Set Theory and its Applications

Lecture Notes in Mathematics Edited by A. Dold, B. Eckmann and E Takens

1401 J. Stepr~ns S. Watson (Eds.)

Set Theory and its Applications Proceedings of a Conference held at York University, Ontario, Canada, Aug. 10-21, 1987

Springer-Verlag Berlin Heidelberg NewYork London ParisTokyo Hong Kong

Editors

Juris Stepr~ns Stephen Watson York University, North York, Ontario M3J 1P3 Canada

Mathematics Subject Classification (1980): 54-06; 04-06 ISBN 3-540-51730-8 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-51730-8 Springer-Verlag N e w Y o r k Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reprod'uction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1989 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr, 2146/3140-543210- Printed on acid-free paper

Preface

The Set Theory and its Applications Conference at York University was held in Toronto, Canada during the two weeks of August 10-21, 1987. It was attended by 80 mathematicians from 12 different countries. Financial support for this conference was provided by the Natural Sciences and Engineering Research Council under the auspices of the Canadian Mathematical Society and by York University through both the President's Office and the Office of the Dean of the Faculty of Arts. The organizers would like to express their thanks for this financial support without which the conference would not have been possible. The conference featured both contributed talks and a series of invited lectures on topics central to the study of set theory and its applications, particularly to general topology. The organizers would like to thank the invited speakers James Baumgartner, Arnold Miller, Andreas Blass, Neil Hindman, Stevo Todorcevic, Ronald Jensen, Hugh Woodin, Jan van Mill, Boban Velickovic, Menachem Magidor, A. V. Arhangel'skii and Mary Ellen Rudin for a series of highly informative and stimulating lectures. We would also like to thank the many speakers who contributed lectures to the conference, in many cases announcing new and important results for the first time. The organizers would like to thank the staff of the Mathematics Department of York University for their help in the organization of the conference. We would like, in particular, to thank Joan Young for her knowledgeable assistance. All of the articles in this volume have been refereed. We would like to thank the referees for their many helpful comments which have greatly improved the quality of the articles which make up this volume. We would also like to thank Eadie Henry for her tireless work in the preparation of one manuscript from handwritten original draft through several series of revisions to a well-written and clearly presented final copy. This volume of Springer Lecture Notes in Mathematics constitutes the proceedings of this conference. We would like to thank the editors for accepting the proceedings.

Alan Dow Donald Pelletier Juris Steprgms Stephen Watson

Dedication T w o weeks before the conference began, we received the sad news that Eric van Douwen had died. We had been musing about the conference for more than two years and even from the beginning we had decided that Eric would be invited and that he would give a series of lectures. We were curious a b o u t what topics he would choose because we knew that when Eric lectured, people paid attention to his ideas and mathematics would advance a little bit in his direction. We knew we would see a solution to some well-known problem, a completely original proof for some well-known result, and we knew that we would all aim a little bit higher. We also simply looked forward to seeing Eric again. Eric wrote over seventy papers in sixteen years b u t we always felt that the greater part of what he knew had never been written down. We wanted to talk to him and listen to him until we tired from sheer stimulation. General topologists form a c o m m u n i t y and Eric Karet van Douwen was an elder who possessed the knowledge of its oral traditions, but was denied the o p p o r t u n i t y to pass it down. T h e loss to us was devastating, for we lost not only a m a t h e m a t i c i a n b u t a friend.

T a b l e

Murray Spaces

of

C o n t e n t s

G. B e l l of I d e a l s

of P a r t i a l

J a m e s E. B a u m g a r t n e r Remarks on Partition

Functions

Ordinals

..............................

......................................

Andreas Blass Applications of Superperfect

18

D o n n a M. C a r r a n d D o n a l d H. P e l l e t i e r T o w a r d s a S t r u c t u r e T h e o r y for I d e a l s

o n P~(l) . . . . . . . . . . . . . . . . . . . . . .

41

Alan Dow Compact Spaces

in t h e C o h e n M o d e l

55

Tightness

Fons van Engelen, Kenneth Kunen Two Remarks about Analytic Sets Frantisek Franek Saturated Ideals C o l l a p s e of H u g e Nell Hindman Uitrafilters

a n d its R e l a t i v e s

5

............

of C o u n t a b l e

Forcing

1

..........

a n d A r n o l d W. M i l l e r ...................................

68

Obtained via Restricted Iterated Cardinals ........................................

73

and Ramsey

Pierre Matet Concerning Stationary

Theory

Subsets

- an U p d a t e

........................

o f [ ~ + + and q N r ~ #, we see that each

~({r})

To Q1 thru Q3 add Q5:

such that

denote closure

b~l(1).

f-l(1)

there exists

to one map.

ly the same w a y that van Douwen and K u n e n proved that C£

f ~ 2 e,

otherwise

We conclude,

as foll-

then you get the total Q-

For each

A first countable compact ccc space without property

c = e 1. A

P0

0 < u < el; < n}.

can be identified

Together with Len~a 3.1, we see that

cellularity.

(CH)

then

an Eberlein compact.

there exists

Example

with

6({q})

is a countable

C o n d i t i o n Q3' implies that

of Corson compact,

n

and ~ 8

In other words, we have a canonical Hausdorff

I(f,~) = I(f,8).

set; hence

a sigma-disjoint

B~-

and choose

= {q c Q : q ~ f

such that if if

Bn c

We were unable to complete the inductive

7 < @, with

Ae

step while insist-

ing that we also satisfy condition Q4 but if the reader only carries along the weaker Q4': for each completing

s < 8,

A

N B8 =

the inductive step.

separability

of

and

Condition Q5 implies that

Hence,

J(Q)

4.

N A8 =

~;

he will have no difficulty

to prove Q3 and Q5, use the hereditary

2 e. ~0

does not have an uncountable

and so it follows from Le~m~ 3.2 that

have p r o p e r t y

B

Upon completion,

is a first countable,

P

has no uncountable

Corson compact

disjoint

disjoint

space that is

ccc

subcollection

subcollection. but does not

K.

References

i. M. Bell, A normal first countable ccc nonseparable space, Proc. Amer. Math. Soc. 74(1), 1979, 151-155. 2. M. Bell, First countable pseudocompactifications, Topology Appl. 21, 1985, 159-166. 3. M. Bell, G K subspaces of Hyadic spaces, submitted manuscript. 4. E. van Douwen and K. Kunen, L-spaces and S-spaces in ~ ( ~ , T o p o l o g y Appl. 14, 1982, 143-149. 5. T. Jech, Nonprovability of Souslin's hypothesis, Comment. Math. Univ. Carolinae 8, 1967, 293-296. 6. I. Jubasz, Cardinal Functions in Topology, Mathematical Centre Tract 34, Amsterdam, 1971. 7. K. Kunen, A compact L-space under CH, Topology Appl. 12, 1981, 283-287.

Remarks on partition ordinals by James E. Baumgartner 1

Abstract.

After a brief survey of the theory of partition ordinals, i.e., ordinals ~ such

that a --~ (c~, n) 2 for all n < w, it is shown that MA(R1) implies that

wlw

and

wlw 2

are

partition ordinals. This contrasts with an old result of ErdSs and Hajnal that c~ 74 (c~, 3) 2 holds for both these ordinals under the Continuum Hypothesis. 1. P a r t i t i o n o r d i n a l s . The theory of ordinary partition relations for cardinal numbers is fairly well understood at present (see [EHMR], for example), but the corresponding theory for non-cardinal ordinal numbers seems still to be in its infancy. This paper begins with a survey of results and problems concerning ordinals (~ satisfying the partition relation a -~ (c~, n) u for all n < w and ends with the proof from Martin's Axiom + -~CIt t h a t this relation holds when a is wlw

or

0J1w2`

As usual, if X is a set and n is a cardinal then IX]'* denotes the collection of all n-element subsets of X . If a, ;3, and 7 are ordinals, then the partition relation c~ -~ (;3, 7) 2 means that for all f : [a] ~ --* 2 there is X C a such that either X has order type ;3 and f is constantly 0 on IX] 2 or else X has order type 7 and f is constantly 1 on IX] 2. A convenient alternative definition says that ~ --* (;3,7) 2 iff every graph on a either has an independent set of type ;3 or a complete subgraph of type 7. Here, of course, the set E of edges of the graph is identified with f - l ( 1 } in the other definition. 1 Preparation of this paper was partially supported by National Science Foundation grant number DMS-8704586.

For cardinal numbers there is a strong result due to Erdfs, Dushnik and Miller (see [Wi] for a treatment of it), namely if ~ is a cardinal then ~ --+ (~,w) 2. The machinery of ordinary partition relations for cardinals yields strengthenings of this result for various ~. For example, if ~ is regular and s > w then ~ ---, (~,w + 1)2; if in addition A~° < whenever A < ~ then ~ -* (~,wl + 1)2; and so on. For non-cardinals the situation is quite different. The following result is well-known: Proposition I.I. Ira • [a] then a 74 (]a] + l,w) 2. Proof. Let < be the usual ordering on a and let 1 let F] = { y e Fj : {zl,Y} e E l } .

By

inductive hypothesis we may find xj e Fj~ with [{x2,..., zk}] 2 C El, and now it is clear that [ { z l , . . . , xk}] 2 C E l as well. To treat E = E2, use Le~mna 2.3(5) and choose F C Fk.

3. A p p l y i n g M a r t i n ' s A x i o m . Let us emphasize t h a t Theorem 2.1 was proved in ZFC. Of course, since one application of the theorem is to the case a = w and ~I'

M.

In any model o b t a i n e d by a d j o i n i n g u n c o u n t a b l y m a n y C o h e n

or r a n d o m reals to any model of ZFC,

g = ~I"

26

Proof.

The preceding

The case

of m o r e

adjoining ground

previous as

all

~I

~1

of

but

model,

adjoining

one.

section

with

The

first

remark by

is a n e a s y a r g u m e n t than

~

sets

of

is that

that

shows

that

is s e n t

f:~ --~ ~.

of c a r d l n a l i t y

< ~.

I Between

dense

this

f(n) of

=

X,

function

imply

the

remark

gives

information

Formulation

(d) of N C F

first

for

the [8],

consistency does

not

We show FD, from

the

was

next and

generated

unpublished these

that shown

and

so

u < g

showed

implies

to i m p l y N C F

that

fewer

Then

than

the

successive B ~ ~.

though

results than

So

filter.

weaker

of K e t o n e n

b

sets

is a

and Laflamme

ultrafilters. NCF

implies

shortly

filter

in S e c t i o n

in

t h a n NCF.

dichotomy This

The

he proved

described

weaker

1.

u < b.

before

this model,

is s t r i c t l y the

elements

to the c o f l n i t e

of C a n J a r

that

~ ~ b

on every

by fewer

by Shelah

u < b

are

u < b,

related

fewer by some

B}

between

because

work

1 shows

obtained

CanJar

NCF,

S

there

is a f i l t e r

large

there

values

inequality

interest,

in S e c t i o n

of NCF.

satisfy

introduced [7].

is that

•

of

As

by

u < ~,

family

on intervals

generated

from

to t h e m all.

many

by

filter

that

the

dense). common

finitely

filter

, < b

follows

Suppose

two sufficiently

constant

about

generated

is a n e l e m e n t

but

Recent

filter

The

this

In fact,

to the c o f i n l t e

every

X

u < g.

inequality,

above.

there

any ultrafilter

P-polnt.

model

to t h e

the same

stronger"

B e ~,

groupwise

all

non-Ramsey more

is e x a c t l y

inequality.

[24].)

is of c o n s i d e r a b l e

that

this new

reduces

this

every

each

is a n

IX n nl,

sends

u < g,

[16]

there

takes

The second than

(but n o t

families,

function members

X

about

(This a l s o

in

For

of

these

over

case

reals.

first

but we close

discussed

i.e.,

proof

is c l e a r l y

random

as

principle

section,

"next

diagram

function

of

reals

the

the H a s s e

Solomon's

(X e [~]~

for C o h e n

and a theorem

together

~B =

~i

then,

so t h i s

the n e x t

finlte-to-one

base

and

reals,

the c o m b i n a t o r i a l

occupy

is f e e b l e ,

with

reals

random

study

will

two remarks

is i n c o n s i s t e n t ,

the r a n d o m

of

c a n be v i e w e d

o

our

proof

the c a s e

reals

the proof

reals,

We now begin

covers

random

~1

Finally,

for r a n d o m

consistency

discussion

than

principle, result

is

27

Theorem

3.

Assume

flnlte-to-one

f:~ --~ ~

the c o f l n l t e

< g.

For

Let

We s h a l l

~

such

for e v e r y

B e ~,

For

I There

some

Clearly, finite ~B

is not

assume

~B

set

each

Case

which there S

=

Y

must

or

in

~

f-l(y),

[x,y)

and,

being

infinite

every

by

is d i v i d e d (because for

an ultrafilter,

SUPERPERFECT

section,

by M i l l e r

countable

This

result

was

shown

that

X

an

but

be

can

the

the

f-l(y) x < y

are

f-l{f(n))

H,

B.

so

is a in

is

But

it c a n n o t

coflnite

be

filter.

dense. to all

of

the

intervals X.

with

f(~)

For f(A)

~ f(~).

~B'S.

Let

[x,y)

into

each

B E S,

G f(B). Since

Since f(~)

is

u < g. superfect

a n d we s h o w

over

into we

is f i n l t e - t o - o n e ,

if

is the

as

o

we d e f i n e

[7],

A e ~

of

Suppose

~, m e e t s

on the

x,y

~

then

of

common

that

YIELDS

[18],

of

n

Then

indeed, n,

f(9)

B).

under So,

this

f(~).

intervals

is c o n s t a n t

= f(~).

in

s a y at

meets

intervals,

filter.

_ y) E ~B;

are

sets.

f:~ --~ ~

to n;

is g r o u p w l s e

it f o l l o w s

supports

is f r o m

set

the e l e m e n t s

FORCING

H

x < y

closed

cofinite

Let

interval

is an

f

X 6 ~B)

~, f(~)

~B

there

~

with

that

dense.

cofinlte

of

shows

B e ~,

so

an

union

contradiction

IS I < g,

In this

such

Ix,y)

It is a l s o

all

B.

of

is the

then

By m e r g i n g

meets

f-l(~

exists

f

an u l t r a f i l t e r .

be a p a r t i t i o n

~B"

be a c o i n f l n i t e

IX N [O,n] I ,

same model

is a

of c a r d i n a l i t y

whenever

A,

subsets.

is in

f(~)

[x,y)

is a n

introduced

there

that,

meets

contains

meets

is a b a s i s

3.

•

that

For

•

= f(~),

groupwise

the n th i n t e r v a l

that

let

This

2.

under

and

in

f(~)

[x,y)

witnessing

Since f(n)

sends

and

_ y)

~B"

there

an u l t r a f i l t e r

a base

such

is not

because

interval

show

f-l(~ _ y) in

is c l o s e d

dense,

that

included

Then

~, a f i n i t e - t o - o n e or

A e ~

~B

of w h i c h

A E ~

f-l(w

filter.

is e i t h e r

with

filter

and

no u n i o n

and we shall contrary,

X

groupwise

that

function

f(~)

filter

is

B e ~,

modifications

intervals,

be a n y

let

in I.

~

that

is the c o f l n l t e

~B = (X e [w] W

Case

let

be a n u l t r a f i l t e r

find,

f(~)

each

and

filter.

Proof.

either

~ < 9

forcing,

that

a model

of G C H y i e l d s

most

the p r o o f

to s a t i s f y

of

NCF.

a notion

iterating

it

a model

is as

in

of

~2 of

[10]

forcing times u < g.

where

the

28

B y a tree, ~, o r d e r e d

we mean

by end

many

immediate

with

at

successors

least

two

is s u p e r p e r f e c t We consider

(of s u p e r p e r f e c t

in

G

is a s i n g l e W of

trees this of

that

w.

trees),

W

of

as n o d e s ,

intersection

the

can be viewed

A straightforward [23]

and

It w i l l trees

fusion

in f a c t

this

the set

G

of all

of a l l

initial

are

the

is a g e n e r i c

branch

the

is a n

trees infinite

superperfect

segments

as adjoining

be useful

if

~

[ao,al),

trunk)

of

at t i m e s

max(a)

e Ik,

member

from

a u {n}

tree

so

that

J > k

of

W.

a generic

so t h e

of

The

Thus,

subset

W

into

concept

a

superperfect with

interval

following

in c h o o s i n g

interval

a path

through

tree amounts

to a s p e c i f i c a t i o n ,

Ij,

by the

indexed

stage

and

l's

interval

[0,a0),

(also called n o d e a,

consists

the if

of o n e

branching),

and,

a u (n}

has an

e 1 I.

It

to o n e w i t h

structure

of

j-tuples

the climbing

set.

At

decisions (1 for

by this

J-tuple

choices

that

stage

are

is n o t

if

hard

interval dense

0, w e m u s t

0

is o f f e r e d

contains

for

a tree

for e a c h

in the

take

the

are

"leave"), to

infinitely

trunk.

coded

and

take

j > O, of

whether

or

often

m a y be

of w h a t

("climbing

l's t h a t

we decide

stages

to us,

"take"

such

I0,Ii,-.-

description

of O ' s a n d

process,

from previous

"take",

structure

anthropomorphic

of

J

I0 =

then

max(b)

tree down

The

the

to s u p e r p e r f e c t tree has

is a n o d e }

with

is

[2].

for e a c h b r a n c h i n g

n e Ii;

b

forcing A

node

is i n f i n i t e l y

with

of a t r e e w i t h

by the

a certain

intervals

I a u (n}

successor

conditions

attention

branching

I O,

a

Axiom

forcing.

clarified involved

of

this

that

A superperfect

first

(so

of

branching

an arbitrary

structure,

the

(n > m a x ( a ) 1 I,

is a s u c c e s s o r

to p r u n e

to c o n f i n e sort.

is a s u b s e t

then each

(infinitely)

notion

simple

shows

Baumgartner's

can be partitioned

etc.,

the

argument

satisfies

of a particularly

structure

each

of

If

A tree

successor.

the c o n d i t i o n s

the

as

a node

branching

subtrees.

of

infinitely

branching.

are

G

subsets

branching;

infinitely

the u n i o n

all

finite

~.

proper

I1 =

of

a node with

is c a l l e d

in w h i c h

then

and

determines

forcing

tree

infinitely

has an

forcing

branch,

contain,

notion

node

of

the

successors

and extensions

set

subset

is c a l l e d

if e v e r y

trees

of

In s u c h a tree,

immediate

the n o t i o n

superperfect

a subtree

extension.

to t a k e At s t a g e

of

subsets

with

as a j - t u p l e

leave.

tree").

