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se s. se s i f and only i f {s e S:
t h a t the d e f i n i t i o n s agree with the e a r l i e r c o n s t r u c t i o n , f i r s t I e p-lim
Then {t e S: t • s e A} e p.
To see
l e t p e ~S, s e S and
That is A/s e p.
Hence A e p * s
Thus p-lim
S i m i l a r l y , one sees t h a t t e T l i m
A/s e p} e q. l n o t h e r c h a r a c t e r i z a t i o n is from the f o l k l o r e , t o l d to me by Blass. ultrafilter
(The
p ® q defined below i s the same as t [ q • p] as defined in [16, p. 157].)
2.2 Definition. on Y, define
Given sets X and Y, p an ultrafilter on X, and q an ultrafilter
p ® q : {A C X ~ Y: {y e Y: {x e X: (x,y) e A} e p} c q}.
100
Now, given our d i s c r e t e semigroup S, we have • : S . S ~ S [ flS so t h a t i t has a continuous extension .fl: fl(S x S) ~ ~S. P * q = *flip ® q)"
I t is then a f a c t t h a t , f o r p,q e flS,
To see t h i s , l e t B e *fl(p ® q) and pick A e p ® q such t h a t
• fl[~] ~ B. Let C = {y e S: {x e S: ix,y) e A) e p). Then C e q and, given y e C, {x e S: (x,y) e k) ~ B/y so t h a t C £ {y e S: B/y e p}. That i s , B e p • q as required. However i t is constructed, we have that (flS,.) is a compact (Hausdorff) l e f t topological semigroup. There is a fundamental s t r u c t u r e theorem f o r compact l e f t t o p o l o g i c a l semigroups due to Ruppert [48, Satz 2]. (Or see [9], where the theory is derived in reasonably elementary d e t a i l . ) 2.3 Theorem (Ruppert). Let T be a compact l e f t topological semigroup. (i) T has a unique smallest two-sided ideal K. ( i i ) K has idempotents and, given an idempotent e e T, the following are equivalent:
Then
(a)
e E K;
(b)
K : TeT;
(c) (d) (e)
eT is a minimal r i g h t ideal of T; Te is a minimal l e f t ideal of T; eTe (with the inherited operation) is a group (and a maximal subgroup
of T). (iii) (iv) (v)
Each minimal r i g h t ( r e s p e c t i v e l y l e f t ) ideal of T is of the form eT ( r e s p e c t i v e l y Te) for some idempotent e e K. K = U{eTe: e is an idempotent in K} = U{eT: = U{Te: e is an idempotent in K}.
e is an idempotent in K}
All maximal subgroups of K are a l g e b r a i c a l l y isomorphic and maximal subgroups of K which are contained in the same minimal l e f t ideal are isomorphic via a homeomorphism.
(k r i g h t ideal of T is a subset A $ ~ such t h a t k • T ~ A. ideals are defined analogously.)
Left and two-sided
Since flS is the Stone-Cech compactification of S, we have t h a t , i f T is any compact l e f t t o p o l o g i c a l semigroup and ~ is a (necessarily continuous since S is d i s c r e t e ) homomorphism from S to T then there is a continuous extension ~ :
flS ~ T.
I t is f u r t h e r quite easy to show that i f , f o r each s e S, P~(s): T ~ T is continuous, then ~ is a homorphism. I t was shown in [9, Theorem 4.51, that a similar universal object e x i s t s f o r any Hausdorff semitopological semigroup S. This r e s u l t was s i g n i f i c a n t l y improved [38, Theorem 2.10] by dropping any separation assumptions and any assumed r e l a t i o n s h i p between the topological and algebraic s t r u c t u r e of S.
101
2.4 Theorem (Hindman and lilnes). topological space.
Let S be a semigroup which is also a
There exist a compact Hausdorff left topological semigroup ~S and
a continuous homomorphism e: S ~ ~S such that e[S] is dense in 6S and Pe(s) is continuous for each s e S. Further if T is a compact Hausdorff left topological semigroup, ~: S ~ T is a continuous homomorphism, and P~(s) is continuous for each s c S, then there is a continuous homomorphism 7: ~S ~ T such that ~ o e = ~.
A recent observation of Ruppert [47] is of considerable interest in the context of ~S because its hypotheses hold whenever S is commutative, as can be routinely verified. 2.5 Theorem (Ruppert).
Let T be a left topological semigroup with dense center.
I f L is a l e f t ideal in T then cl(L) is an ideal. Thus, in particular, if T is compact and L is any minimal l e f t ideal, then cl(L) is the closure of the minimal ideal of T. (See Section 5.)
3.
light Continuity at Points of ~S\S -- Some Contributions of Eric van Douwen.
In 1978 and 1979, van Douwen wrote a manuscript 121] dealing with ~S where S was a discrete cancellative semigroup.
Several of the results in this paper were
published, with his permission, in [37]. results soon.
At that time he was promising to publish the
However, by 1985 when I spoke to him about this matter he had decided
not to publish these results; he felt that most of them were already known or bad been improved on by others in the meantime.
This judgement was largely true.
Ve present
in this section the most significant exceptions. These results respond to a question asked by Raimi in correspondence. concerned with (/~N,+), knew that if p e ~N\N, then pp is not continuous. example [37, Theorem 10.12].)
Raimi, (See for
He asked whether one could show that the restriction of
pp to ~N\N is not (ever or always) continuous.
Eric answered this question rather
nicely in a fairly general context. Unfortunately I only have his answers in the form of a letter which included no proofs.
Accordingly the sequence of lemmas which I present here are my own and I was
unable to prove one of his results in the full generality which he had. Recall that p ~ ~S is a uniform ultrafilter if and only if for each i c p, IAI = ISI. Recall also that an u l t r a f i l t e r is ~- complete provided whenever A ~ p and IAI < ~ one has nA ~ p.
3.1 Definition. Let S be an i n f i n i t e discrete semigroup. (a) U(S) = {p ~ ~S: p is uniform}.
102 (b)
For p e U(S), R(p) = {q e U(S):
(c)
For p e U(S), J(p) = {q e U(S):
PqIU(S) i s c o n t i n u o u s at p}. the r e s t r i c t i o n of * to ~(S) - ~(S) i s
continuous at (p,q)}. 3.2 Lemma.
Let S be a right cancellative semigroup with ISI = ~ > ~ and let
<2 enumerate S.
Let D e [S] ~.
such that, if for ¢ < 2 one has B y a ~ B. Proof.
There is a one-to-one ~-sequence
Construct
[B~]I -< I~t • I~1 (by r i g h t c a n c e l l a t i o n ) so t h a t
each y < ¢, IP
Pick y, e D\Ur/<_~rpal[B],
l~l
Given B , one has ]B] < [el •
so for
l U < ~ Pa [Bfl I < 2.
n
q
3.3 L e n a .
Let S be a with IS[ = 2 > ~ and l e t g<~ enumerate S.
such t h a t q i s not a+-complete.
Let q e flS
There e x i s t s g: 2 ~ w such t h a t f o r each n < ~,
{a : g(~) > n} e q. Proof. a
Pick {An: n < w} c q such t h a t fin<w An ¢ q.
e nn< W An and o t h e r w i s e g(a) = n where n i s the f i r s t
n < w and suppose {a : g(~) > n} ¢ q. with {a : g(~) = m} e q. S\Am e q.
¢ An .
Let
Then Um
I f m = 0 one has (On<~ An) 0 (S\Ao) e q.
Otherwise one has
I n e i t h e r case one has a c o n t r a d i c t i o n .
3.4 Lemma. enumerate S.
Let S be a c a n c e l l a t i v e semigroup with ISI = ~ > w and l e t <~
Let D e [S] 2 and l e t
p e U(S) with {y: q < ~} e p. then p is a P-point of U(S). Proof.
Define g(~) = 0 i f index with a
Let
If there is some q e R(p) which is not w+-complete,
Let such q be g i v e n .
Pick g: 2 ~ w as g u a r a n t e e d by Lemma 3.3.
t h a t p i s a P - p o i n t of U(S), l e t {Cn: n < ~} £ p.
To see
Ve need t o produce E e p such t h a t
IE\Cn[ < 2 f o r each n < ~.
For each ~ < 2 let B be as in Lemma 3.2 and let H = {Yr: r > ~} N Nm
Ve may presume that B ~ {y : ~ < ~}.
Suppose not and pick ~ < ~ with IE\H I = 2 and pick r e U(S) with E \ H n = g(~).
Now A e r * q, since r e 6, so {x e S: A/x e r} e q.
since q is uniform.
And {a : g(r) > n} e q by the choice of g.
e r.
Let
Also {at: r > ¢} e q Ve may thus pick a
103 point in {x e S: A/x ~ r} 0 {ar: r > ~} 0 {at: g(r) > n}. That i s , we have r > a with g(r) > n such t h a t A/a r e r. Since r > ~ and g(r) > n = g(~) we have Hr c Ha. Since E \ H ~ r we have E\H r ~ r. We show that (A/ar) o (E\Rr) ~ {YT: 7 ~ r}, contradicting the choice of r as a uniform u l t r a f i l t e r . Suppose instead we have 7 > r with Y7 e (A/at) 0 (E\Hr). Now Y7ar e A so pick # with Y7 " a r e H# • a#. Pick ~ > ~ such that y~ ~ H# and Y7 ~ ar = Y~ • a#. Now, i f 7 > ff then Y7 * a r e B7 while i f ff > 7 we have yff . a e By. In e i t h e r case we c o n t r a d i c t Lemma 3.2. Thus we must have 7 = Y. But then by l e f t canceIlation we have a r = a and hence r = ~. But then Y7 = Y~ e ~# = Hr while Y7 e E\~r, a contradiction. Thus we have t h a t f o r each ~ < r, [g\H ] < ~. Now given n, pick a such that g(~) > n. Then ~ £ Cn so E\C n £ E \ ~ so [E\Cn[ < ~ as required. Recall that a cardinal ~ is Ulam-measurable if and only if there is an #+-complete non-principal ultrafilter on A. (See ~16, Chapter 8] for discussion of this notion.) The assumption that A is not fflam-measurable in the following theorem is mine. It was not in van Douwen's letter and is presumably not needed. (At any rate it is consistent that there are no Ulam-measurable cardinals.) 3.5 Theorem (van Douwen). Let S be a c a n c e l l a t i v e semigroup with ISI = 2 > 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
so is not a P-space [26, Problem 4K].
p ¢ E 0 U(S) such that p is not a P-point.
Pick
Suppose R(p) # # and pick q e R(p).
Since
is not Ulam-measurable,q is not ~+-complete. But t h i s c o n t r a d i c t s Lemma 3.4. o Notice in p a r t i c u l a r that since (N,+) is c a n c e l l a t i v e and w is not Ulam-measurable, Raimi's question is c e r t a i n l y answered by Theorem 3.5. I quote now (with s u b s t i t u t i o n of notation) from E r i k ' s l e t t e r announcing these results. "One would perhaps think one must be able to get a l l of U(S) in Theorem (3.5), at l e a s t in a special case like S : (N,+) or at l e a s t get the r e s u l t that J(p) = ~ f o r a l l p. Not so:" 3.6 Theorem (van Douwen). Let S be a c a n c e l l a t i v e semigroup with ]S I = w. following statements are equivalent. (a) (b) (c)
{p ¢ U(S): J(p) = U(S)} is dense in U(S). {p e U(S): R(p) # ¢} is dense in U(S). ~ \ N has a P-point.
The
104
Therefore (a) and (b) a r e c o n s i s t e n t with but independent from ZFC. Proof.
That (a) i m p l i e s (b) i s t r i v i a l .
To see t h a t (b) i m p l i e s ( c ) , enumerate S as
Let E = {Yn: n < ~} and pick
By Lemma 3.4 and the f a c t t h a t w i s not
Ulam-measurable p i s a P - p o i n t of U(S).
Since U(S) and ~N\N a r e homeomorphic, ~N\N
has a P - p o i n t . To see t h a t (c) implies ( a ) , we show t h a t i f p i s a P - p o i n t of U(S), then J(p) = U(S).
( I t i s easy to see t h a t i f ~ \ N has a P - p o i n t , then the s e t of P - p o i n t s
of U(S) i s dense in U(S).)
Let p be a P-point of U(S) and l e t q e U(S).
To see t h a t
IU(S)xU(S) i s continuous at ( p , q ) , l e t I e p * q. Let B = {x e S: A/x e p}. ~ e q. Since p i s a P - p o i n t of U(S), pick C e p such t h a t , f o r each x e B, IC\(A/x)I < ~.
¥e claim *[(C x ~) n (U(S) x U(S))] ~ ~.
( r , s ) e (C x B) n (U(S) x U(S)). t h i s end, l e t x e B.
We show B ~ {x e S:
Then [C\(A/x)[ < w.
Then
To t h i s end l e t
A/x e r} so t h a t A e r * s.
To
Since C e r and r e U(S), A/x e r as
required. The l a s t remark about c o n s i s t e n c e and independence r e f e r s of course to the famous r e s u l t s of V. Rudin [46] and Shelah [50]. [] In c l o s i n g t h i s s e c t i o n we turn to another c o n t r i b u t i o n of van Douwen's -- in t h i s case not a r e s u l t but a q u e s t i o n . In [34], I had p r e s e n t e d a proof t h a t , i f the Continuum Hypothesis holds and the (then unproved) F i n i t e Sum Theorem was v a l i d then t h e r e e x i s t e d p e flN\N such t h a t , f o r each A e p, {x e A: A - x e p} e p.
Since we
now know t h a t t h i s simply says p + p = p, and since idempotents are known to e x i s t in any compact l e f t t o p o l o g i c a l semigroup [22], the e x i s t e n c e of such p i s known with no s p e c i a l assumptions. theorem.)
(See [4] f o r a d i s c u s s i o n of the o r i g i n s of the idempotent
I f e l t t h e r e was no longer anything of i n t e r e s t in [22].
However, in
conversation in J u l y of 1985, Eric noted t h a t the proof in [22] a c t u a l l y e s t a b l i s h e d the e x i s t e n c e of an u l t r a f i l t e r B e [N] ~. ZFC.
with a b a s i s c o n s i s t i n g of s e t s of the form FS(N) f o r
He asked whether one could prove the e x i s t e n c e of such an u l t r a f i l t e r
in
This question d i r e c t l y i n s p i r e d t h r e e papers [32] (by me), [11] (by Blass) and
[12] (by Blass and me).
The l a s t of these papers included the answer to E r i k ' s
question. We found out, a f t e r o b t a i n i n g t h i s r e s u l t , t h a t the q u e s t i o n had been independently asked - - and answered -- by ~ a t e t in 1985. ( l a t e t ' s r e s u l t s will appear
in [42].)
105
3.7 Theorem.
(Matet and, independently but l a t e r , Blass and Hindman).
I f there
is p e fin with a base of sets of the form FS(B) for B e [N] W, then t h e r e e x i s t s a P - p o i n t in ~ \ N .
4.
The Semigroups (~N,÷) and ( ~ , - ) .
For some time a f t e r the Glazer-Galvin proof of the F i n i t e Sum Theorem (see [37, Section 8]) i t was an open question as to whether there was some p e ~N\N with p = p + p = p • p.
The e x i s t e n c e of such an u l t r a f i l t e r
was known to imply t h a t ,
given r e N, i f N = Ui
That conclusion i s now known to be f a l s e .
In f a c t , l e t t i n g
PS(B) = {x + y: x,y e B and x ~ y} and PP(B) = {x • y: x,y e B and x ~ y} we have: 4.1 Theorem ( [ 3 0 ] ) . There e x i s t
There do not e x i s t p and q in flN\N with p + q = p • q.
This leaves a very annoying question ( f i r s t asked by van Douwen in correspondence):
Do there e x i s t p , q , r , and s in ~N\N with p + q = r • s?
ks we have already seen, combinatorial statements f r e q u e n t l y have a corresponding a l g e b r a i c statement i n ~ .
4.3 Theorem ( [ 3 0 ] ) .
For example, Theorem 4.1 may be rephrased as:
There i s a p a r t i t i o n of ~N\N i n t o seven open-and-closed
subsets so t h a t , f o r no p e ~N do p + p and p * p l i e i n the same c e l l of the partition. By comparison, I know of no nice combinatorial statement e q u i v a l e n t to the s o l u t i o n of p ÷ q = r • s.
I t i s simply one of those q u e s t i o n s which is annoying
because we c a n ' t answer i t . Several r e s u l t s of van Douwen e x h i b i t i n g "bad" behavior of (~N,+) and (~N,o) were presented i n [37]. 4.4 Theorem. (~,e)}.
In [28], we showed t h a t l e f t c a n c e l l a t i o n is much b e t t e r behaved. Let C = {p e ~N\N: l e f t c a n c e l l a t i o n holds at p in (fiN,÷) and in
Then C has dense i n t e r i o r i n ~N\N.
106
As we have seen in Section 2, the minimal i d e a l of a compact l e f t t o p o l o g i c a l semigroup i s the s m a l l e s t two-sided i d e a l . i d e a l s of (fiN,+) and ( ~ , o ) r e s p e c t i v e l y . c a n c e l l a t i o n in ( ~ , + ) f a i l s a t p. 4.5 Theorem ( [ 2 8 ] ) .
Let us denote by M and K the minimal I t i s easy t o see t h a t i f p e M, then l e f t
There e x i s t s p e c l i such t h a t l e f t c a n c e l l a t i o n holds a t p.
4.6 Theorem ([28] and [33t).
I N K = ¢ but c l l n K ~ ¢.
R e c a l l from Section 2 t h a t M i s the union of p a i r w i s e isomorphic groups. an obvious q u e s t i o n , f i r s t groups look l i k e .
I t is
r a i s e d in correspondence by Karl Hofmann~ as to what these
In [39] we obtained the following r e s u l t .
For l a t e r r e f e r e n c e we
note t h a t the subgroups produced are a l l contained in NneN ~ . 4.7 Theorem (Hindman and Pym).
Let p be an idempotent in (~N,+).
p + fN + p c o n t a i n s a copy of the f r e e semigroup on 2 c g e n e r a t o r s .
Then
I f p e M, then
p + fin + p c o n t a i n s a copy of the f r e e group on 2 c g e n e r a t o r s . This r e s u l t has r e c e n t l y been extended [41]. 4.8 Theorem ( L i s a n ) .
There e x i s t c p a i r w i s e d i s j o i n t t o p o l o g i c a l and a l g e b r a i c
copies of nne N N2n which miss c l L
In p a r t i c u l a r t h e r e e x i s t copies of the f r e e group
on 2 c g e n e r a t o r s which miss clM.
Ve remark t h a t Pym [44] has r e c e n t l y shown t h a t the e n t i r e s t r u c t u r e of OneN N2n a r i s e s in a n a t u r a l way in fS f o r many d i f f e r e n t S, where S i s assumed to have much l e s s s t r u c t u r e than a semigroup. (Ve a r e prevented from being more p r e c i s e by the amount of terminology which would have to be i n t r o d u c e d . )
5.
The Ideal Structure of flS.
In [35] we were a b l e to c h a r a c t e r i z e the minimal i d e a l of ~S and i t s c l o s u r e , with no s p e c i a l assumptions on S. 5.1 Theorem.
Let p e fS.
Then p i s in the minimal i d e a l of flS i f and only i f
f o r each A e p, t h e r e e x i s t s F e [SI <W such t h a t f o r a l l y e S, (Ute F A / t ) / y c p. Also p i s in the c l o s u r e of the minimal i d e a l i f and only i f f o r each A ~ p t h e r e e x i s t s F e IS] <W such t h a t , f o r a l l G ~ IS] <W t h e r e e x i s t s x c A with
107
x • G [ UteF(l/t ). I f S is commutative, i t is an immediate consequence of Theorem 2.5 that the closure of the minimal ideal of ~S is again an ideal of ~S. is not needed.
In f a c t t h i s assumption
5.2 Theorem ([35]). The closure of the minimal ideal of 85 (for any d i s c r e t e semigroup S) is again an ideal of ~S.
However, we showed [36] that T = nne N N-~ is a compact subsemigroup, the closure of whose minimal ideal is not an ideal. (~S\S) * (~S\S) is usually an ideal of flS, hence must contain the minimal ideal. In [53] using the notion " i n f l a t a b l e " (or [54] using the weaker notion of "m i n f l a t a b l e f o r some a ' ) Umoh showed t h a t (BS\S) • (BS\S) does not have to be much bigger than the minimal ideal. (The notion of " i n f l a t a b l e " is too complicated to l i s t here. I t l i e s s t r i c t l y between "cancellative" and " r i g h t c a n c e l l a t i v e " . ) 5.3 Theorem (Umoh). Let S be a countable i n f l a t a b l e semigroup with i d e n t i t y . There e x i s t points in the closure of the minimal ideal which are not equal to q * r for any q,r e #S\S.
I f A is a f i l t e r 85 are of t h i s form.
on S then ~ is a closed subset of 85 and a l l closed subsets of In [17], Davenport obtained the following simple
c h a r a c t e r i z a t i o n of when ~ is a subsemigroup.
5.4 Theorem (Davenport).
Let A be a f i l t e r on S.
Then ~ is a subsemigroup of 8S
i f and only i f f o r each I e A and each B e P(S)\A there e x i s t s F e IS\B] <W such that Ute F A/t e A. The v i r t u e of a c h a r a c t e r i z a t i o n such as t h i s is t h a t i t r e f e r s only to properties of S and A which are usually easy to check. Davenport also obtained a reasonably simple c h a r a c t e r i z a t i o n of the minimal ideal of ~ given t h a t A s a t i s f i e d c e r t a i n reasonably common conditions.
6.
Some Useful Notations Involving U l t r a f i l t e r s .
In [10], Blass introduced the "generalized q u a n t i f i e r " (p x)~(x), where p is an u l t r a f i l t e r on a set S, to mean ~(x) is true f o r p-almost a l l x. That is (p x) ~(x)
108
if and only i f {x e S: ~(x)} c p. This q u a n t i f i e r has the nice property that it "commutes with negation and conjunction and therefore with a l l p r o p o s i t i o n a l connectives." I will not be concerned here with the applications in [10], but r a t h e r with some other a p p l i c a t i o n s which Blass told me about. These a p p l i c a t i o n s depend on the simple f a c t t h a t , given p,q e ~S and a formula ~, i f (under a l l i n t e r p r e t a t i o n s of the unlisted f r e e variables in ~(x)) one has (p x) ~(x) *-~ (q x) ~(x), then p = q. (To see t h i s l e t ~(x) be "x e A". Then, i f A [ S, the statement (p x)~(x) is the statement t h a t A e p.) Now as usual assume S is a semigroup and p,q e ~S.
Observe that
(p.q x) ~(x) +-~ (q y)(p z ) ~ ( z . y ) . Indeed, {x e S: ~(x)}/y = {z e S: ~(z ° y)}. (p*q x)~(x)+-~ {y e S: {x e S: ~(x)}/y c p} e q
~-4 {y e S: {z e S:
Thus:
~(z.y)} e p} e q
{y ~ s: (p ~) ~(~.y)} ~ q
+-~ (q y)(p z) ~(z*y) Consequently we have a simple proof of associativity. ((p*q)*r x) ~(x) ~-~ (r u)(p.q z) ~(z • u) ~-~ (r u)(q v)(p w) ~((w*v)*u) +-~ (r u)(q v)(p w) ~(w,(v.u)) ~-~ (q*r y)(p w) ~(woy) ~-~ (p*(q.r) x) ~(x).
Let p,q,r e ~S.
Another application is a proof of the f a c t from Section 2 that .~(p ® q) = p.q. To see t h i s observe f i r s t that for f: S ~ T and p e flS, one has (f~(p) x) ~(x)
(p y) ~(f(y)).
(If B = {x + T: ~(x)} and B + f~Cp), pick A e p with f~[~] ~ B.
Then A [ {y e S: ~ ( f ( y ) ) } .
Thus (ffl(p) x) ~(x) ~ (p y) ~ ( f ( y ) ) .
Since t h i s
q u a n t i f i e r commutes with negation, the r e s u l t follows.) Also, d i r e c t l y from Definition 2.2 we have (p ® q x) ~(x) ,-* (q y)(p z) ~ ( ( z , y ) ) . We thus have (*~(p ® q) x) ~(x) ~-~ (p ® q u) ¢(*(u)) ~-~ (q y)(p z) ~(z*y) *-~ (p*q x) ~(x). The other notation and i t s applications were t o l d to me by Scott Villiams. did not know the o r i g i n , but said i t is "well known in Prague".
Ke
Given a compact Hausdorff space X, and f : I ~ X l e t fn be n-fold composition. (That is f l = f , fn+l = fn o f . )
(See Definition 2.1.)
Given p e ~N and x e X, define fP(x) = p-lim
109
6.1 Lemma.
Let X be a compact Hausdorff space and l e t f : X ~ X be c o n t i n u o u s .
I f p , q e ]~N, t h e n fq o fP : fP+q.
Proof.
Let x e X and l e t y = f q ( f P ( x ) ) .
To see t h a t y = fP+q(x), l e t U be an
open neighborhood of y and l e t A = {n e N: fn(x) e U}. f n ( f P ( x ) ) e U}. required.
Then B e q.
Let n e B.
Let B = {n e N:
Ve show t h a t B [ {n e N: 1 -
Since f i s c o n t i n u o u s and f n ( f P ( x ) )
neighborhood V of fP(x) with fn[v] ~ U.
n e p} so t h a t A e p + q as e U, pick an open
Let C = {m e N: fm(x) e V}.
g i v e n m e C, f n ( f m ( x ) ) e U so t h a t n + m e t .
Thus C ~ t -
The assumption of c o n t i n u i t y i n Lemma 6.1 i s needed.
n so 1 -
Then C e p and n e p. []
To see t h i s l e t
I = Z U {-®, +®}, t h e u s u a l two p o i n t c o m p a c t i f i c a t i o n of t h e i n t e g e r s and d e f i n e f : X ~ X by f ( n ) = - n -
1 i f n > O, f ( n ) = - n i f n < O, and f(+®) = f(-®) = O.
8bserve t h a t i f n e N t h e n f 2 n ( o ) = n and f 2 n + l ( o ) = - n - 1 and hence f2n+l(+®) = n. Now l e t p , q e ~N with N2 e p and N2 + 1 e q (so t h a t N2 + 1 e p + q).
Then
f q ( f P ( o ) ) : fq(+®) : +® ~ -® : fP+q(o). R e c a l l t h a t x e X i s a r e c u r r e n t p o i n t of f provided x i s i n t h e forward o r b i t c l o s u r e of f, i . e . 6.2 Theorem. continuous.
x e cl{fn(x):
n
e N}.
Let X be a compact Hausdorff space, l e t x e X and l e t f : X ~ X be
The f o l l o w i n g are e q u i v a l e n t .
(a)
x i s a r e c u r r e n t p o i n t of f .
(b)
There e x i s t s p e / ~ with fP(x) : x.
(c)
There e x i s t s p e ~N with p + p = p such t h a t fP(x) = x.
Proof.
To see t h a t (a) i m p l i e s ( b ) , l e t I = {{n e N: f n ( x ) e U}: U i s a
neighborhood of x}. A ~ p.
Then A has the f i n i t e
i n t e r s e c t i o n p r o p e r t y so pick p e ~N with
Then fP(x) = x.
To see t h a t (b) i m p l i e s ( c ) , l e t T = {p e ~ :
fP(x) = x}.
Then T i s c l o s e d .
p e /~N\T, t h e n f o r some neighborhood U of x, {n e N: f n ( x ) e U} ~ p.
For t h i s
U, i f B = {n e N: f n ( x ) ¢ U}, then B i s a neighborhood of p m i s s i n g T.) subsemigroup of (l~N,+).
Also T i s a
Indeed, i f p , q e T, then fP+q(x) = f q ( f P ( x ) ) = fq(x) = x.
Thus by [22], t h e r e i s an idempotent p e T. To see t h a t (c) i m p l i e s ( a ) , pick p with neighborhood of x.
fP(x) = x.
Let U be an open
Then {n e N: f n ( x ) e U} e p so i s non-empty. D
(If
110
7.
Density Results in lIalsey Theory.
A new area of Ramsey Theory was opened in 1974 with the proof [51] of Szemeredl s Theorem. t new powerful t o o l in this area was provided by Furtenberg's proof of Szemer~di's Theorem using ergodic theory [24]. (See [25] and [3] f o r extensive descriptions and additional references.) Recall t h a t the ordinary upper density of a set A ¢ N is ~i(A) = lim sup { [ t n { 1 , 2 , . . . , n } I / n :
n e N}.
A more natural notion of density from
n-~
the point of view of ergodic theory is what, following [251, we have agreed to c a l l the Banach density of A. That is, d*(A) = sup{a: there e x i s t increasing sequences <Xn>neN and
x n + t n } l / t n = a.
n-~m
Several subsets of fin are definable in terms of density.
As with many other
combinatorially defined subsets, these are definable in the form {p ¢ fN: for a l l A e p, ~(A)}. Any such set is automatically closed. ( I f A e p such that ~ ~(A), then is a neighborhood of p missing the defined s e t . ) Another useful notion is that of a " d i v i s i b l e " statement introduced by Glasner [27]. That is a statement ~ about subsets of N is d i v i s i b l e i f and only i f (i) ~(N) and ~ ~($); ( i i ) i f I c B C N and ~(A), then ~(B); and ( i i i ) i f A,B C N and ~(A U B), then ~(A) or ~(B). I t is easy to see that if is a d i v i s i b l e statement, then {p e fN: f o r a l l A e p, ~(A)) $ $. Theorem 6.7] or [36, Lemma 2 . 5 ] . )
(Or see [37,
7.1 Definition. (a) A = {p e fiN: for a l l A e p, ~(A) > 0}. (b) A* = {p e fiN: f o r a l l A , p, d*(A) > 0}. (c)
A1 = {p e ~ :
f o r a l l A e p, there e x i s t s k e N with d*(O~= 1 A - t) = 1).
By the above remarks, A and A* are non-empty. By Theorem 5.1 we have 41 =clM, where M is the minimal ideal of (fiN,+). ~e also have A is a r i g h t ideal of (fiN,÷) and of (fN,.) [37, Theorem 10.8] (due to van Douwen), 41 is an ideal of (fN,+) and a right ideal of (~N,.) [28, L e n a 3.5, Theorem 3.9], and 4" is an ideal of (BIN,+) and a right ideal of (fiN,.) [33, Theorem 7.12]. In [2], Bergelson introduced the following notion. 7.2 Definition.
