Analytic quotients: theory of liftings for quotients over analytic ideals on the integers

of two homomorphisms. If D: P(N)/ Fin -* P(N)/Z and IF: P(N)/ Fin - P(N)/,7 are homomorphisms, we define a homomorphism

®1Y: P(N)/Fin--* P(N x {0,1})/(Z®9) by letting its lifting be A i-- D. (A) x {0} U T. (A) x {1}. In other words, 4D ® T is a homomorphism which makes the following diagram commute: P(N)/T

P(N)/Fin

®1p

P(N)/,7

P(N (D N)1(1 ®,7)

1.3. PREORDERINGS ON ANALYTIC QUOTIENTS

13

where i1: P(N)/I - P(N (D N)/I ® J is given by its lifting A i A x {0} and i2: P(N)/3 - P(N ® N)/I ® 3 is given by its lifting A - A x {1}. Note that ker(4) ®1Y) = ker(') fl ker(T). This operation will be very important in Chapter 3, where it will be used to give a description of arbitrary homomorphisms of quotients over analytic P-ideals under OCA. We note that (a) below is true for arbitrary ideals on the integers.

(a) I
PROPOSITION 1.3.1.

implies I
BE 3

(b) If 3 is a P-ideal, then I
PROOF. (a) We will show only that I N is a reduction of I to 3, and define the mapping 4)h: P(N) P(N) by

4h(A) = U h-1({n}) = h-1(A). nEA

This is a homomorphism of the Boolean algebras P(N) and P(N), and therefore, a lifting of the monomorphism of the quotient algebras P(N)/I and P(N)/3 7 since its kernel is equal to I by the assumption on h. Now assume I
hi 1(B)Ah-1(B)9AE3for all B C N. (c) If P: P(N x {0,1})/I ® Fin -+ P(N)/37, then B = P(N x {0}) satisfies I
that another natural ordering coincides with
LEMMA 1.3.2. If a homomorphism 4D: P(N)/Fin -> P(N)/I has a Baire-measurable lifting then it also has a continuous lifting. PROOF. Let l%: P(N) -i P(N) be a Baire-measurable lifting of (D. Then there is a sequence {Ui} of dense open subsets of P(N) such that C is continuous on G= n,,=, U. Recursively choose a sequence 1 = n1 < n2 < ... and ti C [ni, ni+i) (for i E N) such that A fl [ni, ni+1) = ti implies A E U3 for all A and j = 1, ... , i. Let

Ao = U[n2i,n2i+1), I

Al = U[n2i+1, n2i+2), i

co = Ut2i, i

C1 = U t2i+1. i

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

14

Define a mapping 0 by Go(B) = 1,.((B n AO) U C1) n D.(Ao),

e1(B) = (P. ((B n A1) U Co) \,P.(Ao), E) (B) = ®o (B) U G1(B)

Then a is a stabilization of 4?, by {ni} and {ti}. It is continuous because (BnAE)U CI _E is in G for e = 0, 1. For B C AE we have 4?.(B) =z 4 (B U CI _E) n ,(A,). It is therefore easy to see that T is a lifting for D.

The stabilization O as defined above and its variations will play a prominent role in proofs of our lifting theorems (see Theorem 1.5.2, §3.11). V. Kanovei and M. Reeken ([72]) and D.H. Fremlin ([47, Proposition 1C]) have proved an analogue of Lemma 1.3.2 for Lebesgue-measurable liftings: Every homomorphism between two

quotients P(N)/I that has a Lebesgue-measurable lifting has a continuous lifting. Let us finish this section by stating several simple facts about the structure of Recall that IIAIIw = liminfn(A\n) _ quotients over the ideals of the form limn p(A \ n). LEMMA 1.3.3. If cp is a lower semicontinuous submeasure supported by N, then (a) IIw satisfies the inequality I1AAB1I,o < JAI[, + I1BI1,p. (b) ADB E Exh(cp) implies I1AIIW = 11B11,II

(c) P(N)/

is complete with respect to the metric defined by d, (A, B) _

min(1, 1AAB1I,o).

PROOF. Since (a) and (b) are straightforward, we shall prove only (c). Let {Ai} Find n1 < n2 < n3 < ... such that be a Cauchy sequence in

co((AniAAn,) \ [1,ni)) < 2-i for all j < i. Then A = Ui(Ani n [ni, ni+1)) is the limit of the sequence {Ai}. The above lemma was proved by Just and Krawczyk (Lemma 3 of [65]) in the case when cp is a submeasure corresponding to some EU-ideal. By a result of Solecki (Theorem 3.3 of [112]), Exh(cp) is an Fv ideal if and only if the metric induced by

can be Ii, is discrete. This implies that the quotient over an ideal I = countably saturated only when I is an Fo ideal. The converse is also known to be true-all quotients over FQ ideals are countably saturated. This was proved in II -

[65, §1]; the fact that P(N)/ Fin is countably saturated is consequence of an earlier result of P. Olin (see [59]).

1.4. The Radon-Nikodym property The notion of Radon-Nikodym property is related to the following conjecture posed by Todorcevic in [130, Problem 1] (recall that for h: N -* N mapping

4: P(N) -, P(N) is defined by 4ih(A) = h-1(A)). CONJECTURE 1.4.1 (Todorcevic's conjecture). Suppose I is an analytic P-ideal on N and that 4? is a homomorphism from P(N)/ Fin into P(N)/I with a Baire lifting. Then 4' has a completely additive lifting.

A positive answer would imply that
1.4. THE RADON-NIKODYM PROPERTY

15

answer to Todorcevic's hypothesis would be a form of the Radon-Nikodym property of quotients and their Baire homomorphisms. (Compare with the Radon-Nikodym property of Banach spaces and their vector measures, [16, Definition 111.1.3]). DEFINITION 1.4.2. An ideal Z has the Radon-Nikodym property if every Baire

homomorphism oP: P(N)/ Fin -P(N)IT has a completely additive lifting. Todorcevic's hypothesis can clearly be rephrased as follows. CONJECTURE 1.4.3 (Todorcevic's Conjecture). Every analytic P-ideal has the Radon-Nikodym property.

If 3 is not a P-ideal, then there is a homomorphism 4): P(N)/ Fin

P(N)/3

with completely additive lifting but with no lifting of the form 4)h for a finite-to-one function h. For example, it is not difficult to see that for g: N x N N defined

by g((n,rn)) = n mapping (1,9 is a lifting of a homomorphism of P(N)/Fin into P(N x N)/ Fin x0 with no lifting of the form 4h for a finite-to-one function h. LEMMA 1.4.4. The following are equivalent for every ideal ,7 on the integers: (a) ,7 has the Radon-Nikodym property.

(b) Every Baire homomorphism 41i: P(N)/I - P(N)/,7 has a completely additive lifting for every ideal I D Fin. (c) Every Baire monomorphism 4D: P(N)/I -+ P(N)/,7 has a completely additive lifting for every ideal I D Fin. PROOF. To see that (a) implies (b), assume (a) and fix I and 4D as in (b). Since

I D Fin, the formula 'y([A]Fin) = 41([A]z)

defines a Baire homomorphism ': P(N)/Fin -> P(N)/3. (In the diagram below Tr stands for the natural projection of P(N)/Fin onto P(N)/I.)

,

P(N)/ Fin

P(N)/I

P(N)/.7.

By the Radon-Nikodym property of 3, ' has a completely additive lifting. But this mapping is also a completely additive lifting of (P. (b) implies (c) is trivial. To prove that (c) implies (a), assume (a) fails, and let %F: P(N)/ Fin - P(N)/3 be a Baire homomorphism with no completely additive lifting. Let I be the kernel of W; then 1D([A]z) ='I'([A]Fin) defines a Baire monomorphism ': P(N)/I -# P(N)/J. Every lifting of 4D is also a lifting of 41, and therefore 4) has no completely additive lifting, and (c) fails.

COROLLARY 1.4.5. If 3 has the Radon-Nikodym property, then I
It is not clear whether the assumption that I includes Fin in (b) and (c) is important; some recent results of V. Kanovei and M. Reeken suggest that it is not.

Namely, in [72, 70] they have proved that in all known cases when 3 has the

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

16

Radon-Nikodym property, every Baire homomorphism D: P(N) - P(N)/J has a completely additive lifting.

It would be natural to ask whether after replacing "monomorphism" in (c) of Lemma 1.4.4 with "epimorphism" (or "isomorphism") one obtains an equivalent statement. I would conjecture that it is not so; this would follow from an (expected) positive answer to Question 1.14.4. Recall that by Z* we denote the dual filter of 1, namely the filter of all sets whose complement is in Z, and by Z+ we denote the coideal of all sets, namely all sets of integers which are not in Z. An isomorphism of quotients (b: ?(N)/-T -+ P(N)/,7 is trivial if there are sets A E Z*, B E J* and a bijection h: B -> A such that (Dh is a lifting of (P. Note that (a) below gives a satisfactory answer to the basic question (see §1.2). The part (b) of Proposition 1.4.6 below for I = Fin was first proved by Velickovic ([141]). (In Definition 1.2.7 it was defined when two ideals are isomorphic.) PROPOSITION 1.4.6. If ideals I and J have the Radon-Nikodym property then every Baire isomorphism of quotients P(N)/T and P(N)/,7 is trivial. In particular, (a) The quotients over I and J are Baire-isomorphic if and only if the corresponding ideals are isomorphic. (b) All Baire automorphisms of P(N)/Z are trivial.

PROOF. Let D: P(N)/Z --> P(N)/,7 be a Baire isomorphism and let h1i h2 be such that 4ih, is a lifting of (D and'Dh2 is a lifting of lD-1. If g = h1 o h2 is the identity on a set in 1*, then (Dh, is a lifting of (D and h1 witnesses that I and J are isomorphic. Let B be the set of all x such that g(x) 54 x. Then we can split B into disjoint sets B1, B2, B3 so that g"Bi is disjoint from Bi for i = 1, 2,3 (see e.g., [14, Lemma 9.1]). This easily implies each Bi is in Z, and therefore the set of fixed points for g is in Z*, as required. Both (a) and (b) follow immediately.

The main results of this Chapter are that all the natural examples of analytic P-ideals (in particular, all the ideals from Examples 1.2.3) do have the RadonNikodym property (Theorem 1.9.1), although in general this is not the case (Theorem 1.9.5).

1.5. Asymptotically additive liftings In this section we give a partial positive answer to Todorcevic's hypothesis reducing it to a problem of analyzing non-exact homomorphisms, discussed in §1.8 and [33]. The following definition essentially appears in [106] and [141], and Theorem 1.5.2 below was essentially proved by Just ([62]) in the case of ErdosUlam-ideals (for definition see Examples 1.2.3 or §1.13). DEFINITION 1.5.1. If {n%} is a strictly increasing sequence of integers, {vi} is a sequence of disjoint finite sets of integers, and Hi is a function mapping subsets of the interval [ni,ni}1) to subsets of vi, then define 1YH: P(N) - P(N) by

`I'H(X) = U Hi(X n [ni,ni+1)). i=1

We say that maps of the form 'PH are asymptotically additive. This is because we have

TH(S U t) = TH(s) U TH(t)

1.5. ASYMPTOTICALLY ADDITIVE LIFTINGS

17

whenever s < ni < t for some i. Observe that a completely additive lifting, cFh, is a special case an asymptotically additive lifting-let ni = i, Hi({i}) = h-1(i) and Hi(0) = 0. Already this form of a lifting is sufficiently simple to be useful in the situations when we do not necessarily have an asymptotically additive lifting. For example, see [130, Theorem 8]. Part (b) of the following theorem will be used in the proof that the ideal Fin x0 has the Radon-Nikodym property (Theorem 1.6.2).

THEOREM 1.5.2. If I is an analytic P-ideal and ' : P(N)/Fin -* P(N)/I is a homomorphism which has a Baire lifting, then it has an asymptotically additive lifting. Moreover: (a) The sequence {ni} can be chosen to be a subsequence of any strictly in-

creasing sequence {mi} of integers given in advance. (b) The function WH can be chosen effectively in code for ,.

PROOF. By Lemma 1.3.2 we can assume has a continuous lifting L. Note that the proof shows that this 4o. can be chosen effectively in a code for a given Baire lifting of 4b. Let cp be a lower semicontinuous submeasure such that I = Exh(cp), as guaranteed by Solecki's theorem (Theorem 1.2.5). We need a notion of a stabilizer,

first extracted by Just ([63]) (though implicit in an earlier work of Shelah ([106]) and Velickovic ([141])).

DEFINITION 1.5.3. For n < n' let c C In, n') be an (n, n')-stabilizer of (P. if there exists k E (n, n') such that for all s, t C [1, n) and all X, Y C [n', oo):

(Si) o((I).(sUcUX)0(D,(tUcUX))\[1,k)) <2-' (S2) min((D.(sUcUX)0lP.(sUcUY)) > k. CLAIM 1. For all n and all large enough n' there is an (n, n')-stabilizer.

PROOF. Fix an n and assume there is no (n, n')-stabilizer for all n'. By the uniform continuity of 1)., for all large enough k > n the condition (S2) is satisfied for all c C In, n'), and therefore (Si) fails for all large enough n'. Recursively pick

c1 C C2 C ... and n = mo < m1 < m2 < ... so that cj fl In, mi) = ci for i < j and for some si, ti C [1, n) we have

P((,.(si U ci)AD.(ti U ci)) \ [1,mi-1)) > 2`. Let (s, t) be a pair which appears as (si, ti) infinitely often; then s U U°° 1 ci = T U U° 1 ci, but the set

F

4D.(NU0C)AID. (iuuci)

is not in I, because co(F \ [1, m)) > 2_n for all m E N. This contradiction finishes the proof.

By using Claim 1 pick sequences ni, ki and ci C [ni, ni+1) so that ci is an (ni, ni+1)-stabilizer of 4P., with ki witnessing this fact. At each step we can pick the pair (ci,ni+i) to be the minimal among all pairs for which ci is an (ni,ni+i)stabilizer of % in some fixed recursive well-ordering of Fin xN in order type w. For E = 0, 1 let AE = U [n2i+E, n2i+E+1) i=1

CE = U C2i+E+1 i=1

18

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

We can assume D. (AE) n 4. (CE) = 0 for E = 0,1, possibly by replacing 4). with the mapping

X-

((t.(Ao) n c (co)) u

n i'.(C1))).

This mapping is a lifting of (D because AE n CE = 0 for E = 0, 1. Moreover, it is easily seen from the definitions that ci is still a (ni, ni+)-stabilizer for this mapping. Define functions f, : P(AE) --* P((P. (AE)) for e = 0,1 by fE(X) _ ' (X U CE) n %(AE). Again, we can assume D. (Ao) n

(AI) = 0. Note that 4). (AE) n 4P . (CE) = 0 implies

fE (0) =0 fore = 0,1. Since CE n A, =0 for e =0,1 the set fE (X) A * (X) is in I for every X C AE, and therefore the mapping 4% defined by 4D .(X) = fo(X n Ao) U f, (x n A1)

is a lifting of 4). Define functions Hi: P((ni,ni+1)) ---* P([ki-1,ki+1)) by

Hi(s) _ (s) n [ki-1, ki+1) and let 'H be as in Definition 1.5.1. Note that sets vi = Hi ([ni, ni+ 1))

are pairwise disjoint, by the assumption that sets (D.(Ao) and %(A1) are disjoint. CLAIM 2. The function 'H is a lifting of (D. PROOF. Fix X C AO; we will prove that

W(('D.(X)A H(X)) n [ki-1, ki+1)) < 2-nc+1

for all even i. Let xi = X n [ni, ni+i ); then for some s < ci_1 < xi < ci+1 < Y and s' < ci_1 < xi < ci+1 < Y' we have C(X) = fo(X U Co) = fo(s U c2_1 U xti U ci+1 U Y), Hi(xi) = fo(s' U ci_I U xi U ci+1 U Y')

Then by (S2) applied to (ni+1,ni+2) we have:

-D.(X) n {1,... , ki+1} = fo(s U ci_1 U xi), Hi(xi) n {1, ... , ki+1} = fo(s' U ci_I U xi). Applying (Si) to (ni_1i ni) we get the desired conclusion. By symmetry the same

is true for X C AI. So for any X C N and all i we have cp(('FH(X)05.(X)) n [ki-1, ki+1)) < 2-ni+1 and therefore 00

W((`PH(X)0`f'.(X)) n [ki-1, oo)) < E 2-"'+1 < 2-n:+2 j=i

and `yH is a lifting of 4P.

This ends the proof of theorem. For part (b), note that sequences {ni}, {ci} were chosen effectively and {Hi}, {vi} are defined by simple formulas.

1.6. THE RADON-NIKODYM PROPERTY OF Fin AND Fin x0

19

1.6. The Radon-Nikodym property of Fin and Fin x O We have already developed enough theory to prove that some ideals have the Radon-Nikodym property. Theorem 1.6.1 below was essentially proved by Velickovic ([141]) and Just ([63]). THEOREM 1.6.1. The ideal Fin has the Radon-Nikodym property. Moreover, if 1D is a continuous homomorphism of P(N)/ Fin, then an h for which (Ph is a lifting of ' can be chosen effectively in (D.. THEOREM 1.6.2. The ideal Fin xl1 has the Radon-Nikodym property. PROOF OF THEOREM 1.6.1. Let

': P(N)/Fin - P(N)/Fin be a Baire ho-

momorphism. Then by Theorem 1.5.2, -D has a lifting of the form 'PH with some parameters {ni}, {vi} and {Hi} chosen effectively in 1).. We claim Hi is a ho-

momorphism of the Boolean algebras P([ni,nitj)) and P(vi) for all but finitely many i.

Note first that Hi(s) C vi for all s C [ni,ni+1). Suppose there are infinitely many i such that for some si, ti C [ni, ni+1) one of the following happens:

Hi(si) n Hi(ti)

Hi(si n ti)

or

vi \ Hi(si) Hi([ni, ni+1) \ si). Assume that the first possibility happens infinitely often, say for a pair si, ti for all i in some infinite A C N. Let X = UiEA si and Y = UiEA ti. Then sets 'YH(X) n T H (Y) and 'PH (X nY) are not equal modulo finite, because their intersections with vi are distinct for all i E A. This is a contradiction. The second case is handled in a similar manner. Therefore we can assume all Hi's are homomorphisms (possibly after deleting finitely many of them). Since TH(N) = U°°1 vi the set vo = N \ U°11 vi is finite. Define a finite-to-one function h: N --> N as follows: r1, if m E vo, h(m) = Slk, if k E [ni, ni}1) and m e Hi({k}) for some i > 1. 'Ph(X) for all X, because: e= kEvi and k E Hi(X n [ni,ni+1)) kEvi and k E Hi({l}), for some l E X n [ni, ni+1) t* k E h-1({l}), for some 1 E X. This ends the proof, and it is clear that'Ph is effective in '1 H and in (P..

Then 'PH(X) k E -5.(X)

PROOF OF THEOREM 1.6.2. Let'P: P(N)/Fin P(NxN)/ Fin xO be a Baire homomorphism. Recall that for a function f : N -* N we have

rf = {(m,n) : n < f(m)}. Then, since the ideal Fin x O f r f is isomorphic to Fin for all f, the mapping

V(A) _ (.(A) nrf is a homomorphism of P(N)/Fin into P(rf)/Fin and therefore by Theorem 1.6.1 it has a completely additive lifting,'Ph, for some h = h(f) : r f -* N which is chosen effectively in f, so that the function f -* h(f) is Borel. In particular, the set

X = {(f,h(f)) : f

E NN}

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

20

is Bore!. Define a partition [Xj2 = Ko U K1 by letting a pair {(f, h(f)), (g, h(g))} in Ko if h(f)(m) # h(g)(m) for some m E Ff n Fg. This partition is open in the natural separable metric topology of X (see §2.2 for an explanation and terminology), therefore by the Principle of Open Coloring for analytic sets (see Theorem 2.2.12) applied to this partition we have the following two possibilities:

The set X can be covered by a countable family of KI-homogeneous subsets. Since the partial order (N5, <') is a-directed there is a KI-homogeneous set 7l such that its projection to NN is cofinal in (N5, <*). Then CASE 1.

h = U h(f) fE7{

is a function because 7-1 is K1-homogeneous. We claim'Dh is a lifting of ', namely

that'Dh(A)A'.(A) E Fin xO for all A. Assume this is not true, and let A be such that B = 4Dh(A)AOD.(A) is not in Fin xO. Let f E 71 be a function such that I'fnB is not in Fin x0; but h(f) [ (FfnB) = h [ (FfnB), and Dhlfl(A)A4.(A) E Fin xO, contradicting the assumption that'Dh is not a lifting of 4D. CASE 2. There is a perfect, Ko-homogeneous 7-l C X. Then we can find a g E NN such that If < g : (f, h(f )) E 7-1} is perfect. But since h(f) [ r f =' h(g) [ rf for all f E 7-l, for some i. the set of f E 71 for which m > n implies

h(f) (m, i) = h(g) (m, i) for all (m, i) E r f is uncountable. Such a set includes an uncountable Kl-homogeneous subset, and therefore Case 2 is not possible, and Case 1 applies. O

1.7. A reformulation of Todorcevic's hypothesis The results of §1.5 show that a canonical Baire-measurable lifting of a homomorphism between analytic P-ideals is asymptotically additive, i.e., it is obtained by putting together finite maps Hi. In §1.6 we have seen that when in the range we have the quotient P(N)/Fin, then all but finitely many of Hi are homomorphisms. In general the situation need not be that simple, and these Hi can be only approximate homomorphisms (see Definition 1.7.1 below).

Note that the method of the proof of Theorem 1.6.1 essentially gives that 0 x Fin has the Radon-Nikodym property. This is because the natural generating submeasure p for this ideal satisfies the formula cp(AUB) = max{cp(A), W(B)}, and

this implies that the mapping

'Di(s) = U Hi({j}) jEs

is an ei-approximation of Hi, if Hi is an ei-approximate homomorphism (see Definition 1.7.1 below). In this section we shall see that Todorcevic's hypothesis turns out to be equivalent to a statement that such non-exact homomorphisms can be approximated by exact homomorphisms modulo a constant error. Recall that a map w: P(X) -* [0,oo] is a submeasure supported by X if FP(M) = 0,


1.7. A REFORMULATION OF TODORCEVIC'S HYPOTHESIS

21

for all A, B C X. We write supp(cp) = X. The norm of co is defined by Doll = W(X )

A submeasure cp is pathological if there is A C supp(cp) such that p(A) < cp(A) for every measure µ on X = supp(cp) dominated by cp. This corresponds to the notion of an E-pathological submeasure for e > 0 (see [136]) but it is different from the standard definition of a pathological submeasure as a submeasure which does not dominate a nonzero measure (see [66], [13], [122]). However, we are mainly interested in submeasures with finite supports, and a nonzero submeasure with a finite support always dominates a nonzero measure. A submeasure is non-pathological if it is not pathological, i.e., if it is equal to the pointwise supremum of the measures dominated by itself. For a submeasure cp the function cp defined by

b(A) = sup v(A) (here v ranges over measures dominated by cp) is a maximal non-pathological submeasure dominated by cp. Let

P(
cp(!1)

1(A)#o ,o(A)

DEFINITION 1.7.1. If P([1, n)) is equipped with a submeasure cp a mapping H: P([l,m)) -* P([l,n)) is an E-approximate homomorphism (with respect to cp) for some e > 0 if for all s, t C [1, n) we have

cp((H(s) U H(t))AH(s U t)) < E, cp(H(s')AH(s)C) < E.

A homomorphism b: P([1, n)) -* P([1, m)) is a 6-approximation for H

cp('(s). H(s)) < 6 for all s C [1, m). For a submeasure cp defined on a finite set X let K,s denote the supremum of all K such that there is an E > 0 for which some E-approximate homomorphism H: P(Y) -* P(X) cannot be K E-approximated by a homomorphism.

If D is a class of metrics, we say that non-exact homomorphisms with respect to metrics in D are stable if there is a universal constant K such that every --approximate homomorphism with respect to some metric in D can be Keapproximated by an exact homomorphism. THEOREM 1.7.2. The following are equivalent (1) The approximate homomorphisms between finite Boolean algebras with respect to arbitrary submeasures are stable, (2) Todorcevic's Conjecture (1.4.1).

Before proving Theorem 1.7.2, let us prove its special case which shows the idea behind the proof that (1) implies (2) more transparently.

THEOREM 1.7.3. If the approximate homomorphisms with respect to measures are stable, then all summable ideals have the Radon-Nikodym property.

PROOF. The following measure supported by P(N) will be useful

µf (A) _ E f(n) nEA

22

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

Note that If = Exh(if) (see §1.2). Let -D: P(N)/Fin -+ P(N)/Zf be a Bairemeasurable homomorphism. By Theorem 1.5.2 we can assume that [ni,ni+1), vi P(vi) are such that `FH (see Definition 1.5.1) is a lifting and Hi: P([ni,ni+1)) of a homomorphism of P(N)/Fin into P(N)/I9. Define Ei to be the minimal E for which Hi is an E-approximate homomorphism (for definition see §1.8, and note that Ei is, being equal to a maximum of a finite set of real numbers, well defined). By so we shall denote either Ini, ni}1) \ s or vi \ s, depending on whether s C Ini, ni+1) or si C vi for some i. It will always be clear from the context which case of the definition applies.

CLAIM 1. E = E 1 Ei < oo. PROOF. Assume the contrary. For every i there are si, ti C [ni, ni+1) such that either µf ((Hi(Si) U Hi(ti))AHi(Si U ti)) = Ei or

ei.

Let M1 be equal to the sum of all Ei for which the first possibility applies. Then sets X = US si and Y = Ui ti satisfy fif('FH(X UY)AOFH(X) U'FH(Y)))) = M1, therefore M1 is finite, and the sum of all Ei for which the second possibility applies is infinite so we can find a set X such that f"f('FH(X)CA'FH(XC))) = 00.

This contradicts to the assumption that 'FH is a lifting of a homomorphism and ends the proof.

By applying the stability assumption to Hi, we obtain a function hi : vi [ni,ni}1) such that Dh, is a KEi-approximation (with respect to µf) of Hi. Let h = Ui hi. Then for A C N we have 1. if (4h(A)DWFH(A)) < E KEi < 00

and therefore 'h is the required lifting of 4D.

PROOF OF THEOREM 1.7.2. We shall first prove that (1) implies (2). Let ,P: P(N)/Fin -> P(N)/Z be a homomorphism with a Baire lifting. Then Lemma 1.3.2 implies that 4) has a continuous lifting, so let 4). be a fixed continuous lifting of k. We shall use the following. For A C N let 1 A denote the restriction of I to the set P(A). Let

JRN = {A :

[ A has a completely additive lifting }.

We will prove this is an improper ideal. CLAIM 2. The ideal JRN is a EZ set of Teals.

PROOF. For A C N we have A E JRN if and only if

3h: N -+ AVB C AVn 3m lo((4).(B)O(Dh(B)) fl [m, oo)) < 2-", and therefore JRN is as required.

1.7. A REFORMULATION OF TODORCEVIC'S HYPOTHESIS

23

In particular, the assertion of theorem, "the homomorphism ' has an asymptotically additive lifting," is a E2-statement, and therefore it will suffice to prove that it follows from Martin's Axiom (see [82]]), since by Shoenfield's absoluteness theorem ([110]) this additional assumption can be removed from the proof. CLAIM 3. The ideal JRN is nonmeager and therefore it is either improper or it does not have the property of Baire.

PROOF. By a result of Jalali-Naini ([58]) and Talagrand ([121]) (Corollary 3.10.2), it will suffice to prove that Fin RB RN, namely that for every strictly increasing sequence mi of integers there is a set in TRN which includes infinitely many of the intervals [mi,mi+1) Let mi be an increasing sequence of integers. By Theorem 1.5.2 this sequence has an infinite subsequence ni such that the homomorphism 4) has an asymptotically additive lifting, %PH, with parameters {ni}, {vi} and {Hi} for some sequences {vi} and {Hi}. For every i E N let m(i) be the minimal m such that Hi is not a 2-m-1-approximate homomorphism with respect to V. We claim that lim m(i) = oo. i

00

Otherwise we would have an fn E N such that m(i) = m infinitely often, and this would give us sets A, B C N such that either co(('I'H(AU B)A('PH(A) U'I'H(B))) n [k, oo)) ? co(WH(AC)ATH(A)') n [k, oo)) >_ 2--

for all k, contrary to the fact that `I'H is a lifting of P. (This is proved exactly like in the proof of Theorem 1.6.1, where all but finitely many Hi's were homomorphisms.) Therefore we can find an infinite set C such that the sequence {m(i)}iec is strictly increasing. Note that, by our assumption, for each caj = cp ( vi we have P(cpi) < M and therefore by the stability assumption (1) for each i E C there is a homomorphism H2: P([ni,ni+1)) --i P(vi) which is an M M. 2-"mil-approximation to Hi. Each Hi' is of the form 4h; for some hi : vi ---* [ni, ni}1) and if we let h = Ui hi then ' h is a witness that UiEC [ni, ni+1)

is inJRN. Recall that Martin's Axiom implies that all E2 sets of reals have the property

of Baire ([91]), and therefore by Claims 2 and 3 the ideal JRN is, under MA, improper. In particular, N E JRN and therefore oD has a completely additive lifting. As noted before, since this conclusion is absolute, this concludes the proof that (1) implies (2). Now we prove that (2) implies (1). Let us assume that (1) fails. By using this assumption, we recursively construct

(a) sequences 0 = no < nl < n2 < ... and 0 = mo < m1 < m2 < (b) a submeasure cpi on [mi, mi+1 ),

....

(c) an 1/2'-approximate homomorphism Hi: P([ni, ni+I)) - P([mi, mi+l)) which cannot be 1-approximated by a homomorphism. Define a submeasure cp on N by M

o(A) = rwpi(An [mi,mi+1)) i=O

24

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

and let J be Exh(W). Consider the asymptotically additive mapping W H determined by the parameters {ni}, {vi = [mi,mi+i)}, and Hi, as in Definition 1.5.1. CLAIM 4. WH is a lifting of a homomorphism which has no completely additive lifting 'Ph .

PROOF. It is straightforward to check that WH is a lifting of a homomorphism from P(N)/Fin into P(N)/,7 and it is obviously continuous. Assume h is such that 'Ph is a lifting of the same homomorphism as TH. Let nk(i) be a subsequence of ni such that

(1) h-'([l, nk(i))) S [1,nk(i+l)), and (2) h"[l,nk(i)) S [1,nk(i+1)) Such a subsequence can be chosen recursively,using the fact that h is finite-to-one. Let H2: P([nk(i),nk(i+l))) be defined by "joining" Hi's, namely by k(i+l)-1 (3) Hk(i)(s) = Uj=k(i) Hj(s)(Clearly, Hk(i)( s ) =

H(s )) Let also v k(i) = .

I

Ik( k(i)-1

Uk(i+l)

vj> so that the range of Hk(i)

is included in P(vk(i)). Note that the map WHk(X) = U Hk(i)(X n [n'k(i),nk(i+1))) i=1

coincides with %PH. Also, it is straightforward to prove that SUBCLAIM. The map Hk(i) cannot be 1-approximated by a homomorphism.

We claim that the set B = {j : j E Vk(i) and h(j) [nk(i),nk(i+l)) for some i} is in the ideal J. Assume otherwise, and let C = h"B. Since B \ lph(C) E J, C is

not in ker(4)h) = ker('YH). As usually, let Ao = U[nk(2i),nk(2i+1)) and

Al = U[nk(2i+1),nk(2i+2))i=1

i=1

Assume that C n A0 is not in ker(4h). Then the set 'Dh(C n Ao) \ n Vk(2i+1) _ Dh(C n Ao) \ `PHk (Ao) i=1

is J-positive, a contradiction. Therefore C n A0 is in ker(4)h), and an analogous proof shows that c n Al is in ker(4Dh). Therefore, since B C (bh(C), this proves

that B E J. Let us now define Hk(i): P([nk(i),nk(i+l)) -+ P(vk(i)) by Hk' (i) (s) = -D h(s) n Vk(i)

Since the set B defined above is in ,7, the mapping Hk(i) can be 1/2-approximated

by a homomorphism (namely, the restriction of 4 h) for all but finitely many i. Therefore, by the Subclaim above, Hk(i) cannot be 1/2-approximated by Hk'(i), and H,(i) is not a 1/2-approximation to Hk(i) for all large enough i. Let si g [ni,ni+1) be such that co(HH(si)AHi(si)) > 1/2. Since h is finite-toone, we can find an infinite D C N such that UiED H, Si) is not in J,

1.8. APPROXIMATE HOMOMORPHISMS AND NON-PATHOLOGICAL SUBMEASURES 25

and therefore'Dh((JiED Si)DWA(UieD s,) is also not in J. Therefore'Dh cannot be a lifting of this homomorphism. Therefore the ideal 17 does not have the Radon-Nikodym property.

We shall now give a version Theorem 1.7.2 which will turn out to be more useful. We say that an analytic P-ideal I is non-pathological if it is of the form for a non-pathological lower semicontinuous submeasure s. THEOREM 1.7.4. The following are equivalent (1) The approximate homomorphisms between finite Boolean algebras with respect to arbitrary non-pathological submeasures are stable, (2) Todorcevic's Conjecture for non-pathological analytic P-ideals.

PROOF. The proof is a minor modification of the proof of Theorem 1.7.2. In direction (1) implies (2), one only has to notice that if W is non-pathological, then so is its restriction to a subset of its support, and this follows immediately from the definition of a non-pathological submeasure. In direction (2) implies (1), one only needs to note that if every cp, is non-pathological, then so is En cps. LEMMA 1.7.5. If an analytic P-ideal I is equal to Exh(ip) for a lower semicontinuous submeasure W such that K, < oo, then I is non-pathological.

PROOF. If I = Exh(cp) and P(cp) = M < oo, then cp < cp < Mcp, therefore I = Exh(o) and I is non-pathological. Therefore we can say that an analytic P-ideal is pathological if for every lower semicontinuous submeasure p such that I = Exh(cp) we have P(ip) = oo.

1.8. Approximate homomorphisms and non-pathological submeasures Our motivation for studying non-exact homomorphisms is explained in the previous section, but they were studied in their own right, for example in the context of so-called Hyers-Ulam stability. See, for example, [139, §VI.1], [140, V.4], (44], also [67], [33], [37], [72] for more on non-exact homomorphisms in various contexts. The upper bound for the constant KM claimed in [33] is 521, which is slightly

better than the constant that can be obtained from the proof of Theorem 1.8.1 below. Max Burke and David Fremlin have noticed two mistakes in a proof in [33], the correcting of which has caused the numerical bound to increase. Anyway, in [37] this bound was improved to 160, and it can be improved further by combining the results of [72] and [37, §3]. THEOREM 1.8.1. The non-exact homomorphisms with respect to measures are

stable. Namely, there is a universal constant KM such that K,,, < Km for every measure µ on some P([l, n)). THEOREM 1.8.2. The non-exact homomorphisms with respect to non-pathological submeasures are stable. Namely, the constant K also satisfies K., < Km P(co) for every non-pathological submeasure W. In our proof of Theorem 1.8.1 we shall need a result which says that a family of subsets of some finite set which almost everywhere looks like an ultrafilter must be close to some principal ultrafilter, (k) _ {s E P([1, m)) : k E s}. Let v denote the

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

26

uniform probabilistic measure on P([1, m)), and let v2 denote the product measure

on P([1,m)) x P([1,m)). It should be noted that the following theorem remains true even when the condition (Al) is omitted, possibly at the expense of increasing the numerical constant.

THEOREM 1.8.3. If S > 0 is small enough (say, S < 1/156) and A C P([1,m)) satisfies

(Al) v2((A x P([l, m)) U P([1, m)) x A)0{ (s, t) : s U t E A}) < 5, (A2) v2((A x A) 0{(s, t) : s n t E A}) < 5, (A3) v{s : s, so A or s, SC E Al < 5,

then there is a k E {l,. .. , m} such that v{s E A : k E s} > (1 - 285)/2. In particular v(AA(k)) < 295. PROOF. For I C {l, ... , m} and s we define

sI=(Ins)u(Ic\s), A2(I) = {s : s, sI E Al, EI = v(A2(I)). The number El is, in some sense, a measure of how A concentrates on I. For example, if A is an ultrafilter, then El is equal to either 1/2 or 0, depending on whether I is in A or not. Note that (A3) implies (A)

E0 < 5

and E{I,...,r } = v(A) >

The set of all s E P([1, m)) such that s, so

A or s", (s') 0 A is, by (A3), of measure < 25, and for each s outside of this set we either have s E A2 (I) U A2 (IC ) or so E A2(I) U A2(IC). (To see this, consider the possible cases: if s, sI E A then s E A2(I), if s, (s')C E A then s E A2(IC), and so on). Therefore (B)

EI+EIc > 1-25 2

We claim that

EI Eic < 55.

(C)

To see this, note that the left-hand side is equal to A2(IC)) and that C = A2(I) x A2(IC) = C0 U C1 U C2, where

v2(A2(I) x

Co = {(s, t) E C: snt,si ntic E Al,

C1={(s,t)EC:snt0A}, C2={(s,t) EC: sI nt1C ¢A}. It suffices to show that v2(Co) < 35, v2(C1) < 5 and V2 (C2) < 5. The latter two inequalities follow immediately from (A2), and Co has the same size as the set

{(u, v) : ((u n I) u (v n ll ), (v n l) u (u n lC)) E C and v n u, vc n u E A}. To see that the measure of this set is < 35, we split it into three pieces, depending on whether v 0 A, vC A, or v, vC E A. Each of these pieces has a measure less than 5 (by (A2) and (A3)), and therefore v2(C2) < 35. Let -y =

(1/2 - S)2 - 206.

1.8. APPROXIMATE HOMOMORPHISMS AND NON-PATHOLOGICAL SUBMEASURES 27

By using (B) and (C), we get E1(1/2 - El - 5) < 55 and therefore

El < 1 - 25 - 2y

(D)

or

4

1-25+2y 4

< E1.

Let Ek = E{1,...,k}. By (A) there exists a minimal k < m such that Ek > (1 - 2b+ 2y)/2, and by (D) for this k we have

Ek - Ek_1 > y.

(E)

This means that A concentrates on {1, ... , k}, but not on {1, ... , k - 1}. We shall prove that the principal ultrafilter (k) _ {s E P([1, m)) : k E s} is close to A. In order to do so, we have to prove that the set

G={sEA2({1,...,k}):k0s} is small. Consider the set

F= fs:sEA and sulk} VA}. Fix s E G \ A'(11,..., k - 1}). Then both s and 8{1,...,k) are in A but s{1, ...,k-1} s{1,...,k-1} = s{1,...,k} is not. Since k 0 s (because s E G), this implies U {k}, and therefore s{ 1,...,k} E F. Hence we have

v(G) < Ek-1 + v(F).

(F) We claim that

83 v(F)=v{s:sEA and sU{k}VA}< 1-2b'

(G)

To see this, let L = {t : k V t and either t or t U {k} is in Al. We have

{t:t,tC OA}D{t:k

t and

;{t:k¢t

t,tU{k},tC\{k},tC$A} t,tC \{k}¢L}

= M, say, and therefore by (A3) we have (H)

v(L) >

v{t:k0t}-v(M)

>

1-25 4

2

Consider sets

Fo={(s,t)EFxL:snt=sn(tU{k})¢A}, F1={(s,t)EFxL:snt=sn(tu{k})EA}. For t E L define

Jt, t U {k},

if t E A, otherwise.

Thus (A2) implies v2{(s,t) : (s,t) E Fo} < 5 and v2{(s U {k},t) : (s,t) E F1} < 5.

Since s -+ s U {k} is a 1- 1 mapping on F and t H t is a 1- 1 mapping on L,

28

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

this implies that v2(F x L) = v2(Fo) + v2(F1) < 26. Therefore (H) implies that v(F) < 86/(1 - 25), as required. Finally we have A

A

= Ek - v{s E A2({1, ... , k}) : k ¢ s}

> Ek - Ek-1 -

86J

8J

> 't

(by (F) and (G))

1

(by (E))

26

Therefore

v(A2({k})) > v{s : k E s and s, so U {k} E A}

>2v{sEA:kEs}-1 _

86

>2 y 125

_

1

2

1

> 2(y - 85) -

2

We claim that v(A2({k})) > (1- 2ry - 25)/4. By the above, this inequality reduces to 10y > 3 + 625, which is satisfied since 5 < 1/156 and y = (1/2 - 5)2 - 205. By (D), we have v(A2({k})) = E{k} > (1 - 25+ 2y)/4 and

v{sEA:kEs}>v(A2({k})-v{sE A:sU{k} 0A}> 1- 285 2

Also, v(AA(k)) = v(A)+v((k))-2v(An(k)) < 1/2+5/2+1/2-(1-285) < 295.

