Partition Calculus for Top ological spaces

Soheil Homayouni

A thesis submit t ed in conformity wit h the requirements for the degree of Ph.D.

Graduate Department of Mathematics University of Toronto

Copyright @ 1997 by Soheil Homayouni

National Library 1*1 of Canada Bibliothèque nationale du Canada

Acquisitions and Acquisitions et Bibliographie Services senrices bibliographiques

395 Wellington Street 395. rue Wellington OîîawaON K1AON4 Ottawa ON K1A ON4 Canada Canada

Yow lSb Votre nikirna,

Our iYe Narre rel4rnco

The author has granted a non- exclusive licence allowing the National Library of Canada to reproduce, loan, distribute or sell copies of this thesis in microform, paper or electronic formats.

The author retains ownership of the copyright in this thesis. Neither the thesis nor substantial extracts fiom it may be printed or otherwise reproduced without the author's permission.

L'auteur a accordé une licence non exclusive permettant à la Bibliothèque nationale du Canada de reproduire, prêter, distribuer ou vendre des copies de cette thèse sous la forme de microfiche/film, de reproduction sur papier ou sur format électronique.

L'auteur conserve la propriété du droit d'auteur qui protège cette thèse. Ni la thèse ni des extraits substantiels de celle-ci ne doivent être imprimés ou autrement reproduits sans son autorisation.

Abstract Title: Partition Calculus for Topologicai spaces

by Soheil Homayouni, Depart ement of Mat hematics,

University of Toronto, 1997

The subject matter of this exposition is the study of partition relations for particular

topological spxes . For topological spaces X and Y, and for a positive integer n and a

cardinal k the notation X -t (Y); means that for any partition f of [XIn into k pieces,

there is a subspace H of X homeomorphic to Y and f is constant on [Hln. In contrast

with the ordinary partition calculus in this area very few positive results are known. For

example, it is consistent that for n > 3 and for any space X, we have X 74 (Y); for

every topological spaces Y in which not dl the points are isolateci. Even in the case of

n = 3 if the space Y is compiicated enough the relation fails to be positive. It seems that

the cases n = 1 and 2 are the cases where there exïsts some positive relations for non-

trivial space Y. The spaces considered in this exposition are the foliowings: Ordinal spaces,

Re*, Rationals, Cantor Cubes, C-products, cech-~tone cornpactification of the integers,

Aiexandroff compactification of discrete spaces and the countable sequences converging t o

a point in the sense of an arbitrary ultrafilter of N, denoted by w U ( p j . 'We wiU present

and review many techniques for proving topological relations when n = 1. Then we shall

use a stepping-up procedure to obtain results for higher dimensions n > 1 when possible.

Using ordinary axioms of set theory (ZFC), we will establish some positive relations for the

space 2"= such as Y t (Y, Q)', 2C t (w u { p l ) ; and CH, tf (kt2):, where Ex, denotes

CH, {O, l}, the sigma-product of NÎ copies of the two point d ic re te space {O, 1). Beyond

ZFC, we introduce and apply the method of elementary submodels and use the SPFA and

some of its consequences to establish the consistency of 2W1 -t (wt + 1. B(wl)) ' as weU

as the consistency of 2wl -t (wi + 1, ul) l . The additional axiom SPFA has a considerable

large cardinal consistency strength, but we shall establish that the large cardinal assurnption

is necessary for the consistency of the positive relation ZN, + (wl)& The techniques of

forcing and iterated forcing are used to define generic partitions for some spaces leading to

the independence of the partition of the form Ex, t (wl)$.

Acknowledgement

I would like to thank my supervisor professor William Weiss, for his patience in teaching me both Set theory and problem solving, for the time he spent on me and for his valuable guidances. The support of the Toronto Set theory Seminar and its members individually has been w m ; for that 1 am thadfid. The staff of the mathematics department gave me technicd help and administrative guidances as well as providing a friendly environment ; 1 felt at home in the department.

Special thanks to my mentor professor S tevo Todorcevic. Many conversations with him were illuminating and developed interest as well as insight in me. 1 thank him also for long hours of reading and rereading the drafts of this thesis, for making correct ions and offering useful suggestions.

Contents

0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

O 1 . 1 A list of the topological partition relations . . . . . . . . . . 9

0.1 -2 A chart of topological partition relations . . . . . . . . . . . . 11

1 Definitions and simple corollaries 12

1.1 Stepping-up methods . . . . . . . . . . . . . . . . . . . . . . . . . . 12

. . . . . . . . . . . 1.2 Partition relations for uncountable ordinal spaces 14

1.3 Baire Category axguments . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 C-products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.5 pw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.6 4(n) and B(wl) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Partitioning Cantor cubes 2 4

2.1 Cantor cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Special subsets of P ( q ) . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Finite partitions of Cantor cubes . . . . . . . . . . . . . . . . . . . . 27

2.4 Infinite partitions of 2" . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 The method of elementary submodels 34

3.1 Elementory submodels . . . . . . . . . . . . . . . . . . . . . . . . . . 35

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Reflection Principle 37

3.3 A correspondence between [wzIHo and P(wl ) . . . . . . . . . . . . . . 40

3.4 SPFA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Chang's conjecture 47

. . . . . . . . . . . . . 3.6 Partitioning the C-product into three pieces 51

4 Negative Partition relations in forcing extensions 54

. . . . . . . . . . . . . . . . . . . . . . . 4.1 Generic filter as a partition 54

. . . . . . . . . . . . . . . . . . 4.2 Iteration of countably closed forcings 61

5 Homogeneous copies of w u 70

. . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The basic construction 71

. . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Club guessing principle 72

. . . . . . . . . . . . . . 5.3 Argument using heredi tary separability of R 75

0.1 Introduction

Ramsey's theorem of 1930 is a two dimensional version of the standard pigeon-hole

principle and states that whenever the edges of ar? infmite complete graph are parti-

tioned into two pieces there is an infinite complete graph all edges of which belong to

the same piece of partition. Later on, generalized in various ways, this theorem was

expounded to the subjects that axe today known as partition calculus and as Ramsey

theory (see [EHRM] and [GRS]). In these subjects as weU as in the subject matter of

this thesis the Erdàk-Rado awow notation is very usefd:

Notation: Given cardinal numbers n, A, p and a positive integer n, the notation

stands for the following statement :

Given any partition of the n-tuples of a set A of size K into p pieces, f : [A]" + p.

there is a set H c A of size X which is homogeneous for f, that is, f is constant o n

[Hln- In this case it is understood that for some a < p zue have [Hl" C f-'({a)). Such

a set is said to be a-homogeneoas for f. The negation of this statement is denoted

by

Variations of this symbol will be introduced as the occasion arises.

Using this notation Ramsey's theorem becornes w + (w) ; , where w stands for

the f i s t infinite cardinal number. The finite Ramsey's theorem states that for any

triple of positive integers nt, n and k there is a positive integer r such that

As a natural extension, the partition relations for order types such as ordinals

and in general for partial orderings were introduced (see [EHMR] and [Tol]). The

homogeneous set in these cases must be some ordinal (type). One of the f i s t and the

most important such result was wl -t (a): due to J-Baumgartner and A.HajnaI

[BH] which has been extended to ail partially ordered sets by S-Todorcevic [Tol] as

follows: for a partially ordered set P that satisfies P -t (a): we have the positive

relation P (a): for every countable ordinal a and a positive integer k.

Besides as order types the ordinals c m be thought of as topological ordinal spaces

(limit points are required to be limits in the topological sense.) These results have

Ied to first question in the area of partition calculus foi. topo~ogzcal spaces: Galvin and

Laver asked whether one of the partition relations wl --+ (a): or W -t (a): can

be extended to topological ones. ki other words can we always get the homogeneous

set not only of order type a but topologica.lIy homeomorphic to a. (incidentally, this

question remains open tedate. )

The combinatorid nature of many problems in topology is well known and under-

stood (see for example [ToZ] and [To~]). More recently, people have realized that the

arrow notation can dso be used to characterize certain families of topological spaces

(see [ T o ~ ] and [GS]). So, let us introduce the m o w notation for topological spaces.

For topologicd spaces X and Y and the cardinal nurnber 6

means: For any partition f : [XIn -+ L there is a homogeneous subset H of X

homeomorphic to the space Y; we often identify H with Y. From now on, a.ll the sets

X and Y are considered as topological spaces unless otherwise stated. If any risk of

confusion is predicted the prefix top in front of Y may appex to ernphasize that we

are interested in a topologicd copy of Y.

The arrow notation has the following frequently used variations:

x -t (Y, 2)"

to mean

is either

f"([zln)

that for spaces X, Y and Z and for each partition f : [XIn -t {O, 1) there

a copy of Y in X such that f"([Yjn) = {O), or a copy of 2 in X such that

= (1).

The following convention will be used from time to time. The notation

means that for any partition of [XIn there is either a O-homogeneous subspace home-

omorphic to Y or for 1 (1 5 1 < k) there is an 1-homogeneous subspace homeomorphic

to 2. A weakening of the positive arrow relation is the square bracket relation:

which means that for every f : [XIn + k there is H C X homeomorphic to Y and

i < k such that i 6 f"[HIn. It goes without saying that if X t (Y)," and X v Z then Z + (Y): as well.

Similady, if W v Y then it follows that X + (W);. One has similar monotonicity

proper t ies regarding the negat ive relations.

Finally, the subscri~t k in the arrow notation is presently chosen to be finite but the

subscript in general need not be a finite cardinal. We discuss some infinite partitions

into w or wi pieces but as we shall soon see, in partition calculus for topological spaces

we are often concerned simply with the case k = 2. Also notice that in this case the

square and the round bracket notations will coincide.

Simple facts about the topological arrow notation

1. In Partition Calculus a superscript n = 1 signds an easy problem the solution

to which is a pigeon-hole argument together wi th some cofina.lity considerations.

In partitioning topologicd spaces, however, n = 1 is often a problem in topology

that will be, as we s h d soon see, often a difficult one.

2. A negative result cân be extended to one with a larger superscript. Suppose for

example that X and Y are arbitrxy sets with IY 1 > 2. Then X t, (Y): implies

X t, (Y, 3)? To see this we take a partition f : X + {O, 1) witnessing the

negative relation X + (Y):, and define the partition g : [XI2 + {O, 1) by

g(x, y) = O if f f (z) = f (y) witnesses the negative relation X t, (Y, 3)2. This

argument is generalized (see lemma 1.2) to extend negative relations d any

dimension to any higher dimension.

3. There are negative partition relations from the ordinary putition calculus for

n > 1 that one c m use to draw conclusions in the area of partition relations for

topological spaces. One such result is the following: (originally due to K. Godel,

(sec [KaIl)

The proof for this relation is simple: Consider X = ->2 , as the set of all branches

of the binary tree of height w, and let D denote the collection of al1 the nodes

in such a tree. For any two elements f , g E X let d( f,g) denote the separatirtg

node of f and g. Of course size of X is 2 N ~ and size of D is w. Therefore d is a

partition of X into w many pieces, and since no three elements of X can shore

the same branching node, d-homogeneous sets are of size at most 2. This result

clearly restricts the possible positive partition relation for "small" topological

space X and infinite subscripts, in the case of n = 2. Such negative results are

expounded, in partition calculus, to hold for n > 2 by using some stepping-up

lemma, as discussed before (for more on these see section 1.1).

4. The ordinary Stepping-up lemma were extended by Komjath and Weiss to other

parameters of the urow formula. They proved: If a first countable space X

satisfies X -t (w + 1): then for any countable ordinal a we have X -t (a):,

and in the case when character of space X is NI the relation X + (w2 + l):, is independent [KW].

The main purpose of this exposition is to discuss the topological m o w relation

X + (Y);, where X and Y belong to the following list:

- the ordinal spaces,

- the set of rationals with the usual topology Q,

- the Cantor Cubes 2",

- the C-products &{O, 1) hereafter denoted by C,,

- the Baire space B ( w l ) ,

- the cech-~tone compactification of the natural numbers pu,

- the sequences convergent in the sense of some arbitrary ultrafilter w U {p), and

- the one point compactification of discrete spaces, A ( K ) .

The simplicity or the universality of these spaces axe the reasons for studying these

spaces. There are consistency results that assure a negative relation for ail topological

spaces X, which shed light on the project we are just about to undertake. One such

result announces that for n > 2 the positive relation X --+ (Y); implies that the

space Y must be scattered [HJW]. Another result, for the case n > 3 maintains that

the positive relation X -+ (Y); implies that the space Y must be discrete [Wel]. On

the other hand there is a consistency result that ensures the existence of a topological

space X of size N3 that satisfies X -t (non-discrete); [JS]. The impact of these

results on our project is obvious; here is a list of negative relations for any topological

space X and any n > 3 that we consistently have

4. X +- (pu);, Also, lemma 2.1 of [HJW] implies

And of course for the spaces Y that embed w + 1, w U (P), Q and pw we obviously

have X (Y);.

Existing methods in topological partition calculus for n = 1

The earliest result in this area is, perhaps a result of Bernstein (1908):

There is a partition of the reals into two pieces neither piece contains a copy of

the the Cantor set.

The proof of this result is based on the following general scheme. Given topologicd

spaces X and Y, let {Y, : a E n) be an enumeration of all the possible homeomorphic

copies of Y inside X. If for a.ll a E n we have lYPl 2 6 then X t+ (Y): follows easily.

To construct a partition f : X -r {O, 1) witnessing this relation one chooses a pair

{x:, x:} from Y, \ U B < , { z ~ , z;) and let f (si) = i. Then, one extends f arbitrarily

over X. To adopt this scheme to X = 2w and Y = 2" (see section 2.1 for definitions )

we observe that since 2" is separable every copy of Y in X is determined by a countable

set. There are continuum many countable subsets of X = 2w hence there are at most

continuum many homeomorphic copies of the Cantor set. The cardinality condition

is now satisfied. This argument was generalized in different ways (see [Ma], [We3]

and [BSS]) to establish the consistency of X + (2"): with certain size restrictions

on the topological space X.