A set B ~ N is a set of nice combinatorial recurrence if and
only if f o r a l l e > 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
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
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
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>n<¢ in k i so t h a t FS(<Xn>n<w) ~ k i and for each n, given <xj>j
115 has p o s i t i v e upper d e n s i t y . (c) (d)
i i i s piecewise s y n d e t i c . t h e r e i s some k such t h a t f o r each n t h e r e e x i s t s x with x • {1,2 . . . . . n}
(e)
U~=1 l i l t f o r each 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 f a m i l y F of f i n i t e subsets of Z t h e r e e x i s t s F e F with F ~ Ai .
(f)
t h e r e i s a s y n d e t i c s e t B with D(B) [ D ( l i ) .
(g)
f o r each e > 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
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
116
Let D = {y e N: {x e N: (x,y) e C} e p}. Bn ~ D.
Ve choose
Given n > 1 and
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
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2.
V. Bergelson, A density statement generalizing Schur's Theorem, J. Comb. Theory (Series A) 43 (1986), 338-343.
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D. Davenport and N. Hindman, Subprincipal closed ideals in fiN, Semigroup Forum, to appear.
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20.
V. Deuber and N. Hindman, Partitions and sums of (m~p,c)-sets, J. Comb. Theory (Series A) 45 (1987), 300-302.
21.
E. van Douwen, The ~ech-Stone compactification of a discrete cancellative groupoid, manuscript.
22.
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Z. Frolfk, Sums of u l t r a f i l t e r s , Bull. Amer. Math. Soc. 7_33(1967), 87-91.
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25.
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26.
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27.
S. Glasner, D i v i s i b i l i t y properties and the Stone-Cech compactification, Canad. J. l a t h . 32 (1980), 993-1007.
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N. Hindman, linimal ideals and cancellation in ~N, Semigroup Forum 25 (1982), 291-310.
29.
N. Hindman, gn density, t r a n s l a t e s , and pairwise sums of integers, J. Comb. Theory (Series A) 33 (1982), 147-157.
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N. Hindman, P a r t i t i o n s and pairwise sums and products, J. Comb. Theory (Series A) 3A (1984), 46-60.
31.
N. Hindman, Ramsey's Theorem for sums, products and arithmetic progressions, J. Comb. Theory (Series A) 38 (1985), 82-83.
32.
N. Hindman, Su~able u l t r a f i l t e r s and f i n i t e sums, in Logic and Combinatorics, ed. S. Simpson, contemporary lathematics 65 (1987), 263-274. N. Hindman, Sums equal to products in ~N, Semigroup Forum 2_!1 (1980), 221-255.
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N. Hindman, The ideal structure of the space of ~-uniform u l t r a f i l t e r s on a d i s c r e t e semigroup, Rocky lountain J. l a t h . 16 (1986), 685-701.
36.
N. Hindman, The minimal ideals of a multiplicative and additive subsemigroup of /~N, Semigroup Forum 32 (1985), 283-292.
37.
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(1975), 276-290. 44.
J. Pym, Semigroup structure in Stone-Cech compactifications, manuscript.
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47.
V. Ruppert, In a l e f t topological semigroup with dense center the closure of any l e f t ideal is an ideal, Semigroup Forum, to appear.
48.
~. Ruppert, Rechstopologische Halbgruppen, J. Reine Angew. Lath. 261 (1973), 123-133.
49.
I . Ruzsa, Difference sets and the Bohr topology I . , manuscript.
50.
S. Shelah, Proper forcing, Lecture Notes in Lath. 940 (1982).
51.
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] <<
P i e r r e MATET* Freie Universit~t Arnimallee
Berlin,
3,
Institut
1OOO B e r l i n
fur M a t h e m a t i k
II,
33, W e s t G e r m a n y
O. I n t r o d u c t i o n We shall
start by p r e s e n t i n g ,
tions of those Then,
subsets
in S e c t i o n
of
in Section
[l]
version
of the w e l l - k n o w n
ty of w e a k l y
compact
cardinals.
the p r o p e r t i e s
associated
< will limit
a ~b.
such that a S ~ a
that C is closed
T of
{B ~ X
can be
and J e a n - P i e r r e
a fixed uncountable
regular
~<.
We set
[A]
C is closed for
8
(resp.
U a u
directed
closed)
"The author g r a t e f u l l y Forschungsgemeinschaft.
acknowledges
we let
IID=
the s u p p o r t
as 6C,
of size <<. C d 6 [C] <<. We
of those
an u n b o u n d e d and C nT
a£[A]<~
It iswell-kno~.~
nonempty
I o v e r a set X, we set I ÷ = {B ~ X D £ I+
sequence
£ c.
the c o l l e c t i o n
that there e x i s t s
:X - B £ I}. For each
for e v e r y
unions
if Ud C C for e v e r y
strongly
: lal < K}. Let
in case
if f o r e v e r y
we h a v e
SNS~, A) d e n o t e
[A]
Giv e n an ideal
ordinal
closed
(respectively
such that C is c l o s e d
I~ =
denote
iff C is c l o s e d u n d e r
is said to be strongly let NS<, A
Donder
C is said to be u n b o u n d e d
there is a b 6 C w i t h e < y
3 and 4 we inves-
[l]<< w h i c h
some d e f i n i t i o n s .
Let A be a set of size Z<. [A] <~ b e given.
a
proper-
games.
this p a p e r
L e t us now recall
reflection
over
set.
to o b t a i n
discussions.
cardi n a l and X a fixed
C~
unbounded
in S e c t i o n s
ideals
like to t h a n k H a n s - D i e t e r
for h e l p f u l
Throughout
stationary
Finally,
of some normal
with c e r t a i n
The author w o u l d Levinski
a closed
2 we use one of those c h a r a c t e r i z a t i o n s
two-cardinal
tigate
I, some simple c h a r a c t e r i z a -
subsets
C ~ [A] <<
= O.
: B (I}, {B ~ X
and
: B ~ D £ I}
of the D e u t s c h e
120
An
ideal
onal
I over
Finally
we
I. C l o s e d Fix
set ~
Let
unbounded
the
1.1.
Given
Define
is the
increasing
of
if I*
is c l o s e d
under
diag-
j : ~ z I ~I
easy
such
with
that
l, n £ ~
j(O,O)
= O, and
j(~,8) £ a
let E~ 3
for all ~ , 8 6 a
with
, there
g : [l]n ~ l
exists
a 6 E ~ w i t h g[[a] n] ~ a . 3 = g ( { J ( d k , d k ÷ I) : k < n } ) w h e n e v e r
such
infinite
f(d)
enumeration
result
e.
remark.
f : [I] n+1 for all
ordinal
[~]<<
[~]<<
this
g so t h a t
following
dp,
p sn,
of d £ [I] n+l.
is e s s e n t i a l l y
due
to B a u m g a r t n e r
(see
[7],
[I] <<.
Then
115).
PROPOSITION there
1.2.
exists
Let D be a c l o s e d
g : I ~I
a N < 6 <~ and
PROOF.
We
to I. By d~ad,
shall
first
let
d £ [I] n,
and
k
= 8 whenever 1.1
define
on the
fn÷1
f n + 1 ( d U {dn_ I + 1})
Lemma
the p r o p e r t y
equals
k
and
Define
p
subset a £D
of
for all
be the
order
and
: I ~ ~ such
n £ ~,
a £ E~ such 3
the
letting
g1(e)
from
such
= ~ + 1.
conditions.
enumeration
Sup-
of d. T h e n
type
dk; has
and fn+1(dU{dn_1+2k+3}) order
fn+l[[a] n+1] ~ a
type
d k. N o w use
for all
infinite
a £ E~ w i t h [a] c a . S u p p o s e a 6 E ~ is such t h a t a ~ O , a N < £ K, a n d 3 gn+1 -3 gn[a] ~ a for all n £ ~ . Then, c l e a r l y , a = U{a d : d £ [a] <~ - { O } } , and since nally,
D is c l o s e d define
g ( j ( n , n + I + e)) We m e n t i o n COROLLARY
under
g : I ~I
= gn(~)
this
1.3.
directed
so t h a t
whenever
immediate
unions
for all n 6 ~.
corollary.
< NS<, I = SNS<, I I Ej.
1
that
of a d ; f n + l ( d U {dn_ I + 2k + 2}) =
~ Nad_{dk} that
[I] <~,
following
increasing
type
of f u n c t i o n s
a d £ D, d £
gl : I ~ I by
satisfy
~ N a d has o r d e r
8 £ ad_{dk}, gn÷l
gn'
of d, d e f i n e
: [l]n÷1 ~ l
let dp,
e £ ad,
to f i n d
that
a sequence
size
a c ~ a d for c ~ d .
n £ ~*,
whenever
with
unbounded
g[a] ~ a .
induction
and
Given pose
subsets
make
PROOF.
that
to be n o r m a l
for e v e r y
function
f[[a] n+1] ~ a
The
{O}
set of all a 6
us f i r s t
LEMMA
page
is said
= ~-
a one-to-one
denote
that
[I]
intersections.
of size
<<,
~ < l, g ( j ( O , e ) )
we h a v e
a £D.
= ~ ÷ I, a n d
Fi-
121
The
following
PROPOSITION there
exists
is e s p e c i a l l y
1.4.
Let
useful
because
D be a c l o s e d
h : I x I ~I
such
that
of its
unbounded
a 6D
simplicity.
subset
whenever
of
a ~O,
[I]
Then
a D K 6 K, a n d
h[a x a] _ca. PROOF.
Let
define
h so t h a t
h ( ~ , ~ +2) We
g : i ~I
need
PROPOSITION
in the
for e v e r y
= g(~),
shall
be as
and
the
1.5.
statement
e
h(e,e)
h ( ~ , 8 +2)
following Suppose
Let
j : A xA~A
be a o n e - t o - o n e
A.
Then
exists
such
that
a DK £K*,
2. S t a t i o n a r y For We
NWC< with
of t h i s
by r e c a l l i n g
some
the f o l l o w i n g
properties:
a limit
ordinal
e = 8; a n d
successors and
~,B
(5) T h a s
in c a s e
definition
of the w e a k l y [1], w h i c h
with
j(O,O)
a 6D
for all
K--cA and = O, a n d
with
that
be a s s u m e d
to be a c a r d i n a l .
such that
(I) T = A ;
of e a c h
are both
6, a n d
of length
ideal
implies
is n o n s t a t i o n a r y ;
inaccessible
compact
is a tree
(2) ~ S T B
s £T
at level
no b r a n c h
there
NWC
if
(4)
<. K is said
differs
rE for
NShK, I d e n o t e s a 6B,
with
such
that
in case The
the
(3) the
if
6 is
{ y : y < T ~} = { y : y < T B } ,
a n d K ~ NWC<. K
(T,_
e < 8;
to be w e a k l y
Note
that
from that
our
given
all ~ £ A -
g rb ~ f a ~b
that
for all
f
:s~,
B < < such
8.
set of all B c
the p r o p e r t y
[I]
for e v e r y a 6B
g : I ~I,
with
b-ca.
there
there
are
fa : a ~ a ,
exists
b 6 [I] <<
< is s a i d
to be
l-Shelah
[I]
following
[9] ) .
on
is i n c o m p l e t e .
d e n o t e the set of all A _ C K s u c h t h a t t h e r e are K the p r o p e r t y t h a t for e v e r y g : K 4 < , t h e r e e x i s t s
g ~B # f
let
a 6 [A] <<
let N S h
£A,
and
I will
A cK
K is s t r o n g l y
page
91 o f
that
that
definitions.
set of all
immediate
We
paper
the
set of
then
function such
such
1.2.
reflection
denotes
compact
of P r o p o s i t i o n
j[a xa] --ca, and g[a] -ca.
the r e m a i n d e r start
g :A~A
Now
8 >e.
A is a set of o r d i n a l s
UA ~ A .
1.2.
h ( e , e + 1) = O,
whenever
generalization
D£NS~,
there
= ~ +1,
= j(~,B)
easy
of P r o p o s i t i o n
collects
some
well-known
results
(see
[I],
[4],
[5]
122
PROPOSITION and N S h K , I
2.1.
is a n o r m a l
(ii) N S h (iii) The
(i) A s s u m e
K,<
ideal o v e r
which
PROPOSITION
2.2. A s s u m e
be a set of r e g u l a r
~ is w e a k l y c o m p a c t , NSK, I.
}.
inaccessible,
is a t w o - c a r d i n a l
the t h e o r e m of
Then
[l]
= {B ~ [K]<< : B N K 6 N S h
If K is s t r o n g l y
following,
strengthens
< is l - S h e l a h .
then NWC
version
= NSh
K
K
of T h e o r e m
. 2.9. of
[I],
[6]. K is l-Shelah.
uncountable
Let T 6NS~,I,
cardinals.
Then
the
a n d let R 6 N W C ~
set of all a £ [I]
such t h a t a N < £ R a n d T N [a]
Let H denote
H 6NSh~
the set of all a 6 [I] << w i t h a N < £ R .
~. S u p p o s e o t h e r w i s e .
lles in N S h K , I. P i c k ry g : < ~ < ,
f
Then the set K = {a 6 [~]<< : a N < 6 K - R }
:~ ~ u ,
there exists
B
e 6K -R,
a 6K,
8 ~a
that
S u p p o s e D 6 NSh<, I. F i x a b i j e c t i o n
set of all a £ D N E w i t h 1.5, p i c k
b 6 E N [a]
Ua ~ a .
We c l e a r l y
a 6B
with b~a
b £ [a] < a N K a n d g ~b = h a rb,
3. G a m e
8 < <, t h e r e e x i s t s
to a c o n t r a d i c t i o n .
with
j(O,O)
= O, a n d let
D e n o t e by B t h e
h a v e B £ N S h ~I,. _
ha : a ~a
a n d ha[b] ~ b .
w i t h b N K £ K ~ and g[b] ~ b .
~ 6K -R
s u c h t h a t T N [a] < a N K £ N S a N K , a.
j :I x ~ , I
for e a c h a 6 B ,
b NK 6<~
b 6 [l] <<, t h e r e e x i s t s b 6T DE
leads
the set of all a E [ I ]
Proposition
t h a t for e v e -
rB w h e n e v e r
for e v e r y
a n d k r8 = h a rB. T h i s e a s i l y
N o w let D denote~ the set of all a 6 H
E denote
the p r o p e r t y
so t h a t h a ~a N < = faNK" T h e n
one c a n f i n d k : I ~ 1 w i t h t h e p r o p e r t y with
with
such that g r8 ~ f
w i t h ~ Z 8. N o w s e l e c t h a : a ~ a ,
a 6K
We claim that
Now using
such that b ~T
Select g : I ~I
whenever
such t h a t f o r all
a n d g ~ b = h a ~b. N o w c h o o s e
Then one can
find a 6B
such that
a contradiction.
ideals
F o r the d u r a t i o n ular cardinal
of this
a fixed infinite
reg-
l e s s t h a n <.
A s u b s e t D of
[A] << is s a i d to be v - c l o s e d
a n £D,
s
denote
the c o l l e c t i o n
v-closed
section V will denote
if f o r e v e r y
such t h a t a 8 c a s for 8 < e ,
unbounded
of a l l
sequence
w e h a v e ~
set D ~ [l] <<.
T h e e a s y p r o o f of the f o l l o w i n g
is l e f t to the r e a d e r .
123
P R O P O S I T I O N 3.1.
NS~, I is a normal ideal over
[l] << that extends
NS<, 1 • For each T ~ [I]
(T) c o n s i s t i n g of
moves, w i t h p l a y e r I m o v i n g first at limit stages. First I chooses an element a O of
[l]
II answers by playing b O E [I]
then I selects a I £ [I]
II answers b y p l a y i n g b I ~ a 1,
I thus p r o d u c e s a sequence a , ~ <~,
II wins iff ~ J a ~<~
We remark that the games G ~ <(T), T c K , in the literature,
in p a r t i c u l a r
the axiom of d e t e r m i n a c y Since,
such that a s i a ~ for 8 <e.
E T. have a l r e a d y been c o n s i d e r e d
(in the case < = ~1 ) in c o n n e c t i o n w i t h
(e.g. see [10], pp.
26 - 3 2 ) .
for ~ > ~1' the games G~,I(T)_ have u n c o u n t a b l e length, we in-
c l u d e a d e f i n i t i o n of s t r a t e g y to avoid m i s u n d e r s t a n d i n g s . A s t r a t e g y for such a game G H (T) is simply a function, w i t h v a l u e s <,I on [I]
if ~ u
ce
does not b e l o n g to T
(resp. does b e l o n g to T) when-
E [I]
conditions:
(1) c 8 ~ c ~ for 8 <~;
(3) Given n E ~ cy = a ( c
(2) c o = o(O)
(resp. c I = g(Co)); and
and a limit ordinal 8 E ~ with 8 + n ~0, we have
: ~
for y = ~ + 2n
(resp. Y = 8 + 2 n + 1)
We shall need the f o l l o w i n g piece of notation.
Given a strategy T
for II in the game G ~ K,~ (T) we define a f u n c t i o n 9 w i t h the f o l l o w i n g property. Suppose that I plays as, ~
T h e n we let ~(a~ : ~ ~8)
= b 8 for every B < U .
Let NGS~, 1 denote the c o l l e c t i o n of all those T ~
[I]
has a w i n n i n g s t r a t e g y in G u (T) K,~ P R O P O S I T I O N 3.2.
(i) G i v e n T ~ [I]
s t r a t e g y in the game G ~
([I]
Ns ,x NGS , x (iii) NGS~, l is a normal PROOF.
ideal over
[l]
(i): Let T ~ If] << be fixed. Let us first assume that I has a
w i n n i n g strategy g in the game G U (T)
We need to find a w i n n i n g strat-
egy T for p l a y e r II in G D ([I] << -T) Suppose n and 8 are o r d i n a l s such K,I that n £ ~ , 8 E P and 8 = U8. We let ~ ( a : e s S + n ) = ~(cy : y ~ 8 + 2 n + 1 ) , where
(1) c o = a(O);
(2) c I = ao UCo; and
(3) if 8 and p are o r d i n a l s
124
such that ~ = U6, p 6 w and c6+2p+1
and 0 < 6
+p
+n,
then c6+2p = a (c7 : y <~ +2p),
= a6+pL_) c6+2p.
Now for the reverse direction: in G~<,I([I] << -T).
Suppose
T is a winning
We are looking for a winning
strategy
for II
o for player I
strategy
in G ~K, I(T). First put o(O) = ~ (O) Now assume n 6 ~ and 8 6 ~ are such that 8 = U ~ and 8 + n ~O. We set o(c : ~ < 8 +2n) = ~(a : 7 _< B +n), where (I) ~ a7; and a O = O; (2) if 6 <_ B + n is a nonzero limit ordinal, then a 6 = -~J<~ (3) if 6 and p are ordinals such that 6 =U6, p 6 ~ and 6 + p < B +n, then a6+p+ I = c6+2p+1. (ii)
is straightforward.
(iii) : NGS~<, 1 is easily seen to be an ideal over show normality. T
[i] <<. We only
Thus suppose Ty and Ty, y <~, are such that T y c [~]<< and strategy for II in G~,A(T Y ) . . We define a winning strat-
is a winning
Y egy o for II in the game GUK,I(y~<% T ) by letting
@(a
: ~_<~)=
~_~
a
Donder and L e v i n s k i have investigated three ideals NSK,I, where.
NS~, I_
Given an ideal I over istence of a sequence
[8]
[I]
ta~a,
B N a } £ I + for all B ~ I . quence. In
and NGS~, I._
Jech o b s e r v e d
~y(a~ : 6 < ~ _ < B ) .
the r e l a t i o n s h i p
between the
Their results will appear else-
the principle<><,l[I]
asserts the ex-
a 6 [l]
Such a sequence
that OK,I[NSK,I]
is said to be a O w , l [ I ] - s e -
can be forced by adding 1 <<
Cohen subsets of K to the ground model. We now show t h a t ~~, l ][ N G~S ~~, holds as well in the generic extension. Thus let M be a t r a n s i t i v e model of ZFC. A s s u m e 2
in M, of all those functions
dom(p) c { ( ~ , a ) £ I x [l]<<:~ £ a } , clusion.
PROPOSITION
(i)
and ran(p) ~ 2 ,
Jdom(p) J < <,
o r d e r e d by reverse
in-
Define u : P * [I]
a such that
(UG) (~,a)
p such that
(u,a) £ dom(p)
for some e £ a.
3.3. Let G be P-generic over M, and set t a = {e £ a :
= I} for all a £ [l] <<. Then the following hold:
In M[G],
ta, a £ [I]
(ii) A s s u m e that v <~ < K holds in M for all infinite cardinals and that Y is, in M, a stationary
subset of K c o n s i s t i n g
v < K,
of infinite
125
cardinals
of cofinality ~. Set W = {a £ [I]
ta, a £ [I]
(ii), as the proof of
denote the set of all increasing
and e £ ran(h) _cs ÷ 1. Let p £ G
and f,T,T
(i) is similar.
in M[G]
be such that p forces
that f £ 2 l, that T_c [I]
By induction on e < K, define,
Kw
For e a c h u < K,
functions h such that dom(h)£~* strategy
for II in
in M, p u C P, a , gu so
that: (I)
Po = p' and a ° = g o =O;
(2)
Pu+I S P a '
(3)
gu+1 : ae+1 ~ 2 ;
(4)
P~+l
and a + 1 c u ( P = + 1 ) ;
forces that f ~ a
+ 1 = gu+l
and that ae+ ] = ~ hE E
%(u(Ph(8)):
B £ dom(h)) ; (5)
if ~ is an infinite
limit ordinal,
then pe = ~ P s '
as = 8<~u a8
and gu = ~<~ug 8. Set A = ~~< K
a
r
and define a b i j e c t i o n k : A ~ K
all ~ < <. Put C = { ~ < K : k[ae] of K, it is possible q : ap ~ {ap} - 2
to select
(UG) (e,ap)
4. The n o n s t a t i o n a r y This last section
= gp(S).
= f(e)
ideal over
subset
p C Y N C. Now define a function
by letting q(u,ap)
ap C W n T and that
SO that k[a e] £ K for
= ~}. As C is a closed unbounded Clearly,
pp U q
forces
that
for all u C a p .
[l] <~I
is devoted to the special case K = ~I" We shall use games
to obtain several c h a r a c t e r i z a t i o n s of t h e n o n s t a t i o n a r y subsets of
[l]<ml
Let us first make the following observation. PROPOSITION
4.1. Let A be a set of size a~ 1, and let D be a closed un-
bounded subset of [A] <~I. Then there exists h : A x A ~ A whenever a £ [A] <~I is such that a . O PROOF.
By Theorem
Proposition
1.4 of
such that a £ D
and hie x a] _ca.
[2] and the proofs of P r o p o s i t i o n
1.2 and
I .4.
Given To_ [I] <~I , we define two more two-person
games H ~I ,l (T) and
126
K ~I,I(T). ~
Either
game lasts w moves, player
In H~I,I(T), I and II a l t e r n a t e l y a sequence cn • n 6w. Kw
pick m e m b e r s of
the first move.
Ill<W, thus b u i l d i n g
II wins H w1,1(T) w
(T) is defined similarly
~I,1
I making
'
just in case ~ / c n 6 T. The game n£w where now the choices are made from "
and to win II must insure that the set of chosen ordinals PROPOSITION
4.2. Let T ~ [I] <wl be given.
lies
in T.
Then the following are equiv-
alent: (i)
T 6 NS
(ii)
~l,l" T £ NGS~I,I.
(iii)
I has a winning
strategy
in H m ~i,I (T)"
(iv)
I has a winning
strategy
in K e Wl,l (T)"
(v)
II has a winning
strategy
in H ~ ~l,l([ll<Wl - T ) .
(vi)
II has a winning
strategy
in K w~l,l([l] <wl -T).
PROOF.
(i) ~ (ii) (ii) ~ (i)
holds by Proposition : Let T be a winning
G~l,l([l]<ml -T).
a2p+l
collection
for II in the game fn : I n+l ~ [l]<~l
For each n 6 w, we define a function
by letting fn(So,...,en) p
3.2.
strategy
= T(ao,...,a2n),
= T(ao, .... a2p)
where a O = {s O} and for every
and a2p+2 = a2p+l U {~p+1 }. Let C denote the
of all nonempty a 6 [I] <~1 such that fn[a n+l] ~ a w h e n e v e r
n £ ~. Clearly C is closed and unbounded. an enumeration
Given a 6 C ,
of a. Then a = ~ / fn(eo .... ,an), n6w
(v) ~ (i),(vi) ~ (i) and
(iv) ~ (i) are proved as
let s n, n E w, be
and c o n s e q u e n t l y
a ~ T.
(ii) ~ (i).
(i) ~ (vi): Assume T E N S g : A x I ~A
Then by P r o p o s i t i o n 4.1, there exists ~i,I" such that a ~ T for every nonempty a E [i]<~I with g[a x a] ~ a .
We shall define a winning
~l,l( [I ]<wl -T). strategy T for player II in K w
Suppose n E w and Sp 6 I, p S 2n, are given such that S2k+1
= • (So,-..,S2k)
for all k
as a sequence of g's, parentheses
and ek'S.
t, suppress all parentheses of t, and substitute and k + I for each Uk" The resulting
sequences
the image of x under h. For instance, then
(O,O,O,4,6,3,O,I,6)
Eh(x).
For each such sequence in t
O
for each g
of natural numbers
form
if x = g(g(g(~3,s5),s2),g(~o,S5)) ,
Then define
r : ~ / w m+1 ~ w mEm
by letting
127
r (qo' .... qp)
= TT w]~ ,
where w~a, j <~,
is the increasing
enumeration
i_
>I. Finally,
if there exist x 6 a and y 6 h(x) with
then set T(~o' .... ~2n ) = x; otherwise
put T(eo,...,~2n ) =
O.
(i) ~ (v) and Finally,
(i) ~ (iv) are proved
the proof of
(iii)+~(v)
similarly
to
(i) ~ (vi).
is left to the reader.
REFERENCES
[i]
J.E. BAUMGARTNER, "Logic,
Ineffability
Foundations
Butts and Hintikka
properties
of Mathematics (eds),
Reidel,
of cardinals
and Computability Dordrecht
If,
Theory",
(Holland),
1977,
87 -106.
[2]
J.E. BAUMGARTNER, "Handbook
North-Holland, [3]
D.M. CARR, Soc.
[4]
86
[5]
The minimal
D.M. CARR,
D.M. Math.
[7]
C.A. nals,
[9]
31
(1985),
The structure
axiom,
47
filter on P
of weak compactness,
of ineffability
(1986),
"Mathematical
properties
of PKI, Acta
3 2 5 - 332.
Logic
Pure Appl.
Z. Math. Logik
393 -401.
DI PRISCO andW. MAREK,
Fund. Math. 128(1987),197-198.
Some aspects of the theory of large cardiand Formal
Math.
T.J. JECH,
Some combinatorial
cardinals,
Ann. Math.
Logic
Systems",
94, Dekker, problems
5 (1973),
Alcantara
N e w York,
concerning
nateness",
uncountable
165 -198.
[11] T.K. MENAS, Logic
"InfinitaryCombinatorics
Lecture Notes in Math. On strong compactness
7 (1974),
327 - 359.
(ed),
1985, 8 7 - 139.
C.A. JOHNSON, Some partition relations for ideals on PKI,
[10] E.M. KLEINBERG,
(eds),
913 - 959.
A n o t e on the l-Shelah property,
Lecture Notes [8]
normal
forcing
Kunen and Vaughan
316 -320.
Math.
Hungar.
D.M. CARR,
1984,
PKl-generalizations
CARR,
[6]
of the proper
Topology",
Amsterdam,
(1982),
Grundlag.
Applications
of Set-Theoretic
andtheAxiomof
612, Springer,
Berlin,
and supercompactness,
preprint. Determi1977. Ann. Math.
When H e r e d i t a r i l y
Collectlonwlse
Hausdorffness
Implies
Regularity
Zs. Nagy M a t h e m a t i c a l Institute of H u n g a r i a n Academy of Sciences Budapest, V., R e a l t a n o d a U. 13-15, H-1364, Pf. 127 Hungary and S. Purlsch* D e p a r t m e n t of Mathematics, T e n n e s s e e T e c h n o l o g i c a l Cookevllle, T e n n e s s e e 38505, USA
ABSTRACT:
Every c o l l e c t l o n w l s e Hausdorff
Is regular.
(CWT2)
University
first c o u n t a b l e space
G e n e r a l i z a t i o n s of first c o u n t a b l l l t y are considered.
Is h e r e d i t a r i l y
CWT2(HCWT2),
no first c o u n t a b l e S space,
blobular,
then
X
of c h a r a c t e r
ls regular.
Nl'
and contalns
The e x i s t e n c e of a
first c o u n t a b l e S space implies the e x i s t e n c e of a nonregular, lob space of c h a r a c t e r a n a l o g u e s of S spaces ls consistent.
~I"
If It IS c o n s i s t e n t
for all regular ordinals
Every HCWT 2
If X
HCWT2,
that there are no
~,
HCWT 2
then the following
globular space Is regular.
A space ls c o l l e c t l o n v f s e Rausdorff
(CWT2)
If the polnts
In each
d i s c r e t e closed subset can be s e p a r a t e d by a p a l r v l s e disjoint c o l l e c t i o n of open sets. the above holds
So a space ls h e r e d l t a r l l y
HCWT 2
(MOS) subject c l a s s l f l c a t l o n 54A35,
for s c a t t e r e d spaces.
(1980).
Often r e g u l a r i t y
Primary 54A25,
54D10,
54D15,
54E99.
Key words and phases. regularity,
If
was used In c h a r a c t e r l s l n g m o n o t o n e
n o r m a l i t y or total o r d e r a b l l l t y
Secondary
(HCWT2)
for each discrete subset.
In [Pl] and [P2]
AMS
CWT 2
H e r e d i t a r i l y c o l l e c t l o n v l s e Hausdorff,
lob, blobular,
globular,
S space,
rlght separated,
scattered. *
The second author ls p l e a s e d to thank the I n t e r n a t i o n a l R e s e a r c h and E x c h a n g e s Board and the H u n g a r i a n A c a d e m y of S c l e n c e for Support d u r i n g the p r e p a r a t i o n of thls paper.
129
was obtained partlcular which
for
we are
every
p
has
Cp
has
base.
spaces
There
A simple
a linearly
UCp
is
WlU{ p}
and a baslc
neighborhood
where
is a c l o s e d
unbounded
In a s p a c e If t h e r e
1.
Proof.
Let
Let
Every
in
X X
be and
XneOnnC.
dlsJolnt
collectlon
EXAMPLE Is n o t
2.
countable,
polnt.
length
The
There
of
separated
from
such
that
cannot
X a
of
sets.
order {p}U{~+I:~eC}
be
HCWT 2
pev
be separated p.
C
and
CoW.