2 j-I

start

the s u b s e t

is

the

IX

1.

or

j > O, of

O's indexed

Any sequence

represents

an

At

leave

of

infinite

29

path

through

the tree,

a path whose

union

is the u n i o n

of a l l

the

taken

sets. If a s u p e r p e r f e c t we are given of w h i c h

another

tree

partition

is t h e u n i o n

superperfect

to t a k e

most

first

its

interval of

~

with

interval

the subtree

I k.

of

I0,II,"'

intervals

the

I's,

structure

of all

subinterval

structure

into

of o n e or m o r e

subtree

suffices

has

JO,Jl,---,

then

that

J0,J1,... meet

and

if each

tree

has a

Indeed,

nodes

that

We

refer

to s u c h p r u n i n g

each

of

lemmas

it

J in at n as merging

intervals. We now embark consistency Lemma

I.

[~]~,

of

If

then

Proof. with

~

In .

is,in

structure

the union

is

Jn

I's

that

are

J's

that

contain

node has

to a c h i e v e

Then

of

p"

have

comparable color.

have

with

inductively, the

for s o m e

a

the

n

th

Thus, of

p' b y m e r g i n g

as d e s c r i b e d of

X,

every so,

since

p

by

of a s u p e r P e r f e c t all

every

condition

of w h o s e

tree are branching

color.

the second

[]

it

segments

from

that

subtree

p"

intervals

from

initial

conclusion,

nodes

is a s u p e r p e r f e c t

to b u i l d ,

nodes

are

Jo,Jl,...,

the d e s i r e d

dense,

(n + I) st such.

shown

of

finite

Ik'S

be obtained

We have

branching

is if,

that

under

the

to b e a s u b s e t

infinitely

Try

a

W

If the

the s a m e

of

p"

holds.

have

fall

those

structure

forcing

nodes

can

Let

subset

of t h e s e

is c l o s e d

the

just

X.

is g r o u p w i s e

including

conclusion

then there

branching

~

dense

it to a c o n d i t i o n

many)

of

are

of

~

the u n i o n

this

2-colored,

Proof.

not

p" f o r c e s

p"

As

since

interval

clearly

an extension

2.

to t h e

X • ~.

Extend

(infinitely

them.

is a s u b s e t

genericity, Lemma

~ X

of s o m e

be given.

denote

u p to b u t

the

intervals

p

I 0 ~ X,

Let ~ X

is a s u b s e t

of s o m e

the

above.

up

model r a groupwise

IO,Ii,-..

X

We may assume

that

W

condition

modifications. one

the ground

the g e n e r i c

interval

leading

u < g.

Let any

contains

on a sequence

a superperfect

first

color.

node

a, all

color.

But

is s u p e r p e r f e c t

subtree

The only way infinitely

then

and homogeneous

of w h o s e

the c o n s t r u c t i o n

branching

the s u b t r e e

all

successors

of n o d e s for

the s e c o n d

30

Lemma

3.

[18]

generates

I_ff ~

a n u l t r a f i l t e r , in fact

forcing

extension.

Proof.

"In

the

fact

forcing

included

extension.

either

or

Successively (1) node

If

and

decides

if

(2) all

but

about

If

J.

model,

is

to w h i c h

in the g r o u n d

"A ~ ~".

We seek

is a n many

prune

then

the s u b t r e e this

infinitely immediate

the

successor

Call

{j e ~

I "j e A"

that

we

model.

So

in t h e

an extension

such

tree)

of a n

p"

p"

forces

a's

branching

successors

then,

a

have

that:

branching

comparable

opinion

node, of

to a r r a n g e

infinitely

of n o d e s

decision

is t h e o p i n i o n

successors (3)

All

Replacing

or none

A

with

about

a

j.

for e a c h

j e ~,

the s a m e o p i n i o n

by

of

of t h e s e t s

~ - A

of a l m o s t

all

immediate

a}. Aa

are

if n e c e s s a r y ,

in

~;

this uses

we may assume

Lemma

that all

2.

Aa

are

~. (4) As

i.e.,

p ~

included

has

interval

in

Aa

further

(5) (Remember enough

If

successor

(7)

to

If

a ~ ik

of

it w i t h

to The

that

can

f i x a set

branching

to a r r a n g e

node that

B e ~ a

the

is a b r a n c h i n g is f i n i t e ,

node,

of

p.

interval

so we

then

is a l m o s t

Merge

intervals,

structure,

say

just

B - A a C- ik+ I.

need

to t a k e

ik+ 1

large

ik.) is a b r a n c h i n g

node

n > ik+ I,

if

j e A.

and

(Again,

and

branching

node

is

a u (n)

j ~ A a n ik,

just

take

ik. ) first

that

satisfies:

B - Aa

the o p i n i o n

relative

p,

a ~ ik

that

we

for e v e r y

extend

relative

(6)

structure.

is a P - p o l n t ,

[0,i0),[i0,11),...,

has

in t h e g r o u n d

model)

because

Set

Aa =

in

lemma

an ultrafilter

(i.e.,

immediate

j e A.

a

the

in the e x t e n s i o n

generates

forces

p

j ~ max(a),

finitely

of

~

"B ~ ~ - A".

is a n

whether

rest of

is a P - p o i n t ~

then

in the s u p e r p e r f e c t

the

subset

(in t h e g r o u n d

extend

a

from

~

~

that

p

B e ~

"B ~ A"

of

that

So s u p p o s e

and a set

a P-point,

countable

subset

on proving

in the g r 0 u n d m o d e l ,

follows

any

the assumption

we concentrate

p

a P-point"

is p r o p e r ;

in a c o u n t a b l e

can apply

of

is a P - p o i n t

~ i 0.

ik+ 1

is a n

immediate

then

a u (n}

large

enough

31

Since

~

and still

is a n u l t r a f i l t e r , to b e c a l l e d

(of t h e

interval

at most

every

with

the same

from

the

Replace still

then

fourth

trunk as

B

minus

by

B

the proof

there would

exist

Let of

nodes

necessary,

branching by

< ik_l,

and

interval

structure).

the

J e A a.

Thus,

the nodes

that

p

still

fourth Since

minus

the

in

interval B

meets

to a c o n d i t i o n

p"

trunk are disjoint intervals.

of. c o u r s e

of

the proof

turn

CH,

a

the n e w

of

with

the

b

of

in

b

B

is

with

a u (n}

(6)

is,

As

max(a)

if Let

we have

for s o m e

j e B

n a ik+ 2

and

that of

j e A. This

of

J a ik

a u {n) p

But

a

< ik+ 2.

the definition

tells us

j ~ A.

j e B, q

k ik+ 2.

the subtree

forces

forces

J.

If not,

forcing

[ik_l,ik+2),

But

lemma,

has

consisting q

is a n

contradiction

o

of s u p e r p e r f e c t

superperfect

superperfect

p'

from

B - A a ~ i k. and

p"

max(b)

and

That

B ~ A.

of

Extending

has

of

of

a u {n) and

q

a,

iteration

iterate

Because

trunk

choice

j ~ A.

the

forces

containing

is d i s j o i n t

(5) •

condition

to

we

its

j e A a n ik+l,

completes

We now

By

p"

[ik_l,ik+2).

is a s u c c e s s o r

comparable this

interval

from

in t h e

that

extension

support.

every

< i 2.

and an extension

that

As

"last"

the opinion

of

by showing

be t h e

b

of

model

to a set,

the a d j a c e n t p';

predecessor

(7).)

(because

of

nodes

from

t r u n k of

are d~sjoint

we may assume

last

(in p)

can extend

and

j e B

[Ik,~k+l) p"

(This e x i s t s max(a)

B

at m o s t

no members

and whose

the

We complete

the

we

B

~.

"j ~ A".

so

p,

meeting

in

that meets

and has

interval,

intervals

all

be

B,

structure)

we can shrink

forcing

forcing

forcing.

~2

is p r o p e r ,

times

Over

with

Shelah's

a

countable

work

implies Lemma

4 [23,

~2-antichaln Lemma

5.

III.4.1,

Cardinals [23,

see also

condition,

Lemma

6

stage

of c o f i n a l l t y

Lemma

?.

and

V.4.4].

Every

a P-polnt,

Proof.

Use Lemma

Every

The

iteration

satisfies

the

real

are

preserved

by

the

in t h e

iteration

occurs

iteratlon. flrst

at a

D

in t h e g r o u n d

in the 3 and

235].

cofinalltles

~ ~.

P-point

in fact

9, p.

o

iterated

[9,

Section

mqdel

forclng 4].

generates extension. []

an ultrafllter,

[]

32

We are Theorem model

now

4 [7].

of CH,

Proof.

the

By L e m m a

For

7 and

the

rest

of

the

[23,III.3.2

of r e a l s

model

theorem have

the proof,

first

~

in the

[22]

that

in

M 2

has

unbounded then

#

~ n M

for an

given

~l-Closed

unbounded

that

each M .

of

of

the

see

model

~i

to f o r m Mu+1)

has

belongs

of

to all

M

for

to

itself.

Let

closed type

supersets the g i v e n

• n M

R~R

be

the set

By an

ideal,

and

closed

I ~ J

The associated

and

defined

of ~'s,

families.

dense

M

not

is g r o u p w i s e in

M 2

there

is an

these

real

W

only

as a n y

~1

~ < ~2

such

dense

family

(adjoined

restrictions,

Therefore,

family

dense".

then,

to a g r o u p w l s e

generic

to

hence

g > ~I"

Since

o

of n o n - d e c r e a s l n g

under

closed

mean binary

under

therefore

functions

a subset

growth

types,

relation,

of the p r e o r d e r e d in w h i c h

to g r o w t h

from

R/~R maxima.

(hence are

R = ~-{0}

that

also

is

A growth closed

pre-ordered

under

by

the

by

equivalence

diagram,

of

(pointwise)

doubling

f e I)(3 g e J)(¥ s u f f i c i e n t l y

partial d e s c r i p t i o n following

set

families

of

every

for a n

is a g r o u p w l s e

"~ n M

restricts

we s h a l l

ideal

Ideals,

(3 r e R / ~ ) ( V

equivalent

~

intersect,

in e a c h

6 M

from

that

~

If

I, the n e x t

236-237]

TYPES

downward

relation

~2

g = ~2"

a

the e x i s t e n c e

It f o l l o w s

9, pp.

has

unbounded

dense

families

by L e m m a

is a n o n e m p t y

addition).

in

over

g = ~2"

the m o d e l

iteration.

also

M

satisfies

groupwise

g ~ c = ~2' w e h a v e

GROWTH

but

sets

then,

~ < ~2"

~l-Closed

the g i v e n

But

4.

of

as a m e m b e r

If we are

in

set

times

implies

~2 ~l-closed

~2

u < g.

u = ~I a n d

CH

let us w r i t e

III.4.1;

final

iS i t e r a t e d

satisfies

of

u = ~I'

stages

and

the c o n s i s t e n c y

forcing

resulting

immediately

after

properness •

we

the

obtained

to p r o v e

If s u p e r p e r f e c t

then

of P - p o i n t s ,

set

in a p o s i t i o n

types.

underlining

I ~ J ~ I, set

of

large

is w r i t t e n

ideals

is u s e d

n)f(n)

is g i v e n

to i n d i c a t e

~ g(r(n)). I ~ J. by

the

ideals

M

33

???? = {f e ~ / a ~

[ (¥ n)

f(n)

~ n}

I = Uke ~ B k

{ l B k = {f e ~

I (¥ n)

f(n)

~ k}

l l BI =

{I}

i B0 The

only

ideals

in the d i a g r a m . for a n y to a n y

fixed ideal

L

=

that

that

contains

family.

in the d i a g r a m .

Every

The p r e o r d e r i n g in t h e i r

then

the n

study

with

maxk~ n

where

but

is

finitely

was

such

that

is not

If

I

~ g(n)}

an those

by GObel

of a b e l i a n

follows.

I

and Wald

groups.

The

is a g r o w t h

contains

L

is e q u i v a l e n t

to one of

introduced

classes

below

f(n)

L

but

types

the

type,

function

If(k)[ [I] of they

ei

has

many

eI

all

to

group

is

that

there

the a d d i t i v e

clearly

[I]-slender

[I]-slender

components

O.

G6bel

[J]-slender

four

are

contain

if e v e r y

at

therefore

at

least

that,

~

is a n u l t r a f i l t e r ,

if

function

of g r o w t h

proved

[15]

I (¥ n)

generally,

is e q u i v a l e n t

f:~ --~ Z

monotone;

G

indicated

ideal

is as

a subgroup

A group

those

{f ~ ~ 7 ~ More

an unbounded

theory

constitute

e i q Z ~,

.

of s l e n d e r n e s s

are

called

to

are

Borel

group

functions

~-~

~ L

g e ~/~

indicated

[14]

not

is e q u i v a l e n t

unbounded

unbounded

connection

are

group

equal

slenderness then

to

homomorphism and Wald

if a n d o n l y

least

four

0

Such

if

vectors

except

[14]

I ~ J.

inequivalent

ideal

subgroups

unit

el(i)

[I] --~ G

proved

classes. the

Z ~.

the s t a n d a r d

that They

growth

Specifically,

= i.

sends

all

every also

types they

and

showed

$4

I(~) generates strictly

= (f • R/'~R I

(nlf(n)

a growth

(its a d d i t i v e

between

MA,

they produced

open

problem

five,

could

of t h e these

whether

in

[7],

proof

they are

as

of a n u l t r a p o w e r that

all

the

proof

from

First, for

~)

of

this

union

that

NCF

[6] a b o u t

I(f(~))

if

of

the

of

We

this

for

chain

being

in its o w n

therefore

X

and

f(n)

~ n

and

a n

for all

the

the

in a n y b a s e

for all

f • ~/~

right,

observations.

(or

s n

part

modify

information;

X e ~

that

that

to t h e s t a t e m e n t

preliminary

next(X,n)

< ~b~R,

each

interest

it here.

three

the ordering

of t h e n o n s t a n d a r d

is e q u i v a l e n t

next(X,-)

X e ~,

of

n • X

for a l l

so

n • X

~

f(n)

~.

Indeed,

c a n be t a k e n

the

to b e

n,

then

function

r(n)

r

= f(n)

I(~)

required

+ I

~ f(next(X,n))

next(X,--) Third,

Indeed,

if

is m a j o r i z e d

every

finitely

many

t h e m all. because,

ideal

f • (~/~)

the strong

are

Then if

by

then

is

if

infinite),

then

for s o m e u l t r a f i l t e r > g(n))

(i.e.,

I f(n)

by

> g(n))

r(n) • ~,

for

Z

= f(n)

g(n)

~ g(next(X,n))

is m a J o r l z e d

by

~ f(next(X,n)) next(f(X),--)

~ next(f(X),

o r.

~.

g • J

intersections

is a n u l t r a f i l t e r

as witnessed (n

o r.

(nlf(n)

property

so there

X =

+ i)

~ I(~)

the s e t s

intersection

J ~ I(f(~)),

g • J,

next(f(X),-)

J < ~/~ - J

finite

~ next(f(X),f(n)

of

containing + I,

so

f(X)

f(n)

+ I),

and

g

by

because,

then

next(X,n)

have

f E ~/~

for a n y u l t r a f i l t r

the d e f i n i t i o n

so

Just

(for e x a m p l e

ideals

of c h a i n s ,

the r e s t

Indeed,

if

or e v e n

solved

was part

information of all

is of

of

following

I(~). and

part

using

the

below,

from

completion

are equivalent) a small

was

an

f ~ next(X,-). Second,

so

information

some

and

functions

• I(~),

presented

in t h e o r d e r i n g

~,

to a v o i d

generate

next(X,--) then

of

problem

Assuming

it r e m a i n e d

g.

the disjoint

requires the

types,

or the D e d e k i n d

only

[7]

modification

growth

some

Though

I(~)'s

prove

inequivalent

the s o l u t i o n ,

cofinal

ordered

but

This

lies

that

and R/~R.

types,

in ZFC alone.

[7] u s e d

+I(~))

to L)

growth

for d e f i n i n g

a singleton

we shall

this many

I(~). are

closure,

is e q u i v a l e n t

inequivalent

and

in

ideals ideals

either

2c

motivation

The

(which

be produced

negatively original

type +L

~ n) • ~)

e f(~)

35

Theorem R/~

5 [7].

are

Proof. show

!i < ~,

t h e n all

F i x an u l t r a f l l t e r that,

B e S, ~B =

if

J

T

with

(X e [~]~

If all fewer

that,

the

is a n y

sufficiently next(B,--)

g

B • S,

large

~ J,

Suppose, certain

ideal,

betwee n

L

and

of

cardinallty

J • L

or

< g. We

J a I(T).

For

z,

large

z)

~ f(next(X,z))}.

are groupwlse

is

f ~ J

next(B,z)

dense,

a common

then,

member

such

that,

by

for,

there

This

means

for all

~ f(next(X,z));

majorlzed

since

X.

in o t h e r

where

words,

r = next(X,-).

as d e s i r e d .

B e S.

Since there

[al,ai+l),

no

necessary,

we assume

J

•

then

they have

there

therefore,

modifications,

that

~B

of them,

is e v e n t u a l l y

I(~)

a base

I (3 f • J)(V s u f f i c i e n t l y

families

than

for e a c h

Thus,

strictly

let

next(B,z)

are

ideals

eq u i v a l e n t .

first

each

If

that ~B

must

infinite

is c l e a r l y

union

to be g r o u p w l s e

closed

of w h i c h

each

b y the

value

fails

be a p a r t i t i o n

that

is d o m i n a t e d

~B

constant

with

Suppose,

for a c o n t r a d i c t i o n ,

ai+3;

function

B.

J ~ L,

f • J

and

finite

intervals

We shall

on e a c h

that

for a

intervals

Merging

meets

some

subsets

into

~B"

which,

means

that

w

is in

[al,ai+l)

this

under

of

dense

if

show

[ai,ai+l),

is

as d e s i r e d .

were

not

so d o m i n a t e d .

Then X = U an

{ [ a i + l , a ~ + 2)

infinite

witness z, s a y

union

f, in

i

with

f(n)

with

> a i + 3.

X • ~B' This

ideals

J

some

n • [ai,ai+1),

intervals,

to our let

j ~ i+1,

next(B,z) so

of o u r

contrary [a~,aj+l),

some

I For

is n e v e r t h e l e s s of

the

there

since

is B

< a ~ + 2 ~ ai+ 3 < f(n)

So

in

~B'

with

Indeed,

for a n y

x • [ai+l,ai+2)

(by d e f i n i t i o n meets

> ai+3),

intervals.

x = next(X,z).

and

Then,

choice

f(n)

each

of X)

of our

for

n E [al,ai+l)

intervals,

~ f(x)

= f(next(X,z)),

J ~ 5

or

as c l a i m e d .

completes and

the p r o o f

that

for a n y u l t r a f i l t e r

~

J ~ I(~),

generated

by fewer

for all than

g

sets. Now suppose

J ~ L

ultrafilter

generated

preliminary

remark,

and

J < R/~R.

by f e w e r let

~

than

g

Let

Y

still

sets,

and,

be an u l t r a f i l t e r

with

by

be a n the

third

J ~ I(~).