In studying the relation n t (wi): one has access to some parts of each copy

of wl in K by using a combinatorial principle like 0 or O,, (see chapter 5 for an

example of such a situation.) These principles parantee the existence of sequences

that approximate some subsets of n that could be a copy of wl . To define a partition of

rc that admits no homogeneous sets of type wl one will have to construct the partition

based on the information given at that stage by the combinatorial principle. One of

the fkst instances of such a construction is due to K.Prikry and R.Solovay ([PSI) who

proved K f-t ( w l ) i for every ordinal rc using a variation of the combinatorid principle

0, *

Another technique for proving a negative partition relation X t+ (Y): is to

stratifjr the topological space X according to the specifications of the space Y so

that any copy of the space Y has a known place in the stratified X . Then we define

f : X + {O, 11, in a "geometric wayn to preclude the possibility of any homogeneous

copy of Y in X ( for a simple example, see section 1.2.) This technique was used

in [CKS] to establish w* ++ (w U { p l ) (for definition of w U { p ) and w' see section

1.6.) Also, this technique was used in [KW] to establish, assuming 0, that for any

topological space X of size NI, we have X + (wi):. Another technique, used in [HJW], is to inductively define a partition of a space

X starting from a very s m d subset Xo of X for which a negative relation exists, and

trying to carefdy extend the existing partition to larger and larger subspaces in such

a way that no copy of the space Y can be homogeneous for the partition constructed

up to that stage of the iteration. Again to manage the limit stages of the iteration

some assumption about the cardinal arithmetic has tn he made.

Perhaps the fastest way to produce a negative partition relation is to use a generic

se t given by a forcing extension. If a generic set is used in defining a partition of

certain space X, with some care this partition will split a,ll the copies of a certain

specified space Y inside X. This technique is demonstrated in chapter 4 in proving

the consistency of C , f+ ( w l ) i and 2u1 j+ (w& This method was used by W .Weiss

[We4] to prove the consistency of X + (2"): for a metric space X of a fked size.

In the positive direction, the sequences given by the combinatorial principle 0

can be used to approximate ail the the partitions of the space X. and then to try

to assign to each partition a homogeneous copy of the space Y (for a sample of this

construction see 5.1 . )

Another collection of axioms have been useful too. For example, the Semz-Proper

Forcing Aziom (SPFA) (see 3.4) guarantees the existence of elementary submodels

with very usef'ul properties. Similar axioms such as Reflection Princzple, (RP) (see

3.2) and Chang's Conjecture, (see 3.5) guarantee the existence of continuous chains

of elementary submodels extending a given elementary submodel. Such chains are

often used to create a continuous collection of ordinals (hence homeomorphic copy of

some ordinals) or to Usmoothn a given collection of sets or functions (see 3.5). These

techniques are used in chapter 3 to establish ZW1 + (wl + 1, B(ui)) ' , as well as

2w1 + (w* + 1, w ~ ) ~ .

In chapter 1 we introduce the spaces for which we intend to study the topological

arrow notation. In this chapter we also give a s w e y of the existing topological

relations, and use some well-known results from ordinary partition calculus to draw

conclusions in the case n > 1. We also quote several classical results from general

topology and use them to establish relations for n = 1.

Chapter 2 introduces the Cantor Cubes and establishes the positive relations

2" t (2", Q): , and 2" + ( 2 7 , (QIul ) '. Chapter 3 introduces the method of elementary submodels and discusses the re-

flection principle, Chang's Conjecture and SPFA. A lemma of S. Todorcevic's appears

and is used to improve a result of S. Shelah to 2w1 t (wl + 1, B(wl ) ) l . The strength

of a positive relation for C-products is examined and is seen to have a a large cadina1

nature.

Chap ter 4 cont ains a forcing argument establishing some negative partition re-

lation about certain spaces X of size Hz. This forcing argument is iterated to prove

similar result about the space 2w1.

Chapter 5 demonstrate a number of different principles and assumptions to bring

three different arguments in establishing a positive relation 2' + (w U {PI):.

A chart of the partition relations for topological spaces summerizes some existing

results together with the new ones, (see page 11.)

O A list of the topological partition relations

1. @w t) ( w + 1): : see 1.6.

2. W' t-, ( w u {PI): : see 1.6.

3. pw + (2'); : see 1.

4. $b ,4 (Q): : see 1.

5. &J t, (A(X) ) : : see 1.7.

6. wl + (wl,(a),) ' a < wl: see 1 . 2 .

7. K f i (wl + 1, w + 1): see 1.2 .

8. 0, : K + (ut):: see 1.2.

9. SPFA : w* -+ (~1) : : see 1.2.

10. n + (Q):: see 1.3.

11. K + (2w) : : see 1.3.

12. K + (w U { p } ) : : see 1.6.

13. K f-t (pu):: see 12.

1 4 K ( A ( ) ) : see 1.7.

15. 2" f-t (top tc+) :: see 2.1.

16. 2" + (2", 4) ': see 2.3.

17. 2" -t (a,())' cr < K+: see 16 and 2.1.

18. 2" + (a, B ) l cr < R+, 9 < wl: see 16 and the fact that any countable ordinal is embeddable in Q.

19. SPFA:2W' --+ (wl + 1, q ) ' : see 3.4.

21. CON 2Wl + (wi):: see 4.2.

23. 2Y + ( 2 9 : : see Introduction.

24. 2wl t (2F , Q,,) ': see 2.4.

25. 2' t (w U {PI):: see 5.4.

26. 2W + (w U (pl):: see 1.6.

27. 2C (pw):: see 1.6.

28. 22a -t (pn, Q) ': see 2.3.

29. 2" -+ ( A ( K ) ) ; ~ ( , ) : see 1.7.

30. SPFA: C, -+ (ai):: see 3.0.

31. C, + (wl + 1):: see 1.5.

32. CON C, + (wi):: see 4.1.

33. C, -+ (w l , (a),)' a < wl: see 1.5.

34. C, + (wi):: see 3.6.

35. C, -+ (wl): + CC: see 3.5.

36. C, -+ (2w,Q,)1: see 1.5-

37. CON C, + (2,):: see 4.1.

38. CON C, + (Y, q)' : see 4.1.

39. C, ( 2 W , ( Q ) w ) ' : see 1.5.

40. C, + (w U { p } ) : : see 1.5.

41. C , + ( p w ) : : see40.

42. C, + ( A ( q ) ) : : see 1.5.

E h>

1 h

E C

V NI-

X

7 n E L.

W

NI.

Chapter 1

Definitions and simple corollaries

In this chapter we wiil introduce some basic definitions and discuss some partition

relations involving the ordinal spaces and scattered spaces. We will also introduce

the spaces under consideration and extend some of the earlier results to these spaces.

The relations in this chapter are aU simple corollaries of definitions, exis ting relations

in ordinâry partition calculus and some weLl known theorems in general topology.

In this section we prove a basic lemma which enables us to extend any negative

partition relation to a negative relation for larger superscripts. Thereby many trivid

and/or existing negative partition relations from partition calculus or topological

partition calculus will find their topological counterpart for larger superscripts. Such

a lemma is called a stepping-up lemma. We f i s t quote one such lernma from ordinary

partition calculus.

Lemma 1 ( lemma 24.1 (5) [EHMR]) Let n 2 2 be an integer and p be an infinite

cardinal and X be a n arbitrary cardinal, infinite or finite. Then tc t, (A); implies

2" + ( X - i - l ) ; + l .

Using this stepping up lemma, one can, for example, step-up the relation 2* + (3):

given above as follows: for each space X of cardinality at most 22n0

We will prove another lemma that dows us also to step-up from Q = 1 to any larger

n but keeping the space X and Y fixed.

Lemma 2 Let X and Y be infinite spaces, and let rn, n and k be positive integers

such that m < n. Then X t+ (Y);" implzes X + (Y, r)", where r = r(m, n, k )

ES the Ramsey number clssociated with the triple (m, n, k ) , that i s the least positive

integer that satisfies r -t (n)r proof: Choose a partition f : [XIm -+ k witnessing X + (Y);>. We define a

partition g : [XIn -t {O, 1) as follows:

Observe that there can not exist a O-homogeneous subset of X for g, homeomorphic to

Y, as such a space must be homogeneous for f . To show there is no 1-homogeneous

subset of X for g of size r. we assume on the contraxy that there is a subset A

of X of size r with g"([A]n) = 11). By the finite Ramsey's theorem there is a set

{xl ,x2, .. . , x,) c A which is homogeneous for f. This would imply that

g({xl, 22, ..., x,)) = 0; a contradiction that proves that lemma. 0

The instacce of lemma 2 that is used very frequently is for n = 2 and m = 1 :

X + (Y); implies X t, ( Y , k + 1);.

The following variation of lemma 2 d o w s the mixed relations to step up:

Lemma 3 For topologicul spaces X , Y and Z we have that

proof: The proof is similar to the proof of lemma 2. Given a partition f of X into

two pieces, we define a partition g of [XI2 into three pieces as follows:

for i = O, 1 g({z, IJ)) = i iff f (z) = f (y) = i, and g({x, y)) = 2 o h m i s e . UI

To extend the negative partition relations for infinitely many colors the following

ugument is used in [Wel]:

Lemma 4 For infinite topological spaces X and Y, the negatiue relation X t+ (Y):

can be stepped-up to X + (Y): for any n > m.

proof: Let k = Cg, the number of the n-element subsets of a set of size n. and

let {pl, pz, . . .pk) be the fmt k prime numbers. Define a partition g : [XI" -t w as

follows: for a given set {x 1, xî, ... z,) let {al, a*, ..., a i ) denote the n-element subsets

1.2 Partition relations for uncountable ordinal spaces

Since a considerable amount of work in studying the the partition relations for count-

able ordinal spaces has been done (see [Bal]), here we concentrate on the ordinal

space wl, the f i s t uncountable ordinal.

An unbounded set closed in the ordered topology, (hereafter a club ) subset of wl

with the subspace topology is homeomorphic to wt ; any strictly increasing continuous

function mapping wl onto a d u b wodd be a homeomorphism . A s t a t i o n a v subset of

wl is a subset that intersects every club subset of wl. Hence a non-stationary subset

of wl is a set which is contained in the complement of a club. As a consequence we

immediately have:

wl + (closed and unbounded, stationary)'.

A countable intersection of clubs in wl is a club itself, hence a countable union of

non-stationaq subsets of wt is non-stationary. This gives an improvement :

wl t (closed and unbounded, ( s t a t i ~ n ~ ) , ) ~ .

Considering a result of Friedman [Fr], which says a stationary subset of wl contains

a topological copy of any countable ordinals, we arrive at:

4 ( ( ) for every countable ,ordinalcr E ut.

It follows from the ordinaxy Partition Calculus that

If we reduce the number of pieces in the partitions, although in the ordinary

partition cdculus for ordinals we have

(see [To~]), we still don't get a comparable topological result. Indeed a partition

wl = So U SI into stationary sets establishes

which, using lemma (2), gives

WI ( ~ 1 3 3 ) ~ -

This gives us another striking difference between the topological and the ordinal

version of partition calculus. hdeed, it is stiU open if wl -t (a); for topological

copy of the arbitraq countable ordinal a.

To find homogeneous copies of wl + 1 is much harder. Indeed for each ordinal K

This is witnessed by the following partition f : n + 2, defined by

f ( a ) = O iff cr is limit of countable cofinality.

Clearly wl + 1 f+ f - l ({O)), and no continuous function can map w + 1 into

f -' ((1)) with the topology induced from n. This is extended by lemma 3 to

Beyond this point we enter into the realm of independence results.

A result of Solovay and Prikry [SPI uses a version of O, to provide a partition of

any ordinal r; witnessing

t, c 4 : *

On the other hand S.Shelah [Shl] uses SPFA to establish

Applying lemma 2 to Prikry-Solovay result gives us the consistency of

for al1 ordinals n. Recall that here there is no point in considering infinitely many

colors unless we assume CH, since if CH fails, we have wz + (3);. Also, in general

we have ~2 f-+ (4);.

Let's fist remind ourselves of the notions dense in itselfand scattered: A subset A of

a topological space X is said to be dense in itselfif every point of it is a limit point. It

is said to be scattered if it contains no non-empty dense-in-itself subset. Equivalently

A is scattered in X if the cornplexnent of A, denoted by (A)', is dense in X. (see [En])

It is a basic consequence of these definitions that the intersection of two open

dense subsets of a topological space is open dense. This implies that a dense in itself

space c m o t be the union of two scattered subspace. In terms of the topological

arrow notation, this leads to the positive relation

dense in i tself t (dense in i tself):.

To extend this relation for infinite subscript we r e c d the

Baire Category Theorem: In a complete metric space a countable union of

no-where dense sets is a CO-dense set. (Of course this theorem has a more general

statement, but we intend to use i t here only in this special fom, for more on this see

[En] .) It follows that

dense in itself complete metric -t (dense in itself):.

An example of a complete metric space in our work is the space 2&, the Cantor Cube

of countable weight, also known as the Cantor Set. (See 2.1 for definitions and the

equivalence of the the Cantor Set and 2W .)

Another well known theorem that has a flavor of partition calculus and is used in

the subsequent arguments is

Cantor-Bendixson theorem : Any topological space can be represented as the

union of two disjoint sets, of which one is dense in itself and closed while the other is

scat tered.

We s h d also use the following well-known scheme in constructing copies of the

Cantor set in an uncountable Ga subset of 2" :

Lemma 5 Every uncountable Gs subset of a complete separable metric space contains

a copy of the Cantor set 2W.

proof: Notice that such a space X is hereditady Lindeloff. Assume G = n,,& is an uncountable Gs set in X. Consider the collection of open sets

By the Hereditary Lindeloff property of X one can find a countable Wo C W such

that (J Wo = U W. Let W = U W, and observe that W n G is countable since

W n G = (U Wo) n G. Consider F = G \ W. Observe that F is uncountable, and for

any open nbd U of a point x E F7 U n G is uncountable.

Choose points x<o>, x<i, E F n and in separate the points with open - -

and U<l> of radius smaller than 1, with n UCl> = 0. Next we choose new points

x<oo>, x<oi> E U<<o> n F and sepaxote them with balls of radius s m d e r than 112.

Continuing this process we construct a collection of open sets {Us : s E <W2} to

satisfi:

1. Us C I/, and diam(U.) < l / n whenever dom(s) > n.

2. 2 Crt whenever s extends t ,and

- - 3. for ail s in <'2 , Usno n U,-, = 0.