Let
C
from
For e a c h is c l o s e d
separated
Is n o t
a subspace
regular.
b a s e of
cannot
sets,
point
Is r e g u l a r ,
D = (Xn:n<m}U{p}
D
exists
is s c a t t e r e d length
the
is not
~th
of
a modification 3.
regular.
of
this
if e a c h
derived
set
to s h o w
(in fact
be C.
n(w and
by a palrwlse
C W T 2.
first
countable
result
exist
Trlvlally,
a
its n o n e m p t y
space X (~}
X
space
which
and
every
each
The
of
used
[D].
H C W T 2, polnt
ls t h e
CWT 2
another In
sets has least
an i s o l a t e d
ordinal
is e m p t y .
paracompact).
of a t e c h n i q u e
There
of
a scattered
difficult
is r e g u l a r
limitation
EXAMPLE
of o p e n
T 2,
of o p e n
Its u s u a l
and not
neighborhood
is
if e a c h
of
normal.
that It
X
class
and e a c h m e m b e r
form
space
p
general
w I.
T 2,
that
to i n c l u s i o n )
the
W
countable
such
the polnts
([g]).
A space
not
first
has
is of
of
in
space.
wI
and
first
Since
Since
V
~0
sequence
c a n be
CWT 2
peX-C
discrete.
p
sets
{On}n< m be a decreasing
pick
such
open
IC I < P
p
subset
an element
are dlsjolnt
THEOREM
closed
X
of
respect
In
is a s p a c e
is g l o b u l a r
HCWT 2
where
that
in the m o r e
where
transflnlte)
topology, C
(with
A space
a nonregular
investigation.
([N]),
interested
Scott).
(perhaps
exlsts
spaces
ordered
are
subbase
example
to the p r e s e n t
in lob
( n a m e d b y B.
is a d e c r e a s i n g I.
lead
We also
a neighborhood
EXAMPLE
This
interested
point
neighborhood globular
free.
scattered next
theorem
countable, Is a GS.
I.
space
example The
scattered
of
shows
flnlte a
construction
space
which
Is
is
130
In t h e o r d i n a l last
point For
each
+ An_ I
Dewnm
n<@
denote iff
for
AnCP(wn+l
An_ I
has
been
the t r a n s l a t e there
some
exists
nonzero
A = (wn(m+l)
the
topology
at the
of
- (wnu(wn+l)(1)))
defined.
For
An_ I
~n(m+l)
B'~An_ I
to such
as
each
that
follows.
m<w
let
- wnm.
B = {wnm
That
Is,
+ ~;~ee'}.
m<w
- (wnmu(wn+l){l)))Uu{one
w n l + A n _ l :O
redefine
lff e i t h e r
AeA n
I)
define
Assume
+ An_ I
Define
w W + I we will
w ~ = p.
A 0 = {{I}}. wnm
space
A = U{one
A subbaslc
member
of
or
member
of W n m
neighborhood
+ A n _ l : O < m < w }.
of
p
ls
{p}OU{one
member
of A n : n e @ - k } ,
K
fixed. Note all
by
Induction
B cA n
If
Hence,
vlth
neighborhood
I i
IB{
this
of
p
It is e a s y
to s h o w then
= k i n+l,
topology
has order
type
ww + i type
for all (NB)
Is not
ww + I
n < w flAn ~ ~
for
~ ~n+l-R
regular
and
and
since
Is d i s j o i n t
every
basic
from
(ww) (I) To see w i t h
this
topology
HCWT2,
(UAn}C(t~n+l+l)-(wn+l)
and by
induction
Dcwn+l+l
that
there
exlst
Obviously
thls
space
An S space The
it is e a s y
existence
right
separated
segment
is open.
THEOREM
2.
first Proof.
If
countable Let
countable
X
AeA n
X
is
be
T2,
S space,
a n d not
order
X of
that
and
w~
Is
for all
all
n(w,
dlscrete
~.
=
scattered.
separable,
independent
HCWT2,
then
n<w AND
and
note
[ w n + l , w n+l)
SUCh t h a t
is a w e l l
S space,
HCWT 2
for all
hereditarily
Is lob,
lob,
Is
interval
Is c o u n t a b l e
an S s p a c e if t h e r e
= the to s h o w
is r e g u l a r ,
of
w~ + I
of Z F C
on
and not (see
it u n d e r
of c h a r a c t e r
[R]).
which
~I"
LlndelSf. A space
each
Is
initial
and contains
no
is r e g u l a r . character
regular.
Let
N I, c o n t a i n i n g C
and
p
b e as
no
first
in t h e p r o o f
131
of
theorem
I, and
Is a d e c r e a s i n g then
0 = {0
sequence
in the
proof
of
For
each
~<w I
chOOSe
that
separated. C W T 2,
then
Y
X I
¥
space,
contains
cardlnallty separated
~<~
ls not
I,
X
~
Is
is not
~'
Since
X
(perhaps
Y'U{p}
contains not
collectlon
and
~=w,
ordlnal
ls right If
let
no
first
Y
Y
Is not
be
CWT 2.
countable
subset
Y'
Its p o l n t s
of o p e n
which
If
flrst
So
closed)
p
let Z=w I.
Is the
it Is lob.
Is d i s c r e t e
disjoint
SO
Y = ( x ~ : ~ < w I}
since
of
or w I.
C W T 2.
where Then
base
~
and we are done.
a discrete
by a p a l r w l s e
and
countable
HCWT 2
Since
sets
x~¢O~,.
is r e g u l a r .
H I.
be a n e l g h b o r h o o d
x~¢O~,NC
is first
By t h e o r e m Y
theorem
for all
So
}~<~
of o p e n
as
svCh
<w I
let
sets,
S
of
cannot
X
be
Is not
HCWT 2 • EXAMPLE
4.
existence
X = {x
set
For
existence
:~<Wl}
a basic
a neighborhood
fails
space
do.
wlll
We w a n t the p r o o f
wlth
x
In
X
for
((x=;~<m<Wl)XU)O(p}
regularity
HCWT2,
the
neighborhood
of
In t h e o r e m
a first
2 and
countable lob
be a flrst
(Xx(m+l))u{p}
=<w I
NOTE.
of
of a n o n r e g u l a r ,
Let the
The
space
of
and
m<~.
S space.
topology.
<X=,U>
implies
of c h a r a c t e r
countable
following
S space
lS
~I"
Our Xx~
example
Is
is d i s c r e t e .
Ox{n:m
A basic
the
where
nelghbOrhOOd
Of
p
0 is IS
~<w I . example
to h a v e
linearly
to g e n e r a l i z e
Is d i f f i c u l t .
4 If we o n l y ordered
theorem
For
that
require
neighborhood
2 to g l o b u l a r reason
the p o i n t s
theorem
bases,
spaces. 2 was
where
then
any S
Surprisingly
proven
separately. In the d e f i n i t i o n I Cpl has
i 2,
character
THEOREM no
then
first
Proof.
3.
the ~I'
If
X
countable
Let
X
of
space then
a globular Is c a l l e d the
space
Is b l o b u l a r , S space,
be b l o b u l a r ,
space
blobular.
T2,
Note
Ifa
point
p
we have
globular
space
is b l o h u l a r .
HCWT2,
then
If for e a c h
X
of c h a r a c t e r
~l'
and
contains
is regular.
of character
~l'
containing
no f l r s t
132
countable If
p
S space,
has
theorems
a linearly
1 and
So
let
2,
p
not
a neighborhood
are
strictly
not
have
a linearly 00U
for
each
Case {Xn}n< w
i.
n<w
Cn
For
each
is c l o s e d ,
Is n o t
ordered
C n = CO
of
0 ~<W
and
and
C
base,
where
sequences
let
p
be
then
as
by
in t h e o r e m
the
proofs
I.
of
HWCT2.
subbase
n<w
Let
neighborhood
Is not
decreasing
each
regular.
ordered
X
has
For
and
open
sets.
O n O V ~.
Then
Since
C n ~ ~.
discrete,
0 = {On}n< w
base.
and
{Cn}n< m
Then
p
U = {O}~<ml
is d e c r e a s i n g ,
I
Is c l o s e d . n<w
neighborhood
and
X
IS
ChOOSe
cannot
T R,
0 C n = ~. n<w for a l l n(~.
Xn¢C n
separated
be
from
p.
Then
So
x
C W T 2.
Note
If
separated
C
contains
from
p,
then
a countable for
all
subset
D
which
cannot
n<m
CnO D
IS
lnflnlte,
such
that
Cm
= ~,
So
be case
I
holds.
Case H C W T 2,
cannot
suppose be
choose
Then
countable Y'
HCWT 2 .
This
countable We
been
X
any
and
Then
@
Is
right
As
In
the
Since
of
defined.
proof
and
~I" X-{p)
C
can
be
Let
ordinal 6(=)
x~eCn(On+mnU~(~))-{xy:Y<~}.
of
C
Cm
= ~,
strictly ~'
of
X-{p}
Y = { x = : ~ < w I}
is a l i m i t
Is
So
So
Y
theorem is
Is
y
Y'UN
on
~.
n<w,
for
y<~
all
y<~
can
such
exists
Let
be
X
that
a coflnal
For
first
from
each Y
=eW
=
countable contains
contalns
the
since
no
a discrete It
closure
Is n o t of
any
p.
induction and
H C W T 2.
separated
HcC
Hence
Is n o t
he
Is s e p a r a b l e ,
from
6(~)
be
2 since
HCWT2,
Since
= mln{6:for
X-{p}
= mZn{8=~<SeW).
separated
Note
can
X-{p}
decreasing.
HCWT2 .
and
let
there
being
by
If
a countable
separated.
cardlnallty
subset
done.
exists
where
Y
contradicts
are
subset
there p.
from
S space
of
define
we
countable
x efOmOU-OmnU=,)NN
subset
where
m<m
{OmnUON:~eW)
is g l o b u l a r .
first
is
not.
that
{x~:~eW). it
of
separated
such
exists
neither
closure
For
Wee I
There
then
The p.
2.
Let
= ~ + n,
assume
xy¢OmRU6}, defined
~
xy
has
choose because
C m = ~;
133
the set
CO(On+mOU6(~})-{xa:y<~ }
closure
of the c o u n t a b l e
Yu{p}
is d i s c r e t e
XB+neOn+mOu6(B+n) not
{Xy:y<~}
because
c OnnU B.
Y
since o t h e r w i s e
Would
separated fro~
not De
Y-{Xy:y(~}-OmnU6(=+l) cannot
be s e p a r a t e d
the
= {x~}.
from
p.
p.
Slnce
so
X
ls
HCWT 2 .
DEFINITION. every
Por a regular
HCWT2,
discrete
regular,
subspace
Note
for
%
EXAMPLE
5.
set
not
the
limit, HCHT 2
is 8 regular
which
contains
subspace
regular
12
is not
In e x a m p l e
5 can
This
seems
to be q u i t e
ZFC
using
the t e c h n i q u e
neighborhood
SubbaSe
first
a right
~0
A
2~<%,
that
countable
space
-m w
separated b,
subspace
where -w w
X
be chosen
problem.
in e x a m p l e
U 0
contalns
implies
However,
always
a difficult
space
=<%
%
a
D(%)
b
A
with of type
Is an
Is ccc.
So If
A
is
HCWT 2,
lob space of c h a r a c t e r
For a g l o b u l a r
be the statement:
space Of type
of c a r d i n a l i t y
ordinal
-w ~
D(%)
hypothesis.
There
QUESTION.
HCWT2,
separated
i.e.
~w
an S space,
let
a strong
and no d i s c r e t e
uncountable
right
%
%.
(IT])
underlying
ordinal
of c a r d f n a l l t y
is true even without
b
set
is n o n e m p t y
in
ZPC
to be
HCWT2?
If it can he done,
4 there w o u l d
then
in
be a nonregular,
b. and n o n l s o l a t e d
for some n o n e e r o
paX, n<w
p
has a
where
i ( n %i for each
i(n
= {Oi,~}~<%i
"~i
is a d e c r e a s i n g
Generalizing THEOREM
4.
the proof
If for all
is c o n s i s t e n t NOTE.
Is a regular
that e v e r y
Actually
sequence
need
HCWT 2 D(%)
%
if
l+l
and
O.~i
sets.
3 we have
ordinals
globular
we only
of open
of theorem
regular
%i<%i+1
ordinal,
the
D(%) space
following.
is consistent,
then
is regular.
to be c o n s i s t e n t
for g l o b u l a r
spaces.
REFERENCES
[D]
E. K. v a n Douwen, Remote p o i n t s , 1981.
Dissertationes
Math.,
vol.
188,
it
134 (a)
P. J. Nylkos0 O r d e r - t h e o r e t l c base axioms, in: G. M. Reed. ed., Surveys in General Topoloqy0 A c a d e m i c Press, New York, 1980, 367398.
[PI ] S. Purlsch, to appear.
The o r d e r a b l l l t 7 and closed
[P2] S. Purlsch,
M o n o t o n i c a l l y normal s c a t t e r e d spaces,
In preparation.
JR)
J. Roitman, Handbook of 295-326.
Basic S and Set-Theoretic
Vaughn, eds., Amsterdam, 1984,
IT]
S. Todorcevlc, Remarks on c e l l u l a r i t y 57 (1986), 357-372.
[W]
M. L. W a g e , A c o l l e c t i o n v l s e Hausdorff, Can. J. Math. 28(1976), 632-634.
L,
Images of s c a t t e r e d spaces,
in: K. K u n e n a n d J . E. Topoloqy, North Holland0
In products,
non-normal
C o m p o s l t l o Math.
Moore space,
CLASSES OF COMPACT SEQUENTIAL SPACES Peter J. Nyikos Department of Mathematics, University of South Carolina Columbia, South Carolina 29208 Eric van Douwen was a regular contributor to the Problems Section of Topology Proceedln~s, of which I have been the Problems Editor since its inception.
One of the
problems he contributed to Volume 8 was the following: Problem.
In the class of compact Hausdorff spaces, are there any other implications
besides the ones shown?
pseudocompact subspaces are ~ compact
countably compact
hereditarily realcompact ~
/
subspaces are compact
Fr~chet * sequential [For definitions, see Section I.] I happened to know that Zhou Hao-Xuan had shown [Zh] that if every subspace of a compact Hausdorff space is pseudocompact, the space is Fr4chet.
This makes it possible
to straighten out the diagram to a linear sequence of implications. Eric) that the one-point compactification of being Fr~chet.
Y
I also knew (as did
(Example 3.3) was sequential without
The problem of whether "countably compact subspaces are compact" implies
"sequential" had already been posed by me in Volume 3 (C17) in an equivalent form (see below).
So I just made two separate problems out of Eric's, whether the first two
implications in the straightened-out diagram reversed, together with consistency results (discussed below) on each. I found it very interesting that "Fr~chet" and "sequential", which in Tychonoff spaces have little to do with the other three concepts, should come to be sandwiched in between two such similar-sounding properties.
"Pseudocompact" and "countably compact"
are often treated together, as in the Gillman and Jerison text [GJ], and so are "Fr~chet" and "sequential", as in [F] and [AF]. What made it even more interesting is that, as I also knew at the time, "countably compact subspaces are compact" is in turn sandwiched in between "sequential" and "countably tight" (another concept treated in [AFJ) in the class of compact Hausdorff spaces, and at the time it was still a famous unsolved problem whether there is a compact space of countable tightness that is not sequential. is countably tight if whenever
x ~A,
there is a countable
B c A
such that
(A space x ~ ~.)
This problem is now solved, thanks to the combined efforts of D.H. Fremlin, Z. Balogh, A. Dow, and myself (and the older consistency result of Ostaszewski [OSI] whereby axiom <~ implies an affirmative answer):
under the Proper Forcing Axiom, and also in some models
not requiring the consistency of anything more than of countable tightness is sequential [BDFN].
ZF,
every compact Hausdorff space
136
This did not dispose of "Problem C17," however.
In [IN] we had shown that a compact
Hausdorff space is sequential if, and only if, it is sequentially compact and countably compact subspaces are compact, so that the question of whether the last arrow in Eric's straightened-out diagram reverses becomes equivalent to: Problem C17.
If a compact Hausdorff space has the property that all countably
compact subsets are compact,
is the space sequentially compact?
We also showed [ibid~ that the answer is yes if either
MA
or
2 W < 2~i.
seemingly disparate axioms were combined by Eric into one [vD, 6.4]: definition of [vD].)
t
These
2 t > 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
<~,n>
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 <* g
X
be the resulting space and
a
w-space (and a v-space if the Problem ii.
Can
Y
I do not know whether
Y
f=
where
such that
<*
is the eventual domination
f(i) < g(i)
for all
In [NYs] I show
Let Y
is
~o-bisequential?
b = c.
Specifically, we make
meets each column in a cofinite set}
~O-
(~xm) U [®}
Let
F = {A c mxm[ A
i > n.
form a scale).
can even be rigged "in ZFC" to fail to be
bisequential, but it can be done if be bisequential.
n
its one-point compactification.
ever be Y
<*-unbounded, <*-well-ordered family
f : m ~ ~,
means there exists
{n}x~,
and let
fail to
149
F' = {A-(nx~)[A G F}. how
Then
F'
is chosen.
is a filter on
a countable subfamily converging to descending countable family infinite for each Well,
~.
there is an
of subsets of
fm
is infinite for each
listing all such families m
B
by
(B:
~
~
no matter
such that
whose graph meets each
F G F'
infinitely many columns in an infinite set.
B
which converges to
F'
has
This is equivalent to saying that, for each
B = {Bn: n E ~ )
F G F',
Bn U F
~xm
We will choose it so that no ultrafilter extending
m < c)
b=c
is
B n.
if, and only if, each
And if
Bn N F
Bn
meets
we can arrange this simply by
and making sure
fm
hits every member of
.
By omitting the columns from the ADF, one obtains a wD
Y-like space
Z
hereditarily, and its one-polnt compactlfication does also, but if
filter
F
shows it is not
It can be shown that Definition. D, and
b=c
Z
is hereditarily pseudonormal "in ZFC" [vD, ll.4e].
A space is pseudonormal if, given any two disjoint closed sets
one of which is countable, there are disjoint open sets
U
and
V
C
and
such that
C c U
D c V. Now every pseudnormal space satisfies
3.9.
A "barely pseudonormal"
of a Hausdorff gap B ={Bm:
for all
m, 8
m < wl]
wD [vD, 12.1]. Z + ~
where, as usual,
is a
but there is no
c 8 o~ D
In the
for all
A c ~
F, 6
6 G &,
V-like space
Z
satisfies
X
of
A = {Am: m < ~i}
such that
~i
is a o~-ascending sequence
A m o~ A o~ B E C
= A +I\A ~
and any
G, D c w
it must be that
which uses the ADF
G O D
~
such that
for all
for all
m,~.
~,
such that
F
C m o~ ~)-Ax+I
is countable, we can let for all
m > X,
the points associated with the points indexed by X
r
F,
so that
AX+ I
The added
Cy o~ G
for all
is infinite.
{C : m < el)
X = sup F,
A ~ c* B 8
then for each
as in 3.3, this translates
to the fact that no two uncountable subsets have disjoint neighborhoods. hand, if
wD
I have an unpublished ZFC construction
c*-descending sequence of subsets of
pair of uncountable subsets y G r,
So
does also.
Y-like space.
twist in the construction is that if one lets
Thus
the same
~0-bisequential.
hereditarily, and it is easy to see that
and
that satisfies
and then
Cy o~ AX+ 1
On the other for all
X ~ F,
has countable clopen closure containing all
and now it is elementary to get disjoint open sets around
and those indexed by
el-r,
and so these sets will be clopen.
is pseudonormal, but just barely!
It is easy to see that a pseudonormal space in which the nonisolated points form a (closed) discrete subspace is hereditarily pseudonormal. hereditarily and But [GJ,
]
X
X + m
So
X
satisfies
wD
is easily seen to satisfy it also.
is not realcompact.
Disjoint zero-sets can be put into disjoint open sets
and so the co-countable zero sets form a
Z-ultrafilter with the countable
intersection property which is not fixed. I have been unable to ascertain whether
X + ~
is bisequential.
The answer could
well vary from one model to the next, but the question easily reduces to that of whether
150
U {~}
is bisequential, so that if it is not, then
X + ~
is not
Y)o-bisequential
either. Problem 12.
Is it consistent that a compact space satisfying
wD
hereditarily is
~'~o-bisequential? The f o l l o w i n g example shows why we cannot have a ZFC theorem here. 4.
Some topological Games of Gruenhage. Gary Gruenhage has invented a number of related topological games which are good for
analyzing the sequential structure of spaces.
His original one is the one that gave
w-
spaces their name. As usual, the game lasts for
~
plays and involves two players.
I will follow a
terminology I picked up from a recent paper co-authored by Shelah and call one player "the hero" and the other "the villain".
The idea is that the spaces in which we are
really interested, at least in the context of the game, are where the hero has a winning strategy, or at least the villain does not have one.
Or, to paraphrase Gruenhage [Grl],
for every strategy of the villain, the hero has a counter-strategy that will defeat it. It may not be immediately obvious that this is equivalent to the lack of a winning strategy for the villain, but it becomes clear from the usual definition of a strategy [ibid.].
It is a "decision tree" in which the nodes are the various possibilities for
each player, but the player, call him/her
X, whose strategy it represents has only one
choice at each node which represents a turn by
X.
including the ones we consider here, has no draws:
Now almost every topological game, any game played out in full is a win
for either one player or another, and the only issue is whether one or the other has a winning strategy of a certain sort (see Definition 4.8).
Thus "all" the opponent has to
do, on being informed of the strategy [a very valuable piece of information, which alters the nature of the gamel] is to check whether any branch leads to a loss by make the choices at each turn represented by such a branch.
X,
and then
If there is no such branch,
why then that is exactly what is meant by the tree being a winning strategy! Game 1.
(The point-open game.)
Given a point
j
(the "jail") of a space
hero attempts to fence the villain down into the jail in picks an open set inside
U n.
Un
~
moves.
X,
the
On each move he
containing the jail, and the villain must always pick a point
The hero wins if the sequence
converges to
j,
Pn
otherwise he loses.
A slight modification is: Game I'.
Here the villain gets the first move and with it he picks a point
j;
is where the hero must situate the jail, and then we proceed just as in Game I.
Of
course, the villain will try to put the jall in the remotest possible point.
that
It would
clearly be suicidal to put it on an isolated point, and also to put it on a point of first countability.
On the other hand, if there is a nonisolated point to which no
nontrivial sequence converges, all the villain has to do is put the jail there and then he does not need to worry about any strategies because the hero cannot possibly win. Let's look at some other cases.
151
4.1
Lemma.
Proof. j G A~
If
X
is not Fr4chet,
the villain has a winning strategy in Game I'.
Let the villain pick a point
but no sequence from
points of
A,
A
j
for which there is a set
converges to
x.
Ever after,
A
such that
the villain need only pick
and this he can always do.
It is also easy to see that the villain has a winning strategy if
X
is not
~/:
the villain picks a non-~./-point and lists infinitely many sequences converging to it that witness this, then makes sure that on successive moves he hits every one of those sequences at least once.
A nontrlvial result is that these are the "only" spaces on
which the villain has a winning strategy: 4.2.
Theorem [Sh, in effect]:
The villain has a winning strategy in Game i'
iff
is not a w-space. This statement of the theorem uses the definition I gave of "w-space". 4.2 was the definition of "w-space" rather then a theorem. 4.3.
Definition.
Game I'.
A
A
Originally,
Also:
W-space is a space in which the hero has a winning strategy in
W-point is a point which, when chosen for
j,
gives the hero a winning
strategy (for either game). Of course, a space
X
is a
W-space iff each point is a
W-point.
One can also
define "w-polnts" in the same spirit. Of course, every first countable space is a
g-space;
so is the one-point
compactification of a discrete space:
all the hero has to do is pick an open set on the
nth move that excludes all the points
Pk
the villain has picked up to that point.
interesting fact [Grl, 3.5, and 3.6] is that every
An
g-space has the property that
separable subspaces are first countable and hence countable subspaces are metrizable. This extends to arbitrary cardinalitles: by 4.1 Like
g-spaces are Fr~chet,
so that they are v-spaces, and also
5~o-bisequential spaces,
countable products (also
w(X) ~ IXl, X(X) ~ d(X) [ibid.].
Of course,
Z~n-bisequential.
they have the nice property of being preserved by
E-products,
see [A4] and [Grl] respectively).
commented in [Grl] that he did not know of an example of a
Gruenbage
w-space that is not a
W-
space and was surprised to see such a well-behaved (Fr4chet and countable subspaces Ist countable) class sandwiched in between them. Galvin subsequently found a space in this class that was not a
W-space.
The first ZFC example of a countable w-space that is not
first countable was given by Isbell and published by Olson [O1], both of whom thought 2~'~O < 2 ~ i [No2].
subspace, [NY8].
was required for it.
I observed in 1983 that it could be done "in ZFC"
The ZFC portion of Example 3.8 was the first compact w-space with a countable
The
((~x~) U {~}, %'
which is not first countable.
Related results will appear in
variant of Example 3.4 is the first which is also bisequential.
A strange feature of all these examples and any other that might appear is: "The villain's revenge".
If a countable space is a
w-space,
countable, it is impossible to tell in ZFC whether it is a This is a corollary of Theorems i.i. and 1.2.
but is not first
v-space.
I have not been through either proof
X
152
yet but Dow has made me see, using some of my own results [NY8] how the following gives Theorem I.I. 4.4.
Theorem [Do]
A family [An: n e ~ } An\S
S
Laver forcing preserves
of subsets of
m
is called
of infinite subsets of
are both infinite for all
~,
o0-splitting families of subsets of
m.
00-splitting if, given any countable family
there exists
S G S
such that
An 0 S
and
n.
As for Theorem 1.2, it is established using a variant of Miller's rational forcing. Game 2.
This is like Game i except that a subset of
of a point, and for the hero to win, the sequence
X
is used as the jail instead
jail" in the sense that all but finitely many terms must lie in any given open set containing the jail.
(It-is sufficient for it to converge to some point in the jail for
the hero to win, but not necessary: [-I,I]
on the
i y-axis and the space is the union of the jail with the sin ~ curve.)
The concept of a point. 4.5. in
X2 4.6.
consider the case where the jail is the interval
W-set is now defined in the obvious way to generalize that of a
W-
Here are some strong results from [Gr3]: Theorem. <=>
A compact space
every closed subset of
Theorem.
X X2
is Corson compact is a
<=>
the diagonal is a
W-set
W-set.
A closed subset in a compact space of countable tightness is a
W-set
if, and only if, its complement is metalindel6f. 4.7.
Corollary.
Every compact hereditarily metalindel6f space is a
W-space.
Hence every countable subset in a compact hereditarily metalindel~f space is metrizable, and we get two of the implications in Diagram 2.
Related results are given
in Section 5. In [Gr4] some fine-tuning of strategies and more games are discussed. 4.~.
Definition.
Recall:
A strategy for a player is stationary if a play depends only on
the opponent's preceding move.
A Markov strategy is one which depends only on the
opponent's last move and the number of the move. Unless the jail is an isolated point in Game 1 or an open set in Game 2, the villain can foil any stationary strategy by the hero by the simple expedient of landing in Jall in odd-numbered movesl
For then (s)he can pick any point
open set that is not in the jall and keep playing
p
p
in the hero's corresponding
in even-numbered moves.
But if we
alter the game slightly by forbidding the villain to use the jail, then the hero has a stationary winning strategy if the jail is a point of first countability.
All (s)he has
to do is pick a descending local base and then always pick an open set that excludes the last point played by the villain. The hero also has a Markov winning strategy in Game 1 if countability.
j
is a point of first
In fact there is what might be called a winning "chronological" strategy,
in which the hero only needs to keep track of what the number of the move is. The situation is different on the one-polnt compactification of an uncountable discrete space.
Even if the villain is barred from the jail, the hero has no stationary
153
winning strategy.
For each point
the complement of a finite set
p
is excluded from infinitely many outside
P(q),
that the villain plays, the hero has to stick to
P(p)
all through the game.
P(p),
and among these
p's
there is a
so that the villain need only alternate between
does not even have a Markov winning strategy: P(p,n)
Now there is a point q which
for
P(p)
PO
substitute
and
PO q.
which is The hero
Un= 1 P(p,n)
is the finite set whose complement the hero picks if the villain picks
nth move.
The villain may need to pick different
P2n r s
(always outside
p
where on the
P(q,2n-l))
but the hero's strategy is foiled as before. But the situation is different if we let The ~ame play
Pn
G(J,X).
{j} = J
in:
This is Game 2 with the rules altered so that the villain must
in the intersection of the first
n
open sets played by the hero.
the effect of relieving the hero of some bookkeeping. the hero now need only follow up a play of
Pn
with
This has
In the one-point compactlfication, X - {pn }
and thus have a
stationary strategy. It is only with respect to these special strategies that 2;
G(J,X)
differs from Game
the question of whether the hero or villain has a winning strategy,
with "perfect
information" (of the past, not the futurel) is easily seen to be the same in both games. Thus Theorem 4.5 tells us that a compact space winning strategy in 4.9.
Theorem.
G(A, X2). [Gr4]
A compact space
hero has a Harkov winning strategy in The
P(p)'s
and
X
is Corson compact iff the hero has a
The new wrinkle is:
P(p,n)'s
X
is Eberlein compact if, and only if, the
G(~,X2).
mentioned earlier suggest turning things inside out,
giving us: The Game
G(Y).
villain picks
nth turn, the hero picks a compact set
Pn ~ U{Ki: i ~ n}.
the whole space exile.
On the
Y.
The hero wins if the
and then the
have no cluster point in
Here we might say the hero is trying to drive the villain into
For instance,
if
Y
is locally compact and noncompact,
is equivalent to a win in the game the point at infinity. during the game.
Pn'S
K n,
G([-}, Y + ~):
the hero drives the villain out to
The only difference is that in
Had we only required
Pn ~ Kn'
then a win for the hero
G(Y)
the villain cannot land on
this difference could really make
itself felt, as we already saw. But the real point of bringing in 4.10. (a)
Theorem {Gr4]
If
Y
G(Y)
here is to mention:
is locally compact and noncompact, then:
Y
is metacompact if, and only if, the hero has a stationary winning strategy in
Y
is
G(Y). (b)
a-metacompact if, and only if, the hero has a Harkov winning strategy in
G(Z). By the comments in Section 2 on Classification E, it is now easy to formulate necessary and sufficient conditions for a compact space to be hereditarily (~-)metacompact.
154
Further uses of
5.
G(Y)
and a simple variation
G*(Y)
are given in [Gr4].
R e l a t i o n s h i p s among s c a t t e r e d compact spaces One characterization of scattered spaces is that in every subspace, the set of
relatively isolated points is dense. scattered, many of them even
A majority of the examples in Section 3 were
Y-like.