Since

36

u < g,

we

Section

have

i.

function

FD

So

with

one-to-one

by

let f(~)

g,

may

Replacing

assume

that

f

f(n) of

I(f(~))

by

the

second

preliminary

= I(f(r))

as

f(~)

I(Z)

as

f(Z)

J

as

Z

is g e n e r a t e d

by

we

have

the

fact

two

that

neither

depend

equivalent, It

steps,

J,

in

[7],

additive

in

gof

for

for

all

a suitable

n.

Then

we

have

remark

is g e n e r a t e d

invoked

nor

this

J

is

shows

by

fewer

fewer

than

first

~ L°)

that

all

than g

part

sets

sets.

of

Thus,

such

9

the

proof

J ~ I(Z).

ideals

J

and

Since

are

o

is s h o w n

but

out

= f(Z)

I(Z)

on

as p o i n t e d

non-decreasing

with

choice

last

NCF,

a n

our

didn't

I(~)

therefore

a finite-to-one

by

the

the Y

3 and be

= f(T) .

we

J ~ I(~)

(At

Theorem

f:~ ---~ ~

for

filters

closures,

consequence

that

that

the

filter

but

Theorem

there

dichotomy 3 was

implications

in

considering

that the

are

are

used

in

ideals

not

four

the

of

Theorem

(defined

5,

inequivalent

(Thus, proof

I(~)

ultrafilters)

conclusion only

principle.

the

u < g ----~ A l l

by

9

of

Theorem

or

their

or

the

growth

5.)

immediate

types,

5 subsumes

Theorem

like

For

implies

Theorem

all

3,

the

chain ideals

,, ~ O n l y

four

strictly

between

inequivalent

L

growth

and

~/ ~

are

equivalent

types

FD NCF, the

converses

5.

ANOTHER The

are

open

NOTION

following

in a l l

groupwise (a,X)

family The

of

a

pairwise

condition

(b,Y) of

Y

are

such

in

finite

of

X.

(a,X)

the

unions

of

W

of

subsets is

a ~ b members

of

ground

subset

Accordingly, that

is d e s i g n e d

a subset

finite

disjoint

of

forcing

families is a

meaning"

members

of

possible,

dense

of

members

as

where

"intended

union

FORCING

notion

straightforwardly

pair

OF

problems.

and ~

that

a ~ W

an

extension

and of

that

model. ~

of

to a d j o i n , ~

such

that

X.

The

as

has

A

supersets

condition

X

is a n

disjoint

from

and

W - a

of

(a,X)

b - a set

W

and

is a

infinite a.

is a is a all

corresponding

37

to a g e n e r i c

set

of c o n d i t i o n s

the conditions

in t h e g e n e r i c

forces

and

"a ~ W

It is e a s y the s p e c i a l

W - a

to s e e

form

(a,

m a x ( x k)

< min(xk+l)

confine

attention

This step

~,

is a u n i o n adJolnts

If

G

can

of

sort

belong

to

step

Since

both

steps

by Mater

first

of

of

of

~ of

for

X

(a,X) intended. to o n e

of

and

convenience,

iteration.

in

[4]

finite

that

first

finite

member

this

called

of

Y

forcing

a stable

nonempty

under

The

disjoint

if e v e r y

(there

the c l o s u r e s ,

X

subsets

of

finite unions,

~.

of t h e

~.

under

conditions finite

the

two

is c o u n t a b l y condition;

iteration

I shall

Mater

antlchain

first

the

[17],

quotient)

the same

of

at o u r

of p a i r w l s e

shown

is to f o r c e w i t h

the c l o s u r e

consideration, The

families

a base

< min(XO)

as a t w o - s t e p

the s e t

then

as

(a,X)

unions

as above,

is r e q u i r e d

to

~.

considered under

set,

that

X",

of

conditions.

It w a s

on

components

can be extended

So w e may,

of u l t r a f i l t e r

constitute

The second that

X.

of

max(a)

is a n e x t e n s i o n

ultrafilter)

is a g e n e r i c

except

k.

infinite

of m e m b e r s

a special

G

where

first

to c h e c k

of m e m b e r s

condition

be d e c o m p o s e d

Y

of

every

for a l l

where

ordered-union

members

that

of t h e

it is e a s y

to s u c h n o r m a l i z e d

forcing

of

set;

is a u n i o n

{ X k l k e ~}),

is to f o r c e w i t h

subsets

is the u n i o n

were,

as

their

composite,

far as

I know,

first

the f o r c i n g

now

forcing. forcing

closed.

in fact,

component

call

are

notions

The

(or r a t h e r

second

satisfies

it is o - c e n t e r e d compatible.

its s e p a r a t i v e the

countable

since

conditions

wlth

It f o l l o w s

that Mater

forcing

is p r o p e r . If is a n y

~

is a g r o u p w i s e

condition,

~.

Then

(a,Y)

and

therefore,

then

dense

X

has

since

countable-support

after

that

a model

The forcing

We

where

a P-point

(one-step)

forcing.

It f o l l o w s ,

thus

under

just as

iteration g = ~2"

forcing

and

have

another

model

can be clarified

Mater

model

and

whose

union

finite modifications,

model

therefore for

forcing

and

I have

generates also

(a,X) is in

"W - a ~ U Y e ~"

for s u p e r p e r f e c t of M a t e r

"W

forcing,

has that

over a model

of

(independently)

an ultrafilter

after

iterated

Mater

u < g.

forcing

by considering

Y

forcing

Laflamme

in the g r o u n d

between

subset

(a,X)

Mater

similarity

in the ground

infinite of

is c l o s e d

an ~2-step

produces

~".

~

in

GCH

an

is a n e x t e n s i o n

a superset

verified

family

and Miller's

superperfect

the a n t h r o p o m o r p h i c

description

38

of s u p e r p e r f e c t description specified 2 j-l,

subset

initial

segments

branching

above

the

first

than

e

OTHER

|

those

already

of

(a,X)

a u y~ a u y

the branches the same

where

y

is a

themselves but

as

has as nodes

are the

succinctly

of M a t e r

forcing

(selective)

described

of M a t h i a s

ultrafilter

forcing, on

w,

and

it.

INVARIANTS cardinal

invarlants

of

the c o n t i n u u m ,

defined

in S e c t i o n

2, a n d w e e x t e n d

inequalities

in S e c t i o n

2 to

two sets

of

cardinality

infinite in the

~; h e r e

the

family

and

from

the

family

have

of a n y

subsets family

include

the splitting

infinite

number,

the Hasse

these

maximal

"almost

almost

disjoint"

means

intersection. independent

means

that any

finitely

of a n y

finitely

many

family

many

other

sets

sets

intersection.

is the s m a l l e s t

family

~ ~ ~(~);

here

infinite

A ~ e,

there

S e Y

is

infinite

~; h e r e

of a n y m a x i m a l

"independent"

the complements an

of

have a finite

cardinality

splitting

are

to

than Matet

invariants.

from

s,

in w h i c h

decomposition

a

the s t a n d a r d

is t h e s m a l l e s t

of subsets

adjoin

CARDINAL

here

family

that every

decomposition

through

is the s m a l l e s t

disjoint

form

rather

A normalized

essentially

c a n be r o u g h l y

to a w e l l - k n o w n

of p r o v a b l e

additional

the

the s e t s

If that

Mathias forcing. Laver forcing

two-step

a real

We define other

=

generically

APPENDIX:

diagram

of X;

tree

node are

3.

only one

decisions)

forcing.

corresponding

The situation

is a n a l o g o u s

then shoot

of

has

b y the p r o p o r t i o n

that

namely

tree

of s e t s

Mater forcing Miller forcing Notice

The

in S e c t i o n

interval

the previous

to M a t e r

branching

of m e m b e r s

nodes.

summarized

jth

as a superperfect

infinitely

union

structure

the

of

is e q u i v a l e n t

from any other.

finite

interval

so that

can be viewed

from any one those

with

(independent

the result

condition

the

trees

is m o d i f i e d

"splitting" such

that

cardinallty means

that,

both

A n S

of a n y for e v e r y and

A - S

infinite. r

is t h e s m a l l e s t

here

"unspllttable"

such

that

R n A

cardlnallty

means

and

that,

R - A

of a n y u n s p l i t t a b l e

for e v e r y

are not both

A ~ w,

there

infinite.

f a m i l y ~ ~ [e]~; is

R •

39

p every A

is the s m a l l e s t finite

subfamily

s u c h that

A - B

is d e f i n e d inclusion These

cardinallty

modulo

has an infinite

is finite like

finite

cardinals

p

for all

except

in ZFC)

diagram.

I thank Peter

~ ~ Y(~)

intersection

but

such

there

that

is no set

8 e ~.

that

~

must

be l i n e a r l y

ordered

by

sets.

and the ones d e f i n e d

(provably

w h i c h was m i s s i n g

the

of any family

inequalities Nyikos

in S e c t i o n

indicated

for p o i n t i n g

from the d i a g r a m

in the out

the

2 satisfy following

Hasse

inequality

~ s i,

at STACY.

\!/ 9

i I

I P

i

REFERENCES 1,

B. Balcar, N covered

J. Pelant, by n o w h e r e

and P. Simon, The space d e n s e sets, Fund. Math.

2.

J. B a u m g a r t n e r , A.R.D. Mathias,

3.

D. Bellamy, A n o n - m e t r l c 38 (1971) 15-20.

4.

A. Blass, U l t r a f i l t e r s r e l a t e d to H i n d m a n ' s finite u n i o n s t h e o r e m and its e x t e n s i o n s , in Logic and C o m b i n a t o r ! c ~, ed. S. Simpson, C o n t e m p o r a r y M a t h e m a t i c s 65 (1987) 89-124.

5.

, Near c o h e r e n c e of filters, I: Coflnal e q u i v a l e n c e m o d e l s of a r i t h m e t i c , N o t r e Dame J. F o r m a l Logic 27 (1986) 579-591.

I t e r a t e d forcing, L o n d o n Math. Soc.

of u l t r a f i l t e r s on 110 (1980) 11-24.

in S u r v e y s in Set Theory, ed. L e c t u r e N o t e s 87, 1983, pp. 1-59.

indecomposable

continuum,

Duke Math.

of

J.

40

6.

, Near c o h e r e n c e of filters, II: A p p l i c a t i o n s to o p e r a t o r ideals, the S t o n e - C e c h r e m a i n d e r of a half-line, order ideals of sequences, and s l e n d e r n e s s of groups, Trans. Amer. Math. Soc. 300 (1987) 557-581.

7.

and C. Laflamme, C o n s i s t e n c y results about filters and the number of i n e q u i v a l e n t g r o w t h types, to appear in J. S y m b o l i c Logic.

8.

and S. Shelah, to appear.

9.

,

U!trafilters

w i t h small

, There may be simple

the R u d i n - K e i s l e r order may be d o w n w a r d Logic 83 (1987) 213-243. i0.

, consistency

11.

proof,

, Near coherence to appear.

and

P

dlr~cted,

of filters,

and G. Weiss, A c h a r a c t e r i z a t i o n ideals, Trans. Amer. Math. Soc.

operator

P

generating

points

An~.

III:

sets, and

Pure Appl.

A simplified

and sum d e c o m p o s i t i o n 246 (1978) 407-417.

12.

A. Brown, C. Pearcy, and N. Salinas, Ideals of compact on H l l b e r t space, M i c h i g a n Math. J. 19 (1971) 373-384.

13.

E. van Douwen, The integers and topology, in H a n d b o o k of SetT h e o r e t i c Topology, ed. K. K u n e n and J. Vaughan, N o r t h - H o l l a n d , 1984, pp. 111-167.

14.

R. Gbbel and B. Wald, W a c h s t u m s t y p e n S y m p o s i a Math. 23 (1979) 201-239.

15. certain

, slender

und s c h l a n k e

operators

Sruppen,

, M a r t i n ' s a x i o m implies the e x i s t e n c e groups, Math. Z. 172 (1980) 107-121.

of

16.

J. Ketonen, On the e x i s t e n c e of P - p o i n t s in the S t o n e - C e c h c o m p a c t i f i c a t i o n of integers, Fund. Math. 92 (1976) 91-94. Some

filters

17.

P. Mater,

18.

A. Miller, R a t i o n a l perfect set forcing, in A x i o m a t i c Set Theory, ed. J. B a u m g a r t n e r , D.A. Martin, and S. Shelah, C o n t e m p o r a r y M a t h e m a t i c s 31 (1964) 143-159.

19.

J. M i d o d u s z e w s k i , On c o m p o s a n t s of ~R-R, Proc. Conf. T o p o l o q y Measure, I (Zinnowitz), ed. J. Flachsmeyer, Z. Frolik, and F. Terpe, E r n s t - M o r i t z - A r n d t - U n i v e r s i t ~ t zu Grelfwald, 1978, 257-283.

20. Cs~sz~r,

Colloq.

of partitions,

of

to appear.

and

An a p p r o a c h to ~R\R, in Topology, ed. A. Math. Soc. J~nos Bolyai 23 (1980) 853-854.

21.

M.E. Rudln, C o m p o s a n t s and ~N, Proc. General T o p l o g y 1970, pp. 117-119.

22.

W. Rudln, H o m o g e n e i t y p r o b l e m s in the theory of Cech c o m p a c t l f i cations, Duke Math. J. 23 (1956) 409-419.

23.

S. Shelah, Proper Forcing, S p r l n g e r - V e r l a g 1982.

24.

R.C. Solomon, F a m i l i e s 27 ( 1 9 7 7 ) 5 5 6 - 5 5 9 .

25.

E. S p e c k e r , Additive Gruppen Math. 9 (1950) 131-140.

Lecture

of sets and

von

Washington

Notes

Univ.

in M a t h e m a t i c s

functions,

FoJgen

State

ganzer

940,

Czechoslovak

Zahlen,

Conf.

Math.

Portugal.

J.

A STRUCTURE THEORY FOR IDEALS ON P

TOWARDS

DONNA M. CARR AND

k K

DONALD H. PELLETIER 1

Introduction In

their

extended

now

much

classic

of

the

uncountable cardinal Jech (K, ~

(e.g.

ultrafilter

over

for

P A

A

measurable),

that

not

does

there

have

the

is

a

partition

K

property.

Thus

Rowbottom's

this

result.

If

Proof.

Assume

f : [X] 2< --> 2. A

xl

that

over

K

is

= i}.

A ~ U. xl

Moreover,

{X ~ X : A

f[[H]~]

of

and every

P A.

Notice

then

X ~ U,

that

either

and for

x ~ X,

that

let

pick

each

{x ~ X : A

w.l.o.g,

and

Further,

for

every

X ~ (U) 2. < U

be

X ~ U i e 2,

either

A

set

xO

~ U} ~ U

xO B = {x ~ X : A

~ U or

xO

a and

or else

~ U} ~ U,

by

ifa~B

ao

Xa = IPKA Now set

P A generalization

A-supereompact, P A K

x E X

~ U} ~ U. Assume xl and then define (Xa : a E P A) A

a nice

A-supercompact,

over

each

= {y g X : f(x,y)

else

is

U

ultrafilter For

not yield

our notion does:

~

A-supercompact ultrafilter

A-supercompact

does

However,

Theorem.

1.10

property

otherwise. H = B n A A,

bijection

hence

x

{x ~ P X : lxl X K

show that m

{ x e PKA :

(V~ ~ x ) ( ~ - l ( ~ )

< x)} , {x ~ PKA : ~[PK x] ~ x} ~ I .

~ CFKA g I ,

by

Suppose

so

it

remains

to

obtain

way

of

contradiction

that

x

{ x ~ PKA : ~ [ P K x ] g X} ~ I .

Then

x

X = {X ~ P A : ~ [ P

X] ~ x } = { X ~ PKA :

(3y < x)(~(y)

~ x)}

~ I +"

For each

x

x ~ X,

pick

to o b t a i n

Yx < x

y ~ PKA

(¥x ~ Y)(~(y) ~ x) Generic

such that such that

ultrapowers

and then use s t r o n g n o r m a l i t y

Y = {x ~ X : Yx = Y} ~ i +"

thereby c o n t r a d i c t i n g

PMA

where

But

note

that

Y ~ i+ ~ i~A" +

of a transitive

u l t r a f i l t e r over

I-generic

~(yx ) ~ x,

model I

M

[]

of

is (in M) a

ZFC

modulo

an

s t r o n g l y normal,

K

non-principal,

K-complete

ideal

over

F A

will

be studied

in a sequel

to

this p a p e r .

2.

Selective,

I~ey

and quasinormal

ideals

I

over an uncountable

on P A K

Recall

that an ideal

regular cardinal

~

is

f : K --) ~

is an

said to be (i) a A ~ I•

p-point

such that

iff for every

I-small function

(¥~ ~ K)(NA a f-1 [{~}]] < K).

there

47

q-point

(ii) a

iff

(¥~ • K)(If-1[{~}]J (iii) selective A • I"

for

every

< K), there is an

iff for every

on which

f : K --~ K

f

is

A • I"

with

on w h i c h

I-small f u n c t i o n

one-one,

i.e.

iff

the

I

f

property is

one-one.

f : K --> K is beth a

that

there

is an

p-polnt

and a

q-point. (iv) Ramsey

iff

for

every

A ~ (I x I)',

where

= {X g K x K : {~ • K : {~ • K : (~,~) • X} • I +} • I},

I x I

there

is

an

there

is

summarized

in

o

H • I

such that

(H x H) n {(~,~) • K x K : ~ < ~} g A.

(v) quasinormal

iff for every

K-sequence

(X

: ~ < K) • KI

o

an

A • I

such that V(X

Kunen

A) = {~ • K :

: ~ •

(see

[B])

and Weglorz

(3m • ~ n A)(~ • X )} • I.