- Notice that for each f E 2u7 the intersection nnEW Uf is a singleton in G, and denote

it by {x~).

claim: X = {zr : f E w 2 ) is homeomorphic to 2W.

Let H be the açsignment f I+ xf clearly a bijection. We show that H is open

and continuous. To show this note that for s E H f f { f E '2 : f > s) = U. n X. O

And the following lemma will be used in constructing copies of the rationals Q.

Lemma 6 Any countable, first countable, dense in i tself subspace of a regulat space

is homeomorphzc to Q.

proofr The proof follows from the following well-known results(see [En]):

1. a countable f i s t countable space is second countable,

2. a second countable space is metrizable if and only if it is regular (Urysohn), and

3. any countable dense in itself metric space is homeomorphic to Q (Sierpinski).

The following is a list of corollaries of the results mentioned above.

Corollary 1 1. If X is dense in ztself, t h e n X t ( dense in i t s e f l ; . Moreover,

i f X is d e n s e in itseif and Bazre, t h e n

X + ( a dense in i t se l f Gs set , ( d e n s e in itself),)'.

3. If X is uncountable separable comple te m e t r i c space,

t h e n X + ( a dense in itseif Gs s e t , (Q),)'.

4. A n y uncountable Ga subset of 2" con ta ins a copy of 2W. Hence,

2" -t ( t o p 2w,size N*)'.

proof: To prove the second part of (1) suppose {An : n E w ) is o partition of the

space X. By Cantor-Bendixson theorem we partition each A, into a scattered and a

dense in itself. If no A, has a dense in itself component, then (An)' is open dense,

and n , s ,<w(~ )c is a Gs set and dense in itself. (2) and (3) follow from (1) and

lemma (6), (4) follows from lemma (5), and (5) follows from (3) and lemma (5). O

The S.-product of the space {O, 1) is a subset of the space {O, 11% defined as follows:

where, s u p p ( f ) = {a < 6 : f ( a ) #O).

Remark: The space C,{0,1) is Frechet space [En; 3.10.D], that is, any subset

contains a sequence converging to any of i ts limit points. This implies that the ordinal

space wl + 1 does not embed in C,, that is:

Notice also that 2" naturally embeds into C, by restricting the product to w. Similarly

wl embeds into C, if n is uncountable. Therefore, the following positive relations hold

:(see sections 1.2 and L -4 )

To improve these results to C, + (wl):, C, -+ (2W)i or C , -t ( Z W . WI)' would

not be possible in ZFC as the corollary 4.1 announces the consistency with ZFC of

the following relations:

On the other hand we s h d see (section 3-6) that in ZFC, we do have the restriction

Using lemma 2 this result extends to

We s h d also establish that

E + ( ) implies Chang's Conjecture,

thereby showing that the positive partition relation has a large cardinal consistency

strength.

The cech-~tone compactification of the discrete space w , is characterized as the col-

lection of al1 the ultrafilters on W . The space w is embedded in pw via the principal

ul t rdters . For properties of the space pw see the exposition by W. Rudin [Ru]. on

each ordinal K one c m show (see [JHS]) that:

Other observation about the space @w is that it embeds no copy of a convergent

sequence, w + 1, that is

Pw ++ (a + 1):.

If we remove the subspace w from ,Bw we would be left with cech-~tone remainder

w* = ,ûw \ W . It is known (see [Ru]) that elemnts of the space w* have uncountable

character less than or equal to the continum, c.

To motivate our next definition, let us r e c d the definition of the "Frechet filter "

on N :

The convergence of a sequence S to a point s can be rephrased this way: S converges

to s if for any neighborhood U of s we have U n S E Fs. In the sequel we identify

a sequence with its index set. Rather imprecise, the above statement would then

become: S converges to s if for any neighborhood U of s we have U f~ S E 7, taking

3 to be the Frechet filter on K This idea can be generalized for an ultrafilter of w that

is a . element of w*. For a p E w' we defme W U { p ) to be a countable discrete set which

converges to a point in the sense of an ultrafilter p. That is for any neighborhood U

of the limit point we have U n w E p.

If we choose a point p E w' since the topology of w U { p ) is inherited from j3w then

also in the space w U {p) the point p has uncountable character. This fact implies

that w U { p ) does not embed in the space 2W. Hence,

Since the closure of every countable subset of a C-product C , is second countable we

can not have a copy of w U {p) embedded in C,. Hence:

Neither could we have w U {p} embedded in an ordinol space n. This is because in

ordinal spaces a countable set can have limit point of countable cofinality (hence

countable character.) It is established in [CKS] that

From this result one concludes that for any countably compact space Y,

This is so because such a Y must contain a copy of w U {p).

It is also established in [CKS] that for any space Y of size 2' :

Recall that A ( K ) , the one-point compactification of a discrete space of size K is defined

by adding one extra point to the discrete space of size rl and as neighborhood basis

at that extra point we consider all the CO-finite subsets of that set of size K. As such.

A(w) is just a converging sequence. Also, for each X 5 K we have A(X) v A(n).

Therefore, A ( K ) ++ pu because w + 1 f+ pw.

It follows from the definition of A(n) that the extra point of A ( K ) has character K .

Notice that A(n) f+ X for any K and A. Similady, A(wl ) gLt Cs. Findy , A ( K ) + 2A

for each X < K . However, the space A(&) not only embeds in the cantor cube 2" but

also we have positive partition relation about these spaces. There is a result (of Elkes,

Erdos, Hajnal. based on a result of J . Baumgartner, for regular n and of S-Todorcevic

for singular K , see [We2]) as follows:

The space A ( K ) is one of the simplest among the non-trivial spaces of size K or of

character K which makes it a good candidate for positive partition relation regarding

large target spaces. In this regard there is a result from [HJW] which shows that for

any uncountable cardinal K there is a topological space X such that

The Baire space of uncountable weight, B ( w i ) , is the countable power of the set

wl with the discrete topology. We will establish in 3-4 that under SPFA we have

This space is known to be universal for strongly zero-dimensional metrizable spaces

of weight w l . As such, this result c m be interpreted as a result about this class of

metrizable spaces.

Chapter 2

Partitioning Cantor cubes

In this section we study Cantor sets of uncountable weight and concentrate on positive

results about the cube 2'' . In section 2.1 basic properties of 2' are summorized, and

in section 2.3 we consider paxtitions of 2" into finitely many pieces to establish

In section 2.2 we introduce collections of subsets of w l , and rve investigate to which

subspaces of 2"' they will translate. These families a,re then used in 2.4 to prove

where 2i1 is the refinement of the topology of 2W1 by adding al1 the Gs-sets. (for more

on this space see [CN].)

2.1 Cantor cubes

Consider the discrete topology on the set {O, l}. The product topology on IIiEr{O, 1),

where I is an index set of size n, is what we c d the Cantor cube of weight n and will

denote by 2'<. Cleârly if X and Y are two index sets of equal size, then X2 and '2

with the Tychonoff product topology are homeomorphic.

We reserve 2% and 2Nl to denote the cardinality of the continuum and the cardi-

nality of the power set of q. Instead, 2" and 2" will be the Cantor cubes.

As usual, the point set of 2" is considered to be functions with domain n and

range {O, 11, in other words "2.

Some facts and observations about 2"

1. The Cantor middle third set C is homeomorphic to 2? To see this one can

identify elements of C with sequences of 0's and 1's that is the branches of the

binary tree '2, This implies that 2w ~t W.

2. The weight and the charocter of 2" is K. (see [Ho] for definitions and properties

of cardinal functions .)

3. For each a < iç and i E 2, define B&) = {x E" 2 : x ( a ) = i). Then, the

collection {B&) : a E w l , i E 2) forms a subbase for 2". Since the B:'s are

closed and open (clopen,) this topology is zero-dimensional.

4. 2 ̂ is universal for the zero dimensional spaces of weight n, that is, any zero-

dimensional space of weight at most K is embeddable in ZK. In particular for a.ll

v _< 6, 2" L ) S K .

5. Since weight of the space K+ is n+ and weight of the space 2^ is i( we con-

clude that icf does not embed in 2". However, any ordinal a < rcC being

O-dimensional of weight n is embeddable in 2".

6. For a regular n, any non-empty subset F of 2" which is an intersection of less

than n-many open sets of 2" contains a homeomorphic copy of 2 5 Pick a point

x E F and let {LI, : a E X < n) be open sets such that F = na,, U,. For each

a choose Fa E [ K ] ' ~ such that z E n,,, B,(z(y)) c U,. Since I = K \ UaCAFa

has size 6, we have that 2' 2". Therefore,

7. As singletons of 2" are closed, item 6 above implies that

2" -t (top 2", sire K ) ' .

S. 2"' can be partitioned into 2' pieces such that each piece is a eintersection

of open sets. An example of such a partition is

K+2 = U ( A ~ : f E K2) , where ilf = { h ~ " 2 : hl, = f}.

By (2) each piece contains a copy of the 2"'. This irnplies

2"' + ( t o p 2K+,size 2')'.

A simple case of this is 2"' -t (2*, 2 H o ) ' .

2.2 Special subsets of P(wi)

We define certain collections of subsets of wl and use the map 9 : P ( n ) + 2r

defined by @ ( A ) = X A , to translate them to some well known topological spaces.

These collections will be used in later sections.

Definition 1: A 2)-collection of subsets of n is a collection D = {d. : s E <"w)

of subsets of tc such that :

1. s 5 t implies d. 5 dt , and

2. u = S A t implies d, f3 dt = du.

Definition 2: A C-collection is the family of a l l unions U,,,, clla (f E "'2), for

some collection T = {c, : s E <W12) of subsets of n such that

1. s 5 t implies c, 5 q ,

2. u = s A t implies c, n Q = G.

Lernma 1 The image under the map ik of a D-famzly is homeomorphzc to Q.

proofi Let D = {d, : s E < w ~ ) and Z = Q"(Z?j. Clezt-ly, Z is countable and for

each s E r8 = @ ( d 8 ) is a limit point of {z8- , : n E w ) so that 2 is dense in

itself. By lemma 1.4 it is enough to show that each point z E Z has a countable base.

To see this let I = ds , and note that for every z E Z the basic open nbds

of z whose supports are included in the countable set I form a neighborhood base of

z in 2. O

Lemma 2 The image under the map P of a C-collection is homeomorphic to 2F ".

proof: Let T = {t, : s E 2) be the tree underlying the definition of a given

C-collection, C, and we will prove that the map defined by

is a homeomorphism between 2;' and !Y"'. Clearly, Q is one-to-one. Consider a

subbasic clopen set B,(l). Shen, O-'(B,(l)) is the set of all g E 2"l such that for

some c, a E t l lc Note that by definition 2 this is equal to the set of d l g E 2"' which

extend s, where s is the minimal such that a E t,. Clearly, this is a clopen subset

of 2y1. TO show that <P is open consider an s E 2, of successor height and pick

a E t , \ UrZs t,. Then using definition 2 it is easily checked that

@((f E 2w1 : s c f)) = ~ ~ ( 1 ) n \kt%.

This complet es the proof.

2.3 Finite partitions of Cantor cubes

The following proposition is an scheme for producing a sequence converging to a fixed

point, thereby producing a simple first countable subspace in the Cantor Cube 2wl.

Then, we will iterate this scheme to produce a countable, dense in itself f i s t countable

subspace of the space 2'.

Proposition 1 2"' + (2"l, w + 1) l .

proof: Consider a partition {A, : n E w) of wl into uncountable sets. Given a

partition {Xo, Xi) of 2"l , we fix z E XI and for each n E w we define Zn to be the

set of ail x f 2"' such that

x ( a ) = z ( a ) f o r a E U k , , A i ,

x ( a ) # ~ ( a ) for a E A,

x ( a ) arbitraxy when a E Uk,n Ak

As ( Uk,,, Akl = HI, we have Zn z 2w1. Assume 2W1 f+ Xo, hence Zn n Xi # 0 for rtll n. For each n, we choose r, E Zn n Xi, define Zn = {zk : k > n ) and let

Z = { a : n ~ w ) .

Consider &(z) = {x E 2W1 : x ( a ) = z (a) ) . Arbitrarily fix a, E Ai for each

i E w,and define the open sets Un = Ba, (2) n . . . n Ban ( z ) , and Vn = n B:n ( z ) .

Observe that:

2. For al1 n E w, Zn C Vn and (Vm # n)Z, n I.n = 0. As the V,'s are disjoint, Z

is a discrete subset of U{Zn : n E w ) in the induced topology from 2"'.

3. For each n and p E A,, Ba, ( z ) n Zn = B4(z) n Zn. It follows that (VO E

w l ) ( 3 n E w)Un n Zn & Bq(-) n Zn. (Indeed such n is the integer with ,û E A,.)

Hence, 2,'s converge to z in the usual sense of a convergent sequence in the

induced topology from 2Wl .

It is now apparent from the observations that {Un n Z U { r ) : n E w ) forms a local

basis at z , making Z U { z ) into a homeomorphic copy of w + 1 embedded in XI. O

Proposition 2 : 2' -+ (Zn, Q) ' .

proof: Let {Xo, Xi) be a partition of 2", and assume 2" Xo. Fix a t E XI and

by the method of the last proof construct a discrete collection (Z<,, : n E w) of

subspaces of 2" "convergingn to z. Of course, each Z<,> is a homeomorphic copy of

the space 2". As before, choose z,,, E Z<,> n XI, and in the subspace topology of

Z,,, repeat the process to construct a collection {Z<nit> : k E w ) ? and we choose

-<n>t> E Z<nk> n Xi. Similarly we construct open sets (of ZK) {(i<nk> : k E w ) such

that {U<nk> fl Z,,, : k E w) forms a local base at z,,,, in the subspace topology of

,Tc,>. Observe that in the subspace topology of Z<,> the point z<,> is a limit point

of the sequence (qnk> : k E w).

We repeat this process countably many times, to constnict a collection

{ Z s : s E of homeomorphic copies of 2", a collection of points Z = { z S : s E

where z, E 2, n Xi, and a collection of open sets (of 2'7 {Us : s E <"w) with the

following properties:

1. each zs is the a limit point of { z . - ~ : k E w},

2. each collection {Usnk n Zs : k E w) is a locd basis at zs, in the subspace 2,

witnessing that z, is of countable character in 2,.