Nevertheless,
there is quite a lot of
simplification in restricting ourselves to compact scattered (also called dispersed) spaces.
They are sequentially compact [Bali, so from Theorem i.I there follows:
5.1.
Corollary.
A compact scattered space is sequential if, and only if, countably
compact subsets are compact. A pleasantly parallel result is: 5.2.
Theorem.
A compact scattered space is Fr~chet if, and only if, pseudocompact
subsets are compact. Proof.
We need only show necessity.
compact scattered Fr~chet space
X.
there is a sequence of points of
A
If
Suppose x ~[,
A
then by the above characterization,
isolated in
A
discrete sequence of (relatively) open slngetons of pseudocompact. Of course,
So
x e A,
T + ~
which shows
A
is a pseudocompact subset of a
and converging to A\{x},
x.
These give a
which is therefore not
is closed.
is a compact scattered sequential space which is not Fr~chet.
First countability is a very strong property in compact scattered spaces: equivalent to countability, hence to metrizability. the scattered height of derivative
X (m)
X,
is empty.
X (~+I) = the derived set of
i.e. the least
=
such that the
~th Cantor-Bendixson
This derivative is defined by induction, X(~);
if
m
it is
This is easily shown by induction on
is a limit ordinal,
with
X (0) = X
and
X (~) = N {X(B): B < ~}.
This can be done for any space, and a standard exercise is that a space is scattered iff X (=) = 0 X (Y)
for some
and that
~,
X (7)
and that a compact scattered space has a last nonempty derivative is finite.
The greatest coalescence is in Classifications B and E. 5.3. a
Definition.
Recall:
A compact space is strong Eberlein compact if it is embeddable in
v-product of two-point discrete spaces.
A
v-product is a subset of a product
consisting of all points which differ in only finitely many coordinates from a fixed point
x O.
An internal characterization is the existence of a point-finite cover by clopen sets. 5.4.
Theorem.
The following are equivalent for a space
(i)
X is strong Eberlein compact.
(ii)
X is a compact scattered
X.
W-space.
( i i i ) X is a compact scattered hereditarily metalindelbf space. (iv)
x
is a compact scattered hereditarily metacompact space.
(v)
x
is a scattered Eberlein compact space.
T0-separating open
155
Proof. The equivalence of (i) and (ii) is shown in [Gr3]. The rest follow quickly from other results in the same paper. (iii) implies (il) by Corollary 4.7, which is taken from [Gr3] , and (i) and (il) together give (v), and every Eberlein compact space is hereditarily metalindelSf as we have seen. Finally, (iv) obviously implies (ill), while the converse follows from Theorem 4.6 and: 5.5. Theorem.
[Gr3] In a compact scattered space
W-set if, and only if,
X - H
X,
a closed subset
H
is a
is metacompact.
Of course, we are also using the fact (Section i) that these hereditary covering properties need only be checked for open subspaces.
In connection with scattered spaces,
there is additional information on these in [NYl]. The following problem is reminiscent of Problem 12. Problem 13. wD
Is it consistent that every compact, scattered space satisfying Property
hereditarily is bisequential? I have not had the time to investigate the use of locally countable prototypes of the
tangent bundle here. In [NY4] , connectedness is used to establish "in ZFC" that T + + is never bisequentlal. If we leave
wD
out of the picture, we can condense the known relationships between
compact scattered sequential spaces into one diagram:
J
fi~t countable itarily
strong Eberlein
realcom )act
compact
~ - ~ h e r e d i tarily weakly 8-refinable
weakly first countable
/ blsec uentlal
: v-space ,// ~" ~ "
" 0
w-space
Radon
.
#
." ~
o-blsequential
/
(e4+) Fr~chet
sequentiat Diagram 4.
Relations between compact scattered spaces.
156
6.
A few additional classes Three unpublished examples brinK us up to date on Diagrams 1 through 4.
There is a
pair of 1981 constructions by D. H. Fremlin of compact spaces that are not Radon, one of which is first countable and the other scattered and hereditarily realcompact.
The third
space is a very recent (1988) construction by Alan Dow of a compact scattered sequential space which is not hereditarily ~-realcompact, nor Fr~chet.
All are ZFC examples and so,
except for the unsolved problems already listed, complete the story of the relationships between the classes. Of course, we could easily expand the classifications.
For instance, the class of
Talagrand-compact spaces is known to properly contain the Eberlein compacta, and be properly contained in the Gul'ko compacta.
It can be inserted in Diagram 2 without much
fuss, because Reznichenko's space (3.6) is Talagrand-compact. A much more radical expansion was hinted at in defining "hereditarily
wD".
Hereditary normality is a natural strengthening which has attracted much attention over the years.
The following old questions may be ripe for attack:
Problem 14.
Is it consistent that every compact hereditarily normal space is
sequentially compact? Problem 15.
Is every separable, hereditarily normal, compact space countably tight?
An affirmative answer to Problem 15 would imply one to Problem 14 under PFA, and in all models where compact countably tight spaces are sequential.
Even within the class of
sequential spaces there seem to be interesting questions, such as: Problem 16.
Is it consistent that every hereditarily normal compact sequential space
is hereditarily realcompact?
bisequential?
There are also theorems like the recent result of Kombarov |K] reminiscent of Gruenhaze's analyses: 6.1.
Theorem.
Let
X
be compact.
If
X2-6
is normal, then
X
is first
countable. However, the analogy is not perfect: such that
X2-6
is normal, yet
X
in [GN] there is a ZFC example of a compact
is not hereditarily normal, much less
X 2.
X
Also
there we will bring the reader up to date on Kat~tov's old problem of whether hereditary normality of
X2
implies metrizability of the compact space
X.
Example 3.1 explains why Problem 16 asks only for consistency.
Even if we go to
hereditary collectionwise normality and perfect normality, the picture does not change. On the other hand, the following sheds a little light on Problems 14 and 15. 6.2.
Theorem.
If a compact separable space is hereditarily collectionwlse
Hausdorff, then it is countably tight. Proof.
Every separable hereditarily collectionwise Hausdorff space is easily seen to
be of countable spread, and tightness < spread in compact spaces [H, 7.15]. Note that under
2 ~ 0 < 2t~1,
the Jones Lemma shows every separable, hereditarily
normal space is of countable spread, hence the answer to Problem 15 is affirmative there.
157
And Problem 14 then entangles us in
S
and
L
spaces as shown in the Problem Section of
Topology Proceedings, v. 2 and also in [NY3]. A theme related to the one at the beginning of Section 2 is: collectionwise normal +
=> paracompact.
As shown in [Ba2], "locally compact + metalindelSf" works here, giving the corollary that every compact, hereditarily collectionwise normal, hereditarily metalindelSf space is hereditarily paracompact and, if every discrete subspace is of nonmeasurable cardinal, hereditarily realcompact.
This meshes nicely with another, recent result of Balogh:
in
a model obtained by adding "supereompaotly many" Cohen or random reals, every normal, locally compact space is collectionwise normal. If we go all the way to monotonically normal spaces, then "separable" implies both "hereditarily separable" and "hereditarily LindelSf" lOs2].
So Problems 14 and 15 have
affirmative answers for them, but Problem 16 still appears to be open for them. There are other clas~es which similarly mesh well with compact sequential spaces, but the reader will probably profit more from reading the many fine papers of the Russian school of general topology, than anything I could add at present.
REFERENCES
[AI]
A. V. Arkhangel'skii, The frequency spectrum of a topological space and the classification of spaces, Doklady Akad. Nauk SSSR 206 (1972) 265-268 = Soviet Math. Dokl. 13 (1972). [A2] , On invariants of character and weight type, Trudy Moskov. Mat. Obs. 38 (1979) = Trans. Moscow Math. Soc. 1980, Issue 2, 1-23. [A3] , The star method, new classes of spaces, and countable compactness, Soviet Math. Dokl. 21 (1980) 550-554. [A4] , The frequency spectrum of a topological space and the product operation, Trudy Moskov. Mat. Obs. 40 (1979) = Trans. Moscow Math. Soc. 1981, Issue 2, 163-200. [AF] and S. P. Franklin, Ordinal Invariants for Topological Spaces Michigan Math. J. 15 (1968) 313-323. [AL] D. Amir and J. Lindenstrauss, The structure of weakly compact subsets in Banach spaces, Ann. Math. 88 (1968) 35-46. [Bali J. W. Baker, Ordinal subspaces of topological spaces, Gen. Top. Appl. 3 (1973) 85-91. [BAN] Z. Balogh, Paracompactness in locally nice spaces, preprint. [BDFN] Z. Balogh, A. Dow, D. H. Fremlin, and P. J. Nyikos, Countable tightness and proper forcing, AMS Bulletin, to appear. [Be] A. Berner, ~X can be Fr~chet, AMS Proceedings. 80 (1980) 367-373. [BL] H. R. Bennett and D. J. Lutzer, A note on weak 8-refinability Gen. Top. Appl. 2 (1972) 49-54. [Bu] D. Burke, Covering properties, in: Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan ed., North-Holland, 1984, pp. 347"-422. [vD] E. K. van Douwen, The integers and topology, in: Handbook of Set-Theoretic Topolo~m/, K. Kunen and J. Vaughan ed., North-Holland, 1984, pp. 111-167. ,~v,rn^~ ,~.^ Dow, Two classes of Fr6chet-Urysohn spaces, preprint. [Dy] N. Dykes, Generalizations of realcompact spaces, Pac. J. Mat,h. 33 (1970) 571-581. [E] R. Engelking, General Topology, Polish Scientific Publishers, Warsaw, 1977. [F] S.P. Franklin, Spaces in which sequences suffice I and II, Fund. Math. 57 (1965) 107-115 and 61 (1967) 51-56. [Gal] R. J. Gardner, The regularity of Borel measures and Borel measure-compactness, Proc. London Math. Soc. 30 (1975) 95-113.
158
[Ga~] [Gi] [GJ] [GN]
[GP] [Gr I ] [Gr 2 ] [Gr 3 ] [Gr 4 ] [Gr~]
[Gr~] IH] ~ [IN] [J~] [K] ILl [Ma] [Mi 1 ] [Mi~]
[No I ]
[No~] [Nyl]
[NY2 ] [NY 3 ] [NY4] [NY5] [NY6] [NY7] [Ny~]
[Ol~
[0s 11 [0s~] [PY]
[Ro]
, Corson compact and Radon s p a c e s , Houston J . Math 13 (1987) 37-46. R. F. G i t t i n g s , Some r e s u l t s on weak c o v e r i n g c o n d i t i o n s , Canad. J . Math. 26 (1974) 1152-1156. L. G i l l m a n and M. J e r i s o n , Rings of C o n t i n u o u s F u n c t i o n s , D. van N o s t r a n d Co., 1960. G. Gruenhage and P. Nyikos, H e r e d i t a r y n o r m a l i t y i n s q u a r e s of compact s p a c e s , preprint. R. J . Gardner and V. P f e f f e r , Bore1 m e a s u r e s , i n : Handbook of S e t - T h e o r e t i c Topology, K. Kunen and J . Vaughan e d . , N o r t h - H o l l a n d , 1984, pp. 961-1043. G. Gruenhage, Infinite games and generalizations of first-countable spaces, Gen. Top. Appl. 6 (1976) 339-352. , Generalized metric spaces, in: Handbook of Set-Theoretic Topolog[, K. Kunen and J. Vaughan ed., North-Hollagd, 1984, pp. 423-501. , Covering properties in X--a, W-sets, and compact subsets of Eproducts, Top. Appl. 17 (1984) 287-304. , Games, covering properties, and Eberlein compacts, Top. Appl. 23 (1986) 207-316. , A Corson compact space of Todorcevic, Fund. Math. , A note on Gul'ko compact spaces, AMS Proceedings. R. Hodel, Cardinal functions I, in: Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan ed., North-Holland, 1984, pp. i-61. M. Ismail and P. Nyikos, On spaces in which countably compact subsets are closed, and hereditary properties, Top. Appl. II (1980) 281-292. I. Juh~sz, K. Kunen, and M. E. Rudin, Two more hereditarily separable non-Lindel6f spaces, Caned. J. Math. 28 (1976) 998-1005. Kombarov, R. Laver, On the consistency of Borel's conjecture, Acta. Math. 137 (1976) 151-169. V. I. Malyhin, On countable spaces having no bicompactlfication of countable tightness, Dokl. Akad. Nauk. SSSR 203 (1972) 1001-1003 = Soviet Math. Sokl. 13 (1972) 1407-1411. E. Michael, A quintuple quotient quest, Gen. Top. Appl. 2 (1972) 91-138. A. Miller, Special subsets of the real line, in: Handbook of Set-Theoretlc Topology, K. Kunen and J. Vaughan ed., North-Holland, 1984, 201-233. T. Nogura, Fr~chetness of inverse limits and products, Top. Appl. 20 (1985) 59-66. , The product of <~>-spaces, Top. Appl. 21 (1985) 251-259. P. Nylkos, Covering propertie§ on o-scattered spaces, Top. Proceedings 2 (1977) 509-542. Metrizability and the Fr4chet-Urysohn property in topological groups, AMS Proceedings 83 (1981) 793-801. Axioms, theorems, and problems related to the Jones Lemma, in: General Topology and Modern Analysis, L. F. McAuley and M. M. Rao ed., 441-449. Varlous smoothings of the long line and their tangent bundles I: Basic ZFC results, Advances in Math., to appear~ Various smoothings of the long line and their tangent bundles II: Countable metacompactness and sections, Advances in Math., to appear. Various smoothings of the long llne and their tangent bundles III: Consistency and independence results, Advances in Math., to appear. The Cantor tree and the Frdchet-Urysohn property, to appear in the proceedings of the Third New York Conference on Limits. ~o~and the Fr4chet-Urysohn and ~i-properties, preprlnt. R. C. Olson, Bi-quotient maps and countably blsequential spaces, Ge. Top. Appl. 4 (1974) 1-28. A. Ostaszewskl, On countably compact, perfectly normal spaces, J. London Math. Soc. (2) 14 (1976) 505-516. , On spaces having a G6-diagonal, Coll. Math. Soc. J~nos Bolyai (1978). E. G. Pytkeev and N. N. Yakovlev, On bicompacta which are unions of spaces defined by means of coverings, Comment. Math. Univ. Carolinae 21 (1980) 247-261. H. P. Rosenthal, The heredity problem for weakly compactly generated Banach spaces, Compositio Math. 28, Fasc. i (1974) 83-111.
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Special N o n - S p e c i a l l~l-trees. Chaz Schlindwein University of Kansas Lawrence, KS 66045 USA
In [Sch2], the consistency of "ZFC + continuum hypothesis + Suslin's hypothesis + not Every Aronszajn Tree is Special (~EATS)" is demonstrated using proper forcing. In fact, the following stronger statement is shown consistent: (*) ZFC + CH + there is an Aronszajn tree T* and a stationary co-stationary set S* such that T* is S-st-special iff S - S* is non-stationary, and every Aronszajn tree is S~-st-special. See [Sh, p. 286] for the definition of S-st-special. The proof relies heavily on [Sh, chap. V]. Of course the consistency of CH with SH is due to Jensen [DJ]. In this paper, we build a model of (*) + every wl-tree is S*-*-special (and therefore there are no Suslin or Kurepa trees) starting from an inaccessible cardinal. Consequently, every combination of Z F C + ~ H ± S H ± K H ± E A T S not ruled out by EATS==~SH, is attained. The main new ingredient of the proof, compared to [Sch2], is the use of a method of Todorcevic [To]. Familiarity with [Sch21 is not assumed, but familiarity with [Sh, chaps. III& V 1 is. We confuse sequences with tuples and structures with their universes. By "tree," we mean "normal tree." D e f i n i t i o n 1. Suppose T is an col-tree (i.e., height wl and every level countable) and S C_ L i m N w l . Then f S-*-specializes T i f f dom(f) = T ~ S = {x E T:ht(x) E S} and f ( x ) < ht(x) and whenever x
It is easy to show that if T is Aronszajn, then T is S-*-special iff T is S-st-special. Consequently, if T is an wl-tree and S is stationary and T is S-*-special, then T is not SusIin. Also, under the same hypothesis, T is not Kurepa, for if f S-*-specializes T and G(x) is the unique branch of T such that Sx = {y E T : y > x and f ( x ) = f(y)} _C G(x) when S, is uncountable, then by Fodor's theorem each uncountable branch of T contains some uncountable Sz, so G maps onto the set of uncountable branches. By [Sch2] (essentially proved by [Sh, chap. V & IX]) we have the following result; see [Sh, pp. 154, 164, 178-180, 293] for the relevant definitions:
161
T h e o r e m 1. Suppose ZFC proves that for every Aronszajn tree T* and ~ i - t r e e T and stationary
co-statlonary S* C Lim n wl there is a poser P = P(T,S*) such that P is < wl-proper, is ({A _C [~1]~o:wl - S* C_ A}, P)-comptete for some simple R 1-complete completeness system P, is
(T*,S*)-preserving, and 1 tkp "T is S*-*-special." Then Con(ZFC + ~ is inaccessible) implies C o n ( ( * ) + every w,-tree ,s S - -spec,al). We t u r n our attention to building the poset P = P(T, S*) as in the hypothesis of theorem 1. We may assume T, T* • H~,2 . Let B = {b: b is an uncountable branch of T}; fix ~ > card(B) + RI, regular. 3/ is a ~-chain iff 3/ is an •-chain of countable elementary substructures of H~ (not necessarily continuous); L~ = U{3/('y):q • dom(3/)}. Preliminary
d e f i n i t i o n . Q = {(f,S, 3/}:S c L i m a wl, [SI _< N0, f S-*-specializes T, 3/ is a
~;-chain, and wl n 3/(a) • S whenever a • dora(A/)}, ordered by (f,S, 3/) ~_ (f',S',3/t) iff S end-extends S', f _ f ' , and 3/end-extends 3/'. T n = {(xo . . . . . x . - t ) : (3a)(Vi)(xl E T~)}, ordered b y ~ < ~ i f f ( V i ) ( x i < Yi). F is a p r o m i s e i f f t h e r e i s n : n(F) • w and C = C(F) _ wl closed u n b o u n d e d and ~ = min(I') • Tno(r) (where o(F) = n c ) and G = G(F) C_ n and bt = b,(I') • B (t • (7), such that 5 •
r c T"[C andS•
r ==~ ~ > 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 <~ and fl > 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~<,~3/(~),
and T • 3/(0) (unless 3 / = O), and, for all a • dom(3/), ifx • d o m ( f ) - X/(a),
then the following are equivalent:
i) (3y < x)(y • dom(f) N 3/(a) and f(x) = f(y)) ii)
(gb • B n N ( a ) ) ( x • b). For p • P , we give fp, Sp, 3/p, g2p, and ht(p) their obvious meanings; we let Lp = L.% and
162
pv(x) = (least "7 such that x E A/(~)) [or x E Lp, and for b E B A Lv we set ~v(b) = wl A ~p(pv(b)) and #v(b) = (unique x E b such that ht(x) -- Sp(b)) and up(b) = h ( , v ( b ) ) and UN = U ( B N N )
Up = UL~. We order P by p < q iff
and
k~v 2 k~q and Sp - Sq C C(F)
for all F E ~q. This definition should be compared with the Main Definition of [Sh, p. 184] and with the poset of Todoreevic [To] for specializing wl-trees with finite conditions. Notice that if b E B M Lv and ~(b, fp, F) and q _< p, then q~(b, fq, F). D e f i n i t i o n 4. ( M , p , 6, n,5, fi~) is as usual iff M < HA (A large enough, regular)~ IM I = Ro; P ,
T . . . . E M , 6 = wl A M , • E T~; p E P M M ; .;t(i) C_ w~ is finite for i < n; and ~(~, f v , A ) . L e m m a 1. Suppose (M,p, 6~ n,'£,.71) is as usual and 8 E H~ N M . Then there is q < p with 8 E Lq
and ( M , q , 6 , n , ~ , f [ ) is as usual. Proof: First take N - < H~ countable with N E M and p, 0, T, A n 6 r~ in N. B u i l d q <_ p with ~q -- ffJv and )4q = 3/p~(N> as follows. First set $~ = ~1 n N and then choose 6m /
6~ with
sufficient care: namely, for every F E g2v and every ~ E F (1T~<~, there is an infinite W C_ r n T~<~÷, (or W = F n T < ~
if this is a singleton) such that whenever ~ and ~ ' are distinct elements of
W then { ~ ( t ) : t E n(F) - G(F)} is disjoint from { ~ ' ( t ) : t E n(F) - G(F)}, and also disjoint from Up. Then list all ( ~ k , F k , i ~ , m k > with Fk E ffJp and ~k E Fa ~ T <~ ~ N and fi~ C_ (6~) <~ a finite rectangle and rnk < ~, with infinitely many repetitions, and so in order type w one builds the required fq (with Sq -- S v U {6m: m _< w}) by defining, at stage k, fq appropriately on some ~k > ~ with ht(~ k) = 6,n~ with ~"(t) ~ Up for t E ~(r~)-c(r~) and each such ~ ( t ) incomparable with all previously considered notes from T (doing nothing if ht(~ ~) ~_ 6m~). Also we have a listing of T~S~ in order-type w which we use to ensure that dom(fq) = T~Sq. The care in choosing 6m for m < o~ is needed because we are constrained to take fq(x) -- ap(b) for x E b E B (1 Lv (assuming x ~ Lv). Aside from that quibble, the construction is simply an elaborated version of Fact 6.6 [Sh, p. 184], and there is no problem (although we must distinguish between x E UN and x ~ UN when x E T ~ ; in the former case we must take y < x not yet considered, with y ~ T ~ { 6 m : m E w} - [Iv, and set fq(y) = fq(x), at the time we consider x). D e f i n i t i o n 5. W e s a y tha~ H is M-good iff H c_ P ~ M
and whenever (M,p, 6, n,~,.4) is as usual
and D C_ P is open dense and D E M , then there is q <_ p such that q E H ~ D and c)(-£,fq,~). ][,emma 2. Suppose M -< Hx (large enough A, regular), IM] = blo, P E M .
M-good.
Then P ~ M is
163
Proof: Let (M,p,6,n,5,fi~) and D be a counterexample. We may assume A~ C_ ~n hence EM.
Let a = h t ( p ) .
Let ~ < h h a v e h e i g h t
aandlet
A = {~ETn:~iscomparableto~and
there is no q < p such that q E D and ht(q) _~ ht(ff) and ~)(ff, fq, fi,)}. Notice that A is downward closed (i.e.,ff < if' E A = = ~ f f E A). Everyff < 5 i s i n
A, for otherwise the q w i t n e s s i n g y ~
A
could be taken to be in M , which would contradict the assumption that we are dealing with a counterexample to the lemma. Therefore M ~ "A is uncountable." Claim. Suppose A ' _C T k (some finite k) is uncountable and downward closed and every element of A ' is comparable to 21. Then there is a promise F' _C A' with min(F') = 51 Proof of claim: We build G* _C k and uncountable branches (bt:t E G*), and an uncountable downward closed A* consisting of sequences (~*(i): i E k - G*) such that for each ~* E A~, the unique ~ E T~t(~. ) extending
with ~(t) E bt for t E G* is a member of A', and (~* (i): ~" E A*
and i E k - G*} is Aronszajn or empty. The procedure is as follows: start with G* = 0 and A" = A'. If {ff(t):ff E A* and t E dora(y)} is hronszajn, we are done; otherwise, we take b an uncountable branch. Fix t such that bn(y(t): ~ E A ' } is uncountable. Since A* is downward closed, b C_ {p(t):y E A*}. Let bt = b and put t into G* and take the new A* to be {(y(t'):t' E k - G ' ) : the unique ~' extending (ff(t'):t' E k - G * ) with dora(if') = k - G * u { t }
is in the old A*}. Continue
this procedure until we finish (i.e., we reach an Aronszajn tree or G" = k). Let ~* be the restriction of ~ to k - G*. Now, in the case G* ¢ k, the fact that A* is a downward closed, uncountable subset of some Aronszajn (T*) k' (some k'), and every element of A* is comparable with ~*, allows us to assert the existence of a promise F* C_ A* with min(F*) = ~* and G(F*) = 0 (see [Sh, pp. 188-189]). Let F' consist of all ~ E T k such that ~ extends some ~ E F* and ~(t) E bt for t E G*. T h e n F' is the required promise. QED claim. Take r c / , a promise with
rain(r)
= ~. Let p' = (]'v, Sp, )Jp, g/p U {F}). Now suppose r < p'.
We may take ~ E F with ht(~) = ht(r) and ~ ( ~ , f~, 4). Now, since ~ E A, there is no q ~ p such that q E D and ht(q) < ht(~) and ~)(~,fq,2). In particular, r ~ D. But this contradicts the fact that D is dense. QED temma. L e m m a 3. Suppose (M,p,&, n,5,.4) is as usual. Then there is q < p such that q is M-generic
and ht(q) = ~ E Sq and c2(-£,fq, ~i). Proof: Let (Dm:m E w) enumerate the dense open sets of P in M, (0m:m ~ w) enumerates N = H~ ~ M , and (~m,Fm,~i~n) enumerates all triples in M with Fm a promise and y'~ E Frn n T <w n N and A m C_ 6<w a finite rectangle with lh(/]~n ) = n(Fm), with each triple listed infinitely often. We build p _> P0 _> P~ _> ... with p ~ E D~ and 0,~ E L~.,+~ and p,~ E M;
164
simultaneously we construct g: T~ ~ 5 such that in the end q = ([J fp~ Ug, [J Sp. u {6}, [J .Yp ^(N}, U gap,} E P and ~P(~,fq,.4). Because of lemmas i and 2, there is essentially no problem in doing this. Notice that at stage n we may assign finite sets Fi to finitely many nodes xl in T5 and demand that g(xi) ~ Fi and fp~ (y) ~ Fi for all y < x~ and all rn E ~ (if xi E Up. then we can only do this if ~(xi, fp,, Fi)); this is how we take steps to ensure that each F which appears in some ~p~ will be fulfilled by (fq,Sq) at level 6. Also, when we put x E T6 into dom.(g), we check whether x E UN; if so, we immediately strengthen our current p~ to p~ E M such that the unique b E B 7~ N with x E b will be in Lp, and we set g(x) = op, (b); if not, i.e., x ~ UN, take g(x) < 5 unequal to h , ( Y ) for all y < x and demand that fp~(y) ¢ g(x) for a l l y < x and a l l r n E w. This completes the construction. Lemma
4. 1 IF %0 is S*-*-special."
Proof: Let S be the name for [.J{wx n )¢p(c~):p G ¢ and a E dorn()/p)}. is S-*-speciah"
Let C be such that 1 IF '@ = {a < ~ l : s u p ( S N a )
= a}."
Notice 1 I~- "5b Thus 1 tF '@ is
closed unbounded." Suppose p II- "& E C N S*." Take q < p, 13 < wl with q I~- "& = ~." Since q I~- "/3 E C," we have sup({wl n )4q(a):a E dom(Nq)} N/3) = /3 G S ~. By the Main Definition, W l n )4q('7) = 13 for some ~. Thus q IF- "/~ E S." We conclude 1 I~- ' @ n S" C S." This establishes the lemma. Lemma
5. P is (E, P)-comptete for some simple Rl-comptete completeness system P, where
E = {A C [Rl}~o:wl -- S* C_ A}. Proof: First note that lemma 3 holds with Te replaced by any countable collection of branches of T<e by the same argument. Then an especially tedious coding argument gives the completeness system (and the first-order formula defining it); see [Sh, p. 250, lines 8-t3] for a thumbnail sketch. L e m m a 6. P is <wl-proper. The proof of this lemma is identical to the proof of lemma 13 in [Sch2], so we give only an outline. Take {Na: a < p) a countable tower of countable elementary substructures of H~ ; we prove by induction that g n NO+I is N~+l-good, where g = {p E P : (V~)(p q~ N~ =:~ p is N~-generic)}. For successor/3, for p E H M N~+t, by induction we may assume p q~ Np and by lemma 2 we are done. For limit /3, repeat the proof of lemma 3 using the induction hypothesis rather than the M-goodness of M = NO; we thereby see that again we may assume p ~ NO, and again we finish by lemma 2. The following l e m m a corresponds to lemma 12 of [Sch2}.
165
L e m m a 7. Suppose M -< Hx (large regular A), IMI = ~qo, P, T E M , p E P N M , and 6 =
wl A M ~ S*.
Then ~here is q which is preserving for (p,P,T*,S ~) (see [Sh, p. 293] for the
definition), and hence P is (T*, S*)-preserving. Proof: Let Din, ~m, Fro, A-*m (m E w) be as in the proof of lemma 3, and let (zm,Am) enumerate all pairs (x,A) such that A E M is a P-name and x E T~ and (VX E M ) ( x E X ==~ (3y < x)(y E X)). We build p _> 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 ~* <
166
w".th ht(~*) = ht(y) witnesses y e Y. For suppose q* -< p2n+l and ht(q ~) = ht(y) and ~(~*, fq., A) but there is no F with o(F) = ht(q*) and (fq.,Sq.,Nq.,~q. U {F})IF "~ ~ A , ' } . We may assume that q* E M. Let A = {~t E T ' ~ : ~ ' is comparable with ~" and (Vq+ < q*)(if ht(q +) < h t ( ~ ' ) and q~(~',fq+,A) then q+ 1~z "~ • A~")}. Notice that every ~ ' < ~ is in A, hence M ~ "A is uncountable." Also, A is clearly downward closed. Let I' C A be a promise with min(F) = ~ . I claim that q+ = (fq.,Sq.,JC¢.,~q. u {F*}) IF "~) ~ An," which is the desired contradiction. Suppose that q+ I~ "~ ~ An." Take q' < q+ such that q' I~- # E An." Since (fq,,Sq,) fulfills F, we may take ~ t • F with ht(~') > 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<~(*)
M ~ L J8,81
colours occurs).
2 ~ [~l]n, 2
(we get an iso-
Our positive independence
are generalized naturally.
This will be
Later are given generalizations wlth finite conclusion,
but infinite number of colours ; and we improve the bounds for
wsp
T h i s r e s e a r c h was p a r t i a l l y s u p p o r t e d by t h e U n i t e d S t a t e s I s r a e l B i n a t i o n a l Science Foundation (Number 289). We thank Juris Steprans for handling the paper.
168
§ i.
The consistency of the partition theorem from a measurable cardinal. I.I
Notation:
M = (A,<,F)
where
F : [A] <~ ~ ~
~ <
(for
@ < w , ~
a cardinal)
is the class of triples
is a well ordering of the nonempty set
such that
F(~) = 0 . We let
A
and
[A] n = {u : u E A , [u[ = n} ,
[A]
If
A
is a set of ordinals,
<
F
and
will be the usual order and
omit it. We write
f : M ~ N
S = ([M[, <M, FM) , and use
is an
(K~)-embedding if
(Vu e [IM[]
M
for
IMI
sometimes.