[W] established

the

facts

the following theorem. 2.1 T h e o r e m Let

(I)

I

is

normal

{infCf-%[{~}])

(2) If (3)

I

I

I

: ~ •

be an ideal over an uncountable regular cardinal

iff

for

every

l-small

function

K} • I'.

iS normal,

is selective

then iff

Our first objective

I I

is selective. is Ramsey

if£

I

is quasinormal.

in this section is to provide a

P k

analogue of

K

Weglorz's prove

c r i t e r i o n for n o r m a l i t y (2.1(1) above).

that

I-s.an

an ideal

function

Z • PKA, Our

I f :

over

~

P A

-~

is

~x,

In particular,

strongly normal

U~y z : z • ~x~

iff

• I"

result.

is a To

PKA

obtain

version the

of

reverse

Pelletier's

implication

proof

we

need

for every

where

Yz = {y • f-l[{z}] : (Vx < y)(f(x) ~ f(y) = z)}

proof

we shall

for

(2.3

in

[P]

the

of

each

below). Weglorz's

following

lemma

which is interesting in its own right. 2.2

Lemma.



Ig(x) t >

we' d

have

IgCg(x))l

Cn = {x • B : nx = n}. E = U{C2n : n ~ ~}.

an

> ....

For

Further,

Clearly

infinite

B

set

descending

each

n •

sequence ~,

D = U{C2n÷1 : n • ca}

is the disjoint

u n i o n of

D and E.

set

and Note

48

(Vn ~ (a)(Vx ~ C ) ( g ( x ) (~ C

that

n

g~'E : E --> (P ~CA - E). positive

Moreover,

Thus

at

g D : D --~ CPKA - D)

least

one

of

D

or

E

and

must

have []

measure.

Notice the

).

n-1

above

that

the

result

(Vx ~ B ) ( f ( x ) 2.3

(P A,c)

for

version

functions

of the a b o v e

d o e s not y i e l d

argument

f : B ---> P A

with

the

property

that

the

following

¢ x).

For

Theorem.

any

ideal

I

PKA,

over

are

equivalent: (i) I is (ii) for

strongly every

normal,

A ~ i÷

and

I-small

every

function

f : A --gPKA,

e

U(Yz : z ~ PKA} ~ (I[A)

where

Y

~ f(y) = z)}

= (y ~ f-i[{z}]

: (~x < y)(f(x)

for

each and

z ~ PKA,

IIA

is

the

ideal

z

{X ~ P A : X ~ A ~ I}, (iii) for

every

and function

I-small

f : P A --9 P A,

U{7

~

where for every

Y

z ~ PKA,

is defined

: z ~ P A} ~ I z

as in (ii) above.

z

Proof.

(ii)

It

is c l e a r

and ( i i i )

an

(ii)

implies

(iii).

This

leaves

Suppose

I-small

that

function

I

is

strongly

f : A --~ P k.

Set

prove

that

(PKA

is

as

hence

end,

notice

that

required.

To

this

: y ~ PKA}

and

I

is s t r o n g l y

g

:

y

that F-~ x

y

for ,

y

~

A - Z.

normal,

each A

-

Z

Because

it n o w

follows

A - Z e I. (iii)

A ~ I÷

--9 (i)

and

Suppose

an

g = f u id~(PKA

- A).

A n B ~ I ÷,

s u c h that It is c l e a r

that

I-small

h

g~C is

g

that

not

is

strongly

I-small,

I-small,

so b y

~ gCy))}

l e m m a 2 . 2 to

: (Vx < y ) ( h ( y ) If

Choose

an

f : A --9 P A.

Let

(iii), : z ~ P k} ~ I'.

gtA n B Define

so we m a y use

Z' ~ P k - C:

normal.

function

: C --) (P k - C).

Z" = U { { y ~ h-i[{z}] But n o w n o t i c e

is

: (Vx < y ) C g C x )

so we m a y a p p l y

C ~ I÷ a n d that

I

~} have disjoint closures ( such a sequence is called a

{x~:~ c . b) PFA implies t h a t if X is compact separable with t -- w then X has cardinality at most c .

57 A RELATED QUESTION. T h e r e is a related question asked by the a u t h o r and independently by van Douwen . Is every (first countable) initially w l - c o m p a c t space of countable tightness compact? A space is said to be initially wl-compact if every open cover of cardinality at m o s t wl has a finite subcover. A space is initially w l - c o m p a c t iff every subset of cardinality at m o s t wl has a complete accumulation point ( i.e. a point whose every neighbourhood hits the set in full size). Both van Douwen a n d the author have shown t h a t the answer is yes if C H holds a n d very recently Fremlin and Nyikos have shown t h a t the answer is yes if P F A is assumed. A ~no ~ answer is not yet known to be consistent. 3. F O R C I N G

PRELIMINARIES

Cohen reals are added to a model by forcing with a poset of the f o r m Fn(I, 2) = ( p C I × 2 : p is a finite function} ordered by reverse inclusion. An antichain of a poset P is a set of pairwise incompatible e l e m e n t s . A poset is said to be ccc if every antichain of P is countable . W h e n the context is clear we shall use elements of V as n a m e s for themselves in a forcing sentence. Furthermore if G is P-generic over V and A C X is in V[G] while X E V then we shall assume t h a t a n a m e for A , say A , is a subset of X×Pandforeach xEX ( p E P : (x,p) E ,4} is an antichain. Recall t h a t i f P i s e c c , X E V , G i s P-generic over V and A E V[G] is a countable subset of X t h e n there is a countable n a m e for A . REMARK: Suppose t h a t 0 and X are Ostaszewski's space and Fedorcuk's space respectively. If G is Fn(I, 2)-generic over V then we m a y ask w h a t becomes of 0 and X in V[G] . It is not difficult to see t h a t 0 remains countably c o m p a c t and locally c o m p a c t (because it is scattered) and it has been shown t h a t it remains hereditarily separable . Therefore this provides an example in V[G] of a c o m p a c t non-sequential space with countable tightness in which there is a point to which no sequence converges. Recall t h a t Fedorcuk's space is a c o m p a c t subspace of 2 ~1 - but we m a y view it as the Stone space of a certain Boolean algebra. We are then iterested in the Stone space of the s a m e algebra in V[G] . S. Todor6evid has pointed out to the author t h a t it is straightforward to check t h a t any property K forcing (in particular Cohen forcing) will preserve the fact t h a t the Stone space of a Boolean algebra is hereditary separable. This provides an example in VIG ] of a separable space of countable tightness of cardinality 2 wl regardless of the size of the continuum. However the resulting space will have converging sequences. Indeed each Knew" point will be the limit of a sequence of ~old" points. In addition, Todor~evid has noticed t h a t (under very general circumstances) the Stone space will acquire points of first countability. 4. E L E M E N T A R Y

SUBMODELS

AND

FORCING

In the next section we shall frequently be discussing a set X , a Fn(I, 2)-name ~ for a topology on X and occasionally some Fn(I, 2)-name of another subset of the power set of X , say y . T h e r e is some cardinal 0 large enough so t h a t any sentence we wish to discuss a b o u t these objects will be absolute for H($) ( i.e. they hold in H(0) iff they hold

58 in V ) . We shall always assume without mention that 0 is this large enough cardinal. F u r t h e r m o r e , when we speak of an elementary submodel we shall mean an elementary submodel of this H(0) and t h a t the X , # (and perhaps y ) under discussion are in the submodel. Recall t h a t M is an elementary submodel of H(0) iff for each finite sequence < r n l , . . . ,rnn > of elements of M and any formula of set theory ~ ( v l , . . . ,vn)

M ~ ~t~Cml, ... , r~rt )

i~

HC0) ~ ~ o ( ~ i , . . . , Tc~rt)

(and by our assumption on 0 iff V ~ ~ ( m l , . . . . m,~) providing the mi's are things we are going to talk about). When we investigate an elementary submodel M we are interested in the following. From X , ÷ and y we define new names which are in a sense the restriction of these names to M . We then use elementarity to deduce what properties the objects n a m e d by these new names will have in the extension obtained by forcing with P N M . Finally we are then interested in the relationship of this object to the space X in the final extension. If G is P-generic over V we can define, in V[G] , the set M[G] by (val(fi,,G) : E M } . If P is ccc (or if M , G are assumed to have additional properties) it can be shown t h a t M[G] is an elementary submodel of H(6) v[G] (for example see [Sh]) . To best exploit this relationship we use the fact that M[G] can often be (essentially) obtained by just forcing with P ;3 M . Indeed, suppose we are given X a set , P a forcing poset and y a P - n a m e such t h a t 11t-p y C P ( X ) . Let M be an elementary submodel and define X M = X A M and P M ~- P f'3 M . For each P - n a m e 12 E M such t h a t l II-P 12 C X , we can define a PM-name 12M so that, for each p, x E M p l~-p~ x E 12M

iff

p I~-p x • 12.

Indeed, for each x E X M , let Ax be any antichain which is maximal with respect to being a subset of {p e P M : P II-P x E 12} • Then define 12M to be the name U{ { x } × A z : x C X M } • In fact, since we are assuming that we are only working with "nice" names we can (by elementarity) define YM to simply be Y M M . Note t h a t if P is t e e , we have t h a t 1 It-p 12 M M = 12M • Now of course we define YM , a P M - n a m e , such t h a t 1 ]~-pu YM : {12M: 12 e M and 1 ]~-p 12 e y } . In this f a s h i o n , we have, for example, t h a t if 1 ]F-p (< X, ~ >

is a regular topological space )

then 1 I~-p~ (< X M , cM >

is a base for a regular topological space) .

We will often be careless in clistinguishing between a topology and a base for a topology. If we assume now t h a t P is ccc and that G is P-generic over V then it follows t h a t V[GI ~

val(?, G) n M

=- valC12M, G A M ) = valC12M, G ) .

Notice then t h a t we also have that, in V[G], the topology induced on X M by GNM) is the same as that induced by val(~,G) t3M[G] .

val(~M,

59 Now suppose we make t h e additional assumption on M , t h a t M ~' c M (i.e. that M is closed under w-sequences) . Recall that the Lowenheim Skolem theorems give us t h a t for any countable set A e H(0) and any set B E H(0) with IBI _< c there are elementary submodels of H(8) M A and M B such that [MA[ = w , [MB[ = c , A C M A , B C M B and MB is closed under w-sequences. If M is closed under w-sequences and P E M is ccc then P n M is completely embedded in P (i.e. every maximal antichain of P n M is maximal in P ) hence if G is P-generic over V then G MM is PM-generic over V as well • Therefore, in this case, V[G AM] is obtained by forcing over V by the poset PM • Yet another consequence of the fact that P is ccc and M is closed under w-sequences is that we get a kind of w-absoluteness for M . For example, suppose that 1 I~-p y is a countably complete filter on X , then we get 1 t~-pM Y M is a countably complete filter on X M • This is almost like saying that, in V[G AM] , the model M[G] is closed under w-sequences. Henceforth when we say "by w-absoluteness" , we shall mean "since M w C M and M is an elementary submodel of H(8)" . As indicated above we shall be interested in < X M ,val(#M ,GAM) > in both models , V[GAM] and V[G] (we are still assuming M is closed under w-sequences) . One thing to be careful of is that , in general, val(#M ,G) is a strictly weaker topology than that induced on X M by val(#,G) . However we shall make frequent appeals to w-absoluteness to overcome this. For example , if A,/~ are both countable names i n M and I[[-PMcl÷~AAcl÷~/~ # ( = ) 0 , t h e n l[F-pcl÷Ancl~/~ (=)9 where the second pair of closures are with respect to the larger topology. 5. MAIN RESULTS Let us begin by considering initially wl -compact spaces of countable tightness. THEOREM 5 . 1 . If P is a ccc poser , 1 I ~ P < X , ~" > iS initially wl -compact and t = w , and if M is an elementary submodel of H(O) closed under w-sequences , then 1 [k-p~ < X M , 7"M ~> ha8 countable tightness. PROOF: Suppose that pI~-v~< X M ,~M > does not have countable tightness. By w-absolutenss, we know that p [F-< X M , ~M > is countably compact. Therefore, by Arhangel'skii's theorem we may choose PM-names { ~a : s < wl ) so that each x~ E M and p i k p M { ~ : s < w l } f o r m a f r e e s e q u e n c e . But now, since l[F-v < X , ~ :> is initially wl-compact, we may choose a condition q E P with q < p and some x E X so that qil- z is a complete accumulation point o f ( ~ a : s < wl } • But now, by countable tightness and the fact t h a t P is ccc we may choose s o < s l < wl so t h a t qIk x E cl~{ ~ : fl < s0 } N cl÷{ ~ : a0 < fl < a l } . This contradicts, by w-absoluteness, t h a t p l F - c l ÷ ~ { ~ : /~<so) M cl÷~(~ : s0 is an initially wl -compact space with countable tightness and ~r is a countably complete filter on the closed subsets of X . If H E ~r+ (i.e. H n F # 9 for each F E ~r ) then clH' E ~r+ for some countab/e H I C H . PROOF: Suppose t h a t H is a counterexample and choose by recursion on s < wl ha E Hnn( F~ : /~ < s ) a n d F ~ E ~r so that F a N c l { h~ : /~ < a} = 9 . But now by countable tightness F = Uz is completely regular) , we m a y choose a p E P and a P - n a m e ] such t h a t p ]F-p ] is a continuous function from < X, # > into the unit interval such t h a t ]*-'(0)M A and ] ' - ( 1 ) n . 4 are u n c o u n t a b l e . For each a E X , choose (if possible) Pa E P such t h a t Pa < P and Pa II- (a E f~ and ](a) = 1) . Let B = {a E X M : Pa was chosen} ; B is uncountable since p ]F-p ]*-(1) N A is uncountable . Let us say t h a t a family S C P forms a A - s y s t e m , if S ' = {dora(s) : s C S} forms a A - s y s t e m of sets and all functions s E S agree on the root of S I . T h e c o m m o n restriction to the root of S ~ will be called the root of S . By passing to an uncountable subset of B , if necessary, we m a y suppose t h a t {Pa : a E B} f o r m a A - s y s t e m with root pt < p . Next, for each a E XM, choose (if possible) ra E P so t h a t r~ < p' and ra It-p a E fi and ](a) = O. Similarly assume t h a t B ~ is an uncountable set so t h a t {ra : a E B ~} forms a A - s y s t e m with root r . By removing at m o s t finitely m a n y m e m b e r s of B we m a y also assume t h a t dorn(pa - p') M dorn(r) = 0 for each a E B . Let B0 = {a E B : P a - P' C M } and B ~ - - {a e B ' : r a - r C M } . C l a i m : Bu and B~ cannot b o t h be uncountable. To see this note first t h a t

C--{(a, pa-p'):aEBo}

C'--{(a, ra-r):aEB~o}

and

are P M - n a m e s such t h a t r

n A

and

r

c

n A.

Therefore r IF'p ct~C M clrG" = 0 . But then, as noted at the beginning of the proof, it m u s t be the case t h a t r N M I ~-PM £1rMC I-t c l r M C I : 0 • By our a s s u m p t i o n on A , r N M [ ~ - ( e i t h e r C o r C ' is countable) . But since { P a - P ' : a E B o } and {ra - r : a E B~} are A - s y s t e m s (with (empty) root compatible with r) , it follows t h a t either B0 or B~ is c o u n t a b l e .

61 Since the proofs are similar, we shall assume t h a t B0 is countable. By considering B - B0 , we m a y assume t h a t for each a • B , it is not the case t h a t (p n M ) U p' I~-p ](a) = 1 (note t h a t it is the case t h a t (p n M)I~-p~ a • -4 ) . Now, for each a • B , choose some qa • P and na • w so t h a t qa < (P N M ) U p' and dom(qa) D dom(pa) a n d q,~ Ik ](a) < 1 - 1/n~,. Choose an uncountable C c B , n • w and q • P so t h a t {qa : a • C} forms a A - s y s t e m with root q and n~ = n for all a • C . Now let qt = q n M a n d choose {an : n < w} any infinite subset of C . For each n • w , let Pn = Pa,, N M and qn = qa,, ;3 M (observe t h a t p= C qn ) • Since M is closed under w-sequences {an : n • w} , {pn : n < w} and {qn : n • w} are all in M . Let us call a sequence { < p ~ , q ~'' > : n < w ) a A-system pair (for { a , ~ : n < ~ } , {p,~:n•w} and {qn: n • w} ) if there is a P - n a m e ] such t h a t : (i) 1 l~-p ] is a continuous real-valued function on < X, # > ; (ii) for each n • w p~ h, ](an) = 1 , q~ t~-p ] ( a , ) = 0 ; (iii) for each n • w , Pn c p~ n q", q~ c q' ; (iv) { p ' : n • w} forms a A - s y s t e m with root p' ; (v) {q~: n • w} forms a A - s y s t e m with root q ' . T h e sequence { < Pa~,qa~ >: rt • w} constructed above witnesses the fact t h a t there are A - s y s t e m pairs for {an : n C w} which are not in M , hence there m u s t be uncountably m a n y in M . We shall inductively choose an uncountable sequence P a = { : n • w } fora<wl of A - s y s t e m pairs for {an : n • w} so t h a t if

D =

U{dom(pn) U dorn(q~):n

• w}

and for each t~ < wl

Da = U {dom(p~ - p~) u dom(q~ - qn) : n • w} t h e n for each a < ~ < w l DanD=0andD~NDp=O. Indeed, suppose we have chosen p ~ for ¢~ < / ~ so t h a t each p~ is a A-system pair which is a subset of M . Since M is closed u n d e r w-sequences the entire sequence < p a : a < ~ > is a m e m b e r of M . T h e A - s y s t e m pair constructed in the above p a r a g r a p h would serve as the next m e m b e r except t h a t it is not a subset of M . However by absoluteness there m u s t be such a A - s y s t e m pair in M , so we m a y choose p p . For each a < wl let q~ be the root of the A - s y s t e m {q~ : n E w} in the A - s y s t e m pair p ~ . By construction {q~ : a < wl} forms a A - s y s t e m with root qt . For each a < wl , fix a P - n a m e , f~ , of a continuous real-valued function which witnesses t h a t p ~ is a A - s y s t e m pair. We are going to prove the following C l a i m ql ]F-p the closure of H~ is initially wx-compact and [0,1] ~ has weight wl , q~ [}-p the image of < X , T > under H ~ < ~ f~ is compact. However by [2.3] this is a contradiction since it means t h a t qt [~- < X, r > does not have countable tightness. Now to establish the claim. If G is P-generic and ql • G t h e n L = {¢~ < w l : q~ • G } is uncountable . Furthermore, let t • F n ( L , 2) and r • G be a r b i t r a r y For each • dora(t), there is an n~ • w such t h a t [dorn(p~ - pt) U dorn(q~ - q~)] N dora(r) = 0 .

62 Therefore there is an rt E w such that p~ and q~ are b o t h compatible with r for each a E dorn(t) . Furthermore, since Da N D~ = 0 for a < fl < wl we have t h a t [dom(p~ - p,~) U dorn(q~ - q,t)] n [dom(p~n - p,~) U dorn(q~ - qr~)] = 0 . Therefore there is an extension r' of r such that for each a E dora(t) r' Ikp ]~(a=) = t(a) . From this it follows that V [ G ] ~ the map H ] a (a E L) takes { a , : n E w} onto a dense subset of 2L • l By a very similar proof one can show the following. PROPOSITION 5 . 4 .