Observe that the subspace Z is countable, fxst countable and dense in itself. We

apply lemma 4 to conclude that Z is homeomorphic to Q. O

Corollary 1 For each cardinal K

z2* + (pn, Q)'.

proof: Notice that ,On is zeru-dimensional space of weight 2K hence it is embed-

dable in 2*". Now the positive relation 22s + (2**, Q)* implies the corollary. O

2.4 Infinite partitions of

The main book-keeping tool in the proof of the following result is a collection of

O-ideals on P(wl), defined as follows:

Let {Sa : cr < wl) be a partition of wl into uncountable sets. For each C < wi

define a proper O-ideal Ic on P ( q ) as folIows: X E Ic iff

Hence, we have Io > . . . 3 1, > IC > . . . ( O < C). Notice d s o that for each 7 E d l , 1, is a-complete, and that if we choose & E 1,

f ~ r 2 TJ, then U,I,<,I A, E I,.

Lemma 3 Suppose X, Y E Io, and X n Y = 0. Then ,

[ X , Y c ] C = { Z ~ ~ l : X C Z ~ Y c and Z \ X E IC}.

contains a lattice isornorphic to (P(ul), C).

proof: Choose a Z c Ire such that Z > X and Z \ X E Ic and IZ \ XI = NI. For

example, let W = {w, : C, 5 a < wl) where w, E Sa \ (X U Y ) , then let Z = X U W.

Then, ( X U V : V E P ( Z \ X)) is isomorphic to the structure (P(wl), s) , and is contained in [X, YC] c. n

Proposition 3 (CH) Gzven a partition {Pa : a E wl) of P ( w l ) , either there is

a O-homogeneous C-collection o r for some a >_ 1, there is an a-homogeneous 27-

collection.

It follows from this and lemma ( 1) and (2) that we have the following fact.

CoroUary 2 [CH] ZW1 + (2?, ( Q I u , ) '.

30

pmof of proposition:

Given C < wi and s E SC2, we constmct a triple (X,, Y., i ( s ) ) of sets X , 2 wi , and

Y. C w i and i ( s ) E w, that satisfy

1. { X , : s E 5C2) forms a binary tree under E, and for each s E <C2 we have:

3. E = U { X t : s e x t e n d s t ) 5 X , and X, \ E E IC.

4. Y. E Io, and U{I; : s extends t ) C Y,.

5. X, E Pi(,), and for all y < wl , O < 7 < i ( s ) implies that no X E P, exists such

that (1) - (4) hold for some Y, and X replacing Xs.

In addition, if i is a limit, or if C = r ] + 1 for some q, and s = tnO for some

t E '''2, we require that :

6. if 0 # Z E Ic and Z n (Y, U X,) = 0. then (2 U X,) @ Pi(,!.

Assume that Po does not contain a C-collection.

The proof follows from the following three observations.

Observation1 [CH] If at some levef i = 11 + 1 and for some t E '2, and

x-0 and i ( t n O ) can be dejined, then ( X t - I , i ( t - 1 ) ) is also possible to defzne.

proof: Let '2 = {s, : a < wl j and assume that for some ,B < al, and ail 7 < ,B

the XS7;'s are constructed as well as XSgo We will choose a set XSp that avoids

Note that Y E Io as Io is a a-ideal. D e h e

Notice that A # 0, as otherwise for each non-empty Z E IC if Z n Y = 0, then

Z LJ ,Y,, E Po. This means that [X,,, YcIc Po, and lemma ( 3) gives us a contradic-

tion with the assumption that Po does not contain a C-collection,(which can easily

built inside F ( w l ) if CH holds.) Choose i(s,^ 1) = min(A) and choose Z to witness

that fact. Define X.; 1 = X,, U 2, and Y,,- 1 = Y,,. Now Xda 1 satisfies (1)-(5).

Observation2: The constrcrction must be impossible for some (' and some s E (2.

proof: Otherwise, for each f E 2 let Xf = X,. We will show that

X = {XI : f E *'2) C Po, which is a contradiction as X is a C-collection.

To this end, we fLY a f and assume XI E P, for some 7 > 0. For some limit C > y

and s = f Ic, by (3 ) we have Ac = X , \ UVcc{Xt : t = f 1,) E IC. Let

Also, Xf n Y, = 0. Then, (5) implies y 3 i(s). Now, if s' = f for some other

limit 5 > C, then 7 2 i(s') 2 i(s). Furthermore, from ( 6 ) we have y 2 i ( s f ) > i ( s ) .

Since this is true for arbitrary limit C, it follows that y > b for all d E q; a

contradiction that implies 7 = 0.

Observation3: When the constrvction stops, there is some y > 0 such that P,

contains a D-collection.

To prove the observation suppose we stopped at a level with s E C2. Consider

A = U ( X t : s extends t ) and B = U(k; : s extends t ) .

By our assumption [A, BCIC Po Choose D E Ic with D C ( A U B)' and y > 0

such that X, = D U A E P,. Consider Y , = B and notice that (X,, Y,, y) satisfies (1)-

(4) with -y = i (s) . By repeated applications of (5), if needed, we h d a set such that

(X,, y,', i ( s ) ) satisfies (1)-(5). Now by the assumption (that the construction stops)

( 6 ) must fail for this triple. Choose Zo E Ic witnessing this fact, Le. Zo U X , E Pi(,)

We, recursively, build (2, : n < w} by, at stage n considering (X,, y, i ( s ) ) where

Y: = Y, U U{Z, : m < n). Of course for each n, X," = X , U Zn E Pi(,) and satisfies

(l)-(S)-

For each n we continue to find a set witnessing that ( 6 ) fails for the triple

( X ) , k;", i ( s ) ) . We will continue to construct a collection { z ( " * ~ ) : n < u ) such

that X," U Z(n*m) E PI:(.), and a collection {YCn~"> : m E d) such that for ail

m < w , E:cnvm> = U{Z<n,h> : k < m) LJ y, and that the triple (X:, x<n*m>, i ( s ) )

satisfies (1)-(5). Notice that the collection {X," u Z<"*"' : rn E w) forms a A-system

with mot X,". Recursively, we obtain {Zr : r E cww) and {r : r E '"w) and

{X: : r E <"w} such that that {Xi ü Zr-" : rn E w ) forms a A-system with root

X:, X,' U ZrAm E Pi(,), and that the triple ( X i , Y'-", i ( s ) ) satisfies (1)-(5). Together

with -Y,, the family {Zr : r E <"w) f o m s a D-collection. O

Chapter 3

The rnethod of elementary

submodels

In this chapter the C-products of topological spaces are further studied and the con-

sistency strength of &{O, 1) + (wl): is examined. Semi-proper forcing axiom

(SPFA) and the Chang's Conjecture (CC ) are introduced and is shown that

CH^ t (wI ): implies CC.

Also, the reflection principle is introduced and is used in the construction of a struc-

ture that will be translated into B ( w l ) , the Baire space of wezght wi, and it is shown

t hat

S P F A implies 2" +(wl+l,B(w1))'.

Section 1 introduces elementary submodels and some uguments about them. In

section 2 the Reflection Principle and a consequence of it (a key lemma due to S.

Todorcevic ) are introduced. In section 3 a structure is constnicted which is translated

into a Baire space. In section 4 we use this to modify an argument of S. Shelah's and

prove that under SPFA,

Section 5 establishes that C, (wl)i implies CC, and section 6 uses the technique

used in section 5 to establish the negative relation C, + (wi):.

3.1 Elementary submodels

The proofs and the constructions in this chapter are based heavily on the notion of

elementary submodels of some structure of the fom ( H e , E) (see the notations below.)

Definition 1 A submodel of a model N is elementary, denoted by iM 4 N, zf for

any Iormula Q with parameters from M , we have that M 4 i f l N N 4.

The following resdt is used in establishing the elementarity of a submodel of He.

Tarski-Vaught test : For a subrnodel M of N , we have M 4 N i f i f o r each

f o m u l a d(y, xi, ..., x,) with parameters xi, .. ., x, f rom LM

Notations

1. For a cardinal 8, He denotes the collection of al1 sets whose transitive closure

has size less than B. It is known that for a regu1a.r O , (He, E) is a transitive

model of axioms of ZFC other than the power s e t aziorn ( see [Ku]).

2. For any formula 4, dHe denotes the relativization of the formula 6 to Hel and

it means He C 4. It is known that is absolute for transitive models that

contain He, in particular, if X > d , then (HA + iff (see [Ku])

3. We shall frequently and implicitly add a well ordering 5, of He as a predicate

and we consider (Ho, E, &) instead of (Ho, E), so that for a given set A, we

can take Skie (A), the skolem closure of A in (He, E, &) .

4. For elementas, submodels M and N with M 4 N, we say N end-extends M if

As a first elementaxy subrnodel argument we will show that

Lemma 1 Suppose M 4 HA, 8 < X and 0 E M. Then, He E M and M fi He 4 He.

He E &Id :

Since O E HA, we have Ho E HA, so that HA "3y (y = He)". And as M 4 HA,

we have (3y E M ) HA /= "y = Hen, which means that (3y E M) y = He.

M n H e + H e :

Fix a formula $(y, xi, ..., 2,) with X I , ..., xn E M n He such that

He "3yr#(y, xi, ..., x,)', then follow the chain of implications:

HA "3y E He 4He(y, XI, ..., x,)"

* (3y E M ) HA "y E He and dHe(yl XI , ..., x,)"

* (3y E hl) M C "(y E He and xi, ..., x,))"

(3y E M n H ~ ) M t= x,, ..., J,)"

(3y E 1M n He) HA + "#He(yl X I , ... x,)"

* (3y E M n He) 4He(y, xl1 - - * l ~ n ) - ( 3 ~ E M n He) He l= U 4 ( ~ , x i l *.* 1 4 " .

We, now, apply the Tarskz-Vaught test to deduce M fi Ho 4 He.

We shall also need some standard facts about the statzonary sets and closed and

vnbounded sets (clubs) in the structures of the form [AlHo. The following facts will be

occasionally used (see [Je] .)

Facts about clubs and stationary sets

1. The collection of a l l countable elementary submodels of a He forms a club in

[HO]

2. a) If C is club in [AlHo then C* = {X E [B]'o : X n A E C), contains a club in

[BIHo, and if C is club in [BIN0 then Cla = {x n A : x E C}, contains a club of

[A]

b) If A Ç B and S C [AlH0 is stationary, then S* is stationary in [BIHo.

Conversely, if S is stationary in [BI" then SIA is stationary in [AIH0.

3. For any set A we have:

a) If F : [A]<" -+ [.4IW, then the set CF = (x E [Alw : (Ve E [x]'") F ( e ) C x)

is a club.

b) (Kueker's lemma) For every club C in [A]" there is a function

F : [A]<" + [A]" such that CF 2 C.(see [De])

3.2 Reflection Principle

The Reflection Principle (RP) is the following statement: (see [Be])

For any cardinal n, any set A of size n, any stationary S C [A]*o. any cardino1

X > n and any countable Mo 4 HA, there is a continuolls E-chazn {Me : [ < dl)

of countable elernentary subrnodek of HA which starts from Mo, and a n stationary

subset E C wl such that (Vc E E ) ME n A E S.

Given a countable elementary submodel M, and some ordinal (Y @ Ml we often

extend M to aa elementary submodel that extends M and contains a. In doing so,

many new objects that may not be desirable are also added. We use RP to provide an

abundance of some caref ly chosen ordinals so that extending M using these ordinds

does not include undesirable ordinals.

Ln the following discussions we assume B = (2'~)+ and A = )+. and for any

countable elementary submodel M of HA we let

DM = {a < wz : (Vf E EM ri * w 2 ) (f is regressive f (a) E M ) ) .

Lemma 2 (RP ) The collection 7 = {M 4 He : DM iS unbounded in a*)

contains a ch6 subset of [HslH~.

proof: Suppose S = [H~]'O \ 7 is stationary. By R P find a continuous E-

chain (klC : f < w l } of countable elementary submodels of HA with t9 E l&, and a

stationaxy E wl such that ('de E E) Mc n He E S.

Let 6 = UE+.. ( M E f7 w2). Since the c h i n of submodels is an €-chah, we have for

each 6. that Mt n wl < Mc+l fi w ~ , so that wi C UE<wl Mc. Hence, any a < 6 will fall

in M = UecYi ME as M is an elementary submodel and ul C M . Therefore, d is an

ordinal.

Take a countable N 4 HA such that {Me : ( < w l ) E N and 6,d E iV and

1) = N n wl E E. This is possible because there are club many N 4 He which contain

{ M E : 5 < q ) , by fact (l), and from fact (2), E' = { X E [HAIH0 : X n W I E E) is

s tationary in [HA]Xo.

Observation : For each 5 E E we have ME n He E S, hence by the definition of

S, DMtnH, is bounded in q. Let 4 = sup DMenHe, hence 6~ < w2. NOW, E f&+I

and He E Mt+l, so that DMCnHe E MF+l. It foUows that 4 E Mc+I nu2 , hence 6( < 6.

daim 1: N ri Ut,,, MC = M,.

proof: Let q = w l . Then, r ) is a limit ordinal and Ut<, 11/lc = M,. To see this,

observe that if [ < r ) then M, E N, so that M, 2 N n Ut<, ME. On the other hand,

if x E N n Ue<.i ME there is < wl such that x E Mc as follows:

claim 2: 6 = min(N n wz \ M,).

proof: We need to show that if a < d and a E N n W.L, then a E M,.

If <r E Nnd, then N t= "(36 < w i ) a E Me17wz." Hence, (3c < winN) a E M$wz,

that is a E M e n w 2 for some [ < 7. Ol

daim 3: 6 E DMqnH,.

proof: Let f E M, n He n *w2 be regressive then, f E N. As 6 E N , we have

f (6) E Nn6, hence by claim (2) we have f ( 6 ) E M,nw2. It follows that d E D b l v n ~ , O

Cloim (3) implies that 6 < d,, contradicting the observation which maintains

6, < S. a

proof: Let C C 7 be a club. From fact (3 ) we have

HA " ( 3 F : [Hd]'" + [HelW) CF C C."