0

REMARK 1.8.4. Note that the proof of Theorem 1.8.3 shows that (under the same assumptions and using the same notation) for all I D k we have (I)

I v{s:s,$EA}>

1 - 25 + 2,y 4

1 - 25 + 2-y >

4

and lim6.o+(1 - 25+2ry)/4 = 1. PROOF OF THEOREM 1.8.1. Assume H: P([l,m)) P([l,n)) is an E-approximate homomorphism with respect to a measure y on P([1, n)). For j < n let

Aj = {s E P([l,m)) : H(s) 3 j}. Let 5 = 1/156, and for i = 1, 2,3 let Ni be the set of all j < n for which (Ai) fails for the set Aj. Since for all s, t E P([1, m)) we have µ((H(s) U H(t))AH(s U t)) <

e, Fubini's theorem applied to the product (P([l,m)))2 x P([1,n)) implies that µ(N1) < s/5. To get a bound for p(N2), note that p((H(s) n H(t))AH(s n t)) = µ((H(s) n H(t))COH(s n t)C) < µ((H(s)C U H(t)C)A(H(sC) U H(tC)) + µ((H(SC) U H(tC))..H(sC u tC))

+ µ(H((s n t)5)AH(s n t)C) < 4e.

1.8. APPROXIMATE HOMOMORPHISMS AND NON-PATHOLOGICAL SUBMEASURES 29

By Fubini's theorem we have µ(N2) < 4e/6, and similarly, µ(N3) < E/6. Therefore 3

u UNi < a

W)

i=1

For every j E {1..... n} \ U =1 Ni the set Aj satisfies assumptions of Theorem 1.8.3, so for such a j let k(j) < m be such that v(Aj n (k(j))) > 296. Assume for a moment that Ni = 0 for all i, and define a homomorphism ': P([1, rn))

P([1, n))

by

(1 - 25 + 2-y)/4 and recall that (m 1 when 6

Let us write

0.

CLAIM 1. The homomorphism D is a (10/( + 1)E-approximation to H. PROOF. Assume it is not. Thus for some I C 10

m} we have

+1)

E.

\H(I1)) C H(I) AH(IC)Cand the measure of the latter Since (H(I) \ set is not bigger than E, we can assume (possibly by replacing I with IC) that µ(ID (I) \ H(I)) > Using the notation defined in the proof of Theorem 1.8.3, by (I) we have

v{s:s,sl EAR}>( for all j E (D(I) \ H(I). By Fubini's theorem, there are a C1 C -1)(I) \ H(I) and a u E P([l,m)) such that 5E

µ(C1)( -

= 5e

and u, u' E njEC, Aj. Finally we have

0 = µ(C1 n H(I)) > µ(C1 n (H(u) n H(u' ))) - 5e = µ(C1) - 5E > 0, a contradiction.

If the set U3j=1 Ni is not empty, then extend the function k obtained using Theorem 1.8.3 by letting k(j) = 1 for all j in this set, and define as in Claim above. Then the proof of the Claim shows that

v (((s)H(s) n

(P([1, n)) \ U Ni/ i=1

( 10 + 1

/ <\

\ E.

(

This shows that P is a (10/( + 1 + 6/6)-approximation of H, for any 6 < 156. The definitions of Ad's and 4P did not depend on either p or E, and therefore we have proved the following stronger version of Theorem 1.8.1: THEOREM 1.8.5. For every map H: P([1, m))

P([l, n)) there is a homomor-

phism (D: P([1, m)) -, P([I,n)) such that for every measure µ on {1...... .} and every E, if H is an E-approximate homomorphism with respect to µ, then 4, is an e KM-approximation to H.

30

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

PROOF OF THEOREM 1.8.1. Let H: P([1,m)) -> P([l,n)) be an e-approximate (with respect to cp) homomorphism, and let D: P([l,m)) --, ?([1,n)) be as guaranteed by Theorem 1.8.5. To prove that D is as required, fix s c- P([1,m)). let u= H(s)AqD(s) and find a measure p< ep on P([l, n)) such that P(cp) p(u) = cp(u). Such a p exists because P([1, n)) is finite, and we have sup = max. Since µ < cp and H is an E-approximate homomorphism with respect to cp, it is an Eapproximate homomorphism with respect to p as well. Therefore Theorem 1.8.5 implies W(u) = P(cp) µ(u) < E K,e. Theorem 1.8.1 fails for arbitrary submeasures by the following result. THEOREM 1.8.6. The non-exact homomorphisms with respect to arbitrary submeasures are not stable. Namely, for every M < oo there is a submeasure cp such

that K, > M. PROOF. For m > 23M+2 let n = 22'', so we can identify {1, ... , n} with the set N of all subsets X of P([1, m)). We shall denote the elements of N by X, Y, Z. Define H: P([1, m)) --* P([l,n)) by

H(s)=Bs={XEN:sEX}. For k = 1,. .. , m let (k) denote the principal ultrafilter {s E ?([l, m)) : k E s}. Define the submeasure cp = supported by N by letting W{(1), (2)_., (m)} = 0 and for A disjoint from {(1),..., (m)} let
Y nX L(k)nXforallYEAandk=1,...,m}. In other words, if for X E N we define

Cx={YEN:(YA(k))nX¢O forallk=l,...,m}, then cp(A) is equal to the smallest size of X E N such that A C Cx. CLAIM 2. The function cp is a submeasure.

PROOF. The monotonicity of cp is trivial, while its subadditivity follows from the formula Cx U Cy C Cx uy CLAIM 3. The function His a (3 + c) -approximate homomorphism with respect to cp for every e > 0.

PROOF. Note that for every s E P([l,m)) we have H(s)CAH(s0) C C{s,SC}' therefore co(H(s)CAH(sC)) < 2. Similarly for all s, t we have (H(s) U H(t))AH(s U t) C C{s,t,sut}, thus cp((H(s) U H(t))AH(s U t)) < 3.

If K,,, _< M then H can be 3M-approximated by a homomorphism D. For

s E P([1,m)) let As ='(s)AH(s), let Xs* C P([1,m)) be of size 3M such that As C Cx,., and let Xs = Xs * U{s}. Then for k = 1, ... , m we have ,D({ k})

H({k}) \ A{k}

Since any Y E H({k}) satisfying (YA(k)) n X{k} = 0 is not in A{k}, we have Id1({k})J > I{Y E H({k}) : (YA(k)) n X{.} = 01I

_ I{Y E N: {k} E Y and (YA(k)) n X{x} = 0}I

22m-3M-2

1.9. THE LIFTING THEOREM AND THE NON-LIFTING THEOREM

31

The sets 4)({k}) (k = 1, ... , m) are pairwise disjoint, so we have m

n > U D({k}) > m

22m-3M-2 > 22m = n

k=1

a contradiction. Therefore K,o > M, as promised.

1.9. The lifting theorem and the non-lifting theorem In this section we shall use results of §1.5 and §1.8 to prove that for the class of natural analytic P-ideals Todorcevic's hypothesis is true, but that it fails for pathological ideals. An analytic P-ideal I is pathological if for every submeasure we have P(cp) = oo. The following lifting theorem for co such that I = non-pathological analytic P-ideals is the main result of Chapter 1. THEOREM 1.9.1. If an analytic P-ideal.T is non-pathological, then it has the Radon-Nikodym property.

PROOF. This follows immediately from Theorem 1.7.4 and Theorem 1.8.2.

This result shows that Todorcevic's hypothesis (Conjecture 1.4.1) is "true for all practical purposes"; let us list some of its specific instances. COROLLARY 1.9.2. Every summable ideal has the Radon-Nikodym property.

PROOF. Summable ideals are induced by measures and therefore non-pathological.

COROLLARY 1.9.3. Every Erdos-Ulam ideal has the Radon-Nikodym property.

PROOF. Erdos-Ulam ideals are non-pathological by Theorem 1.13.2 and Theorem 1.13.3.

COROLLARY 1.9.4. The ideal 0 x Fin has the Radon-Nikodym property. PROOF. This is because 0 x Fin = Exh(cp) for a non-pathological submeasure W : A ¢ [n, oo) x N}. defined by W(A) = max{2-n

The class of non-pathological ideals is rather extensive since it includes essentially all analytic P-ideals occurring in the literature. As we have seen, all P-ideals given in Example 1.2.3 are non-pathological because natural submeasures inducing them satisfy = cp. A large class of non-pathological ideals was constructed by Louveau and Velickovic ([87]), who proved that the structure P(N)/Fin under the ordering of almost inclusion embeds into the class of analytic ideals ordered by
THEOREM 1.9.5 (Non-lifting theorem). There is an analytic P-ideal without the Radon-Nikodym property. In particular, Todorcevic's hypothesis may be false for ideals generated by some pathological submeasures. PROOF. This follows immediately from Theorem 1.7.2 and Theorem 1.8.6.

32

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

The ideal constructed in the proof of Theorem 1.9.5 is Fo, but there is also an F, ideal which is not F, and without the Radon-Nikodym property (recall that these two classes exhaust all analytic P-ideals). If in the proof that -'(1)

implies -(2) in Theorem 1.7.2 we define co to be the supremum of all cpi's, then the ideal Exh(cp) is not Fo because 0 x Fin is Rudin-Blass reducible to it.

Since the homomorphism constructed in the proof of Theorem 1.9.5 is not a monomorphism from P(N)/ Fin into P(N)/J, it is natural to ask whether every Baire monomorphism from P(IE)/Fin into a quotient over an analytic P-ideal has a completely additive lifting. However, the answer is once again in negative.

This is because if IF: P(N)/Fin - P(N)/,7 has no lifting of the form 4)h and ID: P(N)/Fin --+ P(N)/Fin is a Baire isomorphism then 4' ®W is a Baire monomorphism of P(IN)/ Fin into P(N ® N)/,7 ® Fin with no completely additive lifting.

1.10. Permanence properties of quotients In [130, Theorem 8], Todorcevic proved that if J
LEMMA 1.10.1. If I is an Fo ideal and the quotient algebra P(N)/,,7 is Baireembeddable into P(INY)/I, then ,7 is an FQ-ideal as well. PROOF. By Lemma 1.3.2, the embedding has a continuous lifting, and therefore 3 is Fo as a continuous preimage of I. The above proof clearly shows that the above lemma is true for Fa6 ideals, and the ideals in any other pointclass closed under continuous preimages.

PROPOSITION 1.10.2. If I f is a summable ideal and 3
PROOF. If 4)h is a lifting of a homomorphism of P(N)/9 into P(N)llf, then for the function g(n) P') we have 3 = I. PROPOSITION 1.10.3. If I is a non-pathological analytic P-ideal and 3
PROOF. Let ON be a lifting of a homomorphism from P(N)/,7 into P(N)/I. Let I = Exh(Np1) for some non-pathological submeasure oz (we can assume c07 is non-pathological by the Fact above). Then the pull-back submeasure oj defined

by (PAS) = 01('Ph(s)) is non-pathological as well. To see this, pick s C N and find a measure A < oz such that /-Z (4D h(s)) = c01 (4D h(s)). Then v: P(N) -- [0,oo]

defined by v(A) = p.(Dh(A)) is a measure dominated by o9 and coj(s) = v(s). is equal Therefore the submeasure oj is indeed non-pathological. Also. 0 to the set of all A such that Oh(A) E I, and therefore to J. The following variant of Proposition 1.4.6 gives a strong answer to the basic question for non-pathological analytic P-ideals. PROPOSITION 1.10.4. If I is a non-pathological analytic P-ideal and the quo-

tients P(N)/I and P(N)/9 are Baire-isomorphic, then the ideals I and 3 are isomorphic.

1.11. SIMPLE F. P-IDEALS WHICH ARE NOT SUMMABLE

33

PROOF. By Proposition 1.10.3, the ideal J is non-pathological and therefore both I and J have the Radon-Nikodym property, hence the conclusion follows by Proposition 1.4.6.

1.11. Simple FQ P-ideals which are not summable Pathological submeasures have already been used to construct Fo ideals with peculiar properties. Mazur ([95, Theorem 1.9]) constructs an Fo ideal which is not included in any summable ideal using one of the most frequently rediscovered examples of a pathological submeasure (see [136], [122], [66]). Mazur's ideal is not, a P-ideal, but note that our Theorem 1.9.5 gives an example of an F,, P-ideal which is not summable. This ideal is clearly pathological, and Example 1.11.1 below shows that there can even be a non-pathological F,, P-ideal which is not summable.

EXAMPLE 1.11.1. A non-pathological FQ P-ideal I on N which is not summable. For k = 0, 1, 2,... let Ik = [2k, 2k+1) and define submeasures Ok : P(Ik) -+ R+

0: P(N) --> R+ and an ideal l by +Gk(S) =

mink, Is!) k2

Gk(A n Ik),

, P (A) _ k=0

I = {A : V)(A) < oo}. We claim that I is not equal to a summable ideal If for any function f. Assume

this is not true, and let f : N --* R+ be such that If = I. We claim there is an integer M such that (1)

U{Ik : F(Ik)/iJ(Ik) < M} E I.

Assume this fails for all M; then we can pick a sequence of finite disjoint sets wf such that for all j (recall the notation u f(A) = r-EA f (n)): -,/f' (UiEw; Ii)

'(Ui j2 -

,

I,) >

"

<

(u'Z

j2

t(EWJ

This sequence is constructed recursively as follows: If wl,... , wm are chosen, let 1 > m + 1 be such that 21 > max Uj I w,,,,, and consider the set A,+, of all k > I m + 1. Then this set is not in the ideal, and, since for which 1,b(Ik) < 1/(m+1)2 for all k E A,,,,+1i we can pick a finite w,,,,+1 C A,+1 such that Ik) < 2/(m + 1)1. Let W = Uf w3. Then the set 1/(m + 1)2 X = U li iEW

34

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

is in I \ If. This is because for all j we have

U Ij >i--1 =

, and 1

Ew 1

UU

iYj

7

<

2

This finishes the proof that there is an integer M satisfying (1). So let M be a fixed

integer satisfying (1) and let A be the set of all k for which µf(Ik)/0(Ik) < M; note that >kEA 1/k = on. Then for k E A there is an Sk C Ik of size k such that

Mk _M

Af(Sk) !5T* 2k

2k

and therefore Y = UkEA sk E If \ I which is again in contradiction with our assumption that I and If are equal. The above proof gives a more general fact: PROPOSITION 1.11.2. If {co,} ,,is a sequence of submeasures with pairwise disjoint finite supports, then the following are equivalent:

(a) The ideal Exh(Z. (o,) is summable. (b) There is a sequence of measures {vn} such that supp(vn) = supp(
For a while it was unclear whether there are any Fo P-ideals more substantially different from summable ideals (see [75], [96]), in particular which are not of the form Exh(En fin) for some sequence of submeasures with pairwise disjoint finite supports. The discovery of such ideals (see [113], [36], [39]) required importing ideas from the theory of infinite-dimensional Banach spaces, in particular in [36] and [39] we have used the Tsirelson space, an infinite-dimensional Banach space which does not include a copy of co or any Pr, for p > 1 (see [137]). Let us note that the ideals of [113], [36] and [39] are non-pathological.

1.12. The structure of summable ideals Recall that summable ideals are ones of the form

If = {A :

f(n) < oo} nEA

for some positive function f such that E. f (n) = no. A summable ideal is dense if limn f (n) = 0. Since the summable ideals have the Radon-Nikodym property (Theorem 1.9.2), the relation of Baire-embeddability of quotients coincides with the much simpler Rudin-Blass ordering in the realm of summable ideals. This is why the large part of this section is devoted to the study of Rudin-Blass-ordering. In particular, we shall prove the following:

THEOREM 1.12.1. The structure of all dense summable ideals ordered by
(d) it is a dense ordering, namely for all a < b there is c such that a < c < b.

1.12. THE STRUCTURE OF SUMMABLE IDEALS

35

The proof of Theorem 1.12.1 will occupy a large part of this section. K. Mazur ([94]) has proved (using an idea of Louveau-Velickovic ([87])) that the structure of all Fo quotients ordered by <_BC is complicated in the sense that it includes an isomorphic copy of (P(N)/ Fin, C*). Theorem 1.12.1 (c) above shows that, with respect to the ordering
to f,

µf (s) = Ef(k) kEs

so that Tf = {A : pf(A) < oo} = {A : limn uf(A\ [1, n)) = 0}. The following lemma gives a characterization of the
function h: N -* N is a reduction of Zg to If if and only if there are positive real numbers c < C such that the set

A[c,C]={n: c
is in Z! and h-1(A) is in Zg .

l

PROOF. If h is of this form, then it is easily seen to be a reduction. Assume h is a reduction, yet the above conditions fail. There are three possibilities. CASE 0. If there are 0 < c < C < oo such that A = A[c, C] E Zf but h-1(A) is not in 19, then N \ A is in If but h-1(N \ A) is not in 7g, which is a contradiction. CASE 1. For every c > 0 the set

A[c, ) = n : c < Ag(h-1(n)) f(n)

is not in I. Therefore we can inductively pick a sequence of disjoint finite sets p,,,, C N \ A[1/m2, ) so that 1 < pf(p,,,,) < 2 for all m. Let B = U,.,.=1 pm; then

B is not in If, but A.(h-1(B)) < 2E°=11/m2 and therefore h-1(B) is in I

,

contradicting the assumption on h. CASE 2. For every C < oc the set A(., C] = S n : Ag(h (1)n)) < C,} l n

is not in I. Therefore we can inductively pick a sequence of disjoint finite sets 2 for all m. Furthermore, we can assume that min(p.) is large enough so that f assumes only values less than 1/m2 on p,,,.

p,,,, C t`N \ A(., m] so that 1 < A f

Therefore we can find a disjoint partition mz pi>,,

per,. = i=1

36

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

so that µf(p;n) < 2/m2 for all i. Since µg(h-1(p,,,,)) > m, there is i = i(m) _< m2

such that µg(h-1(p;,i)) ? 1/m. Let C = p'-(m). Then Af(C) < Emm=12/m2 but µg(h-'(C)) O°=11/m, and therefore C is in If but h-'(C) is not in Ig, contradicting the assumption on h. Let us now consider other, not necessarily dense, summable ideals. Recall that

for an ideal I its orthogonal, 1, is defined by

I1={A: AflBis finite for all BEI}. An ideal I is atomic if is generated by a single set over Fin. LEMMA 1.12.3. There are four disjoint classes of summable ideals I f, and each summable ideal belongs to one of them:

(Si) Atomic ideals: For some e > 0, the set A f,+ = {n : f (n) > E} is infinite and µf (N \ Afe+) < oo. (S2) Ideals of the form Fin Dense: For some e > 0 the set AfE+ is infinite, µ f (N \ AfE+) = oo and limngA fE+ f (n) = 0. (S3) Ideals whose orthogonal is isomorphic to Fin xO: A f.1 \ AfE+++, is infinite for all n for some strictly decreasing sequence {En} converging to 0. (S4) Dense summable ideals: limn. f (n) = 0

PROOF. Let us first note that all four classes are nonempty. For example, if f : N -> R+ is given by

f(2n(2m-1))= m

for rn,n EN,

then the ideal If belongs to (S3). Now we prove that these four classes are pairwise

disjoint. If I is atomic, then I obviously cannot satisfy any of (S2)-(S4). To see that conditions (S2)-(S4) exclude each other, we consider an orthogonal I1 of an ideal I. Let I be a summable ideal. Then I is dense if and only if 11 = Fin, I is in (S3) if and only if I1 is isomorphic to Fin 4, and I is of the form Fin Dense if and only if I1 is an atomic ideal different from Fin but I itself is not atomic. It remains to prove that each summable ideal If satisfies at least one of (S1)-(S4). If A f,+ is finite for all a > 0 then lim f (n) = 0 and If is dense. Otherwise, for some E > 0 the set Afe+ is infinite. If there is a small enough E > 0 such that limnA 1E+ f (n) = 0 then the ideal belongs either to (Sl) or to (S2). Otherwise we can recursively pick a sequence {en} like in (S3).

LEMMA 1.12.4. Let II be an ideal in the class (Si), for i = 1, 2, 3, 4. Then (a) I1 FRB 12 FRB 13 FRB 12 FRB 14

(b) I RBI1 fori=2,3,4. (c) I4 FRB Ii for i = 1, 2, 3.

PROOF. (a) The first relation is proved in [93]. The others will make use of the following.

CLAIM 1. If Ig is a dense summable ideal and If is a summable ideal, then there is an Ig-positive set A such that If
nl1 < n2 < ... < 1

,nk(1) 1

< nl2 < ... <

nk(2) 2

< nl3 < ...

1.12. THE STRUCTURE OF SUMMABLE IDEALS

37

such that iµ9({ni,n2,...,nk(Z)}) - f(i)I

< Zi

for all i. This is possible because limn g(n) = 0. Then for A = {n i E N, j < k(i) } mapping h: A --, N defined by h(n3i) = i witnesses that If
Now we prove the remaining relations, that Iz
(b) Assume h: N -, N is finite-to-one and such that A E I if and only if h-1(A) E Fin. This implies I = Fin, and therefore I is not equal to Ii for i > 2. (c) This proof is similar to the proof of (b).

By Lemma 1.12.4, the only class of summable ideals in which we can expect some structure with respect to M . µf([1, ko]),

(A2) g(k2) > M f (ko), and in particular (A3) ko, k1 > M.

(b) If RB I9.

PROOF. First of all, note that we can assume function f (and g as well) is monotonic, possibly by composing it with a suitable permutation of N. This operation obviously leads to an ideal isomorphic to if -

(a)=(b)If If
as guaranteed by Lemma 1.12.2. Let B = N \ A[c, C] and pick M > C + 1 large enough so that µy(B \ [1, M]) < 1 and µf([1, M]) > 1. Then if ko, k1i k2 satisfy (A1)-(A3) and we let t = [k1, k2] \ h-1 ([1, ko]),

we have

p9(t) ? µ9([k1, k2]) - Cµf([l,ko]) ? (M - C)µf([1,ko]) > 1. If I E t n A[c, C] is such that in = h(l) > ko then by (A2) and the definition of A[c, C] we have

µ9(h-1({m}))

M, g(k2) > g(l) f (m) - f (ko) > f (m) a contradiction. Therefore h-1 (m) is disjoint from t for all m > ko, and t is included C

in B. This contradicts to previously proved facts µ9(B \ [1, M]) < 1 and µ9(t) > 1, together with t C_ [M, oo).

(b) = (a). Assume that (a) fails. Then for some M and all ko > M and

k2>k1>Mwehave (7)

µ9([k1i k2]) > M µf([1, ko])

implies

g(k2) < M M. f (ko).

38

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

Choose some integers No, N1 > M such that µ9([M + 1 , No]) > M . µf ([1, N1]).

We recursively choose a sequence No = k1 < k2 < k3 < ... so that ki+1 is the minimal such that

µ9([ki,ki+1)) > Mf(N1 +i)

(8)

for every i. Then we have i

µ9([M+1,ki+1)) > Mµf([1,N1])+MEf(N1+7) =Mµf([1,N1+i]) j=1

and therefore (7) implies that g(ki+i - 1) < M f (NI + i). This implies that (9)

µ9([ki, ki+1)) = 1.z9([k,, ki+1 - 1)) + g(ki+1 - 1) < 2Mf (Ni + i).

Let h: [No, oo) -* [Ni + 1, oo) be a map which collapses the interval [ki, ki+1) to point N1 +i. Since both the domain and the range of h are cofinite and (8) and (9) hold, function h satisfies the conditions of Lemma 1.12.2 with c = M and C = 2M. Therefore h is a witness for Ig
We are almost ready to start the proof of Theorem 1.12.1. It will be helpful to visualize summable ideals as follows. If If is a summable ideal, consider the increasing unbounded sequence of real numbers ak = ak defined by k

f(k)

ak = i=1

Then If is equal to the family of all A for which the set UkEA[ak_1, ak) has finite Lebesgue measure. On the other hand, every unbounded increasing sequence ak of positive reals determines an f = f {ak} by f (k) = ak - ak_1. LEMMA 1.12.6. If {ag}k>,n, is a subsequence of {ak} for some positive natural

number m, then Ig ,,, and {ak}k>1 are isomorphic, we can assume m = 1. Let n(k) be an increasing sequence such that ak = an (k) for all k. Define h: N -* N by (let n(O) = 0)

h-1(k) = [n(k - 1), n(k)). Then µf(h-1(k)) = g(k) = ak - ak_1 (where ao = 0), so h is as required. LEMMA 1.12.7. Assume f,g are nonincreasing and such that {ak} is a subsequence of {ak}. Let {n(k)} be the increasing sequence determined by ak = an (k). Suppose that for arbitrarily large positive integer N there are N < k < k' such that: (B1) ak, = N an(k), and (B2) g(k') > N N. f (n(k)). Then Ig is strictly below If in the sense of the preordering 2M such that aN > 2aM. If N < k < k' are such that (B1) and (B2) are satisfied, then let ko = n(k), k1 = M + 1 and k2 = k'. Let us check (Al)-(A3) are satisfied:

1.12. THE STRUCTURE OF SUMMABLE IDEALS

39

(Al) µ9([k1, k2]) = µ9([l, k2]) - µ9([l, M]) > µ9(f1, k2l) /2 = ak,/2 > N N. a,h(k)/2 > M µf([1, ko]) (A2) g(k2) = g(k') > N f(n(k)) > M - f(ko)

(A3) This is obvious. Therefore Lemma 1.12.5 implies the desired conclusion, that Zf SRS 19. We are now ready for the proof of Theorem 1.12.1. PROOF OF THEOREM 1.12.1 (a). To prove that there are no minimal elements in the structure of dense summable ideals with respect to the preordering 2i ak(i).

The recursive construction of {k(i)} is as follows: If k(1), k(2),... , k(i) are chosen, pick k(i + 1) large enough so that (11) holds and

f(k(i + 1)) < (i+1)3. A sequence {k(i)} constructed in this manner satisfies (10) and (11). For every i find integers li and k(i) = n(i, 1) < n(i, 2) < . . < n(i, li) = k(i + 1) such that .

(12)

f

1 a2

f

2

< an(i i+1)) - an(i,j) < i2

for all j = 1,...,1 - 1. Note that (10) implies this is possible. Let n(k) be the i E N, j < li} and let g be defined by sequence ak = an(k). Then (12) implies that ¢k+1 - ak converges to zero, and by the construction this sequence is monotonic. Given N > 0, let k = k(N), k' = k(N + 1); then (B1) follows from (11) and (B2) is satisfied because by (12) and (11) we have increasing enumeration of {n(i, j)

:

1

1

ak - ak_I > Nl2 = N N3 > af(k)+l - afl(k), therefore Lemma 1.12.7 implies the desired conclusion.

PROOF OF THEOREM 1.12.1 (b). To prove that there are no maximal elements in the structure of dense summable ideals with respect to the preordering
ak(i+l) > 2i ak(i)

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

40

for all i. Define a1 recursively: assume ai , ... , of (i) are defined so that am(z) _ ¢k(i). Then for j E [k(i), k(i + 1)) partition interval [a9, ag+l) into the pieces of equal length less than both (14)

i (ak(i+l) - ¢k(i+l)-1)

and

2 (¢,,,.(i)+1

- a,. (i)-1)'

Let of (i)+1, ... , a (i+l) be an increasing enumeration of endpoints of these intervals; this describes the construction.

Then function f = f {af } is nonincreasing and lim (att+1 - an) = 0. To see that the conditions of Lemma 1.12.7 are satisfied, fix N > 0 and consider k = k(N), k' = k(N + 1). Then (13) implies (B1) and (14) reads as 9 (ak

9 ) f f > an(k)}1 - an(k) - ak-1

which is equivalent to (B2). An application of Lemma 1.12.7 ends the proof. PROOF OF THEOREM 1.12.1(c). For A C N define fA: N -+ IIR+ as follows:

Let 0 = n0 < n1 < n2 < ... be a sequence of integers recursively defined by A

A

1((2k)!)2,

k

nk+1 - nk = ll2k((2k)!)2,

A,

k E A,

and for i E N let k'(i) be the unique k such that i E [nk , n,+,). Let

_ 1/(2k)!, fA(Z)

1/(2k(2k)!),

kA(i) A kA(i) E A.

In particular we have UfA([nkA,nk+1)) = (2k)!. If A, B are such that A C* B, then Al is almost included in the sequence {a2 B }, IfA
(Al) µf([nk,nti+1)) _ (2k)! > k2(2k-2)! > kLk o (2i)!+1 = kE B1 fB(i)+l (A2) fA(n +1) _ (2k)! > k2k(2k)! = fB(nB) Therefore if the set B \ A is infinite then Lemma 1.12.5 implies P(N)IZfA ¢BE P(N)/ZfB, therefore the mapping A' IfA is an embedding of P(IE)/ Fin into the class of dense summable ideals ordered by
vice versa, then there is an f such that If
f: N -+ R+ by f'(n) = µ9(h-1(n'))

Then If, = If, because h is a Rudin-Blass reduction and pf,(A) =µ9(h-I(A)) for all A. Without a loss of generality we may assume that h-1 (k) = [nk, nk+1) for some sequence

0=no
1.13. THE STRUCTURE OF DENSITY IDEALS

41

For B C N let f B be a mapping defined by dom(f B) = (N \ B) x {0} U U [nk, nk+1) x { 1} kEB

fB(i,j) =

i¢B

f (2) 19(i),

andj=0,

iE

UkEB[nk,nk+1)

and j = 0.

Let ZfB be the summable ideal on the index-set dom(f B) determined by f B. Then for A C B C N. If
Let j = {B

:

ZfB
analytic sets (the characterization of orderings <_RB and <--BE from Lemma 1.12.5 is Borel, and in fact Fo) which are downwards (respectively. upwards) closed, closed under finite changes, and not equal to P(N). Therefore these two sets must be meager (by [58] or [121]; see also Theorem 3.10.1 below) hence there is a set B E P(N) \ (.F U 9). Then ZfB is the required summable ideal.

Although Theorem 1.12.1 gives many quotients over summable ideals which are pairwise not Baire-isomorphic, let us give an explicit example of a pair of such

quotients (see also Corollary 3.4.3). Let Zlf f = {A : n9A 1/ f < oo}. PROPOSITION 1.12.8. There is no isomorphism between quotients over Zl/n, having a Baire lifting. and Zl PROOF. By the Radon-Nikodym property of summable ideals and Proposition 1.4.6, it suffices to prove there is no bijection h: A -* B such that N \ A E N \ B E Zlfn, and C E Zl/,,,n- if and only if h"C E Zlfn. By Lemma 1.12.2, we can also assume that for some 0 < p < q < oo and all n E A such function satisfies

p

This implies that h(n) < qvrn- for all n E A, and since h is 1 - 1 this easily implies that N \ A is not in Zlf,,n-, a contradiction.

Note that Lemma 1.12.3 and Lemma 1.12.5 together show that the ordering
Note that if ideals I and j are dense and not isomorphic, then the ideals Fin ®Z and Fin (D.7 are not isomorphic as well: Otherwise let h: N x {0, 1} -, N x {0, 1} be a 1 - 1 mapping such that A E I if and only if h"A E J and N \ h"N E J. It is easy to see that h"(N x {0})A(N x {0}) is in J, and this implies that the restriction of h to N x {I } is an isomorphism between I and J-a contradiction.

1.13. The structure of density ideals Sets of integers of zero density, i.e., sets A such that lim sup

Afl

[1,n I

n

=0

form a well-known ideal Z0 which has many interesting properties. For example, the famous result of Szemeredi ([120]) says that every Zo-positive set contains

1. B:\IB1'-,%1EASUBABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

42

arbitrarily long arithmetic progressions. Another related classical notion is Slog, the ideal of sets A of logarithmic density zero, i.e., sets such that lim sup

EseAnil,nl 1/i

ton g

n

=

0.

The question whether quotients over these two ideals are isomorphic was asked by P. Erdiis and S. Ulam long ago (see [139, Problem 1.12.11]. [28, p. 38-39], [20, Question 48]), and solved by Just-Krawczyk ([65]) and Just ([62], [64]), relatively recently. Just and Krawczyk ([65]) have generalized the ideals Zo and Slog in the following way. We say f : N -. R+ is an Erdos-Ulam function if (recall the notation

µl(9) = E.E. f (n)) we have µ!(N) = oo

and

f(n)

lim µ!({1....171})

0.

The Erdos-Ulam submeasure coj and the Erdos-Ulam-ideal EU1 (also called f -ideal in [65] and [62]) on P(N) associated with an EU-function f are (recall the notation IIAII,p, = hmn sv/(A \ [1, n)))

v f(A) = sup n

µq (A n {1, ... , n})

µ!({1,-..,n})

W, = Exh(cpj(A)) = {A : IIAIIw, = 0}.

In particular, ?.o = WI and Z1g = EUI/n. In this section we shall define a class of density ideals which further extends the class of EU-ideals. For a submeasure ip let supp(ip) = In : gyp({n}) 96 0}. Submeasures W and ik are orthogonal if they have disjoint supports. Let

at+(ip) =

sup

gyp({k}),

kEsupp(1p)

at (gyp) = kEsinf

p({k}).

DEFINITION 1.13.1. For a sequencep = {µi} of orthogonal measures on N each of which concentrates on some finite set define the submeasure apt, by 'Ps, = sup W . i

Then

Z = Fxh(sa,.) = {A : IIAIIw,, = 0} is a density ideal generated by a sequence of measures (or simply a density ideal).

Note that 0 x Fin is an example of a density ideal which is not an EU-ideal (all EU-ideals are dense). If we drop the requirement that measures concentrate on finite sets then we get an even larger family of ideals including all summable ideals. However, EU-ideals themselves are never F, (see Proposition 1.13.14). Since the submeasure (p, is defined as the supremum of measures, all density ideals are nonpathological, and therefore by Theorem 1.9.1 their important property immediately follows:

THEOREM 1.13.2. Every density ideal has the Radon-Nikodym property.

0

1.13. THE STRUCTURE OF DENSITY IDEALS

43

The following result shows that the class of density ideals in our sense extends the class of EU-ideals in a very natural way. We should note that the seeds of (a) can be found in Oliver's note ((100]). The proof of (b) given below is a simplification

of my original proof due to Max Burke, and it is included here with his kind permission. THEOREM 1.13.3.

(a) Every Erdos-Ulam ideal is equal to some density

ideal Z,. Moreover, we can assume each Fc; is a probability measure.

(b) A density ideal Z, is an Erdos-Ulam ideal if and only if (Dl) sup, IIi II < oo, (D2) lim sup; at+(p;) = 0, and (D3) lim sup; Ilpi II > 0.

PROOF. (a) Consider an ideal El 4f. Since limn f (n)/p f({ 1, ... , n}) = 0, possibly by changing finitely many values of f, we can assume n sup

<

1 .

n>2pf({1,... n}) 4' Pick a sequence 0 = no < n1 < n2 < ... so that n1 is arbitrary and n;+1 is the minimal such that pf({n;+1,...,n;+1}) > pf({1,...,n;}). Then for all i we have f(n;+1) < 5 (1) 1 < pf({ni+l,...,n;+1}) < 1+ pf({l,...,ni+t}) -4 pf({1,...,n;+1}) Define the measure a; with the support {n; + 1, ... , n.,

1 } by

i(A) = pf(An {n; + 1,...,n;+l}) pf({1,.... ni+1}) Although a; is not a probability measure, it will suffice to prove that 2a = .614f (because (1) reads as 1 < IIAi1I < 5/4, so taking Za' for A' = A;/IIA,II would lead to the identical ideal). Since A1(A) <

pf(An {1,...,n;+1}) < Gf(A),

we have ,p,\ < ,p f. CLAIM 1. IIAII,,, = 0 implies IIAII,, = 0 for all A.

PROOF. Assume IIAII,,, = 0, or in other words, limp Ap(A) = 0. Pick e > 0,

and let i > 2 be such that Ap(A) < e for all p > i. For m > k such that n; < k let j be such that n j -I < m < ni, then we have

pf(An [k, m)) < pf(A n Ini, ni)) pf(I1,m]) pf(Ini-l,nt-1)) uf(A n (np, np+1)) < i
<

s

max

4 i
pf(An[np,np+1)) pf(Inp,np+1))

5

< 4e. Therefore

IIAIIv, = lim sup k m>k

pf(An [k, ml)) < 5f pf(11,m1)

4

(by (1))

44

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

and since E > 0 was arbitrary, we have A w, = 0. We therefore have IAII,p, = 0 if and only if IIAII,, = 0, and this implies M f = Exh(cpf) = Za. Now we prove Theorem 1.13.3 (b) Observe that (D3) is equivalent to the assertion that Z, is a proper ideal (i.e., that N 0 Z,), and every EU-ideal is such. Therefore it will suffice to prove that the conjunction of (Dl) and (D2) is equivalent to (a), under the additional assumption that Zµ is a density ideal. Assume now that (Dl) and (D2) hold. We can assume IIµiII < 1 for all i (by possibly replacing each ui jwith pi/M for a large enough positive real number M). Without a loss of generality we can also assume that there is a sequence 1 = no <

n1 < ... such that supp(Ai) = Di = {n, + 1, __ ni+1 }. CASE 1.

There is an e > 0 such that the set AE = {i IIµiII < E}

is either finite or limiEAE IIµiII = 0. Then UEA, SUpp(pi) is in Z,, and therefore we can without a loss of generality assume 11/ii II = 1/2 for all i. Define f : N -* R+ by

f(k) = 2'pi({k}),

if ni < k < ni+1e

f(1) = 1.

Then pf({ni + 1,...,ni}1}) = 2' and pf({1,...,nil) = 2' for all i. Therefore for n E Di we have

pf({1,...,n}) and the value of this expression converges to 0 by our assumption (D2). Hence,

f is an EU-function. Note that ni+1 is the minimal integer n > ni such that pf({ni + 1, ... , n}) > p f({1, ... , ni}). Note also that for every set Awe have Pi (A) =

Af(An {ni+ 1,...,ni+1 Af({1,...,ni})

and therefore pi is equal to the measure Ai constructed in the proof of (a) as applied to the ideal Elf f. Hence this proof implies that ideals Zµ and Elf f coincide. CASE 2.

With AE as defined in Case 1, the sequence 11 pill (i E AE) is infinite and

it does not converge to 0 for every e > 0. Since we can assume lim supi Ilpi II > 0 (otherwise Zµ = 2(N)) there is a sequence E1 > E2 > E3 > ... converging to 0 and such that the set

Aj = {i : ej > IµiII > Ej+1} is infinite for all j. By the discussion of Case 1, if Bj = U Di iE A,

then the ideal Zj = I Bj is an EU-ideal. We claim that I = {A : A n B3 E 1, for all j}. The inclusion "C" is obvious, so let us prove the reverse inclusion. Assume A is not in Z. Then for some j we have that limsupj pj(A) > Ej. This implies A n Uk=1 Bk

1.13. THE STRUCTURE OF DENSITY IDEALS

45

is not in Z, and therefore A n Bk is not in I (and therefore in Ik) for some k < j. This shows that replacing µi for i E Aj with vl _ 2-3 1µi11

does not change the ideal Z. Enumerate Aj = {i(j, k) : k = 1, 2.... } and let k

vk =

vi(j,k).

j=1

Then I = Z,,, and Case 1 applies to prove that I is an EU-ideal. Back to the proof of (b). To prove the other direction, let us first assume (D2) fails. Let E > 0 be such that the set A of all i for which there is a ki E supp(ii ) such that pi({ki}) > E is infinite. Then the set {ki : i E A} does not have an infinite subset in I. Since EU-ideals are dense, I is not an EU-ideal. Let us now assume (D2) is satisfied, yet (Dl) fails. By (a), it will suffice to prove there is no sequence vj of probability measures for which limj at+(v3) = 0 such that Exh(supi µi) = Exh(supj vj). Assume the contrary, and let supp(lii) = Di and supp(vi) = Ei. Let

Si={j:EjnDi#O}.