Now
(Vx,y E M)[x <My {=~ f(x)
M C N
(M
and
a submodel of
N) if
the identity is an embedding.
I.IA
Explanation:
quantifier free type of
We are thinking of
M
u , more formally, if
as a model,
FM(u)
as the
u = {a0,...,an_ I} , a0<...
we call A F('"'xi''")iEv vCu the quantifier free type
tpqf
= F(''''ai .... )iEv A x0<...<Xn_ 1 of
u
(this notation happens when we do not
have a fixed model). Below in 1.2,
1.2
d
is thought of as a coloring of
Definition:
for every function
M,N E ~
For
free
The arrow where
eh 1.3
and cardinal
d : [M] <~ ~ 8 , there is an embedding
such that for every non empty quantifier
, ~ ~ w
M .
type of
~sp
u
u E IN] <~ , d(f'(u))
in
8 , Mf
of
N
(N)~~ into
if: M
depend only on the
N .
is defined in 2.1.
eh
First we define the weaker notion
stands for end homogeneity. Definition:
end homogeneous) an embedding
f
M eh of
N
For
M,N E ~
(N)~~ into
if: M
,~ < w
and cardinal
for every function
such that:
if
m < ~,
8
(eh stands for
d : [M] <~ ~ 8 , there is
a
169
N
< -increasing
s e q u e n c e o f members o f
N , and
tpqf((a 0 ..... am_2,am_l),~,N)
= tpqf((a 0 ..... am_2,am),~,N)
then d ( f ( a O) . . . . . f(am_ 2) , f ( a m _ l ) ) = d ( f ( a o ) . . . . .
1.3A
Remark:
f(am_ 2) , f ( a m ) )
There are obvious monotonicity properties.
•
Here
qf
stands for quantifier free.
1.4
Fact:
If
A 0 < Al<...
predicates only and if for (*)m
for some model
t h e n f o r e v e r y model cardinality Proof:
An_1
(N)~n
N 6 K~
such that
N E ~a
of cardinality
M 6 Ka~ of cardinality
lm Am+ 1
holds.
of cardinality
A0
t h e r e i s an model
M E Ka a
of
M ~ (N)~ n .
Remark:
We can define canonization relations - say, in how many
variables the coloring does not depend.
1.5
Lemma:
"sp(~+l)<~i~ p E P iff p(~) = 0
p
Suppose
See [EI~ilR]: [Sh95].
p = p
(see Definition 2.1), let
is a partial function from
P
be the forcing notion defined by:
[A] <~
to
~
of cardinality
< p and
and ordered by inclusion.
Then i n < p+l
VP
f o r some
we have:
1.5A Remark: (2)
is a finite vocabulary with
Trivial.
1.4A
type
L
m < n
for every model
M eh
and
If we let
(I)
M E K~ a , IM[ = ~ , and f o r e v e r y
M ~eh(N)~
P
N 6 Kaa o f o r d e r
.
is really just adding
A
Cohen subsets of
INI < p , the proof is somewhat simplified.
p .
170
P r o o f o f Le~ma 1 . 5 : f E Gp
f(u) = j .
of generality
i s an o r d i n a l
o f t h e Cohen s u b s e t s = N . of
Let
H(X) .
X Let
second phrase
M = (~,<,F~),
~ : [~]<~
Let
{~1
Let
P
def = (xj
(*) T h e r e i s
i < /~
: j < i>)
B C ~ ,
B
f o r some
Without loss
d e p e n d s on a t most loss of generality
and l e t
<* X
(see Definition
a well-ordering
2 . 1 and 2 . 3 )
(for
the function .
of order type
Hu, v , (u,v E [B]
iff
be P - n a m e s .
without
cardinal
By t h e h y p o t h e s i s
in (I) use for
N e ~a
i n two s t a g e s )
be a l a r g e enough r e g u l a r
F(x 0 . . . . , x j . . . . ) j < i
(I)
8 , and
F~(u) = j
e + l < p + 1 , and ( a s N
and d o i n g
p* E P •
where
+ I
and
and
such that:
su~ (H(x),e,<~) , (¥a e [Nul<~)[a e Nu]
(hence
D ~ Nu)
and e Nu
(II)
(b) - (h) o f 2 . 1 $
(hi) Let
p e PNN¢ .
B = {~i : i < ¢}
and
[i < j ~
be the unique order preserving Let f o r
~i < ~j]
function
from
"
First
¢ + 1
assume
onto
¢ < D .
B , i.e.
g(i)
Let
g
= ~i
"
i < e I i = [{~j : j < i
Next let for
u E I i , K(u) = {(u,h) : h
or
j = ~}]<~ .
is a function from [u]
to
~,h(~) = 0) J.
1
Note t h a t
[i < j ~
( U l , h 1) _< ( u 2 , h 2)
if
embedding will take def = {~i : i ~ e} .
that
: u E I.
1
I i C lj] , and
[Explanation:
I
= {(u,h)
and
(u,h)
[i < j ~
E K(u)}
Ji ~ Jj] "
.
We say
u 1 [ u2,h 1C h2 . Note that i
to
we h a v e a l r e a d y
g(i)
decided that
, so the universe
the desired
o f t h e image o f
What we h a v e t o do i s t o f i n d a c o n d i t i o n
the embedding is as required.
Now
~6
is simultaneously
in
N
will
P
forcing
a good
be
171
"approximation" to
{~j : j < i
or
~j
over
g(j) = e}
{~i : i < j )
and we d e f i n e a c o n d i t i o n
by induction on
i , though in
realize different quantifier free type over by dealing simultaneously with conditions We now define by induction on
P(u,h)
for
may
(u,h) 6 Ji "]
i <- e ,
i P* ~ P(u,h)
(c)
if
j < i , (u,h) E Jj
, then
(d)
if
( u l , h 1) < _ (u2,h2)
then
(e)
i P(u,h)
(f)
i+l i puo{~i},h i > HuU{¢},uU(~i} (P(uU{e),he))
(b)
N , Jl ~ J2
{i : i < Jl fl J2 ) " We are saved
i P(u,h) E P n N u , and i
(a)
for
such that:
h C P(u,h)
forces a value to
functions such that
j < i P(u,h) - P(u,h)
"
P i( u l , h l ) = P (i u 2 , h 2 ) r (A O Nul) • ~(v)
for every
v C u .
if
u C i
and
hi,h e
are
(u U ((i),hi) E Ji+l ' (u 0 {~i},<,hi) ~ (u 0 {e),<,h e) •
We shall carry out the definition in detail. [Explanation:
Condition (a) is in order to have control over the condi-
tions and to utilize the indiscernibility. (b), note it is the role of
Condition
deal with the case
i P(u,h)
to ensure our being able to
h = (FN o (g-l)) ~ u .
Condition
(c)
should be clear.
Condition
(d) enables us to form the condition
U xEX P(ux'hx)
for suitable
X .
Condition (e) of
~ ~ we certainly have to have approximations forcing some values for them. Condition (f)
~i,e
a s we want to form a condition forcing the "right" values
are similar
over
value to
~({~i,~e})
The
ion.
Induct Case
A:
i
This comes for the end-homogeneity, we want to say that
= 0
{~j : j < i ) this
.
similarity
, o f c o u r s e t h e m i n u t e we want t o f o r c e a cannot be maintained.]
172
So
I.
= {C}
so
J
I.
1
= {~,
{e}} .
(except
when
~ _< t
which is not so
1
interesting). For every
(u,h)
E Ji
L e t u s enumerate the functions
(so
70 < ~) . We define (i)
i P(u,h)
we h a v e t o d e f i n e
P7
"
Let
u 0 = ~ , u 1 = {~} .
h : u I ~ ~ , h(~) = 0 : {h7 : 7 < 7 O} ,
by induction on
7
such that:
P7 E N{c } ~ P
(ii)
h 7 C P7
(iii)
for every
(iv)
P7
~ < 7 , p~ I N~ < P7
forces a value to
~({~}) .
There is no problem in doing this by i PCuo,¢) =
U
el,e 2
below.
In the end let
P7 I N~
7<70 i P(ul,h)
i = P7 U P(uo,~)
where •l
if
q7 6 P
for
least upper bound of functions
7 < 7(*) < ~ , then
{q7 : 7 < 7(*))
U q 7<7(,) 7
is in
if and only if for any
P
and is the
71,72 < 7($)
the
are compatible qTl'q72
only
if
ql E Nul S P , q2 E Nu2 fl P
if
•2
t
ql
NlflU2,
q2
C a s e B:
i limit.
For any
(u,h) E Ji
then:
ql,q 2
are compatible if and
are compatible [by 2.I d].
t NulflU2
there is
J(u,h) < i
such that
(u,h) E J. O(u,h)
de(
We let
i P(u,h)
=
U{p~ u,h) : j < i
in checking the conditions (note:
and
i P(u,h) E Nu
[eEN uA [aJ < ~ a e N
Case
C:
i
= j
+ 1
.
(u,h) 6 Jj} . because
u] by Z)
There are no problems
173
Let us enumerate 7(*) < ~ •
Note that
:
{(uT'h7)
7 < 7(*)} -
We define by induction on
7 ~ 7($)
a sequence
such that (compare with (a) - (f) above):
Ji) (a)'
7 q(u,h) E P n
(b)'
h C 7 - q(u,h)
(C)'
/3
(d)'
if
< 7
=
Ji
Nu
implies
u,h) -~ qTu, h) (Ul,h 1) ~_ (u2,h 2) then 7 = q7 tN q(Ul,h I) (u2,h 2) uI
if
(e) '
V
(f)'
~
/3
< 7
U
•
then
forces a value to
7 q(u/3, h/3)
~(v)
for every
the parallel of (f) .
Subcase
C
(a)
:
7
=
0
Define
"
(e)
it is
J P(u,h)
(/3)
it is
Hul,u(P~u,h))
if
def v = u\{~i} E I j
q(u,h) 7
as follows:
u EIj if
u E I i , ~i
E u , e ~ u ,
; and we let: def
uI J
(7)
P(uo,hluo )
it is
:
v O{e)
,
J U HUo,Ul(P(uo,ho ))
if
u : v U {~j,¢} , ~j ~ v , c ~ v , and we let
u0 =
v
Subcase C (b): Use unions
q(u,h) 7 :
Subcase
C (c):
such that:
(4u,h)
def v U {Ej)_ :
U {c) .
7
is limit.
U
4 u ,h) "
/3<_~
7 = /3+
Note that the only demand on satisfiedby
U1
1 .
7
in
: (u,h) E Ji)
(qTu,h) : (u,h) E Ji) is (e) ~ for
which is not clearly
/3 . We first choose
r = rV ,
174
(i) ~us'h8 ) (ii)
r
< r e Nup n P
forces
Clearly such
r
a value
exists.
to
~(v)
for every
Now for every
v C u~ .
(u,h) E J.
let
1
q ( u , h ) def = ~(u , h ) OOfr r Nv : v C_ u n u~,h
possible that
Vl,V 2 C u n u~
mad
h t ( v 1 U v 2) # h ~ I (v 1 U v 2)
Now whenever
qTu,h) E P
by
[note that it is
h I v I = h~ I v I , h [ v 2 = h~ ~ v 2
but
.3
el,e 2
and as
(v C u n u~,h t v = h~ r v) . : (u,h) E Jl>
I v = h~ t v}
is as required:
q(u,h)
v = q(v,htv) - q(u~,h~) -
It is easy to check that i.e. conditions
(a)' - (f)' are
satisfied.
So we have defined
can finish Case C:
So we h a v e f i n i s h e d Lastly
the definition
of
< P i( u , h )
= U{P(u,h ) : (u,h) h(v)
Clearly
= FN(g-l(v))
the union is well
defined
> ~ , 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] <~)~ •
and forces t o make
w h a t we n e e d e x c e p t when g
an embedding of
(too large cardinality).
i P(u,h)
i < p , an ordinal such that
the parallel
: j < i ) U {¢} , a n d
i P(u,h)
E J¢ , u C e ,
We c a n n o t u s e t h e d e f i n i t i o n
, and choose by induction on
sup a. < a. < Max B j
then
for
N
to
M
.
e < p .
~ = p .
union will not be a condition
(u,h)
E Ji >
let p
V[Gp]
J (u,h)
and we
of (a) -
(u,h)
End o f p r o o f
E Je
above as the
But we can work in ai ' such that
u C _{aj : j < i) U (e) (f)
and
of le~na 1.5.
above such that:
h(v)
= FN(g-l(v)
and if
for
v e
[u] <~
175
1.6
Conclusion:
Assume that there is a class of measurable cardinals.
Then in some generic extension V m,n < w V 8 V N E K~n(~4)[M E K ~ n A {M l ~ im+l((INl + ~ +e) +) A M - (s)~] Proof: zero)
Iterate the forcing (with e.g. Easton support)
is adding
~6+n+l
Cohen subsets to
~6+n ' where
Q6+n
(6 limit or
~0 = R0 ' for limit +
ordinal ~c~+l
6 , ~6 =
is the first
Iterating
§ 2.
On 2.1
and
U ~ a<6
their
s6
measurable
is singular
> ~
.
s6+ I = ~6 "
In all other cases
By 1 . 5 we g e t e n o u g h i n s t a n c e s
u s e b y 1 . 4 we g e t t h e d e s i r e d
of
eh
conclusion.
~sp " Definition:
n < w .
We d e f i n e
It says that
operations each with u E [A]
, if
< a
if
A
N
. .<#,
is an algebra
where
with universe
places, then there is
are cardinals
A,~,~,~
A E [A]~
and with
A
and
N
< p
for U
such that:
(a)
N
(b)
Nu
(c)
N
is a subalgebra of
U
N
has cardinality NA=u
U
(d)
N u N N v = Nu0 v
(e)
for
(the main point!)
u,v E [A]
of the same cardinality,
N
isomorphism from
N
(f)
H
maps
(g)
Hu, u = the identity,
H
U#V
U~V
t NUl C_ HVl,Ul
onto
U
u
N to
where
the unique
-~ N U
--
V
, order preserving, exists, we call it
V
H
U,V
v Hu2,u3O Hul,u2
vI C v
=
is such that
Hul,u3 Hu,v
and for maps
uI
uI C u , onto
vI
(so
equality holds), (h)
for
u E [A]
2.2
Definition:
(i)
if
(I)
We define
. .
v C u E [A]
then
[Nv[
similarly adding is an initial segment
176
of
INul; (2)
We omit
8
when
2.3 Observation:
(i)
8 = R 0 , i.e. omit (h) in 2.1 . If
A ~
(~)<~,
spn
A sp
8 < A
then
(~)
(2) s,~,n,8
Those a r r o w s h a v e o b v i o u s m o n o t o n i c i t y
.
For
(3)
(See
and
In 2.1,
[Sh3] and
2.4
dsp
§
we can i n c r e a s e
properties:
we can decrease
A .
2 . 2 we can u s e as
N
any a l g e b r a
A c INI
such that
3).
Fact:
(I)
If
A
is measurable,
~ < A , ~ + 8 + ff < A ,
n < w
. .<~,
then
(2)
If
A
2.5
la~mma:
is minimal such that
A ~ (~)~wu ' @ ~ ~
<W,<W A ~spn(N)~,8
then
<w,<3 Proof: H(X )
Let
If
X > 2A
So we have
.
(a)
su
~ > 3 , A ~sp(~)w,D be a regular
<Mu : u E [~]<2>
then
cardinal
A ~ (<)~w r~
and l e t
<*
x
6
be a w e l l o r d e r o f
such that:
~(H(X),E,<;) ,
(h) Mu N M v =Muf ~ (c)
A E Mu
(d)
if
(unique) (e)
[u[ = [v[ , Mu,M v
are isomorphic and let
Hu, v
denote the
isomorphism if
u = {il,i2}
, v = {jl,J2}
, i I < i2
and
Jl < J2
then
H(il},{jl) ~ Hu,v , H{iz},{d2) ~ Hu,v , idMo ~ H{il},{jl }" (f)
M{i } N A # M~ n A .
Let
a i = a(i) = Min(M{i } N J - M#) .
(d),(e)) and
p < @i "
So for all
Also
(This follows from (c) of 2.1.)
~i # ~j
i < j < < , "~.z < a." j
(as
Clearly
H{i},{j}(a i) : ~j
M{i } A M{j} = M~)
for
has the same truth value.
(use
i # j . Since
177
gi # aj
if
(A)
( > ~ , as the ordinals are well ordered: (~i : i < (>
is strictly increasing.
If
we could inverse
( < w
the indexing and also have (A).
Next we shall prove (B)
If
i < j
same type over [Proof:
and
~ 6 M{i ) , then
{7 : 7 < @i } Let
~(~,~)
(in
be a formula,
or
and
H{i},{j)(~)
realize the
(H(x), E, <~) • )
be the following order (of Godel) only if Mex(~) < Max(~)
~
Ig(~) = Ig(~),Ig(y) = n . Let
on n-tuples of ordinals:
Max(~) = Max(~)
and
~
~
<
gd
if and
is smaller than
~
in
the lexicographic order. i
Let that:
B
F~(Xl,X 2) =
the
(n-tuple of ordinals) such
V(Xl,~) ~ ~ ~(x2,y) • Clearly
F
is definable in
(H(x),E,<~)
hence each
M
is closed under U
F
Let
~j = H{i},{j}(c)
defined for some
(2 a l l )
F(Cjl,Cj2) E M { j l , j 2 } .
the set
{Z(Cjl,Cj2)
e.g. t h a t the f i r s t %(Cjl,Cj2)
,
for each
j < ( and assume that
J l < 32 < 3:
otherwise (B) is immediate.
However by a c l a s s i c a l
trick,
~(cjl,cj3),~(cjz,cj3))
= F~(Cjl,Cj3) E M{jI,J2 ) N M { j l , j 3 } = M{31) .
M{j2} . As clearly
(for
(Cjl,Cj2) >_ Mzn{@jl,@J2} " . As
£ = 1,2 , as
~
a. l
J l < J2 < J3
then
F (~jl
Generally (according ) belongs to
M{jI}
~Jm = F{i},{j2} (~)) we can deduce
was any formula, we have finished the proof
(~)]. (C)
So
two are equal, so
F~(Cjl,Cj2) > 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,<~)) .
Now if
fl,f2
are
iI < i2 , we get a
contradiction to (B). Next if
~i
are not strong limit, then there is
E M{i }, 2~ ~ ~i " j , so in ~(~)
M~
~ E M~ , and by the
So
~ < ~i '
H{i},{j } 's , 2~ >_ Gj
for each
there is a (definable from #) one-to-one function from
2~
to
and we get contradiction to (B).] (D)
W.l.o.g.
(E)
For
i < j < ~
M{~} U (Gj N M{i }) [Proof; definable,
with
is the Skolem hull of
M{i } U M{~} .
the intersection of the Skolem hull of Gj
is included in
If not, there are
M{i } .
~ E Gj N M{i } , d E M{j} , y = G(~,d) , G
y E Gj \ M{i } .
Let w.l.o.g. remember
M{i,j )
J < Jl < ~
( > 3 • As
~
{7 : 7 < aj} , clearly
and
(we use that w.l.o.g, ~, def = H{j}, {j1}(~ )
y = G(~,d')
i = 0 , j = 1
and
realize the same type over
too, so
y E M{i,j } n M{i,JI} = M{i ) .] We shall code, for each formula over
{7 : 7 < @i } (F)
For each
by an ordinal ~ , and
~(x0,...,Xn_l,y ) , ~-types of n-tuples
< 2 ~
n > 1 , there is
~ ,n,i E M{i }
such that:
I
(1) whenever
c ,n, i
codes the ?-type of
i < i I <...
(2) H{i),{j)(~,n,i) = ~,n,j
<~i,Gil, .... Gin_l )
over
{7 : 7 < Gi }
179 [Proof: For
For
n = 1
n + 1
if
this is easy.
i < iI ,
think of the meaning) so function (over
~,n,i
%,n+l,i
can be computed from
= G(~i'C~,n,il ) where
a definable
G
m
< ~I
(just
~) .
However, by coding such things naturally hence
ai, ~ , n , i l
(by (C)) .
is an ordinal
%,n+l,i
So it necessarily belongs to
< 2 1 ,
by (E) , so (0),(i)
M{i }
holds. By the way
was defined,
c- ,n+l, i
If there i s
F : [A] <W ~ ~
conclusion of 2.5, then such (H(x),e,< ~) !) , and
F
also (2) holds.]
which is a counterexample belongs to
M~
contradict
to the desired
and is definable over
~
(in
its choice (by (F) above), so
that the lemma 2.5 follows.
§ 3
Refining the combinatorics Definition:
3.1.
(I)
For
x E {sp, spn}
we define r~<~,n ~j ~,8
~ ex(k)
like
(~<~,n
~ --4 x
~p,8
(see definitions 2.1 and 2.2) except that we replace (e), (f), (g) by (e) e
if
u,v E [A]
initial segment of H
u ~k v
(which means that for some
and
v
lu\wl = Iv\wl _< k)
and
then
w , w N u ~= N v
is an and let
be the unique isomorphism
U,V
(f)e
u
and
H
when d e f i n e d
UsV
(g)e
if
u I ~k
then H
maps
o H
= Hul '
Ur% H t Nu~ ~ Hu,I,u,2 Ul,% If
(3)
For
k = 1
onto
v ;
H U~U
= id
u2 ~k u3 ' ui E [A]
%
u2,us
(2)
u
and for any
u~ C u I if
(so equality holds).
we o m i t i t .
x E {sp,
spn,
esp,
espn)
we d e f i n e
:
Hul,u2(Ui) then
180
---4wx [~)~,8" .<~,
(d) w
and (a,~) ~ u = @
(Note that (4)
N u ~ N v C Nun v
then
(~,~) ~ M
now in (g) e q u a l i t y
For any of the
x
. .
just like
U
and if
~ < ~
are from
~ , u E [~]
= # .
does not follow.)
for which
P was defined X
defined as above except that also
A
d : [~]
---* XV
. .<~,
[~)~,8
is
'
is given and (h) is
replaced by: (h) v
For each
h(u) (u E [A] generally,
£ ,
TM)
does not depend on
exv(k)
Ul,U 2 E [A[]
d ~ [A] £ . is constant when
and
max(u)
e
is a c o . o n
x ; and
when it appears and, more
does not depend on the last w
does not appear in
k
members of
initial segment of
u
(i.e. if
Ul,U 2 ]u[ - w] ~ k
then
h(Ul) : h(u2) ) •
3.2. (I)
Observation:
. .<~,
A ---# ~)~,Av
We have
~
. .<~,
A ....~ (~)~,Av
x
(2)
n
or
x
is
If
(e
appears
where:
y
is
x
y
when we omit
in
x
and)
e
or
w
or
v.
t <, a8 , < n (tl.D
J2 --~
and
Al
t
x(k) then (3)
If
x (4)
If
x(~)
...<~,
A2---#
x(k+£)(AO)#,O
A
t . ,<~,
with
e
12---*
and
k > n -I _
then
. . . <~,
[AI)~, 8
x(k)
~I ~ (~0) M then
3.2A R m r k . strong (when
w
A---~
~,8 Y (~)<~,
where
y
is
omitted. , and
£ = n - 1 - k , y ...<~,
and
when we omit
y
By 3 . 2 ( 4 )
~2
~ Y
even
does not appear in
~aO)~,8
A ....
x(0)
x).
is
x
with
e
omitted
"
. <0",
n > 3 , is quite
181
3.3.
Definition.
A
t+ (~)<~
means:
for each club
C C A
and for
#
n < ~ , i < ~ , Fn, i : [A] n ~ A belongs to
A and
not depend on 3.3.A.
there is
m < n , i < ~
~m,...,~n_l Remark:
and
A 6 [C] ~
such that if
Fn,i({a 0 ..... an_l} ) < a m
Neplacing "a club
C ~ A"
satisfying the first definition the 3.4.
Fact.
If
~ < 8 < I , A
be, in fact, any limit ordinal,
3.5.
umm.
by "a final segment
Proof.
A
I =
C C A"
U A i , each i
satisfies the second definition.
~ (~)~
then
A spn~ ~)~,@" "<w'<~
(~
can
w~ = ~ ).
A spnP (~)<w,<3w,~ , ~ >_ 3
If
then it does
.
does not change anything except that in the later version, if A. 1
aO<...<~n_ 1
then
~ --~
(~)~w . we know
Similar to the proof of 2.5 but by the definition of spn
sup(N~ N A) < ~0 = A .
In the end, if there is a sequence
(C, (Fn, i : n < w , i < ~>> definable over 3.6.
~
~a.
as
C E ~
(I)
and easily
For every
k = (2n-l)2)"" such that : if (2)
contradicting the conclusion,
Vn < w 3 k = k 2 < ~ n
s<# = ~
then
such that :
if
A---~ wspn
Part 2 is used for e.g. consistency of
3.
We do n o t t r y Proof.
(1)
Let
define, b y i n d u c t i o n (i)
A 0 : An
(e.g.
• +.
# (~)~,~ wsp
~,~,~ < A
and
A
e
is
(a+k)-Mahlo
, .<~,
2H
t wsp
is stronger than
+ 2
[~ ] 0 , 3
A0 = ~ , A£ + 1 =
A on
n
and at most m < n , a set
~ Am
2n+l(A£)
.
Suppose
functions each with and
~u
--~ . wesp
"
h e r e t o g e t t h e b e s t bound (but s e e 3.8 and s e e
algebra with universe
I
k = kI < w n
k(S) +
Using part I for 4.1 note that
2.
it is
and continue as before.
n < w , there is
strongly inaccessible cardinal then Remark. i.
~i 6 C
wlog
[4]). N*
< e
(u £ [Am]
i s an
places.
We
182
(ii)
Am+l C Am
(iii) II
~_~
IAml :
(i)
N~u ( f o r
(ii)
The a n s w e r t o " i s t h e 7 1 - t h e l e m e n t o f
element of
~u ?"
u E [Am]
where
isomorphism type of (iii)
set
(i,,)
~f
For
m = 0
u U s •
II.(ii) be:
If
if
u , v E [Am+l ]
(u U v, u , v ,
u,v E [Am]
let
< [ (u U v ) )
II (ii)
i.e.
An-m ~ ("An - m - 1 ). 22 ~( n - l ) )
[u I > n - I - m
if
Nm ~
X E <
U
N N~v and
Iwll = I w l , l h l
E wlUv 1)
on
71,72
and t h e
.
Am+1 C Am
--
by
~vv e q u a l t o t h e 7 2 - t h
and
(VC~ E wUv)(~#
(2)
We n e e d 3 . 7 b e l o w i n s t e a d
N~u = Nm+lu
N~u i s t h e s u b a l g e b r a
m < n - 1 , choose
and if
< m .
then
A0 = An ,
(using
of cardinality
depends just
.
Now f o r
N$
g e n e r a t e d by t h e
n ~
such that
u 6 [Am+l ]
: w,v e jAm+l] n
The cardinality of
N m+l
Nm+l u and
is
<
U
*
--
Wl,V l E [Am+l }
: I v l , wl n v
{I,,~1
of
[Am+ll = An_m_ 1
lul < n - I - m , the Skolem hull of u { ~
u = w N v)
N*
u ~ jAm+l] TM , lul >_ n - 1 - m then
If
holds
i s a submodel o f
l:wnv
: Iwlrml A Ivr~l : IVlrml]
then
o f E r d o s - R a d o and t h e n t h e p r o o f i s s i m i l a r
to
that of part i. 3.7.
I~mma:
If
A
is
an algebra with universe is unbounded in
A
A
(a+n)-Mahlo and strongly inaccessible and and
< I
then for every
operations each with arity
/~ < A
there is
~ : /~ < ~ < A , ~
@-Mahlo and strongly inaccessible and there is
A C A 0 [I ~
(*)
A
If for
£ = 1,2
~ < a <...c~n_ 1
<~01
.
are from
1 '~n_l),
realize the same type over
{3 : 7 < ~}
2
then
2
N$ .
< A
unbounded in
N$ and
is A0
is s :
183
3.7A
Historical Remark:
We proved 3.7 in 1968 as part of some research
on transfer theorems in model theory.
As Schmerl was doing parallel research,
it appeared in [ScSh20] but somehow this version does not appear - only the version with a finite conclusion. bound for
A
Subsequently Schmerl found a better lower
(how Mahlo it should be) and proved that it was exact.
Hajnal
independently proved 3.7 and the author wrongly told him it had appeared in [ScSh20]. Proof: For
n = 0
to find M r ~ for
We prove it by induction on there is nothing to prove.
s < I
which is
For
p < ~ , there is is a club of
A0
and (as
n = 1 , let
C =
(N,A r N) ~L
~ E C
A N ~ : 7'
A 0 : {f(#) : ~ E C'}
n - 1 .
Expand
{6 < I : ~
is a strong limit and for each
such that
~ [ INI [ ~} •
Clearly
C
which is ~-Mahlo. f : ~ ~ ~
Clearly
C'
7
by
over
{i : i < ~}}
is a club of
$
Let
and
is as required.
Lemma
Suppose
A = 8+,8 ~ = 8 , ~
Irl < H
and each member of
universe
as there for
apply the induction hypothesis
realizes the type of
C' = {~ < $ : (~G < ~)f(G) < ~} .
use the induction hypothesis
A 0 fi ~
7 E A - ~ , define a function
f(a) = m i n { 7 1 E
3.8.
n > 1
n > I)
(M,A)
I , s o there is
Choose
For
(~+l)-Mahlo and
by a predicate for n = 1 .
n .
A
= ~ , ~<~ = ~ , r
tke, we can find
has arity
T
is a vocabulary
< ~ .
If
M
such that
is a r-model with
6,~,B, ( M s : s E [B] ~2 > , (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
M . ~,a
(d)
Ms n B = s
(e)
For
s,t
, M{i } N B = M{i } n B = {i}
fi [B] ~2
Hs, s = idM
(and
H{i},{j }
maps
Its, t
tsl
All
(g)
M s N M t C Man t .
(h)
Vi < j < k
: Hs, t
1 ' Hs,t = H-t,s
s
M{i }
(f)
= It]
onto
= H Sl'S2
M{i,j } n M{j,k ) = M{j) ,
(7)
M{i,k ) n M{j,k ) = M{k ) .
(i)
For
i < j
(j)
M~ C M { k } , M~ C M s
for
(k)
(a)
w C A , V~ e w
cf~ =
(~)
6 = maxW
(7)
If
(6)
If
from
B , j
s
onto
t .
is the first element of
M{i,j}\ M{i} •
k E S , s 6 [B] ~2
and
k E W
then
i<jEB
then
Remark:
(I)
Va < A [I@I~ < A] .