By elementarity of M and the fact that 0 E M we have

( 3 F E M) (F : [Hs]'" t [Holu and HA + "CF C C.)"

Being closed under F, M fi He E CF, hence it is in C, aad DwnH, is unbounded

in us.

The following argument shows that DM must dso be unbounded in u*.

We observe that any f E M nY ~2 is already in He because Jtc( f ) l < (2W2)+ = 8.

It follows that

Va /01~ DM"& f (a) E M n He Ç M.

This, of course, is true for a.ll f as described in the previous paragaph; so that

DMnH, C D M . This implies unboundedness of DM in y.

Fix a countable elementary submodel M of HA, and let LM{") denote skA(MU { a ) ) .

Remark: (Ve E Du \ M ) e = r n i n ( ~ { ' } n w* \ M).

proof: If 6 < e and d E ~ ( ~ 1 , then there is a formula q5 wit h free variables in

M U {e) that uniquely defines 6. We define a regressive function h E M fl such

that h ( e ) = 6, which is indeed a regressive Skolem function, as follows:

rnin{a < z : $(a, al, ..., a,, r ) ) if not e m p t y

O otherwise

Clearly, h E M n *w2, and h ( e ) = 6. It follows that 6 E M.

Similarly, one can prove that for each e E DM

3.3 A correspondence between [u2]*0 and P(wi)

In translating the structures between [w2IH~ and P(wl ) we use a mapping 9 defined

based on an almost disjoint f ami ly of subsets of q, of size N2. That is, a collection

A = {Aa : a < u2) C P ( w l ) such that for dl distinct a and /3 in u2, IA, n AD] 5 No

The existence of such a family is a fact of ZFC.

The map 9 : ( [ w 2 j U ~ , 2) (P(w1), C), first considered by B. Velickovic(see

[We2]), is defmed as follows:

Notice that almost disjointness of A implies that

And, this implies that @ is an order-preserving injection, hence an isomorphism onto

its image. The range of 8, however, is not all of P(wl), for example wl $ @"([w~]*~).

Clearly, the image under Q of any continuous strictly increasing c h i n of iength wl

in [w2INo is a copy of wl in P(wl). The union of such a chain, however, does not belong

to [u~]'~. For the purpose of finding converging wl sequences in P ( w i ) we will have

to arrange for a copy of wl in [w2lU~ whose image under @ is destined to converge to

a fixed point in P ( w l ) . If we plan to have a copy of wl converging to an uncountable

Y C wl we start with an almost disjoint family of subsets of Y.

In the next section we will be interested in h d i n g homogeneous copies of wl and of

a Baire space of weight N i . The image under O of the following structure y contains

such spaces.

Construction of y (RP)

We start with a countable elementary submodel M . It is a consequence of SP F A

that 2*0 = N2, (see[Be]) . Therefore, IcWtwl ( = N2. Also, (DMl = Nz. We choose an

ordering 5 of <wlwl in type wz that respects the original partial ordering of <'lwl.

Then, using corollary 1, we construct M = {Ms : s E <"lwl) as follows:

2. iIfs = UtCs Mt, for s E "wl, for any lirnit a.

3. 111s-7 = s k x ( M S U {e)), where

e = min LIbf. \ r n a ~ { s u ~ { ( ~ ~ n w2) : t 5 s), sup{(Ms-% 4 : 0 < y)).

For each s E <' lw~ let X' = M' n w2, and Ys = Q(Xs), and define

y = {Yr : s E ' w l ~ ~ ) . Notice that each branch of y is isomorphic to wl. Also,

define Y' = {Yf : f E '"wi), and Z = @"(yt), where Q is the chuacteristic function

assignment map defined in section 2.2. As such, Z is a subset of 2"' and with the

subspace topology we have

Lemma 3 Z is homeomorphzc to B(wt).

proof: R e c d that B ( w l ) is "wl with the product topology, where wl is given the

discrete topology. Define F : "wi -+ 2, by F( f) = xy,.

ctaim 1: F is a bijection.

proof: If f # g then there is n E w such that f (n) # g(n). Let s = f 1, = gl,.

Suppose f (n ) = 7, g(n) = 6, sny 5 9-6, MSe7 = skA(MS U {a)) and M"-' =

skA(MS U { P ) ) , for some cr and p. By the construction, LW' wiU never contain a.

Choose E Aa \ 8(Xg), and notice that F( f )(c) # F(g)([), so that F ( f ) # F ( g ) .

daim 2: F is open.

proof: Choose a subbasic open set of "wl , Say D ( n , a) = { f E Wwi : f (n) = a).

Observe that

F"(D(n, a ) ) = { F ( f) : f(n) = a) = { X ~ J : f(n) = 0)

= { X ~ J : f l n + & ) = a) = {xv : Y' C Y with s (n) = a)

which is open in the induced topology of Z from 2"'.

claim 3: F is continuous.

proof: Choose a subbasic open set in Z , Say for some i E {O, 1) and a < wl,

B a ( i ) = {f E Z : f(n) =il. casel: i = 1. &(1) = {h E Z : h(a) = 1) = {xu : Y E y' and a E Y). Choose

s E such that a E YSnP \ Y for some p E ai. If no such s exists then either

cr E Ys for al1 s or a Ys for al1 S. Hence the set {Y E y' : a E Y) is either empty

or it is the entire 2; so that B&) n 2, is empty, or equals to 2. If such s exists then

B a ( l ) = {xY, : s C f). Then, P1(B,(l)) = (f E "w1 : s c f) which is open is "w l .

casez: z = 0.

&(O) = {f E 2 : f(a) = O ) = {xY: Y E Y and a $ Y ) .

Consider s as before. If no such s exists then either a 4 Ys for a l l s or a E Ys for ail

S . In the former case F-'(&(O)) = 2, and in the latter F-'(B,(O)) = 0, both open

sets. If such a s exists then for some 0

which is open because each componont of the union is clopen.

This establishes the continuity of F, and finishes the proof.

3.4 SPFA

A semi-proper partial order as a forcing notion does not destroy stationary subsets of

q. The Semi Proper Forcing Axiom ( SPFA) is the following statement:

For any collection V of dense open subsets of a semi - proper partial ordering P

there exists a V - generic filter G, i.e. a G 2 P such that:

1. G is af l ter , and

2. for ail LI E D, G nD # 0.

A typical use of S P F A that we shall have here is as follows. Suppose the elements

of some partial order P are structures similar to countable ordinals, and a suit able 2,

guarantees the extension of each element of P into arbitrary large countable structures.

Then, the the D-generic filter G will contain large enough "comaptible" structures,

so that U G becomes an structure similar to wl.

It is known (see [Be]) that SPFA implies the RP.

Proposition 1 (SPFA) The following topological partition relations hold,

proof: It would be sufficient to prove that

where, y is as constructed in section 3.

Given a partition {Xo,X1) of P(wl) we assume that y y% XI. Observe first

that Xo contains an uncountable subset of wl, since otherwise for any uncountable

co-uncountable subset Y of wl, we would have

Note that [Y,wl] as a lattice is isomorphic to P ( w l ) , which implies y ~t XI; a

contradiction.

Choose an uncountable Y E Xo, and let A = {Aa : a < w2) be an aimost disjoint

family of subsets of Y such that Y = U A. As in section 3.2 we define a map @ and

we let h: = W 1 ( X ; ) , for i = 0 , l .

Now, we define P as follows:

p E P iff p = ( X , : a $ 7) C Ko is a countable continuous increasing closed

chain. (Notice that each condition p is considered to be a function with domain y + l for some countable ordinal 7. Similarly, range(p) is the collection {X, : a < y} of

countable subsets of wz. ) The ordering of P is end-extension.

Case 1: P is semi-proper.

Define for each a E wl

Da = { p E P: a E dom p ) ,

and for each a E Y, let

E, = { p E P : (3b E range(p))(3/3 E B) a E 4).

We wil! argue that Da's and Es's are dense in P.

Da 's are dense, as otherwise any extension of some {Xt : 5 y } C [u2]'~ is in

h; . This would mean that [ w ~ ] ' ~ ~t Ki, which implies y XI.

Similady, E,'s are dense because

and for each p E wz, the collection B = {B E [w2IN~ : ,û E B} is isomorphic

to [4'0 and therefore must contain element from Ko extending any given chain

p = {Xa : a 5 Y) from P.

Now let

Z) = {Da : cr E wl)

and

E = {E , : a E Y ) ,

and by semi-properness of P we choose a 2) u E-generic subset G of P.

Observe that g = (J G satisfies:

1. g : wl + Ko is a continous, increasing map. Therefore,

{@(g (a ) ) : a < w l ) = wl, and is contained in Xo.

2. From the E-genericity of G it follows that

Of course, (2) implies that the copy of wl created by (1) will converge to Y ( i . e . its

union is Y;) giving rise to a copy of w l + 1 in Xo.

Case 2: P is not semi-proper. Then, there must exist an stationary subset S of

ol such that ,

IFp "S is not stationary in W1."

Fix a narne r , a stationary S as above, and a condition p such that:

p It- "T is a club in W l , and S n T = 4."

Let X = (22'2)+, a d notice that HA contains all the information discussed this

f a . Take a countable elementary M 4 HA which contains the above information as

well as ik1 n wl = S E S.

Sub case 1: For all the elementary end-extensions N of M we have N n wz E Ki.

Recall that in section 3-3 we constmcted a tree M = {Md : s E <W1 u ) of elementôry

end-extensions of a given elementary submodels and a tree y of the traces of elements

of M on U* Each branch of this tree translates to a copy of wl and a n initial part

of this tree translates to B(wl) (see section 3.3). Indeed the assumption that for any

end-extension iV of M we have N nw2 E K I , implies that Y c KI which is more than

needed in both cases (a) and (b) of proposition.

Subcase 2: For some end-extension N of M T we have N n L J ~ E Ko.

claim : There is a descending chah of conditions { p , : n < w) below p , and an

increasing sequence of countable ordinals {y, : n < w) c o h a l in 6 such that

dom pn = 6, a d

proof: Pick a sequence (6, : n E w) of ordinals less than 6 converging to 6, let

N n wp = {In : n E w), and define

D n = { q ~ P : C n E U r a n g e q , and 6 n ~ d ~ m q ) -

Clearly, for each n E w, D, E N and Dn is dense in P. Since

and so on. Inductively, a descending sequence { p n : n < w) is constructed . such

that (1) holds and for each n E w pn E D, n N. Then, Un,, dom pn = b and

U n E w U range pn = N nu2. O

is a condition, as it is continuous and closed. Notice that,

q Ii- {in : n < W) c r and is unbounded belowd.

Therefore, q IF "8 T " .

But, q 5 p and p Ii- "8 E Sn. It fdows that q Ii- '8 E r fI Sv, which contradicts

JI ~ t - U~ n S = 4." O

3.5 Chang's conjecture

The SPFA used in the last section is a forcing axiom whose consistency proof uses a

strong large cardinal assumption. Would the consistency proof of [u2JH0 t (w&

also need a large cardinal hypothesis? In this section we will demonstrate that this

is indeed the case,(see corollary of proposition 2.)

Chang's Conjecture , hereafter CC, is the statement that every structure of the

fonn (q, w l , <, ...) for a countable language hm an uncountable elementary submodel

B such that B n w l is countable.

It is known(see [Ka]) that CC implies the consistency of ZFC hence it is is a large

cardinal hypothesis.

Assume the negation of CC and choose a counter-example 2l = (w2, wl, h, <, . ..)

to CC that contains as a predicate a function h : [ q I 2 + ~1 with the property that

h ( a , 7) # h(& y) cr < p < 7, and that h(a , 7) E w for a, y E q- To each {a.?}. we

associate

Ba0 = slca({a, B I )

where, of course, &({a, p) ) is Skolem closure of { a , b ) in the structure a. the s m d -

est elementary mode1 of that contains a and 8. Notice that Ba@ n wl is a countable

ordinal, so we can define the function e : [w2I2 -t w l as fo9ows:

It is easily seen that (see, for example, [ T o ~ ] )

Lemma 4 For every uncountable A C w2, the image e"([AI2) is uncountable.

Notation : For a topological space X and a point xo E X let

where, supp(f) = {a E w* :' f ( a ) # xo). This is the C-product of the topological

space X around the point 30. (see [En])

Proposition 2 Suppose X is a topo~ogzcal space such that XN0 contains no copy of

wl. Then zz X + (q): inplies CC.

proof: Assume the negation of CC and consider the function e as above. Con-

sider a partition {Eo, E l } of wl into stationary CO-stationary subsets, and define

O : Cz X -t {O, 1) as follows : for a given g E C z X, let

@(g) = i iff s ~~(e"[su~~(~)]~) E Eil i = 0,1 .

Notations:

1. F = {fa : a < w l ) denotes any copy of wl in x, X.

2. S, = supp( fa), S = {Sa : a < wl) a d for each a < a l , A, = sup (e f ' [~ , ]* ) .

4. A = {a E wz : I, is uncountable ).

5. (Mt : ,f < w l ) is an €-chin of elementary submodels of HA for some large

enough X such that HA contains ail the information discussed so far. Also

assume e, F, X, @,S, A E Mo.

6. C is a club in wl such that Vd E C Ma n wl = S.

claim 1: (V6 's CC) Ss c Md.

proof: For all a < 6 , Sa E M . ; since {fc : { < wl) E M hence Sa C i& as Sa is

countable. By continuity at 6, every P E S s ) must belong to Sa for cohally many a

below 6; which implies that ,8 E Ad''. O

claim 2: (Va < w l ) (3P > a ) Sp \Da # 0. proof: Otherwise, the support of 7, = {fp : /3 > a) is contained in Sa? which

is a countable set, whle Fa is homeomorphic to wl. This means that the countable

product XSa contains a copy of w l , which contradicts o u assumption about X. O

Remark: Notice that if C is a club in wl , then {fa : a E C) is also homeomorphic

to wl so that the above argument produces an ordinal E C as descnbed above.

Lemma 5 (V6 E C) Ss = A n Ms.

proof.. C:

Choose 6 E C and a E Ss. Then, CY E Ms as Ss C Ma, by remark (1). Assume for

a moment that a @ A. Then, as A E Mo c Ms,

Hence,

Md I, is countable.