Let P = F n ( I , 2) and suppose that

1 [~-p< X,# > is a completely regular initially Wl-compact space of countable tightness.

Let M be an elementary submodel closed under w-sequences, and suppose that Ji , are PM-names of subsets of X M such that 1 [Fv~ if B is an uncountable subset of /i then cliMB N cl~MC # 0 • Then 1 [F-p cluB ¢3 cl~C # 0 for each uncountable

B cft

.

THEOREM 5 . 5 . [CH] If P -= F n ( I , 2 ) and 1 [~-p< X,~ > is initially wl-compact and has countable tightness, then 1 Ik p < X , Zr > is compact. PROOF: Suppose indirectly that there is some p E P and some P - n a m e ~ such that p [F-p ( ~ is a maximal free filter on the closed subsets of the initially wl-compact space of countable tightness , < X,~ >) . By [5.2] , p]F-p ~ has a base of separable sets. Let M be an elementary submodel containing < p , X , ~ , ~ > which is closed under w-sequences and has cardlnality w 1 . Let ( F a : a < Wl}C M be a listing of all PMnames of countable subsets of X M such that p i ~ p eliza E ~ . By w-absoluteness p [F-p~ { clrM~'a : a < Wl} is a filter base for a countably complete filter on X M • For each ct E Wl , let ha E M be a PM-name such t h a t pl~-pM a~ E N~ is an initially w l - c o m p a c t space of countable tightness.

65

Therefore, by [5.5], we have t h a t < X M , v a l ( # M , G N M ) > is c o m p a c t in V [ G N M ] . Next we use [5.12] to deduce t h a t gIG N M] ~ < XM,val(#M,G n M ) > contains a point, say x , of character wl • But since M is closed under wl-sequences the base for x will actually be a m e m b e r of M[G N M], i.e. it will have a n a m e in M . We can therefore deduce, by elementarity, t h a t this base will be a base for x even in the full V[C] . t

We finish with a s o m e w h a t surprising reflection l e m m a , however I do not have any applications for it. LEMMA 5 . 1 5 . Suppose V is a model of CH , P = F n ( I , 2) and 1 1~-< X, i" > is a compact space of countable tightness. I f M is an elementary submodel closed under w-sequences and G is P-generic then V[G n M] ~ The Stone-¢ech compactiflcation of < X M , P(~I(TM,G n M ) > has countable tightness. Furthermore if we let < K M , a > be this compactification from V[G N M] then V[G]~ (there is a set K ' such that X M C K ' C X , K ' with the topology induced by val(# ,G) is Lindel6f and there is a unique continuous one-to-one map from K ' to KM which is the identity on XM ). PROOF: Let us begin by working in V[G AM] . Define K M to be the Stone-(~ech compactification of < XM,val(#, G) > . By countable tightness and w-absoluteness each continuous real-valued function on X M extends , in V[G] , to one on clXM ; indeed if f : X M --+ [0,1] is such that, in V[G] , c l f ' - ( 0 ) n clf~-(1) # 0 , then this would be exhibited by a countable s e t . Now t h a t we know the continuous real-valued functions are absolute upward , we can use t h e m to lift any free sequence of KM to one in X . Therefore KM has countable t i g h t n e s s . An i m m e d i a t e consequence of this fact is t h a t K M will have character R 1 since the set of c o m p l e m e n t s of the closures of countable subsets of X M which do not have a given point in their closure will f o r m a base for t h a t point. Next, we know t h a t X M is countably c o m p a c t , hence each zero-set ultrafilter on X M is countably complete. F u r t h e r m o r e we claim t h a t each ultrafilter of zero-sets will have a unique accumulation point in X . Indeed, if not, we can choose, by countable tightness, a countable subset of X M whose closure will contain both the p u r p o r t e d limit points. But then we can assume we are working with a separable space in which case there will only be R1 continuous real-valued functions. A routine diagonalization process allows us to pick a set A which will m e e t each m e m b e r of the ultrafilter in a co-countable set and which will have b o t h of the above accumulation points as complete accumulation points. Of course this contradicts the conclusion of [5.3] . To complete the proof of the claim we note t h a t the hypotheses on A in [5.3] can be weakened to just the assumption t h a t no two uncountable subsets can be completely separated (as is the case here) since this is w h a t was proven. We then define the space K ' and the m a p g : K M ---* K ' in the obvious way. It remains to show t h a t K ' is Lindel6f. We first observe yet another consequence of the Main L e m m a which is interesting in its own right. F a c t T h e G6-topologies on K ' generated by val(~:M, G n M )

and

val(~, G)

are i d e n t i c a l .

66 Indeed, suppose t h a t we have a point x • K ' which is in the G~-closure of a set F with respect to the smaller topology. Let {Wc` : a • wl} be a neighbourhood base for x with respect to val(~M, G n M ) . Working in V[GNM] fix a n a m e ~" for F . Let p • G be a r b i t r a r y and for each a • wz , choose a condition p,~ • F n ( I - M , 2) below p and a point xc` • K M so t h a t pc`l~ g(xc`) • F n

N W~ /~ a, Ct~ \ K is nonempty.

proof This follows easily from a well-known theorem of Sierpifiski that every countable scattered space is isomorphic to an ordinal. For simplicity we sketch a proof here. aResearcla partially supported by the Netherlands organization for the advancement of pure research 2Research partially supported by NSF grant MCS-84017U

69 Let D ( X ) be the derivative operator, i.e. D(X) is the set of nonisolated points of X. T h e n let Da(X) be the usual a th iterate of D, defined by induction as follows.

D'~+I(X) = D(Da(X)) DX(X) = N a < x D a ( X ) if)~ a limit ordinal Define the rank of any X (rank(X)) as the least a such t h a t Da(X) is empty. Then the l e m m a follows easily from the following facts: 1. Every compact subset of Q has a countable rank. 2. If X C Y then D(X) C D(Y). 3. If X C Y then r a n k ( X ) < r a n k ( Y ) . 4. r a n k ( C ~ + l ) = a + 1. [] To prove the T h e o r e m let L = Uneo~ L,~ where each Ln is compact. Let Kn C Q be the projection of L , onto the n 'h coordinate. By the l e m m a there exists C~ which is not covered by any Kn. It follows that H~ is not covered by L. Q We don't know whether the theorem is true for T ° sets ( G ~ ) or even for a set which is the union of a countable set and a II ° (G6). Next we prove the following theorem: 3 Suppose that A is an analytic subset of the plane, R 2, which cannot be covered by countably many lines. Then there exists a perfect subset P of A such that no three points of P are collinear.

Theorem

proof A set is perfect iff it is homeomorphic to the Cantor space 2~. The proof we give is similar to the classical proof that uncountable analytic sets must contain a perfect subset. A subset A of a complete separable space X is analytic iff there exists a closed set C C w ~ x X such that A is the projection of C, i.e. A--p(C)={yEX

] 3zew ~ (x,y)•C}

Every Borel subset of X is analytic. Let A be analytic subset of the plane I~2 which cannot be covered by countably m a n y lines. Let S be the unit square ([0, 1] x [0, 1]) minus all lines of the form x -- r or y -- r for r a rational number. Without loss of generality we m a y assume that A is a subset of S. Since S is a complete separable space there exists C C w~ x S a closed set such t h a t

A = p(C). Give S the basis B8 for s • 4 5 so t h a t 13 forms a &-system with root /XC~r for some ~ < 5. P r o o f : W L O G assume t h a t t.AI = 6. Let A = {tiC5 : fl regular]-, a n d let .A -- (X~ : flEA]-. By L e m m a 4, for each X~ there is some cr~ so t h a t 7 $ c~ < 5 and X~Ccr~. Let B = {flEA : tra < fl}. (a) Assume t h a t B is stationary in &

93

f(fl) = "the least T so t h a t X~NflCv ~. Since [X/3Aj31 < fl and [3 is C C B - 7 and g < S so that ] " C = {~r}. Thus, if/3EC, XZNflCo" and XZCo'~C~, so X~Co'. Thus T~ = {XEA : XC~r} has size ~. Now apply the A - s y s t e m l e m m a t o / ) to obtain a &-system BC1) o f size 6. T h e n / X , t h e root of/3, is a subset o f ~r For e v e r y / 3 C B - 7 define

regular, f is regressive. By Fodor's t h e o r e m there are s t a t i o n a r y

(b)

Assume t h a t B is not s t a t i o n a r y in &

BNC = O. D = ANC is s t a t i o n a r y and ifi3ED, t h e n / 3 ~ B and XylthC~" " By Fodor's t h e o r e m there are a s t a t i o n a r y E C D - 7 and ~ < 6 so t h a t f " E = {¢}. So for all/3EE, X~nflCcr a n d / 3 < eZ. By induction choose a sequence (/3a : ct < 6)CE so t h a t cr~ v. If ~ is not Ulam-measurable then {p e U(S): R(p) : ~} is dense in U(S).

Proof. Let V be open in U(S) and pick D ~ IS] A with D 0 U(S) £ V. Pick a one-to-one ~-sequence 0 and a l l A ~ N, i f d(A) > 0 then there e x i s t s n e B with

111

~ ( A n A - n) ~a(A) ~ - ~. Bergelson showed t h a t i f C is any i n f i n i t e subset of N, then D(C) = {x - y: x,y e C and x > y} is a set of nice combinatorial recurrence. He then established [2] the following g e n e r a l i z a t i o n of Schur's Theorem. 7.3 Theorem (Bergelson).

Let m ~ N and let N = Ui< m A i.

Then there exists i < m

such that a(ii) > 0 and for every ¢ > 0 ~({n ¢ ki: a(k i N I i - n) ~ d(Ai )2 - c}) > O.

Bergelson has r e c e n t l y t o l d me in conversation of the following theorem, whose proof we are presenting with his permission.

7.4 Theorem (Bergelson).

Assume that whenever B e IN] w, one has that

{x2: x e FS(B)} is a set of nice combinatorial recurrence. p + p = p. a(A) 2 -

Then for all A e p and all e > O, {x e A: A e and a(A N A -

Proof.

x) > a(A) 2 -

Let p c ~ such that x E p and a(A N A - x 2)

e} e p.

By [5, Lemma 2.11 we have {x E A: A - x e p and a(A fl A - x)

~(A) 2 - ~} e p.

(re e s s e n t i a l l y duplicate the above c i t e d proof in what follows.)

Let B = {x c N: a(A N A - x 2) > a(A) 2

~} and suppose t h a t S ~ p.

pick (see f o r example [37, Theorem 8.6]) C ~ [N1W with FS(C) K N\B. assumption x e FS(C) such that d(A N A - x 2) > a(A) 2 - e.

Since p + p = p, Pick by

But then x e B, a

contradiction. [] The interest in Theorem 7.3 is strengthened by Bergelson's announcement (in conversation) that he and Furstenberg have proved that the hypothesis is true. consequence, they easily obtain a non linear Ramsey Theory result: then there are some i < m and x,y,z e A i with x + y2 = z. p + p = p and pick i < m with A i e p.

If N = Ui< m A i

(To see this let p e A with

Let c = ~(Ai)2/2 and let B = {x e Ai:

A i- x e p and a(l i N i i - x 2) > a(Ai)2y e B,

As a

e and a(A i N A i - x) > d(Ai)2 - e}.

Pick

Pick x e A i fi A i - y2 and let z = x + y 2 )

In [37] we presented the following Theorem of Raimi [45]: There exists E c N such that whenever m e N and N = Ui< m A i there exist i < m and k e N with [(A i + k) N E l = # and l(Ai + k)\E I = w.

Using properties of a probability space, Bergelson and

Veiss [71 have generalized this result.

112

7.5 Theorem.

(Bergelson and Weiss).

There e x i s t s E ~ N such t h a t whenever A [ N

and ~ ( l ) > O, t h e r e i s some k e N with ~((A + k) 0 E) > 0 and ~ ( ( l + k)\E) > O. C a l l a f a m i l y F of subsets of the set Z of i n t e g e r s t r a n s l a t i o n i n v a r i a n t provided, whenever F ~ F and k e Z one has F + k e Z.

C a l l such a f a m i l y p a r t i t i o n

r e g u l a r i f , whenever m c ~ and N = Ui<m Ai t h e r e e x i s t i < m and F e F with F ~ Ai . In [3], Bergelson made the following c o n j e c t u r e : I f F i s a t r a n s l a t i o n i n v a r i a n t p a r t i t i o n r e g u l a r f a m i l y of f i n i t e subsets of Z and i f A ~ N with d*(A) > O, then t h e r e i s F e F with F ~ A.

(The most famous i n s t a n c e of the v a l i d i t y of B e r g e l s o n ' s

c o n j e c t u r e has F c o n s i s t i n g of a l l length k a r i t h m e t i c p r o g r e s s i o n s . ) Davenport and I made the f o l l o w i n g simple o b s e r v a t i o n :

If F is a partition

r e g u l a r t r a n s l a t i o n i n v a r i a n t set of f i n i t e subsets of Z and A = {p e ~N: f o r a l l A e p t h e r e e x i s t F e F with F ~ A}, then A i s a closed i d e a l of (BN,÷). r e g u l a r i t y y i e l d s t h a t A ~ ¢.

(Partition

Translation invariance yields that A is a right ideal.

To see t h a t A i s a l e f t i d e a l , l e t p e A, q e ~N, and A e q + p.

Pick F e F with

F ~ {x e N: I - x e q}. Since [FI < w, nxe F A - x e q. I f t e OxeF A - x, then t + F ~ 1.) B e r g e l s o n ' s c o n j e c t u r e i s e a s i l y seen to be e q u i v a l e n t to the a s s e r t i o n t h a t f o r any such A, A* £ I .

Since A1 i s the s m a l l e s t closed i d e a l of (~N,+) one does

always get A1 [ A and, f a i r l y e a s i l y , t h a t A1 ¢ A.

F u r t h e r by d e f i n i t i o n , p e A1 i f

and only i f f o r each A e p, t h e r e e x i s t s k with d*(U~= 1 A - t ) = 1.

Also, by [29,

Theorem 3 . 8 ] , p e A* i f and only i f f o r each I e p and each e > 0 t h e r e e x i s t s k with d*(~t= 1 A - t ) > 1 - e. The s i m i l a r i t y between t h e s e d e s c r i p t i o n s l e d us to b e l i e v e t h a t perhaps no closed i d e a l s of (~N,+) could be found s t r i c t l y between h 1 and A*, (so one would have a proof of B e r g e l s o n ' s c o n j e c t u r e ) . Observe t h a t , by Theorem 2.5, i f p e ~ \ N , then cl((~N\N) + p) i s a closed i d e a l of (~N,+).

Call such an i d e a l " s u b p r i n c i p a l " .

The answer which we obtained [18] i s

v a s t l y d i f f e r e n t than the one we wanted: 7.6 Theorem. (Davenport and Hindman). A1 i s the i n t e r s e c t i o n of s u b p r i n c i p a l closed i d e a l s l y i n g s t r i c t l y between i t and A*. Presumably one of the main reasons we were unable to o b t a i n our d e s i r e d r e s u l t is that Bergelson's conjecture is false.

We are g r a t e f u l to Imre Ruzsa f o r permission to

p r e s e n t h i s unpublished proof of t h i s f a c t .

( I t i s i n s p i r e d by his [49, Theorem 1 ] . )

Kecall t h a t a s e t B ~ N i s s y n d e t i c i f and only i f B has bounded gaps; t h a t i s , k there e x i s t s k e N with N = Ut= 1 B - t . Also B i s piecewise s y n d e t i c i f and only i f there e x i s t s k with d*(U~= 1 B - t ) = 1.

113

7.7 Lemma.

Let t be piecewise s y n d e t i c .

Then t h e r e e x i s t a s y n d e t i c s e t B ~nd

an i n c r e a s i n g sequence neN such t h a t {Yn + x: n e N, x e B, and x < n} ~ A.

Proof.

Pick k such t h a t d*(U~: 1 A - t ) = 1.

{x n + 1, x n + 2 , . . . , of <Xn>neN so t h a t (1) (2)

For each n pick x n e N with

x n + n} ~ U~=1 A - t and Xn+1 > x n.

Choose a subsequence n,N

f o r each n e N, {Yn + 1, Yn + 2 , . . . , Yn + n} c U~=1 A - t and For n,m, and s in N and t e { 1 , 2 , . . . , k } , i f s < n < m, then Yn + s + t e A f f and o n l y i f Ym + s + t e A.

(See t h e proof of [28, Lemma 3.4] f o r a d e t a i l e d d e s c r i p t i o n of how to do t h i s . ) Let B = {n e N: Yn ÷ n e A}.

Then by (2) we have immediately t h a t {Yn + x:

n e N, x e B, and x < n} c_ I .

k To see t h a t B i s s y n d e t i c we show N = Ut= 1 B -

m e l~ and p i c k t e { 1 , 2 , . . . , k }

with Ym + m + t e A.

so n

e B.

Thus m e Ok

t=l

7.8 Theorem (Ruzsa).

B -

t.

Let n = m + t .

Let

Then Yn + n e A

[]

Bergelson's conjecture is false.

with d ( l ) > 0 and a p a r t i t i o n

t.

regular translation

That i s t h e r e e x i s t A

i n v a r i a n t f a m i l y F of f i n i t e

subsets

of Z such t h a t no member F of F i s c o n t a i n e d i n A.

Proof. Pick any I with d(A) > 0 such that A is not piecewise syndetic. (The sets constructed in Section 11 of [37] are such sets. For a simpler example consider {n e N: for all k > 3 and all m, if 2k-1 < m < 2k-1 + k, then n ~ m (mod 2k)}.) Since A is not piecewise syndetic we have (by simply negating the definition) that there exists b: N ~ N such that for all g,x e N there exists y c {x + i, x + 2,...,

x + b(g)} with {y + 1, y + 2 , . . . ,

y + g} 0 A = ~.

g i n t can be found w i t h i n b(g) of any p o i n t . )

(That i s a gap of l e n g t h

Ve may presume b i s an i n c r e a s i n g

function. Let F = { { a l , a 2 , . . . , a k } : k e N\{1}, each a i e Z, a 1 < a 2 < . . . < ak, and b(max{ai+ 1 - a i : 1 < i < k}) < k}. F i s c l e a r l y t r a n s l a t i o n i n v a r i a n t . To see t h a t F is partition

r e g u l a r , l e t m e N and l e t N = Ui<m Ci .

piecewise s y n d e t i c .

Pick j < m such t h a t Cj i s

(For example, l e t p e A1 and pick j such t h a t Cj e p.)