CLAIM 2. For each m, there are infinitely many i for which there is ti g Di such that µi(ti) > 1 but supj vj(ti) < 1/m. PROOF. We will prove that for infinitely many i there are aj c EjnDi (j E Si)

such that vj(aj) < 1/m and Iii(UjEs, aj) > 1. Take S = 1/(m + 1)2, so that (m+1)(1/m-6) > 1. Consider any of the infinitely many i such that µi(Di) > m+1 and supjES. at+(vj) < S. For j E Si, partition Ej n Di into (at most) m+ 1 pieces each of measure < 1/m. (Take all but (at most) one of the pieces to have measure in the interval [1/m - 6, 1/m).) Let aj be the piece of largest µi measure. Then 1

lµi(Ej nDi), pi(aj) > 7Yt and therefore µi(ti) > pi(Di)/(m+ 1) > 1.

Using Claim 2, we can choose an increasing sequence of i(1) < i(2) < ... so that the sets Si(,) (m = 1, 2, ...) are pairwise disjoint, and find tr,,, C Di(,,,,) so that µi(,) (t,,,,) > 1 but supj vj(t,) < 1/m. Then Ui ti is in Z, \ Z,. Note that the discussion of Case 2 in the proof of Theorem 1.13.3 (b) gives the following general fact:

LEMMA 1.13.4. If I is a density ideal, then there is a sequence to = {µi} of measures such that I = Z. and 11µi11 > 1 for all i. Theorem 1.13.3 also implies the following.

COROLLARY 1.13.5. If I is an Erdos-Ulam ideal and the set A is I-positive, then I I A is an Erdos-Ulam ideal as well. PROOF. By Theorem 1.13.3 (a), ideal I is equal to some Z, such that IIpi II = 1

for all i. Then I I A is equal to Z,,, where v = {µi [ A}. By Theorem 1.13.3 (b), this is an Erdos-Ulam ideal.

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

46

COROLLARY 1.13.6. If N = Un A, is a disjoint partition of N into infinite sets and ZN,,, is an Erdos-Ulam ideal on the set An, then

Z={B : Bf1A,EZµ foralln} is an Erdos-Ulam ideal as well. PROOF. By Theorem 1.13.3, for every n there is a sequence {Ani}iEN of orthogonal measures supported by An, such that µn = supi = FA.ni, and IIA,.ill = 1/n for all i. Then Z = {B : supra lira supi Uni(B) = 0} = {B lim Sup,_i..'Uni(B) = 0}, and Theorem 1.13.3 implies that Z is an Erdos-Ulam ideal.

The following proof of a result stated in [65] was pointed out to me by Max Burke.

COROLLARY 1.13.7. Under the Continuum Hypothesis all quotients over EUideals are homogeneous.

PROOF. In [65] it was proved that under CH all quotients over EU-ideals are pairwise isomorphic. Corollary 1.13.5 implies that all quotients over EU-ideals are homogeneous under CH. Let us state yet another corollary for future reference (see the end of §3.14). COROLLARY 1.13.8. Under the Continuum Hypothesis the quotient P(N)/Zo is isomorphic to its infinite power, (P(N)/Zo)N.

PROOF. By Corollary 1.13.6, a quotient over an Erdos-Ulam ideal is isomorphic to a countable product of quotients over Erdos-Ulam ideals, f°_-1 P(N)/Z ,. Since by [65] all quotients over Erdos-Ulam ideals are isomorphic under CH, the statement follows.

LEMMA 1.13.9. There are four disjoint classes of density ideals Z, and each density ideal belongs to one of them: (Z1) Atomic ideals: infi at- (µ,) > 0. (Z2) Nonatomic ideals which are not dense:

infi at-(.i) = 0 and limsupi at+(µi) > 0. (Z3) Erdos-Ulam ideals: limi at+(µi) = 0 and supi 11µil[ < oo. (Z4) limi at+(µi) = 0 and supi 11µi[1 = oo. PROOF. It should be clear that these four classes are disjoint, because ideals in (Z3) and (Z4) are always dense, and in Theorem 1.13.3 it was shown that these two classes are disjoint. All four of these classes are nonempty-e.g., the ideal 0 x Fin is in (Z2). So it will suffice to prove that every ideal ZN, belongs to one of these four classes. If Zµ is not dense, then it belongs to either (Z1) or (Z2), so let us assume it is dense. By Lemma 1.13.4, we can assume 1[µij[ > 1 for all i, and therefore Z. is in (Z3) or (Z4), depending on whether supi 11µill is finite or not.

The following shows a difference between the EU-ideals and the summable ideals (compare with Theorem 1.12.1), and it moreover shows (together with Theorem 1.13.12 below) that The Schroder-Bernstein property (see remark at the end of §1.12) fails in the realm of quotients over density ideals. LEMMA 1.13.10. £U f
1.13. THE STRUCTURE OF DENSITY IDEALS

47

A special case, Zo
£Uf = Z. and £U9 = Z,,, as guaranteed by Theorem 1.13.3 (a). Then

limg(n)/µy({1,...,n}) = 0 n

translates as limi at-(vi) = 0, therefore we can find 0 = mo < m1 < m2 < ... such that for every i and all mi < j < mi+1 we have

a at-(µi) ? at+(vj)

(*)

mi+i Let ")i = sup.,<j<mi+i vj, Di = supp(lli), Ei = supp(vi) and Fi = Uj=-i+1 Ej.

CLAIM 3. For every i there is a function hi : Fi --* Di such that

1-2i


\i(v)

z

for allvCDi. PROOF. For simplicity assume that Di = {1,. .. , n} and let e = at+(vi) and J = at-(µi), so that (*) reads as e

1

b - i

For each j (mi < j < mi+1) find pairwise disjoint subsets Fil (1 = 1, ... , n - 1) of Ej such that for all k < n:

vj(FjI-pi({l}))I<2E

and

vi

(UFii) -µi({1,...,k} <2e. t=1

These sets are constructed recursively in a straightforward manner. Note that if w e set Fin = Ek \ U 11 Fil then the above formulas remain true for k = n because vj(Ej) = pi(Di). Define hi: Uj Ej -+ Di by

h-1({k}) = U Fjk. j=m. +1

Then for all v C Di and j we have (let b = at- µi): yj(UkEV Fjk) < l.i (y) + 2EIvl < 1 + E d Ai(v) N'i(v)

-

2,

v[ <- 1 + t'

Ivl

which implies 0i(h-1(v))/Ai(v) < 1 + 2/i. The other inequality is proved in a similar way.

If hi (i E N) are as guaranteed by Claim 3, then the function h = ui hi witnesses that £U f
As mentioned before, a question of Erdos and Ulam whether the quotients over Zo and Zing are isomorphic was the motivation for the introduction of ErdosUlam ideals. In [65], Just and Krawczyk used CH to prove that all quotients over Erdos-Ulam ideals are isomorphic. Just ([62], [64]) proved that under a different set-theoretic axiom the quotients over Zo and Ztog are not isomorphic. (Let us

48

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

remark here that by Lemma 1.13.10 this is best possible, since Zo 0 define the functions Fµ, Gµ6: N -> N by

FN(n) = supp(pn) Gµ6 (n) = max{ j

:

Aj(s) > 6 for some s of size < n}.

The function Gµ6 is well-defined because limi at+(µi) = 0.

For example, if Zo = Z. and Zlog = 2v for natural sequences of measures {µi} and {vi} induced by EU-functions like in Theorem 1.13.3 (a), then for 6 > 0

we have FN(n) = 2", Gµ6(n) = O(log(n)), F,(n) = O(a2') (for some a > 1) and G,6 (n) < Klog(n), for a fixed K > 0. THEOREM 1.13.12. If the ideals Zµ and 2 satisfy

limat+(µi) = limat+(vi) = 0 and Gvs o F, = o(n) for all J > 0, then their quotients are not Baire isomorphic. PROPOSITION 1.13.13. Quotients over Zo and Zlog are not Baire-isomorphic.

PROOF. By the above, for Zo = 2µ and Z1og = 2 we have

G,6 o F,(n) < Klog(2") = o(n) for all 6 > 0, and therefore Theorem 1.13.12 implies the desired conclusion.

PROOF OF THEOREM 1.13.12. Assume the contrary, that the quotients are Baire-isomorphic. Since the density ideals have the Radon-Nikodym property, by Proposition 1.10.4 ideals Z. and Z. are isomorphic as well. Let h be a 1-1 partial function witnessing the isomorphism, such that A E Zµ if and only if h"A E 2v. Let Di = supp(µi).

CLAIM 4. There is a 6 > 0 such that for all but finitely many i there is j for which vj(h"Di) > 6. PROOF. Assume otherwise, that for every m there are infinitely many i such

that for all j we have vj(h"Di) < 1/m. Since supp(vj) is finite for every j, we can find a sparse enough subsequence of {Di} whose union A is such that lim supi vj (h"A) = 0, which contradicts the choice of h.

For i E N let J(i) denote the minimal j guaranteed by Claim 4. Then

CLAIM 5. There is an m such that the function J is at most m-to-1 (i.e., the size of J-1(k) never exceeds m). In particular, J(n) # o(n). PROOF. Assume otherwise, that for every m there is a k for which IJ-1(k)I > m. Since limk at+(vk) = 0, we can find such k and sk C supp(vk) so that vk(sk) > 6

and oµ(h-1(sk)) < 2/m. Now it is easy to find an infinite set A such that the set UkEA sk is not in 2 but its h-preimage is in Zµ.

1.13. THE STRUCTURE OF DENSITY IDEALS

49

Now observe that J(n) < G, b o FF(n), and therefore by Claim 5 the function G, j o F, cannot be o(n).

Let us now relate summable and density ideals. Proposition 1.13.14 below shows that these two classes are quite distant in the sense of the
Before starting a proof of Proposition 1.13.14, let us prove a lemma. LEMMA 1.13.15. 0 x Fin
equivalent statement is: There is a partition of N into disjoint I-positive sets F, (n E N) such that I is equal to {B : B n Fn E I for all n}. We shall prove a more general version of Lemma 1.13.15. PROPOSITION 1.13.16. Assume = supi Wi, where {cpi} is a sequence of submeasures with disjoint supports such that limsupi 0 and limi at+(cpi) = 0.

Then 0 x Fin
PROOF. It will suffice to prove that 0 x Fin
Condition limi 0 is equivalent to the assertion that I is a dense ideal, therefore it is not destroyed by applying Lemma 1.13.4. By going to a subsequence of coi, we can moreover assume that at+(cpi) < 2-2i, and use this to find a disjoint

partition si = ti Ut?Ci ... lit such that

ipi(ti) -

<

1

2f - 2i

for all j < i. Let A3 = U°f ti. Then each Af is I positive and we have

BnA3EIforallj,

BEI

{j} x N is an RB-reduction of I [ A3 to Fin, then h = Uj h3 is an RB-reduction of I to 0 x Fin. so if h3 : Af

PROOF OF PROPOSITION 1.13.14. One direction follows from Lemma 1.13.15,

because an Fv ideal cannot be RB-reducible to 0 x Fin, since the latter ideal is not F. Let us now prove another direction. Assume the contrary, and let h be such that A E If if and only if h-1(A) E Z,,. Find a sequence 1 = n1 < n2 < n3 < ... and ti C {n,. .. , ni}1 - 1} such that for all i, j (let Dk = supp(Ak )): (1) 1/i < Af(ti) < 2/i

(2) h-1(ti) n Dk 0 0 and h-1(tf) n Dk 0 for some k implies i = j. The recursive construction of these sequences proceeds as follows: Assume n1,. .. , nk-1 and t1, ... , tk_1 are already chosen. Since the set k-1

E = U ti U h-1(ti) l=1

50

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

is finite, there are only finitely many j such that Dj fl E # 0, therefore we can pick an nk so that {1, ... , nk - 1} includes E and all such Dl's. Now choose tk satisfying (1) (this is possible because Z, is a proper density ideal) and such that min(tk) > nk. This describes the recursive construction, and it is easy to see that (1) and (2) will be satisfied.

Assume that ti (i E N) are chosen to satisfy (1) and (2). Then Uiti is, by (1), not in If, and therefore limb supi uj (h-1(tj)) = e > 0. By (2)

limsupµf (Uti I =limsup(p3(ti)), i

i

I

j

i

and there are subsequences ti of ti and p of Ai such that u'(t') > E/2 or all i, in particular UiEA t'i is Tf-positive whenever A is infinite. But by (1) there is an infinite set A such that UiEA t' is in If-a contradiction.

Classes of density and summable ideals do not exhaust all non-pathological analytic P-ideals-for example, consider ideals of the form T f ® Z. To end this Chapter, let us give an example of a non-pathological analytic P-ideal which is substantially different from both density and summable ideals. EXAMPLE 1.13.17. Let T be the ideal on the set {0,1}
(

A)

sup

1

xE{01}N xFnEA n

This is obviously a non-pathological submeasure. Let .7br be the ideal on {0, 1}
(1) A 0 (Ybr,T). (2) T I A is not summable.

(3) There is a B E T+ ( A such that I [ B is a proper density ideal.

PROOF. For a set in A E Jbr a function f : A - R+ by f (A) = ExfnEA 1/n satisfies I [ A = If F A. Therefore if (1) fails, (2) fails as well. Since (3) obviously implies (2), it remains only to prove (1) implies (3). So let A be such that for every B E Jbr set A \ B is not in T, and let

T(A) = It E {0,1}<1 : A n [t] 0

1)}

The tree T(A) is infinite, so by Konig's lemma it has an infinite branch C. Then for some E > 0 we have (let {0,1}5n be the set of all t E {0, 1} E. n-00 Now recursively pick a sequence 5k of finite chains of {0,1}1N included in A\C such

that E/2 < cp(sk) < E for all k and every t E sk is incomparable with every u E sI for l ¢ k. This is done as follows: If S1, ... , sk are already chosen and satisfy the conditions, pick an n such that Uk 1 si C 10, 11:5'. Let t E C be of length > n and such that [t] fl (A \ C) 0 (Ybr, T). Find s C [t] n (A \ C) such that cp(s) > E/2, and let sk C s be a finite chain such that cp(sk) > E/2. This describes the construction.

1.14. REMARKS AND QUESTIONS

51

If sk are as above, then Uk sk is not in I, µk = (P t sk is a measure for every k, llmk at-(µk) = 0, and cp(D fl Uk Sk) = Supk ILk(D), therefore I I Uk sk is a proper density ideal. By Claim 6, every A such that A) B is .1-positive for infinitely many branches B of {0, 1}<" has subsets A0, Al such that .T [ A0 is a dense summable ideal and I Al is a proper density ideal.

To conclude this section we state an appropriate open problem. An ideal J is co-like if J = Exh(supi cpi) for some sequence of pairwise orthogonal submeasures 'pi. Thus every density ideal is co-like, but co-like ideals form a larger class, including, for example, many pathological ideals. QUESTION 1.13.18. Is there an analytic P-ideal I such that

(i) I is not isomorphic to Fin, (ii) If J
comes from the "basis problem for turbulent actions," and the reader may consult [40] for the background and a closely related Question 12.10.

1.14. Remarks and questions To end this section let us point out to, in our opinion, most important open questions related to this work. QUESTION 1.14.1. Is there a simple characterization of analytic P-ideals with the Radon-Nikodym property?

We do not know whether all pathological analytic P-ideals lack the RadonNikodym property. Let us recall Question 11 from [33]. This is a question of a finitary nature closely related to Question 1.14.1. For a submeasure (recall that (p is the maximal non-pathological submeasure dominated by cp; see §1.8 or §1.9) let C(AP) =

_-

___holl

11011

Then 0 < C(W) < I.

QUESTION 1.14.2. Is limt.1- inf{K, : C(W) = t} = oo? A positive answer to this question can be interpreted as "for every sufficiently pathological submeasure there is a non-exact homomorphism which cannot be approximated by a homomorphism." Therefore, such a positive answer would in

turn imply that the class of pathological ideals coincides with the class of analytic P-ideals which lack the Radon-Nikodym property (by a construction of Theorem 1.9.5). The following weaker form of Todorcevic's conjecture would, if true, still be very useful. QUESTION 1.14.3. Is it true that every isomorphism of quotients over analytic P-ideals which has a Baire-measurable lifting has a completely additive lifting?

52

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

Note that a positive answer would imply that basic question has a satisfactory answer in the realm of analytic P-ideals: Two quotients over analytic P-ideals would be isomorphic if and only if the corresponding ideals are isomorphic. It is interesting

that this question has a finitary version. Let us say that an E-approximate homomorphism H: P([1,m)) -* P([1, n)) (see §1.8) is an E-approximate epimorphism if for every s E P([l, n)) there is a t E P([l, m)) such that o(H(t)As) < E. QUESTION 1.14.4. Is there a universal constant K such that every E-approximate epimorphism can be K E-approximated by a homomorphism? The proofs of Theorems 1.9.1 and 1.9.5 easily give that Question 1.14.4 has a positive answer if and only if Question 1.14.3 does. Some of results of Chapter 1 can be summarized in the following diagram which represents the ordering of Baire-embeddability between analytic quotients: (Z3)

(Z4)

(F, P)

(Z2)

(S4)

r

T

0 x Fin

(S2), (S3)

(T)

EFin

The ordering
(S2) Summable ideals of the form Fin Proper (Lemma 1.12.3). (S3) Summable ideals whose orthogonal is countably generated (Lemma 1.12.3). (S4) Proper summable ideals (Lemma 1.12.3). (T) Fo P-ideals which are not
for every analytic ideal I which is not a P-ideal. By Lemma 1.13.15, 0 x Fin is Rudin-Blass reducible to every density ideal. By a result of Solecki ([112, Theorem 3.3]), it is moreover Rudin-Blass reducible to any analytic P-ideal which is not Fa. Since the ideal 0 x Fin belongs to the class (Z2), it is the minimal element of this class in the Rudin-Blass ordering. By Lemma 1.13.10 for every pair Z,,, and Z, of

1.14. REMARKS AND QUESTIONS

53

Erdos-Ulam ideals we have 2,2
It is interesting that we get a similar picture under any of the five other preorderings defined in §1.2 and also under the ordering <EM (see §3.4) if we assume OCA. For example, one gets a very similar picture even in the rather distant ordering of Borel-cardinality of quotients; see [75]. Let us turn to the problem of determining which analytic ideals which are not necessarily P-ideals have the Radon-Nikodym property? Recall that by a result of Mazur ([95]; see Theorem 1.2.5) every FQ ideal is of the form Fin(cp) _ {A : cp(A \ n) < oo}

for some lower semicontinuous submeasure cp: P(N) -, [0, oo]. Therefore it is nat-

ural to say that an Fo ideal is non-pathological if it is of the form Fin(1o) for a non-pathological lower semicontinuous submeasure cp. The following result from ([70], [72]) nicely supplements Theorem 1.9.1. THEOREM 1.14.5 (Kanovei-Reeken). All non-pathological FQ ideals have the Radon-Nikodym property.

In particular, this result confirms a conjecture from [32] that the Fo ideals constructed by Mazur in ([94]) have the Radon-Nikodym property. It also gives an alternative proof of Theorem 1.6.2, the Radon-Nikodym property of the ideal Fin x 0.

Let us describe another important class of analytic ideals. Recall that an ordinal a is additively indecomposable (or simply indecomposable) if a cannot be represented as the sum of two strictly smaller ordinals. For an indecomposable countable ordinal a let an ordinal ideal Ia be theideal of all subsets of a of strictly smaller order type. Note that the Fubini product Fin x Fin is isomorphic to All these ideals have the property that Ia r A is isomorphic to Ia for every positive set A; we say that such ideals are homogeneous. The following answers a question from [32] and it was proved in [72], where the ideals T. are called generalized Frdchet ideals.

THEOREM 1.14.6 (Kanovei-Reeken). The ideals Ia have the Radon-Nikodym property. In particular, 1,2 Fin x Fin has the Radon-Nikodym property.

An important role in proofs of these two Kanovei-Reeken theorems plays the notion of a Fubini submeasure. These are exactly the submeasures to which the proof of Theorem 1.8.2 applies. Let us say a word about the relationship between

1. BAIRE-MEASURABLE HOMOMORPHISMS OF ANALYTIC QUOTIENTS

54

Fubini and non-pathological submeasures. From results of Christensen ([13]) it follows that every Fubini submeasure is non-pathological. On the other hand, it was proved in [72] that every lower semicontinuous non-pathological submeasure is Fubini.

Theorem 1.14.6 may be related to a question of Galvin ([50]) concerning partially ordered sets P(a) = (P(a)/ZQ., Ca). Theorem 1.14.6 implies, for example, that there is no definable Boolean algebra monomorphism of P(w3) into P(w"). Galvin ([50]) asked whether there is (provably without using any additional settheoretic axioms) a strictly increasing mapping from P(w3) into P(w'°) (see [32, §3], [30] for more details). A related class of topological ordinal ideals was suggested by W. Weiss ([144]).

Let W. be the family of all subsets of a which do not include a subset which is homeomorphic to a in the ordinal topology. Then an application of Ramsey's theorem shows that Wa is an ideal if and only if a is a countable ordinal of the form w'"0 (i.e., a multiplicatively indecomposable ordinal). In [70] the authors prove that all of these ideals have the Radon-Nikodym property, answering a question of author. A similar theory of liftings can be developed in contexts other than the quotients of Boolean algebras; see [38] for a rather general setting. For example, analogous

results to lifting theorems have been proved for Baire homomorphisms between analytic quotients considered as groups (see [38], [72], [70], also [71]). Theorems 1.14.5 and 1.14.6 were first proved in this context. Again, in the heart of all these results lies the Ulam stability of non-exact (group) homomorphisms (see [37]). Groups have "less" structure than Boolean algebras, and this fact has three curious consequences:

(a) The results about liftings of group homomorphisms tend to be stronger than the results about liftings of Boolean-algebraic homomorphisms (see [37, §4], [70]).

(b) The proofs of the stability of group homomorphisms turn out to be simpler than the proofs of stability of Boolean-algebraic homomorphisms (compare §1.8 and [37]). (c) It is impossible to develop a theory of group homomorphisms with arbitrary liftings analogous to Chapter 3 (see [38, §7]).

Even in the context of the ordering of the Borel-cardinality of quotients, where we do not have any algebraic structure at all, it is possible to say a lot about the structure of the liftings-see [40].

CHAPTER 2

Open Coloring Axiom and uniformization In this short chapter we gather some results about the uniformization of coherent families needed in proofs of lifting theorems of Chapter 3 and Chapter 4. Although the needed prerequisites on the Open Coloring Axiom are given in this chapter, it is not intended to be an introduction into OCA; see [125, §8] or [135, §10] instead.

Let us recall the statement of the Open Coloring Axiom. Let X be a separable metric space, and by [X]2 denote the family of all unordered pairs of its elements,

[X]2 = {{x, y} : x# y and x, y E X }. Subsets of [X]2 can be naturally identified with the symmetric subsets of X x X minus the diagonal. A partition (or coloring) [X]2 = Ko U Kl is open if K0, when identified with a symmetric subset of X x X, is open in the product topology. We say that a subset Y of X is Ki-homogeneous if [Y]2 is included in K, (i = 0,1). (OCA) If X is a separable metric space and [X]2 = Ko U Kl is an open partition, then X either has an uncountable Ko-homogeneous subset or it can be covered by a countable family of K1-homogeneous sets. We shall usually say that a set covered by countably many K1-homogeneous sets is a-K1-homogeneous. We should first say a word to clarify our use of the phrase "open coloring." The spaces like P(N), NN, FinN, and the finite products of such spaces, will always be considered with their natural separable metric product topology. In

order to be able to apply OCA to some partition [X]2 = Ko U K1, it suffices to know that there is a separable metric topology r on X which makes Ko open. For example, for X C P(N) and for each x E X we fix an ff E N' consider the partition [X]2 = Ko U Kl defined by {x, y} E Ko if and only if fx(n) # fy(n) for some n E x fly. This Ko is not necessarily open in the topology inherited from P(N), but if we identify X with a subspace of P(N) x NN via the embedding x (x, fx), then this partition becomes open. We shall use this fact tacitly quite often and say only that the partition Ko U Kl is open, meaning that it is open in some separable metric topology.

Indeed, all of our open partitions will be variations of the above, and it can be proved (see (c) of Proposition 2.2.11) that every open partition is isomorphic to one of this form. (Two partitions Ko U Kl and Lo U Ll are isomorphic if there is a bijection between their domains which sends pairs in Ko into pairs in Lo.) We shall frequently say that a coloring [X]2 = Ko U Kl is open if Ko is equal to a union of countably many rectangles, Ko = Ui Uz x Vi. This is equivalent to Ko being open in some separable metric (not necessarily Hausdorff) topology on X (see Proposition 2.2.11). 55

56

2. OPEN COLORING AXIOM AND UNIFORMIZATION

OCA has a strong influence on essentially any structure closely related to the real line (see e.g., [125, §8], [135, §10]). In the case when X is an analytic set, OCA is true even if we replace the word "uncountable" with "perfect" (see Theorem 2.2.12). This fact was already used in §1.5 to prove the Radon-Nikodym property of Fin x 0. A word on the evolution of OCA. The roots of OCA can be traced back to the work of Baumgartner [3, 4] who has discovered the phenomenon that assuming the Continuum Hypothesis one can force, in a ccc fashion, uncountable homogeneous

sets for certain open partitions on sets of reals. For example, one such partition considered by Baumgartner is the incomparability relation of (P(N), c) (see the proof of Theorem 2.2.1). This idea was pushed further by Abraham, Rubin and Shelah who have isolated several axioms stated in terms of the existence of large homogeneous sets for open partitions ([1]). One of these axioms was called OCA and another, SOCA, had the asymmetric form of what we presently call OCA. However, SOCA, as well as all other axioms of [1], applies only to sets of reals of size X21 lacking thus the following crucial component of Todorcevic's OCA: If X is not a-Kl-homogeneous, then it has a subset of size ZZI which is not a-K1-homogeneous.

This new reflection component of OCA does not appear (nor is hinted at) in the work of Baumgartner ([3, 4]) or Abraham, Rubin and Shelah ([1] ). It is in fact this component of OCA that is responsible for many of its fundamental consequences needed for our purpose here, e.g., its effect on the gap spectrum of P(N)/Fin (Cororollary 5.2.5). Furthermore, the axioms of [1], even when taken in conjunction with MA, are consistent with the existence of various `nonreflecting' objects incompatible with Todorcevic's OCA, such as for example (w2iw2)-gaps in P(N)/Fin. This can be easily shown e.g., by using the methods of [81]. A discussion on the evolution of OCA can also be found in [125, 8.17], the place where this axiom has been first introduced.

The organization of this chapter. In §2.1 we give a brief survey of the results of this and the following chapter and put these results in a broader context. In §2.2 we prove standard results about OCA and uniformization of coherent families of functions. These results are extended to uniformization of coherent families of partitions in §2.3, and in §2.4 we note how the properties of the uniformizing function (or a partition) reflect to the members of the coherent family. These results will be very useful in the proof of the OCA lifting theorem (Theorem 3.3.5). In the concluding §2.5 we give some remarks and an open question.

2.1. Why Open Coloring Axiom? In Chapter 3 we study arbitrary homomorphisms between analytic quotients. We have already remarked that the Continuum Hypothesis oversimplifies the problem by making many analytic quotients isomorphic regardless of the structure of the corresponding ideals (i.e., they are isomorphic simply because they are saturated; see [12]). Therefore, in order to hope for any nontrivial result about the rigidity of analytic quotients in the presence of the Axiom of Choice, we have to work under

an alternative to CH. It turns out that the right alternative is a version of twodimensional Perfect-Set Property known under the name of Open Coloring Axiom (OCA). This axiom, since its first exposition by Todorcevic in [125], has found many applications, especially in problems related to the structure of real numbers.

2.1. WHY OPEN COLORING AXIOM?

57

The original purpose for introducing OCA was a study of Hausdorff-completeness

of the structure P(N)/ Fin (more on this in Chapter 5). However, it turned out that OCA implies that P(N)/Fin is complete in many other ways (see e.g., [34]). We should remark that an axiom called OCA (and several of its variations) have been previously considered in [1], but these axioms do not seem to be as useful to us as the axiom we presently call OCA. We will prove that the theory of Baire-measurable liftings, as developed in

Chapter 3, can be extended to arbitrary liftings assuming OCA (modulo some obvious limitations-see §3.2). The reader may be inclined to think of these results as relative consistency results which are merely saying what happens in yet another model of ZFC. We assert that this is not quite the case. Consider, for example, the following result:

THEOREM 2.1.1. For every two given analytic P-ideals T and J the following are equivalent: (1) There is a transitive model M of a large enough fragment of ZFC containing

T, 17, and all countable ordinals such that the quotients over I and 3 are nonisomorphic in M. (2) The quotients overT and 3 are nonisomorphic in every transitive model of a large enough fragment of ZFC containing T, 3 and all countable ordinals and satisfying OCA and MA.

PROOF. It suffices to prove that (1) implies (2). Let be two analytic (and therefore F,6) P-ideals. Then statement "There exists a continuous map P(N) which is a lifting of an isomorphism between P(N)/1 and F: P(N) P(N)/J" is, being E2, subject to Shoenfield's Absoluteness Theorem ([110]) and therefore the nontrivial direction follows from Theorem 3.4.1.

Roughly, Theorem 2.1.1 is saying that if the usual axioms of Set theory are not strong enough to show that the quotients over given two analytic P-ideals are isomorphic, then OCA and MA imply that they are not. In fact, a more precise version of this result (see Theorem 3.4.1) states that two quotients over analytic P-ideals are isomorphic under OCA and MA only when there is a simple lifting witnessing this. This is an instance of a frequently encountered phenomenon in mathematics, that connecting maps between definable structures are usually definable themselves (see [46]). A result analogous to Theorem 2.1.1 in the case of embeddings rather than isomorphisms is the following: THEOREM 2.1.2. For every two given analytic P-ideals T and 3 the following are equivalent: (1) There is a transitive model M of a large enough fragment of ZFC containing

T, 3, and all countable ordinals such that the quotient over I does not embed into the quotient over 3 in M. (2) The quotient over I does not embed into the quotient over 3 in every transitive model of a large enough fragment of ZFC containing T, 9 and all countable ordinals and satisfying OCA and MA. Theorem 2.1.2 follows from Proposition 3.2.2 and Corollary 3.4.7 proved in this Chapter, followed by an application of Shoenfield's absoluteness theorem ([110]). Consider now a topological interpretation of the similar phenomenon (X 3X \X is the Cech-Stone remainder of X):

2. OPEN COLORING AXIOM AND UNIFORMIZATION

58

THEOREM 2.1.3. Assume X, Y are countable, locally compact, topological spaces'. Then the following are equivalent: (1) There is a transitive model M of a large enough fragment of ZFC containing

X, Y, and all countable ordinals such that X* and Y* are not isomorphic in M. (2) X* and Y* are not isomorphic in every transitive model of a large enough fragment of ZFC containing X, Y and all countable ordinals and satisfying OCA and MA.

(This can be deduced from Corollary 4.7.2 using Shoenfield's absoluteness the-

orem, in the same way that Theorem 2.1.1 is deduced from Theorem 3.4.1.) A result analogous to Theorem 2.1.1 is also true in this context: If the usual axioms of Set theory are not strong enough to prove that X* maps onto Y*, then OCA and MA imply that it does not (follows from Corollary 4.7.4). A deeper principle lying behind these two results, called the weak Extension Principle, or wEP, (see §4.1), roughly says that maps between Cech-Stone remainders can often be extended to maps between the corresponding compactifications. Clearly, the dual statement merely says that the existence of a given homomorphism between clopen algebras is witnessed by a simple lifting. The following metatheorem summarizes a part

of the above discussion (where Z and 9 are analytic P-ideals, and X and Y are countable locally compact spaces).

THEOREM 2.1.4. Assume cp is a statement in the language of Set theory saying that

(1) The quotients over .1 and j are not isomorphic, or (2) The quotient over I does not embed into the quotient over ,7, or (3) The remainders X* and Y* are not isomorphic, or (4) The remainder X* does not map onto the remainder Y*. Then is either false in every transitive model of a large enough fragment of ZFC which contains all countable ordinals or cp is true in every transitive model of a large enough fragment of ZFC which contains all countable ordinals and satisfies OCA and MA.

A remarkable consequence of this result is that all sentences that can be put into one of the forms (1)-(4) above and which are not outright false hold simultaneously in a single model! It is well-known that the effect of the Continuum Hypothesis

is quite the opposite from the described effect of OCA and MA. Under CH all quotients over, say, Fo ideals are saturated and therefore isomorphic (see [65], also §1.2 and §3.14), but the liftings which make them isomorphic are very far from being simply definable. Similarly, under CH many remainders of rather different topological spaces are homeomorphic to N*, and N* maps onto any compact space of weight < 2H0 (see [101], also §4.1). Both statements fail badly under OCA and MA. A metamathematical explanation of this effect of CH is provided by a result of Woodin ([145]; see also [146] for a complementary -CH-result).

THEOREM 2.1.5 (Woodin). Assuming large cardinals, if a E'-statement is true

in one forcing extension of the universe then it is also true in any other forcing extension which satisfies CH. 'I.e., countable limit ordinals

2.2. COHERENT FAMILIES OF PARTIAL FUNCTIONS

59

Note that the negations of (1), (2) and the negations of the duals of (3), (4) of Theorem 2.1.4 are all of the form Ei: each one of these statements asserts the existence of a set of reals (a lifting) which satisfies certain projective statement. Theorem 2.1.4, together with Theorem 2.1.5, show that if we want to know the relationship between two analytic quotients (whether they are provably/consistently isomorphic, or whether one provably/consistently embeds into another) we need to

use only two different tests, the CH-test and the (OCA and MA)-test. Note that, while the CH-test requires substantial large cardinal assumptions, the (OCA and MA)-test does not require any. Let us emphasize that Theorem 2.1.4 summarizes only a part of what is true in this context. Another major point is that under OCA and MA the relation between two structures (analytic quotients, or Cech-Stone remainders) has to be

witnessed by a simple lifting, and the assertion that such a lifting exists is of much lower complexity-E2, instead of E1, and therefore absolute by Shoenfield's theorem (see [110])! (Sometimes the complexity of this statement is as low as the complexity of the involved ideals, see Lemma 1.12.5.) This brings the theory of analytic quotients as close as possible to their natural setting in the "Borel world," as discussed in Chapter 1. All this seems to suggest that OCA and MA is, in some

sense, the optimal test for studying not only analytic quotients, but also many related structures. The class of statements to which Theorem 2.1.4 applies can be extended to treat other ideals (Theorem 3.3.6), finite and infinite powers of CechStone remainders (§4.1 and [41], respectively) a larger class of topological spaces (§4.9, §4.10), preservation of Hausdorff gaps by embeddings (§5.9), .... We believe that this is only the beginning of a very promising line of research.

2.2. Coherent families of partial functions Let us recall some results of Todorcevic about OCA and uniformization of coherent families. There results will be slightly generalized in the following section. A family Y' of partial functions on some countable set S is trivial if there is a

function h: S -* S such that f (n) = h(n) for all but finitely many n E dom(f), for every f E F. We also say that h trivializes T, or that F is trivialized by h. The family -F is or-trivial if it can be covered by countably many trivial subfamilies. Two partial functions f,g on some countable set are incompatible if f (i) # g(i) for some i E dom(f) fl dom(g). Equivalently, f and g are incompatible if f U g is not a function. THEOREM 2.2.1 (Todorcevic). Assume OCA. Then every family F of partial

functions on some countable set is either o-trivial or it includes an uncountable subset of pairwise incompatible functions.

PROOF. We can assume that the domain of each function in .P is a subset of the natural numbers. Define a partition [.P]2 = Ko U KI by letting { fA, fB} in Ko if fA and fB are incompatible, i.e., if fA(n) 0 fB(n) for some n E A n B. If we identify functions with sets of ordered pairs and consider .7 as a subset of P(N) x P(N), then this partition is open in the Cantor-set topology, and we may

apply OCA. (Alternatively, note that Ko = Ui Uio x Ui1, for Uij = { fA =; i E

2. OPEN COLORING AXIOM AND UNIFORMIZATION

60

dom(f ), fA(i) = j}, and use Proposition 2.2.11.) Therefore OCA implies that either there is an uncountable set of pairwise incompatible functions, or.7 = Ui Xi, where each Xi is K1-homogeneous. But then the union of all functions in Xi is a function which trivializes Xi. Therefore F is u-trivial. The open partition used in the above proof has played an important role in most

of the later applications of OCA found in the literature; for example, see [142] and Chapter 3. This fact is not surprising, since it is essentially the only open partition (see the proof of (c) implies (b), Proposition 2.2.11). We will state several consequences of Theorem 2.2.1, after recalling some terminology. A coherent family of partial functions indexed by A, for some A C P(N), is a family hA (A E A) such that

(2) hA: A - N, and (3) hA(n) = hB(n) for all but finitely many n E A n B, for all A, B in A. Recall that a partial ordering P is directed if for every pair of elements p, q E P there is r _> p, q. It is a-directed if for every sequence pn (n E N) of elements of P there is r E P such that r > pn for all n. More generally, a partially ordered set is k-directed if all of its subsets of size at most k are bounded. (a) If a directed partial ordering Q is partitioned into finitely many pieces, then one of these pieces is cofinal. (b) If a a-directed partial ordering Q is partitioned into countably many pieces, then one of these pieces is cofinal. LEMMA 2.2.2.

PROOF. (a) Assume Qi C Q for 1 < i < n. If Qi is not cofinal in Q, pick a qi E Q such that qj ¢ q for every q E Q. By the directedness, there is a p E Q

such that qt

U Q.

The proof of (b) is analogous.

In this section, we consider subsets of P(N) as ordered by C*, the inclusion modulo finite. Therefore we can talk about cr-directed, N1-directed subsets of P(N). Variants of the following lemma are given in Lemma 3.8.5 and Proposition 3.13.3 (KI is defined as in the proof of Theorem 2.2.1, i.e., by {f , g} E Ko if f (n) = g(n) for all n E dom(f) n dom(g)).

LEMMA 2.2.3. Let F be a coherent family of partial functions indexed by a or-directed family A: Then the following are equivalent (1) F is trivial,

(2) F is a-Kl-homogeneous.

PROOF. (2) a (1). If h trivializes F = {hA : A E Al, then for s E Fin and

p: s -*Ntheset

.PS,p = {hA : p C hA

and

(hA r (N \ s)) C h}

is K1-homogeneous.

(1) a (2). If F is o-KI-homogeneous, then by Lemma 2.2.2 for at least one of the K1-homogeneous pieces, say f, is the set

{AEA: hAE?-l} is cofinal in (A, C*). Then h = U? is a function which trivializes Y.

2.2. COHERENT FAMILIES OF PARTIAL FUNCTIONS

61

COROLLARY 2.2.4. Assume OCA. Then every coherent family of partial functions indexed by a a-directed set is either trivial or it includes an uncountable subset of pairwise incompatible functions. PROOF. By Theorem 2.2.1, the family 17 either includes an uncountable subset

of pairwise compatible functions or it is a-trivial. But by Lemma 2.2.2, if F is atrivial then it is trivial, and this concludes the proof. COROLLARY 2.2.5. Assume OCA. Then every coherent family F of partial functions indexed by an 111-directed set A is trivial. PROOF. By Corollary 2.2.4, it suffices to note that every bounded subset of a coherent family of partial functions is trivial.

We shall consider the set NH = f f : f : N --* N} with two natural orderings. The first one is <, the pointwise dominance: f < g if f (m) < g(m) for all m. The second one is the ordering of eventual dominance, <*, namely the ordering < taken modulo Fin: we let f <* g if f (m) < g(m) for all but finitely many m. Note that (NH, _<*) is a-directed. By b we denote the bounding number. the minimal cardinal k

such that there is a subset of NH of size k which is unbounded in the ordering <*. Equivalently, b is the minimal cardinal such that 0 x Fin is not b'-directed. If a family of partial functions F is indexed by A = {I' f : f E NN}, then we say that F is indexed by NH and denote a typical element of.F by hf instead of hF f . THEOREM 2.2.6 (Todorcevic). OCA implies that b = w2.

PROOF. This follows by [125, Theorem 3.4] and [125, Theorem 8.5]. See also Corollary on page 87 of [135]. The following was proved in [125, Theorem 8.7]. COROLLARY 2.2.7. OCA implies that every coherent family .77 of partial functions indexed by NN is trivial.

PROOF. This follows immediately by Theorem 2.2.6 and Corollary 2.2.5. COROLLARY 2.2.8. Assume OCA and MA. Then every coherent family of func-

tions indexed by an analytic P-ideal is trivial. PROOF. A result of Todorcevic (see Lemma 5.6.3) implies that every analytic P-ideal is k-directed for every cardinal k less than the additivity of the Lebesgue measure. Since Martin's Axiom implies that the latter is at least l 2 (by [91], see also [82], or [135, Proposition 7.3]), the result follows from Corollary 2.2.5.