It fouows that 1, C n/f6 ll ui = 6, so 6 @ I,, a contradiction.

Choose a E A n Md, then l& + a E A, so that

1 ' ' 1 is uncount able " .

This transfers to:

This together with the fact that "3 is continuous at 6" gives that a E Ss. This

finishes the proof. O

Clairn 1: A is uncountable.

proofr Fix a d E C. Since A E fils, then if it were countable it had to be a subset

of Ms. Hence, to prove A is uncountable it is enough to prove that A \ MJ # 0. To

this end, choose a 7 > d in C such that S, \ Ss # 0. The existence of such a 7 E C is

guaranteed by claim (2) and the comment following it. Choose a E S, \ Ss. Because

S, = A n M, and Ss = A n Ms , we have a E A \ Ms. This finishes the proof.

R e c d that A, = ~u~(e"[S,]~). We are planning to prove that A = {A, : a E C)

contains a club subset of w ~ ; so that

A n E; # @ for every i = 0,1,

which implies

G 1 ' ( 3 ) = {O, 1).

Since .F has an arbitruy copy of wl this means that 8 witness C Z X t., (ai):

finishing the proof of proposition 2.

Claima : For all 6 E C, As = 6.

pmof: 5:

For d a and 4 in Ss we have e(a , /? ) E At8 because Ss C M,,. Hence. e ( a , 8) < 6. Therefore, Ad 5 6.

>: - suppose Ag < 6, hence Xa E Ms. Now it follows from lemma 4 that

that is,

( 3 q 13 E Md fi A) e(cr, /3) > As.

By lernma (5) we have that a, P E S, which contradicts As = sup (e"[S,I2).

It follows fiom claim (2) that C Ç A and this finishes the proof.

Corollary 2 C L -t (wl): irnplies CC. Or more precisely,

[4*0 + (wl ): irnplies CC.

3.6 Partitioning the C-product into three pieces

In this section we modi@ the proof of section 3.5 to establish the following:

Proposition 3 Suppose X is a topological space such that XH0 contains no copy of

Before we start the proof with the notation as in the previoüs section we will define

for each g E x4 X ,

The following observation about D ( g ) are usefull:

1. For each g, D ( g ) fi cc.1 is an ordinal.

2. If Ç = {g, : a < w l ) is a sequence of elements of C Z X such that

(supp(g,) : a < w l ) is increaszng and continuow (that is for a < 0,

supp(g*) C supp(g'3) ~ U P P ( % ) = UT<, supp(g,) for limit a, then

{D(g , ) n wi : cr < w l ) is a n increasing continuous collection of ordinals in wi.

3. For each g and h with supp(g) C supp(h) if D ( g ) n wl = D(h) f i wl then

min(D(h) \ D ( 9 ) ) n w 2 s u p ( D ( 9 ) n 4-

proof of observation 3: Let a E ( D ( h ) n s u p D ( g ) ) n y . We daim that a E D ( g ) .

Choose y E D ( g ) n wz such that 7 > ct and consider the ordinal

Since D ( g ) is an elementary submodel of (w2,wl. f , ...) which contains ( and y

and since a is the unique solution to the equation f = f (a , y) if follows that

a E D ( g ) - O

proof of proposition: W e define maps

Claim: Suppose the supports of the elements in Ç = {g, : a < wl ) form an

increesing and continvous chah of sets, then ot least one of the <P"Ç or Q"Ç contains

a club.

proof of claim: It follows fiom observation (2) that &"' is increasing and con-

tinuous, as the supports are increasing and continuous. If it is unbounded, @"F will

be a club in wl , and if it is bounded then there must be some a0 such that

In this case, by the observation (3) we have D(gli) end-extends D(gJ whenever

p > cr 2 ac. This property of the D(g,)'s will guarantee that ik(g,) (ao 5 i < q),

fonn an increasing continuous collection of ordinals in wl of length wl . It follows that

in this case \E"Ç contains a club subset of wl. O

Now we d e h e a coloring c : E z X ) -t (0 ,1 ,2) as follows. Let {So, SI , Sz) be

a partition of wl into stationary sets, then

c(g ) = min{i : Q ( g ) @ Si and Q ( g ) @ Si).

To show that this partition does not allow a homogeneous copy of dl we assume

3 = { f, : a < w l ) is m y copy of wl in x P X . Using the arguments in the previous

section we can select a subcollection Ç = {g, : a < wl) of 3 such that the supports

{szlpp(g,) : a < w l ) form an increasing continuous chain of subsets of wz.

It follows from the daim that one of the Q"Ç or i k " ~ will be a club, which intersects

al1 the Sis, that is, either for some go, gl and g2 in Ç, we have O (gi ) E Si for i = 0,1,2,

or t hat for some gl , gz and g2 in 8, *(si) E S, for i = 0,1,2. In arîy case we have

This means that 3 is colored by c in more than one color.

Remark: It follows from proposition 3.6 that the negative result above is

strongest possible, in the sense that the number of colors cannot be decreased to

two.

Chapter 4

Negative Partition relations in

torcing extensions

In this chapter we establish the consistency of some negative relations. In the first

section we use a c-closed forcing to partition any ground mode1 topological space X in

such a way that for any uncountable first countable space Y we have X +-+ (Y):. In

the second section we use an iteration of O-closed forcings to establish the consistency

of the the relation 2W1 t) (uI)ii

4.1 Generic filter as a partition

First we bring a definition and a lemma from general topology.

Definition 1 A topological space X is said to concentrate around -4 c X whenever

any open set about A contains al1 but at most countably many points of X.

Cemma 1 Uncountable first countable countably compact Hausdorff spaces cannot be

concentrated around their countable subsets .

proof: First observe that:

Any topological space concentrated about a countable set is Lindelof, hence it

is enough to prove : "no uncountable, h s t countable compact space is concen-

trated about count able subsets."

No uncountable, first countable space c m be concentrated about one point.

Let B be the set of all x E X such that every open neighborhood of x contains

uncountably many points of X. Then, B is closed. Also, if X is compact first

countable then B is dense in itself.

This cas be justified as follows. Let xo E B be isolated and h d open set Cr such

that U n B = {xo). Fix a countable decreasing collection of open sets {Un : n E

w ) with E U and n,,, Cr. = {xO) Since Uo \ {xo) = UnEw(Un \ Un+1) is

uncountable for some no, Un0 \ Un,+* must be uncountable. Let y be a complete

accumulation point of CS, Then, y E G\ u,+~ C U. But, by definition,

y E B; a contradiction.

If B # 0 and X is compact first countable, then B is uncountable.

This is an immediate consequence of cech-~os~ki l ' s lemma (see [En] )and of

the previous observation on B.

If X is uncountable and concentrated about A = {a, : n E w ) , then B n A # 0,

(in particular B # 0.) To see this, suppose A f~ B = 0 and choose countable

open neighborhoods Un of a,. But, U is countable, A c Cr = UnEw Un, and as

U is open, X \ U is countable; a contradiction.

To prove the lemma , we assume X is an uncountable, compact, first countable

space concentrated about a set A = ( a , : n E w). Consider B as in observation (3).

Notice that by observations (4) and (5) , B is fkst countable, compact dense in itself.

Hence, we may assume without loss of generality that X has these properties.

Aiming towards a contradiction, we will h d an open set U > A whose complement

is uncountable.

We use (2) to h d an open set Uo 3 a0 whose complernent in uncountable. Then,

we choose an uncountable open set vo) whose closure is disjoint from closure of Uo.

Next, we apply (2) again to find an open set Ul 3 al together with uncountable open

sets ViOvi), i = 0 , I such that :

Having constructed open sets Un 3 a,, n E y, and V = {vi : s E ''2) such that

1. c K, whenever s extends t ,

2. = 0, for ail n and al1 s E "2, and

- - 3. V.-onV.-I =0,

we define

Let U = Un,, Un and notice that we have:

3. Z is uncountable.

Thus, U is the desired open set.

Given a cardinal rc and a topological space X = (n, 7), over a rnodel of ZFC + C H

we force with P = Fn(n, 2, wl). For any generic filter G, U G : n + 2 is a partition

of n, hence a partition of X into two pieces.

Proposition 1 With respect to the above geneRc partition, there is no homogeneous

copy of any countably compact uncountable j î r s t countable space inside X .

56

Before proving the proposition we mention a basic consequence of countable

closedness of P:

Remark 1 Wzth the above notation, zf A is countable, and C is a n a n e such that

p Ii- '<C is a subspace o f x wwith -4 c C, and (Va E A) ~ ( a , C) = No", then:

1. these is a condition q 5 p and there is a countable collection of open sets W A

such that for any name v:

q I k "Y iS open and(Va E A)a E v (3W E Wa4) a E WnC c un[."

2. There is a condition q < p such that for a n y name u,

q I t - "v is open in C and A c u (3W E AC WTi C c v."

pro0 f:

1. Let A = { a , : n E LJ). There are names un for n E w , such that,

p IF "un is a countable local ba is at à, with respect to C."

Since 7 is a b a i s for the topology of X in the extension, one can find a condition

po 5 p and a countable collection, Wo, of open sets such that

po Il- "(Y open v 3 ào)(3W E w O ) ü E W n C c v."

We call this process deciding a local b a i s for ao, below p.

Next, we decide a local basCs for ai, below PO, and continue to cons tmct a de-

scending chain of conditions p,, n € w, along with countable collections of open

sets W , which are decided by the p, 5 to be the ground model local b a i s for the

a, 'S.

The desired collection Wa is the unzon of W, 's and the desired condition is

Q = Un<u ~ n -

2. In this part we use the condition q and Wa as above. Gzven a name v , =ch

that q Il- ' A c v and v is open," we know frorn (1) that

q Ik "(Vn E W)(3Wn E w&, E Wn n C c v." i.e.

But W E 7 as zt i s a countable union of elements o f 7 and Our partial order is

a -corn pl et e . a

proof of proposition: Fix names and r, and a condition p E P such that

p l f is an uncountable f i s t countâble countably compact, ond T : f v U r-'[O).

Define for r 5 p,

Ar = {x E dom(r ) : (39 5 r ) q IF "(3a E { ) ~ ( a ) = 2")

and

A, = {x E X : r Ii- "(3a E E ) r (a) =I").

Claim 1: With the notation as above, for any condition r there exists q 5 r such

tâht q "Aq C ~ " ( c ) . " proof of daim: Let qo = r. For each positive integer n, having chosen q, as

dom q, is countable, by countable closedness we choose a condition q,+, 5 q, such

that q,+l It- "Aqn 5 rt'(()". Define q = q, = inf {q, : n E w ) and notice that

Aq = UnEv Agn, and q It- "A9 C rff(c)". O

We are planning to find a z E A and a condition below p which contains (z,l) .

This, of course, implies that the image of T meets the color 1 as well; hence the desired

contradiction is reached.

Notice that whenever, in the following construction, for some r 5 p it happens

that A, \dom r # 0, then we may choose r E A, \dom r and extend r to r U { ( z , 1))

to get the required condition. Thus, we assume throughout, that for any r 5 p we

have A, S dom r .

Claim 2: Given a condition q 5 p there exists r 5 q and a W, c 7 such that

r Il- "Y is open in rn([) and A' C v" (3W E w,) A' C W n r'(e) c v."

proof: By daim 1 extending q we may assume that q Il- "-4s i ( 5 ) ." We use

remark 2 to find a condition qo 5 q and a ground model local basis Wo for Aq in the

subspace rl'([). We repeat to find qn+l 5 qn and a ground model local basis Wnci

for 119". We let r = qn and W, = Un,, W,. nl

Claim 3: There exist:

1. a sequence { q , : n < w} of elements of < asd a countable descending chain of

conditions p, 5 p n < w ,

2. a sequence of open sets (LJn : n < w ) such that .4Pn c Un, and

3. a countable collection of points (x, : n < w ) c X such that x,+1 $ Un for - n n < rn and pn IF ?(vn) = x n .

proof: fix a name qo, and find qo 5 p and xo E X to satisfy

qo lt- "77 E rn( ( ) and r(qo) = I o and AqO c rrr([)".

Since qo I t "r"(tJ is first countable countably compact," then by proposition 1,

qo It- (3U open in ~ " ( t ) ) Aq c U and ~"(5) \ U is uncountable."

Claim 2 guarantees that there is a condition p, 5 po and an open set Uo with

Am C Uo and po It- Üo C U." It follows that po It- ''rl'(C) \ Uo is uncountable."

Choose another name rll, another point x l @ A \ Uo and q, 5 po such that

ql IF ? ( I ) ~ ) = IIn. Then, we apply claim 1 to find pl such that pl IF ".4P1 Ç r"(C)".

and an open set Ul > AP1 such that pl IF "rf'(() \ (Uo u Ui) is uncountable ."

This describes the inductive steps of the construction, and finishes the proof of

claim 3. a

Define pw := Un,, pn- We have

pw lt- is fkst corntable and countably compact",

so that,

CIaim 4: There is a point r E X and a point q E such that the sequence

(x, : n < w) accumulatex to z, and such that p, li- ?(i l ) = 2.''

proof: It follows from (*) and the maximality principle that there is a name q

and a sequence of names {qnk : E E w ) C {qn : n E w ) such that

Since X is countably compact, there must be a subspace of the {x, : n E w) Say

{x,, : k E w ) that converges in X to a point 2. Then z and q are as required. 0

It follows, on the one hand , that z E *Ap,, and on the other hand, r @ APn for

all n E W. This is because APn c Un, Un is open, ând x, 6 Un 'dm > n, while some

infinite subsequence of {x,) converges to r. By Claim 4 and the definition of APn we

must have that z $ dom(pn) for all n. Therefore, z @ dom pw. Now, we me in the

situation where a E A,, \ d m p,. The sought after condition is pw U ((2, l)), and

this proves the proposition. O

Corollary 1 For euery topological space X there is a forcing notion P which forces

that X + (Y): for any first countable uncountable countably compact space Y. In-

deed, one has: X + (x, h)' for f i s t countable uncozmtable countably compact

spaces Yi and YÎ.

