = satisfies our requirements because, accord-ing to (2), condition ()0 holds trivially for each < .

Next assume that the construction has been done and the inductionhypothesis has been established for n. For each < choose a functiong F() witnessing ()n+1 and then write K = K

g(hn g) andpick (, + 1) \ K. Clearly the set

L = { : |{ < : / K}| < }


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is countable and so we can pick n \ (L dom hn); then the set

J = { < : n / K}is of size .

Now set g = g {, n} for every J. For every such then, n / K implies g

| K = g | K, hence hn g N(g

) by (3).

Since < + 1 < (g) = (hn g), we can now apply (4) to get such that (hn g) {,

} N(g).We can then fix n such that Jn = { J : = n} is of size

and let hn+1 = hn {n, n}.If Jn then hn+1g

= (hng){, n} N(g

) and (g

) > ,

so g witnesses ()n+1 . But Jn is unbounded in , hence the inductive

step is completed.

By ()n0 , for each n < there is gn such that dom hn = ran gn andhngn N(gn). Hence, by (1), (gn, hngn) is A-dyadic, and so hn is A-dyadic according to observation 2.4. Consequently h =

{hn : n < }

is as required: it is A-dyadic and infinite. 2.9

Given any infinite set I we denote by CI the poset Fn(I, 2), i.e. thestandard notion of forcing that adds |I| many Cohen reals.

Lemma 2.10. Let = (2)+. Then for each we have

VC |= If D M(, ) is both -determined and m.d.-extendible then there is I

such that D

= D,n : , n I is nicely extendible.Proof. Assume that

1CD M(, ) is m.d.-extendible.

Let be a large enough regular cardinal and consider the structure

H =

H, ,,,, D

, where H =

x : | TC(x)| <

and is a

fixed well-ordering of H.Working in V, for each < choose a countable elementary sub-

model N of H with N. Then there is I

such that themodels {N : I} are not only pairwise isomorphic but, denoting

by , the unique isomorphism between N and N, we have(i) the family {N : I} forms a -system with kernel ,

(ii) ,() = for each ,(iii) ,() = .

For each < and n < let D,n be the -minimal C-name of the

, nth entry of D. Since is in H and ,() = we have

Claim 2.10.1. ,(D,n) = D,n for each , I and n .


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Let G be any C-generic filter over V. We shall show that

V[G]|=

D

=D

,n :

, n

I

is nicely extendible.

For each f F(I) define the bijection f : as follows:

f() =

,f()() if N for some I, otherwise.

In a natural way f extends to an automorphism of C, which willbe denoted by f as well. Clearly, we have

Claim 2.10.2. If f F(I), f() = , p C N then ,(p) =f(p).

For f F(I) let Gf = {1f (p) : p G} and then set

N(f) = {s Fn(I, ) : s is D[Gf]-min-dyadic} =

{s Fn(I, ) : q Gf qs is D-min-dyadic}.

To define Kf, for each s N(f) pick a condition ps G such that

1f (ps)s is D-min-dyadic

and letKf(s) = { I : (N \ ) domps = }.

Note that Kf(s) as defined above is finite, although 2.8.(3) onlyrequires Kf(s) to be countable.

To check property 2.8.(3) assume that f, g F(I) and s N(f)with g | Kf(s) = f | Kf(s). Then 1g (ps) =

1f (ps) and so

1g (ps)s is D-min-dyadic,

hence s is also D[Gg ]-min-dyadic , i.e. s N(g).Before checking 2.8.(1) we need one more observation.

Claim 2.10.3. Df(),n[G] = D,n[Gf] whenever f F(I),

dom f, and n < .