By Lemma

7.7 pick a s y n d e t i c s e t B and an i n c r e a s i n g sequence neN such t h a t {Yn + x: n e N, x e B, and x < n} £ Cj.

Since B is syndetic, pick g e N such that N = U~= 1 B - t.

Let k = b(g) + i.

Pick a I e B and inductively for 1 < i < k, pick ai+ 1 e {a i + 1, a i + 2,..., a i + g} 0 B. Let d = max{ai+ I - ai: 1 < i < k}. Then d < g so b(d) < b(g) < k so {al,a2,...,ak} e F. Let n = ak. Then {Yn + al' Yn + a 2 ' " " {y + a , y ÷ a , . . . , y + a } e F.

Yn + ak} ~ Cj and

114

Now suppose we have some F ~ F with F ~ A.

Pick k such t h a t F = { a l , a 2 , . . . , a k }

with a I < a 2 < . . . < a k. Let x = a 1 - 1 and l e t g = max{ai+ 1 - a i : 1 < i < k}. Note b(g) < k by the definition of F. Pick y ~ {x + i, x + 2,..., x + b(g)} with {y + I, y + 2,..., y + g} 0 A = #.

Now y < x + b(g) ~ x + k -

the least i such that y < a i and note i > 2.

i = a I-

1 + k-

1 < ak.

Pick

Now a i ~ A and {y + l, y + 2,..., y + g}

o A = # so a i > y + g + 1. Since ai_ 1 < y we have a i - ai_ 1 > g, a contradiction. Since the proof of Theorem 7.8 works on any A which is not piecewise syndetic, one obtains counterexamples with density arbitrarily close to 1.

However, the size of

the finite sets involved is always unbounded. The following result of Krlz [40] is much stronger since only pair sets are used. Its proof is also much more complicated.

7.9 Theorem

(Kfi~).

Let E > O.

There e x i s t a set A with d(A) > 1 / 2 -

c and a

p a r t i t i o n r e g u l a r t r a n s l a t i o n i n v a r i a n t family F of two element subsets of Z such that no F e F i s contained in k.

8.

New Combinatorial Applications of U l t r a f i l t e r s .

In 1982 Tim Carlson proved a remarkable theorem, whose proof u t i l i z e s ultrafilters,

and which has as c o r o l l a r i e s numerous e a r l i e r r e s u l t s in Ramsey Theory.

This theorem i n i t i a l l y c i r c u l a t e d in notes by Prikry. 3 of [13].

I t now can be found as Theorem

U n f o r t u n a t e l y , and perhaps unavoidably, one must develop a large amount of

terminology to s t a t e C a r l s o n ' s Theorem and we w i l l not do t h i s here. A recent r e s u l t [4] addresses the issue of whether one can f i n d s o l u t i o n s to d i f f e r e n t Ramsey type problems a l l l y i n g in the same c e l l of a p a r t i t i o n .

(For

example, if m ~ N and N = Ui< m A i one can certainly find i < m and j < m so that d(li) > 0 and l{x ( N: x 2 ~ lj} I = ~. i = j.)

Dn the other hand, it is easy to prevent

The result extends earlier work of mine [31] and joint work with Deuber [20~.

The proof of this result is very simple, producing an ultrafilter every member of which has the listed properties.

It utilizes the simple fact, using alternatively

(~N,+) and (/~N,.), that if L is a left ideal of a semigroup and R is a right ideal, then L 0 R @ @.

Given I ~ N, D(I) = {x- y: x,y ~ I and y < x}.)

8.1 Theorem. (Bergelson and Hindman) exists i < m such t h a t

(a)

Let m c N and l e t N = ui< m A.. 1

There

Ai c o n t a i n s s o l u t i o n s to a l l p a r t i t i o n r e g u l a r systems of homogeneous l i n e a r equations with i n t e g e r c o e f f i c i e n t s .

(b)

One can i n d u c t i v e l y choose a sequence <Xn>nn<w) ~ k i and for each n, given <xj>j O, d({n e Ai: a ( l i n t i - n) > a(Ai )2 - e}) > O.

(h)

For each k e N, d({m e Ai: ~(n~= 0 t - tm) > 0}) > O.

(i)

There e x i s t s B e IN] W with FP(B) [ t i .

Our f i n a l a p p l i c a t i o n u t i l i z e s an old method of proof of Ramsey's Theorem using ultrafilters. (See [14, page 39].) When I f i r s t saw t h i s proof over ten years ago I was q u i t e unimpressed. I t e s s e n t i a l l y t a k e s a standard p r o o f ' a n d r e p l a c e s appeals to the pigeon hole p r i n c i p l e ( i f m E N and N = Ui<m k i , then some I i i s i n f i n i t e ) with r e f e r e n c e s to a n o n - p r i n c i p a l u l t r a f i l t e r p ( i f m e ~ and N = Ui< m k i , then some t i e p). However, Bergelson pointed out t h a t we could probably get s t r o n g e r r e s u l t s i f we used s p e c i a l u l t r a f i l t e r s . Indeed, t h i s i s so. For example u t i l i z i n g p such t h a t p + p = p, one o b t a i n s the M i l l i k e n - T a y l o r Theorem ([433, [52]).

In [6] we

d i s p l a y the r e s u l t s when we u t i l i z e a " c o m b i n a t o r i a l l y l a r g e u l t r a f i l t e r " . an u l t r a f i l t e r

(That i s ,

used to produce Theorem 8 . 1 . )

I will illustrate

the method here with a simple r e s u l t u t i l i z i n g an u l t r a f i l t e r

every member of which c o n t a i n s a r b i t r a r i l y by van der Vaerden's Theorem.)

long a r i t h m e t i c p r o g r e s s i o n s .

(These e x i s t

The method of proof d i f f e r s somewhat from [6] because

we u t i l i z e here the product ® introduced in D e f i n i t i o n 2.2.

8.2 Theorem.

(Bergelson and Hindman.)

Let m e N and l e t [N] 2 = Ui< m i i .

Then

t h e r e e x i s t i < m and neN such t h a t each Bn i s an a r i t h m e t i c p r o g r e s s i o n of length n and {{x,y}: t h e r e e x i s t n , t e N with t ¢ n and x e Bt and y e Bn} ~ t i . Proof.

Pick p e ~N such t h a t each member of p c o n t a i n s a r b i t r a r i l y

arithmetic progressions.

given y, {x e N: (x,y) e L} i s c o f i n i t e , hence in p. (x,y) e L} e p} = N ~ p . )

long

Observe t h a t L = { ( x , y ) : x , y e N and x > y} e p ® p.

(For,

Thus {y e N: {x e N:

Now given i , l e t Ci = { ( x , y ) e L: {x,y} e l i } and pick i

such t h a t Ci e p ® p. I t thus s u f f i c e s to show t h a t whenever C e p ® p, t h e r e e x i s t s a sequence neN with each Bn an a r i t h m e t i c p r o g r e s s i o n of length n and { ( x , y ) : x > y and t h e r e e x i s t n , t e N with t ¢ n and x e Bt and y e Bn} 5 C.

116

Let D = {y e N: {x e N: (x,y) e C} e p}. Bn ~ D.

Ve choose neN inductively with each

Given n > 1 and ~ ~ we l e t a = max Bml and l e t En = D N

Nt=ln-1NzeBt{x e N: (x,z) e C} N {x e N: x > a}. Pick a length n arithmetic progression Bn ~ En.

Then since each Dt & D we have En e p. Then neN is as required. D

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E. Szemer~di, On sets of integers containing no k elements in arithmetic progression, Acta. Arith. 27 (1975), 199-245.

52.

A. Taylor, A canonical p a r t i t i o n relation for f i n i t e subsets of w, J. Comb. Theory (Series A) 21 (1976), 137-146.

53.

H. Umoh, Ideals of the Stone-Cech compactification of semigroups, Semigroup Forum 32 (1985), 201-214.

54.

H. Umoh, The ideal of products in flS\S, Dissertation, Howard University, 1987.

CONCERNING STATIONARY SUBSETS OF [X]
c.

(For the

and many other cardinals important to set-theoretic topology,

The issue is whether a negative answer is consistent,

At any rate, every since Eric sent me the diagram,

see

and that is still unsolved.

I have been interested in what

sorts of implications hold between various classes of compact, countably tight spaces. From now on, "space" will mean "Tychonoff space", since all our spaces will be subspaces of compact countably tight spaces and even have the property that countably compact subspaces are compact.

Since no negative answer to Problem C17 has been found in any

model, I have taken the liberty of simplifying the title of this paper. The first two sections are devoted to greatly expanding van Douwen's diagram, to the point where it seemed best to split it into three.

Section 3 gives examples (and some

theory as well) to justify the absence of arrows between the various classes, although some questions still remain in this regard.

Section 4, devoted to some topological games

of Gruenhage, will also justify some arrows that do appear.

In Section 5, it is shown

what happens when one is restricted to the compact scattered spaces, associated with superatomic Boolean algebras via Stone duality.

In anticipation of this, as many

examples in Section 3 as practical are scattered.

Section 6 points the way to further

expansions of van Douwen's diagram.

I.

Additional classes and implications. Most of the classes of compact spaces dealt with here fall into five informal

classifications: A.

Classes definedby convergent sequences:

bisequential,

G-bisequential,

Sequential, Fr4chet,

(weakly) first countable,

B.

Banach space classes:

C.

"Rings of continuous functions" classes:

=i"

Eberlein, Gul'ko, and Corson compact spaces. hereditarily realcompact,

pseudocompact subspaces are compact, countably compact subspaces are compact. D.

Measure classes:

E.

Hereditary covering and separation properties:

Radon, hereditarily a-realcompact. (weakly)

~-metacompact, weakly

O-refinable, metalindel6f, Property wD. Let us look more closely at each in turn. Classification A. sequence in

A

A subset

A

of a space

converges to a point outside

X A.

is sequentially closed in

X

A space is sequential if every

if no

137

sequentially closed subset is closed. whenever

x G A,

there is a sequence from

if for every point subset

{An:

x

An .

A

and every ultrafilter

n G ~o} of

finitely many

A space is Fr4chet-Ur~sohn

~

converging to ~

x.

converging to

X

there is a countable x

contains all but

A space is weakly first countable if to each point

containing

such that

A space is bisequential x,

such that every neighborhood containing

associate a countable weak base, i.e. a countable collection of

(or simply Fr~ehet) if

x

such that a set

Bn(X ) c S.

S

{Bn(X):

is open iff for each

A compact space is

_~_~0-bisequential

x

one can

n ~ o~} of subsets

x E S

there exists

if it is countably

n

tight and

every countable subspace is bisequential. A sheaf at x space

X

sequence ~I:

is a countable collection of sequences converging to

is an g

=i-point (i=1,2,3,4)

converging to

ran ~n o~

ran

g

x

iff for each sheaf

x.

A point of a

{~n: n G ~0} at

x,

there is a

such that:

for all

n;

[As usual, we write

A o~ B

if

A\B

is

finite. ]

=2: 0%:

ran ~n 0 ran a

is infinite (equivalently,

ran ~n 0 ran a

is infinite for infinitely many

n.

=4:

ran ~n 0 ran ~

is nonempty for infinitely many

n.

A space is an The

=i

=i-space if every point is an

nonempty)

for all

=i-point.

properties are primarily of interest when the spaces in question are

Fr~chet, and that is the context in which we will consider them. bisequential or strongly Fr4chet if it is Fr4chet and and

=2,

n.

and a v-space if it is Fr4chet and

a

A space is countably

w-space if it is Fr~chet

~'i" Actually, except for

were originally given completely different-looking definition and equivalences,

=4'

definitions.

"v-space",

see [Mill, [AI, 5.23] and [Sh], [NOl] respectively.

important theorem is that eygr~ qqmpact Fr4chet space is countably bisequential The

these

For the original An [Mill.

~i-properties are especially important in product theorems, as are

(~o)-bisequential.

For instance,

the countable product of countably compact Fr4chet

~i-spaces is again such a space if

i=1,2,3 [Nol, 3.12], and any countable product of

(~o)-bisequential "hisequential"

spaces is again one [A4].

The first definition of

in [A4] is incorrect.)

The equivalence in the definition of is granted,

(Caution:

the implications

=i => ~ + l

and first countable => bisequential => [A4, 6.23] that

~o-bisequential

in the definitions,

~/

above is an easy exercise [No2].

are obvious;

so are

~o-bisenquential.

implies both Fr~chet and

neither of "bisequential"

Once this

Fr~chet => sequential,

Less easy to see is the fact ~3"

Despite some resemblance

or "weakly first countable" implies the

other.

In fact, a space is first countable iff it is weakly first countable and Fr~chet

[Sil].

However,

Problem I.

the following is still unsolved: Is there a weakly first countable compact space that is not first

countable? An affirmative answer is consistent

(Example 3.11).

138 Every weakly first countable space is sequential [Sill, [NY2].

Thus for compact

spaces we have the implications => in:

first

countable

. v -ip a~e

bisequential ~'~0 - bisequential~:

w-space

weakly first countable

~ + Fr~chet (~4 ÷) Fr~chet => sequential

Diagram 1

A remarkable recent result of Alan Dow is: I.i.

Theorem.

[Do]

In the Laver forcing model [L] every

w-space is a v-space.

For more on this and the following result, see Section 4. 1.2.

Theorem.

[Dow and Steprans]

It is consistent that every countable v-space is

first countable. A corollary is that it is consistent that every compact v-space is ~o-bisequential. dotted lines.

I have indicated consistent implications in the above diagram by

No other implications, consistent or otherwise, are possible, except those

embodied in: Problem 2.

Is it consistent that every compact,

"'J"~o-bisequential?

or that every compact w-space is

a3,

Frdchet space is

~o-blsequential?

(See Example

3.8.) Classification B. real lines.

A compact space is Corson compact if

(not the one making it Banach). Cp(X)

it

embeds in a

Z-product of

It is Eberlein compact if it embeds in a Banach space with the weak topology A compact space

X

is Gul'ko compact if the space

of continuous real-valued functions with the product topology is a Lindel6f

Z-space.

(For the concept of a

Z-space see [Bu] or [Gr2].)

Related classes of compact

spaces and many equivalent conditions may be found in [Ne], as well as the implications Eberlein => Gul'ko => Corson and examples showing the arrows do not reverse.

See also the discussion of

Classification E. It is easy to see that every separable subset of a Corson compact space is metrizable, and not too difficult to show that a Corson compact space (in fact a .-product of first countable spaces) is Frdchet [Gr I, 4.6].

Corson compact ~ ~ O - b i s e q u e n t i a l ~v-space

Hence we also have:

139

Classification C.

I assume readers are familiar with the concept of a countably

compact space, but here is a characterization which helps relate it to some of the other classes:

a space is countably compact iff every filterbase of closed sets has the

countable intersection property.

A space is pseudocompact if every real-valued

continuous function is bounded, and realcompact if it can be embedded as a closed subspace in a product of real lines. Although it may grate on some set theorists' nerves, I will follow here the custom of calling a maximal centered subcollection and calling

F

F

of a family

A

of sets an

"A-ultrafilter",

"fixed" if it has nonempty intersection, and "free" otherwise.

[I never

could understand why logicians persist in using the cumbersome expression "non-prlnclpal ultrafilter" when "free ultrafilter" is available.] set if it is of the form Z

f-l{0)

stand for the collection of all zero-sets in 1.3.

(B)

Theorem [GJ,

].

A space is

pseudocompact iff every

A subset of a space

X

for some continuous real-valued function

(A)

X,

is a zerof.

Letting

we recall:

compact iff every

Z-ultrafilter has the c.i.p.

Z-ultrafilter is fixed;

(C)

realcompact iff every

Z-ultrafilter with the c.i.p, is fixed. From this it is obvious that a space is compact iff it is pseudocompact and realcompact, accounting for one arrow in van Douwen's diagram.

Another is accounted for

by the fact (also clear from the above) that every countably compact space is pseudocompact.

We have already provided references for the other implications in his

diagram, as straightened out by Zhou's theorem. Classification D. class

K

if

A Borel measure

U

is inner-reGular with respect to a certain

~(B) = sup {~(K): K c B , K ~ K }

Radon if every finite (i.e.

u(X) < + m)

for each Borel set B.

A space

X

measure on the Borel sets is Radon,

inner-regular with respect to the compact subsets.

A space is

~-realcompact (also known

as closed-complete) if every "closed ultrafilter" (= C-ultrafilter where

C

is the

collection of closed sets) with the countable intersection property is fixed. Radon space is hereditarily

is

i.e. is

Every

o~-realcompact; this follows from 6.8, 7.4, and 8.12 of [GP]

and the diagram on p. 992 of [GP], which gives a number of concepts intermediate between these (but some are equivalent to Radon for compact spaces, cf. [GP, 7.9]). Classification E. open refinement.

A space is metalindel~f if every open cover has a point-countable

A collection

A

is

~-pointlfinite if it is the countable union of

point-finite collections, and weakly ~-point-finite if point

x,

A

members of

is the union of all the An .

a-point finite open refinement.

x

such that

A = U{An: n e ~ } x

A weakly 8-refinable space is one where every open cover

U = U{Un: n G ~]

where for each point

x

there is an

is in at least one, but no more than finitely many, members of

addition each

Un

where, for each

is in at most finitely many

n A space is [weakly] a-metacompact if every open cover has a [weakly]

has an open refinement that

A

can be a cover, then we have the definition of a

Un.

n

such

[If in

8-refinable space.]

140

It is easy to see that the following implications hold: metalindelSf o-metacompact => weakly a-metacompact

~weakly e-refinable Of course, every compact space is

~-metacompact, so that these properties only are

interesting in our context if they are satisfied hereditarily.

Now, as is so often the

case with covering properties, it is enough to check that every open subspace has the respective property. in

Y,

then each

The idea is that, if

V ~ U

is of the form

and if we refine the collection

W

U

is a cover of a subspace

W O Y

for some

W

Y

by sets open

open in the whole space,

of all these expansions on the open subspace

the appropriate way, the traces of the refinement on

Y

t~

in

will also behave as desired.

This is especially worth noting in the context of compact spaces since every open subspace of a (locally) compact space is locally compact, and these hereditary covering properties are neither created nor destroyed in passing to one-point compactifications. It also leads to such simplifications as the following theorem alternate characterizations of weak e-refinability in 1.4.

Theorem.

[BL]

A locally compact space is weakly

of

X

U An

is

which uses

[Bu].

e-refinable iff every open cover

has a G-relatively-discrete refinement by compact subsets. space

[NY5] ,

and

~-relativel~-discrete if it is of the form

[A collection of subsets of a U{An: n G ~]

has a neighborhood meeting at most one member of

where each point

An. ]

It also helps in the analysis of the hereditary metalindelhf property.