The problem with the above proof is that we can only guarantee that for the

ground model topological space X the above relation holds. Since our forcing notion

was a a-closed notion, we know that certain spaces wiLl remain unchanged in the

extension. An example of such spaces is C-products of any topological space. Hence

we have

Corollary 2 there is o forcing eztension in which we have

4.2 Iteration of countably closed forcings

In the above proposition we have shown how to generically putition each copy of a

topological space X that appears in the ground model. In general, new copies of such

spaces can appear in the generic extension. In this case in order to produce a result

of the sort X + (Y): in some generic extension, we need to iterate the process of

the last section until it is guaranteed there are no new copies of such space produced

and the existing ones are a l l generically partitioned at some stage of the iteration.

We will demonstrate this process in establishing the following result.

Theorem 1 There is a forcing extension in which 2w1 + (w&

proof: Clearly we may assume GCH. We define the countable support iteratzon

and a collection {Ca : a < w Z ) , of names as follows:

1. Po = 0, and Q = (2W1 )V.

2. For alI cr E wz, Ca is a P, name for (2'~ \ U4<* ( 2 W ~ )V[Ge] , m d 7~~ wiU

be a Pa- name such chat IFPa "na = Fn(to, 2, wl).

We will show that forcing with P produces a generic partition of 2"' which admits

no homogeneous copies of wl.

Remarks :

1. In defining the iteration we adopt Baumgartner7s approach [ Ba2). We use the

following convention in switching back and forth between Pa and 8, for each

a < u2: A condition p E Pa is a function with domain w2 with support in a.

Hence a Prname r would also be considered a Pa- name for 4 < a.

2. For simplicity of notation we let P = P,, we use Il- to mean IFpw2, and by name

we mean P-name . For each 7 with a < y 5 wz we use PI,,,1 to denote the

remainder of PT after forcing with Pa, that is a partial ordering such that P,

is isornorphic to P, * Pla,,l where * stands for iteration of the partially ordered

sets.

3. We d l prove with the help of lemma 2 that each Pa is forcing quivolent to

Fn(u2, z7 ~1 ), and as a corollasy we will have that each Pa is u-closed and h a

the N2-chain condition, and hence preserves cardinds and the GCH. Similarly

Pt,,l has O-closed and ha. the Hz-cc.

4. In establishing the next lemma we use the partial order noCa Fn(w2, 2, wl ),

that is the product of the part idy ordered sets Fn(wz, 2, wl ) with countable

supports. Elements of this partial order are as usual the funct ions f with domain

Lemma 2 L e t (Qa7 p,; a 5 wz) be a countable support iteration such that

Il-, " p , = Fn(w2,2,wl)", then each Q is forcing equzvalent to F n ( ~ ~ ~ 2 . u ~ )

and hence epuiualent to F,(w2, 2. wl ).

proof: First notice that &<, Fn(wz, 2, wi ) is isomorphic to Fn (u2 x {p), 2, )

which in turn is asomorphic to no,, Fn(u2 x (Y, 2, W I ) via the isomorphism

Finally, np<, Fn(u2 x a, 2? ul) is isomorphic with F'(W~~ 2, ~1 ).

To prove the f is t equidence, we conveniently let 114 stands for &<p Fn(u2, 2, ~ 1 ) .

W e will show that II; is densely embedded into û& for each /3 5 a. Each element of

IIa is a function with domain ,O and range Fn(w2, 2, wl ); such function belongs to the

ground mode1 because its support is countable. Elements of Q axe functions with

countable support but their values are names of functions rather than the functions

thernselves. Of course the elements of IIB c m also be found arnong those of Q,

and we c d such elements detennzned conditztions. Hence, we may consider IIg as a

subordering of Q via identification with the set of ail determined conditions.

We will show that the determined conditions are dense in a, that is

Let p E Q be given. First assume = 7 + 1. By definition of pl

Notice that p ( y ) is a Q,-name for a countable function from wl to (O, l),and by

O-closedness, one can decide such a funetion. So that, (3r E Q?) r 5 pl, and

3 f E [ * R ~ ] < ~ o such that r It- f = p ( ~ ) . " Now, by induction, choose r' 5 r, r' E IIo,

and define s as follows: S I , = r' asd s ( 7 ) = f. Clearly, s E and s 5 p. If

c f ( P ) = w , then we choose an strictly increasing sequence of ordinals (Pn : n E w)

cofinal in p. Define a descending sequence ( r , : n E w ) of determined conditions such

that rn E IIg, and r, 5 plp,. Let p, E IIp be an extension of { p , : n E w ) . Clearly

Pu 5 P-

If cf ( P ) is uncountable then, as usual, there is some 7 < P such that

V6 > y IF " p ( 6 ) = 0." We choose r < pl, and extend r to s by 4 6 ) = 0, V6 > 7. In order to establish lemma 2 it remains to show that each Pa is forcing equivalent

to Qa. This is done inductively and by considering the fact that at stage cr we have

that implies

At stage cr of the iteration, of course, U4<o - is 2"', but in the final extension

Uo<a b is just some subspace of 2W1. This subspace, because of 0-closedness of the

p u t i d orders, is countably compact.

Lemma 3 (Va < <iJ IF "UB<a - 6 iS countably compact."

proof: Let G be a P-generic set and r and p be names such that

It- r = U &mdV[G] + "pG is a countably infinite subset ofrGn. P<Q

By O-closedness of Py,a] we know that pc E V[G,]. But, V[G,] TG = {O, l)"', so

t hat

V[Ga] 3C E TG Ç is an accumulation point of pc.

Hence there is a Pa-name a such that

V[G,] + OG E TG and uc is an accumdation point of TG. (t)

We now claim that

V[G] + OG E TG and OG is an accumulation point of TG.

O therwise,

As f E V[G,], we have V[Ga] "uG E Ut and Uj n TG = 0" which contradicts (t). This proves that V[Ga] + is countably compact." O

for all cr E u)2 we let +a be the a Pa- name for the generic partition of ta and

choose a name ik such that It- "8 = U,,, Gan. Then,

Remark 4: IF "9 : 2"l + {O, 1) is a partition".

Claim: IF "wl y% 9-1{O}".

proof of claim: Assume, for a contradiction, that for some p E P and some narne

T we have

p I l "r : wl v @-l{~) ."

We will try to find some condition q 5 p, some ordinal a E wz and Pa-names t and

X such that ql, IFp, "A E 5, n P ( w 1 ) aad (A, 1) E q(cr)". Then we have

qlt- Q(X) = 1 and A E r (w1 , )

which is, of course, a contradiction to the assurnption of p Il- "rf'(wi ) c Q-' {O)". To

construct such a condition q we proceed as follows.

Remark 6: The Cech-Stone Compactification of the space wl with the order

topology is the space wl + 1.

proof of ~t?maTk: R e d that if every continuous function from a Tychonoff space

X to the closed intenml I is continuously extendible to a compactification a ( X ) of X ,

then a ( X ) is equivalent to the cech-~tone cornpactification of X , (see [Enl].) Also

notice that every continuous real-valued function form wl is eventually constant,( for

example see [SS] .) Clearly, such function can be trividy extended to wi + 1. It follows

that the space wl + 1 is the Cech-~tone compactification of the space wl with the

order topology. 0

Therefore we can extend T to the (unique ) function 5 defined on wl + 1. we choose

a name B for ~ ( w ~ ) , and we have p IF = r(wl) E 2w?

Worlcing in the ground model, we will construct the following collections:

1. a strictly increasing collection of ordinals (7, : n E w) c wl ,

2. an increasing collection of ordinals (a, : n < w ) c ~ 2 , and Ba,-names A,,

3. a descending chah of conditions (pn : n < w), with po 5 p,

4. An extending collection { fn : n < w ) of countable functions with domain count-

able ordinals and range {O, 11,

such that the following properties hold:

(for simplicity Un will denote Clf, = {g E 2"' : fn C g ) and notice that Un is closed

Gt5 7

a) pn IF ''0 E Un , ~ ( 7 , ) = An E Un and An E fa,", and

Assume such sequences are constructed. We let a = l i n a,, 7 = StLPn<yTn,

p, = inf{pn : n < w ) and X a Pa-name such that p, IF "A = ~ ( 7 ) = limn<,hnW.

This limit exists because T is continuous at 7. It follows that:

Remark 7: (Vp < w2) pu IF "A 6 d o m pW(/3)".

proof of remark: If for some p E wz, pw Il- ''A E d o m pW(p)", then there is an

n E w such that p, IF "A E d o m pn(@)", which implies that pw Il- "A @ Unn! ( see (b)).

On the other hand,

pu Il- '9 = lim A,, and (Vn E w) A, E Uf, 2 Uj,,-,".

This implies that pu IF "A E nnEw Un = O,,, Unn, that is (vm E #)pu Il- "A E k". which contradicts p, Il- "A Unv. al

Next we define a condition q E P as follows :

On the one hand, q Il- "A E r M ( w l ) and $ J ~ + ~ ( A ) = 1." O n the other hand q 5 p?

which implies q IF "r"(wi) c V1 { O ) , " and this is a contradiction. Q

Construction of the sequences {a, : n E w ) , : n E w) and { p , : n E w)

R e c d that 0 stands for the 7(wl) in the previous discussions.

Lemma 4 Given a countable fvnction f and a condition p E k with p IF "9 E Uj',

then, there is a countable fvnction g with g > f and a condition q 5 p sueh that

V@ < w2 q Ii- "0 E Us and dom p ( B ) n U, n rU(wl) = 0" .

proof of Iemma: As the iteration has countable support, the support of p, that

is {y < w2 : p ( 7 ) # O), is a countable set. Also, each coordinate of a condition is a

countable function, so that we can enurnerate Up<, dom p ( P ) = Upea d o n p ( @ ) as

{ b , : n < w } . Let po = p and fo = f. For each n 2 1 choose a pn 5 pn-1 S U C ~ that

pn decides whether b, is in rn (wl ) or not, and choose fn > h-1 S U C ~ that bn 9 (i/, d i l e p , IF "8 E Crfn". Findy, we let q = in f { p , : n < w) ând g = UnEu Un. Clearly.

(Qp < w2) q Il- E Cig and d o m p ( P ) n r"(wl) n Ug = 0" . O

Now we construct the sequences as follows:

Choose a po < p, a countable function fo and cro < wz such that a* is the least a

such that there is a -y < wl with po IF '~(7) E ta n Uon , and let 7 0 be the least such

7. Also let Xo be an 00-name such that po I i - " ~ ( 7 ~ ) = Xon.

For each n 2 1, having chosen pn , f,, y,, and a, such tkat yn > yn-l, a, 2

n - n n - and fn 2 fn-i satiseing

we proceed as follows:

We apply lemma 4 to p, and fn to obtain q 5 pn and fn+l > fn such that

q IF "8 E UA+, and for all p < wz q Il- "dom(p(0) ) f~ UL+, fl r"(w1 ) = 8, " and consider

the set = {y > 7 n : pn Il- ~ ( 7 ) E UBSan (0)-

Case 1 : N is countable. We choose y,+( 2 sup N , and p,+l 5 q and an+, to

be the least cr > O, such that

Case 2 : N is uncountable. In this case we let a, = an and choose y, >

for d l m > n, with y, E N. Cleady, the condition (a) is satisfied. Similady, condition

(b) is satisfied since by the lemma,

Notice that by countable compactness of the UOEan 6, N must be closed. It follows

that p, Ii- "A E N." To check the condition (c), observe that since pu Il- "A E 6" Y -

we need only to prove that pw IF "A @ UOCo b" . Assume the contrazy. If case 1

happened d the time, then pw It- "A fZ 6 for all ,O < a by the the choice of the ynYs

and the fact that (Vn) 7 > y,, which implies that 7 > supN. Therefore X @ for

any p < (Y.

If case 2 happened at some point in the construction, then for some n we have

a = a,, and p, IF 'A E Up,,,, b " for some ordinal 7. Of course we have q < (Y, = a, -

"nd T ( % ) E (JO<., 6 eventudy for all m which contradicts the minimality of a,. - This finishes the construction of the sequences and the proof of the theorem. a

Chapter 5

Homogeneous copies of w U { p )

For a fixed non-principal ultrafilter p on w ,

Definition 1 A poznt x in a topologzcal space X is the p-limit of a sequence

{x, : n < w ) whenever: U is an open nezghborhood of x iff { n : x, E LI) E p .

w u denotes a sequence converging to a point in the sense of a p-limzt.

In [CKS] it is established that for each p E w- = Du \ w, c~' t, (w u { p ) ) : . There

are points p E w' whose character is the continum, so that for such p the character of

the limit p in the space w LJ {p) would also be the continuum, c. Heence the smallest

Cantor Cube that embeds w U {P} is 2'.

In this chapter we present some techniques for proving

We shall give two different proofs of this result, each using different assumption and

technique. Section 1 demonstrates the basic construction used in other sections and

establishes -t (w u The theme of other sections is as follows. To prove a

positive partition relation for a space A we will List al l the possible partitions of the set

A into countably many pieces, Say {fa : a < n), then recursively we d e h e topologies

r, on initial pieces of A in such a way that in r, it is guaranteed that fa will have a

homogeneous copy of the space w U { p ) . In section 2 we use a combinatorial principle

for club subsets of w2, as a means of 'guessing " or "predictingn any possible countable

partition of a given set. Together with the negation of CH this wouls establish

establish 2' + (w u { p l ) : . In section 3 we will use the hereditarily separability of

the reals to give a ZFC argument establishing the same relation ZC + (w u { p ) ) A .

5.1 The basic construction

Proposition 1 For a f ied p E w*, we have 2' + (w U (P)):.

proof: We define a O-dimensional topology rP, of weight at most continum on

w 2 + 1 which refines the usual topology and (w2 + 1, T,) + (w U { P I ) : . Then. since

of course (w2 + l,rp) embeds into 2=, the proposition follows.

Recursively we construct (T, : a 5 w2 + 1) in such a way that:

1. r, is a O-dimensional topology on CY refining the usual topology;

3. each Limit point is a plimit point.