Proof of claim 2.10.3. Let k . Then k Df(),n[G] iff p G p

k Df(),n iff p G Nf() p k Df(),n iff q Gf Np = ,f()(q) k Df(),n iff q G

f N q k D,n iff

q Gf q k D,n iff k D,n[Gf]. 2.10

Now let f F(I) and s N(f). By the definition of N(f), s isD[Gf]-min-dyadic and so by observation 2.6 s is D[Gf]-dyadic over .But it follows from 2.10.3, that s is D[Gf]-dyadic over if and only ifthe pair (f, s) is D[G]-dyadic over .


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2.8.(2) is clear because is trivially A-min-dyadic for any A M(, ). Finally 2.8.(4) follows from the definition of N(f) because

D[Gf] is m.d.-extendible. 2.10

Now, to complete the proof of theorem 2.7, first apply lemma 2.10to get I

such that

D = D,n : , n I

is nicely extendible. Then applying lemma 2.9 to D we obtain aninfinite D-dyadic function h :

p . Since the matrix D is -

determined the function h is D-dyadic, as well. 2.7

3. Cardinal sequences of regular and 0-dimensionalspaces

For any regular, scattered space X we have |X| 2|I(X)|, henceht(X) < (2|I(X)|)+ and |I(X)| 2

| I0(X)| for each . This implies thatfor such a space X its cardinal sequence s satisfies length(s) < (2|I(X)|)+

and s() 2s() whenever < . We shall show below that theseproperties of a sequence s actually characterize the cardinal sequencesof regular scattered spaces.

In [1], for each < (2)+, a 0-dimensional, scattered space of height and width was constructed. The next lemma generalizes that

construction.For an infinite cardinal , let S be the following family of sequences

of cardinals:

S =

i : i < : < (2)+, 0 = and i 2

for each i <

.

Lemma 3.1. For any infinite cardinal ands S there is 0-dimensionalscattered space X with CS(X) = s.

Proof. Let s = : < S. Write X =

{} : <

.

Since |I| 2 we can fix an independent family {Fx : x X}

.The underlying set of our space is X and and the topology on X

is given by declaring for each x = , X the set

Ux = {x} ( Fx)

to be clopen, i.e. {Ux, X\ Ux : x X} is a subbase for .The space X is clearly 0-dimensional and T2.

Claim 3.1.1. If x = , U and < then U ({} ) isinfinite.


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Proof of the claim. We can find disjoint sets A, B

X\ {x}

> m1 and for all i < m sequences si Si such that s =s0

s1 . . . sm1 or s = s0

s1 . . . sm1

n for some naturalnumber n > 0.

Proof.(1)= (3)

By induction on j we choose ordinals j < ht(X) and cardinals j suchthat 0 = 0 and 0 = | I0(X)|, moreover, for j > 0 with j1 infinite

j = min

ht(X) : | I(X)| < j1},

and j = | Ij(X)|. We stop when m is finite. For each j < m let

j = j+1.

j . Then the sequence sj =

| Ij+(X)| : < j

is in Sj .Thus CS(X) = s0

s1 . . . sm1 provided m = 0 (i.e. Im(X) = )

and CS(X) = s0s1

. . . sm1 m when 0 < m < .


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(3)= (2)First we prove this implication for sequences s of the form s0

s1 . . . sm1

by induction on m. If s So then the statement is just lemma 3.1Assume now that s = s0

s1 . . . sm1, where 0 > 1 > >

m1 and si Si for i < m.According to lemma 3.1 there is a 0-dimensional space Y with cardi-

nal sequence sm1. Using the inductive assumption we can also fixpairwise disjoint 0-dimensional topological spaces Xy,n for y, n I0(Y) , each having the cardinal sequence s

= s0s1

. . . sm2.We then define the space Z = Z, as follows. Let

Z = Y

{Xy,n : y I0(Y), n < }.

A set U Z is in iff

(i) U Y is open in Y,(ii) U Xy,n is open in Xy,n for each y, n I0(Y) ,

(iii) ify I0(Y) U then there is m < such that

{Xy,n : m < n

istvan juhasz, saharon shelah, lajos soukup and zoltan szentmiklossy- cardinal sequences and cohen...

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