~ < k [ 7 < k
For simplicity, 8,~,D,a
(3)
We can replace MO ~L
~+)
By the choice of
M0
> ~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
<
is a relation
L +
is an ordinal Na 4L
Nb ~L
+M 0 ~,p
of
M .
of cardinality 6a< A
+M , 6 a E N a ,
/~ .
of cofinality INal
and an isomorphism
= s; .
f
from
185
Na onto Nb o v e r {N* I = p
N : M ~({Na{ 0 {M0l) •
be such that
6 a E N*
M{6a} = N* t INB{ and
and
Let 6b = f(6 a) .
{N, NB,Nb,f} E N*
in some coding.
Let
~
he minimal element of
(V~)[~ £ Mi~b} A (37 E N) ~ < 7 ~
~ < (~]
Now we define by induction on
to he the
¢-th
and
H{~},{(} ,
member of
conclusion holds, and
We let
Let
h : IM{6b} I ~ INal : h(~) = min{7:7 E INal , ~ < 7) •
(~ < ~) , M{6~,6a }
M,
= N* t {Nb{ ; M{6b} = N* ~ {M0{ , M{6b,6a} = the M{6h}
Skolem hull in M of M{~B} U M{6b} .
W = range (h) - {6a} .
Let N* ~L
N
such that
~ : supW .
~ < ~+, 6j , M{6~}, M{6(} , M{6~,6~ } H{~,6a} ,{~,6a}
B) such that:
M{6~} C N ,
i.e.
Let
for
for
~ < ~
for
(understand
the relevant cases of the desired ~ < ~ , M{6~,6~ } C N , M{6~,6a ) C N a ,
etc. and lemma 3.8 is proved. 3.9. Le~m:
Suppose
GCH
for simplicity
~
~<~
~
< ~ < ~ < ~+ < A I
II
A.
P
8re regular. There is a forcing notion + is strategically ~¢ -complete.
B.
P
preserves cardinalities and cofinalities.
c.
{Pl
~.s < "*
There are
S* C S [ ~ , {C6 : 6 E S}
and for
: s~ [s]<2>. ("~.s: s e [,~]i>. ~ : <
2
.~, ~, C8,
6 E S* ,
.t : s . t ~
~"
The relevant conclusion of 1.1 holds for each 6 E S* with B6 an
unbounded subset of
(B)
such that:
~ •
Is{ : {t{> . (A)
P
• > ~++
( I n VP)
(*)
If
~6 < ~++' ~0 = min W < ~+
~6(I) = C6(2) t h e n there i s a f u n c t i o n
U M6(1),s U W6(1) U C6(i) s
H6(n),6(2 )
from
onto U M6(2),s U W6(2) U C6(2) which i s orderS
preserving end preserves all relevant properties and the domain and range are disjoint.
6~
186
(C)
6 E S* ~
cf 6 = ~
of cardinality (D)
if
< ~
(follows from (A)) and for
6 E S
C6
is a club of
6
and
6 E S* , @
is an accumulation point of
C6
~ E S A Ca = C6 N
then
(follows from (B)). (E)
For
6 { S*, W~ [ C 6 . ++
Proof.
If
A = ~
If we succeed to force cardinality
~
+
we shall force by approximations of cardinality for
For
A ,
A = ~
+n
we can force for
A+
, we iterate this, for
~ .
by approximations of A > ~
+w
we have to take
care of the singular case.
§ 4.
Eliminating the Measurables 4.1.
(I)
Lena.
Suppose
~----~
("+1 ~ ' < ~ ( * ) Let
P
of cardinality
(2)
(see def. 3.1)
'p,O
wesp
and
p = p<~ , p < 0 < ~ < A , ~ < w , ~(*) <
be the forcing action as in 1.5. ~
In a p p l y i n g
for every
N E KG v
Then in
of power
VP
for some
~ , M-.--q eh ( N ) ~ (*)
~ we can weaken i t r e p l a c i n g wesp
(d) in 3 . 1 (3) by
[lul,Ivl <
n-l]
Nu O A = u !
but
(d)-: if u U v U{@,~) C A , (Vi E u U v)[i < ~ A i < ~] A then
NuO{@ } N N 0 ( H )
A =
~(,)_2(~ ++) Proof:
replace 3.1 (3).
N A C Nu N N
suffices for
However we still need
~+
We indicate the changes in the proof of 1.5.
"(b) to (h) of 2.1" Defining
i P(u,h)
Of course, we
by the appropriate variants from definition for
(u,h) E Ji
by induction on
i
we change (d)
(u2,h 2)
(both from
J), are compatible (i.e.
to:
(d) hI
If
(Ul,hl),
r (ul O u 2) : h2 r (ul O u 2)) i P(ul,h I)
M 6 I~
t~e.
t N NN uI
NA u2
and in case (C) we change (d) l to
i = P(u2,h2 )
t N NN uI
NA u2
187
(d)'
If
(Ul,hl),(u2,h2)
are compatible
7 t (Nul 0 0 ~) : 7 t (NulN Nu2fl ~) • q(Ul,h 1) Nu2 q(u2,h 2) In subcase (C)(a), we use (d) above ( t h i s influence 47) there) and in the proof of subcase (C)(c), we l e t , for or
w E don ~(u,h)
qifu, h) (w) (a)'
-
is defined and
~(u,h)(W)
is
(f)'
r(w)
7 (w) (u,h) E Ji ; q(u,h)
The value of
w E INu N ~]<~(*)
when defined and
r(w)
otherwise.
Let us check
.
(a)'
Trivially
q(u,h)7 E P as
q(u,h)7 C Nu
definition) clearly
7 q(u,h) E Nu by the demand in the beginning of the proof of 1.5). (b)
~s defined i f f
'
as
h C
(c) l fll < 7
assuming
(d)'
(as a set of p a i r s , by i t s
((Va E [Nu]
from
,h) C q(u,h) implies
fll < fl or
(Ul,hl),(u2,h2)
fll = fl and check are compatible we have
q~(Ul,hl) t (NUlfl Nu20 ~) = q~(u2,h2)(Nul0 Nu2o ~) • AS c l e a r l y ,
dem q(ue,he) = dom
e,he ) U(Dom (r) n Nue)
the e q u a l i t y of the
domains is easy, s i m i l a r l y check e q u a l i t i e s of values. (e)t (f)s immediate.
4.2.
Conclusion:
Assume, for s i m p l i c i t y only, t h a t
V satisfies
GCH.
Then in some generic extension, not collapsing cardinals nor changing cofinalities,
k < w
(a)
2R~
(b)
for every
< ~+w
for every n < w and model
and model M , JMJ <
N E K
k(HNJ{ + ~ + 8)
and M ~ (N) m{7
remark) and 4.1. ) Proof:
m < w
Like 1.6 using 4.1 instead of 2.5.
and .
(By
0
for some 3.6
(i)
(see
I
188
§5
(3)~0
K4 C G ~
K
The question we address is an old one of Erdos and Hajnal.
is the n
complete graph with Question:
n
vertices.
Is there a graph
G
which embeds no
K4
G ~ (3)~
such that
? 0
We still d e a l
We get here the consistency of a slightly stronger statement. with graphs although the proof says something more general.
More on the case we
are interested in (forbidden infinite subgraphs) will appear later. 5.1. Leman: A ~
Suppose
(2k(*)) W'<3) ~,~
wsp
forcing
notion
P
p < I < ~ , ~ 2 < m < ~I
'
and
is measurable (or just A = A
For some
s > l(A)
or
A+-c.c. A-complete
-
o f power
~
,
f p " 2A = ~
and f o r some g r a p h
G
o f power
2 G ~ (Kk(,)) P
(i) (ii)
G
embeds no
Proof:
Zk(,)+l ."
The forcing
of vertices of
G
P
introduces just the graph
G .
]G[ , the set
Let
be m
[m]inc = {(~0 .... ' ~ - 1 ) : ~O<'''<~m-1 < ~} " m
We say
~/ = (c~0 ..... ~m-I ) ' ~ = (~0 ..... ~m-I )
connected if Let
s O < ~0 ( ~I ( ~I < ' " < a m - I
P = {G : Kk(,)+l m
[dom(G)]in c G 1 < G2
where
dora G
is a subset of
and
]P[ = ~ .
such that
Let
~r
It is a graph of the right form.
By the choice of [U I = A
(U
Is = {s E U : Is[ < 2m}
~
~r
to
#
[~]inc
and
are potentially
(or interchange them). and
of power
m
function from the set of edges of enough.
s
G
G 1 = G 2 t [dom Gl]in c .
fp "2 A = ~"
U{L : L E ~p} .
< ~m-I
is not embeddable into
if and only if
P ~ A-c.c.,
from
G
is a graph as above on < A} .
Clearly
P
We say is A-complete,
be the P-name of Let
~
p E P •
be a P-name of a Let
X
be large
and the partition theorem, we can find
is really larger but this does not help). and let
{M s : s E Is}
be such that
U C
Let
U 0 Ms = s ,
189
(Va,~)[a < ~ h
a E u ^ ~ E U~
(a,~) N MS = ~] ; M s N s t = MsN t
ss N M tcssn t) (VaCSs)(lal < A Isi
s,t e
Itl
=
[p,P,A,p,le.,~r,dE
As
Now we want to find Let
=
We
Hs,t(s) = t
an isomorphism onto, so that
diagrams commute and
Kk(,) .
aeS s) and llMsl I=A andfo
we have
Mt
Hs, t : M s
(or just
e
p _( q E P
e
a l l the
M s] •
such that
e
(aO,a I, .... am_ 1 ) E [U]m
q
for
forces a monochromatic
e < k(*)
such that
2 .aok(*)-I < aO < a l < . . . a k ( * ) - l < . . . < a . e E U , aO < aol < ao<." t e = range
We
We shall find a condition r E MtoUt I N P
and
q > p .
i.
r ~ (~O,~l) E edges of
2.
r ~ Sto ~ q
3.
r t Mtl >_ hto,tl(q)
4.
r ~ ~ (WO,~I)
5.
r F Vx,y E vertices (q)~ i f
6.
if
Gr
q" - ~
and As
P
i - 4 and:
(x,y) E edges
P ~ qo E P N Mto
above for
~ = ~0 " Note is
~r].
q' | M~ = q" r M~ r'
[x E vertices
rI ~
is A-complete also
N~ 0
and
then we can find
such that
~r N M ~
x,y ~ ~
(x,hto,tl(Y)) ~ edges of
r t Mto _< q' E P N Mto
satisfies
then we can find
=
r | Mtl K q" E P N Mtl rI
q E P N Mto , p ~ q
~ < p such that i. - 6. below holds, where
[{x,y} ~ {TO} ~
and
If
such that
P N Mto
and:
ql q, K r' E MtoUt I
q' - M~
(xy) = (WO,WI)
and
and
y E vertices
r' F "~(WO,~I) = ~"
is p+-complete so there are
Vq : qo ~ q E P N Mto
t "the distance in
~r
of
70
we can find
~0 < ~
and
qo,~o , r
as
from vertices in
> 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
$
isomorphic t o
Kk(,)+l .
for
i < k(*)).
So n e c e s s a r i l y f o r some
t h e d e f i n i t i o n o f " p o t e n t i a l edge" and as
~i(1)(1)) from
M~ .
i s disjoint to So
Do,,
i(1)
, ~i(1) E ~ •
(m ~ 2 and) the i n t e r v a l
M~ , we have: ~i(1)
~ N M~ = ~ .
and t h e r e s t r i c t i o n s
Hence i t has no subgraph Now by (~i(1)(0),
i s not connected to any vertex
Now c o n s i d e r t h e sequence
u j
o f t h e graph
q
t o them.
: i Easily the first
i s in
P , and
in each s t e p we use e d g e l e s s amalgamation (we c o u l d have s t a r t e d with t h i s
192
argument) so we finish.
Concluding Remarks: 9
5.2
Easy variants:
subgraphs of
G
We can have
is
S
G ~ (H)~
such that the family of finite
(up to isomorphism) where for some
n :
i. s ~ ¢ 2.
S
closed under edgeless 8~nalgamation
3.
If L I .... ,LIH I e S ; i ~ j ~ ×i E L i
and the distance of
Li n L J xi
from
= L ; L
in
Li
is
> n
then
n
L* E S
where:
edges(L*) =
vertices (L*) =
U vertices (Li) i=l
n U edges(L i) U edges(L* ~ {x i : i = 1 ..... [HI}) i=l
L* r {Xl,. 'x'H')l I ~H 5.3
Easy Remark:
Instead of graphs we can have a model where relations are a
partition of the singleton and of the pairs. 5.4
Note that the proof of 5.I tells us that in 4.2 for
n = 3 (i.e. coloring
of singletons and pairs) we do not need 1.4 but can directly prove hence lowering the required cardinal. 5.5
On generalizing 5.2 to relation and colorings with more places see later
works.
193
References [I]
A. Hajnal and P. Komjath, Embedding graphs and colored graphs.
[2]
S. Shelah, Notes on combinatorial set theory.
Isr=el J. K61~. 14 (1973)
262-277.
[3]
S. Shelah, Was Sierpinski right ? I.
[4]
S. Shelah, Was Sierpinski right ? II.
IS]
J. Nesestril and V. Rodl, Partition (Ramsey) theory, a survey.
Isr=el ]. 14tk. 62 (1988) 355-380. Preprint. Colloq.
Math. Soc. Janos Bolyais Vol. 18, North Holland, Amsterdam, (1978),
75-192. [Sh95] S. Shelah, Canonization theorems and a p p l i c a t i o n s , J. o f Symb. Logic 40 (1981) 345-353.
[EI~tR] P. Erdos, A. Hajnal, A. Mate and R. Rado, Combinatorial Set Theory: Partition relations for cardinals.
D i s q u i s i t i o n e s Mathematicae
Hungaricae 13, Akademiai Kiado Budapest, 1984, North Holland. [Sc Sh20] J. Schmerl and S. Shelah, On power-like models of h y p e r i n a c c e s s i b l e c a r d i n a l s , J. of Symb. Logic 37 (1972) 531-537.
Topological Problems for Set-theorists Franklin D. Tall I University of Toronto This short note is aimed at set-theorists who have heard there are interesting applications of set theory to topology but are perhaps deterred byanoverabundance of terminology e.g., is every weakly ~8-refinable space with a regular G6-diagonal submetrizable? We shall introduce some of the classic problems of set-~heoretic topology, explaining the concepts involved, briefly outlining what is known, and referring the reader to relevant literature. We make no attempt to be comprehensive. We assume no more topology than the reader can be expected to recall from the one course taken in graduate school many years ago. a) b) c) d) e)
Our criteria for selection (with occasional exceptions) are that a problem should have been around for at least a decade, have been worked on bymore than one strong researcher, be well-known t o set-theoretic topologists, require a minimum of topological knowledge to state and work on, be apparently set-theoretic in nature.
In addition, we have attempted to select problems from a variety of areas within set-theoretic ~opology. Those who wish to see large numbers of additional problems may consult the Problems Section in back issues of Topology Proceedings. I should like to thank Jim Baumgartner for inviting me to speak in his seminar on this topic, which led me to compile this assortment. Input from the members of the Toronto Set-theoretic Topology Seminar has also been helpful. At the su~estion of the referee, I have omitted well-knownproblems which have been surveyed elsewhere: see [To2] and [R], [Wi] and [vD], [Ru], INs] and IV] for L ~ S spaces, box products, Dowker spaces, countably compact spaces respectively. 1.
The Cardinality of Lindel6f T9 spaces with points G~, and a related problem. Arhangel 'skii proved that Lindelgf T2 first countable spaces have cardinality
< 2R0. A natural question is whether first countability can be weakened to "points G6". (This condition is equivalent to first countability in compact T2 spaces.) It is not difficult to show that even if only "TI" is assumed, such spaces cannot have weakly compact cardinality, or cardinality _> 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
_<~} +
~o.
This cardinal function (i. e. function from the class of topological spaces into the class of cardinals, invariant under homeomorphism) is known as the Lindel6f degree.
195
Shelah's example (or rather, two versions of it) partially solve another problem on LindelOf spaces: how big is L(X x Y),where X and Y are LindelSf? There is an easy example of a (hereditarily) LindelOf space X such that L(X~ = 2~°: X is the Sorgenfrey line, i.e. the real line equipped with a basis of all [a,b). The line y =-x in the plane is closed discrete; its complement plus the open sets witnessing discreteness formanopen cover with no subcover of size <2~0. One might conjecture that that is as bad as matters get, that if X and Y are LindelSf, L(X x y) < 2R0 . This may well he consistently true, but it is consistently false. See [J2] for the example where L(X x y) - ~2 > 2~0 = ~I. Indeed, the only upper bound known for L(X × Y) at present is the first strongly compact cardinal. (Proof: generalize Tychonoff's Theorem.) Surely this can be improved. 2.
Linearly LindelSf $PaC~.
Definition. A point x is a complete accumulation point of a set Y if for each open U containing x, 3U n Y3 = 3Y3. It is not difficult %o show that a space is compact if and only if each infinite set has a complete accumulation point. ~fini~ion. A space is finally compact in the sense of complete accumulation points (FCCAP), if every uncountable set has a complete accumulation point. It is not difficult to show that LindelOf implies FCCAP, but the natural attempt to prove the converse encounters difficulty with covers of size R . Indeed, there is a completely regular counterexample [HI. However the assumption that every sequence {Fn}n< ~ of closed sets such that F n D Fn+ 1 and n Fn = ® can be fattened to a sequence -
n<~
~Un}n< ~ of open sets such that Un D Un+l, Un 3 Fn, and O U n = ® is sufficient to n<~ reduce covers with size of countable cofinality to ones with regular size, and hence get that FCCAP implies Lindel6f. The open question is whether there is a normal FCCAP space that is not Lindel~f. For those who know the terminology, this is asking for a special kind of Dowker space [Ru]. It is interesting that FCCAP has an equivalent form that appears even closer to LindelOf. Definition. A space is linearly Lindel6f if every well-ordered by inclusion open cover has a countable subcover. To work on this problem~ the minimal topological background can be found in [M] and [H]. 3.
Non-ArchimedeansDaces.
Definition. A space is non-Archimedeanif it has a basis ~s.t. B,B' ~ implies B ~ B ' or B' C B or B n B ' =®. Acollection ~/ of subsets of aspaceis poin$-countable if each point is in at most countably many elements of ~'. A space is perfectly normal if i¢ is normal and closed sets are G~'s.
196
The linear order obtained by squashing a Souslin tree is anexample of a perfectly normal non-Archimedeannon-metrizable space with a point-countable base of size ~,. Todordevi6 [To,] proved that, assuming MA plus the non-existence of weak Kurepa [also known as Canadian] trees, every perfectly normal non-Archimedeanspace of weight <~! is metrizable. The question is whether MM, for example, allows us to drop the weight restriction, even adding "point-countable base". For further information, also see [NI] and [N2]. 4.
Reflection problems.
In recent years there has been a growing interest in problems that ask whether, if all small subspaces of X have a certain property, then X does. Here is a typical one: Problem. If X is first countable and every subspace of size R 1 is metrizable, is X metrizable? A non-reflecting stationary set of ~-cofinal ordinals is a counterexample [HJI]. The non-existence of such sets (which holds in the model obtained by L~vy-collapsing a supercompact to w2 [B] implies there is no counterexample in which each point has a neighbourhood of size ~RI [Do]. Replacing . R1 ,, by . <2 ~o,, yields a positive answer if supercompact many Cohen reals are adjoined [DTW]; in the given problem we are missing a preservation lemma. Another problem of the same sort presumably requires less topological knowledge. Definition. A space is g-collectionwise Hausdorff if for each closed discrete subspace {x }~
Omitting Cardinals.
There is a problem-generating machine manufactured in Hungary, into which one inputs a cardinal function and two cardinals and outputs a problem. We mention only one such: does every Lindel6f space of cardinality 22R0 have a Lindel6f subspace of cardinality 2R°? See [JW]. 6.
Partition problems.
W. Weiss and others have extended the partition calculus to topological spaces, thereby creating a host of new theorems and problems. For a survey, see [We]. One of the more interesting questions is the following:
197
Problem. Can every Hausdorff space be partitioned into two pieces, neither of which includes a Cantor set? V = L implies "yes" [HJW], but perhaps it's just true in ZFC.
7.
Can ~* be homeomorphic to ~T*?
This problem and the next one are purely set-theoretic but arise frequently in . topological contexts, w is the Stone-Cech compactification of w, with the natural copy of ~ that lies within removed. Similarly for Wl*. By Stone duality, the question of whether the two spaces can be homeomorphic is equivalent to asking whether the Boolean algebras ~w)/Fin, 5~Wl)/Fin can be isomorphic. In all familiar models the answer is no; in particular MAand 2~0 < 2R1 provide negative answers. Although many people have worked on this, little is known. The completions of the two algebras are isomorphic [BW], but that doesn't seem to help. For pairs of distinct infinite cardinals ~,I, it is known that ~* and ~i are the only infinite ~ • and could conceivably be homeomorphic [BF].
8.
Can there be a small dominating family in The structure of ~
AS
that
wlw?
is by now reasonably well understood, but the same cannot be
said for ~ , ~ an uncountable cardinal.
In particular, it would be interesting to
known if it's consistent that there be a family ~9"of fewer than 2~I functions from w! to ~, such that any function from wt to w is exceeded everywhere by some member of J. An example of the topological import of this question can be found in [W2]. Stepr~ns [S] and Jech-Prikry [JP] showed that there is no such small dominating family if cf(2 ~0) rain (2 ~1 , ~Wl), while the existence of such a family implies the existence of a measurable cardinal in an inner model. 9.
Are locally compact normal metacompact spaces paracompact?
There used to be many problems in general topology which asked for the implications among rather basic properties. Virtually all have been settled now, often being shown to be undecidable. One of the few that remains is the one mentioned above. Recall Definition. A space is metacompact (paracompact) if every open cover has a point-finite (locally finite) open refinement. Several things are known. To obtain a positive answer, it suffices to prove collectionwise Hausdorffness. If "metacompact" is strengthened so that for each open cover there is an n ~ w such that it has a point-n refinement, again the answer is positive [D]. Under V = L, the answer is also positive [WI]. If there is a counterexample, there is one which is a subspace of PR(C) for some g [D]. C is with thecofinitetopology.
PR(C ) is the topology on the collection of non-empty
198
finite subsets of ~ with basis of sets [A,U] = { B ~ [~]<~: ACB_CU}, where Ac [~]<~ and U is open in C ,
10.
Compact spaces with hereditarily normal powers.
A curious theorem of Katevov [K] states that compact spaces with hereditarily normal cubes are metrizable. One naturally wonders whether "squares" suffices. Under MA (with or without~CH), the answer is "no", but whether the question is undecidable is unknown. [GN] surveys the situation. In conclusion, I hope that this list of easily stated problems will stimulate even more set theorists to lend their talents to the problem-strewn field of set-theoretic topology.
Footnote 1.
The author acknowledges support from Grant A-7354 of the Natural Sciences and Engineering Research Council of Canada.
References
[A]
C.E. AulI, Some base axioms for topology involving enumerability, 54-61 in Gen. Top. and its relations to Modern Anal. and Alg. (Proc. Kanpur Top. Conf., 1968), Academia, Prague, 1971.
[AT]
U. Abraham and S. Todor£evid, Martin's Axiom and first countable S-and L-spaces, 327-346 in Handbook of Set-~heoretic Topology, ed. K. Kunen and J.E. Vaughan, North-Holland, Amsterdam, 1984.
[B]
3.E. Baumgartner, A new class of order types, Ann. Math. Logic 9(3) (1976) 187-222.
[BF]
B. Balcar and R. Frankiewicz, To distinguish topologically the space m , II, Bull. Acad. Polon. Sci. S4r. Mat. Astronom. Phys. 26 (1978) 521-523.
[BW]
S. Broverman and W. Weiss, Spaces co-absolute with ~N-N, Top. AppI. 12 (1981) 127-133.
[D]
M. Daniels, Normal locally compact boundedly metacompact spaces are paracompact- an application of Pixley-Roy spaces, Canad. J. Math. 35(1983) 827--833.
[Do]
A. Dow, An introduction to applications of elementary submodels to topology, Top. Proc., to appear.
[vD]
E.K. van Douwen, Covering and separation properties of box products, 55-130 in Surveys in General Topology, ed. G.M. Reed, Academic Press, New York, 1980.
199
[vDTW] E.K. van Douwen, F.D. Tall, W.A.R. Weiss, Nonmetrizable hereditarily Lindel6f spaces with point-countable bases from CH, Proc. Amer. Math. Soc. 64 (1977) 139-145.
[DTW]
A. Dow, F.D. Tall, W. Weiss, New proofs of the consistency of the normal Moore space conjecture, Top. AppI., to appear.
[FI]
W.G. Fleissner, Separation properties in Moore spaces. Fund. Math. 98 (1978) 279-286.
[F2]
W.G. Fleissner, The normal Moore space conjecture and large cardinals, 733-760 in Handbook of Set-theoretic TopoloKT, ed. K. Kunenand J.E. Vaughan, North-Holland, Amsterdam, 1984.
[F3]
W.G. Fleissner, Left-separated spaces with point-countable bases, Trans. Amer. Math. Soc. 294 (1986) 665-678.
[GN]
G. Gruenhage and P.J. Nyikos, Normality in X2 for compact X, preprint.
[H]
R.W. Heath, Screenability, pointwise paracompactnessaad metrization of Moore spaces, Canad. J. Math. 16 (1964) 763-770.
[Ha,]
A. Hajnal and I. Juh~sz, On spaces in which every small suhspace is metrizable, Bull. Acad. Polon. Sci. S4r. Math. Astronom. Phys. 24 (1976) 727-731.
[Ha2]
A. Hajnaland I. Juh~sz, LindelSf spaces ~ la Shelah, Coll. Math. Soc. J. Bolyai 23 (1978) 555-567.
[HJW3
A. Hajnal, I. Juh~sz, W. Weiss, Ramsey type theorems for topological spaces, in preparation.
[Ho]
N.R. Howes, Ordered coverings and their relationship to some unsolved problemsin topology, 60-68 in Proc. Washington State U. Conf. on Gen. Top., March 1970, Pullman, Washington, 1970.
[J]
I. Juh~sz, Cardinal Functions II, 63-110 in Handbook of Set-theoretic Topology, ed. K. Kunenand J.E. Vaughan, North-Holland, Amsterdam, 1984.
[aP]
T. Jechand K. Prikry, Cofinality of the partial ordering of functions from ~I into u under eventual domination, Math. Proc. Camb. Phil. Soc. 95 (1984) 25-32.
[aWl
I. Juh~szand W. Weiss, A Lindel6f scattered space that omits ~, Top. Appl. (to appear).
[K]
M. Katetov, Complete normality of Cartesian products, Fund. Math. 36 (1948) 271-274.
[L]
L.B. Lawrence, The box product of countably many copies of the rationals is consistently paracompact, preprint.
[M]
A. Mi~enko, Finally compact spaces, Sov. Math. Dokl. 145 (1962) 1199-1202.
[N,]
P.J. Nyikos, Some surprising base properties in topology, 427-450 in Studies in Topology, ed. N. Stavrakis, Academic Press (New York), 1975.
IN2]
P.J. Nyikos, Order-¢heoretic basis axioms, 367-397 in ~urve¥s in General fToDology, ed. G.M. Reed, Academic Press, New York, 1980.
200 [N3]
P. Nyikos, Progress on countably compact spaces, 379-410 in General Topology and its Relations to Modern Analysis and Algebra VI, Proc. 6th Prague Top. Symp. 1986, HeldermannVerlag, Berlin, 1988.
[R]
J. Roitman, Basic S and L, 295-326 in Handbook of Set-theoretic Topology, ed. K. Kunen and J.E. Vaughan, North-Holland, Amsterdam, 1984.
[Ru]
M.E. Rudin, Dowker Spaces, 761-780 in Handbook of Set-theoretic Topology, ed. K. Kunenand J.E. Vaughan, North-Holland, Amsterdam,.1984.
IS]
S. Shelah, Remarks on l-collectionwise Hausdorff spaces, Top. Proc. 2 (1977) 583-592.
[st]
J. Steprhns, Some results in set theory, Thesis, University of Toronto, 1982.
[TI]
F.D. Tall, Normality versus collectionwise normality, 685-732 in Handbook of Set-theoretic Topology, ed. K. Kunenand J.E. Vaughan, North-Holland, Amsterdam, 1984.
[T2]
F.D. Tall, Topological Applications of Supercompactand Huge Cardinals, 545-558 in General Topology and its Relations to Modern Analysis andAlgebra VI, Proc. 6th Prague Top. Symp. 1986, Heldermann Verlag, Berlin, 1988.
IT3]
F.D. Tall, Topological applications of generic huge embeddings, Trans. Amer. Math. Soc., to appear
[Tol]
S. Todordevid, Some consequences of MA + ~wKH, Top. Appl. 12 (1981) 187-282.
[ToD
S. Todor£evid, Partition problems in general topology, American Mathematical Society, Providence, 1988.
IV]
J.E. Vaughan, Countably compact and se@uentially compact spaces, 569-602 in Handbook of Set-theoretic Topology, ed. K. Kunen and J.E. Vaughan, North-Holland Amsterdam, 1984.
[Wl]
S. Watson, Locally compact normal spaces in the constructible universe, Canad. J. Math. 34 (1982) 1091-1096.
[w2]
S. Watson, Separation in countably paracompact spaces, Trans. Amer. Math. Soc. 290 (1985) 831-4342.
[We]
W. Weiss, Partitioning topological spaces, in Mathematics of Ramsey Theory, ed. J. Nesetril and V. P~dl, to appear.
[wi]
S.W. Williams, Box products, 169-200 in Handbook of Set-theoretic Topology, ed. K. Kunen and J.E. Vaughan, North-Holland, Amsterdam, 1984.
A Beginning For Structural Properties of Ideals on
P K
William S. Zwicker Department of Mathematics Schenectady, New York 12308 U.S.A. §0 Introduction This paper flows from the observation that coding sets, introduced in provide a means by which one can begin t o carry out for
[Zl],
PK~ a program analogous
to that which Baumgartner, Taylor, and Wagon developed for an uncountable cardinal K in t h e i r paper, "Structural Properties of Ideals" is provided by D. Carr's papers ([CI],[C2],[C3]),
[BTW]. Essential background
which also develop the basic
notation for ideals on PK~, and of course by Jech [ J l ] . Our approach w i l l be to view PK~ as a set p a r t i a l l y ordered by ~ ; w i l l always be cardinals with
K regular and ~ # ~.
be assumed to be K-additive extensions of sets of
IK~, the ideal of "not unbounded" sub-
PK~, where "unbounded" means cefinal.