Suppose for each P < cr r d is defined. If cr is a limit ordinal, then we define

r = (U(ro : a < a)). The notation (X) stand for the topology generated by X,

that is the subbasic open sets are taken to be elements of X. If a is the successor

of a successor ordinal, Say cr = 0 + 2, then we declare { B + 1) to be open: that is

r, = ( T P + ~ U {B + 1)). For a = + 1 where ,û is a limit ordinal, the basic neighborhoods

at f l will be defined as follows: We fix a sequence of ordinals P,., increasing to ,LI (if

P = w2 we require that the B ' s be limit ordinals.) Then we define u,,p = (@,-i, Bnjr and for a fixed element A E p we let WAb = { P ) u U { U ~ , ~ : n E A). Finally, we declare

W@ = {W*J : A E p ) to be a neighborhood basis at ,û, and define T O + ~ = (Q U WB).

Notice that T, = q,,2+1 is a O-dimensional topology: this foUows from the fact that

for a l l limit B and for all A E p, the W A , ~ i s closed as it is a union of closed un's

together with the limit of the collection of un's.

Also, Ip( 5 c, which implies that at each limit point we have a basic neighborhood

of size at most c. Therefore w((r,, w2 + 1)) 5 c.

It remains to show that (w2 + 1, r,) -+ (w U { p ) ) : . Let w2 + 1 = Ko U Ki, be a

partition. Consider a sequence {P, : n < w ) of limit ordinals converging to w 2 , and

consider (&,, /3,] = un. If for some n, un C Ki, we have a copy of w u in h; . This

is because p,, is a limit point, hence there is a sequence fiom (Pn-l, &,] converging to

13, in the sense of p. Of course, by the construction, any sequence converges in the

sense of p. On the other hand , if for al1 n un n Ki # 0, for i = 0,1, then we can choose

pairs {y:, -y;) C u, such that -y: E Ki. Both sequences (2 : n < a}, i = 0,1, have

w2 as their plimit. Therefore, if w2 E Ki, then we have {ri : n < w ) C Ki, a copy of

w u {PI- O

5.2 Club guessing principle

We introduce a combinatorial principle and Ive reconstruct a proof of its existence

due to SShelah ( presented by UAbraham at the Toronto set theory seminar.) Then.

we use this principle and -CH in order to show that 2' --i (w U { p ) ) : .

Definition 2 Reeall the notation S: = {A E w* : cf ( A ) = w ) .

1. A Ladder System on S is a sequence (a : 6 E S) = r l , where each i ) ~ is an

w-sequence converging to 6.

2. A Ladder system q is club guessing if for each club C C S, there is some 6 such

that the range(r)s) C C.

Theorem 1 (Shelah) [SM?] There ezists a club guessing ladder system o n S&

proof: Fix any ladder system r) = (w : 6 6 Si). Notation: Given a club C C Si, we define CR(r), C) to be ladder system

p = {ps : 6 E S,2), as follows:

Observe that :

2. if for some 6, (V i E w ) sup(C fl r ~ ( i ) ) = ~ ( i ) , then range(%) E C.

Define a decreasing sequence (CC : Ç E w l ) of clubs in w~ and ($ : [ E w l ) of

ladder systems as follows :

- Ce+' is a club witnessing that rfC1 is not a club guessing Iadder systern,

- for a limit (, r f is defined as follows: &i) = min{$(i) : 7 < c). (it follows

fiom observation 1 that each ( $ ( i ) : y < () is a non-increasing sequence of

ordinals. )

- Ce = ni,€ Ca, f a lirnit ordinal ;

We claim that this process must end at some < < and hence, $+' is a club guessing

ladder system. Assume for a contradiction that the process of choosing CC's is not

interrupted and let C = ne,,, Ce. Clearly C is a club in w2.

Claim: V6 E C 3es E wl V i < w Ve > €6 a(i) = q?(i). proof of clazm: Since

from observation 1 we know that for al1 6 E Si and for each i E w, { q l ( i ) : 7 < N I }

is a non-increasing sequence of ordinds we c m inductively choose C E wl such that

(VÇ > t j ) & j ) = q F ( j ) , and let e = s u p {C: j < w) . CI

It follows from the claim that r)' = r)'+', which means by observation 2 that

range(& CE+', ço that CC+' would not witness that tlgfl is club guessing. This

means that the process of o u construction was interrupted. This proves the theorem.

We are ready to prove:

Proposition 2 There is a O-dimensionai topology rp on Si such that

(SO7 T P ) + (w u {PH:-

Let 7 = {rp : 6 € A) be a club guessing ladder system on S;.

proof of proposition: Define a topology rp on Si which is O-dimensional and is of

weight 5 N2 that refines the usual topology.

for each X E S: and each A E p we define the following open neighborhood of X

as follows:

W A , = { [ ~ x ( n ) , VA(^ + 1)) : n E A) U {A).

Notice that each WAVA is the union of a discrete collection of closed sets together with

a singleton. In the usual topology, such a set is closed. By declaring every such set

open in the new topology we are defining a clopen basis at each point A.

Then by adding open sets of this form at each point of Si, we get a refinement rP

of the usual topology. Clearly, rp is O-dimensional. In this topology, each point has a

basis of cardinality Ipl 5 c. It follows that w ( ( S , T ~ ) ) = c.H2.

To show (502, T ~ ) -t (W U { p ) ) : , we fix a partition of Si, Say f : Si -+ W . We

consider Al = {n E w : f-'{n) is unbounded inSi), and for each n E Af we let

C; = (0 E S: : (f I4)-l{n} is unbounded below ,3)?

and d e h e the club Cf = n,,,, C'; \ sup( f -'(w \ n ) ) . We then choose X such that

I)A C Cf. Let n = f (X) . Cleaxly, n E A!, because otherwise X $ CI, whereas E Cf

implies that X E Cf.

Now, for all k, qx(k) E range(qx) C Cf, which means that f J,*(&z) is unbounded

below r)x(k). We choose -yk E (f l,,)+(n) n [ r ) ~ ( k - l), q ~ ( k ) ) , for dl k .

Now we have : k E w ) +, X, which finishes the proof. 01

proof: 4 H implies that c 2 N2, so that (Si, T,) being O-dimensional of weight at

most c is embeddable in 2'. O

5.3 Argument using hereditary separability of R

Proposition 3 There is a O-dimensional topology r, of weight at most continum on

Z such that

( b J + ( w u {PI):*

Since such a topology is embeddable in 2', we have:

Corollary 2 2' + (w U { p l ) : .

proof of p~oposition: Consider an enumeration of the irrationds I = { a , : a < c ) ,

and let {fa : a < c ) be an enumeration of all the w-partitins of the countable sets

of irrationals in such a way that each f appears cofinally often indexed by a limit

ordinal.

Recursively, we defme topologies (r, : a < wi), each T, a O-dim topology of weight

at most continum on {ap : p < a) refining the Euclidean topology on Z. In defining

r, we take care to assign a homogeneous copy of w U {P) to fa, if possible. Then of

course, r, = C,,,T, would be a O-dim topology on I of weight at most continum.

.4t stage a, suppose rg is constructed for ail ,O < a, as required. If cr is limit. let

7, = m d if a is the successor of a successor ordinal, Say a = 0 + 1, let r, be

the topology generated by r p U ( { a B ) ) .

The main case would be when a = B + 1 for a lirnit P. We start with a countable

subset E of the reals such that fa iç an w-partition of E and a, E P. If no such

E is found we declare {a,) open. If a, 6 dom f, or a, 6 fgl({n))& for any n , we

let (a , ) to be open, otherwise we proceed as follows. Let M denote the collection of

all n E w such that a, E fgl({n))& and for each n E M let Sn = (a; : n 5 rn < w )

witness that a, $ f;l({n))&. Also choose two sequences of the rationals: (q: : k E w)

strictly increasing and converging to ap from left and (q i : k E w ) strictly decreasing

and converging to ap from right. We reduce the Sn's to subsequences such that in

the usual ordering of the reals, am < am+l < ag or a d < a:+, < a:. At this point we

introduce a partition of M = Mr U M i , where Mr denotes the set of all the n's in i1.1

with elements of the Sn approaching ao from right in the usual ordering of the reals.

Similady Mi stands for approaching as from left. If n E M' we would like the am's

to belong to [6,, q!,,+l) and sirnilarly, if n E Mr we require that an E [qk, qm+l ).

Now,to a@ we assign neighborhoods that make it the plimit of all these sequences.

To define the subbasis at ap, let u i = [k, sk+l) u [ap7 m ) and for each n E M' let

U; = (-00, ap] U (q;,, , q:]. Then to each t E {r , 1 ) and to each A E p assign

WA = U{& : m E A) ,

and let W = {Wi : t E {r,Z), A E p}. Finally we define r, to be the topology

generated by TD U W.

Clearly, r, refines the usual topology on r, and since each uk is cloed in the

induced topology of the irrationals, in T, they are also open, making the T, zero-

dimensional. This topology is of weight at most continum as 1 W 15 wl .

To show that such a topology will satisfy (Z, T,) + (w U { p ) ) : , we proceed as

follows. Given a coloring f : Z .-t w , we consider for each n < w ? a countable

dense Dn C f - ' ( { n ) ) and D = Un,, D.. Then, D is countable. Choose a large

enough limit ordinal P and a countable E 3 D such that fp = f l E and ao @ D.

If f(ap) = no, then a0 6 D implies that ap is a cluster point of (f-'({no))) Tabe

the sequence Sn0 = {a, : n E w ) as in the construction at the stage p + 1 of the

construction, and of course { a , : n E w ) U {ap } is homeomorphic to w U { p ) . Kl

[Bal] J. Baumgartner, Partition relations for countable topological spaces. J . of combina-

torid theory, Ser. A no. 4 p 17û-195, 1986.

[Ba21 Iterated forcing, Surveys in set theory. London Math. Society Lecture

Note Series 87.

[BH] J. Baumgartner, A. Hajnal, A proof (including Martin's Axiom ) of a partition rela-

tion. Fund. Math. 1973.

[BSS] Bergman, ~ a ~ i r o v s k i i , Sostak, On partition of topological spaces. Casepis pro pest.

Math, 109 (1984) 27-53.

[Be] M.Bekkaii, Topics in set theory, Lecture notes in math, no. 1476. Springer-Verlag

1984.

[CN] W.Comfort, S.Negrepontis, The theory of ultrafilters, Springer-Verlag 1974.

[CKS] W.Comfort, A-Kato. S.Shelah, Topological partition relation of the form w' t, (Y): .

[De] K.Devlin. Costructibility, Springer-Verlag, 1984.

[Do] A. Dow, an introduction to applications of elementary submodels to topology, Top.

proc. 13, 1988, 17-72.

[En] R. Engelking, General Topology, Sigma Series in Pure Math. vol 6. Heldermann

Verlag, Berlin.

[En11 cor 3.6.3 .

[EHRM] Erdos, Hajnal, Rado, Mate, Combinatorid set theory, Partition relations for cardi-

nal~, Akedemiai Kiado, Budapest 1984.

[Fr] H. Friedman, On closed sets of Ordinals. Proc of AMS vol 43 No. 1, March 1974.

[GS] J.Gerlits, Z.Szentmikiossy. A Ramsey-type topologicd theorem. (to appear )

[GRS] %Graham, B.Rothschild, J.Spencer, Ramsey Theory.John Wiey and Sons Pu blishing

Co. 1980.

[HJS] A.Ha jnal, 1. Juhasz, SShelah, Splitting almost disjoint families. Transactions of the

AMS vol. 295, no.1 May 1986.

[HJW] A.Hajna.1, LJuhasz, W.Weiss, Partitioning the pairs and triples of topological spaces,

Topology and its applications, 35 (1990) 177-184 North Holland.

[Ho] R. Hodel, Cardinal functions. Handbook of set theoretic topology.

[JS] 1-Juhasz, S.Shelah, On partitioning the triples of topological spaces.

[Je] T. Jech, Multiple forcing. Cambridge tracts in mathematics, 88, 1987.

[Ka] A.Kanamori, The higher Infinite, Springer-Verlag, 1995.

[Ku] K. Kunen, An introduction to independence proofs , Norht Holland 1980.

[KV] K.Kunen, J.Vaughan, Handbook of set theoretic topology. North Holland, 1984.

[KW] P. Komjath, W. Weiss, Partitioning Topologieal spaces into countably many pieces.

Proc. AMS, 1988.

[Ma] V. Malyhin, On Ramsey Spaces. Soviet Math. Dold. vol. 20, 1979, No. 4.

[PSI K.Prikry, R.Solovay, On partitions into stationary sets. J. of symbolic logic, vol 40

no.1 march 1975.

[Ru] W. Rudin, Hornogenity problems in the theory of cech compactification, Duke math.

J. 23 (1959) 409-416.

[Shi] S.Shelah, Proper forcing, Lecture notes in math. vol. 94 Springer-verlag 1982.

[Sh2] - Cardinal Aritbmetic. Oxford university press, 1995.

[SSl] L. Steen, J. Seebach, Counterexamples in topology, 2nd ed. Springer- Verlag.

[SS21 (p. 66-68)

[Tol] S. Todorcevic, Partition relations for partially ordered sets. Acta Math. 155 Aug.

1985.

[To2] - Partition problerns in topology. Arnerican Math. Soc. Providence 1989.

vol. 84.

[ 'W - Some applications of S and L combinatorics, Ann. of New York Academy

of Sciences. Dec. 28, 1993.

~ 0 4 1 A partition property of spaces with point-countable basis, notes of June

1996.

[To~] Forcing positive partition relations. Trans. AMS. 280 (1983) 703-730.

Po61 Some partitions of three dimensional combinatorid cubes. Journaal of

Cornbinatorial theory, Series A, vo1.68, No2. November 1994 pg 415.

[Wel] W. Weiss, Partitioning the quadruples of topological spaces. A tribute to Paul Erdos,

Cambridge University press, 1990.

[We2] -- Partitioning topological spaces. Mathematics of ramsey theory (J.Nesetri1,

V.Rodle, eds) Springer-Verlag, 1990.

IMAGE NALUATION TEST TARGET (QA-3)

APPLIED & IMBE. lnc 1653 East Main Street -

,=: Rochester. NY 14609 USA i I -- - - Phone: 7 16i48Z-WOO -- -- - - Fa: 71 6/288-5989

O 1993. Applied Image. Inc.. Al1 Rights Reserved