In Section 4

we will see that "hereditarily metalindelSf" implies both "v-space" and " ~ o-hisequentlal " " for compact spaces.

See also [PY] for a quick proof that if

point of a compact space

is metalindelSf and

sequence from

A

X

and

converging to

Finally, a space discrete subspace

X

S

each of which meets

X-{m]

satisfies Property

wD

wD

in exactly one point.

X)

collection

(Recall that a collection

U

of open sets, A

is discrete

A.

is the weakest in a hierarchy of properties extending to normal (and

beyond to collectionwise normal, etc.)

[vD, Section 12], [Vl].

because compact sequential spaces which satisfy

wD

realcompact space satisfies is countably compact.

of pseudocompact spaces might be said that normal

wD [VII;

It is included here

hereditarily happen to fit nicely

into Eric's diagram and illuminate some of the relationships.

wD

then there is a

if for every countably infinite closed

if each point has a neighborhood meeting at most one member of Property

iS a

x.

there is an infinite discrete (in S

x G [,

x

On the one hand, every

on the other, every pseudocompact space satisfying

This is easy to see if one considers another characterization [E, 3.10.23]: => wD

every discrete family of open sets is finite.

It

is the "real" reason for the fact, initially surprising

to many students, that every normal pseudocompact space is countably compact. rate, we can squeeze "sequential and satisfying

wD

At any

hereditarily" in between the first

two classes of Eric's straightened-out diagram, because of what happens on the other end:

141

sequential implies countably compact subsets are closed in any space, hence compact in a compact space.

2.

More implications The covering properties in Classification E all imply compactness in a countably

compact space.

This is part of a theme carried considerably further in IV2, Section 6],

[A3] and [T]: "countably compact +

=> compact."

The simplest thing to put in

the blank is "LindelSf," but the weaker the property, the better. So, if one of the Classification E covering properties is satisfied hereditarily in a compact space, it implies all countably compact subspaces are compact.

But, except for

a-metacompactness, I do not know how much further we can go (see Problem 6, and Example 3.6 below). 2.1.

We do have the following 1984 result of Uspensky:

Theorem.

[U]

Every ~-metacompact, pseudocompact space is compact.

Gardner [GP, 10.2-3] showed a hereditarily weakly 8-refinable, locally compact space is Radon iff it has no discrete subspace of a real-valued measurable cardinal, and that a weakly 8-refinable space is =-realcompact iff it has no closed discrete subspace of a measurable cardinal. Gruenhage [Gr3] gave the following unexpected characterizations: 2.2.

Theorem.

(i)

X

(ii)

X2

The following are equivalent for a compact space

X:

is Corson [resp. Eberlein] compact. is hereditarily metalindelSf [resp. ~-metacompact]

(iii) X 2 - A

is metalindel6f [resp. ~-metacompact]

He also showed [Gr6] that

X2

is hereditarily weakly ~-metacompact if

X

is Gul'ko

compact and conjectured that the converse is true, and also asked: Problem 3. metacompact?

If is

X2 - A X

is weakly ~-metacompact, is

X2

hereditarily weakly a-

Gul'ko compact?

So far, we have the following implications,

for compact spaces where

H. stands for

"hereditarily": Eberlein compact

=>

Gul'ko compact

=>

H. ~-metacompact

=>

H. weakly

=>

pseudocompact subspaces are compact

H. weakly ~refinable " l!

~

~ ~

~ ~ H. ~realcompact

H. metalindelSf

~0-~sequential v - s p a c e

~

Radon

Corson compact

~

~

x # ~. ~

2 Diagram 2

142

Here

.... >

cardinals, and

means the implication holds if there are no real-valued measurable +++>

means it is consistent that the implication does not hold, but that

it is not known whether it fails in ZFC. Problem 4.

If a compact space

X

See Problem 7 below, and:

is hereditarily metalindelSf, and no discrete

subspace is of measurable cardinality, is consistent

that

X

I n [GP, Example 1 1 . 2 0 ] constructed cardinal.

X

hereditarily e-realcompact?

Is it

i s a l w a y s Radon? t h e r e i s a compact, h e r e d i t a r i l y

u s i n g CH, i n which no d i s c r e t e

m e t a l i n d e l S f non-Radon space

subspace is of a real-valued

G a r d n e r [Gd2] h a s a s k e d w h e t h e r

MA + -CH

measurable

i m p l i e s no s u c h s p a c e e x i s t s .

I do n o t e v e n know t h e answer t o : Problem 5. cardlnality,

is

If

X

X

i s m e t a l i n d e l B f , and no c l o s e d d i s c r e t e

T h i s i s p a r t o f a theme r e l a t e d section:

"No d i s c r e t e

~-realcompact."

subspace i s of measurable

o~-realcompact? to t h e one m e n t i o n e d a t

subspace of measurable c a r d i n a l i t y

I f one l o o k s a t t h e c h a r a c t e r i z a t i o n s

compact i n S e c t i o n 2, i t

is clear

serve for the other one.

the b e g i n n i n g of t h i s

+

of

=> ~ - r e a l c o m p a c t and c o u n t a b l y

t h a t a n y t h i n g one can t r u t h f u l l y

put i n t h i s

But this theme has not progressed nearly so far:

blank will

As I said,

weak 8-refinability works, but it is the weakest covering property I have seen so far that does.

But "metalindelSf" is a reasonable candidate to try since metalindelSf spaces

that are not weakly 8-refinahle are still in short supply. described by Gruenhage (Example 3.5) and they are subspaces of compact Radon spaces if one assumes

The only "real" ones were

~-realcompact; c

in fact, they are

is not real-valued measurable.

It might be said that this second theme is the "real" reason why normal e-refinable spaces

[Z]

and normal, countably paracompact, weakly 8-refinable spaces [Gall with no

closed discrete subspaces of measurable cardinality are realcompact: paracompact space is realcompact iff it is a-realcompact [Dy]~

a normal,

countably

also every 8-refinahle

space is countably metacompact [Gi] and every normal, countably metacompact space is countably paracompact

JR,

] and [RUl, I.I, (i) => (ii)].

A big unknown as far as Diagram 2 is concerned is: Problem 6.

Is a compact space sequential if it is any of the following:

(a)

hereditarily weakly 8-refinable

(b)

Radon

(c)

hereditarily a-realcompact?

Of course, these are special cases of Problem C17.

There are also several problems

about what implies hereditary a-realcompactness, also involving Diagram 2. Problem 7.

Is there a

m-realcompact but is sequential

(d)

(a)

ZFC

example of a compact space which is not hereditarily

Frdchet

(b)

~0-bisequential

Example 3.10 includes a construction using ~ Problem 8. ~-realcompact?

(c)

hereditarily

wD

and

such that every pseudocompact subspace is compact? that satisfies all these properties.

Is there a weakly first countable compact space which is not hereditarily not Radon?

not hereditarily weakly 8-refinable?

143

Here I do not know of any consistency results, not even under large cardinal hypotheses, nor am I aware of bounds on the cardinality of weakly first countable compact spaces. It is quite easy to show that every realcompact space is =-realcompact [Caution:

[DY]"

the collection of all zero-sets in a closed ultrafilter with the c.i.p, is not

always a

Z-ultrafilter, but that is all right:

compare the proof of 2.3 below.]

For

compact spaces satisfying these properties hereditarily, we have an interesting interpolant, 2.3.

independently noticed by Reznichenko and myself.

Theorem.

Proof.

Let

Every compact, hereditarily realcompact space is bisequential.

Y

ultrafilter on

Y

be hereditarily realcompact and compact. converging to a point

F = {F e U: Then

F

F

extends to a unique

disjoint zero-sets such that continuous function such that

y.

Let

is a zero-set of

Z1

X = Y - {y}

Z-ultrafilter

for

H

in

Z 0.

be a free

and let

X.

Indeed, if

F,

let

It is now easy to see that

i,

and so

Zi,

i=O,l

f: X ~ [0,I]

i=0,1 [G3, 1.15].

a base for an ultrafilter that can only converge to is disjoint from

U

X}.

meets every member of

f~Z i = {i}

Let

The image of fell/2,1[

H = {Z ~ Z: Z O F # #

are

be a UIX

is in

is

F

and

for all

F e F}. H

is free because

are all in

H.

intersection.

For each

F n = Cly hi=in f~[0,~] • of

Y

y

has a base of zero-set neighborhoods in

So there is a countable descending family n,

let

Then

H n = f~[O}

A~= I F n = [y}

for a continuous and each

Y

{Hn}~_ I,._

Fn

with empty

H

fn: X ~ [0,1].

is in

as in [E, proof of 3.3.4], we see that the filterbase

and their traces

in

U.

Let

Using compactness

{Fn}n= I

converges to

y.

Hereditary realcompactness is not affected by adding one point, so Theorem 2.3 extends to locally compact spaces. 2.4.

Theorem.

Proof.

Let

Every compact bisequential space is hereditarily =-realcompact.

Y

be a compact space.

there is a free closed ultrafilter regularity) adherent point. extending it converges to in

F,

so we can pick

Let subset of

N

Then y.

x e X

Let

F F

So

Y

Let

is a filterbase on Un

U

for each

Y,

n e ~.

X

y e Y - X

of

Y

on which

be its unique (by

and any ultrafilter The sets

U

(clyUn) N X

are

in their intersection.

be a closed neighborhood of N.

Suppose there is a subspace with the c.i.p.

y

that misses

x.

None of the

Un

can be a

is not bisequential.

Theorem 2.4 does not extend to locally compact spaces. bisequential, as is any first countable space, but not

The ordinal space

oh

is

~-realcompact.

Bisequentiality also interposes between hereditarily realcompactness and another Classification C property: 2.5.

Theorem [A3, in effect].

subspace is compact.

In a compact bisequential space, every pseudocompact

144

Proof.

Theorem 6' of [A3] states:

for any

x G X

and any filterbase

{Pn}~=l_

of regular closed subsets of

meets every member of

~.

Now if

E

Y

A (regular) space is bisequentlal if and only if with X

which we may take to be

which converges to

P

C-decreasing.

intersection of the closures of the

as an adherent point, there is a sequence and such that every

x

is a pseudocompact subspace of

closure, the set of all,lnteriors of the Y

x

Pn'

X

with

x

Pn in its

trace a sequence of relatively open sets on n Then [GJ,9.13] there is a point of Y in the

and this must be

x.

Thus

Y

is closed.

Also hereditary realcompactness interposes between bisequentiality and another classification 2.6.

A

property:

Theorem [GJ, 8.15]

Every first countable realcompact space is hereditarily

realcompact. And so we have the implications => in the following diagram, spaces.

again for compact

The other arrows have the same meaning as in Diagram 2.

f i r s t countable realcompac t

bisequential

H. wD and

\\

\\

H. ~-metacompact

pseudocompact ~

sUb o Z'dre /

~ ~/// H. ~-realcompact Problem 9.

Eberlin compact

Diagram 3

If a compact space has no discrete subspace of measurable cardinality,

it bisequential if it is (a) Eberlein compact?

(b)

hereditarily

is

~-metacompact?

Note that the one-point compactification of a discrete space of measurable cardinality is not even hereditarily

=-realcompact,

but it is Eberlein compact.

clear from Rosenthal's characterization of Eberlein compacta [Ro]: the compact spaces F -sets.

K

which admit a

a-point-flnite,

If one inserts "weakly" in front of

To-separating cover by open

"v-point finite" here, one has Sokolofffs

characterization of Gul'ko compact spaces [So]. And if one has "point-countable" instead, one has the Corson compact spaces [MR].

3.

That is

they are precisely

Counterexamples. We begin this section with a space van Douwen felt could supplant the more

complicated Tychonoff plank in most elementary texts.

145

3.1

Thomas's plank.

Let

W

its one-point compactificatlon. where the underlying set of X

W

be an uncountable discrete space and let Let

X = (W + ~) x (~ + 1).

W + ~

denote

Thomas's plank is the case

is the set of real numbers.

is not hereditarily normal.

corner point does not satisfy

wD.

In fact, the subspace

Y

obtained by removing the

Indeed, no infinite set of points

expanded to a discrete collection of open sets.

Afortiori,

X



can be

is not hereditarily

realcompact. On the other hand,

X

is both bisequential and Eberlein compact.

the significance of the classes of

productive and hereditary [Hil] , [A4]. countably productive:

Indeed, part of

(~'¢0-)bisequential spaces is that they are countably The "Banach" classes of compacts are also

see [AL| for Eberlein compacta and [ ] for Gul'ko compacta.

For

Corson compacta it is trivial. Since every separable subset of

X

is metrizable, it is a

v-space.

Fr~chet but not first countable, it is not weakly first countable. then

X

Since

X

If we make

is

IXI = ~I'

is Radon.

3.2.

Alexandroff's "two arrows".

order topology.

Let

As is well known [E,

X = [0,I] x {0,I}

] [SS],

X

with the lexicographical

is hereditarity separable and

hereditarily Lindel6f, so that it satisfies all the Classification C and E properties. Since every discrete subspace is countable,

X

is Radon.

Since

X

is first countable,

it also satisfies the Classification A properties. On the other hand,

X

satisfies none of the "Banach" properties since it is

separable and nonmetrizable. 3.3.

The Hr6wka-lsbell ¥(+~).

Let

family (HADF) of infinite subsets of topologized by letting the points of a base for the neighborhoods of

A

A

~. ~

be an infinite, maximal almost disjoint The underlying set of

is

e U A,

be isolated and letting {{A} U (A-n): n e e}

for each

be

A e A.

AS is well known [GJ, Exercise 5I] and easy to prove, pseudocompact.

T

T

is locally compact, and

Being a countable union of closed discrete subspaces, it is hereditarily

(weakly) 8-refinable, and so is

T + %

the one-point compactification.

On the other

hand, no separable non-LindelSf space can be metalindel6f, and it is even more obvious that

{~} U {A U {A}: A e A}

has no point-finite open refinement.

T + ~

satisfies none

of the "Banach" properties for the same reason as 3.2. T + ~ sequential: A

is not Fr~chet since no sequence from unless a subset of

e

~

converges to

®.

But it is

has compact closure, it has infinitely many points of

in its closure, and these have a sequence converging to

~

since

A U [~}

is the

one-polnt compactification of a discrete space. From the discussion surrounding Yakovlev's space (3.11) it will be evident that T + ~

is not weakly first countable.

It is Radon whenever

c

is not real-valued

measurable.

Special versions of it are Radon even in models where

measurable.

In fact, as far as I know, no negative answer is known to:

c

is real-valued

146

Problem i0.

Is

a,

the least cardinality of an infinite HADF of subsets of

~,

always less than the least real-valued measurable cardinal? Eric van Douwen [vD] has called a space

Y-like if it is locally compact, has a

countably infinite dense set of isolated points, and the nonisolated points are a closed discrete subspace. x

If

x

is a nonisolated point and

in which all other points are isolated,

is clopen and the sets disjoint family.

N x N W,

where

W

Nx

is a compact neighborhood of

then the Hausdorff condition tells us each is the set of isolated points,

It is maximal iff the space is pseudocompact [vD, 11.6].

Conversely,

any almost disjoint family of infinite subsets of a countable set gives rise to a space as a MADF gives rise to

Nx

form an almost

T-like

V.

Every one-point compactification of a

V-like space is hereditarity weakly

O-refinable, being a countable union of (closed) discrete subspaces, but not hereditarily metalindelBf if the space is uncountable. 3.4.

The Cantor tree + ~.

of height

w + 1

Let

T

denote the Cantor tree, i.e. the full binary tree

with the interval ("tree") topology.

nonmetrizable, hence not metalindelSf.

Unlike

V,

fact, it has a coarser compact metric topology, = {t': t' > t}

where

t

It is a

V-like space and is

it is hereditarily realcompact.

In

for which a base is the set of wedges

is on a finite level of

T,

and their complements.

Vt

And any

space with a finer topology than a first countable realcompact space is hereditarily realcompact [GJ, 8.17]. Todorcevic' has pointed out that coarse wedge topology",

T

with this latter topology, which I call "the

is "really" the tree of all initial segments (which he calls

"paths") in the full binary tree of height discussion of this theme, see [Grs].

m,

with the product topology.

For a fuller

In [NY7] I show a very natural embedding of

T

with this topology in the plane, with the points of the top level topologically identified with the Cantor set. Thus

T + =,

bisequential,

with

etc.

In

T

[NY7]

topology, that it is not a top level of

T

having the interval topology,

I show, u s i n g t h e B a i r e c a t e g o r y theorem on t h e c o a r s e r

v-space.

Also [ibid.], if one removes all points from the

except those corresponding to a

k'-set [Mi2] and then takes the one-

point compactification of what remains, one has a 3.5. of

~i

Let

T

Give

The Todorcevic-Gruenhage space. and let

T

is hereditarily realcompact,

Let

S

w-space. be a stationary, co-stationary subset

be the tree of all compact subsets of

S,

ordered by end extension.

be obtained by adding a point at the end of each branch (maximal chain) of T

the coarse wedge topology (see 3.4).

compact, hence so is Radon if

c

~2,

but

~2 _ ~

is not weakly

is not real-valued measurable.

paracompact, hence it t h e Thomas p l a n k ( j u s t

is hereditarily

In [Gr5] it is shown that O-refinable;

It is also shown that

realcompact, etc.

However,

t a k e t h e s u b s p a c e of p o i n t s of l e v e l

sized subset in each copy).

< 1

T

also that T

@2

T.

is Corson ~2

is

is hereditarily

c o n t a i n s a copy o f

and an a p p r o p r i a t e -

On t h e o t h e r hand, Gruenhage m e n t i o n s a f i r s t

147

countable variant,

and t h a t has a l l c o u n t a b l e powers f i r s t

c o u n t a b l e and so h e r e d i t a r i l y

realcompact, etc. 3.6.

Reznichenko's space.

d i s c o v e r e d i n 1987.

Gruenhage.

This i s a s p a c e w i t h some amazingly s t r o n g p r o p e r t i e s ,

Uspensky p r i v a t e l y communicated a d e s c r i p t i o n

i n E n g l i s h to

Since a full treatment is not likely to appear in print in English anytime

soon, I thought it worthwhile to at least give the definition here. Define sets A With

A8

For each as e A a

and F8

finite sets

defined for all

F

for

B < ~,

= {as: S

F

let

{k

A = U {Aa: ~ < e l } ,

as follows.

Note t h a t the s e t s by i n c l u s i o n .

= U {Fs: ~ < ~},~ A= = U [As: 8 < a}.