"nubound" - an element of
K and
All ideals on PK~ w i l l
Thus for any
PK~ that no element of
X £ IK~,
X has a
X properly contains.
on ~ w i l l likewise be assumed to be K-additive extensions of
Ideals
IK, the ideal of
bounded subsets of K. The f i r s t conundrum facing a theory of ideals on
PK~ is that any reasonable
definition of "selective" (which has occurred to any one!) f a i l s to be satisfied by the ideal
NSK~ of non-stationary subsets of
PKg. Since NS ~
is the proto-
typical normal ideal on P ~ , and one certainly want normal ideals to be selective, K this ~ a problem. One solution is to strengthen the definition of "normal" on PK~ so that i t does imply selective; the cost is that stronger property.
NS~
cannot have the
T h i s approach begins to seemmore natural when one recalls that
while every normal measure on K has the partition property, such is not the case on
PK~ [rl] - this result had shed doubt on the traditional definition of
PK~
normality some time ago.
§I A regressive function
Normalitz and Selectivity f
on a set
each non-empty x E A,
and an ideal
sive function on a set
A in
I
A ~ PK~ is one satisfying
f(x) E x
for
on pK~ is (point) normal i f every regres-
I + is constant on some set in
I +.
These are the
usual definitions for "regressive" and "normal" found in the l i t e r a t u r e . Theorem l . l of
[D. Carr]:
N~S~I ~B i s (point) normal for any stationary subset,
_PKX, and every normal ideal on
PK~ extends NS ~
(see [Cl]).
B,
202
Thus, the traditional notion of normality on PK~ does y i e l d the analogue to Neumer's theorem on K.
For A and B subsets of
to be set-relressive i f
f(x) ~ x
some element of If
I
for
B.
P~
we w i l l define a set
B c P ~ to be a witness index
if (1)
and
B E I* ,
(2)
For every set
I ~ , every set-regressive f : A ÷ B is
A in
constant on some set in and
(3)
An ideal
I
I+ ,
{y E BIU(B~y) = y} E I*. for which a witness index exists is said to be set-normal.
Proposition 1.2 ~roof:
f: A + B
A which properly contains
For B c PK~ and y E P ~, B~y w i l l denote BO{x E PK~Ix~y}.
is any ideal on I
PK~ we w i l l define
for each element of
If
I
is a set-normal ideal, then ~ i s (point) norm~l.
Use property (3) above to replace a (point) regressive function
set-regressive function g satisfying
f
with a
f(x) E g(x), then apply K-additivity of
I.
The definition of set-normality goes back to Jech who, in [J2], noted that a (point) normal measure on PK~ is set-normal i f and only i f i t has the p a r t i t i o n property.
(He could afford to leave condition
An ideal domain P ~
I
(where f
there is a set for every
(3)
off.)
on PK~ is selective i f for every if
Y E I*
I-small means that
on which f
f-l[f(x)] E I
is comparably l : l ,
x,y E Y which are comparable in the ~
terms of partitions is that for any partition exists a set
A in
we to demand that
I*
I-small function
P of
with
for each x E Pm~)
which m6ans that f ( x ) # f(y)
ordering.
The equivalent in
PK~ into sets in
which intersects each piece of
f
I
there
P in an anti-chain,
Were
A meet each piece in a singleton, the property would be impossi-
ble to satisfy when K < ~, as witnessed by the partition
p~ =C]K'I(~),
where
K(x___L): U (x n ~). Proposition 1.3 proof:
If
=lany
Let
is a set-normal ideal, then I
The proof is just as in the
index B and f k(y)
I
K case.
If
I
is set-normal with witness
is any function with domain PK~k ,
x~ y
with
is selective.
x E B and f(x) = f ( y ) ,
define
k
as follows:
i f such an x
exists
lLundefined, otherwise.
A = {y E P ~Ik(y)
on B - A E I*.
If
is defined}.
If
AE I
A E I +, choose D E I + with
constant on D, and so
f
is not
I-small.
then f
is comparablyone-to-one
k constant on D.
Then f
is
203
Proposition 1.4
When
> m , NS ~
is not selective, hence not every (point) nor-
mal ideal is selective. proof:
Baumgartnerand Velleman, independently, have shown that when ~ > K, the
m function (introduced above) can not be comparably one-to-one on a cub (closed and unbounded subset of P X - see K on an unboundedsubset of P ~.
[ZI]).
T h i s function can clearly not be constant
We'd l i k e a theorem for set-.normality analogous to theorem I . I , and coding sets appear to provide the key. quently exist.
In particular, they show that set-normal ideals fre-
A subset A of
one-to-one function
P~
is a coding set i f i t comes equipped with a
K
c : A + ~ such that for
A stationary coding set, or Velleman and Stanley
x,y E A , x ~ y
[V]
and Shelah [Sl and $2] have shown that
tently exist for a broad variety of
K's
and
~'s
SC's exist for
cardinal ~ ~3"
in conjunction with
Note that
1.3
SC's consis-
under various hypotheses. Fur-
thermore, Shelah has shown that for
i f and only i f c(x),E y.
"SC", is a coding set that is stationary.
P ~+ whenever K is a successor 1.5
and 1.4
implies that
~ > 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<
B
for some SC, B. .Co.rol.lary l.G
If
strictions of
NS ~
PK~ has an unbounded set of size ~, then any set-normal reis equal to
NSK~I B f o r some SC,B. n
The extra hypothesis is f a i r l y mild; i f
~ = ~ ,K
then PK~ has an un~ounded subset of size ~. naturally in the proof of ... ~
i
f
A E I*
We w i l l call an ideal
given any
(we'll say A outstrips
i f there exists a set
. . . , for
n Em,
I
on PK~....
f : PK~ ÷ PK~ there is a set
such that for each x,y E A,
f(x) ~ y . . . lean
1.5.
.... ~
Someinteresting definitions arise
A E I*
... and K-footed i f there exists a set
if
x~ y
then
f), with
JAJ = ~,
A E I* such that
JAqxl < K for each x E PK~ . The last property is closely related to one clause in the definition of simplified morass (see [V2]). Proof of 1.5
If
B is a SC t h e n NSK~ I B is (point) normal.
To see that i t
is set-normal, l e t
A E (NSK~I B)+ and l e t
tion.
is a (point) regressive function on B h A, so i f ( c o f ) ( x ) = c ( yo)
Then c o f
f : A + B be any set-regressive func-
204
for each x
in
D E (NS ~ I B) +, f
Now assume that final subset of
I
is constant on
D with value Yo"
is set-normal with witness index B and that
PK~ with
)Y)= ~.
there would be some x £ PK~ and
B must witness i : K ~
I's
B I x.
Y is a co-
K-footedness, for i f not
Then i o K would be set-
regressive on B ~ R (where ~, the cone a~ove x, is
{y E P ~I X c Y } ) , but i o K
can never be constant on a cofinal set. I t follows that
I
is lean, since
of cardinality less than K. be arbitrary and define k any k(y) =
x~
y
Also,
I
B = U B1x is the union of h-many sets, each mustxEY be outstripping; l e t
f : PK~÷ P~
by with
x E B
proper subset o f
and
y, if
f(x)
not a
such an
x
exists
undefined, o t h e r w i s e . Then
A = {y
k(y)
is undefined}
must be in
could apply s e t - n o r m a l i t y to get an possible.
Now A ~ B
To f i n i s h , l e t and such that
outstrips
X E I÷
I*,
for if
on which
k
P ~-A
were in
I +, we
is c o n s t a n t , which is im-
f.
c : B + ~ be any 1 : I
A outstrips the function
function and choose A c B with
h(x) = x U{C(X)}.
where C = {y E PKXI(V~ E l ) ( c - l ( ~ ) ~ y ) } ,
then as
I f we set
A E I*
E= A~ C
C is a closed and unbounded
subset of
P ~ and as I is (point) normal, E E I* must be stationary. K coding set by construction and I extends NSK~IE, so we are done.
E is a
This proof breaks down into a series of lemmas. Rather t~an state them we express the relationships with a diagram. N.B. In Figure I , ,below, a single arc joining two arrows indicates that the co~j~no.txLo~of the two arrow sources implies the target. Some Properties o f
PK~X Ideals (Figure I)
K-footed (point) normal
outstrfppin9
~
lean
concentrates on a coding set extends NSK~ K-f0oted and lean ( t )
extends
NSK~1B for some SC,B
* i f PK~ has an unbounded set of cardinalfty ~. (+) This is immediate since the coding set B must have )B[ < ~ and )Bixl < K for each x ( PK~
205
§2 The Q-Point Puzzle Recall that an ideal, I , on ~ domain ~
is
l : l
K into sets in singleton.
on some set in
IK,
is a q-point i f every IK-small function f with I*.
there is a set
Equivalently, given any partition
A in
I*
intersecting each piece of P in a
I f we specialize the d e f i n i t i o n of outstripping to
K in the obvious
way, by requiring for each f : m ÷ m the existence of some set for each m,B E A with ideal
I
A E I*
then, to define an ideal partition of
such that
m < B,f(m) < B, then we w i l l show in Theorem2.1 that an
on K is outstripping i f and only i f i t is a q-point.
with domain PK~
P of
I
on
PKX to be a q-point i f every
is comparably l : l
on some set in
PK~ into sets each of which is in
intersecting each piece of
I ~-small function f
Equivalently, given a
IK~ , there exists a set
P in an anti-chain.
be a stronger property than q-point.
I*.
I t is natural,
A in
I*
On PK~, outstripping appeoJts to
Is i t actually stronger?
This is the q-point
puzzle, and i t is open. Attempts to solve i t produced much of the rest of this paper as spin-off. Theorem 2.1
On K , ~-p0intedness is equivalent to outstripping.
On P ~ , an out-
stripping ideal is necessaril Z a q-point. Theorem 2.2
If
P, is an outstripping ideal on ~
I
well-founded subset of Theorem 2.3
If
I
then
I
concentrates on a
P
extends NSK~I B for some SC, B, then I
is outstrip~i.ng.
(This is the analogue of the theorem on K which says that any extension of
NS
i s a q-point). Corollary 2.4 extension of A in
(Immediatefrom 2.3 and figure I ) . NS~,
I
For an ideal
I
that is a lean
is outstripping i f and only i f there exists some coding set
I*.
The reader who attacks the q-point puzzle might wish to f i r s t think about the following, presumably easier, question:
Can "q-point" replace outstripping in 2.2 or 2.4?
In this connection, a tempting conjecture is that an ideal which is both a q-point and concentrates on a coding set must be outstripping, but I have been unable to show this. Proof o f 2.1
Let
no , nl . . . . . q~ . . . . f
I
be a q - p o i n t on
enumerate a closed
(~ < q~ + f ( ~ ) < n~).
s m a l l , and i f
A
Let
K
and
f:K
be a r b i t r a r y .
Let
and unbounded set o f clo~ure p o i n t s f o r
g(~) = the l e a s t
i s any s e t on which
+ K
g
is
~
1 :I,
with A
~ < n~.
Then
clearly outstrips
g
is f.
I
-
206
If
I
is an outstripping ideal on P ~ and f
fined on PK~, define A E I*
is an
I .-small function de-
h :PK~ ÷ P ~ by h(x) = any nubound for
which outstrips
h.
Then f
is comparably l : l
f" I f ( x ) ] .
on A.
Choose
The proof in the
K case is the obvious specialization. Proof of 2.2 subset
Y of
First note that i t is always possible to build a well-founded cofinal P~.
Enumerate PK~ by its cardinality.
meration, throwing an element into ready in
Y.
Now l e t
I
by f(x) = any element y strips
f
be an outstripping ideal on PK~ and define f : PK~ ÷ PK~ of
Y with
x ~ y.
If
A is a set in
then A is well-founded, since a descending chain in
interlace with a descending chain in Proof of 2.3
I*
which out-
A would perforce
Y.
I t is easy to see that any extension of an outstripping ideal is out-
stripping (and any extension of a q-point is a q-point). is outstripping when B is a NSK~ I B
Then walk through the enu-
Y only i f i t is not a subset of any element a l -
SC,
for some SC, B, and l e t
We already know NSK~ B
so we are done. Alternately, assume I extends f : PK~ ÷ P ~.
Choose ~
to be any function
satisfying ~ : P ~ + B and ~(x) ~ f(x) for each x E P ~. Define k : ~ + ~ by K k(~) = c ( f ( c - l ( ~ ) ) ) , where c is B's coding function and ~ E c"B. Let C c PK~ be the cub of a l l elements of is in
P ~ closed under k. K
Then CC~ B outstrips
f
and
I*. ~3 Quasi-Normality and P*-points Recall that an ideal
I
on K is a p-point i f for every A in
i f for every
with domain K there exists a set
which
f
I -small function K is one-to-one, and I
domain K there exists a set
f
I*
is selective i f for every A in
I*
on which f
a p-point ideal allows, for any p a r t i t i o n of of a set
A in
I*
for any partition of
on which f
A in
I-small function
K into sets in
f
I -small, a q-point
is one-to-one.
intersecting each p~ece in a set in K into sets in
is
I-small function
with domain K there exists a set
I* on f
with
Equivalently,
I, for the existence
IK, a q-point ideal allows,
IK, for the existence of a set
A in
I* in-
tersecting each piece in a singleton, and a selective ideal allows, for any p a r t i t i o n of
K into sets in
piece in a singleton.
I , for the existence of a set
A in
I*
intersecting each
Thus i t is immediate from the definitions that an ideal on K
is selective i f and only i f i t is both a p-point and a q-point. ideal
I
on PK~ to be a p-point i f for every I-small function
P ~ there exists a set
A in
I*
on which f
is
I f we now define an f
with domain
IK~-small, i t is equally imme-
diate that an ideal on P ~ is selective i f and only i f i t is both a p-point and a K q-point. These relationships prompt several natural questions. Outstripping appears to be a strengthening of q-pointedness in the
PK~ context; are there analogous
207
strengthenings of p-pointedness and of selectivity? Were they already known in the context?
Will they shed l i g h t on the q-point puzzle?
There are indeed such analogues; in
[W]~ Weglorz defined quasi-normality for
an ideal on K and proved i t equivalent to s e l e c t i v i t y .
This turns out to be the
K-analogue of the "strong" version of selectivity.. I have found no direct reference in the l i t e r a t u r e to "p*-point", the
K-analogue of the "strong" version of p-point.
Its introduction is probably worthwhile, because i t suggests a reconsideration ( s t i l l in the
K context) of Weglorz's proof that s e l e c t i v i t y implies quasi-normality,
yielding a new proof on K which seems to me to be more transparent than the o r i g i nal. In Figure 2, below, a double arc linking arrows indicates that the conjunction of the two arrow sources is equivalent to the target.
In the
K-context, the new
P~perties o f Ideals on K (Figure 2)
Ipoiog
selective
proof of
c
and its converse consists of the observation (immediate from the d e f i n i -
tions) that
p*
and outstripping f i t together to form quasi-normallty (in much the
same way that p-point and q-point f i t together to form s e l e c t i v i t y ) , together with proofs of
a,b
and t h e i r converses.
In the
PK~ context,
a
and b
(and hence c)
are proved just as on K, while a l l three converses are open questions.
Thus the
attempt to shed l i g h t on the q-point puzzle has yielded additional puzzles. Definitions and Proofs for
K:
An ideal
I
on K is quasi-normal i f for every
sequence ~ B ~ < K of sets in
I
~,B E A with
( I t is equivalent to ask for a set
~ < B, B ¢ B .
there exists a set
A in
I*
such that for every A'
in
I*
for
which
V B E I , where 9 B = {~ < KI for some ~ E A' with ~ < B, B E B }; sEA' ~ ~EA' ~ this is Weglorz's original formulation). An ideal I on K is a p*-point i f for every sequence ~ B ~ < K of (not necessarily disjoint) sets in set with
A in
I* and a progressive function
f(~) < B,B E B , where f
immediate, then, that . . .
I , there exists a
f : K ÷ K such that for every ~,B E A
is progressive i f
f(~) ~ ~ for each ~.
I t is
208
Observation 3.1 pin 9 and~ ~ Theorem 3.2 proof:
An ideal, ~, on K is quasi-normal i f and only i f i t is outstrip~ .
An ideal
Assume I
B'
on m is a p-point i f and only i f i t is a p*-point.
is a p-point, and l e t
each of which is in the
I
I.
D i s j o i n t i f y the
as a partition of
B by setting
m into sets in
I
A with
B'(%
:
_
B
subsets of
8~J B8.
Viewing
and applying the formulation of
p-pointedness in terms of partitions gives us a set section of
Bi
A in
I*
such that the inter-
is bounded for each m. Let g(m) be any bound for AliBi,and
f(m) = mU( ~<~ LJ g(~)) Theni t is easy to see that for m,B(A with f(c~)
sets each of which is in apply
I , define
f :m ÷ m such that for each I
K
B = the set which m is an element of, and C~
p*-pointedness to m<m , to get a set
for any m,
Corollary 3.3
since
~,~ E A
with
A in
I*
and a progressive
f(~) < ~,~ ~ B~.
Then A F~B
is in
f ( m ) bounds A ~ B . C~
(Weglorz) An ideal
I
on ~ i s s e l e c t i v e i f and onl# i f i t is
quasi-normal. Definitions for
PK~: An ideal
of sets in
x~'y,y ~
.
I
I
on
PK~ is ~uasi-nQrmal i f for every sequence
there exists a set
A in
I*
I t is equivalent to ask for a set
V B EI where V B = {Y ( P~I for some x ( A ' x EA' x ' x~./~' x " I t is immediate that on P ~ an ideal every sequence
A'
in
with
I*
for which
x ~ y, y EB } X "
is outstripping i f and only i f for
I ~ there exists a set
A'
in
I*
with
In one direction just replace the sequence
f(x) = any nubound for
Bx; in the other replace
I
such that for every x,y (A
z ~y}).
f : PK~ + P ~
Co-cone(z) is defined to be
by the sequence
{Y ( P~I
it is not the
This observation lies behind the claim that quasi-normal is to
s e l e c t i v e as o u t s t r i p p i n g is to q - p o i n t . An ideal in
I
on
PK~ is a p * - p o i n t i f f o r every sequence
I , there e x i s t s a set
such t h a t f o r each progressive i f
A
in
x , y £ A with
f(x) ~x
f o r each
I*
and a progressive f u n c t i o n
f ( x ) ~ y, x.
y ~ Bx.
A function
"
o f sets
K
f : PK ~ + PK~ f : PK~ ÷ PK~
is
I n c i d e n t a l l y , the requirement t h a t a func-
t i o n be progressive can be dropped in several places in t h i s paper; i t s presence simplified the exposition. Observation 3.4
An ideal
I
on P ~ is quasi-normal i f and onl~ i f i t is both
209
outstrippinB and a p,,point. Theorem 3.5
I f an ideal
I
on P'K ~ is a p*-point, then i t is a p-point. ''
(The
converse is open - the "p-point puzzle"). The proof of 3.5 is just as in the proof of this implication in the Corollary 3.6
I f an ideal
I
K-context.
on PK~ i s quasi-normal, then i t is selective. (The
converse is open). The proof for
K that a p-point is a p*-point entailed d i s j o i n t i f y i n g a non-
d i s j o i n t collection of sets via set difference, applying p-pointedness to the resulting p a r t i t i o n , adding back in the parts of sets that had been subtracted o f f , and f i n a l l y observing that this " r e j o i n t i f i c a t i o n " leaves the intersections s t i l l in The obstacles to the carrying out of this argument for
I .
PK~ seemto be the f i r s t
and last steps. Fortunately, there is a second way to d i s j o i n t i f y a non-disjoint collection: instead of handling overlap between two sets by subtracting one from another, we can merge the two into a larger set, and continue to remerge any two sets that overlap until the sets have no more overlap. on K that a q-point is outstripping. intervals by f's
[~, f(~)]
In effect, this is what happens in the proof The original sets, which do overlap, are the
white the merged sets are, roughly, the intervals marked o f f
closure points.
The f i r s t type of d i s j o i n t i f i c a t i o n could never have been
used in this proof, since there is no hope that r e j o i n t i f i c a t i o n would leave the intersections as singletons. In the next section this second type of d i s j o i n t i f i c a t i o n is applied to both the p-point and q-point puzzles on P ~. However, in the course of merging overlapping K sets, they might get too big - i . e . , no long l i e in IK~ (in the case of q-point) or in
I
(in the case of p-point).
So we posit additional properties, the "scandina-
vian properties" designed to control the growth of merging sets. §4 The Scandinavian Properties Address the P-point and Q-point Puzzles Lets re-examine the proof that, in the
K context, a q-point is outstripping.
Given a function, f , to be outstripped, f i r s t closure points for
f
are found, then
these are used to decompose K into d i s j o i n t blocks, and then q-pointedness is applied to this p a r t i t i o n .
Both of the f i r s t two steps are d i f f i c u l t for
Velleman observed that we wish to put i t is not the case that
x
and y
in the same block i f
P ~.
Dan
xc y
and
f(x) ~ y, and that we want to keep the blocks as small as
possible, since they must each be not unbounded sets for' q-pointedness to apply. For any two place relation, R, on a set, define (R)~ to be the smallest equivalence relation containing R, and given any function lation
Rf
by xRfy i f both
xcy
f : P ~ + P X define the reK
and i t is not the case that
K
f(x) ~ y .
Dan's
210
suggestion was that we decompose PKX into the equivalence classes of than into some analogue of closure point intervals.
(Rf) ~ rather
In fact, i t is straightforward
to check that, on K, any closure point interval is a union of such equivalence classes. The following example, then, probably explains why there is no straightforward l i f t i n g of the proof for K (that a q-point is outstripping) to the Assume ~ has c o f i n a l i t y ~ ~, and for any x E P~ and l e t
t : P Z ÷ PK ~ by t(x) = x L}{top(x)}.
set top(x~) = sup {B+IIB E X}
Then (Rt)~
turns out to be
PK~ × PK~, SO there is but one equivalence class, which is a l l of thing l i k e this can happen, since
(see 4.2)
{top(x)} that
x
and w are any two elements of
(Rt)~
To see why (Rt)~ blows
PKZ, and we set u = ( w U x ) -
and reverse
Rt.
Fortunately, we can attempt to hinder the blowing Rt
to some set in
out some of the elements that appear in the middle of the
Rt
exists a set
A in
I*
Theorem 4.1
An ideal on P
IK~.
is outstripping i f and only i f i t is both danish and
K
proof:
If
I
f.
Then the equivalence classes of
is outstripping and
tons, hence in
f : P ~ ÷ P ~ let K
is a danish q-point, and l e t
such that each equivalence class of
the resulting partition to get a set class in an anti-chain. Ev~
This A'
ideal on ~ ~
A'
in
I*
outstrips
f.
danish.
~roof sketches: Note that on K, the set
I*
which out-
I Z.
A in
Apply q-pointedness to
intersecting each equivalence
On ~ , !~
is not ~anish, but every
On P Z , ~K~ is not danish.
A we r e s t r i c t
the result follows by a simple cardinality argument. to be not danish, while the
We already knew i t was
f : P ~ ~ PKX. Choosea set
(R~A) ~ is in
~-~oint and every dual to an u l t r a f i l t e r is.
shows I
A be a set in
K
(R~A) ~ are easily seen to be single-
I K ~ . Thus, an outstripping ideal is danish.
a q-point. Now suppose I
not danish.
is in
K context is the obvious specialization.
strips
Theor~ 4,2
to be R~(A x A).
f : PKZ + PKZ there
such that each equivalence class of (R~A) ~
The d e f i n i t i o n of a danish ideal in the
a q-point.
I * , thus tossing
chains.
R is any two place relation and A any set, define P,~
An ideal, I, on PKX w i l l be said to be danish i f , given any
n + l
Note
relates elements betweenwhich there exists a f i n i t e chain of elements Rt
up of equivalence classes by f i r s t restricting
I*
(On K, no-
R produces
and v = w - {top(x)}, then xRtu, vRtu and vRtw, so x(Rt )~ w.
linked by
If
P X.
every "small" relation
(R) ~ equivalence classes which are "small" - i . e . , in IK.) up, note that i f
P ~ context.
t(x)
R to can be a l l of K, and
I t is easy to see that
f(n) =
discussed e a r l i e r shows IK~
The proofs that both q-points and u l t r a f i l t e r s on ~ are danish were
pointed out to me by Andreas Blass, and are presented in the next section.
211
I t turns out that there is a second property that makes up the difference ( i f there is one) between p-point and p*-point.
Like danishness, i t asserts that equiv-
alence classes can be prevented from blowing up. PK~ and
f : PK~ ÷ PK~ any function, define
~(x) c y.
For
R~f
R any two place relation on •
by x(R~f)y
if
xRy and
Let us call a two place relation, R, on P ~ r-small (where I
ideal on PK~) i f for each x E PK.~, xR = {y E PK~IxRy} is a set in ideal, I, on
P ~ is said to be finish i f for every
there exists a progressive alence class of
danishness of the ideal set
A in
I*
I-small relation
f : PK~ ÷ PK~ and a set
((RXf)~A)~ is in
I.
A in
K
I : for each IKs-small relation
K
Theorem 4.3 An ideal
I*
is an Now an
R on
P
such that each equiv-
Observe that the following is equivalent to
such that each equivalence class of
we define i d : P ~ ÷ P X by
I.
id(x) = x
R on
PK~, there is a
((R]id)~A) ~ is in
Imp, where
for each x.
is a p*-~oint i f and only i f i t is both finish and
I, on P ~K
-point ...... ° ~roof:
If
I
is a p*-point and R is an I-small relation, then applying p*-point-
edness to <XR>xEP ~ yields a progressive
f : PK~ ÷ PK~ and a set
A in
I*
such
K
that for each x,y E A with alence classes of Now, i f
f(x)~y,
y ¢ xR.
The same f
((RIf)]A) ~ into singletons, so
I
and R make the equiv-
is finish.
is a finish p-point and
I
f: P ~ ÷ P ~ K
of
and a set
A in
I*
such that the equivalence classes
I's
p-pointednessto the partition
K
( ( 4 f ) ] A ) ~ are each in
I.
Next, apply
determined by these equivalence classes, to get a set which intersects each equivalence class in a set in
A' ~ A,
with
h: P X ÷ P ~ defined by h(x) = any nubound for the intersection of K
A'
in
I*,
Imp. This induces a function A'
with
x's
K
equivalence class. x,y E A'
with
I f we now define
g(x) ~ y, y ~ Bx,
g
and
by g(x) = f(x) U h(x) g
then for any
is progressive because f
is.
Hence I
is a p*-point. I f we now define an ideal
I
to be scandinavian i f i t is both danish and
finish, the following corollary is immediate: Corollary 4.4
An ideal on
P X i s quasi-normal i f and only i f i t is both selective
and scandimavian. The results of section 3 and 4 are summarized in
Figure 3t
next page. At thi's
point, the q-point and p-point puzzles can be rephrased (and remai'n open): Is every q-point on
PK~
danish?
Is every p-point on PKX finish?
212 Is the Q-point Puzzle Resolved?
(Figure 3)
As before, a double arc l i n k i n g arrows indicates that the conjunction of the two arrow sources is equivalent to the t a r g e t . finish
"~ ~-"" .... outstripping
selective --.
s ~ ~
.-scandinavian
quasi-normal Thus, the q-point and p-point puzzles can be rephrased (and remainopen): Is ever~ q,point on ~ danish? Is every p-pointon P~_~ finish?
§5
G a l v i n ' s Negative Results - Discussion
An a l t e r n a t e approach to the r e s o l u t i o n o f the q - p o i n t puzzle is suggested by the f o l l o w i n g p r o o f , due to A. Blass, t h a t on filters
are danish.
by
no = 0
by
Bo = {0}
let
and
Let
f:
nj+ 1 : sup(f"{0,1 . . . . , n j } )
and
Bj+ 1 = [ ( n j )
E or
0
for
+ I, nj+l].
E = BoU B2 U . . . U B2k U . . .
be whichever o f
m both q-points and duals to u l t r a no < nI < . . . . < n3. < . . .
m ÷ m and d e f i n e integers
is in
and I*.
j ~ O.
Now i f
I
Decompose m i n t o blocks is a dual to an u l t r a f i l t e r
0 = B1 U B 3 U . . . U B 2 k + I U . . . . Then
(R~A) ~
and l e t
A
s a t i s f i e s t h a t the equivalence
class o f any i n t e g e r is a subset o f the block t h a t the i n t e g e r belonged t o , since we've deleted a l t e r n a t e blocks too l a r g e for
f
to s t r e t c h across.
q - p o i n t , consider two p a r t i t i o n s of m, one i n t o sets o f the form into
B°
and sets o f the form
partition,
g e t t i n g sets
AI, A2
Bj
This means A
outstrips
I
is a
B2j U B2j+I
and
B2j+I U B2j+2. Apply q-pointedness s e p a r a t e l y to each each in
t h e i r respective p a r t i t i o n s in s i n g l e t o n s . tersects each
If
I*
and each i n t e r s e c t i n g the pieces o f Now i f we l e t
A = A1 N A 2, then
A
in-
in at most one p o i n t and i n t e r s e c t s no two successive blocks. f
and, in p a r t i c u l a r ,
(R~A)~'s
eRuivalence classes are
singletons. Thus
m (and K)
have the p r o p e r t y t h a t the range o f a f u n c t i o n on a bounded
subset is bounded, and t h i s property is enough to get o u t s t r i p p i n g from q - p o i n t . While in
P ~ K
Im~
c l e a r l y cannot have'the property t h a t the range o f a f u n c t i o n on a set
is always in
I ~,
i t seems reasonable to ask i f an ideal
have the following property ( * ) : A
in
I*
such that for each set
Given any function W in
IK~
I t is f a i r l y easy to modify the proof for
with
f :
P~ + P<X
Wc A, f"W
I
on
PKX can
there is a set
is in
IK~.
m to show that any q-point on
P~
213
wlth the extra property
(*)
must be outstripping, but this leaves open the ques-
tion of whether outstripping ideals need have this property (*)
(*)
and even whether
is possible. Fred Galvin answered a question I posed at the Toronto meeting whose proceed-
ings this paper appears in.
A consequence of his results is that there are reason-
able, circumstances under which outstripping ideals exist on PK~ , yet no ideal on PK2 has property unbounded set
(*)
becausethere exists an f : PK2 ÷ PK~ such that for any
A there exists an
Since any A in any
I*
X~ A with
X in
I ~ and f"X
must be unbounded, this does rule out
(*).
unbounded. Thus (*)
cannot replace danish as the "difference" between outstripping and q-point on PK2. Galvin's results are listed l a t e r in this section. The property
(*)
is related to a question about functions which preserve the
order almost everywhere on PKg. Given any I I*
i t is possible (see Theorem 5.10) which dominates f
then h(x) ~ h(y).
f : PK~ ÷ PK~ and any K-footed ideal
to find a function
h defined on a set
on A and which satisfies that for each x,y
We w i l l say that the function
forwards order-~[eservin9 property.
h, and the ideal
if
A in
x~ y
I , have the
For most of the properties we look at, i f one
can "handle" a function dominating a given function on a set of measure one, one can handle the given function.