So far we have considered only the coherent families of functions with range equal to {O,1}, and we shall now see that this was not a loss of generality. The statement (1) below has been considered before, in particular in connection with Milnor's additivity axiom for strong homology (see [90], [25], [125, 8.7], [129]). It is also known that (1) does not follow from MA ([129]) and that it is inconsistent with CH ([90]). The following is a consequence of a result of Todorcevic reproduced in [25, Theorem 4.1] (or rather, its proof). THEOREM 2.2.9. The following are equivalent: (1) Every coherent family of functions with range N indexed by N5 is is trivial.

62

2. OPEN COLORING AXIOM AND UNIFORMIZATION

(2) Every coherent family of functions with range {0, 1} indexed by NN is trivial.

PROOF. It will clearly suffice to prove that (2) implies (1). Assume (1) fails, and let g f (f E NN) be a nontrivial coherent family of functions whose ranges are included in N. For f E NN define f E NN by f(m) = 2f (m)+1 - 1.

Family f (f E NN) is clearly cofinal in NN, therefore to refute (2) it will suffice to construct a nontrivial family of coherent functions hf : rf --, {0,1}. Let

Af={(rn,2'(2gf(m,n)+1) : mEN}nrf, and let hf: rf - {0,1} be defined by hf1(0) = Af and hf'(1) = rf \Af. We claim that this is a nontrivial coherent family. Let us introduce some notation. For g f : r f

N and gf' : rf' --, N let

A'(gf,gf') = min{n : gf(n,i) = gf'(n,i) whenever n < n and i < min(f (n), f'(n))}, This is clearly well-defined if functions gf are coherent. Family hi (f E NN) is coherent because A'(hf, h f,) < A'(g f, gf, ). To see that this family is nontrivial, assume for the sake of getting a contradiction that it is trivialized by some F : N2 --,

{0,1}. Define F: N2 - N by F(m, n) = min{i : F(m, 2'(2i + 1)) = 0}

if such i exists and F(m,n) = 0 otherwise. Then O'(gf,F [ rf) < p'(hf,F [ rf), and therefore F trivializes family {gf}. COROLLARY 2.2.10. OCA implies that every coherent family of functions with range N indexed by NN is trivial.

PROOF. This follows by Corollary 2.2.7 Theorem 2.2.9.

A nontrivial coherent family indexed by a P-ideal can be obtained from a Hausdorff gap (see [5, pages 96-98) and Chapter 5, in particular §5.12). A uniformization theorem for partial functions into the reals can be found in [135, Theorem 10.6]; see also [135, Theorems 10.3* and 10.3**] for consequences of Corollary 2.2.5. Related

results about coherent families of partial functions from countable subsets of an arbitrary cardinal a can be found in [134] and [21). The remaining part of this section, in which we give two reformulations of OCA,

will not be used later in this text. A subset A C [X]2 is a rectangle if it is of the

formA=U®V={{x,y}:xEU,yEV and x#y}. IfAC P(N) wesay that a family 7 of functions fA : A --, N (A E A) is indexed by A. Such family is

trivial if there is g: N -, N such that fA =* g [ A for all A E A. We say that g trivializes F. We say that F is u-trivial if it can be partitioned into countably many trivial subfamilies. In the following result, only the implication from (c) to (a) and (d) is due to the author; all the other implications are Todorcevic's. PROPOSITION 2.2.11. The following are equivalent:

(a) OCA. (b) For every partition [X]2 = Ko U KI such that Ko is equal to the union of countably many rectangles, X either has an uncountable Ko-homogeneous subset or it can be covered by countably many KI-homogeneous sets.

2.3. COHERENT FAMILIES OF PARTITIONS

63

(c) Every family of partial functions { fA: A -> {0,1}} indexed by some A C P(N) is either a-trivial or it includes an uncountable subfamily .T' such that for all a bin A we have fA(n) 0 fB(n) for some n E An B. (d) The same as (c), but with fA: A -+ N.

PROOF. Note that if Ko is open and X is separable metric, then Ko is equal to some union of countably many rectangles; therefore (a) implies (b). To show the reverse implication, assume that Ko C [X]2 is such that Ko = Ui Uoi (D U1i. Define f : X P({0, 1} x N) be defined by

f(x)={(i,j):xCUUj}. Let Y = f"X and define a partition [Y]2 = Lo U LI by letting {a, b} E Lo if (0, i) E a and (1, i) E b, or (1, i) E a and (0, i) E b for some i. This partition is open (when P({0, 1} x N) is given its natural Cantor-set topology). Note that the f-preimages of LI-homogeneous sets are K1-homogeneous, and therefore if Y is a-LI-homogeneous then X is a-K1-homogeneous. So by OCA we can assume that there is an uncountable Lo-homogeneous subset of Y; let y£ (Z; < w1) be its 1 - 1 enumeration. Pick xt such that f (x£) = y£; then the set {xt : l < ci1} is Ko-homogeneous, and therefore (b) is true. (a) implies (d) will be proved in Theorem 2.2.1. (d) implies (c) is trivial.

To see that (c) implies (b), we fix a partition [X]2 = Ko U KI such that Ko = Ui Uoi ® U1i and prove that one of the alternatives given by OCA is true. Note that each set Uoi n U1i is Ko-homogeneous, and therefore we can assume that all these sets are countable. Since the set Ui Uoi n U1i is countable, it is a-K1 homogeneous and we can assume that it is empty without a loss of generality. For x E X define a(x) C N and fa(x): a(x) ---> {0, 1} by

a(x)={i:xCUoiUU1i} fa(x)(i) = j, if x E U. The two alternatives given by (c) clearly translate into two OCA-alternatives. We shall frequently say that a coloring is open whenever it is equal to a union of countably many rectangles.

Variations on (c) of Proposition 2.2.11 are the main theme of this and the following section, and its slight weakening is equivalent to a statement about gaps in P(N)/ Fin (see the remark after the proof of Theorem 5.2.4). One of the reasons why OCA can be considered a natural axiom is the fact that it has a definable version, the following Principle of Open Coloring for analytic sets.

THEOREM 2.2.12. Assume that X is an analytic set of real numbers and that [X]2 = Ko U KI is its open partition. Then exactly one of the following applies: (1) X has a nonempty perfect Ko-homogeneous subset, or (2) X is a-K3-homogeneous. PROOF. See [135, Theorem 10.1], also [43].

2.3. Coherent families of partitions By [A]NN we denote the family of all finite subsets of a set A. A coherent family of partitions indexed by NN is a family Pf (f E NN) such that

2. OPEN COLORING AXIOM AND UNIFORMIZATION

64

(1) Pf : [rf]
E, = In, oo) x N. A coherent family F of partitions is trivial if there is an n and a partition

P: [En]" -, N such that (3) Pf(s) = P(s) for all but finitely many s C [rf n E,]<', for every f E NN. We also say that P trivializes F on E. THEOREM 2.3.1 (The OCA uniformization theorem). Assume OCA. Then every coherent family of partitions indexed by NN is trivial.

PROOF. Let {Pf: [r'f]
Pf(s) 0 P9(s) for some s c r f n r, This partition is open, if we identify each f with the pair (f, Pf). (Alternatively, note that Ko = Us,m#n Us,m x Us,n, for Us,. = { f s E dom(f ), f (s) = m}, and apply Proposition 2.2.11.) By Lemma 2.2.3 (or rather, its version for coherent families of partitions) and Theorem 2.2.6, every subset of NN of size l is o--KI:

homogeneous. Therefore there are no uncountable Ko-homogeneous sets. By OCA, NN is o-K1-homogeneous. By Lemma 2.2.2, there is a KI-homogeneous subset 7-l of NN which is cofinal in the <`-ordering. By Kunen's lemma 7-l is cofinal in Nln'O°> for some n. Then

P= U Pf fan

is a function whose domain includes [[n, oo) x N]
hgf: [rfl`N x [r9]
LEMMA 2.3.3. A family of partitions indexed by an ideal (or, a set directed under C) I is coherent if and only if the coherence condition holds for all A C A'

in I. A family A generates the ideal I if for every B E I there is an A E A such that B C A. LEMMA 2.3.4. If a coherent family F of partitions is indexed by a set A which

generates an ideal I, then F can be extended to a coherent family of partitions indexed by I which is trivial if and only if F is trivial.

2.4. E2-REFLECTION FOR COHERENT FAMILIES

65

2.4. E2-reflection for coherent families The following two lemmas are extracted from proofs of lifting theorems. LEMMA 2.4.1. Assume F = {h f : If --* N} is a trivial coherent family indexed by NN and [N2]
(1) hf1({i}) is Ko-homogeneous for all f and i.

Then we can choose the trivializing function h for F so that h-1({i}) is Kohomogeneous for all i.

LEMMA 2.4.2. Assume F = {h f : r'f - N} is a trivial coherent family indexed by NN and IN']
We shall prove these two lemmas at the end of this section. The "real" result behind them seems to be Theorem 2.4.4 below. Recall that a formula is E2 if it is of the form 3xdyv)(x, y)

for some quantifier-free formula Vi.

A proof of the following lemma, as well as its strengthening, can be found in [80].

LEMMA 2.4.3. If a subset 7-L of NN is cofinal in the <*-ordering, then there is n E N such that If I In, oo) : f E 7-l} ordered by the strict dominance, <. is cofinal in THEOREM 2.4.4 (E2-reflection for coherent families). Assume R: [N2] N

is a partition, and

Pf:

[]pf]
is a coherent family of partitions indexed by NN. If this family is trivial, then we can choose the trivializing partition P: [[n, oo) x N]
possibly with parameters-true in the structure (let A = [n, oo) x N) (A, N, R I [A]< N, P

[A]
is also true in (let Af = A n F1) (A1, N, R [AfI <", Pf f [Af] `N) for <*-cofinally many f E NN.

PROOF. Let P' be any trivializing partition. By Lemma 2.2.2 and Lemma 2.4.3, there is an n such that the set

7L={f : Pf(s)=P'(s)forallscrffl([n,ooxN)} is cofinal in Nln'O°). Let A = In, oo) x N. Then if (Bi1, ... , ik)(Vil, ...

ik, jl, ... ,.7t, m)

2. OPEN COLORING AXIOM AND UNIFORMIZATION 66

is a formula formula

with parameters m in A U N, let i1, ... , ik E A U N be such that the

is satisfied in (A, N, R F [A]`", Po F [A]`").

Then every f E R such that i1 ,.. . , ik, m E r f satisfies the same formula. To prove Lemma 2.4.1 and Lemma 2.4.2 we shall need only a version of Theorem 2.4.4 for coherent families of functions. PROOF OF LEMMA 2.4.1. For every k c N the statement "there is a k-element

set which is not Ko-homogeneous but the uniformizing function h is constant on

this set" is El. If h is as guaranteed by Theorem 2.4.4 and h-1({m}) is not a set, there is a subset of this set of size k which is not in K0. Then Ko-homogeneous is not Ko-homogeneous for cofinally many f. hf 1({m})

PROOF OF LEMMA 2.4.2. For a fixed pair (i, j) # (i', j') and every k E N the statement "m = h(i,j) 54 h(i',j'), and for every k-element subset s of h-1({m}) the set s U {(i,j), (i',j')} is in Ko" is n1. If h is as guaranteed by Theorem 2.4.4 and h-1({m}) is nonempty and not a maximal Ko-homogeneous set for some k, h-1({m}) such that {(i', j')} U h-1({m}) is Ko-homogeneous. we can find (i', j') Fix (i, j) E h-1({m}) and note that by Theorem 2.4.4 the set hf'({m}) will not

be maximal Ko-homogeneous for cofinally many f.

2.5. Remarks and questions It is natural to ask whether the results of this Chapter can be generalized to

families coherent modulo an arbitrary analytic P-ideal 1. We shall show that this is not the case. Let us say that a family F of partial functions is coherent modulo I if

for all f, 9 E F the set {n E dom(f) n dom(g) : f (n) # g(n)} is in Z. This family is N such that {n E dom(f) : f (n) # h(n)} trivial if there is a single function h: N is in Z for every f E F. It is a-directed if the family {dom(f) : f E F} is a-directed modulo Fin.

THEOREM 2.5.1. There are Fo P-ideals Z and C and a family F of functions coherent modulo I indexed by C such that

(1) F is a-directed (2) F is not trivial, (3) every subfamily of .F of size less than the additivity of Lebesgue measure is trivial, moreover

(4) F is an analytic subset of P(N) x 10, 1}, PROOF. This is a reformulation of Theorem 5.10.2, where it will be proved that there are Fo P-ideals Z, A, and B such that

(a) A fl B E Z, for all A E A and all B E B (A and B are I-orthogonal),

(b) there is noCCBsuchthatA\CEIforallAEAandBnCElforall B E B (A and B are not separated in P(N)/Z).

2.5. REMARKS AND QUESTIONS

67

By Mazur's theorem (Theorem 1.2.5(a)), let coo and cpl be lower semicontinuous submeasures such that A = Fin(coo) and B = Fin(cp1). Let

/(0 ={ACN:cpo(A)<1}, I(1 = {B C N cp(B) < 1}, and let D = {(A, B) : A E ICo, B E K.1, A n B = 0}. Finally, let .F be the family of all f (A,B) for (A, B) E D defined by

f(AB) ({0}) = A,

.f(,; B)({1}) = B.

Then F is clearly closed. It is also coherent, by (a). Since dom(f) (f E F) generates the Fo P-ideal generated by A and B, .F is v-directed. By (b), F is not trivial. Let us now show that Theorem 2.4.4 cannot be extended to cover the coherent families indexed by other analytic P-ideals. PROPOSITION 2.5.2. If I is an analytic P-ideal different from Fin and 0 x Fin, then there is a trivial coherent family of functions {hA: A -> N U {*}} indexed by I and a partition [N]ON = Ko such that hA1({j}) is Ko-homogeneous for all E N U {*}, but for no trivializing function h the set h-1({*}) is Ko-homogeneous.

PROOF. By Corollary 1.2.11, we can assume that the ideal I is dense. By Solecki's theorem (Theorem 1.2.5), let cp be a lower semicontinuous submeasure such that I = Exh(cp); note that limn cp({i}) = 0, since I is dense. We can assume that limticp(N\ [1,k)) > 2. Let Ko = Is : cp(s) < 1}. For A E I let kA be the minimal such that cp(A \ [1, kA)) < 1, and let hA be equal to the identity on A n [1, kA) and to the constant * on A \ [1, kA). Then since hA(n) is equal to * for all but finitely many n E A, the constructed family is coherent. Also, hAl({i}) is K1-homogeneous for every i E N U {*}. But if some h trivializes {hA} then the complement of h-1({*}) has to be in I. Therefore co(h-1({*})) > 1, and h-1({*}) cannot be Ko-homogeneous.

The uniformization and reflection results of this chapter will play an important

role in the proof of the OCA lifting theorem (Theorem 3.3.5). The main result of §3.8 also relies on uniformization of a particular coherent family of functions indexed by a P-ideal. However, the proof that there are no Ko-homogeneous sets is much more complex in this case, in particular it requires invoking Martin's Axiom. PROBLEM 2.5.3. Generalize uniformization results of this Chapter so that they have Lemma 3.13.4 as a consequence.

Let us note that E2-reflection is the most that one can hope for in Theorem 2.4.4, since already the 112-reflection fails in this context. Namely, there is a partition R of N2 and a 112-sentence cp which is true in ([n, co) x N, R) but fails in all (r f n ([n, oo) x N, R) for all f and n. These R and cp are constructed using the fact that in In, oo) x N every vertical section is infinite, while in r f no vertical section is infinite.

OCA is a consequence of the Proper Forcing Axiom, PFA, and the structure of the continuum in all known models of set theory in which OCA is true resembles the structure of the continuum under PFA (modulo the obvious differences implied

by the fact that OCA does not have any large cardinal strength); see [31]. An important and difficult open question about OCA along these lines is the following (see [127]):

68

2. OPEN COLORING AXIOM AND UNIFORMIZATION

QUESTION 2.5.4. Does OCA imply that the continuum has size 112, or equivalently, that every subset of NN of size less than continuum is bounded in the ordering of eventual dominance?

One may say that this subject has started with Shelah's ground-breaking consistency proof ([106]), of the statement "All automorphisms of P(N)/ Fin are trivial." In this proof Shelah introduced oracle-chain condition forcing construction, and his ideas were later systematized by others (see [63], [10]). Later on, the axiomatic approach was suggested by Shelah and Steprans ([107]) who adapted the original proof into the PFA-context. After Todorcevic has isolated the Open Coloring Axiom ([125]), Velickovic ([142]) used OCA and MA to prove that all automorphisms of p(N)/Fin are trivial. We believe that Chapter 3 and Chapter 4 of this monograph show the advantages of the axiomatic approach. This gives an additional interest to the following question (see e.g., [45] or [10] for terminology, and note that the word "lifting" has different meaning than elsewhere in this monograph): QUESTION 2.5.5. Does OCA (or PFA) imply that the Lebesgue measure algebra has no Borel lifting?

CHAPTER 3

Homomorphisms of analytic quotients under OCA 3.1. Introduction The results of this Chapter can be put in proper mathematical context only in the light of §2.1, and the reader may wish to consult this section before proceeding further. We prove, working in the context of OCA (see §2.2) and familiar Martin's Axiom (MA), that arbitrary homomorphisms of quotients over analytic P-ideals are close to ones with Baire-measurable liftings. Most efforts of this Chapter are put in the proof of the result (Theorem 3.3.5) to the effect that under OCA every homomorphism P(N)/ Fin -* P(N)/Z between quotients over analytic P-ideals is obtained as an amalgamation of two homomorphisms, (b1 and (D2:

P(A)/1 id

P(N)/I

P(INY)/Fin d

P(N \ A)/Z so that D1 has a Baire-measurable lifting, while the kernel of 412 is nonmeager (even

ccc over Fin; see §3.3), and therefore not included in any analytic ideal. We shall refer to this result as the OCA lifting theorem. An analogous result is proved in the case when .T is a countably generated ideal (Theorem 3.3.6). A homomorphism (D is an amalgamation of two homomorphisms 11 and 412

if and only if it has a Baire-measurable almost lifting, i.e., a map F such that is in I for all A contained in some nonmeager ideal (Lemma 3.3.4). The OCA lifting theorem immediately implies that (under OCA and MA) every isomorphism between two analytic P-ideals has a Baire-measurable lifting. By the rigidity results of Chapter 1, this fact has a number of consequences. For example, under OCA and MA the ideals Fin, Fin x0, 0 x Fin, II/n,, Zl1Vjj, Zo and 2105 have pairwise nonisomorphic quotients (Corollary 3.4.5). More generally, in the class of non-pathological analytic P-ideals our basic question has a satisfactory answer: Two quotients are isomorphic if and only if the corresponding ideals are isomorphic themselves. Moreover, a version of the basic question stated in terms of the embeddability (instead of isomorphism) relation between quotients has a satisfactory answer as well: P(N)/1 embeds into P(N)/j if and only if there is a reduction of I to J (Corollary 3.4.7). Some of the rigidity phenomena extend to the arbitrary ideals whose quotients embed into some quotient over an analytic P-ideal. If P(IY)/Z embeds into P(N)/J, 69

70

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

and J is an analytic P-ideal, then I is an amalgamation of an analytic P-ideal (reducible to J) and a nonmeager ideal (Corollary 3.5.1). This implies that there are many quotients P(N)/T that are not embeddable into any quotient over an analytic P-ideal (Corollary 3.5.4). We also prove that if P(N)/T embeds into P(N)/ Fin, then I and its orthogonal, 1', do not form a gap (Corollary 3.5.2). Needles to say, all of these results require OCA and MA. An another consequence of the OCA lifting theorem is that all automorphisms of a quotient over a non-pathological analytic P-ideal are induced by 1-1 mappings of the integers (this was proved by Shelah for P(N) /Fin in [106]). Automorphism groups of analytic quotients have proved to be objects worth of study, especially in situations when automorphisms have simple liftings (see e.g., [18]). In §3.6 we give a complete description of Aut(P(N)/Tf) for summable ideals Tf, and prove that it always has a subgroup isomorphic to the free group with continuum many generators. We will construct a summable ideal If such that every inner epimorphism of P(N)/Tf is automatically an isomorphism and prove some other curious properties of these automorphism groups. We also investigate which quotients over analytic P-ideals are homogeneous

under OCA and MA, and prove several results suggesting that P(N)/Fin can be the only one (§3.7). This stands in sharp contrast to the fact that under CH all the quotients over the FQ-ideals and all the quotients over the EU-ideals are homogeneous (see [65]). More consequences of the OCA lifting theorem and its applications to topology can be found in Chapter 4. The proof of the OCA lifting theorem spans from §3.10 through §3.13, but it is presented as a sequence of self-contained results, some of which are interesting in their own right. In §3.10 we characterize hereditary nonmeager subsets of P(N), extending the well-known characterization of nonmeager ideals by Jalali-Naini and Talagrand ([58], [121]). Many of of the results from §3.11 are really results about the complete normed group (P(N)/T, A, 11.11,) (see §1.2) and the stability and completeness properties of partial approximate homomorphisms between such groups.

These objects deserve to be studied in their own right. In the proof of Theorem 3.8.1 and Lemma 3.13.5 Martin's Axiom is applied to "multiply" an uncountable set homogeneous for a certain open partition. This approach is likely to be of use in the future study of liftings. The OCA lifting theorem, together with some results of Todorcevic, implies that the homomorphisms between the analytic quotients frequently preserve their gaps. This result is proved in Chapter 5 (Corollary 5.9.2). A striking consequence of this Chapter's main result is that OCA is, in some sense, an optimal assumption for investigating quotients over analytic P-ideals. This is because, roughly speaking, two such quotients are provably (without using any additional set-theoretic axioms) isomorphic if and only if they are isomorphic under OCA and MA (see §2.1). Finally, let us point the reader's attention to David Fremlin's exposition of a proof of the OCA lifting theorem in [47].

The organization of this chapter. We start by giving examples of homomorphisms with no Baire-measurable liftings in §3.2. In §3.3 the notion of an almost lifting is introduced, and the OCA lifting theorem is stated. The following four sections, §§3.4-3.7, are devoted to proving various consequences of this

3.2. HOMOMORPHISMS WITHOUT BAIRE-MEASURABLE LIFTINGS

71

theorem. In §3.4 (§3.5, respectively) consequences related to the rigidity of analytic (arbitrary, respectively) quotients are investigated. In §3.6 we investigate the automorphism groups of analytic quotients, and in §3.7 their homogeneity. In §3.8 and §3.9 we prove the instances of OCA lifting theorems for homomorphisms into quotients over Fin and Fin x O, respectively. The former section also serves as an introduction to the proof of the general theorem, as it introduces some of the ideas. Sections §3.10 and §3.11 prepare the grounds for the proof of this theorem in its general form. A characterization of hereditary subsets of P(N) which are nonmeager is given in §3.10. In §3.11 the stabilizers (introduced in §1.5) are modified and utilized in study of partial approximate homomorphisms of the complete normed group (P(N)/Z, A, 11 11,o). A `local' version of the OCA lifting theorem is proved in §3.12, and the proof is completed in §3.13. In §3.14 we discuss possible directions for a further study and list some open problems.

3.2. Homomorphisms without Baire-measurable liftings The two examples presented in this section are most likely already well-known. As we have already remarked, all automorphisms of P(N) / Fin can be trivial. However, the situation with homomorphisms rather than automorphisms is by far more

complex. If there is a maximal nonprincipal ideal I on P(N), then P(N)/.T is the two-element Boolean algebra and therefore it is embeddable into P(N)/Fin. This homomorphism, (DO, is not trivial in the sense of §1. It is, however, in some sense close to being Baire, because it has a lifting which is equal to the union of two continuous (even constant) functions. It is neither 1 - 1 nor onto, but we will see that there are nontrivial homomorphisms with either one of these properties and therefore demonstrate the sharpness of Shelah's result by showing that only automorphisms can be expected to be trivial. Recall the ordering P(N)/j and IF: P(N)/Fin --* P(IE)/Z, which is a

homomorphism of P(N)/Fin into P(N®N)/I 3 such that the following diagram commutes:

P(N)/I P(N ®N)/(Z ®9)

P (N) / Finn

P(N)/ where i1: P(N)IT P(N ® N)/2 ® j is given by its lifting A --* A x {0} and i2: P(N)/,7 P(N ® N)/Z ® j is given by its lifting A --* A x {1}. EXAMPLE 3.2.1. A monomorphism between two analytic quotients with no Baire-measurable lifting. Let 4D1: P(N)/Z -+ P(N)/371 be a monomorphism between analytic quotients which has a Baire lifting and let ,72 be an analytic ideal.

Let /C be a maximal ideal extending Z and let 'D2: P(N)//C -- P(N)/,72 be a

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

72

monomorphism. Let J = ,71 ®,72 and let I)

='I (D 4D2: P(N)/I

P(N ®N)/9.

Then OD is a monomorphism of P(N)/I into P(N)/,7 which has no Baire lifting. If we assume otherwise and let 4* be a continuous (see Lemma 1.3.2) lifting of ', then we would have

K = {C : 4*(C)AF(C) E ,7} hence K would be a nonprincipal maximal analytic ideal, which is impossible by a classical result of Sierpinski (or by, say, Theorem 3.10.1).

Note that the method of the above example implies that one cannot hope that the Baire-embeddability and embeddability of analytic quotients coincide. However, the embeddability of quotients over analytic ideals is closely related with the simply definable ordering
I
This, for example, implies that every quotient over a proper summable ideal (see §1.12) is embeddable into any other such quotient; recall that this is false in the case of Baire-embeddability (Theorem 1.12.1). EXAMPLE 3.2.3. An epimorphism between two analytic quotients without a Baire-measurable lifting. Let {U} be a sequence of ultrafilters on N. Define

Fu: P(N)

P(N) by

Fu(A) = In : A E UU}. This mapping is clearly an endomorphism of P(N). If the sequence {Un} is discrete, of pairwise disjoint subsets of N we have A, E U, for i.e., for some sequence every n, then Fu is an epimorphism. To see this, pick an arbitrary B C N. Then

there is C C N such that A. C* C if n E B and A. I C if n ¢ B. This proves that Fu is an epimorphism of P(N), and therefore a lifting of an epimorphism 4): P(N)/Fin P(N) /Fin. Now assume that moreover each U. is a nonprincipal ultrafilter. Then '' does not have a Baire lifting: Assuming otherwise implies it has a continuous lifting as well (Lemma 1.3.2), therefore ker(4) is an Fo ideal. But ker(') is nonmeager because it includes an intersection of countably many maximal nonprincipal ideals (see e.g., Theorem 3.10.1 (b)), so it does not have a property of Baire-a contradiction.

Maps of the form Fu are fairly general. For example, if each U, is a principal

ultrafilter, then the map Fu is of the form 4h (where h(n) = nU,,,). We do not know whether there are homomorphisms of P(N)/Fin other than those constructed in Example 3.2.3 (see Question 3.14.2).

3.3. Almost liftings and lifting theorems In this section we formulate main results of this Chapter and give some of their

applications. Proofs of main theorems will be postponed to latter sections. The following notion of largeness will be of crucial importance in the proof of the main

3.3. ALMOST LIFTINGS AND LIFTING THEOREMS

73

result of this Chapter. Recall that two sets of integers are almost disjoint if their intersection is finite. DEFINITION 3.3.1. An ideal I is ccc over Fin if there is no uncountable family of I-positive, pairwise disjoint modulo Fin, sets. An ideal which is ccc over Fin is never included in an analytic ideal. Moreover, it can never have the property of Baire, as the following simple Lemma shows. LEMMA 3.3.2. Let ,7 be an ideal on N which includes Fin.

(a) If P(N)/,7 is ccc then ,7 is ccc over Fin. (b) If 9 is ccc over Fin and I
(d) If ,7 is ccc over Fin and A E J+, then ,7 [ A is ccc over Fin. (e) An intersection of two ccc over Fin ideals is itself ccc over Fin. (f) If I is a P-ideal, then it is ccc over Fin if and only if the quotient P(N)/I has ccc.

PROOF. (a) Follows from the definitions, since 3 D Fin. (b) Let h: N N be a Rudin-Blass-reduction of I to J. If A£ (l; < wI) are Ipositive sets which are pairwise almost disjoint modulo Fin, then h- I(At) (t; < wl) are ,7-positive sets which are pairwise disjoint modulo Fin.

(c) If 3 is meager, then by [58], [121] (see also Theorem 3.10.1) we have Fin P(N) is an almost lifting of a homomorphism 4D: P(N)/Fin -* P(N)/I if it serves as a lifting of (D on a ccc over Fin ideal, namely if the set (c, is some lifting of c) {A C N : F(A)A,%(A) E I} includes some ccc over Fin ideal. The following lemma connects the notion of an almost lifting with the notion of amalgamation of homomorphisms introduced in §3.2, and it is a simple consequence of Lemma 3.11.6 which will be proved later on.

LEMMA 3.3.4. A homomorphism ': P(N)/Fin --> P(N)/I has a Baire-measurable almost lifting if and only if D = 4)I ®'CD2 so that 'D2 has a continuous lifting and ker(,PI) is ccc over Fin. PROOF. The reverse implication is obvious. To prove the direct implication, assume there is a Baire-measurable mapping F which is a lifting of 4D on a ccc over Fin ideal J. Then Lemma 3.3.2 (c) implies that 3 is nonmeager, and Lemma 3.11.6 implies that 'D = (PI ®4D2 as required.

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

74

An important consequence of Lemma 3.3.4 is the fact that the Radon-Nikodym Property of an analytic P-ideal applies not only to Baire-measurable liftings, but also to Baire-measurable almost liftings. The main results of this Chapter are the following two lifting theorems, showing that every homomorphism into a quotient over an analytic P-ideal or a countably generated ideal is an amalgamation obtained in a manner similar to Example 3.2.1.

THEOREM 3.3.5 (OCA lifting theorem). Assume OCA and MA and that I is an analytic P-ideal. If 4?: P(N)/ Fin P(ty)/I is a homomorphism, then 4? has a continuous almost lifting. Equivalently, 4i can be decomposed as 4)1 E£ 4?2 so that 4?1

has a continuous lifting, while the kernel of 4?2 is ccc over Fin.

Recall that an ideal I is countably generated if there are sets An (n E N) such

that I = {B : B D An for some n}. Note that Fin and Fin xO are the only nontrivial examples of countably generated ideals (up to the isomorphism).

THEOREM 3.3.6. Assume OCA and MA. If I is a countably generated ideal and 4?: P(ly)/ Fin -+ P(IN)/I is a homomorphism, then 4) has a continuous almost lifting. Equivalently, 4? can be decomposed as 4?1 04)2 so that 1)1 has a continuous lifting, while the kernel of 4'2 is ccc over Fin. The proofs of these two theorems occupy §§3.11-3.13 and §3.9 (the reader may also want to consult the exposition in [47]). We shall first investigate some of their consequences.

3.4. Applications: rigidity of analytic quotients This section is devoted to applications of the above theorems, which will be proved in the remaining sections of this Chapter. We shall first treat the problem of isomorphisms of two quotients over analytic ideals.

THEOREM 3.4.1. Assume OCA and MA, and that 1_7 are analytic ideals and at least one of them is either a P-ideal or countably generated. Then every isomorphism 4?: P(N)/I P(N)/,7 has a Baire-measurable lifting. In particular, quotients over I and J are isomorphic if and only if they are Baire-isomorphic.

PROOF. Assume that J is a P-ideal or a countably generated ideal, and let 4i: P(ty)/I P(N)/,7 be an isomorphism. Decompose 4i = 4?1 E)4'2 so that 4?1 is Baire and ker(4?2) is ccc over Fin. Since ker(4)2) is an analytic ideal which is ccc over Fin, it is improper. Therefore P1 coincides with 4, so the latter has a Baire-measurable lifting, as required.

Let us recall the basic question of this monograph: How does a change of the ideal I effect the change of its quotient? The following gives an answer for a rather rich class of analytic ideals (see the definition of when two ideals are isomorphic in §1.9).

COROLLARY 3.4.2. Assume OCA and MA. If I and 9 are analytic ideals and at least one of them is a non-pathological analytic P-ideal or a countably generated ideal, then their quotients are isomorphic if and only if I and 3 are. PROOF. If the quotients are isomorphic, then by Theorem 3.4.1 they are Baireisomorphic. By Proposition 1.10.4 the ideals are isomorphic as well.

3.4. APPLICATIONS: RIGIDITY OF ANALYTIC QUOTIENTS

75

Recall that Erdos and Monk (see [201) have constructed an isomorphism between P(N)/Fin and the summable ideal P(N)/11/n (see §1.12) using the Continuum Hypothesis. Later on, Just and Krawczyk ([65]) observed that all quotients over F, ideals are countably saturated. Therefore, under CH, Cantor's back-andforth argument is applicable to construct an isomorphism between any two F, and in particular summable, quotients (see e.g., [12]). The following shows that the situation under OCA is different, and it also answers a question of Koppelberg ([79]). COROLLARY 3.4.3. OCA and MA imply that ideals 71/n and Zlf,,n- have nonisomorphic quotients.

PROOF. This follows by the Radon-Nikodym property of summable ideals, Corollary 3.4.2 and Proposition 1.12.8.

The following gives an alternative solution to a problem of Erdos and Ulam about density ideals solved in [65] and [62] (see §1.13 for more details). COROLLARY 3.4.4. OCA and MA imply that ideals 20 and Aog have nonisomorphic quotients. PROOF. This follows from Proposition 1.13.13 and Theorem 3.4.1. COROLLARY 3.4.5. OCA and MA imply that the ideals Fin, Fin x O, 0 x Fin, Zl/n, Zl f, 20 and Z1og have pairwise nonisomorphic quotients.

PROOF. By Corollary 3.4.2 it suffices to prove that these ideals are pairwise nonisomorphic (in the sense of Definition 1.2.7). Neither Fin not Fin xO can be isomorphic to any of the other ideals, since Fin and Fin x 0 are countably generated and the others are not (since a countably generated ideal has to be generated by a single set over Fin). The ideal 0 x Fin is the only one of the remaining ideals whose orthogonal is isomorphic to Fin x0. It follows from Proposition 1.13.14 that a dense summable ideal Zf is never isomorphic to a density ideal, therefore neither of Zi/n is isomorphic to either 20 or 2105. (This can also be proved by observing and that the complexity of a summale ideal is always FQ, while nontrivial density ideals are complete Fob subsets of P(N).) The proof is concluded by applying Corollary 3.4.3 and Corollary 3.4.4. Zlf,/r,-

As we have pointed out earlier, the question of triviality of automorphisms of P(N)/Fin was the starting point of the line of study of analytic quotients to which this work belongs. Recall that an automorphism of P(N)/-T is trivial if it has a completely additive lifting 4h for a 1 - 1 partial function h. By applying Theorem 3.3.5 (or Theorem 3.3.6 and Proposition 1.4.6 (b), we get the following. COROLLARY 3.4.6. OCA and MA imply that if I is either a non-pathological analytic P-ideal or a countably generated ideal, then all automorphisms of P(N)/-T are trivial.

Therefore if I is a non-pathological ideal then under OCA tae automorphism group of P(N)/Z reduces to its subgroup of trivial automorphisms. This will be used in §3.6, which is devoted to study of automorphism groups of analytic quotients. Let us now exploit more power of Theorem 3.3.5 and Theorem 3.3.6 by considering the impact of OCA on the embeddability relation on analytic quotients. Let I and 9 be ideals of sets of integers. The symbol

P(N)/ Z - P(N)/,7

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

76

denotes the fact that P(N)/Z is isomorphic to a subalgebra of P(N)/J (a fact frequently rendered as P(N)/Z is embeddable into P(N)/,7) Let us define another preordering on analytic ideals. (05) Z <EM ,7 if and only if P(N)/Z This preordering, unlike those defined in §1.2, is not absolute. For example, CH implies Z,/. <EM Fin, yet we shall see that this is false under OCA and MA. Also note that Example 3.2.1 shows that we cannot hope that I <EM ,7 is equivalent

to I
PROOF. In Proposition 3.2.2 we have proved that I <_BE 7 implies I <EM .7 Let us therefore assume I <EM ,7, and let qP: P(N)/Z -* P(N)/,7 be a monomorphism. By Theorem 3.3.5, we can decompose 4> as 1P1 ®'D2, where 4)1 has a Bairemeasurable lifting, 4D1.. The set A = 1)1.(N) is not in ,7 because ker(') # ker(
We claim that ker(4)) = ker(4)1). Otherwise, since ker(4)) C ker(qp1), we have B E ker(,D) \ ker(D2). Therefore the ideal I f B is ccc over Fin, which is impossible A, as because it is analytic. So we have I = ker((D) = ker(lD1) and I BE required.

As we have pointed out before, the orderings <EM and
Z
3.5. Applications: quotients embeddable into analytic quotients Now we shall see what can be said about arbitrary ideals on N whose quotients

can be embedded as a subalgebra of some analytic quotient. The following is a slight refinement of Corollary 3.4.7.

COROLLARY 3.5.1. Assume OCA and MA, that ,7 is either an analytic P-ideal

or a countably generated ideal and that Z is an arbitrary ideal on N such that P(N)/Z -+ P(N)/,7. Then l is of the form -To or Zo ®Z1 for some analytic ideal Zo such that 10
For an ideal Z, its orthogonal , 11, is defined as

Z1 = {b : b is almost disjoint from all a E Z}. We say that I and Z1 are separated (or equivalently, they do not form a gap) if there is a c C N such that c almost includes all members of I and is almost disjoint from all members of Z1. The following result was suggested by Todorcevic, and it is related to his results about analytic gaps ([126]).

3.5. APPLICATIONS: QUOTIENTS EMBEDDABLE INTO ANALYTIC QUOTIENTS

77

COROLLARY 3.5.2. Assume OCA and MA and that I is an arbitrary ideal on N

and that P(N)/I - P(N)/Fin. Then I can be separated from its orthogonal, 11. PROOF. Let 4): P(N)/1- P(al)/ Fin be a monomorphism. By Theorem 3.3.5 and the Radon-Nikodym property of Fin, there is a finite-to-one function h: N -> N

such that (Ph is a lifting of D on a nonmeager ideal I. Every nonmeager ideal is dense, so I1 = (I f Ib)1. Now note that (I f1 T(p)1 is an ideal generated by N \ h" N and Fin, and therefore the set h" N separates I from I-i-. Recall that two families A, 13 of subsets of N are countably separated if there are sets Cn (n E N) such that for every pair A E A, B E 13 there is an n such that A C* Cn and Cn fl B is finite. COROLLARY 3.5.3. Assume OCA and MA, that I is an arbitrary ideal on the

integers such that P(N)/I -

and that J is either an analytic P-ideal or

a countably generated ideal. Then I can be countably separated from its orthogonal.

PROOF. By Corollary 3.5.1, 1=10 ®T, where I1 is nonmeager, and therefore dense, and TI is an analytic P-ideal or Fin x O. Therefore by Lemma 1.2.9, 1 is countably separated from 11. COROLLARY 3.5.4. OCA and MA imply that there are quotients P(N)/I that are not embeddable into any quotient over an analytic P-ideal or Fin x 0. PROOF. Fix a Hausdorff gap, i.e., two C*-increasing w1-chains of sets of integers

which cannot be separated by a single subset of N. Note that this implies these two chains cannot be countably separated. Let 1 be an ideal generated by one of these two chains. Then I cannot be countably separated from its orthogonal, and therefore, by Corollary 3.5.2, P(N)/I cannot be embedded into any quotient over an analytic P-ideal.

Another application of Corollary 3.5.1 is its topological reformulation. All results about the structure of homomorphisms of quotients P(N)/I have their topological duals which talk about the continuous maps of closed subsets of the CechStone remainder N* of the integers (equivalently, the Stone space of P(N)/Fin; see [97]). Recall that A C N* is a P-set if for every sequence of open neighborhoods {Un} of A set n. Un includes an open neighborhood of A. In the case when A

is closed, this reduces to fact that the ideal on N corresponding to A is a P-ideal. Just ([60], [641) proved that OCA and MA together imply that no nowhere dense P-subset of N* is homeomorphic to N* itself, answering a question of van Mill ([97, p. 537]). We shall now give a slight strengthening of this result. Note that Example 3.2.3 shows that the assumption of A being a P-set cannot be dropped from the following Corollary.

COROLLARY 3.5.5. Assume OCA and MA. Then (a) If a subset of N* is a continuous image of N*, then it is equal to the disjoint union of a clopen set and a nowhere dense set. (b) Every P-subset of N* homeomorphic to N* must be clopen.