C = U~=0 Cn.

|A] 1 U 0

F:}

with

F

Or r a t h e r ,

Tn = {k G F : n e k }

a r e added, the r e s u l t i n g

r e g a r d e d as a s u b s e t o f l e t us r e g a r d

B of

showed t h a t

~

2A

in

2A, t o p o l o g y and

subspace

form a p a r t i t i o n

of UB,

Cn

of

topology is the coarse

2A

is closed.

I t i s not hard to s e e t h a t

C U [A] 1

and each i s a t r e e

c o n v e r g e s to t h e p o i n t Its

By a theorem o f Gruenhage -e[Gr5]~ Cn

i s G u l ' k o compact.

F

Tn

~=

He a l s o showed t h a t

and i f t h e s e p o i n t s

i s E b e r l e i n compact.

C U [A] 1 U 0,

i s t h e o n e - p o i n t c o m p a c t i f l c a t i o n of the d i s c r e t e

Cech c o m p a c t i f i c a t i o n o f

define new elements

F~,

A.

Each b r a n c h

wedge ( " p a t h " ) t o p o l o g y .

A0,= ~, FO = [~]I.

as

F = U {F : a < e l } ,

as t h e power s e t o f

Let

F

is a disjoint sequence in

U {as}: n e ~ ,

t h e n a t u r a l way, w i t h the p r o d u c t t o p o l o g y . all,

a < ~i

~v-sequence s = < kS: n eel> of disjoint members of n with as # as, if s # s'. Let A

Let

and

subspace

and o f c o u r s e [A] 1.

Reznichenko

8(C U [A] 1) = ~, i . e .

i s the o n e - p o i n t c o m p a c t i f i c a t i o n !

Thus

the StoneC U [A| 1

i s pseudocompact (and noncompact). Of a l l

the r e s u l t s

i n t h i s s u r v e y , t h i s was f o r me the most u n e x p e c t e d .

I was

f a m i l i a r w i t h t h e c o n s t r u c t i o n s o f A. Berner [Be] and H.-X. Zhou | Z h ] , which gave s p a c e s w i t h s i m i l a r p r o p e r t i e s under the axioms little role of

a=c

and

b=c

respectively,

and t h e r e seemed

hope o f c a r r y i n g out e i t h e r c o n s t r u c t i o n or a n y t h i n g r e s e m b l i n g i t Cn

was taken o v e r i n Z h o u ' s c o n s t r u c t i o n by the Cantor s e t ,

c o n s t r u c t i o n by a f i r s t

" i n ZFC".

The

in B e r n e r ' s

c o u n t a b l e compact s p a c e i n which e v e r y s e t o f c a r d i n a l i t y

< c

was nowhere d e n s e , and in both c a s e s the r e s t of the s p a c e was the o n e - p o i n t

compactification of a discrete space, with the whole space minus the extra point being pseudocompact.

Their spaces, however, were built by transfinite induction, like the

well-known examples of Ostaszewski [OSl], Juh~sz, Kunen, Eudin [JKR], and many others, including 3.8 and later examples below.

These constructions leave room for a lot of

optional details, whereas Reznichenko's space, as can be seen, is really just one space. (Of course, it is easy to construct variations on it.) Another difference is that Zhou's space is separable, and Berner's seems inevitably to involve separable pseudocompact noncompact subspaces, while in Reznichenko's space,

148

every countable subset has metrizable closure.

Nevertheless, until I saw it, I would

have guessed that the classes "Fr~chet" and "every pseudocompact subspace is compact" were destined for another independence result. 3.7.

Peter Simon's "barely Fr4chet" compacta.

This is a pair of one-point

compactifications of

V-like spaces whose properties are intermediate between those of

and the Cantor tree.

On the one hand, their one-point compactificatlons fail to be

but on the other hand they are Fr~chet, and compactness gives

o~3,

=4"

The product of the two spaces fails to be Fr4chet}on the other hand, the product of an

~3-Fr~chet and a countably compact Fr4chet space is Fr4chet [A4, 5.16], thus both

compactifications fail to be

~3"

These properties of Simon's spaces answer quite a few

questions in [A4]: 5.21, 5.22.1, 5.22.4, 6.12, 6.13, 6.14. There is a close interplay between the construction and Y-like space, with continuous Y + ~

its set of isolated points.

~ Y + =,

is Fr~chet iff

family PI'

f: ~

~

P

f+(Y-~)

of subsets of

such that

with

~

taken onto

(Y-e) U [®}.

is regular open in

~ .

Y ~

be a noncompact induces a

Malyhin [Ma] showed that

Simon was able to find a MAD

which was the disjoint union of two subfamilies

U {A* : A ~ Pi}

denotes the remainder of

~0"

~ko. Let

The identity on

was a regular open subset of

A, i.e.

(cl B~0A) - m.]

~0"

for

PO

i=O,l.

and

[Here

A*

In other words, here we have two

families of disjoint clopen sets, with the union of each family regular open, and the , union of these two regular open sets dense in ~ . It is the fact that the union of these two regular open sets is not regular open that is behind the product of the associated spaces not being Fr~chet. I have called this setup "Petr Simon's checkerboard", with the remainder of each A G PO

a "red square" and that of each

A ~ PI

a "black square".

One might paraphrase

Simon's closing note in [Sil] by saying that every infinite MAD family traces a "checkerboard" on the remainder of some infinite subset of 3.B.

~.

A consistent (b=c) example of a compact w-space which is not

We begin with a ZFC construction which, like the an uncountable

~-bisequential.

k'-modification of the Cantor tree, is

V-like space whose one-point compactification is a w-space.

It is the

example "For later use" in [vD, 12.2], but used for a different purpose there. The set of isolated points is

m×m,

and the ADF is the set of all columns

together with the set of the graphs of a {f : = < b}

of increasing functions

order, i.e.

f 5 a n d i f

then there is W C_ F M T~ such that ~ • W ~

a < fl are in C a n d S •

F N T ~n

~ > ~, and if ~ and ~r are distinct elements of

W then { ~ ( t ) : t • n - G} is disjoint from {Ygt(t):t • n - G}, and for all ~ • £ we have ~(t) • bt for t • G, and W is infinite unless G -- n. A is a finite rectangle if fi_ is a finite sequence with A(i) C_ wt finite for i • dom(.4). V ( ~ , f , A ) iff (Vi < lh(~) = lh(A))(Vy _< ~(i))(f(y) ~ A(i)).

( f , S ) fulfills F iff whenever ~ • r M T a and /3 • C(F) and ]l a finite rectangle, then there is W C_ r n T ~ °' such that W is infinite unless G = n and if ~ and ~ are distinct elements of W then { ~ ( t ) : t • n(£) - G(F)} fq { ~ ' ( t ) : t • n(F) - G(F)} = 0 and, for all ~ • W, _> # and (¢2(#, f, .4) ===~ £)(~, f, A)). For b an uncountable branch of T, we let C2(b,f,F) iff (Vx • b)~)(x, f, F). M a i n D e f i n i t i o n . ( f , S , 3/,g2) • P iff:

( f , S , N ) • Q and • is a countable set of promises

that ( f , S ) fulfills, and if {wl M 3/(/~):fl < a} is unbounded in "~ • S* then a • dora(N) and 3/(a) =

U~ p0 _> pl _> ... such that pi E M and p2~+1 E D , and {Pm:m E ~} is bounded below by some q E P, and in choosing P2n+2 we either have P~,~+2 IF "9 E A , " for some Y < x,~, or we have taken certain steps, described below, towards ensuring q f~- "~, ~ -4n ." Unlike the situation in Iemma 3, we need not have fq defined on T~; however, we still must take steps to ensure that each F E Ume~ ~ w is fulfilled at level 6. Thus, at stage n, we assign finitely many finite sets Fo . . . . Frn. C_ 6 to nodes Xo. . . . , zm. in T~ and declare that fpk (Y) ~ Fi for all y < xi and all k < w. We do this according to our ordering of ( ~ m , F m , . ~ ) (doing nothing, of course, if r ¢

or ht(~ n) > ht(p.) or - ~ ( ~ ,

h . , fi*~)) thereby causing each promise in UnE~ ~v- to

be fulfilled at level 6, The restrictions thus imposed do not hinder us from choosing P2~+1 E D,~ by lemma 3; however, they introduce a further complication towards choosing P2.+2. What we require is the following claim: Claim: Suppose ~ E T ~ and fi~ C_ 6 m a finite rectangle and c3(~,f v. . . . . fi~), and ~ # < ~ and h t ( ~ #) = ht(p2.+l). Then either:

(*) (3q < Pzr,+l)(~y < x.)(ht(q) < 6 and c2('~,fq,fi~) and q IF- "~) E ~{."); or (**)

m ~(..:.::, is comparable to ~ # x,~ E Y = {y E T*:(3~* E T ht(y)jxw

and (Vq < p2,~+,)(if

ht(q) = ht(y) and q)(~*, fq,.~i) then there is a promise r such that o(F) = ht(q) and (fq,S~, J¢~,

• ~ u {r}> t~ "9 ~ 2 . ' ) } . We show that the claim implies the lemma. Suppose (*) holds and q _< P2.+~ and y < x,~ witness (*). Then certainly (~q' _< p ~ + l ) ( h t ( q ' ) = ht(q) and ~)(~',fq,,A) where ~ ' < E and ht(E') = ht(q') and q' It- "9 E ~{,~'); and since ht(q') and ~ ' are in M, we may take P~n+~ to be such a q' in M. If instead (**) holds, we set p~+~ = p:.+~ and we take ~* to be a witness to (**) and demand that h~ (Y) ~ A(i) whenever y < E ' ( i ) and k E w. Having done so, we know by (**) that there is F,~ such that (Ukeo~ fv~, Uke~ SPk, UkEw ~;'~' Ukew ~P~ U {F~}) I~- "~r~ ~ An;" thus, taking q = (U f w , U sp~, U Nw, U kop~ u {F,~: (**) holds at stage 2n + 1}), we satisfy the demands of the lemma. Thus, it suffices to prove the claim. Proof of claim: Suppose (*) fails and y < x,~. We show that y E Y; by choice of x,~ (i.e.,

(VX E N ) ( x . E X ==~ (~y < x . ) y E X)) this suffices. In fact, we show that the unique ~*
ht(q') such that ~ ( ~ ' , fq,, A). Since ~ ' E A, there is no r _< q~ with ht(r) < ht(~') and ~ ( ~ , f~,_A) and r IF "~) • -4n;" but q' witnesses the opposite, a contradiction. Thus q+ 11- "~ ~ An" and we are done. We have proved: T h e o r e m 2. Con(ZFC + ~ is inaccessible) iff Con(ZFC + CH + there is an Aronszajn tree T* and a stationary co-stationary S" such that T* is S-*-special iff S - S* is non-stationary, and every wl-tree is S*-*-special).

References

[Sa}

Baumgartner, James, "Iterated Forcing," in Surveys in Set Theory, A. R. D. Mathias (ed.), London Mathematical Society Lecture Note Series, vol. 87, Cambridge University Press, Cambridge, 1983.

[DJ}

Devlin, K. J., and H. Johnsbr£ten, The Souslin Problem, Lecture Notes in Mathematics, vol. 405, Springer-Verlag, New York, 1974.

[Schll

Schlindwein, Chaz, Club Sandwich Forcing, PhD. thesis, University of California, Berkeley, 19xx.

[Sch2] Schlindwein, C., "Proper Forcing, Aronszajn Trees and the Continuum," 19xx.

[sh]

Shelah, Saharon, Proper Forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.

[To]

Todorcevic, Stevo, "A Note on the Proper Forcing Axiom," in Axiomatic Set Theory, J. Baumgartner, D. A. Martin, S. Shelah (eds.), Contemporary Mathematics, rot. 31, Amer. Math. Soc., Providence, 1984.

Consistency o f positive partition theorems f o r graphs and models by

Saharon Shelah

Department of Mathematics Rutgers University New Brunswick N.J.U.S.A.

Institute of Mathematics The Hebrew University Jerusalem, Israel

Recently A. Hajnal, P. Komjath [I] have dealt with the partition relation H ~ (G)~ :

if we colour the edges of a graph

induced subgraph isomorphic to the same colour). consistent (with for no graph

G

H

by

~

which is monochromatic

ZFC) that there is a graph

G

of

cardinality

R1

such that

H : H ~ (Gi~ .

is consistent.

We g i v e h e r e an a f f i r m a t i v e

answer (even for much stronger partition relations). class of measurable cardinals (in §I, §2). We can also generalize

result

(i.e. all edges get

They prove (generalizing a proof from Shelah [2]) that i% is

They a s k w h e t h e r t h e n e g a t i o n

morphism of

colours, there is an

N

in which only

([3],[4])

like

discussed elsewhere.

2

R0

M ~ (N) 8(*) < 81

We first prove it using a

In §3, §4 we eliminate this. to

rNl ~ , t h e n we h a v e t o a d d i n f o r m a t i o n So we h a v e f i n i s h e d Secondly,

the case

we a s s u m e

6 J

u C {aj

satisfying

and

E ~p •

for

i < c .

and

v E [u] sup(~jln aj2n M~) = sup(~jN M{j~})

of

if

is

has only two members. Assu~e

to which of the three possible equalities holds) or to

F(Cjl,Cj2)

is strongly inaccessible.

178

[Proof:

Note that all

If each

~i

singular,

fl(6)

cf(a i) E M{i }

realize the same type in

is singular, there is is a club of

clearly

6

Let

fl E M~

cf(6) •

hence for some

f2 E M#

8

onto

As

#

(in the model

and

cf(~i) < @i

6 < A

of cofinality

fl(6) . So easily

fl(@i) N M{i } = fl(ai) N M~ = {f2(@i)(7) : 7 E 8 n M#} ; w.l.o.g, definable over

6 < A

8 E M~ ,

be such that for

is a one-to-one function from

(H(x),E,< ~) •

such that for

of order type

cf(~ i) E ~

(Vi < ~)[cf(~ i) = 8] . 8,f2(6)

~i

(H(X),6, n -I _

then

. . . 1

n > I)

(M,A)

I , s o there is

Choose

For

(~+l)-Mahlo and

by a predicate for n = 1 .

n .

A

= ~ , ~ , (M;i } : i E B)

(Hs, t : [s[ = Itl ; s,t E [B] ~2)

and

W

such that:

+

(a)

6 < A , cf~

(b)

B

is a subset

actually (c)

=~

.

of

b u t then

Ms ~L M ~,~

for

6

of order

M{maxB} s E [B] 2

type

~ + (we c o u l d g e t

~+ + i ,

i s not d e f i n e d ) . and

M{i} ~L

M{i} ~L ~,~

M ~,~

for

i E B

and

184

Me ~2

so

there

+M

are regular and

where = 8

A

is inaccessible

D .

by any fragment of

I1%11

6 .

"

We can instead " A = 8 + " assume

Similarly for

8 , (or at least

is increasing converging to

~{i},{j}(~l)

L D,a

then ~#

, ~e = min{~ 6 W : Be < ~} , ~l # ~2 ' and

(2)

Let

and

o Hs0,sl)

eM{i )-r~,ieB, ~:min{~ : ~ e w , ~ < ~ }

If

Mt

onto

S

M{i,k ) = M(i ) ,

Hs,t(fl) = 7

and

M

B

(~)

Proof:

H s0's2

mops

s,t

Mfi,j }

3.8A

and

H

(a)

(c)

is an isomorphism from

M{j} .

are compatible;

from

, M~ fl B = ~ .

IMoI .

is a model

Let


. ~ :
m'.

Now we can find

and

(q~ : ~ < k(*)) such that

190

(i)

q~ E Mt£

(iii) hto,ti(q) ( q~ if qO El ~ q, E Mt

(iv) for £1 < £2 < k(*) we have: 0 q£2 -< q" E M t£1 and q' I M~ = q" r M~

then we can find

[Why? We define, by induction on i < k(*),

and

£0

r as above. such that

(q~,i : £ < i} satisfies (i),(ii),(iii),(iv) above with the natural restrictions.

For

i = 0 , q~,O = qo ' For

i = j+1 apply the assertion above

(before I. - 6.) so with ht£,to(q~'J ) here standing for q there; get there r and

let q~,i = htoUtl,tjUti(r t Mtl) q~,i = htoUtl,tjUti(r ~ Mto) ,

and for £ < j, q~'i = q~,j . In the end let q~ = q~,k(*)-i .

Let

{(~E,7£) : £ < (k~*)) = m}

list the increasing pairs.

by induction on E < (k~*))

{q~ : ~ < 1.

El

£2

q~ ~q~

for £ 1 ~ £ 2

z. q~ e st~ 3.

4.

q£

= q£

E r

r £+1 "HE

s.

r~,7 r Mt < £

s.

r~,~ F ~ ( ~ , % ) = ¢0

k(*)) ,

r~£,7E such that:

Now we define

191

7.

If

e~.l

i s an edge o f

r~.,75" 1

(MtgE. x MtgE. ) U (Mt

x Mt 7[.

I

then 8.

I

eN,e I If

7 g

7[.

I

not in 1

) U {(W~ ,WT~.)} for i I

6 0 ~ £I

have no vertex in common.

{~,~}

then edges(q~+l) = edges(q~)

Now we d e f i n e dom q =

U

edges o f

dom q ~ U U

E+I

~ M~) .

is tailor-made for this.

dom

q = union o f the s e t o f edges o f

k(*) +I ~

,~

.

is connected.)

are pai~ise compatible.)

The least trivial is to show be a set of

q~ , r

dom r~ ,7 \(Mt U My ) i i Hi 7i

(Note that the q~ . ~ . ~

Assume that

.

edges(qot [

U

q :

(Note that any node in

~

and

I

There is no problem in this - qo

Let

i E 0,I

Kk(,)+l

is not embeddable into

q .

vertices.

is a complete graph fin

q) and we shall derive a

contradiction. I f we omit the edges

{(~i,~j):

i < j < k(*)}

from

q , the resulting

graph is o b t a i n e d by s u c c e s s i v e e d g e l e s s amalgamation (look a t the r e s t r i c t i o n to

i

$ the first measurable. Shelah showed it consistent with GCH that there exist an example of size R2, and, assuming the consistency of a weak compact, the consistency with CH that there exists no example of size R2. One certainly expects that a supercompact should suffice to obtain the consistency of there being no examples of size greater than RI, but as usual, the difficulty is in proving a suitable preservation lemma. All one needs to know about this problem appears in [J]. Shelah's example (reworked and improved in [HJ2]) is a graph constructed by forcing or by a morass with built in o. Definition. L(X) = min{~: every open cover of X has a subcover of cardinality

_ K, an SC can never be cub.

Theorem 1.5

NS ~I B is set-normal for any SC,B, and under the hzpothesis theft

P ~ has an unbounded subset of size ~, every set normal ideal extends ~I~S~

-'i