Thus, i f we view ideals which are not ~-footed as being
pathological, i t is safe to assume that any given function is already forwards orderpreserving.
Now on a 2 ~ L n ~ g ordered set, such as K, i f
h(~) < h(B) ÷ ~ < 8. satisfies
m < B +-~ h(m) < h(B) and we know that any function can be replaced on a
set of measure one (which is actually a l l of K) which dominates i t . -
that
by an order-preserving function
Needan ideal on PK~ have the backwards order-~reservin9 p[op-
e [ t y - that an arbitrary function from P ~ to in
~ < B ÷ h(~) < h(B), then
Hence, we speak of a function as being order-preserving i f i t
K
I*, by a function
P 2 can be replaced, on a set
A
K
h which dominates i t and which satisfies, for each x,y E A,
h(x) ~ h(y) ÷ x ~ y? I t turns out (see Observation 5.11) that a backwards order-preserving ideal
must have the property
(*).
Thus Galvin's results t e l l us that "reversing the
arrow" does not come for free on PK~, and that we cannot assume,without loss of generality, that a given function is backwards order-preserving. Because Galvin's theorems state that, under a number of often-occurring cardinal arithmetic situations, ideals with property to view (*)
(*)
do not exist, my reaction has been
as less basic than the properties discussed e a r l i e r in this paper.
Results of Galvin and W. Fleissner Let A c P~
SK(~) assert that for every function such that for every B ~ A with
SK(2) f a i l s then no ideal on of
functions from m to
B in
f :PK~ ÷ PK~ there is a cofinal set I K ~ , f"B
P 2 can have property
(*).
is in
I K ~ . Note that i f
We'll say a family
K is d£minatin 9 i f given any g : m ÷ K there is some
F
214
f E F satisfying that
g(~) ~ f(~)
for each 6 < K, and define
dK to be the
minimum possible cardinality of such a family. Theorem 5.1 ~roof:
[F. Galvin]
In fact, i f
is cofinal in
size
~l
-Y~d:
d = ~l
that for every cofinal f"B
If
~l
then
S (~l)
fails.
then there is a function
A~ P ( ~ l )
Pm(~l )"
there is a set
To construct
f:P
( ~ l ) ÷ Pm(~l )
B ~ A such that
isfying that for each g : m + Pm(~l ) Next define
m~ B and
f, f i r s t use the dominating family of
to construct a sequence of functions, ha :m÷ P ( ~ l )
for each n E ~.
such
for
m < ~l'
there exists an m such that
f : P~(~I ) ÷ P~(~I )
sat-
g(n) c h (n)
by
f(x) = U{h~0)l~ E x } U U { h (n + l ) I ~ E x and n E x ~ } . Let
A be any subset of
Bn = {x E AIn E x}. not l e t
satisfying U A = ~l "
Then for some n E ~, f"Bn
g: ~ ÷ Pm(~l)
g(n) ~ ha(n)
P (~l)
i . e . for each n E m and each x E A, ~ E x.
if
Theorem 5.2
If
n + l E x.
then
then x ( Bn,
n E x.
and for each n E m, i f
Hence ~
dK = ~+ then S ( + )
BB = {x £ AI~ + l ~ x } .
x, contradicting
notice that for each B < K, i f
[F. Galvin]
Noting that i f
h (~) c f(x)
for
nE x
then
x E P (~l).
then
ha
~ + l ~ x,
Also, for
~< K
dominating
g
thus obtaining the
K replacing m.
For regular K, i f
2K = K+ then
i.e.
f a i l s (K regular, as always)
With g defined as before and
same contradiction as above, with Theorem 5.3
with
Now choose any x E A
~roof: Modify Galvin's argument by defining f :P ~ + P~ by f~x) = U{h (0)]~ E x]U U { h (~)I~ < ~ and ~ x and ~ E x}. set
P ( ~ l ) , for i f
f"B n. Choose an m E ~l
ha(n) ~ f(x)
ha(n) c f(x)
As ha(0) c f ( x ) , 0 E x,
h (n + l)c_ f ( x ) , so
let
Then for each n E ~ and x E Bn, ha(n) ~ f ( x ) ,
each n E m and each x E A, i f with
must be cofinal in
by g(n) = any nubound for
for each n E ~.
For each n E ~
§K(~___ +) then
SK(~ J.
+ d = K , and applying 5.3
(proof omitted)
in the contrapositive
yields, for example, that . . . Corollary 5.4 n E m.
If
V = L then
Hence no Pc~(n)
S ~.~
f a i l s for
+
++
bears an ideal with property
Corol_!_arX 5.5 erty
(~).
V = L then
K regular, hence outstripping ideals exist for
I t i s consistent with (*)".
F. Galvin has also shown . . .
for each
SC's exist
PKK+"
Thus . . .
ZFC that °utstripPin~ ideals not have prop~
Hence ZF__CC ]~_"An ideal on
R-point and has ~roperty
....
(*).
The reader should recall, in this connection, that i f on each PKK+ for
(n)
~ = K , K ..... K
P ~ i s outstrippinB i f and only i f i t is a
215 Theorem 5.6 I f
~
{~ < X[S ~I~[)
fails}
i s regular and
X has uncountable c o f i n a l i t y
is stationary in
X, then S ( ~ )
The following positive results are also of interest.
> K and
if
f a i l s . (proof omitted)
I do not know whether they can
be extended to show SK(X) consistent for uncountable regular K, or whether the
consistency of
SK(X) implies t h a t
(*)
can c o n s i s t e n t l y hold o f some ideal on
P),. Theorem 5.7 Let
IF. Calvin]
If
MAx holds, then S (~X). Theorem 5.7
D(~) denote the discrete topological space of cardinality X.
has been strengthened to: Theorem 5.8
[W. Fleissner]
If
D(~) ~
nowhere dense sets,
is not the union of
Additional results include: Theorem 5.9 d < X< ~
--CO
[F. Calvin]
~
or
.
If
~ is ~
i n f i n i t e cardinal, and ~ = X
D(~)~ is the union of
.
.
.
.
.
.
.
~ compact sets, then
.
.
.
.
or
S (X)
fails.
The Order Preservin~ Properties Theorem 5.10
~[oof:
If
I
is a K-footed ideal on
Given f:PKX ÷ PKX and a set
x, E PKX , f i r s t enumerate A as and h: P X + P X in stages. K
A in
{x~l~ <
At the
defined on a l l elements of the set
PX, then ~ i s forwards order-~rese[v-
I*
IAI}.
~th
IAIx]
such that
Next define functions
stage, f i r s t
A~ for which
t
t
t : PKX + m
and then h w i l l be
and h have not already been
defined at an earlier stage, where A~ = {x~} U A~x~. First define any way such that the following two conditions are satisfied: (1) for which t
< K for each
t : A ~ + K in for each z E Aa
{t(w)lw E Aa and t(w) {f(w)lw E A~}, and (2) t
was not defined at an earlier stage, t(z) E
was defined at a stage e a r l i e r than
~} and t(z) E
a one-to-one function when restricted to those elements of was not defined at an e a r l i e r stage.
Next, define
is
A8 for which t ' s value
h on Aa
by
h(z) = U { f ( w ) L?{t(w)}{w E{z} U A~z}, noting that i f
h(z)
were defined at an earlier stage, the new d e f i n i t i o n w i l l
agree. Since h's
cumulative d e f i n i t i o n immediately guarantees that for which i t x ~ y ÷ h(x) c h(y), we need only check that x ~ y ÷ h(x) # h,yj, '
suffices to show that stage m ,
x ~ y ÷ t(y) E h(x).
i t is easy to check that
But i f
t(y) ( h ( x )
h(y)
were f i r s t defined at
by breaking into cases depending
on whether t(x)
were f i r s t defined at stage m or at an earlier stage.
Qbservation 5.11
If
~
i s backwards order-~reservin~, then
I
has ~
(*).
216 ~0of:
Let
f : PK~ ÷ PK~ be an arbitrary function, and apply backwards order-
preserving to obtain a set
A in
satisfying for each x,y E A that that
I*
and a function
h dominating f
h(x) ~ h(y) ÷ x ~ y.
on A and
For any x E P ~ recall
CQ~cone(x) equals {y E PK~I i t is not the case that
x ~ y}.
Note that
a set W is in IK~ i f and only i f there exists an x in PK with Wc Co-cone (x). Thus the backwards order-preserving property of h, read in contrapositive form, states that for any x E A, i f y E A NCo-cone (x) then h(y) E Co-cone (h(x)), in other words that h" (A ~Co-cone (x)) c Co-cone (h(x)), so that h" (A NCo-cone (x)) is a set in Imp. I t follows that for any set W in Im~ with W~ A, h"W is in Imp, from which the same holds with h replaced by the original function f. Thus I has property (*). ~6 Solution to a Problem of Menas (Added in proof) In
[M], T.K. Menas asks whether isomorphs of a partition measure need have the He allows, as isomorphisms for a fine measure ]J on PK~ only
partition property. those functions (1)
k
(2)
k,~
k: P ~ -~ P ~ K
is
l ;l
on some set in
is fine
He shows that i f
~
is a fine partition measure on P ~ and k is such an isomorK has the partition property i f and only i f (using our terminology)
phism, then k,~ k
that satisfy two conditions:
K
is backwards order-preserving on some set
Note that i f
h
is
l : l
A E ~ ( i . e . V x,y ~ A,k(x) c k ( y ) ÷ x._cy).
on A then changing both c ' s
to ~'s
does not change
the meaning. Menas then produces an example of a ~ > K and an isomorphism k of a normal partition measure ~ on P ~ such that because k
k,~
does not have the partition property,
f a i l s to be backwards order-preserving on any set in ~.
In the last
l i n e of this paper he asks whether this f a i l u r e of isomorphisms to preserve the + partition property can also occur when ~ = K . To see that the answer is yes, we need only assume that Galvin's function
f(x)
2m = K+, and modify
from the proof of 5.2 , so that i t meets the two conditions
above, while remaining a counterexample to SK(K+). The new function k(x) cannot then be backwards order-preserving on any set in IK~+ - this follows from the proof of 5.11.
Hence, as Menas points out, k,~
could not have the partition property,
and the violating partition is given by
F(x,y) =
f~
if if
x cy x~ y
and and
k ' l ( x ) c k-l(y) k'l(x) ~ k-l(y)
for
x,y E k"P ~. K
note that one can dominate an arbitrary
a
To modify f , f i r s t 1 : 1 function k: P ~ ~ P ~. K
k inductively by chuosiflg k(x ) is easily seen to be non-empty.
Simply enumerate
PK~ ~ a s
to be any element of
- - ~ ( x~ -
g: PK~ + P ~
by
{x6}~
217 I f we derive such a
k
from the function
g(x) = f ( x ) u x
k dominates f .insures that i t serves as a counterexample to k(x) ~ x
insures that
k,(~)
is fine whenever ~
is.
then the fact that SK(K+)
while
Thus we have proved:
Theorem 6.1 I f ~ is m supercompact and 2m K+ then every fine measure on + P K is isomorphic, in the sense of Menas to a measure lacking the partition property.
As i t is known that normal measures on
i t follOws that isomorphisms on
+
P~
P K+ do have the partition property,
can f a i l to preserve the partition property.
References
[BTW]
Baumgartner, J. D., Taylor, A. D., and Wagon, S., Structural Properties of Ideals, Dissertationes Mathematicae CXCVII, 1982.
[Cl ]
Carr, Donna M., The Minimal Normal F i l t e r on Soc., 86 (1982),--~T6---3-20~20.
[c2]
Carr, Donna M., P ~ P a r t i t i o n Relations, Fund. Math. (to appear).
PK~, Proc. Amer. Math.
[C3]
Carr, Donna M., A Note on the ~-Shelah Property, Fund. Math. (to appear).
[C4]
Carr, Donna M., and P e l l e t i e r , Donald H., Towards a Structure Theory for Ideals on P~, in this volume. Jech, Thomas~., Some Combinatorial Problems Concerninq Uncountable Cardinals, Ann. Math. Logic 5 (1973), 165--~T98-/.
[Jl] [J2]
[.] [Sl ], [s2] [Vl ]
Iv2] D~]
[Zl] [Z2]
Jech, Thomas J. Set Theory, Academic Press, 1978. tlenas, T. K., A Combinatorial Property of P<~, J.S%m. Logic~41(1976~5~TC34 Shelah, S., The Existence of Coding Sets, and More on Stationary Coding, Lecture NoteT-Tn rlat~ema'tic-s, no. ll~-2~-,Springer Ver-rTag, Berlin, T g 8 6 . 188-202 and 224-246. Velleman, D., Souglin Trees Constructed from Morasses, Axiomatic Set Theory (Baumgartner, F ~ , Shelah,'ed.) Contemporary Mathematics, vol. 31, A.M.S., 1984. Velleman, D., Simplifie d Morasses, J. Sym. Logic, 49 (1984) pp. 257-271. Weglorz, B., Some Properties of Filters, Lecture Notes in tlathematics, no. 619, Sprin-~-Verlag, BerTTn-,-~--l-O"fT. 311-329. Zwicker, ~$. S., P ~ Combinatorics I , Axiomatic Set Theory (Baumgartner, Martin, Shelah, e~J Contemporary M~thematics, vol. 31, A.M.S., 1984, 243-259. Zwicker, W. S. Lecture Notes on the Structural Properties of Ideals on PK~, handwritten notes, July 1984. All results in these notes are included ~n the current paper.
Schedule M O N D A Y , A U G U S T 10th 9:00 Registration, Coffee and Doughnuts at Curtis Lecture Hall I 9:30 t o 10:30 Jim Baumgartner - Partition Relations on the uncountable 10:55 t o 11:40 Arnold Miller - Two results on Analytic sets 11:45 t o 12:30 Shai Ben David - Successors of Singular Cardinals may have no Aronszajn trees LUNCH 2:00 t o 3:00 Andreas Blass - Some Applications of Super-perfect set forcing and its relative 3:20 t o 3:50 Dennis B u r k e - On Generalized Metric Spaces 4:00 t o 4:30 Y. Kimchi - Consistency relations between measurability order of cardinals and partition properties DINNER 7:30 Wine and Cheese at the Faculty Club TUESDAY, AUGUST llth 9:10 Coffee and Doughnuts 9:30 t o 10:30 ...Jim Baumgartner concludes 10:55 t o 11:40 Petr Simon - Thin-tall superatomic Boolean algebras 11:45 t o 12:30 Neil Hindman - Ultrafilters and Ramsey Theory : an update LUNCH 2:00 t o 3:00 Andreas Blass continues... 3:15 t o 3:30 Jiang Shou Li - The Strict p-space Problem 3:35 t o 3:50 Ingrid Lindstrom- A Construction of non-well-founded sets within Martin-Lof type theory 4:00 t o 4:30 Chaz Schlindwein - Special Non-Special Trees
219
WEDNESDAY,
AUGUST
12th
9:10 Coffee and Doughnuts 9:30 t o 10:30 Neil Hindman continues... 10:55 t o 11:40 Sabina Koppelberg - g-cofinalities of Boolean algebras 11:45 to 12:45 ...Andreas Blass concludes LUNCH 2:15 t o 3:00 Stevo Todorcevic - Ramsey Problems for the Uncountable 3:15 t o 3:30 Steven Cushing - A Set-Theoretic Argument in the Semantics of Natural Language 3:35 t o 3:50 Claude L a f l a m m e - Rate of'Convergence of Series 4:00 t o 4 : 3 0 Carlos diPrisco - Normal Closure for Filters DINNER 7:30 Problem Session at the Faculty Club
THURSDAY,
AUGUST
13th
9:10 Coffee and Doughnuts 9:30 t o 10:30 Menachem Magidor - Reflection Properties and Applications of Compactness 1 0 : 5 5 t o 12:30 W. Just (with A. Krawczyk) - Triviality conditions for meetpreserving functions on certain Boolean algebras LUNCH 2:00 t o 3:00 ...Neil Hindman concludes 3:15 t o 3:30 Qi Feng - On Reflection of Spaces of Uncountable Sets 3:40 t o 4:10 Pierre Matet - On Jensen's Diamond 7:00 Thai Banquet
220
FRIDAY,
AUGUST
14th
9:10 Coffee and Doughnuts 9:30 t o 1 0 : 3 0 Menachem Magidor continues... 1 0 : 5 5 t o 1 1 : 2 5 J. Pawlikowski- Parametrized Ellentuck T h e o r e m 1 1 : 3 5 t o 1 2 : 2 0 Alan Mekler - On Abelian groups LUNCH 2:00 t o 3:00 Stevo Todorcevic continues... 3:15 t o 3:30 Alain Gaudefrey - A Language Criterion for some Representation Theorems 3:35 t o 3:50 Steve Purisch - Hereditarily Collectionwise Hausdorff and Monotone Normality 4 : 0 0 t o 4 : 3 0 Efim Khalinsky - Curves, Surfaces, Boundaries and the J o r d a n Curve T h e o r e m in Finite P r o d u c t Spaces
MONDAY,
AUGUST
17th
9:10 Coffee and Doughnuts 9:30 t o 1 0 : 3 0 ...Menachem Magidor concludes 1 0 : 5 0 t o 1 1 : 3 5 Zoltan Balogh - Tile Moore-Mrowka Hypothesis - Coloring Axioms and Related Matters 1 1 : 4 5 t o 1 2 : 3 0 Peter Nyikos - Some classes of sequential spaces LUNCH 2 : 0 0 t o 3:00 A. V. Arhangel'skii - Cardinal Invariants in Function spaces, general spaces and Topological Groups 3:15 t o 3:30 Frantisek Franek - Infinite Steiner Triple Systems 3:35 t o 3 : 5 0 A. Blaszczyk - On Some Constructions using Inverse Limits 4:00 t o 4 : 3 0 Donna Carr - Toward a Structure Theory for Ideals on P~(A)
221
TUESDAY,
AUGUST
18th
9:10 Coffee and Doughnuts 9 : 3 0 t o 1 0 : 3 0 Mary Ellen Rudin - Metrizability of Manifolds 1 0 : 5 0 t o 1 1 : 3 5 Bill Weiss - Topological Partition Relations 1 1 : 4 5 t o 1 2 : 3 0 J. P. Levinski - Chang Games, Large Cardinals and the Core Model LUNCH 2:00 t o 3:00 A. V. Arhangel'skii continues... 3:15 t o 3:50 ...Stevo Todorcevic concludes 3:55 t o 4 : 4 0 Jacek Cichon - T w o Cardinal Properties of Ideals
WEDNESDAY,
AUGUST
19th
9:10 Coffee and Doughnuts 9:30 t o 1 0 : 3 0 Jan van Mill - The Works of Eric K. van Douwen 1 0 : 5 0 t o 1 1 : 3 5 William Mitchell - Speculations on Core Model for Extenders 11:45 t o 12:30 Tomek Bartoczynski - On Measurability of Filters on the Natural Numbers LUNCH 2:00 t o 3:00 Hugh Woodin - Determinacy and Large Cardinals 3:15 t o 3:50 Max Burke - Some Applications of Set Theory to Measure Theory 4 : 0 0 t o 4 : 3 0 Alexander Sostak - Fuzzy Topological Spaces DINNER 7:30 P r o b l e m Session at the Faculty Club
222
THURSDAY,
AUGUST
20th
9:10 Coffee and Doughnuts 9:30 t o 1 0 : 3 0 Mary Ellen Rudin continues... 1 0 : 5 0 t o 1 1 : 3 5 ...A.V. Arhangel'skii concludes 1 1 : 4 5 t o 1 2 : 3 0 J a n van Mill LUNCH 2:00 t o 3:00 Hugh W o o d i n - Saturated Ideals 3:15 t o 4 : 1 5 Boban Velickovic - Forcing Axioms and Stationary Sets 4 : 2 5 t o 4 : 5 5 Scott Williams - Set-Theoretical Problems in Topological Dynamics 7:00 Barbeque at the Island Yacht Club
FRIDAY,
AUGUST
21st
9:10 Coffee and Doughnuts 9:30 t o 10:30 ...Mary Ellen Rudin concludes 1 0 : 5 0 t o 1 1 : 2 0 J. Burzyk - F-spaces 1 1 : 3 0 t o 1 2 : 2 0 Murray Bell - Spaces closely associated to Ideals of Partial Functions LUNCH 2:00 t o 2:30 William Boos - Q u a n t u m Theory in Boolean Extensions by Measure Algebras 2:35 t o 3:05 William Zwicker - Ideals: p-points,q-points and selectives 3:10 t o 3:25 S. Thomeier - Remarks on the Number of Mother Structures and an interesting Operation on the Power Set
List of Participants • Arhangel'skii, A.V.; Moscow State University Mech.-Math. Faculty, Moscow, USSR • Aull, Charles E.; Dept. of Mathematics, Virginia Tech., Blacksburg, Va., 24061, USA (te1:703-961-5409) • Balogh, Zoltan; Dept. of Mathematics, Kossuth University, Debrecen H4010, Hungary (teh913-864-4028(o) 841-0757(h)) • Baloglou, George; Dept. of Mathematics, Univ. of Kansas, Lawrence, KS 66045, USA • Barbanel, Julius; Dept. of Mathematics, Union College, Schenectady, NY 12308, USA (teh518-370-6526) • Bartoszynski, Tomek; Dept. of Mathematics, University of California, Berkeley , Calif. 94720, USA (e-mail:[email protected]) • Baumgartner, James E.; Dept. of Mathematics, Dartmouth College, Hanover, NH 03755, USA (te1:603-646-3559; e-maihjeb@dartmouth) • Beaudoin, Robert E.; Division of Mathematics, Dept. of FAT, Auburn Univ., Auburn Alabama 36849, USA • Bell, Murray; Dept. of Mathematics, Univ. of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada • Bilaniuk, Stephan; Dept. of Mathematics, Dartmouth College, Hanover, NH 03755, USA (te1:603-646-2565; e-maihstefan@dartmouth) • Blass, Andreas; Dept. of Mathematics, Univ. of Michigan, Ann Arbor, Michigan, USA (teh313-763-1183; [email protected]) • Blaszczyk, Aleksander; Silesian Univ., Inst. of Math., ul. Bankowa 14,40007, Katowice , Poland • Boos, Bill; Dept. of Mathematics, Univ. of Iowa, Iowa City, IA 52240, USA • Burke, Dennis; Dept. of Mathematics, Miami University, Oxford, OH 45056, USA (teh513-529-3508) • Burke, Max; Dept. of Mathematics, Univ. of Toronto, Toronto, Ontario M5S 1A1, Canada (te1:416-978-4794)
224
• Burzyk, J.; Inst. of Mathematics, Polish Academy of Science, Katowice, Poland • Cichon, Jacek; Dept. of Mathematics, Univ. of Wroclaw, 50384 Wroclaw, Poland • Cushing, Steven; Dept. of Mathematics, Stonehill College, North Easton, MA 02357, USA • diPrisco, Carlos A.; Dept. of Mathematics IVIC, Apartado 21827, Caracas 1020-A, Venezuela (02)746376 • Dordal, Peter; Dept. of Mathematics, Univ. of Arizona, Tucson, Arizona, USA • Dow, Alan; Dept. of Mathematics, York University North York, Ontario M3J 1P3 Canada (te1:416-736-5250; e-maih dow@yorkvml) • Eklof, Paul; Dept. of Mathematics, University of California-Irvine, Irvine, CA 92626, USA (e-maihpceklof@ucicp6(bitnet); teh814-238-6652(h)) • Feng, Qi; Dept. of Mathematics, Pennsylvania State University, University Park, PA 16802, USA • Fleissner, Bill; Dept. of Mathematics, University of Kansas, Lawrence, KS 66045, USA • Franek, Frantisek; Dept. of Computer Science, McMaster University, Hamilton, Ontario, Canada (teh416-525-9140/3233; e-mail:franyaQmacs on usnet) • Frankiewicz, Ryszard; Dept. of Mathematics, Polish Academy of Science, Warsaw, Poland • Gaudefrey, Alain; Dept. of Physics, York University, North York, Ontario M3J 1P3 Canada • Goffart, J.S.T.; Dept. of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, USA • Gorelic, Isaac; Dept. of Mathematics, University of Toronto, Toronto, Ontario M5S 1A1 Canada (tel:416-925-1358(h)) • Grant, Douglass; Dept. of Mathematics, University College of Cape Breton, Sydney, Nova Scotia, Canada • ttechler, Steve, Dept. of Mathematics, Queens College, Flushing, NY 11367, USA (teh718-520-7014(o) 516-462-5509(h)) • Hindman, Neil; Dept. of Mathematics, Howard University, Washington, D.C. 20059, USA (teh202-636-7989(o) 301-593-1755(5))
225
• Jackson, Steve; Dept. of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA • Jensen, Ronald; Dept. of Mathematics, All Souls College, Oxford, England • Just, Winnefred, Dept. of Mathematics, University of Toronto, Toronto, Ontario M5S 1A1, Canada (te1:416-978-5162) • Kato, Akio; Dept. of Mathematics, National Defense Academy, Yokosuka, Japan • Kimchi, Yechiel; Dept. of Mathematics, Ohio State University, Columbus OH 43210, USA • Koppelberg, Sabine; Institut der FU Berlin, Arnimallee 3, 1000 Berlin 33, West Germany • Laflamme, Claude; Dept. of Mathematics, University of Toronto, Toronto, Ontario M5S 1A1, Canada (te1:416-978-3462(o) 416-783-2727(h)) • Landver, Avner; Dept. of Mathematics, University of Wisconsin, Madison, WI 53706, USA (te1:608-263-7939) • Larson, Jean; Dept. of Mathematics, University of Florida, Gainesville, Florida 32611, USA (te1:904-392-6172) • Leary, Chris; Dept. of Mathematics, Oberlin College, Oberlin, OH 44074, USA (te1:216-775-8380; e-mail:cclQoberlin.edu) • Levinski, Jean-Pierre; Dept. of Mathematics, Dartmouth College, Hanover, NH 03755, USA (te1:603-646-2293) • Lim, Chor ttoon; Dept. of Mathematics, University of Wisconsin, Madison, WI 53706, USA • Lindstrom, Ingrid; Dept. of Mathematics, Uppsala University, Trunbergsv. 3, S-75238, Uppsala, Sweden (tel:018-404792(h), 018-183191(o)) • Lubarsky, Bob; Dept. of Mathematics, Cornell University, Ithaca, NY 14853, USA (tel:607-255-4738(o),607-277-5215(h)) • Magidor, Menachem; Dept. of Computer Science, Hebrew University, Jerusalem, Israel (te1:972-2-584123; e-mail:menachem@humus) • Matet, Pierre; Freie Universitat Berlin, Institut fur Mathematics, Arnimallee 3, 1000 Berlin 33, West Germany (te1:30-831-2530) • Mekler, Alan; Dept. of Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada (e-mail:[email protected])
226
• van Mill, Jan; Vrije Universiteit, de Boelelaan 1081, 1081 HV Amsterdam, Netherlands • Miller, Arnie; Dept. of Mathematics, University of Wisconsin, Madison, WI 53706, USA (te1:608-262-2925(o) 233-0876(5)) • Mitchell, Bill; Pennsylvania State University, State College, PA 16802, USA (te1:814-238-7916) • Moran, Gadi; University of Haifa, Haifa 31999 Israel, 972-4-254193, RSMA309@haifauvm. • Nyikos, Peter; University of South Carolina, Columbia, South Carolina, USA (te1:803-777-5134) • Pawlikowski, Janusz; Univ. of Wroclaw, Poland • Pelletier, Don H.; York University, North York, Ontario M3J 1P3, Canada (tel:416-736-5250)(e-mail:DHPELL@Yorkvml) • Purisch, Steven; Box 5054, Tennessee Technological University, Cookeville, TN 38505, USA (te1:615-372-3441) • Qiao, Y.Q.; University of Toronto, Toronto Ontario M5S 1A1 Canada (te1:416-978-3201) • Roitman, Judy; University of Kansas, Lawrence, KS 66045, USA (te1:913842-7010) • Rudin, Mary Ellen; University of Wisconsin, Madison, WI 53706, USA • Scheepers, Marion; University of Kansas, Lawrence, KS 66045, USA (tel: 913-843-9011(b),913-864-4315(h)) • Schlindwein, Chaz; University of Kansas, Lawrence, KS 66045, USA (tel: 913-841-1693)(e-mail:[email protected]) • Shouli, Jiang; University of Wisconsin, Madison, WI 53706, USA (te1:608262-3601) Shandong University, Jinan, Shandong, China • Simon, Petr; Matematicky Ustav Larlovy University, Sokolovska 83, 18600 Praha 8, Czechoslovakia, (te1:422-2316000) • Smith, Martin; University of Toronto, Toronto, Ontario M5S 1A1, Canada (te1:416-978-2967) • Stanley, Lee; Lehigh University, Bethlehem, PA 18015, USA (tel:215-7583723(b) 867-6431(h)
227
• Steprans, Juris; York University, North York, Ontario M3J 1P3, Canada (teh416-736-5250)(e-maih steprans@yorkvml) •
Szeptycki, Paul; University of Toronto, Toronto, Ontario M5S 1A1, Canada (teh416-978-3201)
• Tall, Frank; University of Toronto, Toronto, Ontario M5S 1A1, Canada (teh416-978-3318(o) 762-4829(h) • Thiele, Ernst Jochen; T. U. Berlin, Breisganer Str 30, 1000 Berlin 38, 8017876, West Germany • Thomeier, S.; Memorial University, St. John's, Newfoundland A1C 5S7, Canada • Todorcevic, Stevo; University of Colorado, Boulder, Colorado 80309, USA • Vaughan, Jerry; University of North Carolina at Greensboro, Greensboro, NC 27412, USA (tel:919-334-5891)(e-maihvaughanj at uncg) •
Velickovic, Boban; California Institute of Technology, Pasadena, CA 9 1 1 2 5 , USA
• Watson, Stephen; York University, North York, Ontario M3J 1P3 Canada (teh416-736-5250)(e-mait watson at yorkvml) • Weiss, Bill; University of Toronto, Toronto, Ontario M5S 1A1, Canada (te1:416-978-3324) • Williams, Scott; State University of New York at Buffalo, Buffalo, NY 14214, USA (teh716-831-2144(o)838-3998(h) • Woodin, W. Hugh; California Institute of Technology, Pasadena, CA 91125, USA * Zwicker, William S.; Union College, Schenectady, New York, (teh518-3706197)