PROOF. (a) This is just the topological dual of Corollary 3.5.1 (see [97]).

(b) We shall prove the dual statement: If I is a P-ideal and P(N)/1 embeds into P(N)/ Fin, then I = Fin ®11 so that the quotient P(N)/Il is ccc. By Corollary 3.5.1, 1 = 10 ®11 where 10
3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

78

Fin. Since a P-ideal is ccc over Fin if and only if the corresponding quotient is ccc (see e.g., [60]), this ends the proof.

An extension of Corollary 3.5.5 and more applications of Theorem 3.3.6 to topology can be found in Chapter 4.

3.6. Automorphism groups As remarked earlier, the fact that the automorphism group Aut(P(N)/I) of an analytic quotient under OCA reduces to its subgroup of all trivial automorphisms can be interesting in its own right. For example, van Douwen ([18]) has noticed that the group Aut(P(N)/Fin) is not simple, under the assumption that all automorphisms of P(N)/Fin are trivial. This gives an example of a homogeneous (see below) Boolean algebra whose automorphism group is not simple. It is still unknown

whether a Boolean algebra with this properties can be constructed without using any additional set-theoretic assumptions (see Question 1 of [20]). This section is devoted to the study of automorphism groups of analytic quotients, with the particular emphasis on summable ideals. All results of this section talk about arbitrary isomorphisms between analytic quotients and their automorphism groups under OCA and MA, so we shall not explicitly state these assumptions. Alternatively, these results can be regarded as results about the subgroup of Aut(P(N)/I) consisting of those automorphisms which have continuous liftings. By Theorem 3.4.1 (see also Theorem 2.1.1 in §3.14), there is no difference between these two groups. Let us first prove that an automorphism group of a quotient over a summable ideal is rather large. Recall the notation µf (A) = ZiEA f(')' PROPOSITION 3.6.1. If If is a summable ideal, then Aut(P(N)/If) has a subgroup isomorphic to the free Abelian group with continuum many generators.

PROOF. Let us first assume If is a proper summable ideal. Then we can moreover assume that f is nonincreasing. Find a sequence 1 = nI < n2 < ... such that for all i

(1) pf([ni,ni+1)) > 1, and (2) µf({ni+l - 1 i E N}) < oo. For B C N let hB : N - N be defined by :

hB (k)

k + 1,

=

k,

if k E [ni, ni+1) for i E B, if k E [ni, ni+I) for i ¢ B.

CLAIM I. $he is a lifting of an automorphism DB of P(IN)/If. PROOF. For £ > 0 let

AE = S n : f(n(-)1) < E

If µf (AE) = oo for every e > 0, then we can find a sequence of finite sets of integers si such that for all i

(3) si CAI/i, (4) max si < min si+l, and (5) Af (si) > 1.

3.6. AUTOMORPHISM GROUPS

79

Let B = Ui si, and enumerate B increasingly as {k3 }. Then by (3), (4) and the

monotonicity of f we have limf f(k3)/f(kj_1) = 0 which implies pf(B) < no, contradicting (5). Therefore we can find an e > 0 such that p. f(A,) < oo. Then for

all kEN\AEwe have 0<E
f(hB'(k)) < 1+ 1 f(k) E

Therefore Lemma 1.12.2 implies that (DB is a homomorphism of P(INY)/If. Moreover, hB is 1 - 1 on the set N \ {ni+l - 1 i E N} and by (2) this set is in the dual filter of If. Therefore 4DB is an automorphism of P(N)/If. :

Let F be the free Abelian group with generators {g£ l E R}, and let A = l; E ]R} be a family of infinite almost disjoint subsets of N. Note that the subgroup G of Aut(P(N)/If) generated by ''B (B E A) is Abelian, since for :

{B£

:

B, C E A we have

Bo4'C=DBUC This is because B fl C is finite, and therefore in If. Let fl : F -* G be a homomorphism uniquely defined by Q

4'B,

for C E IR. We claim f2 is an isomorphism. Since it is an epimorphism, it will suffice

to prove that it is a monomorphism. To see this, we pick a = gl' g12 ... gn in G different from unity and prove that 4D = Q(a) is not the identity on P(N)/If. By the definitions, for h = h' o... h2 ohi', mapping (Ph is a lifting of C Since a # 1F, we can assume l1 # 0 and gi # gl for i = 2, ... , n if n > 1. This implies that h has no fixed points in the set C = Uf9B, [nf, of+1). Therefore, by Lemma 9.1 of [14], we can write C = Co U C1 U C2 so that h"Ci and Ci are disjoint for i = 0, 1, 2. By (1), one of these sets is not in If. Since this set is moved by 1), (D is not the identity. This proves that Il is an isomorphism.

If If is not a proper summable ideal, then Aut(P(N)/If) has a subgroup isomorphic to Aut(P(N)/ Fin), and it is easy to modify the above construction to prove that this group has a subgroup isomorphic to the free Abelian group with continuum many generators. If in the proof of Proposition 3.6.1 instead of a family of pairwise almost disjoint

subsets of N we use a family of independent subsets of N, then by essentially the same proof we have the following:

PROPOSITION 3.6.2. If If is a summable ideal, then Aut(P(N)/If) has a subgroup isomorphic to the free group with continuum many generators.

For a function f : N -* R+ such that µ f (N) = no, define the following two subgroups of the infinite symmetric group S. G f = { IT E S : the set A f, [p, q] = {n : p < f (7r(n))/ f (n) < q} is in the dual filter 1; of If for some 0 < p < q < oo}. Hf = { 7r E S.,) : the set {n : f (7r (n)) = f (n)} is in If}. Obviously H f is a normal subgroup of Cf. THEOREM 3.6.3. Assume OCA and MA and let If be a summable ideal other

than Fin. Then Aut(P(N)/If) is isomorphic to the quotient Cf/Hf.

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

80

PROOF. We define a transformation

A: Sr,

Aut(P(N))

by letting A(ir) be -D,r-, (the homomorphism with the lifting A ti it"A). This transformation is obviously an isomorphism between groups S. and Aut(P(N)). By Lemma 1.12.2, A(ir) = -D,_, is in Aut(P(N)/If) if and only if it is in Gf. Moreover, every automorphism of P(N)/If has a lifting of the form A(ir) for some

it in Gf. To see this, let 4D: P(N)/If --* P(N)/If be an automorphism. By the Radon-Nikodym property of summable ideals and proof of Proposition 1.10.4, 4

has a completely additive lifting 'h for a 1-1 partial mapping h: N - N. Since I is not Fin, we can assume that the domain and the range of h are cofinite and therefore extend h-I to a permutation it of the positive integers. Lemma 1.12.2 implies that it is in Gf. Therefore A maps Gf onto Aut(P(N)/If), and it will suffice to prove that ker(A) = Hf. It is obvious that Hf C kerA, so let us prove the reverse inclusion. Assume that it 0 Hf is such that A(ir) is a lifting of the identity of Aut(P(N)/If). Therefore the set A = {n : ir(n) n} is not in If. By [14, Lemma 9.11 we can split A into sets Ao, AI and A2 such that 7r"Ai is disjoint from Ai for i = 0, 1, 2. One of these sets is not in If, and this set is moved by A(ir), contradicting the assumption that A(ir) is the identity of Aut(P(N)/If). THEOREM 3.6.4. Assume OCA and MA and let I be an analytic P-ideal. Then Aut(P(N)/If) is isomorphic to a quotient of a Borel subgroup of Sx over its Borel ideal.

PROOF. If there is an infinite set in I, then A defined in proof of Theorem 3.6.3 maps some subgroup GZ of S,, onto Aut(P(N)/I), and its restriction to G2 is a lifting of the desired isomorphism by the proof of Theorem 3.6.3. In the case when I = Fin, the ideal N ® Fin is isomorphic to Fin and the above proof applies to it.

Now we shall see that Boolean algebras of the form P(N)/If can have rather curious properties.

THEOREM 3.6.5. Assume OCA and MA. There is a summable ideal If such that every epimorphism I: P(N)/If - P(N)/If is automatically an isomorphism. Before we prove Theorem 3.6.5, we shall prove another property of the ideal If which satisfies its conclusion.

PROPOSITION 3.6.6. Assume OCA and MA. There is a summable ideal If such

that Aut(P(N)/If) is isomorphic to some quotient of the group H,'=, S,!. PROOF. Let f be a nonincreasing function uniquely determined by the following conditions: (i) range(f) = {1/n! : n E N}, (ii) the set

In =

{jl

:.f(i) = 1

n!

is an interval of N, and (iii) p f(In) = 1 for all n. Let A be as defined in the proof of Theorem 3.6.3. It will suffice to prove that every automorphism 4) of P(N)/If has a lifting of the form A(ir) for some it E 11°°_I SI,,. So assume ID E Aut(P(N)/If)

3.6. AUTOMORPHISM GROUPS

81

and let T be such that A(T) is a lifting of (P. Let p, q be such that A f,r [p, q] E 1;, pick k > max(q, 1/p) and n such that

M-1)=

and

ki

f(n) _

(k + 1)!

Then for all i, j which are bigger than k and distinct we have 7r"I%fllf = 0. Therefore

the restriction of T to Af,, \ [1, n] can be extended to an element 7r of rj' 1 Sjn. This 7r is as required. PROOF OF THEOREM 3.6.5. We shall prove that the ideal defined in proof of

Proposition 3.6.6 satisfies the requirements. Assume that ID: P(N)/If -, P(N)/If is an epimorphism. By Theorem 3.3.5 it has a completely additive lifting 'h, and by proof of Proposition 1.10.4 applied to the ideals 9 = If and I = ker(OD) we can assume h is 1 - 1, and therefore that P has a lifting of the form A(ir) for some it E S.. Since ker($) If, there is a q < oo such that the set A(.,q] =

{i:

f f(i))) < q}

is in the dual filter I f of If. This was proved as Case 1 in the proof of Lemma 1.12.2. By the same lemma, to prove that ' is an isomorphism it will suffice to find a p > 0 such that

A[p, ) =

{

i

:

f( f (i)

>p

is in If. We claim that p = 1/q works. So let us assume this is not true, and the set A[p, ) is not in If, namely p f(A[p, )) = oo. Find n > q/(q - 2) large enough so that for some k we have

f(n- 1) = 1 ,

f(n) = (k+1)!

and µf(,r

A(q, ) \ [1, n)) < 1.

q] \ [1, n] be such that pf(t) > ti, and find m > Ti so that t c [n, fn] Let t C and for some k' we have

=f(M)>f(m+1=

(k' + 1)!

Define s C [n, m] to be s = {i E [n, m] : f (i) # f (7r-I (i))}, and note that for every l E [k, k'] we have IIlnsi > IItfltl because it is a permutation. In particular, we have pf(s) > p f(t). Also, since for i E t we have f (7r(i)) < f (i)/k, we have

Af(ir"t) <

of(t) < Qµf(t)

Similarly, since f (7r(i)) < f (i)/k for i E [1, n) such that f (i) > it, we have [l, n]) n [n, m))

k

Therefore we have

'Uf(S \ (rr'[1,n] U t)) > lif(t) - gkf(t) - q > nq

2 q

> 1.

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

82

But i E s \ (ir"[l, n] u t) implies f (7r-1(i)) > i. Since f(i) < 1/(k + 1)! for all i E s, and therefore µf((-7r"A(q, )) \ [1,n]) > 1, contradicting the we have s C choice of n.

Boolean algebras 13 such that every epimorphism 4): P(N)/Zf - P(N)/If is automatically an isomorphism are sometimes called Hopfian (see [20]). Therefore Theorem 3.6.5 says that a quotient over a summable ideal can sometimes be a Hop-

fian Boolean algebra. (Note that this cannot happen under CH, since P(N)/Fin is not Hopfian and all quotients over F, ideals are isomorphic under CH.) Not all quotients over summable ideals share this property. The conclusion of Theorem 3.6.5 is equivalent to saying that P(N)/.Tf is not isomorphic to P(N)/,7 for every analytic ideal J 1, and for Z1/n there are many such ideals J. For example, the ideal generated by 2N over Z11,,. Recall that a Boolean algebra B is dual Hopfian (see [20]) if every monomorphism D: P(N)/If -* P(N)/If is automatically an automorphism. So the following statement (which should be compared to Lemma 1.12.5) says that P(N)/Zf is never dual Hopfian. PROPOSITION 3.6.7. If If is a summable ideal, then there is always a monomorP(N)/If which is not an automorphism. phism ID: P(N)/If PROOF. It suffices to prove this for proper summable ideals, so we can assume f

is nonincreasing. Define an increasing sequence 1 = n1 < n2 < n3 < ... such that

2f(i) I f([ni,ni+1)) < 3f (i) This is possible because we have ni > i and therefore f (ni) < f (i) for all i. Note that this inequality also implies that the size of each interval [ni, ni+1) is at least two. Let h be a mapping which collapses [ni,ni+1) to i. Then 4)h is a lifting of a monomorphism 4): P(N)/Zf -> P(N)/Zf. If it was an isomorphism, then there would be a set A E 11 such that the restriction of h to A is 1 - 1. Since there is no such set, ID is not an isomorphism.

Recall that Zo is the ideal of sets of zero density (see §1.13). QUESTION 3.6.8. Is the group Aut(P(N)/Zo) always simple?

In an earlier version of this monograph it was claimed that Aut(P(N)/Zo) is never simple. David Fremlin has pointed out to an error in the proof, and moreover to the proposition below whose proof is included with his kind permission. PROPOSITION 3.6.9. Assume CH. Then Aut(P(N)/Zo) is not simple.

PROOF. Continuum Hypothesis implies that the algebra P(N)/Zo is homogeneous (Corollary 1.13.7). CH also implies that P(N)/Zo is isomorphic to its countable power, (P(N)/Zo)1V (Corollary 1.13.8). By a result of P. Stepanek and M. Rubin ([143, Corollary 5.9a]), if a homogeneous Boolean algebra B is isomorphic to its countable power, then its automorphism group is simple.

Let us end this section with a question an (unlikely) positive answer to which would simplify some of the above proofs.

QUESTION 3.6.10. Does I
3.7. HOMOGENEITY OF ANALYTIC QUOTIENTS

83

3.7. Homogeneity of analytic quotients Boolean algebra B is homogeneous if for all a, b E B distinct from OB and 113 there is an automorphism of B sending a to b. It is weakly homogeneous if for all a, b E B distinct from OB there is an automorphism 4? of B such that 4? (a) fl b # OB.

By a result of Just and Krawczyk ([65, Theorem 1]; see also [20, Question 48]), Continuum Hypothesis implies that every quotient over an F, ideal is isomorphic to P(N)/ Fin, and therefore homogeneous. Similarly, CH implies that all quotients over Erdos-Ulam ideals are homogeneous (Corollary 1.13.7). Once again, the impact of OCA is in the opposite. PROPOSITION 3.7.1. Assume OCA and MA. If If is a summable ideal other than Fin, then the algebra P(N)/I is not weakly homogeneous.

PROOF. Since If is not isomorphic to Fin, there is a positive set C such that If I C is a proper summable ideal and therefore it will suffice to prove the statement

assuming If is a proper summable ideal. Let si be disjoint finite sets of integers such that for all i < k we have

1
3Es;

jEsk

Let A = Ui s2i and B = Ui 52i+1. Then it is easy to see that Lemma 1.12.2 implies If I A' is not Baire-isomorphic (and therefore not isomorphic) to If [ B', whenever A' C A and B' C B are If-positive sets.

By a proof similar to that of Proposition 3.7.1, Theorems 1.13.3 and 1.13.12 can be used to prove the following (see §1.13 for definitions). PROPOSITION 3.7.2. Assume OCA and MA. If Z, is a density ideal not isomorphic to Fin or to 0 x Fin, then the algebra P(N)/Z,, is not weakly homogeneous.

There is an analytic P-ideal different from Fin whose quotient is weakly homogeneous; the ideal 0 x Fin verifies this assertion. One can say more (recall that an

ideal I has the Frechet property if (I')' = I) LEMMA 3.7.3. If I has the Frechet property, then its quotient is weakly homogeneous.

PROOF. If I has the Frechet property, then every positive set A has a subset B such that I [ B is isomorphic to Fin. Therefore if C, D are I-positive sets, then we can easily define a bijection h: N -+ N (we can assume that I # Fin) which sends I into itself and such that h"C fl D is infinite. Then 4)h is an automorphism of the quotient algebra as required. We do not know whether there are analytic P-ideals other than Fin and 0 x Fin whose quotients are weakly homogeneous (Corollary 1.2.11 suggests that a variation of the above proof of Proposition 3.7.1 may apply to give a negative answer), so let us turn to the homogeneous ones. PROPOSITION 3.7.4. Assume OCA and MA. If I is a non-pathological analytic P-ideal different from Fin, then the algebra P(N)/I is not homogeneous.

PROOF. Let I = Exh(cp) be a non-pathological P-ideal whose quotient is homogeneous, where cp is a lower semicontinuous submeasure guaranteed by Solecki's

84

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

theorem. The ideal I is dense, since otherwise the ideal I I A would be isomorphic to Fin for some positive set A, contradicting the homogeneity and the assumption that I itself is not isomorphic to Fin. Therefore we can assume that I is dense, or equivalently that (1) limi cp({i}) = 0. Now construct sequences ui, vi (i E N) and f : N ---> N so that for all i (2) uI < u2 < ... are finite subsets of N, (3) VI < v2 < ... are finite subsets of N, (4) 1w(ui) - 1J < (5) ko(vi) - <

(6) f (i) = E'=I jui j, (7) cp({k}) < 1/(2f (i)), for all k E vi, (8) cp({l}) < 1/(2ilvil) for all l E ui+i. Knowing that (1) holds, this construction is straightforward, so let us assume that ui, vi and f are as above. Let us write i

CLAIM 1. For every i and every 1 - 1 map h: vi -* N there is s C vi such that h"sflUU =0 and cp(s) > 1/4.

PROOF. Let t = h(UU). Since h is 1 - 1, (7) implies cp(t)

2f (i)

l ud = 2

and therefore s = vi \ t is as required.

CLAIM 2. For every h: vi -+ U'I uj we have cp(h"vi \ Ui) < 2-i PROOF. We can assume that h"vi fl Ui = 0. By (8), p(h"vi) :5 E p({h(k)}) < 2ily I Ivi = 2-1, kEv;

and this completes the proof.

Let A = U. ui and B = Ui vi. By (4) and (5), both sets are positive. Since 2(N)/T is homogeneous, there is an isomorphism D: P(B)/T -> P(A)/T. By Corollary 3.4.2, there is a bijection h: A B which defines an isomorphism between

I B and I f A. (Recall that, although in general we cannot assume that h is a bijection, in this case we can since both I [ A and T f B are dense.) By Claim 1 and Claim 2, for every i there is si g vi such that cp(si) > 1/4 and cp(h"si) < 2-'. Therefore Ui si is a positive set which 4 h sends to a set in T, contradicting our assumption on h.

CONJECTURE 3.7.5. Proposition 3.7.4 remains true even when the assumption that I is non-pathological is dropped.

This would follow from OCA lifting theorem, an (expected) positive answer to Question 1.14.3 and Corollary 1.2.11.

In ]65, Question C] Just and Krawczyk asked whether all homogeneous quotients over ideals which are F,b but not FF are pairwise isomorphic. This question

may turn out to have a positive answer trivially. By the above, Fin seems to be the only Fob P-ideal whose quotient is homogeneous, and it turns out, at least when

3.8. ALMOST LIFTINGS OF EMBEDDINGS INTO P(N) / Fin

85

the ideal has the Radon-Nikodym property, that the quotient is homogeneous only when the ideal itself is homogeneous (under a natural definition). The only homogeneous Fob ideal which is not Fo known to me is Fin x Fin. This ideal coincides with the ordinal ideal 1,2 (see §1.14), and it is clear that all these ideals ZQ are homogeneous. The topological ordinal ideals W. (for o multiplicatively indecomposable) of Weiss (see §1.14) are also homogeneous. QUESTION 3.7.6. Are there any other homogeneous analytic ideals?

3.8. Almost liftings of embeddings into P(N)/ Fin This section includes a proof of a special case of Theorem 3.3.5, when the range

of the homomorphism is P(N)/Fin. Our reasons for proving this special case in a separate section are twofold: (i) Its proof shows some of the ideas involved in the (quite long) proof of Theorem 3.3.5 in a more transparent way, and (ii) This is the only instance of Theorem 3.3.5 used in topological applications given in Chapter 4; these applications rely on Theorem 3.3.6, which is proved using Theorem 3.8.1.

THEOREM 3.8.1. Assume OCA and MA. If 4>: P(N)/Fin - P(N)/Fin is a homomorphism, then D has a continuous almost lifting. Equivalently, 4' can be decomposed as D1 ®4'2 so that 4)1 has a continuous lifting, while the kernel of 4>2 is ccc over Fin.

Before we start the proof, let us note the more useful reformulation of this result:

COROLLARY 3.8.2. Assume OCA and MA. If 4': P(N)/ Fin -+ P(N)/ Fin is a homomorphism, then 4' has a completely additive almost lifting. More precisely, there is h: A --+ N for some A C N such that 4>h is an almost lifting of 4'. PROOF. Apply the Radon-Nikodym Property of Fin to the homomorphism 41 obtained by Theorem 3.8.1. PROOF OF THEOREM 3.8.1. We will use the following local version of Theorem 3.8.1 taken from [64, Theorem 11] (in Proposition 3.12.1 we prove a more general statement). LEMMA 3.8.3. The ideal

Jcont = {A : D [ P(A) has a continuous lifting} is ccc over Fin.

By Theorem 1.6.1, the ideal 7,ont equals the set of all A for which 4 [ P(A) has a completely additive lifting 4'hA for some finite-to-one hA: N --+ N. Let us consider another ideal associated with D: ,71 = {A : 4i [ P(A) has a completely additive almost lifting}. Then clearly Ji includes Jcont and therefore it is ccc over Fin as well. Recall that a family .F of partial functions on the integers is coherent modulo Fin if for all f, g E .F

there are at most finitely many n E dom(f) fl dom(g) such that f (n) 54 g(n). For A E ,7i let 9A: A -+ Fin be defined by 9A(n) = h,y'(n) LEMMA 3.8.4. The family {9A : A E 3i} is coherent modulo Fin.

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

86

PROOF. Assume the contrary, that for some A, B E ,71 the set of all n E A n B

such that gA(n) # 9B (n) is infinite. Assume 9A(n) \ gB(n) # 0 for infinitely many n. Let P. be a lifting of (P. Since both {C 'NA (C) =' C(C)} and {C 4hB(C) =` %(C)} include a ccc over Fin (and therefore dense) ideal on :

:

A n B, we can find an infinite C C A n B such that gA(n) \ gB (n) # 0 for all n E C and

'hA(C) =" 4).(C) =' 'PhB(C) Since gA(n),gB(n) are finite for all n, we can furthermore assume that 9A(n) n 9B (M) = 0 for all distinct n and m in C. Then '%A (C) \ 4?hB (C) is infinite, contradicting the above formula. The case when 9B (n) \ 9A (n) # 0 for infinitely many n is treated in an analogous way. This completes the proof. For D C N define a partition [,71]2 = Lo(D) UL1(D) by letting {A, B} in Lo(D) if and only if there exists n E A n B n D such that 9A(n) 9B(n) By identifying A with the graph of 9A, we see that OCA applies to this partition.

LEMMA 3.8.5. The following are equivalent

(Dl) the ideal J, is c-L, (D) -homogeneous, (D2) the set D is in ,71, for every D C N.

PROOF. Assume D E ,71 with hD: N

D as a witness. For n E N let

Xn = {A E J1 : gA(m) = gB (m) for all m E (A n B) \ [l, n]}. Then each Xn can be covered by countably many L1(D)-homogeneous sets, according to what is the behavior of gA on [1, n].

Assume now ,71 is u-L1(D)-homogeneous. We can also assume that D = N, and therefore that 3i is u-L1 (N)-homogeneous. Then there is a nonmeager L1(N)-

homogeneous set X C .71. Note that if A C B then the map hA witnesses that and therefore we can assume that X is hereditary, i.e., closed under BE taking subsets of its elements.

CLAIM 1. There are at most finitely many integers m such that hA(m) # hB(m) for some A,B E X. PROOF. Suppose otherwise, let {mi} be a sequence such that no and n; are in {hA(mi) : A E X} for some n° # n, and all i. By L1(N)-homogeneity of X, sets {n°, nz } (i E N) are pairwise disjoint. Since X is nonmeager and closed under taking subsets of its element, there is an A E X which includes infinitely many of {n°, ni } (i E N) (this follows by Theorem 3.10.1 (a) which will be proved later). But by the

homogeneity we have both hA(m) = n° and hA(m) = n', a contradiction. By Claim 1 and making finite changes to some of hA for A E X, we can assume that h = UAEX hA is indeed a function.

CLAIM 2. For every B E ,pant the set {m : hB(m) # h(m)} is finite. PROOF. If this set was infinite, then since X is nonmeager there would be an A E X having an infinite intersection with this set. But then the pair A, B would contradict the conclusion of Lemma 3.8.4.

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87

This implies that the set {n : h-1(n) # hA'(n)} is finite for all A E 7cont, and therefore I includes .7cont LEMMA 3.8.6. The ideal .71 is a P-ideal. PROOF. Assume ,71 is not a P-ideal, so for some sequence AI C A2 C A3 C .. .

of sets in J1 there is no A. E 9i which almost includes all An's. For convenience, we assume 7cont, ,71 are ideals on N x N and that Ak = [1, k] x N. For an increasing function f : N N consider the set

rf = {(k, m) : m> f(k)}. By the choice of {Ak}, the set N2 \ r f is ,71-positive. We claim there is a f : N -> N

such that for all f >* f the set r f \ r1 is in ,7oont. Otherwise, we can recursively find { ff}{<,, such that rfE+, \ rff, .7cont, contradicting the fact that ,loom is ccc over Fin (Lemma 3.8.3). Since rf fl An is finite for all n, we can assume that f is identically equal to zero. Then F = {9r f f E N' J is a coherent family of functions. By OCA and Theorem 2.2.7, there is a function g: N2 --> Fin which :

trivializes Y.

CLAIM 3. For every A E 91 there is a k(A) E N such that

{(i, j) E A : g((i, j)) # gA((i, j))} C [1, k(A)] x N. PROOF. If this was false, there would be an infinite C = {(mi, ni) i E N} C A in 7cont such the sequence {mi} is strictly increasing and for all i we have :

9((mi,ni)) 7' 9A((mi, ni)) Since Y is cofinal in NN, there is an f E Y such that (mi, ni) E r f for all i. But this implies that gr f and 9A differ on infinitely many places, contradicting the fact that the family .7 is coherent. By Claim 3, for every k E N the set 71k = {A E ,7i : k(A) = k} is LI ((N2)\Ak)homogeneous. For A E 7-lk we can replace function gA with

9Ak((i,j))I (i,j) :--> 9AWIMI

if i < k, otherwise.

After this adjustment, each 7-lk becomes a-L1(Nt2)-homogeneous, and therefore Lemma 3.8.5 implies that N2 E .71, contradicting our assumption. Now that we have proved ,71 is a P-ideal, we can assume Jcont,.71 are ideals on N. Assume 91 is not a-LI(N)-homogeneous, and fix an uncountable Lo(N)homogeneous X C .71. Since 71 is a P-ideal, by increasing some of the elements of X we can assume X forms a chain of type wi in the ordering g* of almost inclusion. Let P be the poset of all (s, k, F), where (P1) k E N and s is a subset of k, (P2) F is a finite Lo(s)-homogeneous subset of X. Define the ordering on P by letting p < q if (recall that p = (8P, kP, FP)) (P3) sP fl k8 = sq and FP Q Fq. We will prove that P has a strong form of a countable chain condition that will assure ccc-ness of a certain amalgamation P, of uncountably many copies of P. A pair of uncountable subsets X, Y of P such that every p E X is incompatible

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA 88

with every q E Y is called an uncountable rectangle of incompatible conditions. One similarly defines the notion of an uncountable rectangle of compatible conditions. CLAIM 4. There are no L1([n,oo))-homogeneous rectangles with both sides un-

countable, for any n E N.

PROOF. Suppose the contrary, and let Y x Z be an uncountable L1 ([n, oo) )homogeneous rectangle; we will prove that this implies there is an uncountable

LI([n oo))-homogeneous subset of X. Let

h= U hA. AEY

Since Z is cofinal in (X, C*), the set of all n E A for which h-'(n) # hAl(n)

is finite for all A E X. This implies that for some k the set of all A E X for hA'(n) = h-1(n) for all n > kin A is uncountable. This set is Li([k, oo))which and therefore a finite union of L1 (ICY)-homogeneous sets; but X is L0(NI)-homogeneous, a contradiction. homogeneous,

A working part of p E P is (sP, k1'). Working parts of p, q are compatible if

5Pn[1,k9]=s9n [1,kP]. CLAIM 5. If X,Y are uncountable subsets of P such that for all p, q E X U Y their working parts are compatible, then there is an uncountable rectangle X' C X,

Ir C y of compatible conditions.

PROOF. Let k be such that for uncountably many p E X and uncountably many q E Y we have kP,kQ < k. Let X = {pot} and Y = {p,£}. By going to an uncountable subset, we can assume that the size of each Fot = FPRR (each Fig = FP11) is equal to some fixed l (m, respectively), and let Fo£ _ {Ao1, ... , Ao£ } and F11 = {A31, ... ,All }.

1 for all 6, i, and j, and (again by This enumeration is chosen so that Az£ uncountable subset) we can assume that there is a k, > k such that for going to an uncountably many and all i, j we have .£

14j++ 1 \ A,

C [l, ki]

for all n E Ay+ i \ [l, ki].

and gAi" (n) = gA,4 (n)

To a condition p£ E X we can associate the 21-tuple Ai

AI 04

a

in P(N)l x (FinN)'. Since this is a separable metric space, we can assume (by going to an uncountable subset of X) that the set of these 21-tuples is 111-dense in itself, i.e., that every open set which contains one of these 21-tuples contains uncountably many of them. We can also assume that the set of 2m-tuples associated to elements of yin an analogous way is an l11-dense in itself subset of P(N)t x (FinN)t. Finally, by Claim 4 there are 1', rl such that {AO1, Ale} E Lo([k1, co)). Since the partition Lo([k1, oo)) is open, there are uncountable sets of Vs and rl's for

which this happens. By the choice of k1, for all these l;, rl and all i, j we also have

{A'01, Ail} E Lo([ki, co)).

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89

Since the working parts of these conditions are compatible, the conditions are compatible themselves. The poset Pw, is defined as an amalgamation of 1 many copies of P as follows: a typical condition is p = (I, k, s(e)(1; E I), F(l;)(Z; E I)),

(k, s(l;), F(i;)) is in P for all f E I. An where I is a finite subset of wl and ordering is defined by letting p < q if (P4) p 1q and p() p q() for all e E 1q and (P5) the sets sP(1;) \ {1, ... , k} ( E Iq) are pairwise disjoint. CLAIM 6. The poset P,, is ccc.

PROOF. Let pa (a < w1) be an uncountable subset of Pw, . We can assume sets la = IP- from their first coordinates form a A-system with root I. It follows from the definitions that p and q are compatible in P,, if and only if the conditions

(I, s£, k',F : eEI)

and

(I, s£, k£,FC : CE I),

where I=IPnIq,

are compatible. Therefore we can assume that

for all pa. Uniformizing further, we can assume that for every fixed i E {1, . the working parts of all conditions pa(Ci) (a < w1) are equal, say

.

.

, l}

s£ =9i and k'i = ki, for all a andi E

We can, moreover, assume that sets F9 = FP,' from the third coordinates of pa (e;i) Is

form a A-system with root F. Now observe that by the definition of P the conditions p, q E P are compatible if and only if the conditions (sP, kP, FP \ F)

and

(sq, kq, F8 \ F),

where F = FP n Fq,

are compatible. Therefore we can assume that the sets F£ (a < w1) are pairwise disjoint for every fixed i E {1, ... , l}. We can apply Claim 4 to p. (C,) (a < w1) and get subsets X1, Yl of w1 such that pa (C1) (a E XI) and pp (1;1) (,Q E Y1) form an uncountable rectangle of compatible conditions. Let k'1 > k1 and sl C [1, ki) be such that

(si,k',F',2UFA) is a joint extension of pa (C1) and pp (C1) for all a E X1 and p E Y1. Now we can extend every Pa (C2) (a E X1 UY11) to C P(C2) = (92, k', F'a(S2)) so that k2 > k1. Another application of Claim 4 gives sets X2 C X1 and Y2 C Y1 such that p'a (f) (a E X 1) and pp (C2) (,3 E Y2) is an uncountable rectangle of compatible conditions. Note that, if

(s2, k2 ,

F62 )

is a joint extension of p.(1.2) and p0' (C), then by our choice of k2 the condition (P7)

is satisfied between si and 92. By continuing this construction for i = 3, ... ., 1we get an uncountable rectangle of compatible conditions in Pw,

P.

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90

Let G be a sufficiently generic filter of P,, and for l; < w1 define

Dt = U SPC and X£ = U F? . pEG

PEG

Then the sets D£ are pairwise almost disjoint and X4 is Lo(Dg)-homogeneous for all e. By ccc-ness of P,1 there must be a condition p in it which forces there are uncountably many e's with uncountable X£'s. Then we can choose a suitable family of R1 many dense open subsets of Pw, so that if C is a filter of P,,,, containing p and intersecting all these dense open sets, then for uncountably many the set X4 is uncountable. By Lemma 3.8.5, for such l; the set D£ is not in ,71 and therefore not in T,.nt But this is in contradiction with Lemma 3.8.3, and this concludes the proof of Theorem 3.8.1.

3.9. Almost liftings of embeddings into P(IN)/ Fin x0 This section is devoted to a proof of Theorem 3.3.6, which will be of crucial importance in topological applications given in Chapter 4. We shall first prove its special case for the ideal Fin xO, but we will need a lemma first. LEMMA 3.9.1. Assume b > t . If (D: P(N)/Fin - P(IN)/ Fin xm is a homomorphism, and he N2 -* N is such that 4hrrf is an almost lifting of Vf for every f E NN then 4h is an almost lifting of D.

PROOF. Let J = {A : 4?.(A) =Fin x0''h(A)}. Assume J is not ccc over Fin, and let A be a family of l many pairwise almost disjoint, .1-positive sets. For A E A let fA in NN be such that the set

(4.(A)o'Dh(A)) nrfA = 4'.(A).AhlrfA (A) is ,7-positive (equivalently, infinite). By Theorem 2.2.6 find g in NN which eventu-

ally dominates all fA (A E A). But'h;re is, by the assumption, an almost lifting of V9, and therefore for all but countably many A E A the set (,.(A)A4h(A)) n r9A ? ((P.(A)o(P h(A)) nrfA is finite; contradiction. THEOREM 3.9.2. Assume OCA and MA. If 4D: P(N)/Fin P(IN)/ Fin xO is a homomorphism, then (D can be decomposed as 4)1 ®(2 so that has a continuous lifting, while the kernel of c2 is ccc over Fin. Equivalently, a continuous lifting of'D1 serves as a lifting of 4D on an ideal which is ccc over Fin. PROOF. This proof is similar to the proof that Fin xO has the Radon-Nikodym property (Theorem 1.5.2). Assume D: P(INY)/ Fin - P(N2)/Fin xO is a homomor-

phism. For f : N - N let 4)f: P(N) -* P(rf) be defined by 4'f(A) = 4?.(A) n rf.

Then 4Pf is a lifting of a Baire homomorphism 4?rf: P(N)/Fin .- P(rf)/Fin. Since the restriction of Fin x O to the set r f is equal to the ideal Fin on r f, by Corollary 3.8.2 there is an E f C r f and a finite-to-one hoo : E f - N such that the ideal

,7f = {A : 4)f (A) _'hf(A)} is ccc over Fin. To simplify the notation, let us define

hf:rf - NU{0}

3.10. NONMEAGER HEREDITARY SETS

91

by letting h f coincide with h f on E f and be equal to 0 otherwise. Note that uhf

is still a completely additive almost lifting of 4rf, and that {hf: f E NN} is a coherent family indexed by NN (by Lemma 3.8.4, or rather by its proof). By Theorem 2.2.7, pick a function h: N2 NU{0} which trivializes this family,

i.e., such that

h[rf='hf

for all f. Then clearly `kh f rf is a completely additive almost lifting of 4 rf for every f, and by Lemma 3.9.1 h is a completely additive almost lifting of 4). PROOF OF THEOREM 3.3.6. By Proposition 1.2.8, every countably generated

ideal is isomorphic (see Definition 1.2.7) to either Fin or Fin 4. Therefore the result follows from Theorem 3.9.2 and Theorem 3.8.1.

3.10. Nonmeager hereditary sets In this short section we study a largeness property of subsets of P(N) which will play a key role in the proof of the OCA lifting theorem (see §3.11). Recall that a set of reals is meager if it can be covered by a countable union of nowhere dense sets. A set of reals is comeager if its complement is meager. If X is a family of subsets of N by X we denote the downwards closure (or, the hereditary closure)

of X, i.e., X = Uaex P(a), and we say that X is hereditary if X = X. THEOREM 3.10.1.

(a) If X is a hereditary subset of P(N), then it is non-

meager if and only if for every sequence si of disjoint finite sets of integers there is an infinite a C N such that Uila Si E X. (b) The family of nonmeager hereditary subsets of P(N) is closed under taking finite intersections of its elements. (c) The family of nonmeager hereditary subsets of P(N) which are closed under finite changes of their elements is closed under taking countable intersections.

PROOF. (a) This proof is similar to those in Jalali-Naini ([58]) and Talagrand ([121]), where a slightly weaker statement (Corollary 3.10.2 below) was proved. For m E N and s C [1, m) define [s; m] = {a C N : a f1 [l, m) = s}. Assume first X is meager, therefore it is covered by a countable union U°°=1 Fm of closed nowhere dense sets. Recursively find 1 < n1 < n2 < ... and ui g [ni, ni+i )

so that [t U ui; ni+i] n U Fm = 0 m
for all t C [1, ni). This is possible because each Fm is nowhere dense. Then any infinite union of si = [ni,nitl) has a subset which is equal to an infinite union of the ui's and therefore avoids all Fm's. On the other hand, if there is a sequence si such that no infinite union of its elements lies in X, then for all i the set

Ui={A: snCAforsome n>i}. is a dense open subset of P(N), and I I1 Ui is a dense Gb set. But this is the set

of all subsets A of N which include infinitely many of the si's, and X is included in the complement of n°° 1 Ui.

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

92

(b) Let fl, ... , xk be nonmeager and hereditary. Since the set nk H is clearly hereditary, we need only to prove that it is nonmeager. Let si (i E N) be a sequence of pairwise disjoint subsets of N. By (a), we can find infinite sets D Ak such that

Al ? A2 D ...

U sj E xi

jEA;

for i = 1, ... , k. In particular, UjEAk sj is in nz Hi, and therefore by (a) this set 1

is

nonmeager.

(c) Assume ni Xi is meager, and let {sk} be as guaranteed by (a). Using (a), A2 ... such that UkEA; sk E Xi and recursively pick infinite sets N = Al

ni = min Ai+I > min A, for all i. Then Ui s,, is in ni Xi-a contradiction. COROLLARY 3.10.2 (Jalali-Naini, Talagrand). An ideal I which includes Fin has the property of Baire if and only if it is meager if and only if Fin
Let us note that the above proof also gives the following well-known characterization of comeager subsets of P(N), which we state for future reference. LEMMA 3.10.3. A subset X of P(N) is comeager if and only if there is a sequence 0 = no < ni < ... of naturals and si g [ni, ni+I) (for i > 0) such that X includes the set {A C N : A n [ni, ni+1) = si for infinitely many i}. Nonmeager, hereditary sets closed under finite changes of their elements are also called groupwise dense, and the minimal cardinality of the family of groupwise dense sets whose intersection is not groupwise dense is called groupwise density

(note that Theorem 3.10.1 (c) says that this cardinal is uncountable; of course, this fact is well-known). These notions were introduced in [9] in order to provide a succinct combinatorial formulation of a statement with strong implications to rather diverse mathematical structures (see [8]).

3.11. Approximate homomorphisms; more on stabilizers In this section we present two results which do not require any additional settheoretic axioms and which will be used in the proof of the OCA lifting theorem, Theorem 3.3.5. One of them is Theorem 3.11.3, a technical statement used to uniformize a sequence of Borel-measurable partial liftings of a given homomorphism. Another one is Lemma 3.11.6, which makes the notion of an "almost lifting" more useful, by making the results of Chapter 1 (in particular, the Radon-Nikodym

Property) applicable to almost liftings. The proofs make use of the stabilizers (see §1.5 and Definition 3.11.5) in a situation where we do not necessarily have a Baire-measurable lifting.

Let Z be an analytic P-ideal and let be a lower semicontinuous submeasure (see §1.2) supported by N such that (recall that IIAI[, = limsupk o(A \ [1, k))) we have

Z = Exh(,p) = {A : IIAIIw = 0}. is guaranteed by Solecki's theorem (Theorem 1.2.5). Pos-

The existence of such sibly by replacing p with min(1, cp) we can assume co, and therefore

Note that this does not change the ideal. Recall that dw(A,B) = IIAOB[I,.

Iw, is finite.

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93

is a complete metric on P(N)/I (Lemma 1.3.3). Some results of this and the following section could be regarded as results about the complete normed group (P(N)/I, II 11.). In fact, we shall be interested in the space of all homomorphisms 4): P(N)/ Fin --* P(N)/I equipped with the natural uniform metric induced by d,,. DEFINITION 3.11.1. Assume cp is a lower semicontinuous submeasure, E > 0, P(N) is an E-approximation

X C P(N), and F: P(N) -> P(N). A a map F: P(N)

of Fon Xif

IIr(A)AF(A)II, <_ E

for all A E X. We say that a homomorphism ': P(N)/ Fin -+ P(N)/ Exh(cp) is E-tame on 7i if its lifting 4), has a Baire E-approximation on 7-l.

Note that the E-tameness of ' depends on the choice of W. It can, however, be shown that the truth of M) is 6-tame for all c > 0" does not depend on the choice of cp (see [47]).

We shall use the following classical theorem (see, e.g., [74, 18.A]).

THEOREM 3.11.2 (Jankov, von Neumann). If R C P(N) xP(N) is analytic and

X = {a (a, b) E R for some b}, then there is a function f : X -> P(N) such that the graph of f is included in R and the f -preimage of every open subset of P(N) :

belongs to the o--algebra generated by the analytic subsets of P(N).

The following theorem gives the conditions which imply that D satisfies a slightly weaker statement than the conclusion of Theorem 3.3.5, but without assuming any additional set-theoretic axioms. THEOREM 3.11.3. Assume I = Exh(cp) is an analytic P-ideal and

4i: P(N)/ Fin - P(N)/I is a homomorphism. If there is a nonmeager ideal J which for every E > 0 can be covered by countably many hereditary sets such that ' is E-tame on each one of them, then lP can be decomposed as 4?1 ®(D2 so that -D1 has a Baire-measurable

lifting and ker(4)2) includes J, in particular it is nonmeager.

Before we give a proof of this theorem, let us note that the requirement that the sets on which (P is a-tame be hereditary may seem to be artificial. However, dropping this requirement would make Theorem 3.11.3 false: If '' is a homomorphism given in Example 3.2.1, then P(N) can be split into two sets (U and its dual, U) such that has a continuous lifting on each one of these two sets, but nevertheless 4D has no Baire lifting. This also shows a difference between Theorem 3.11.3 and Theorem 2 of [142], which states that if an automorphism of P(N)/Fin has a o-Borel lifting, then it is trivial.

LEMMA 3.11.4. Assume I = Exh(ep), 4), and j are like in Theorem 3.11.3, but that J can only be covered by countably many hereditary sets such that is E-tame on each one of them for some fixed E > 0. Then % 5.as a continuous 28E-approximation on J. PROOF. Let e, (i E N) be a sequence of Baire-measurable functions from P(N) to P(N), and for each i let H2 be a hereditary set on which Oi is an e-approximation

of 4) so that ,7 = U°°1 Hz. Recall that for in E N and s C [l, m) we write

[s; m]={aCN : an[1,m)=s}.

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

94

CLAIM 1. We can assume that functions ej are continuous 2e-approximations of 4). on 7{i.

PROOF. To this end, pick a dense G4 set G C P(N) such that ei [ G is continuous for all i and that G n 7-i is empty whenever 'Hi is meager. Now find an increasing sequence {ni} and si C [ni, ni+1) for all i so that G includes

{a C N : a n [ni, ni+1) = si for infinitely many i E N} (see Lemma 3.10.3). Since j is nonmeager, we have UkEC Sk E 3 for some infinite

C = {k(1), k(2).... }. Fore=0,l let a, = U [nk(2j+,), nk(2j+e+1)) and

C. = U

Sk(2j+e).

j=0

7=0

For i, j E N define a function eij by sijo(a) = ei((a fl ao) U ci) n 4) .(ao),

ei3i(a) = ej((a n al) U co) \ 4).(ao),

eij(a) = sijo(a) U eij1(a). Therefore eij is an amalgamation of eijo and eijl. Moreover, eij is continuous, and for every a E 3 we have i, j such that a belongs to the set Nij = {a : (a n ao) U c1 E Hi

and

(a n al) U co E 7{j}.

(This is because both co and cl belong to J.) Each 7-lij is clearly hereditary. For a E 7-lij we have:

eij(a)A,%(a) = ((e ((a n ao) U cl) n'

(ao))

U (ej((a n a1) U co)

=7 ((e1((a n ao) U cl)0D.(a)) n

-D.(ao)))A,%(a) (ao))

U ((ej((a n a1) U co)0D.(a)) \

(a0))

=Z ((ei((a n ao) U cl)A,%((a n ao) U c1)) n D.(ao)) U ((ej((a n a1) U co)04).((a n a1) U co)) \

=7 (ei((a n ao) U

*(ao))

((a n ao) U cl))

U (ej((anal) Uco)04).((ana,) Uco)), and therefore jj©ij(a)A(%(a)[j,p < 2e. Hence each Bij is a continuous 2e-approximation to D. on 7dij. By reindexing 7-(ij (i, j E N) we can assume that each ei is a continuous 2e-approximation of D. on Hi and that each 7-Ci is hereditary. Recall that a set is everywhere nonmeager if its intersection with any nonempty open set is nonmeager. CLAIM 2. We can furthermore assume each 7-1i is either meager or everywhere nonmeager.

PROOF. Assume 7-(i is nonmeager. Let [u] be a basic clopen subset of P(N) such that [v] n 1-ti is nonmeager for every [v] C [u]. Since 7{i is hereditary, we can assume that [u] = [O; p] for some p c N. Let

e4(a) = ei(a \ p)

3.11. APPROXIMATE HOMOMORPHISMS; MORE ON STABILIZERS

95

Then clearly the set

7-I ={a:a\pE7-I} is everywhere nonmeager, hereditary, includes 7-li, and Je (a)A

(a))JJ,, < 2e, for

all aE71. By replacing Oi with O; and 7-Gi with

we get the desired claim.

I]

We need to redefine the notion of stabilizer given in Definition 1.5.1.

DEFINITION 3.11.5. Assume S2: P(N) - P(N). For n < n' and e > 0 we say that c C [n, n') is an (n, n') -e -stabilizer for S2 if there exists k E (n, n') such that for all s, t C [1, n) and all X, Y C [n', oo): (S'1) min(S2(s U c U X)OS2(s U c U Y)) > k, (S'2) co((SZ(s U c U X)AQ(t U c U X) \ [1, k))) < e.

CLAIM 3. Assume that S2: P(N) -> P(N) is continuous and there is an everywhere nonmeager, hereditary set 7-t such that for all A, B E 7-l satisfying ADB E Fin we have IIci(A)AQ(B)IIw < S.

Then for every n and all large enough n' there is an (n, n') -6-stabilizer for Q. PROOF. Assume this fails for some n. By the uniform continuity of S2 for all

large enough n' > n the condition (S'1) is satisfied for all c C In, n'). Let us fix

a k such that this is true for all n' > k. Note that (S'2) fails for all n' > k. By the continuity of i and lower semicontinuity of p the set X which witnesses that (S'2) fails can always be chosen to be finite. We shall choose a sequence n = nl < k = k1 < n2 < k2 < n3 < ... , and wi C [ni, ni+1) so that for every l E N and uw C [1, l) the parameters

c=Uwi,

k=kl,

=n1+1

iEw

satisfy (S'1) for all u, X and Y, but X = w1+I witnesses that (S'2) fails for some s, t C [1, n). To construct w1+1, we need a few auxiliary objects. Let aj (j = 1, ... , 21) enumerate all subsets of [1, l - 1). We shall find:

(1) ni =m1 <m2 <... <m21+1, (2) X, C [mj,mj+1) for j E {1,...,21},

U=

so that w1 = 1 X3 is as required. The construction proceeds as follows. Since UiEa, wi is not an (n, n1+,)-6-stabilizer with k1 as a witness, we can find sl, t1 C [1, n) and X1 C [ni, oo) witnessing that (S'2) fails. By the uniform continuity of S2 we can assume X1 is finite. Pick an arbitrary m2 > max X1. Since s = UiEa, wiUX, is not an (n, m2)-b-stabilizer with ki as a witness, we can find $2, t2 C [1, n) and a finite X2 C [m2i oo) witnessing that (S'2) fails. In this manner we construct U= wi = 1 X1 such that for all j < 21 set XI+1 witnesses that UiEa; W. U Up=1 X9 is not an (n, mjt,)-b-stabilizer with ki as a witness. It is easy to see that wi is as required. The recursive construction of sequences {ni}, {ki} and {wi} is now obvious-if we have n1 < k1 < . < ni, then find k1 > ni and wi in the manner described above and pick n1+1 to be larger than max(wi). Assume these sequences are chosen. For u C [1, n) let 7-tu = {B C In, oo) : u U B E 7-L}.

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

96

Then by (b) of Theorem 3.10.1 the set nuc(l.n) Nu is nonmeager and we can find an infinite C C N such that B = U3EC wj is in this set. We may assume that some fixed fl, ,D C [1, n) appear as witnesses that (S'2) fails for s = B fl [no, np), k = nP and x = wp for infinitely many p E C. Therefore II5l(u U B)Afl(v U B)11,E > b,

contradicting the assumption on 1 and 9-(, for both u U B and v U B are in N.

Continuing the proof of Lemma 3.11.4, note that if Oi is continuous and a 26-approximation to a homomorphism on some set Ni which is hereditary and everywhere nonmeager, then by the triangle inequality for 11w, the function e, satisfies the assumptions of Claim 3 with b = 46. Therefore we can construct m1 < k1 < m2 < k2 < ... and si C [mi, mi+1) so that for all is (3) si is an (mi,mi+1)-4e-stabilizer for ©j with the witness ki for all j < j. 11

By Theorem 3.10.1 (a), find a b E 9 which is equal to some infinite union UiEc Si, say c = {l(j)} and 1(1) < 1(2) < 1(3) < .... Since J I (N \ b) is a nonmeager ideal, we can find an m such that {a C N\b : bUa E 9-1,,,}

is a nonmeager subset of P(N \ b). We can assume m < 1(1). Note that (5) si(i) is an (mt(i), m1(i+i))-8-stabilizer for e3 for all j < i. Now we let T: P(N) --> P(N) be the stabilization of 0f, by {ml(j)} and {st(j)} (j E N), namely let ao = U [mt(2j), mt(2j+l)), j=o

a1 = U [mI(2j+I), mt(2j+2)), j=o

co = U s1(2j), j=o Cl = U 51(2j+1), j=o

and let

To(a) = G,,,((anao) U cl) fl4 (ao), T1 (a) = e , ((a fl a1) U co) \ * (ao),

T(a) = To(a) U Ti(a) We claim that the continuous function T is a 286-approximation of 4). on J. Assume not, then either To is not a 14&-approximation of (P. on ,7 (ao or T1 is not a 146-approximation of % on ,7 f a1. Assume the first possibility applies, and let

d E 3 ao be such that II,P.(d)ATo(d)11, = IIL(d)A(e (dU cl) n l.(ao))) > 146. Denote the set dUc1 by d'. Since e,-,,,(d')A(P.(d') J' 4).(d)0(ei,,(dUc1)n D.(ao)), we have

(6) ®m(d')0$.(d') > 146. Let n be such that (7) 114).(d')0©e(d')11,o < 2e, in particular d' E N. Note that for every j there is a large enough j' > j such that (8)

n [j,j')) > 14e.

Therefore we can find a subsequence {1'(i)} of {1(2i + 1)} such that

3.11. APPROXIMATE HOMOMORPHISMS; MORE ON STABILIZERS

(9) cp((O,r

97

n [kt,(z),kt'(ti+1)-1)) > 14e for all i.

Let us define (10) Ii = [mt'(i), mt,(i+l)), (11) pi(X) = o(X n [kr(i), kt'(i+1)-1)), so that (9) becomes (12) pti(e,,,(d')A ,(d')) > 14e, for all i. A careful reader may recognize that a version of Claim 4 below was the main idea behind the use of stabilizers when we were transforming an arbitrary continuous lifting into asymptotically additive one in Theorem 1.5.2.

CLAIM 4. If anli=bnli, j
PROOF. Note that (Oj(a)z e (c)) n [kp(ti), kt'(i+1)-1) = 0, cpi(ej(c)DOj(b)) < 4e,

by (S'1) and (S'2), respectively, since (aAc) n maxli = 0 and cAb C- [1, min 1j). Recalling that oi(X) = cp(X n [kt,(i), kr(i+1)-1)), we have wi(Cj(a)A(9i(b)) < oi(oi(a)Ae.i(c))+iPi(ej(c)0®.i(b)) 5 4e, completing the proof. By Theorem 3.10.1 (b) and (a), find an infinite C C N such that (13) d" = Uieo(d' n Ii) E N, . (Also note that d" C d' E 7-fn, and therefore d" E 7-ln by hereditarity.) Let i be a large enough element of C. Then since by everywhere nonmeagerness and hereditarity all finite sets are in 7-1,,,,, we have (14) d", d' n [1, ml, (j)), d" n [1, ml, (j)) are in 71,, and since d E 7{n we also have

(15) d"Cd'E7-ln, and therefore (we need i to be "large enough" in (18), (19) and (21) below): (16) Wi(6,,,(d)0©,,,(d' n [1,mt,(i)+1)) = 0 (since this set is, by (S'1), empty),

(17) cpj(©,n(d' n [1,mt,(i)+1))DOm(d")) < 4e (by d n li = d" n Ii, (14) this follows from Claim 4), (18) (19)

2E (since d" E 7-1,,,,,

II I of this set is < 2e),

Ae (d")) < 2e (since d" E 7-(n, of this set is < 2e), (20) (pi(On(d")DOn(d')) < 4e (by d' n Ii = d" n Ii, (15), and Claim 4), (21)

2E (since d' E 7in, 1 Jew of this set is < 2e),

E)(d")

<2E <2E

©n(d") <4E

(d')

<2e

en(d)

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

98

and all this implies coj(0,,,,(d')0(b.(d')) < 14E, contradicting (12). This completes the proof of Lemma 3.11.4. PROOF OF THEOREM 3.11.3. Using Lemma 3.11.4, from now on we can assume that for every n there is a continuous 1/n-approximation Tn to (D. on J. For a C N let Sa, = {b C N : for every n there is an m for which cp((bATn(a)) n [m, oo)) < 1/n}.

Then Uacx{a} x Z. is a Borel subset of P(N) x P(N), hence by Theorem 3.11.2

there is a Baire-measurable function T: P(N) - P(N) such that T(a) E Z. for all a satisfying Z. 0. Since T, is a 1/n-approximation to D. on j for all n, the analytic set {a : Za # 0} includes J. It follows that E Z for all a E 9, so we can say T is a lifting of of 4 on J. LEMMA 3.11.6. If D has a Baire lifting T on a nonmeager ideal 3, then t can be decomposed as 4)1 ® '2, where obI has a continuous lifting and ker4t2 3.

PROOF. First of all, like in the proof of Lemma 3.11.4, we can assume T is continuous. We shall prove that there is an asymptotically additive mapping S2 (see §1.5) such that III (A)AT(A)jI,, = 0 for all A E J. This construction is almost identical to the construction of an asymptotically additive lifting in Theorem

1.5.2. By using Claim 3, we find a sequence 1 = n1 < k1 < n2 < k2 < ... and si g [ni,ni+1) which is an (ni,ni+i)-2-i-stabilizer for every i with ki witnessing this fact. Since 3 is a nonmeager ideal, we can assume Ui Si E J. Since I is a P-ideal, we can assume T maps finite sets into finite sets. Let Co = U 82i, i

Ao = U [n2i, n2i+1 ), i

AI = U[n2i+1,n2i+2),

Cl = US2i+I

i

i

Define functions fo, fl, T: P(N) - P(INY) by

.fo(X)=T((XnAo)UCI)n1.(Ao) f1(X) = T((X n AI) U Co) \,P.(Ao)

T(X)=A(X)Uf1(X) Let Hi: P([ni,ni+i)) -* Fin be defined by

Hi(s) = T(s) n [ki-1, ki+I) Then, by the proof of Theorem 1.5.2, we conclude that the mapping Il = WH with parameters {ni} and {Hi} satisfies

1(B)AT(B) E I for all B E and in particular it is a lifting of 4) on J. Let A = 1(N) and (recall that (DA(B) _ ,D(B) n A): ,D1 =.pA,

'P2 =,pN\A.

We claim S2 is a lifting of 41. Pick a, b C N and assume (11(a) U 1(b))AQ(a U b) is not in Z, namely 11(1(a) U f2(b))Af2(a U b) 11, > E

3.12. A LOCAL VERSION OF THE OCA LIFTING THEOREM

99

for some e > 0. Since l is of the form WH, there is a subsequence na of nz such

that p((Q(a n [n', n'+1)) U 1l(b n [na, n;+l)))OS2((a U b) fl [n', nz+1))) > E

for all i. But we can find an infinite C such that e = UiEC [n', ni+1) is in J, and then letting a' = a fl e and b' = b fl e, it follows that S2(a') U fl(b') 0-7 Sl(a' U b'), contradicting the fact that l is a lifting of 4? on J. A similar proof shows that S?(B)A(A \ Sl(N \ B)) E I for all B C N, and therefore Sl is as required. This implies that 4)I has a Baire-measurable lifting 4?I*, and also that ker(4)2) ? {a : 4)*(a) =Z 4?1.(a)} D J. Therefore mappings 4?I and 4'2 satisfy the requirements. This ends the proof of Theorem 3.11.3.

3.12. A local version of the OCA lifting theorem The main result of this section is Proposition 3.12.1, a "local" version of Theorem 3.3.5 which already suffices to prove many of its consequences. We should remark that Proposition 3.12.1 was proved by Just ([64]) in the case when I is an Fo-ideal, who also proved it is consistent that the conclusion of Theorem 3.10.1 is true for all analytic ideals ([63]). We do not know whether this stronger conclusion follows from OCA and MA. Recall that if we are given a homomorphism 4>: P(N)/Fin -, P(N)/I with a lifting 4?*, then for B C N by pB we denote a homomorphism of P(N)/Fin into

P(B)/I with lifting 4?B(c) = 4i*(c) fl B.

PROPOSITION 3.12.1. Assume OCA and MA. If I is an analytic P-ideal and 4>: P(N)/ Fin - P(N)/I is a homomorphism given by a lifting L : P(N) -, P(N), then the ideal ,?coot = {a

:

4?O'(°) has a continuous lifting}

is ccc over Fin.

Before we start the proof, we need some definitions. Recall that infinite sets A and B of integers are almost disjoint if their intersection is finite, and that A almost includes B, A D* B, if B \ A is finite. The following notion will be very useful in setting up an open partition (see [106, II.4.1], [142]). DEFINITION 3.12.2. A family A of almost disjoint sets of integers is tree-like if there is an ordering -< on its domain D = U A such that (V, -<) is a tree of height w and each element of A is included in the unique maximal branch of this tree.

Equivalently, A is tree-like if there is a 1 - 1 mapping f of N into {0,1}
100

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

CLAIM 1. For every B E B there is an e > 0 such that there are no Borel e-approximations to (P. on P(B).

PROOF. Assume the contrary, and let B E B be such that there is a Borel 2-"-approximation e" to 4P. on P(B) for all n. For a C B let Ya = {b C N : JbAe"(a)J[,p < 2-" for every n}.

Then Y. is equal to {b : b =Z D. (a) } and nonempty because,. (a) E Y a. Moreover

UacB{a} x ya is Borel, and by Jankov, von Neumann uniformization theorem (Theorem 3.11.2) there is a Baire measurable function O: P(B) -* P(N) such that 0(a) E ya for all a C B, and therefore 0 is a Baire lifting of a homomorphism 4' on P(B). By Lemma 1.3.2 there is a continuous lifting of D r P(B) and therefore there is a continuous lifting of 4' (E), contradicting our original assumption that B is Jc.nt-positive.

Thus we can assume from now on that there is an n such that for no B E B there is a Borel 2-"+6-approximation to %. on P(B). By MA, we can find an almost disjoint family A' of size 111 such that every element of A' almost includes infinitely many members of B. Use MA again to find an uncountable A C A' and

a partition A = Ao U AI of each A E A such that Ai = {Ai : A E A} is a treelike almost disjoint family for i = 0, 1 (see [142, Lemma 2.3]). Let us concentrate at A0 for a moment. Let AO = UaEA0 [a]', and let Bb be the unique element of Ao including some b E Ao. Consider the set

X={(a,b) : bCaEAo}. Now we shall apply OCA to an open partition of [X]2 which is a variation of a partition of Velickovic ([142, p. 6]; see also [64, p. 286]). Define [X]2 = Ko U K( by letting {(a, b), (a', b') } in Ko if

(K1) Ba # Ba-

(K2) b'na=bfla' (K3) W((4'.(b') n 4.(a))A(4.(b) n ,%(a))) > 2-" Since the family A0 is tree-like, this is an open partition. [To see this, fix a bijection

f : N -+ {0,1}
CLAIM 2. There are no uncountable Ko -homogeneous sets.

PROOF. Assume the contrary and let N be such a set. Let c = U(a,b)Ef b. Then by (K2) we have cna = b for every (a, b) E H, and therefore 4. (c) n4'. (a) =Z 4D. (b). So we can find s, t C [l, m) for some m and an uncountable 9-l C H so that for all (a, b) E N: (1)
3.12. A LOCAL VERSION OF THE OCA LIFTING THEOREM

101

Then for (a, b) and (a', b') E N we have

d = (4 .(b) n' (a'))A(4'.(b') n -D. (a)) (,D. (b) n 4).(a'))A(4).(c) n 4'.(a) n (D.(a')) U ((D .(b') n -D.(a))A(-D.(c) n 4.(a) n 4D .(a'))

(4'.(b)A(4).(c) n 4'.(a))) U (-D.(b')A((D.(c) n 1D.(a )).

Thus cp(d n [rn,, oo)) < 2-n-1 + 2-'i-1 = 2-n but (2)-(3) imply d n [rn, oo) = d, and (K3) implies 2', a contradiction. Therefore OCA implies that X can be covered by a countable union U1 xz of Kl -homogeneous sets. Let DZ be a countable subset of Nz such that the set

{(a,b,(a),1)..(b)) : (a,b)ED } is dense in {(a, b, 4D. (a), 4D. (b))

:

(a, b) E Nz }. Fix B E A0 which is different from

all B. ((a,b) E Dn for some b and i E N). Recall that a function is said to be a-Borel if its graph can be covered by the union of countably many graphs of total Borel functions.

CLAIM 3. There exists a o -Borel 2-"+2-approximation rB : P(B) --> P(N) to D. on P(B).

PROOF. Fix m E N. For pairs (a, b) and (a', b') in 'P (N) x P(N) and k E N we say that (a, b) and (a', b') agree below k, and write (a, b) f k = (a', b') f k, if min(aAa') > k and min(bAb') > k. This terminology and notation extend to n-tuples in an obvious way. For m E N and (d, e) E D, t let m+(d, e) = min{max(d n B) : (d, e') E E)fftn and (d, e, (P. (d), 4'. (e)) f m = (d', e', 4). (d'), 4'. (e')) f m),

m+ = max{m+(d, e) : (d, e) E D}. Then both m+(c, d) and m+ are well-defined, since (1) m+ (d, e) < max(dAB), and this value is finite by the choice of B and d, (2) m+ (d, e) depends only on (d, e) m, and therefore m+ is equal to a maximum of a finite set. Now recursively define a sequence {mj = mj (B, Ti, m)} by picking m1 = 1 and

mj+1 = mj + 1. Let

00

00

B0 = B n U [m2j, -2j+1), j=0

B1 = B n U [7n2j+1, m2j+2). j=o

We now prove there is a a-Borel 2-f+'-approximation r0: P(B0) -* P(N) to L. Towards this end, we shall first define a set Y. of "candidates" for ro(a). For a C Bo let Ta = Ta(Bo, n, fn) be the set of all pairs (s, k) such that (Tal) kENand sC [1, k) n (D. (Bo), and there exists (d, e) E D' such that:

(Ta2) enBo=and, (Ta3) A (4,. (Bo), 41. (d)) > k, and (Ta4) o((,%(e)Os) n 4'.(Bo) n [ l , k)) < 2 ".

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

102

Note that if (s, k) E T. then for some (d, e) E D such that A(1P* (Bo), 1% (d)) > k the pair {(Bo, a), (d, e) } satisfies (Kl), (K2) and s approximates I ). (e) n 4D. (B0) up to k. Consider T. as a tree ordered by the end-extension: (s, k) _< (t, 1) if k < l and t n [1, k) = s. Note that Ta is downwards closed: if (s, k) E T. then (s n [1, k'), k') is in T. for all k' < k. Let Ya = Ya(Bo, fi, 7n.) _ {b

:

(b n [1, k), k) E Ta for all k E N}.

Since T. is a downwards closed subset of a binary tree, the set Ya is closed for all a and the function a -# Ya is a Borel mapping of P(B0) into the exponential space of P(N). Then . (a) Ya = Ya (Bo, n, 7n) is nonempty, and 2-"-1 for all b E Ya. (b) p(b0(' (a) n `).(Bo))) < SUBCLAIM. Assume (B0, a) belongs to 7-l

PROOF. (a) Since Df is dense in hf'fin (in a rather strong way-see the paragraph before Claim 3) and N is a Ki -homogeneous set, for every m2j there is a

(d, e) E D such that (*) (d,e,,]'*(d),4*(e)) I m2j+1 = (BO, a, 4). (BO), 4). (a)) ( m2j}1.

Then by the choice of m2J+2 there is a (d, e) in D, satisfying (*) and such that, moreover, d n Bo C [l, m23+2). Since Bo is disjoint from [m2j+1, m-2j+2), we automatically have d n Bo C [1, m2j+1). This implies e n Bo C [1, m2 j}1) and d n a C [1, m2 j+1) and therefore enBo = and,

also cp('.(e)A(C(a) n' .(Bo))) = 0, and the conditions for putting (,%(a) n ,P. (B) n [1,m2i),m23) in T. are satisfied. Since m33 was arbitrarily large, we have L (a) n % (Bo) E ya, completing the proof of (a). (b) Assume b E Ya, fix k E N, and let (d, e) E D be a witness that (b, k) E Ta, 771 i.e., such that

(1) enBo=and, (2) A(C (Bo), 4). (d)) > k, and (3) o((,P.(e)Ob) n 4.(BO) n [l, k)) < 2-n.

Note that F = (,D * (e) n 4) .(Bo))A(,D .(a) n 4) *(d))

((,D *(e) n ID *(Bo))A('*(a) n ID *(d))) n [1, k) _ ((,D *(e) n P*(Bo))A(4) *(a) n 4, *(Bo))) n [1, k) _ (ID *(e)04) *(a)) n D*(BO) n [1, k),

and we have '.(a)Ab C F U (,D.(e)Ab). Moreover, by (K3) we have cp(F) < 2-", and therefore o((bA,D*(a)) n

*(Bo) n [1, k)) < 2-" + 2-" = 2-"+1

The submeasure cp is lower semicontinuous and the inequality is true for all k, so we have co(bA(,D*(a) n 4).(Bo)) < 2-7+1

Ya is Borel (as a mapping of We have already noted that the mapping a P(B0) into the exponential space of P(N)), and that each T. is a binary tree. It follows that {a : Ya 54 0} is a Borel set, and therefore the mapping Eo,,;,, defined

3.12. A LOCAL VERSION OF THE OCA LIFTING THEOREM

103

on P(Bo) by (by -in Y. we denote the lexicographically minimal element of Ya, corresponding to the leftmost branch of T,): o,;n (a) _

if Ya

fmin Ya,

0,

if Y. = 0,

is Borel. By Subclaim (b), Eo,,,,, is also a

2-n+'-approximation to 4), on 9-I f1P(Bo).

By letting fn run over all positive integers, we find such E0,m for all m E N. Therefore the mapping Eo: P(B0) - P(N) defined by Eo(a) = Eo,m(a),

whenever (Bo,a) E 7-L;, \ U xk k<m

is a a-Borel 2`+'-approximation to 4), on P(Bo). In a similar fashion we define mapping El: P(B1) - P(N) which is a o-Borel 2-n}1-approximation to 1). on P(B1) and finally define

EB by

for a C B. E(a) = Eo(a fl Bo) U E1(a fl BI), 2-n}2-approximation to 4), on P(B). Then E is a s-Borel

0

We are now in a position to conclude that for all but countably many B E Ao there is a a-Borel 2-fi+2-approximation to 4), on P(B). Similarly, for all but countably many B E Al there is a a-Borel 2-t2+2-approximation to 4), on P(B). It follows that for all but countably many B E A there is a a-Borel 2-n+3-approximation to D. on P(B). Now we have to relate the existence of a o-Borel approximation to 4), with the existence of a Borel approximation to 4).. Regarding this, we should note that in Theorem 2 of [142] it was proved that, in the case of automorphisms of P(N)/ Fin, the existence of a a-Borel lifting implies the existence of a Borel lifting. In our case, when we have a homomorphism instead of an automorphism, this is no longer true (see Example 3.2.1). Let ,70 = {a C N : 4), can be 2-'-approximated on P(a) by a a-Borel function}, font = {a C N : 4), can be 2-"-approximated on P(a) by a continuous function}. It is straightforward to check that these are ideals on N which include Fin. We shall need the following diagonalization lemma which relates 7o and ,7.ont (compare this with [64, Lemma 6a] and [142, pages 4-5]). LEMMA 3.12.3. For any given integer n there is no infinite sequence of pairwise disjoint 7con3t -positive sets whose union is in J,'.

PROOF. Assume the contrary, and let ak (k E N) be pairwise disjoint ,7cont

positive sets whose union a is in J. Let 0m: P(a) --* P(N) (m, E N) be Borel functions which 2-"-approximate 4), on P(a). We shall recursively find sequences

a'j,xj such that for every j E N:

(Al) xjCa'j='aj, 2 n. (A2) II(ej(Uixi) If we succeed in finding such sequences, let j be such that

0, (UX) 0C (

< 2-".

Xi w

104

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

Since a'j n Ui xi = xj, we have

(ei

(u) i

n (b. (aj )) A(D* (xj ) w

(e3(yxi)*(Uxi))

n-P*(a')

<2",

contradicting (A2). Therefore it will suffice to construct a3' and xj satisfying (Al) and (A2). In order to make the recursive construction possible and to assure (A2), we shall need some auxiliary objects satisfying the following for all j < 1 in N:

(A3) cj = a \ Uj=1 az (A4) yj = {y g cj I1(8j(Ui1 xi u y) fl D.(a

w > 2-"} is relatively comeager in P(cj). (A5) Ujl Uj2 ... are dense open subsets of P(cj) such that ni Uji g yj. (A6) Yj,i = {y g ci : Ui=j+1 xi U y E yj} is relatively comeager in P(cj). (A7) {y C cj : y n U=j+1 ati = Ui=j+1 xi} is included in Uji. Assume these objects are chosen so that (Al) and (A3)-(A7) are satisfied and let x = Ui xi. For m E N, (A7) implies that +1 xi E nIE Umi E ym, so (A4) implies (A2). Therefore it will suffice to assure (Al) and (A3)-(A7). We shall now describe the recursive construction of these objects. Assume xk, a', and {Uki}O01 are chosen for k = 1, ... , m - 1 to satisfy (Al) and (A3)-(A7) for :

U00

j
Xm = {x C am : Hm(x) is relatively comeager in P(cm)} and note that if x,,,, 0 Xm, then y,,, is relatively comeager in P(cm) and (A4) is satisfied. The following claim will make it easier to pick xm 0 Xm. CLAIM 4. The intersection of the complement of X,,, with any nonempty cloven subset of P(am) is relatively nonmeager.

PROOF. Assume the contrary, and let v C u C am be finite and such that Xm, is relatively comeager in the set

Ix Cdm : xflu=v}. Then there is a disjoint partition am \ u = a, 00a' and t' C a,, (i = 0, 1) so that for every xCam both and (x\(a°mUu))UvUt° are inXm (see e.g., [141] or proof of Lemma 1.3.2). Define B;,: P(a) - P(N) by Bm0(x) = Bm((x \ (a,i». U u)) U v u tI) n (%(am), Bml (x) = em.((x \ (am. U u)) U v U t°) Gm' (x) = Bm0(W ) U Bml(x)

(am),

3.12. A LOCAL VERSION OF THE OCA LIFTING THEOREM

105

Note that 0,, is a Borel function. For x C am and y g N, define m-1

Hm(x, y)

J(

zg

U xi u x u z n4D.(a,n) Ay

Illl

i=1

' < 2-n+11.1

We claim that for any x Cam the set H;n(x, 4).(x)) includes

H=Hm((x\(a,1, uu))uvutl)nHm((x\(a.°rUu))UvUt°), and in particular it is comeager. This is so because H is the set of all z C c,,, for which

m-1

((e( U xi U X U Z I n .(dm) 041.(x)

<2 -n

i=1

for both a = 0,1 and O; is defined as an amalgamation of O; 0 and e 1. Since the set H;, is Borel, the set Z = {(x, b) E P(am) x P(N) : H; (x, b) is relatively comeager in P(c,,,,)}

is an analytic subset of P(am) x P(N) (again by [74, 29E]). Note that (x, b) E S implies that for some (actually, comeager many) y C cm we have

r IN4?.(x)AbMjW S

m-1

4?.(x)O I ©,n (U xi u x u \i=1

n 4?=(a.,) W

m-1

< 2-n+2

U xi U X U y n D. (am) Ab

(o;,,

i=1

W

and therefore by Jankov, von Neumann theorem (Theorem 3.11.2) applied to S there is a Baire 2-+2-approximation E to 4D, on P(am). Since E is continuous on a dense G6 subset G of P(a,,,,), we can modify E to obtain a continuous 2-n+3approximation to 4?, on P(a,,,,) (like in Lemma 1.3.2 or the definition of o;, above).

Since am =' a,,,, this gives a continuous 2_n+3-approximation to 4>, on P(am), This concludes the proof. contradicting the fact that am is not in

Fix k < m and recall the inductive assumption (A6), that the set yk.n-1 is relatively comeager in P(Cm_1). By the Kuratowski-Ulam theorem (see e.g., [74, 8.K]) applied to the product P(cm._1) = P(am) x P(cm) the set m-1

(*) Zk = {x C am

yC

: U xiUxUy E Ykm_1

Illll

is comeaer in P() }

i=k

is comeager in P(am). Therefore n.-II Zk is comeager in P(am) as well. By Claim 4 we can find

m-1

xm E (P(am) \ Xm) n n Zkk=1

Since we shall choose a;,, _' am and xm =' tm, this takes care of (A4) (see definitions of ym and Hm(x)), and also of (A6). The set Hm(2m) is not comeager in P(c,,,,), because 2m ¢ Xm. Also, for every k < m the vertical section

{crn

m-1

: U xiU2mUyEUkm i=k

106

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

is an open subset of P(C,n), which is moreover dense since it includes m-1

yCC,n: U Xiu

UpEYk

i=k

and the latter set is dense by the choice of 2,n. Therefore we can find a finite t C u C En such that H,n(2m) is relatively meager on {y C C,n y fl u = t} and that XiUUtUyE U, for allyc Cm \ U. This, together with (*), assures (A6) and (A7) for l = m, if we let a'm dm U u, xm = tm U t, and pick a sequence {Umi}°_1 of dense open subsets of :

P(cm) whose intersection is included in Ym. So (A5) is taken care of. Conditions (Al) and (A3) are obviously satisfied, so this describes the recursive construction. Objects constructed in this way satisfy (Al) and (A3)-(A5), and it was remarked before that this implies (A2). This ends the proof of Lemma 3.12.3. Recall that every element of A has infinitely many disjoint subsets which belong to B and that all members of B are Tcon6-positive. Recall also the conclusion that all but countably many elements of A belong to 7,,-3 . All this is in contradiction with Lemma 3.12.3. This shows that the assumption that 7cont is not ccc over Fin leads to a contradiction, therefore finishing the proof of Proposition 3.12.1.

3.13. The proof of the OCA lifting theorem for the analytic P-ideals We are now in a position to start the proof of the main result of this Chapter, Theorem 3.3.5, so let us first recall its statement. THEOREM 3.13.1. Assume OCA and MA and let I be an analytic P-ideal. If

4): P(N)/Fin -1 P(N)/I is a homomorphism, then oD has a continuous almost lifting. Equivalently, ' can be decomposed as 4D1 (D 4)2 so that 4)1 has a continuous lifting, while the kernel of 4D2 is ccc over Fin.

By Proposition 3.12.1, it will suffice to find a single mapping SZ which uniformizes all 4a for a E 7cont (for definitions see §3.12). OCA has proved to be extremely useful in similar situations (see [125, §8], [135, §10], [142, Lemma 2.4]). PROOF OF THEOREM 3.3.5. The key object in this proof is the following ideal extending the ideal ,coat: ,7i = { a C N : there is a partition (D. (a) = Co UCl such that 1Co has a continuous lifting and the kernel of 4)Ci is ccc over Fin} Hence Theorem 3.3.5 translates as ",7i is an improper ideal." We claim ,7i is equal to the following ideal ,72 = { a C N : there exists a continuous lifting',.: P(a) -* P(4). (a)) of a homo-

morphism of P(N)/Fin into P(Ta(a))/I such that the ideal Ka={bCa : %(c)=1%Fa(c) for all cC b} on a is ccc over Fin}.

To see that ,71 C ,72, take some a E ,7i and let W2 = DC'.. Then for b E ker(o;' ) we have 1Da(b) =Z %(b). Therefore ker((D.' ) includes Ka, so it it ccc over Fin. To see that 92 S 71 (a fact which we will actually not need), assume a E ,72, and let

Co=Ta(a). For b C a let Db = (I.(b)ATa(b)) fl Co =z ('.(b) \ Ta(b)) fl Co

3.13. THE PROOF OF THE OCA LIFTING THEOREM

Then {c C a

:

107

xP a(c) fl Db E Z} is an analytic ideal and it includes the nonmeager

set {c C a : 'D.(c)O'Pa(c) E Z}. Therefore b belongs to this ideal, and Db E Z. So (-P.(b)O'I'a(b)) fl Co is in I for all b C a, and T. is a Baire lifting of the homomorphism I)c0 . On the other hand, if b E /Ca, then C (b) \ Co E Z, and therefore ker(4P°') is ccc over Fin. CLAIM 1. For all a E ,72 we have /Ca _ 7cont I a. PROOF. Pick c E .7cont [ a and consider Wa, a continuous lifting of 4). on P(c).

The ideal

{dCc: W(e) =' Wa(e) for all a C d} is a coanalytic subset of P(c) and it includes a nonmeager ideal Ka [ c of subsets of c so it is improper and therefore >ya(c) =' T,(c) =7 %(c).

By Lemma 3.11.6 and Theorem 1.5.2 we can assume each Wa for a E ,72 is asymptotically additive, namely of the form `yH for some strictly increasing sequence of integers {ni = na}, sequence of disjoint finite sets of integers {vi = va} and functions {Hi = Ha} from subsets of the interval [ni, ni+1) to subsets of vi (see Definition 1.5.1). The following coherence property is an important consequence of this representation of functions Wa.

CLAIM 2. For all a, b E ,72 and all n there are m, m' such that for all s C a n b we have

cp((Wa(S)AWb(S)) \ [1,m)) < 2-n

p((Wa(S \ [1,m))AWb(S \ [1,m )))) _< 2-n. PROOF. Assume the first inequality fails for some n and for all in. Then, using

the fact that T. and `fib are asymptotically additive, we can pick subsequences {ma} of {na} and {mb} of {nb} such that

0=ma=Mb <mi <mb <m2 <m2 <... and for every i there is an si C [m2i+ m2i+1) fl a fl b such that W((Wa(Si)DWb(Si)) fl [m2i, oo)) > 2-n.

By Proposition 3.12.1 ideal 7cont is ccc over Fin and therefore nonmeager, so by (a) of Theorem 3.10.1 and we can furthermore assume s = U°° 1 si E .7cont Therefore Claim 1 implies WYa(S) =Z 4.(s) =' WYb(s). By the choice of {si} we have

Wa \U Si = U Wa(Si) and Wb CU Si') = U'kb(Si), i=1

i=1

z-1

i=1

therefore limk- cp((Wa(s).AWb(s)) \ [1, kJ) > 2-1 and Wa(S)DPb(S) is not in Z-a contradiction. To see that m' can be found so that the second inequality is satisfied, find j such that m < nj and pick m' bigger than max(Ui<j(Ha([ni,ni+1)) U H' ([ni, ni+1))) This m' is obviously as required.

The following partition is the key tool for applying OCA in order to obtain a single function which uniformizes all T. (a E 32).

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

108

DEFINITION 3.13.2. For D C N and n E N let [32)2 = L0 -(D) U Li (D) be a partition defined by letting {a, b} into Lo (D) if there is an s C a n b n D such that

c0('ya(s)0'I'b(s)) > 2-n. We shall write Lo, L1 instead of L0 (N), Li (N). Since each T. is continuous and co is lower semicontinuous, these partitions are open in a natural separable metric topology on ,72 obtained when each a E 72 is identified with the pair (a, Ta). Observe that the proof of the following statement does not make use of either OCA or MA. PROPOSITION 3.13.3. For D C N the following are equivalent: (L1) The ideal ,72 is a-Li (D)-homogeneous for all n, (L2) D is in ,72.

PROOF. Assume (L2) and let WD be a witness for D E ,72. By Claim 2, for every a E 32 there is an ma such that for all s C [ma, oo) n D n a we have 1.

IP(WD(s)0'ya(s)) < 2-n This makes it easy to cover ,72 by countably many Li (D)-homogeneous subsets. Now assume (L1). Without a loss of generality, we can assume D = N, so there are L'-homogeneous sets {7-lz }°°1 covering ,72. Recall that 7-l stands for the downwards closure under C of a subset N of P(N).

CLAIM 3. If h C 72 is Li -homogeneous, then there is a Borel-measurable mapping 0: P(N) -> P(N) which is a 2-n+2-approximation to 1, on 7-l n Jcont. PROOF. Let X be a countable subset of N such that the set {(a, Ta) : a E X} is dense in {(a, TO : a E N}. For b E 7N fix some element x(b) of N such that b C x(b). Recall that for d E X the function Td is asymptotically additive, and let {n4}, {vd} and {Hd} be the parameters determining it (see Definition 1.5.1). For d E X and a C N let i(d, a) be the minimal i such that a n [nd, n }1) is not included in d, and let

00

k(d, a) = min U va. j=i(d,a)

(We are allowing i(d, a) and k(d, a) to be equal to oo.) These definitions imply that Wd(a) is a reasonable approximation of (P (a) up to [1, k(d, a)) for a E ,7cont, more precisely that for all a and d E X we have (*) min(Pd(a)OWd(a n d)) > k(d, a). For a C N let Ta = {(s, k)

s C [1, k) and ip((Td(a n d) n [1, k))As) < 2' for some d E X such that k(d, a) > k}, ya= {b C N : (bn [1,k),k) E T. for all k c N}. :

We would like to prove that Ya is a set of good approximations for 0(a), i.e., that any function 0 such that 0(a) E Ya for all a E 7-l n 7cont is a 2-n+2-approximation to 4), on that set. Since X is dense in N in a strong sense and H is L'-homogeneous, we have Wx(a)(a) E Ya for every a E 7-l n 7cont By Claim 1 and ,72 Q Jcont for every a E 7-l n ,7cont there is an in such that W((1b.(a)A'yx(a)(a)) n [m, no)) < 2-n+l

3.13. THE PROOF OF THE OCA LIFTING THEOREM

109

Fix a E 7-l n Jcont and consider a b E) a. For k> m find d E X which witnesses (a)) fl [m, k) (b n [l, k), k) E Ta, thus in particular k(d, a) > k. Hence the set is included in the set (recall (*)) ((bA%Pd(a fl d)) U (Wd(a fl d)Agld(a))

U ('I`d(a)A1P.(a)(a)) U (T.(a)(a)0D*(a))) n [m, k), and therefore, since {d, x(a)} E Li , by using (*) we have p((b04)*(a)) n [m, k)) <

This is true for all large enough k and cp is lower semicontinuous, hence every

mapping 0: 7-l fl Jcont -> P(N) such that 0(a) E Y. is a 2-' 2-approximation to 4)*. But the set Y. is closed and a - ya is a Borel mapping of P(N) into its exponential space, and therefore the mapping O defined by 0(a) = inf ya if 0 and 0(a) = 0 otherwise is a Baire-measurable function. This ends the Y. construction.

Claim 3 implies that for every n the ideal 3cont can be covered by countably many hereditary sets 7-ln n front and that on each one of them there is a Borel 2'2-approximation to (D*. Therefore by Theorem 3.11.3 we have N E ,72.

By OCA and Proposition 3.13.3, it only remains to prove there are no uncountable Lo-homogeneous subsets of 72 for every n. We shall apply MA to prove this. The proof that the partial ordering we use has the countable chain condition requires the following consequence of OCA. LEMMA 3.13.4. OCA implies that 72 is a P-ideal.

PROOF. Assume 72 is not a P-ideal, and let Al C A2 C A3 C ... be a sequence in 72 such that no A E 72 almost includes all Am. For the convenience of notation, assume Jcont and 72 are ideals on N x N and that A. _ [1, m) x N. For f E NN let

rf={(n,m) : m* f such that r9 \ r f is not in .7cont, then an easy recursive construction shows that ,Jaunt is not ccc over Fin. Therefore using Proposition 3.12.1 and going to a set of the form N2 \ r f for some f we can assume rg is in 7Q0nt for all g E NN.

Fix A. We claim there are no uncountable Lo -homogeneous subsets of r f (f E NN). Assume the contrary, let I C NN be of size N1 and such that {rf : f E I} is Lo-homogeneous. Then by OCA there is a go E NN which almost dominates all f E I (see Theorem 2.2.6). By going to a subset of I and possibly making a finite change to g, we can assume that r90 includes all {r f : f E 11, and that., therefore, there is an uncountable Lo(r9o)-homogeneous set. This contradicts Proposition 3.13.3. Applying OCA we conclude that {rf : f E NN} is covered by countably many Li-homogeneous sets. Since NN is a-directed under <*, for one of these sets, say X = {rf : f E J}, the index set J is cofinal in (NN, <*) (Lemma 2.2.2). Then by Lemma 2.4.3 we can find mo such that for every g E NN there is an f E J such

that g(n) < f (n) for all n > mo.

CLAIM 4. For every a in 92 there is an m = m(a) > rr,) such that for all d E X and all s C (and) \ A,,, we have p(Wa(s).A a(s)) < 2-n+3

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

110

PROOF. Assume this fails for some a E ,72. Recursively find dk E X and finite Sk C (a n dk) \ Ak, so that

P(Ta(Sk)ATdk(Sk)) > 2-n+3

for all k. There is a function g E NA' such that I79 almost includes U. sn,, so we can find d E X which almost includes Uk Sk, therefore Uk Sk E Jcont By Claim 2, for all but finitely many k we have cc(WYa(Sk)IWYd(Sk)) < 2-", and therefore by the Li-homogeneity of N we have
Since J dominates NN on [mo, oo), by Claim 4 we can easily deduce that for every m > mo the set H,,, = {a E 72 : m(a) = m} is Li -2 (1` 2 \ A,)-homogeneous. Note that for a E %,,, we can replace the function Ta by the function

bHTa(b\A,,,,)UWAm(bfA.),

for bCa

and still have that the ideal /Ca (see definition of 32) is ccc over Fin. With this adjustment each R, becomes L1-2-homogeneous. Hence we can assume that J,0nt is covered by countably many hereditary Li -2-homogeneous sets for every n. From this and Proposition 3.13.3 we conclude that N2 belongs to ,72, contradicting the assumption that no element of ,72 almost includes every set A,,.

LEMMA 3.13.5. Martin's Axiom implies that for every n E N there are no uncountable L0 '-homogeneous subsets of 32

PROOF. Assume the contrary, that N = {aa : a < wl} is an uncountable Lohomogeneous subset of ,72 for some n. Since ,72 is a P-ideal, we can find b£ (6 < wl ) such that for all t7 < 6 < w1 we have

(a) an C (b) bn C b£. Now redefine each'Pb1 so that it coincides with Tat on P(ae) and that it still witnesses that bt E 32. This can be done in a straightforward manner, using the original'Pb1. Then the set {be < w1} is a Lo-homogeneous (where Lo is computed using the new 'b{'s) subset of 32. Therefore, from now on we can assume that the elements of H form an C'-increasing w1-chain. Our forcing terminology is standard. For the undefined notions see ([82]). The partial ordering we shall use is a descendant of Kunen's partial ordering for making

gaps in P(N)/Fin ccc-indestructible ([81]; see also [125, p. 74]). Let P be the poset whose conditions are of the form p = (s, k, F), where (P1) k E N, s C [1, k), and F is a finite Lo+2(s)-homogeneous subset of H. We shall add superscript "p" to elements of p when needed, like e.g., in (P2) below. Ordering on P is defined by letting p < q if

(P2) kP>k4,sPn[1,k4)=s9,andFPDF9. If G is a sufficiently generic filter in P and D = UPEG SP, X = UPEG FP, then X is an L0'+2 (D)-homogeneous subset of H. Recall that 32 includes the ideal 7cont which is ccc over Fin, and is, therefore, ccc over Fin itself. We will apply MA to an amalgamation Pw, of uncountably many copies of P to violate this fact, but first we have to prove P has a chain condition strong enough to assure the ccc-ness

of P,,.

3.13. THE PROOF OF THE OCA LIFTING THEOREM

111

DEFINITION 3.13.6. A pair of uncountable subsets X, Y of a poset P such that every p E X is incompatible with every q E Y is called an uncountable rectangle of incompatible conditions. One similarly defines the notion of an uncountable rectangle of compatible conditions.

Note that the conclusion of the following claim gives an uncountable rectangle of compatible conditions. CLAIM 5. For every uncountable rectangle pot, pit (( < w1) of conditions in P with equal working parts and such that sets Fot, Fit from their third components are all pairwise disjoint there exist uncountable Xo, X1 C w1, k E N and s C [1, k) such that (s, k, Foe U F1,)

is a joint extension of pot and pi, for all l; E Xo and 77 E X1. PROOF. For s C [1, k) let

Ps,k = the set of all p E P of the form (s, k, F). We shall first prove that no subset {pe} (l; < w1) of P.,k such that Fe's are pairwise disjoint includes an uncountable rectangle of incompatible conditions. Assume the contrary, and let pot, pit (l; < w1) be a counterexample. We can assume all Foe are of the same size l and all Fit are of the same size m, say (1) Foe = {aot,...,al t}

(2) Fit = {ait,...,al}.

For 1 < w1 and j < l let n°ei, v°t3 and H9t3 (i E N) be the parameters of the v t' and H1 t3 (i E N) be the function 'f'oe = Wa'04 and for j < m let ny ,

parameters of the function `Fit = Waif (see Definition 1.5.1). Possibly by refining two sides of the rectangle of conditions by Claim 2 we can assume there is k > k such that for all 1; (recall that a Ck b stands for a n [k, oo) C b):

(3) aotCka2 and for e = 0,1:

(4) cp((IYEt(t)OIYE,(t))n[k, co)) < 2for all finite t C aEtna Et and j, j' < 1. We can assume there is an r E N and n°j, v°', H°* (i <_ r, j < 1) such that for all e

and alli
i,

(5) n°t3 = tii' , v°ti = HP' i = v 0j and H°ti t z

a

(6) n°3- > k for all j < 1. Fix e < 71 < w1. Since 7-1 is Lo-homogeneous, there is a t C aot nap, such that

(7) P(`yoe(t)0`yon(t)) > 2-n. Note that, by (5), we can assume that mint > k. Assume

(8) `p('yit(t)0`f'oe(t)) > 2-n-1. We claim that pot and pl,, are compatible. First we prove that for all j < m and j' < 1 we have

(9) P('I/it(t)Avoe(t)) > 2 -n-2.

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

112

Assuming (9) fails for some j and j', by lg(t)A'Yog(t) c ('P1

U ('Wig(t)A'yog(t))

1

we have cp( ig(t)O'og(t)) < 2-"-3 +2 -n-2 + 2-"-3 < 2-n-1, contradicting (8). Therefore (8) implies (s U t, max t + 1, Fig U Fog) is a joint extension of pog and p1g.

Similarly, the assumption

(10) a(W (t)oWOn(t)) > 2-n-1. implies, by the same argument as above, that pig and pon are compatible. If (8) and (10) both fail, then we have og(t).Won(t) c

og(t)o Sg(t))

and cp(1Yog(t)OtYo,(t)) < 2-n-1 + 2-"-I = 2-n contradicting (7). Therefore at least one of (8) or (10) happens, and this proves that pog, pig ( < wi) is not a rectangle of incompatible conditions.

To finish the proof, let pog, pig ( < wi) be conditions from P with equal working parts and such that sets Fog, Fig are all pairwise disjoint. We can assume Fog (Fig) are all of the same size 1 (rm, respectively), and that the set of all 21-tuples (aog, dog,

'Ypg)

is an l1-dense in itself subset of P(N)1 x (NN)I (the functions'og are assumed to be asymptotically additive, and therefore naturally represented by elements of FinN - Na, the set which, as usually, we identify with the irrationals). Similarly we can assume the set of all 2m-tuples (ait, Ti Since we have already proved is an li-dense in itself subset of P(N)' x that pog (C < wi) and pig ( < w1) cannot be an uncountable rectangle of incompatible conditions, there are compatible pog and p1, Let (s, k, F) be the joint extension of these two conditions. Since the compatibility relation of P is an open subset of P x P (where P is taken with the separable metric topology induced from FinxN x ®°°1(P(N) x N5)2' when (s,k,{a1,...,ak}) is identified there are uncountable sets Xo, X1 C w1 with (s, k, ai, IF,,, a2i 'a ... , ak, (NT)m.

such that (s, k, Fog U completes the proof.

extends pog and pin for all t E X0 and 77 E X1. This

Let P,, be defined as follows: A typical condition is p = (I, sg, kg, Fg

E I),

where

(P3) I is a finite subset of w1i (sg, kg, Fg) is in P for every C E I. (P4) The ordering is defined by letting p < q if (P5) IP D 19,

CLAIM 6. The poset P,,,, is ccc.

3.13. THE PROOF OF THE OCA LIFTING THEOREM

113

PROOF. Let p, (a < w1) be an uncountable subset of Pa,, . We can assume

sets I' = IT from their first coordinates form a A-system with root I. It follows from the definitions that conditions p and q in P,, are compatible if and only if the conditions (I, s£, k£, F

:

e E I) and

(I, s£, 0, F

E I),

where I = Ip fl Iq,

are compatible. Therefore we can assume that I' = I = {51'..., 0

for all pa. Uniformizing further, we can assume that for every fixed i E {1, . . . (a < w1) are equal, say the working parts of all conditions

, l}

s{=siandk=ki,forallaandiE{1,...,1}. We can, moreover, assume that sets F = FP* from the third coordinates of p° (Ci )'s

form a A-system with root Fe; . Now observe that by the definition of P the conditions p, q E P are compatible if and only if the conditions where F = FP fl Fq, (sp, k", FP \ F) and (s1, kq, F9 \ F), are compatible. Therefore we can assume that the sets F (a < wl) are pairwise disjoint for every fixed i E {1,...,1}. We can apply Claim 5 to p. (CI) (a < w1) and get subsets X1, Y, of w1 such that p, (c1) (a E XI) and pp(1'I) (i3 E Y1) form an uncountable rectangle of compatible conditions. Let k' > ki and si C [1, ki) be such that

(si,ki,F£' UF£ ) is a joint extension of pa(C1) and p,(C1) for all a E X1 and 3 E Y1. Now we can extend every p. (b) (a E XI U Y1) to p'a (C2) = (92, k2, Fa(S2))

so that k2 > k1. Another application of Claim 5 gives sets X2 C X1 and Y2 C Y1 such that p,r(C2) (a E XI) and p"3(C2) (/3 E Y2) is an uncountable rectangle of compatible conditions. Note that, if

(s2,k2,FeZ aF{ ) is a joint extension of p'. (C2) and p'' (l;), then by our choice of k2 the condition (P7) is satisfied between si and 92. By continuing this construction f o r i = 3, ... ., 1we

get an uncountable rectangle of compatible conditions in P, Let G be a sufficiently generic filter of Pw1 and for C < wi define

De = U sp and X£ = U F£ . pEG

pEG

Then Dc's are pairwise almost disjoint and X£ is Lo}2(Dc)-homogeneous for all By ccc-ness of P, there must be a condition p in it which forces there are uncountably many 1''s with uncountable Xe's. Then we can choose a suitable family of 111

many dense open subsets of P,, so that if G is a filter of Pw, containing p and intersecting all these dense open sets, then for uncountably many C the set XX is uncountable. (Note that this implies the corresponding Dg are infinite.) By Proposition 3.13.3, for such the set D£ is not in J2 and therefore not in .7cont But this is in the contradiction with Proposition 3.12.1, and this completes the proof of Lemma 3.13.5.

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

114

By Lemma 3.13.5 and OCA, the ideal .72 can be decomposed into countably many Li-homogeneous subsets for all n. By Proposition 3.13.3 this implies N E ,72. This ends the proof of Theorem 3.3.5.

3.14. Remarks and questions Let us start with a diagram of the embeddability relation between analytic quotients under OCA and MA. (Z4)

(Z2), (Z3)

(non-P) T

T

Fin x0

0 x Fin

(S)

(T)

The ordering <EM under OCA and MA.

This diagram is very similar to one given previously, which presents the
PROBLEM 3.14.1. Find more analytic ideals I with the property that under OCA and MA every homomorphism of P(N)/Fin into P(N)/Z has a completely additive almost lifting.

In particular, do the Fo-ideals, or the ordinal ideals Z. and W., (see §1.14) have this property?

It is also relevant to know that Just has proved ([63]) that a local version of the lifting theorem (Theorem 3.12.1) is consistent for all analytic ideals. It moreover follows from OCA and MA, by [64], for Fo ideals (the open coloring in the proof of this fact is defined using Mazur's Theorem 1.2.5(a)).

There is an open question related to our work which can be regarded as a version of Todorcevic's conjecture for arbitrary connecting maps. Note that every additive lifting of a homomorphism is of the form Fu defined in Example 3.2.3,

3.14. REMARKS AND QUESTIONS

115

since if F: P(N) -> P(N) is a homomorphism then U, = {A : F(A) E) n} is an ultrafilter on N. Since nonprincipal ultrafilters on N never have a property of Baire,

a Baire-measurable additive lifting has to be completely additive and therefore Todorcevic's conjecture (see § 1.4) can be rephrased as: Can every Baire-measurable homomorphism of quotients over analytic P-ideals be turned into an additive one? All examples of nontrivial homomorphisms of P(N)/Fin constructed in §3.2 using the Axiom of Choice (or rather, the existence of a nonprincipal ultrafilter on P(IN) have additive liftings.

QUESTION 3.14.2. Do OCA and MA imply that every homomorphism

: P(N)/ Fin -+ P(N)/ Fin has an additive lifting? Or, is this statement at least consistent with the usual axioms for Set theory?

A part of the motivation for asking this question comes from Chapter 4, (see Question 4.11.4). By Theorem 3.3.5 an answer to this question requires analysis of homomorphisms of P(N)/Fin into itself whose kernel is ccc over Fin. The methods developed here do not seem to be relevant to this question. Throughout this monograph we have been looking for the strongest possible rigidity results, and we have seen (§2.1) that OCA and MA form the right setting for such investigation. Let us say a word on some different lines of research. According

to P. Erdos [28], the initial motivation for the question whether quotients over ideals Zo and Zlog are isomorphic (see §1.13) was the question of finding a large class of ideals on N whose quotients are pairwise nonisomorphic. This was later solved by Monk and Solovay ([98]) who have found such a class of 22N0 many ideals,

which is the best possible. Of course, their ideals are not analytic. QUESTION 3.14.3. Are there infinitely (or even uncountably) many analytic Pideals whose quotients are, provably in ZFC, pairwise nonisomorphic?

In [65] the following was proved (see also the paragraph after Lemma 1.3.3). THEOREM 3.14.4 (Just-Krawczyk).

(a) P(N)/ Fin is never isomorphic to a quotient over an Erdos-Ulam ideal. (b) All quotients over F0. ideals are countably saturated and therefore isomorphic under CH. (c) All quotients over Erdos-Ulam ideals are isomorphic under CH. O We know only a few analytic P-ideals whose quotients are provably pairwise nonisomorphic. Theorem 3.14.4 implies that quotients over the following ideals are pairwise nonisomorphic: (i) Fin (whose quotient is under CH isomorphic to the quotient over any other Fo ideal, by Theorem 3.14.4 (b)),

(ii) 0 x Fin (whose quotient is isomorphic to (P(N)/ Fin)', (iii) Zo (whose quotient is under CH isomorphic to a quotient over any other EU-ideal by Theorem 3.14.4 (c))), (iv) Zo E) Fin (whose quotient is isomorphic to (P(N)/Zo) x (P(N)/Fin)), (v) Zo ® 0 x Fin (whose quotient is isomorphic to (P(N)/Zo) x (P(N)/ Fin)M), (vi) The ideal from Example 1.13.17.

116

3. HOMOMORPHISMS OF ANALYTIC QUOTIENTS UNDER OCA

The fact that the quotients over the above ideals are pairwise nonisomorphic follows from the facts that Zo r A is an Erdos-Ulam ideal for every positive set A (Corollary 1.13.5), that P(N)/Fin is homogeneous, and Theorem 3.14.4 (a) above.

Recall that, by Corollary 1.13.8, CH implies that (P(N)/So)N ^ P(N)/Z0. An another line of research which has attracted some attention is concerned with the possible structure of the automorphism group of P(N)/Fin, and in particular in finding models of set theory whose impact on this structure is intermediate between CH and OCA and MA in some prescribed way. For example, Velickovic proved in [142] that MA alone does not imply that all automorphisms of P(N)/Fin are trivial. In [108] Shelah and Steprans proved that there can be nontrivial automorphisms of P(N)/Fin, while all automorphisms are locally trivial. For more recent results on the possible structure of the automorphism group of P(N)/Fin see [109, 119]. Let us mention a question along similar lines. QUESTION 3.14.5. Is OCA a sufficient assumption for the lifting theorems of this chapter?

Note that by the already mentioned result from [142], MA alone is not sufficient. It is known that OCA alone does imply some local versions of lifting theorems; see [64]. There is certain interest in finding OCA's limitations, but the author finds the program outlined in §2.1 much more attractive.

CHAPTER 4

Weak Extension Principle 4.1. Introduction In this Chapter we use some results of Chapters 1, 2 and 3 to analyze CechStone remainders of countable, locally compact spaces. By a classical result of Parovicenko ([101]), Continuum Hypothesis (CH) implies that all these spaces are homeomorphic and that their finite products are continuous images of w*. Van Douwen has proved that, if X is a countable locally compact space, then two finite powers of X* are homeomorphic if and only if their exponents are equal ([17]). Later on, Just ([61]) has described a situation in which (w*)(d) does not even map onto (w*)(d+1) (this result was later reformulated into OCA-context, see [64]). More recently, A. Dow and K.P. Hart ([23]) have proved that under OCA the only CechStone remainder of a o-compact, locally compact, space which is a surjective image of w* is w* itself. The key part in their proof is the fact that (w2)*, the remainder

of an ordinal w2, is not a surjective image of w*. (Note that in [23] the ordinal w2 is denoted by IID.) Before [23], van Mill ([97, Theorem 2.2.1]) proved that w*

and (w2)* are not homeomorphic assuming that all autohomeomorphisms of w* are trivial, the assumption which we now know to follow from OCA and MA ([142]). We shall give a uniformized treatment of these results and extend them. Our main tool will be the following weak Extension Principle (o: and -y will always denote countable ordinals and d, l are positive integers): wEP(a, y) For a continuous F: (a*)I') -, (ry*)(t) there is a clopen partition (a*)(d) = Uo U U1 such that F" U0 is nowhere dense inside (-y*) (1) and F f U1 extends to a continuous map F1 : (/3a)(') -* (,Ory)<<1 so that F1a(d) C y(t).

By wEP we shall denote the statement "wEP(a, y) for all countable ordinals a, y." We shall prove that it follows from from OCA and MA. The set of natural numbers is in this Chapter denoted by w instead of N, because we are working with countable ordinals. Note that a product of d many copies of X is denoted by X(!). This is done in order to avoid confusion and distinguish the ordinal wd from the finite power w(d). A subset A of w(a) is d-compact if it can be decomposed into sets A1.... , Ad

such that the i-th projection of Ai (i = 1, ... , d) is compact. Let Rd denote the algebra of subsets of a(d) generated by clopen rectangles. We shall usually denote

the algebra Rw by Rd. Clearly (a*)(') is the Stone space of the quotient of Rd over the ideal of d-compact sets. It is understood that, whenever we talk about the Cech-Stone remainder X*, the space X is assumed to be non-compact. By X -y Y we denote the fact that Y is a continuous image of X.

The organization of this chapter. In §4.2 we prove a combinatorial lemma about the maps h: w2 -, w which is later used to decompose maps f : (Ow) (2) 117

/3w.

4. WEAK EXTENSION PRINCIPLE

118

(This lemma is generalized in [42].) In §4.4 we analyze continuous mappings between finite powers of Jaw, extending some work of van Douwen ([17]) and partially confirming two of his conjectures ([17, Conjectures 8.3 and 8.4]; see Conjectures 4.3.1 and 4.4.1 below). In particular we give a complete description of autohome-

omorphisms of finite powers of Jaw. These results do not use any additional settheoretic assumptions. Using wEP, we also give a description of autohomeomorphisms of a finite power of w*. In §4.5 wEP is used to characterize when a* maps onto y* for countable ordinals a and -y, and to analyze the question when some finite power of a* maps onto some finite power of y*. These results are extended to locally compact countable spaces in §4.7. In §4.8 we prepare for the proof of wEP from OCA and MA, given in §4.9. Some extensions of wEP are stated in §4.10, and other remarks and open questions can be found in §4.11. The results of this Chapter can be put in proper mathematical context only in the light of §2.1.

4.2. Dependence of functions on their variables In this section we prove a combinatorial result about the dependence of functions on their variables (see [26] for a related study of functions). This result is the essence of Theorem 4.3.2 from the following section. Let us introduce some definitions. For a finite power X (d) of a set X and j < d let Tri : X (d) X be the projection mapping to the j-the coordinate. Mapping F: X(d) -* Y depends on at most one coordinate, (or depends on at most one variable) on U C X (d) if there is i < d such that F t U = (G o 7ri) F U for some G: X -* Y. THEOREM 4.2.1. For every mapping h: w(2) _ w one of the following applies: (1) There is a partition of (2) into finitely many rectangles such that on each one of them h depends on at most one variable.

(2) There are infinite sets X = {m2} and Y = {ni} and f : w - {0,1} such that

f o h(m2, ni) =

I0,

1,

ifi<j, ifi=j.

Before we start the proof, let us note that (1) and (2) exclude each other: If h satisfies (2) and w(2) is split into finitely many rectangles, then one of these rectangles contains (mi, ni) and (mi, ni) for some i < j. But h(mi, ni) differs from both h(mi, ni) and h(mi, n3), and therefore h depends on both coordinates on this rectangle.

PROOF OF THEOREM 4.2.1. Fix for a moment a pair of nonprincipal ultrafilters U, V on w. If there are no sets A E U and B E V such that h depends on at most one coordinate on A x B, we shall recursively pick sequences mi, ni satisfying (2). It will clearly suffice to assure

(3) h(mi, ni) # h(mk, nk) whenever i < j and k E w. Let us recall a standard notation. If U is an ultrafilter and T(x) is a statement of the language of set theory, then

(Ux)r(x) stands for "the set {x : r(x)} is in U."

4.2. DEPENDENCE OF FUNCTIONS ON THEIR VARIABLES

CASE 1.

119

Suppose that

Al = {m : (Vn) (Vn')h(m, n) # h(m, n')} E U and

B1 = {n : (Um) (Um') h(m, n) ¢ h(m', n)} E V.

Let m1 = minA1, B} = {n c B1 (Vn')h(m,1,n) # h(m1,n')} E V, and n1 = < Mk-1 and n1 < n2 < . < nk_1 are chosen to minBi. Assume m1 < m2 < satisfy (3), and moreover that mi E Al and ni E B1 for all i. Let Bk = {n E B1 : h(mi, n) 34 h(m nj) for i, j < k}. Then mi E Al implies that Bk is in V. [If Bk is not in V, then for some fixed i,j < k the set C = {n h(mi,n) = h(mj,nj)} would be in V and we would therefore have VnVn'(h(mi,n) = h(mi,n'), contradicting mi E A1.] Let nk = min(Bk \ {1, ... , nk_1}), so that we have h(mi, nk) # h(mj, nj) for all i, j < k. :

:

Then nk E B1 implies

Ak={mEA1 : h(m,nk)54 h(mi, nj) for all i <j
but there are no A E U and B E V such that h r A x B depends on at most one coordinate. Define g: Al - w by

g(m) = lim h(m,n). n-V If limn.u g(m) is an integer, then let A2 E U be such that g is constant on A2. Since for every B E V the function h does not depend only on the first coordinate on the rectangle A2 x B, we can recursively find mi, ni so that h(mi, n1) = lim g(m) and h(mi, ni) lim g(m) for all i < j,

n-U

m.-U

therefore (3) is satisfied. If limrn.u g(m) is not an integer, consider the function g': Al --* w defined by

g'(m) = h(m, n), if n > m is minimal such that h(m, n) # g(m). If limm-u g'(m) is an integer, then we can recursively choose mi, ni such that

h(mi,ni) = lim g'(m) and h(mi,nj) = g(m) 0 m=u lim g'(m) for all i < j, n-U and this assures (3). Finally, if neither limn.u g(m) nor limrn...,u g'(m) is an integer

then by construction similar to that of Case 1 we obtain sequences in, and ni as required. CASE 3.

Suppose that

Al = {m (Vn)(Vn')h(m, n) h(m, n')} E U B1 = {n (Um) (Um')h(m, n) = h(m', n)} E V and there are no A E U and B E V such that h I A x B depends on at most one < mk_1 and n1 < n2 < . < nk_1 are chosen coordinate. Assume m1 < m2 < to satisfy (3) and moreover (4) mi E Al and ni E B1 for all i. .

4. WEAK EXTENSION PRINCIPLE

120

Let

Bk = {n E B1 : (3m E (A1 \ {1,..., mk_1))

(h(m, n) # h(mi, n)) for all i < j < k}. If Bk was not in V, then we could easily find A E U and B E V such that h depends on only one coordinate on A x B, so we may assume Bk E V. Let

Bk={nEB' : h(mi,n)54 h(mj, nj) for all i,j
Let us note a variation on Theorem 4.2.1. If I is an ideal on w, then a subset A of w(2) is 2-1-small if we can split A into two pieces Al and A2 so that the i-th projection of Ai (i = 1, 2) is in Z.

THEOREM 4.2.2. Let I be a countably generated ideal on w. Then for every mapping h: w(2) .-> w one of the following applies: (1) There is a partition of w(2) into finitely many rectangles such that each one

of them is either 2-1-small or on it h depends on at most one variable.

(2) There are infinite sets X = {mi} and Y = {ni} and f : w -* {0,1} such that

foh(mi,n,)

if i <j,

r 0, Sl

ifi =j,

1,

and both sets {mi : i E w} and {ni

:

i E w} are in I.

PROOF. Exactly like the proof of Theorem 4.2.1, only that the ultrafilters U and V have to be chosen to avoid Z. If (1) fails, then the required sequences mi and ni can be easily chosen to satisfy the requirements since the ideal Z is countably generated.

Theorem 4.2.1 is generalized to maps h: X(d) --' X, for an arbitrary X and a natural number d, in [42].

4.3. Prime mappings In [17, Prime Mapping Lemma], van Douwen proved that for every continuous

map f : ('8W) (d) --> 13w there is a clopen set U such that f [ U is elementary. He has actually proved this result for a larger class of so-called 1w-spaces. In [17, Conjecture 8.4] (reproduced as [52, Question 42]), he has conjectured the following:

CONJECTURE 4.3.1 (van Douwen). Let X be w* or Ow. If f is any continuous

binary operation on X, then there is a disjoint open cover U of X such that the restriction of cp to any set in U depends on at most one coordinate. The following theorem confirms van Douwen's conjecture in case when X = 13w.

4.3. PRIME MAPPINGS

121

THEOREM 4.3.2. Assume F: (/iw)(d) -+ (8w)(1) is a continuous function. Then there is a partition ()3w)(d) = U1 U U Uk into clopen sets such that lrj o F depends on at most one coordinate on each U i ( i = 1 , ... , k) for all j = 1, ... , 1.

Before we prove Theorem 4.3.2, let us note that, together with wEP, it gives a partial positive result for the case X = w* of van Douwen's conjecture. THEOREM 4.3.3. Assume wEP and that F: (w*)(d) -* (w*)(t) is a continuous function. Then there is a partition (w*)(d) = U1 U ... U Uk U V into clopen sets

such that 7rj o F depends on at most one coordinate on each Ui (i = 1,... , k) for all j = 1, ... , l and the image of V is nowhere dense. PROOF. By wEP, we can decompose (w*)(d) into two clopen pieces, U and V,

such that the image of V is nowhere dense and f [ U continuously extends to a map from ()3u)) (d) into /3w. Now apply Theorem 4.3.2 to this extension. PROOF OF THEOREM 4.3.2. Define h: /3w by h = F [ w(2). If for some A, B C w the map h depends on at most one variable on A x B, then F depends on at most one variable on 3A x )3B. Therefore by Theorem 4.2.1 it suffices to show that (2) does not apply.

Assume otherwise, and let mi and ni be the sequences such that h(mi,nj) ¢ h(mk, nk) for all i < j and all k. Define a map g: w(2) -* Qw by g(i,j) = h(mi,nj). This map extends to a continuous map G: (/3w)(2) -* /3w. We shall need a slight extension of [17, Lemma 14.1], but let us recall a bit of a notation first. For A C w define three subsets of w(2) as follows: DA = {(n, n) : n E A},

[A]2 = {(m,n) : m < n are in Al, (A) 2 = {m, n} : m < n are in A}. LEMMA 4.3.4. If f : (3w)(2) -# /3w is a continuous map, then the sets f"[w]2 and f"Aw are not disjoint. PROOF. The proof relies on the following fact, due to van Douwen. CLAIM 1. If s,L are disjoint finite subsets of w such that 1sn1 > n for all n, then the closures of the sets U[sn]2 and A U sn are not disjoint. PROOF. See [17, Fact 14.2].

We shall consider three cases. The closure of the set X = f"Ow in ,3w is countably infinite. Then this CASE 1. is a countable compact subset of /3w, which is a contradiction. The closure of the set X = f"Ow in /3w is uncountable. Then we can CASE 2. find U E w* such that f ((U, ... ,U)) 0 f"Ow, and in particular

(**) f((U,...,U)) 0 f"(w)2. Let U1 D U2 D U2 D U3 D ... be clopen neighborhoods of U such that n. U, fl f"(w)2 = 0. By the continuity there are sets A. E U such that f"[An]2 C U, for every n. Let sn C An be pairwise disjoint and such that ISnI = n for all n. By Claim 1, the closures of the sets X = Un[sn]2 and Y = A(Un sn) have a nonempty intersection, and therefore the closures of their f -images intersect as well. On the other hand, the set T = f"X U f"Y is relatively discrete in /w since T \ Un is finite

4. WEAK EXTENSION PRINCIPLE

122

for all n, and therefore all the limit points of T lie outside of nn Un D f"[w]2 D T. But this is a contradiction. The closure of the set X = f"Aw in /3w is finite. By going to an infinite subset of w, we can assume that for some fixed z E /3w we have f (m, m) = z for all m. Note that by Claim 1, closures of the sets [A]2 and DA have a nonempty intersection for every infinite A C w. Therefore we can derive a contradiction by finding an infinite A such that the closures of f"[A]2 and f"AA are disjoint. Recall that for A C w by [A]3 we denote the set of all unordered triples. This can be identified with the set of all strictly increasing, ordered triples from A. By {l, in, n}< we denote a triple {l, m, n} such that l < m < n. CASE 3.

Consider the partition [w]3 = Ko U K1 U K2 U K3 defined by: Ko, E

K1 i

K2, K3,

if f (l, m) = f (l, n) and f (l, n) = f (k, n), if f (l, m) = f (l, n) and f (l, n) 54 f (k, n), if f (l, m) # f (l, n) and f (l, n) = f (k, n), if f (l, m) # f (l, n) and f (l, n) f (k, n).

Recall that a set A is homogeneous for this partition if [A]3 is included in Ki for some i < 3. Then we say that A is Kt-homogeneous. By R.amsey's theorem, let A be an infinite set homogeneous for this partition. If A is Ko-homogeneous, then f"[A]2 = {z'} and f"DA = {2}, and z # 2'-a contradiction.

If A is K1-homogeneous, then let zm = f (m, n), for m < n in A (note that by the assumption z,,,, indeed does not depend on n). Note that z has an open neighborhood U such that the set C = {m : z,,,. 0 U} is infinite. But this implies that the closures of f"AC and f"[C]2 are disjoint, a contradiction. If A is K2-homogeneous, define zn = f (m, n) for m < n in A. Like above, derive a contradiction by finding an infinite subsequence of {zn} which does not accumulate to 2. Now assume A is K3-homogeneous. Define f1: w(2) -* /3w by

f1(m,n) = f(m,n+1). Then f1 continuously extends to f2: (/3w)(2) -* /3w. Since f2'Ow is infinite, either Case 1 or Case 2 applies to obtain a contradiction. This completes the proof of Lemma 4.3.4.

CLAIM 2. Every point (U1,U2,...,Ud+1) E (/3w)(a+1) has a clopen neighborhood U such that F depends on only one coordinate on U. PROOF. For every n E w consider the function Fn: (/3w)(d) -> /3w defined by

Fn(V1,... , Vd) = F(V1, ..., Vd, {n}). By the inductive assumption, there is an i = i(n) _< d, sets Bnj E U j (j = 2, ... , d) and a function gn: w -* /3w such that for all Vj E Bnj (j < d) we have Fn(V1,... , Vn) = {A : gn,1(A) E Ui(n)}.

If i = limn-u,,, i(n), then in some clopen neighborhood of (U1,. .. , Ud}1) the function F depends only on two coordinates, i and d + 1. Therefore the case d = 2 applies to reduce this further to a single coordinate on some clopen neighborhood of (Ul,...,Ud+l).

4.4. AUTOHOMEOMORPHISMS OF FINITE POWERS OF Ow AND w'

123

By the compactness, Claim 2 implies that (0w)(d+1) can be partitioned into finitely many clopen sets as required. This ends the proof of the case 1 = 1. Now assume l > 1 and fix F: (/3w)(d) ((3w)(l). Consider maps Fi = Sri o F, /3w is the i-th projection (i = 1, ... , 1). By the already proved where 1r : (/3w)(1) case, for each i we can find a clopen partition law = Ui U U2 U . U Ui(i) such that

the restriction of Fi to U,' depends on at most one coordinate for j < k(i). Then for s E jai=1{l, ... , k(i)} the sets

U$_nUs(i) i=1

form a clopen partition such that 7rj o F depends on at most one coordinate on U., for all s and j. This completes the proof.

4.4. Autohomeomorphisms of finite powers of /3w and w* In this section we shall give a simple description of all autohomeomorphisms of some finite power (/3w)(d). Every such autohomeomorphism F is determined by its restriction h to w(d), so it will suffice to describe those h: w(d) _-+ W(d) whose unique extension to (/3w)(d) is an autohomeomorphism. The simplest way to induce an autohomeomorphism is to take permutations (PI, ... , cd of w, fix a permutation

ir of{1,...,d}, anddefineh:w(d)-*w(d) by h(nl,...,nd) = ((pi(ri,r(1))>..., Pd(n,r(d))) We say that a function h of this form is elementary. Not all autohomeomorphisms of (/3w)(d) are induced in this way-for any positive integer m we can split w(d) into m many disjoint rectangles in two ways, say w(d) = R1 U U R, and W(d) = (1)

U P,,,, and define h so that it maps Ri into Pi and that is elementary on Ri, for all i < in. Let us say that an autohomeomorphism of (&)(d) is piecewise elementary if it is induced in this way. In [17, Conjecture 8.3] (reproduced as [52, Question 41]), van Douwen has PI U

conjectured the following (see also Remark 11.4 of [171): CONJECTURE 4.4.1 (van Douwen). All autohomeomorphisms of (/3w)(2) and all autohomeomorphisms of (w*)(2) are piecewise elementary.

In terminology introduced above, this conjecture says that all autohomeomorphisms of (OW) (d) and all autohomeomorphisms of (w*)(d) are piecewise elementary.

The following theorem, confirms this conjecture in the case when X = /3w. THEOREM 4.4.2. All autohomeomorphisms of (/3w)(d) are piecewise elementary. PROOF. This follows immediately from Theorem 4.3.2. We shall now use wEP(w, w) to prove that all autohomeomorphisms of any finite power of w* are piecewise elementary. A mapping 7r: A -+ B between two infinite sets is an almost bijection if it is finite-to-one, and h-1 (m) is a singleton for all but finitely many m E B. Following the case of /3w, we say that an autohomeomorphism F of (w*)(d) is a trivial autohomeomorphism of p*) (d) if it is induced by an h which is defined as follows: We can split w(d) into m many disjoint rectangles in two ways, say w(d) = RI U ... U R,,, and w(d) = P, u ... U Pm so that h maps Ri into Pi is an elementary map whose components co are almost bijections.

4. WEAK EXTENSION PRINCIPLE

124

THEOREM 4.4.3. Assume wEP. Then all autohomeomorphisms of (w*)(d) are trivial, and therefore piecewise elementary.

PROOF. Let F be an autohomeomorphism of (w*)(d), and let (w*)(d) = Uo U U Uk U V be a clopen decomposition guaranteed by Theorem 4.3.3, such that F is elementary on each Uz and that F"V is nowhere dense. Then V clearly has to be empty, and (w*)(d) = Uo U .. U Uk is the desired decomposition. Ul U

Theorems 4.4.3 and 4.4.3 together confirm van Douwen's Conjecture 4.4.1, but only assuming that wEP(w,w) holds. The following result, appearing in [41], confirms both Conjecture 4.3.1 and Conjecture 4.4.1 (here d and l stand for arbitrary natural numbers). THEOREM 4.4.4. Every map F: (w*)(d)

(w*)(1) is piecewise elementary.

4.5. tech-Stone remainders of countable ordinals In the study of Cech-Stone remainders of countable ordinals we can restrict our attention to indecomposable ordinals, i.e., those a such that a = ao + al and al < a implies al = 0, without losing generality. This is because a = ao + al and al 54 0 implies a* pt a,*, and every ordinal a can be written as ao + al, where a1 is indecomposable. We have the following picture (see the proof of Theorem 4.5.1 below):

(Recall that indecomposable ordinals are exactly those of the form we.) In [23] it was proved that the rightmost arrow is irreversible under OCA and MA. We shall extend this result by using the weak Extension Principle. Recall that a continuous map between regular topological spaces is perfect if and only if preimages of compact sets are compact (see [27]). THEOREM 4.5.1. wEP implies that the following are equivalent for indecomposable countable ordinals a and y:

(1) 7
(2) There is a continuous perfect surjection h: a -+ y.

(3) a + 1 -' y + 1. (4) a* -a, y*

Before starting the proof, let us note that the only implication that requires wEP is (4) implies (1), (2) and (3), that under CH (4) is true for all a and y (see [101]), that (2) implies (3) for any pair of topological spaces (by the proof below).

PROOF. (1) implies (2). Assume this implication fails, and let (a, y) be the lexicographically minimal pair for which this happens. We clearly have a > y. Let {an} be an increasing sequence of ordinals converging to a such that otp([a. + 1, an+1)) is an indecomposable ordinal less than a for all n. Similarly let {y,} be an increasing sequence of ordinals converging toy such that otp(['yn + 1, yn+i)) < y for all n. By the assumption on a and y, there is a continuous perfect surjection hn : [an + 1, an+l) ---i [yn + 1, yn.+l )

4.5. CECH-STONE REMAINDERS OF COUNTABLE ORDINALS

125

for all n. Define h: a -, -y by hn(6),

h(1:) = 'Ynl

E domhn,

if

1ffan

Then h satisfies (2). It is clearly continuous in all e # an, and in an it is continuous because hn_1 maps sets unbounded in [an_1+1, an) into sets unbounded in [yn_1+ 1, yn). Also, h is perfect because it maps unbounded sets into unbounded sets and it is onto because h" [an, an+1) = [Yn, 'yn+1) (2) is obviously equivalent to (3).

(2) implies (4). Let FO: ,Qa -, 0y be the unique extension of h to 6a, and let F = F0 a*. Then F maps a* into y* because h maps non-compact sets into non-compact sets. Then F"a* covers y* because Fo is onto and Fo a = h"a = -y. (4) implies (2). Fix F: a* --, y*. Let a* = Uo U U1 be a clopen decomposition as guaranteed by wEP. Since the set y* \ F,"U1 ? -y' n F"Uo is open and nowhere dense, it is empty. Therefore we can assume that F extends to ,3a (by possibly replacing a with a', the trace of U1 on a, or with its end-segment

if a' is not indecomposable). Set -y \ h"a is bounded in a because otherwise it includes an infinite closed discrete subset P, and P* is disjoint from the image F"a,

contradicting the fact that F maps a* onto y*. Therefore there is a y' < y such that h"a D [y', y), and mapping h: a -, -y defined by

JF(e),

.y,

if

> 7',

if<'y,

verifies (2). (3) implies (1). Recall that every indecomposable ordinal is of the form w£,

and that w4 + 1 is a scattered space of rank + 1 (see [27]). Finally, a continuous image of a compact scattered space is a scattered space whose rank is not bigger than that of the original space. A mapping h: a(d) -, y(l) induces a continuous mapping F: (a*)(d) --, (y*)(1) if there is a continuous extension F0 of h to (i3a)(d) such that F = F0 f (a*)(d). A subset A of a(d) is d-compact if it can be written as A = Al U A2 U U Ad so that

the i-th projection of Ai is a compact subset of a. A mapping h: a(d) -, y(I) is (d, 1) -perfect if preimages of 1-compact sets are d-compact and images of d-compact

sets are 1-compact. It is R-measurable if the preimages of sets in Rl are in Rda.

LEMMA 4.5.2. A map h: a(d) -+ y() induces a surjection F: (a*)(d) if and only if it satisfies the following three conditions: (Al) h is R-measurable, (A2) h is (d, l)-perfect, and (A3) the closure of the set y(l) \ h"a(d) inside y(I) is 1-compact.

(,Y*) (1)

PROOF. Condition (Al) is clearly equivalent to the fact that h continuously extends to a mapping FO: (8a)(d) -+ (0y)(I). Condition (A2) (which essentially says that the preimages of compact sets are compact) is equivalent to the fact that Fo (a*)(d) is included in (y*)(l). Finally, (A3) assures that (ry*)(t) is covered by Fo (a*)(d).

4. WEAK EXTENSION PRINCIPLE

126

COROLLARY 4.5.3. wEP implies that the following are equivalent for countable indecomposable ordinals a and ry:

(1) there is a continuous surjection F: (a*)(d) - (y*)(t) (2) there is an R-measurable, (d, l)-perfect surjection h: a(d) -. ry(t) PROOF. Let us prove only the nontrivial direction. If h satisfies (Al)-(A3), then let 6 < y be such that y(t) h"a(d) C {(7)i ...,pt) min77t < } and define h': a(d) ---,

by

h'((r11, ... -77d)) = h((max(771, ), ... , max(rld, C)))-

Since the interval [t;, ry) is homeomorphic to ry, h' satisfies the requirements.

4.6. Some Parovicenko spaces under wEP Let us now see what is the relationship between finite powers of Cech-Stone remainders of w and w2. It is easy to see that we have the following relations: (wa)*

... -) ((w4)*)(2)

... - )

(w4)*

((w3)*)(2)

(w3)

... - ((w2)*)(2)

...

(w2)*

-- (w*)(d) _-, ... -y (w*)(2) , w*

By a classical result of Parovicenko ([101]), the Continuum Hypothesis implies that all of these arrows are reversible; moreover for every fixed d E w spaces ((w°)*)(d) are homeomorphic for all countable a. We shall now use wEP to show that many of the above arrows are irreversible. Namely, we shall prove that for all d E w and all countable additively indecomposable a > w: (1) (w*)(1) 74, (w*)(d}1) (Theorem 4.6.1), (2) a* f" (w*)(d), (Lemma 4.6.2), and (3) (w*)(d) a* (Theorem 4.6.4). These results are proved using Ramsey's theorem and Corollary 4.5.3. In September 1999 the author has proved, using different methods, that all arrows in the above diagram are irreversible (see [41]). The following result was originally proved by Winfried Just ([61]; see also [41, Lemma 5.2] for a generalization).

4.6. SOME PAROVICENKO SPACES UNDER WEP

127

THEOREM 4.6.1. wEP(w, w) implies that the following are equivalent for all positive integers d and l: (1) (w*)(d) - (w*)(t), (2) d > 1.

PROOF. The direct implication is true because the projection

(1. 11 i... ,Ut,Ui}1....Ud) - (U1,... ,ui) is surjective. Note that by results of §4.4 variations of this map are essentially the only surjections. (2) implies (1). By Lemma 4.5.2, it suffices to prove that there is no R-measurable, (d, d + 1)-perfect surjection h: w(d) ---> w(d+l) To prove this in case d = 1,

note that if h: w -* w(2) is onto then the set h-1({l} x w) is infinite so h is not (1,2)-perfect. To prove that a continuous image of w* inside (w*)(d) is nowhere dense, note that every clopen subset of (w*)(d) is homeomorphic to (w*)(d), and therefore a subset of (w*)(d) with a nonempty interior which is a continuous image of w* gives rise to a surjection of w* to (w*)(d), implying d = 1. To prove Theorem for arbitrary d > 2, assume that mapping F: (w*)(d) --> (w*)(d+1) is continuous. By wEP we can assume that F: (Qw)(d) . ((3w)(d}1) and by Theorem 4.3.2 there is a partition (w*)(d) = U1 U U2 U . U Ut into finitely many clopen sets and

m: {1,...,d+1} x {1,...,1} --* {1,...,d} such that aj o F depends only on m(i, j)-th coordinate on Ui, for all i = 1, ... , l and j = 1, . . . , d + 1. Fix i < 1. By the pigeonhole principle there are j j' such that m(j, i) = m(j', i), and therefore the set F"Ui is nowhere dense. Hence the image of (w*)(d) under F is equal to the union of finitely many nowhere dense sets, and this finishes the proof. 13 LEMMA 4.6.2. wEP implies that (w*)(d) is not a continuous image of (w2)*.

PROOF. It is sufficient to prove the case d = 2. Let us assume the contrary. By Lemma 4.5.2, there is an R-measurable, (1, 2)-perfect surjection h: w2 --; w(2). Define a partition [w]3 = Ko U K1 by letting {i, j, k}< E Ko if and only if

h-'(j, Assume C is an infinite Ko-homogeneous subset of w. Then if i = min C and C' = C \ {i}, we have h-1(C' x C') C i x w, contradicting (1, 2)-perfectness of h. Therefore by Ramsey's theorem there is an infinite K1-homogeneous set C. Define a partition [C]2 = Lo U L1 by letting {i, j} < E Lo if and only if

h-1({1,...,i} x (C \ {1,...,i})) C w j. Since the preimages of 2-compact sets are compact, for every i there are only finitely

many j such that {i, j} 0 Lo, and therefore there is an infinite Lo-homogeneous C1 C C. This, together with the fact that C1 is K1-homogeneous, implies that if m1 < m2 < m3 < ... is an increasing enumeration of C1, then for every i there is an ni E [mi,mi+1) such that h-1(mi,k) fl ({nil x w) is nonempty for infinitely many k E C1. Now we can easily find D C C1 such that there are infinitely many n for which both sets (w

(n - 1), w n) fl h-1(D x D)

and

(w (n - 1), w n) \ h-1(D x D)

4 WEAK EXTENSION PRINCIPLE

128

are infinite. Therefore h-1(D x D) is not clopen, and this contradiction completes the proof.

As pointed out in the introduction, in case when d = 1 a result stronger than the following was previously proved by A. Dow and K.P. Hart. THEOREM 4.6.3. wEP implies that the following are equivalent for a countable ordinal a and a positive integer d: (1) a* is a continuous image Of (W.) (d) (2) a = ao + w for some ordinal ao. In particular, if a > w is an indecomposable ordinal then (w*)(d) 74* a*.

We shall prove a more precise version of this theorem, namely

THEOREM 4.6.4. wEP implies that if F: (w*)(d) _ a* is continuous, then the range of F is equal to the union of a set homeomorphic to w* and a nowhere dense set.

PROOF OF THEOREM 4.6.3. Assume Theorem 4.6.4. For the nontrivial direction note that a* is, by Lemma 4.5.2, homeomorphic to w* if and only if (2) applies.

PROOF OF THEOREM 4.6.4. Let us prove the following lemma first.

LEMMA 4.6.5. If h: w(2) -* a is an R-measurable, (2,1)-perfect map, then its range is equal to the union of a relatively compact set and a discrete set.

PROOF. Assume this fails and let h be a counterexample. We can assume that h is onto. This is because the range of h is homeomorphic to some ordinal ao < a and this ordinal is, by our assumptions, not of the form ao+w. We can also assume that ao = w2, possibly by composing h with a continuous perfect surjection of ao to w2. For = w m + n < w2 fix (xmn, y,nn) E h-1(e) so that the following condition is satisfied: min(xmn, ymn) = max{min(x, y) : (x, y) E h-1(l;)}. Note that this maximum exists because set h-1(1;) is 2-compact. Also note that the choice of (xm,n, y,,,,n) assures that (let k = max(xmn, ymn)):

(X) h-'(m w+n) C {1,...,k} xwUw x {1,...,k}. Define a partition [w]3 = Ko U K1 u K1 as follows:

{m, n, k}<E

1KO

if Xmn = Xmk,

K1

if Ym, = Ymk, otherwise.

1K2

CLAIM 1. There are no infinite K2-homogeneous sets.

PROOF. Assume otherwise, that A is an infinite K2-homogeneous set and let m = min(A). Then the set

m
4 6 SOME PAROVICENKO SPACES UNDER WEP

129

Let us now assume A is an infinite Ko-homogeneous set. Define a partition [A]4 = Lo U L1 U L2 by

{m,n,k,l}< E

if (xmn, ykt) E h-1(u) m + n),

Lo, L1, L2,

if (x,nn,yki) otherwise.

CLAIM 2. There are no infinite L2-homogeneous sets. PROOF. Assume the contrary, that B C A is such, let B = {mi } be its increas-

ing enumeration and let Si = w m2i + m2i+1, xi = xf:, Define a partition [w]3 = MO U M1 U M2 U M3 by

{i,j,k}< E

and yi =

M 2,

if (xi,yj) E h-1(Ck), if (x3,yk) E h-l(ei), if (xyI i,k E L-1 Kj),

M3,

otherwise.

M0, M1,

h_1(SCi)

are pairwise disjoint, there are no infinite M,-homogeneous Since the sets sets for j = 0, 1, 2, so there is an infinite M3-homogeneous set C. By the definition, all Si are isolated points in w2, and therefore by R-measurability h-1({l;i : i c C})

is equal to the union of finitely many rectangles, say Y x Zt (l = 1,. .. , k). Let

i < j and l < k be such that (xi, yi) E Y x Z1

and

(xj, yj) E Y x Z1.

This implies (xi, y.7) E Y1 x Z1 C h-'QC, : i E C}). But by the L2-homogeneity and

the M3-homogeneity (xi, yj) is not in h-1(ln) for any n E C-a contradiction. Therefore there is an infinite B C A which is either Lo- or L1-homogeneous. Let us first treat the case when B is Lo-homogeneous. CLAIM 3. The projection of h-1 ({w - m + n}) to y-axis is infinite whenever

m
PROOF. Since the set h-1({w k + I : n < k < 1 E B}) is not relatively 2-compact, (X) implies that set {ykl : n < k < I E B} is infinite. By the Lohomogeneity of B, this set is included in the projection of h-1({w m + n}) to El y-axis, and this completes the proof. Let m = min(B). Since h-1 ({w m+n : fn < n E B}) is relatively 2-compact, Claim 3 implies that we can find n1 < n2 in B such that x,,,,n, = x,n2 = x. Pick 1 > k > max(ni, n2) in B; then by the Ko-homogeneity we have h-1 (w fn + n1) fl h-1 (w m + n2) E) (x, ykl),

contradicting the fact that these two preimages are disjoint. Therefore B cannot be Lo-homogeneous. Now assume B is L1-homogeneous and let m < ft be the first

two elements of B. Then

n.
4. WEAK EXTENSION PRINCIPLE

130

To complete the proof of Lemma, it remains to prove there are no infinite K1-homogeneous sets. But this reduces to the case of Ko-homogeneous sets by considering the function h'(m, n) = h(n, m). Lemma clearly implies the case d = 2 of Theorem. The proof of general case uses the following variation of Lemma 4.6.5.

LEMMA 4.6.6. If h: W(d) -> w2 is a R-measurable, (d, 1)-perfect map, then its range is equal to the union of a relatively compact set and a discrete set.

PROOF. (Sketch) Assume h is a counterexample. We can assume that h is onto. For l = w m + n we fix (x,1nn, ... , xmn) E h-1(e) so that mm(x+nn) = max{min (x') i
:

(x1,

... , xd) E

Partitions K, L, M are defined analogously to the case d = 2. The only difference is that now we have more choices for the distinguished vertex of the hypercube with (xk...... xk1), but proofs are the same. diagonal vertices (x x,d,bn) and An application of Lemma 4.6.6 ends the proof.

4.7. Remainders of locally compact, countable spaces In this section we shall reformulate some of our results by using the following well-known fact.

PROPOSITION 4.7.1. If X is a countable, locally compact, non-compact space, then it is homeomorphic to some ordinal.

PROOF. Let X U {oo} be the one-point compactification of X; by a classical result of Mazurkiewicz and Sierpinski (see e.g., [102]), X U fool is homeomorphic to some ordinal, and therefore X itself is homeomorphic to an ordinal.

In Theorem 4.9.1 we will prove that OCA and MA imply wEP(a, ry) for all countable ordinals a and ry.

COROLLARY 4.7.2. OCA and MA imply wEP(X, Y) for all countable locally compact spaces X and Y. COROLLARY 4.7.3. wEP implies that the following are equivalent:

(1) X* -» Y* (2) there is a continuous perfect surjection h: X -+ Y. for all locally compact countable spaces X and Y. COROLLARY 4.7.4. wEP implies that the following are equivalent:

(1) X* u Y*

(2) there are compact K C X and L C Y such that X \ K and Y \ L are homeomorphic

for all locally compact countable spaces X and Y.

Let us now consider a problem of having a surjectively universal remainder of a countable locally compact space. COROLLARY 4.7.5. wEP implies that for every countable locally compact X there is a countable locally compact space Y such that Y* -» X* but X* f» Y*.

4.7. REMAINDERS OF LOCALLY COMPACT, COUNTABLE SPACES

131

Note that under CH, w* is a counterexample to the conclusion of Corollary 4.7.5. Moreover, under CH all spaces of the form X* for X a countable, locally compact, non-compact space are homeomorphic ([101]). Corollary 4.7.5 says that under wEP there is no surjectively universal object in the class of remainders of locally compact countable spaces. However, one does not have to go very far to find a space which maps onto all these remainders. Let G be a closed G6 subset of w* which is not open. By the compactness and zero-dimensionality of w* we can write G as an intersection of some strictly decreasing sequence of clopen sets, and therefore G is homeomorphic to the Stone space of the algebra P(w (2)) / Fin x 0.

(a) The family of continuous images of G includes all Cech-Stone remainders of countable locally compact spaces. (b) This family is closed under taking finite products of its elements. PROPOSITION 4.7.6.

PROOF. (a) For an ordinal a let cl(a) denote the algebra of clopen subsets of a. Note that a* is the Stone space of cl(a)/(a), where (a) stands for the ideal of bounded subsets of a, i.e., the ideal generated by a = {5 5 < a}. Assume a > w is a countable ordinal and let {an} be an increasing sequence of ordinals converging to a so that every interval [an, an+1) is infinite, and let h,,: w(2) -+ a be a bijection which sends {n} x w into [an, an}1). Let Ira: G -* a* be defined by :

7ra(U)={h"U : UEU}ncl(a). Then Ira is a continuous surjection. (b) It will suffice to prove that G -» G(2). Let h: w(2) -+ (w(2))(2) be a bijection

such that A E Fin xO if and only if we can write h-1(A) = Al U A2 so that the first projection of Al and the second projection of A2 are both in Fin x0. Define a mapping F: (w(4))* -' (w(2))* by F(U) = {h"U : U E U} n R2(2). Then F maps G onto G(2). COROLLARY 4.7.7. If X is a countable, locally compact, non-compact space, then all finite powers of X * are continuous images of G. An another way to look at consequences of wEP is the following. PROPOSITION 4.7.8. If F: a* -+ y* extends to F1: via --* Qry so that F1"a C -y,

then there is a continuous F.: G -* G such that the diagram GF.G

commutes.

PROOF. Let h = F r a, and let hG: w(2) -* a be such that the diagram w(2)

h.

(2)

h. a

h., h

7

4. WEAK EXTENSION PRINCIPLE

132

commutes, namely let h. = h7 I oh o ha. Then h* sends the ideal Fin x O into itself, and therefore mapping F*: (w(2))* , (W(2))* defined by

F*(U)_{h''U: UEU} sends G into G. Since the above diagram commutes, F* is as required.

4.8. Almost liftings and duality This section contains some prerequisites for the proof of wEP from OCA and MA, given in the next section. Let a be a countable indecomposable ordinal, and let (a)(d) denote the ideal of d-bounded subsets of a(d), namely those A C a(d) U Ad so that the i-th projection which can be partitioned as A = Al U A2 U of Ai is a bounded subset of a for all i < d. For a topological space X let cl(X) denote the algebra of clopen subsets of X, and let 1C(X) denote its ideal generated by compact open sets. One can also define the rectangle algebra of cl(X)(d) and its ideal of d-compact sets, but we shall not need these objects. Let us recall a basic fact about the topological duality. Let Stone(B) denote the Stone space of a Boolean algebra B.

(1) (a*)(d) is homeomorphic to Stone(Rd/(a)(d)).

PROPOSITION 4.8.1.

(2) If X is a zero-dimensional, locally compact space, then X* is homeomorphic to Stone (c]. (X) / IC (X)).

Assume that X = Stone(BX) and Y = Stone(By). Y uniquely determines a homomorphism (3) Every continuous map f : X

4?f:By-*BX by 4?f(b) = f-I(b).

Moreover, f is a surjection if and only if 4? f is a monomorphism; f is an injection if and only if 4? f is an epimorphism. (4) Every homomorphism D: BX -+ By defines a continuous map fI, : Y -* X by

f(y)=

n

1(a).

aEBy,yEa

Moreover, f is a surjection if and only if 4' f is a monomorphism; f is an injection if and only if P f is an epimorphism. PROOF. See [14).

If 1_7 are ideals of a Boolean algebra B, then we say that I is ccc over ,7 if there is no uncountable family {At} of I-positive sets in B such that At fl An is in 9 for all t; # i. If 4?: cl(X)//C(X) -+ cl(Y)/1C(Y) is a homomorphism then a mapping e: cl(X) -* cl(Y) is an almost lifting of 4? if the family {A E cl(X) : 1% (A) ='c (Y) O(A)} includes an ideal which is ccc over )C(X).

LEMMA 4.8.2. If (b: P(w)/Fin --' Rd/(a)ldl has a completely additive lifting 4'h, then h-'(A) is equal to a set in Rdd for every A C w. In particular, dom(h) is equal to a finite union of rectangles in a(d).

4.8. ALMOST LIFTINGS AND DUALITY

133

PROOF. We shall prove that h-1(w) is equal to a set in Rd; the proof of the general case is identical.

Let us first prove the case when d = 2. If E C a(t) and ho: E - w is a map such that Oho is a completely additive lifting of 4', define g: (2) -+ {0, 1} by 77) =

f1,

if

77) E E, if (l , rl) E.

0,

Apply Theorem 4.2.2 tog and Z = (a). If w e can split af2> into finitely many rectangles RI, ... , Rk, Rk+1, ... , R, so that Ri E (a)(2) for i = 1, ... , k and g depends on Rj on at most one coordinate f o r j = k + 1, ... , 1, then the restriction of h to Rk+1 U . . . U Rt is as required. Otherwise, there are sequences X = and Y = {77i} of ordinals converging

to a and such that either 7)i) E E for all i but (1 i, m) 0 E for all i < j, or (a) (b) (6i, 77i) 0 E for all i and (L;i,r7j) E E for all i < j.

If (a) applies, then A = {h((ei,rli))

i E w} has the property that h-1(B) is not equal to a set in R modulo a set in (a)(2), for every infinite B C A. But since :

4ih0 is an almost lifting of 4), for one of these B we have 4?.(B)0$ho(B) E (a) (d), a contradiction. If (b) applies, then we find finite subsets si, ti of w satisfying the following conditions: (1) Is,, = i, (2) ui = {i;j : j E si}, vi = {7)j : j E ti}, (3) the sets Yi = h"(ui x vi) (i E w) are pairwise disjoint. This can be done by a straightforward recursive construction, using the fact that h-1({ n}) is in (a)(') for every n and that bi - a and vii -* a as i -* w. Since 4?ho is an almost lifting of 4?, by Theorem 3.10.1 there is a set B C w which is equal to an infinite union of ui and such that 4?. (B)04?ho (B) E (a)(d) . Let X C a(d) be a set in Rte. Then X is equal to a finite union of rectangles, say X = `i=1 Ri. It will suffice to prove that XA4?h,, (B) is not in (a)(d). But this follows from (b), the choice of ui and vi, and the following lemma.

LEMMA 4.8.3. For every in there is an m' = f (m) such that the set D,,,- _ {(i, j)

:

1 < i < j < m'} cannot be covered by m many rectangles disjoint from the

diagonal.

PROOF. Let us first prove an infinitary version:

If the set D. = {(i, j) E w2

:

1 < i < j} is covered by finitely many

rectangles, then at least one of these rectangles intersects the diagonal. Suppose the contrary, and let k be the minimal such that D. can be covered by k many rectangles Ai x Bi (i < k) each of which is disjoint from the diagonal. Since w = Uk 1 Ai, at least one Ai has to be infinite. But this implies that Dm fl (Ai x Ai) is covered by k - 1 rectangles each of which is disjoint from the diagonal, contradicting the minimallity of k. Lemma 4.8.3 now follows by a standard compactness argument.

This concludes the proof in the case when d = 2. The general case is proved by a simple induction.

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LEMMA 4.8.4. If a homomorphism D: P(w)/Fin + has a completely additive almost lifting defined by h: a(d) - w U {0}, then the set

D= {s E a (d)

h(s) # 0} is equal to a set in Rd, modulo a set in (a)(d). Also, h-1(A) is equal to a set in Rd (modulo (a) (d)) for every A C W. :

PROOF. By applying Lemma 4.8.2, we can assume that h depends on only one coordinate, namely it will suffice to treat the case when d = 1. We need to prove that D is equal to a clopen subset, modulo a bounded subset of a. Assume that D is not equal to an open set modulo (a)(d). Then we can find an increasing sequence of limit ordinals an converging to a such that an E D but sup(an \ V) = an for all n. Consider the set {A C w : h(an) E A for at most finitely many n}. This is a Borel ideal, and since {A C w : t*(A) =C 4)h(A)}

includes an ideal which is ccc over Fin, we can find A C w such that the set 4)*(A)O(Ph(A) is in (a), yet h(an) E A for infinitely many n. But this implies that 4). (A) is not equal to a clopen subset of a modulo (a), a contradiction. Therefore it remains to prove that D is equal to a closed set modulo (a); assume it is not. Then there is an increasing sequence of limit ordinals an such that an D yet sup(D n an) = an for all n. Then {A C w : sup(h-1 (A) fl an) = an for at most finitely many n} is a Borel ideal. Like before, we can find A C w such that *(A) =(«) 4)h(A) but 4h(A) is not equal to a closed subset of a modulo (a); a contradiction.

The moreover part follows from the already proved part of the lemma, by applying it to the restriction of ' to P(A)/Fin. LEMMA 4.8.5. Assume that X is a locally compact zero-dimensional space such

that every homomorphism 4?: cl(X)/K(X) - P(w)/I has a completely additive almost lifting whenever T is a countably generated ideal on w. Then wEP(X, a) is true for all countable ordinals a. (X*)(d) is PROOF. Let us first treat the case l = 1 of wEP. Assume F: a* continuous and let 4?: CI(X)/K(X) Rdpa)(d) be the dual homomorphism. Also, the ideal (a)(d) is countably generated (by the sets {l; l; < rl}, for rl < a), and therefore the algebra cld(a)/(a)(d) is isomorphic to a subalgebra of P(w)/Z, for a countably generated ideal I. Let h: a(d) -* X be such that 4h is a completely additive almost lifting of D, and let A = h-1(X) g a(d). By Lemma 4.8.4 we can assume that A is in Rte. Then U1 = A* and Uo = (a*)fd) \ A* is the desired clopen decomposition. Now assume l > 1 and F: (X*)(d) (ry*)(1) and consider maps Fi = iri o F, where 1ri: (ry*)(1) -* y* is the i-th projection (i = 1,... ,1). For each i we can find a partition (X*)() = Uo U Ul into clopen sets such that Fi [ Uo continuously extends to (/iX)(d) and Fi"UU is a nowhere dense subset of ry*. This implies that :

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135

for Uo = fit), U and U1 = Uy`)1 U, the image F"U1 is a nowhere dense subset of (y.)(l) and function F [ U0 continuously extends to ()3X) (d). El

4.9. OCA and MA imply wEP In this section we shall deduce wEP from OCA and MA, using results of previous chapters. Recall that a topological space is Polish if it is separable and completely metrizable, and note that a space is Polish, locally compact and zerodimensional if and only if it is second countable, zero-dimensional and locally compact ([27]). Such spaces include all countable ordinals, the space w x 2", and countable topological sums of such spaces. THEOREM 4.9.1. OCA and MA imply wEP(a,X) for every zero-dimensional locally compact Polish space X and every countable ordinal a. PROOF. By Lemma 4.8.5, it will suffice to prove Theorem 4.9.2 below.

THEOREM 4.9.2. Assume OCA and MA, and let X, d and a be as in Theorem 4.9.1. (1) If .T is a countably generated ideal on w, then every homomorphism

4): cl(X)/)C(X) -* P(w)/I has a completely additive almost lifting. (2) Every homomorphism 1P: cl(X)/)C(X) --* R'/(a)(d) has a completely ad-

ditive almost lifting 'h for a function h whose domain is moreover an element of Rte.

PROOF. We shall prove (1) and (2) simultaneously. Since Rd is a subalgebra of p(a(d)) and and the ideal (a) (d) is countably generated, we can fix a natural injection

W: Rd/(a)id) , P(w(2))/Fin x0 which has a completely additive lifting, induced by a bijection h: w(2) -> a(d). This simple observation shows that (2) is a refinement of (1). Fix a countable basis B of X which consists of compact sets and moreover has the following property:

(131) For every set U E B the set of all V E B such that V D U is finite and linearly ordered by the inclusion. We also fix a partition 0C

X=UWi i=Q

of X into pairwise disjoint compact open sets from B. We may assume that B has the following additional property: (1B2) For every set U E B there is (the unique) n such that U C Wn. Let A denote the family of all partitions of X into pairwise disjoint compact open

sets from B. Let A, B be partitions in A. We say that A refines B, in symbols A -< B, if for every U E A there is (the unique) V E B such that U C V. A partition A almost refines B, in symbols A <* B, if for all but finitely many U E A there is (the unique) V E B such that U C V. For every A E A consider the mapping gA : X -+ A uniquely defined by for every x E X. x E 9A(X)

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This mapping defines a homomorphism 'PA: P(A)/ Fin -* cl(X)/)C(X) by its completely additive lifting, 4'yp. (Note that Fin here stands for an ideal of finite subsets of A.) For A -< B consider a mapping gAB : A

UC9AB(U),

B uniquely defined by

forUEA.

This mapping defines a homomorphism TAB : P(B)/ Fin -+ P(A)/ Fin by its completely additive lifting, oDeAB These definitions make the following diagram commute (for all A -< B in A):

X

Note that 'DA = 4) o'PA is a homomorphism of P(A)/Fin into P(w(2))/Fin A For A E 0 x Fin let (PA : P(A)/ Fin - P(A)/ Fin be defined as usual, by its lifting f1 A,

and let idA: P(w(2))/Fin --+ P(A)/Fin be a homomorphism determined by its lifting, C H C n A. These definitions make the following diagram commute (for A-< B in A and A D B in 0 x Fin):

\

P(B)/FinABP(A)/Fin

B cl(X)/)C(X) I cp

P(w(2))/Fin xO

P(B)/Fin A-P(A)/Fin By Theorem 3.8.1, for all A E A and A E 0 x Fin, the homomorphism 4) A has a completely additive almost lifting, given by a finite-to-one mapping hA : EA -+ A. We shall apply the OCA uniformization theorem (more precisely, its Corollary 2.3.2) to {hA} (or more precisely, to a coherent family { fA} defined below) to obtain the desired completely additive lifting. For U E B let CU be its upwards cone,

Cu={VEB:V_DU}. Note that every CU is finite, by (131). Consider the ideal I on B generated by the sets

A=UCU UEA

forAEA. CLAIM 1. The ideal I is isomorphic to 0 x Fin.

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137

PROOF. Since 0 x Fin is equal to the orthogonal of a countably generated ideal

Fin x0, it will suffice to show that I = ,71, if 3 is the ideal generated by the sets k

Zk= UEB:UcUW, . i=1

We first show that I C J. Fix A E A. Then for every n by compactness there are at most finitely many U E A such that U C Un 1 W. Since each CU is finite, A is in Now we show that j1 c Z. Fix a set C in ,71, and fix n E w. Let Ui , ... , Uk(n)

enumerate all sets in C included in Wn (recall that by (82) every set in C is equal to some U). We can find a natural number 1(n) and a disjoint partition Wn = Ui=1 Vjn such that for every Ui there is (the unique) Vj c Uti. Then A= is an element of A such that k D C, and this completes the proof. For A E A and A E 0 x Fin define a mapping fA: A X A {0,1} (we shall think of the ideal 0 x Fin as an ideal on w instead of w2, to simplify the notation) fA(U, i) _

(1, 5

0,

if hA(i) c U, if

U.

We claim that the functions fA (A E A, A E 0 x Fin) form a coherent family. By Lemma 2.3.3, it will suffice to check the coherence condition for A -< B (note that this is equivalent to A : B) and A D B. Note that

(9AB ° hA) B =' hB, since (P(9ABOhAA)IB and (DhB are completely additive almost liftings of the same homomorphism, 4)B. For all i such that gAB(hAA,(i)) = hAA,(i) and all U E B we have

fA(U,i) = fB(U,i) On the other hand, note that for every j E B there are at most finitely many U E B such that fB (U, j) 0 0 (these are the elements of the finite set ChB (j))). Similarly, for every j and all but finitely many U E B we have fA(U, j) = 0. But we can have fB(U, j) # fA(U, j) only when one of them is equal to 0 and j < n, and we have just proved that there are at most finitely many such pairs (U, j). Therefore {f} is a coherent family of functions indexed by (0 x Fin) x (0 x Fin) and by Corollary 2.3.2 there is f : (w(2) x B) -+ {0,1} such that for all A, A, for all but finitely many i E A and U E A we have fA(U, i) = f (U, i). Consider a partition [13 x w]<w = Ko U K1 defined by letting a nonempty s E K0 if and only if (K1) n{U : (U, i) E s for some i} is nonempty, and (K2) {i : (U, i) E s for some U} has exactly one element. Then (fA)-1({i}) is either empty or a maximal Ko-homogeneous subset of dom(fA)

for every i, and therefore by Lemma 2.4.1 we can assume that the trivializing function f is such that f -1({ j)) is a maximal Ko-homogeneous subset of dom(f) _ (B \ Zk) x (w(2) \ [k, oo)) for j = 0,1.

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138

For every i consider the set

S,={UE13: f(U,i)=1}. By the above, each nonempty Si is a maximal centered subset of B \ Zk for all i, and by the compactness, the set n Sz is a singleton for all such i. Let us define a partial map h from a subset of w into X by letting

h(i) En Sz whenever Si nonempty; note that in this case n Sz is a singleton. CLAIM 2. The mapping 'Dh: cl(X) --* P(w) is an almost lifting of 1D.

PROOF. Let us first note that for all A, A and all C C A, we have (*)

F(gA(C)) =* 4h; (C)-

Assume the assertion of the claim is false, and let u£ (1; < w1) be an uncountable family in cl(X) such that UUnU E (X)

but F(ug)04P*(UC) 0 Fin x0

for all C # r). For every l pick Ag E A such that U£ is equal to the union of some subfamily of Ae. Since by Claim 1 the ideal I generated by A is isomorphic to 0 x Fin, there is A E A such that A -< AC for uncountably many (since OCA implies b > ;11, by Theorem 2.2.6). For every one of these l; find A E 0 x Fin such that the set (F(u£)A,%(ue)) n AC is infinite. Again, we can find a single A which includes uncountably many of these AC. Since 'DM is an almost lifting of 4iA, for one of these l; we have

4'h; A) _*

((DA(uE)

But this contradicts (*) and therefore concludes the proof. This concludes the proof of (1). To prove (2) we need to show that the set {s E a(d) : h(s) # 0} is in 1Zd. In a special case when X is an infinite discrete space, this follows from Lemma 4.8.4.

To treat the general case, pick an arbitrary A E A. Since {hA : A E 0 x Fin} is a coherent family of functions indexed by 0 x Fin, by Theorem 2.2.7 we can find a partial function hA from a subset of w into A which uniformizes this family. (Remember that we are thinking of 0 x Fin as an ideal supported by w to simplify the notation; the supporting set is actually equal to a(d).) Then 4ihA is a completely additive almost lifting of 4PA (by a proof identical to that of Claim 2). Therefore by Lemma 4.8.4 we can assume that the domain of hA is in 7Zd.. But since 19Ahiil and '(DhA(i) are almost liftings of the same homomorphism,'A, we can assume that the domains of gAh(i) and hA are equal. Since gA is a total function, we can assume

that the domain of h(i) is equal to the domain of hA, and the latter set is in fl. This completes the proof.

.

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139

4.10. Versions of wEP There is no a priori reason why wEP should be restricted only to a class of countable locally compact spaces. In this section we show that at least a little bit more can be said at this point. Let us first note the following generalization of Theorem 4.9.1.

THEOREM 4.10.1. OCA and MA imply wEP(X,Y) for all zero-dimensional locally compact Polish spaces X and Y. The proof of this result will appear elsewhere. Let us only note that it requires analyzing the liftings of homomorphisms of P(w)/ Fin into cl(X )l IC(X) for a locally compact Polish space X, and this is done by mimicking proofs of Chapter 3. We do not know for which spaces X, Y the principle wEP(X,Y) follows from

OCA and MA, but we note that the assumption of local compactness is not necessary. Consider S(w), a sequential fan with countably many spines, defined as follows. The underlying set of S(w) is w(2) U {oo}, all points in w(2) are isolated and the neighborhood basis of oo is the family of all sets whose complements are in 0 x Fin (see [27]). PROPOSITION 4.10.2. OCA and MA imply wEP(w, S(w)).

PROOF. (Sketch) This proof goes along the same lines as the proof of Theorem 4.9.1, and we shall therefore only give a sketch. First note that the ideal of compact subsets of S(w) coincides with the ideal of finite subsets of w(2) which we shall denote by Fin (note that {oo} V Fin). We have to prove that every homomorphism 4i: cl(S(w))/Fin --+ P(w)/Fin has a completely additive almost lifting. This is proved by considering cl(S(w)) as a limit of algebras isomorphic to P(w)/Fin, just like in proof of Theorem 4.9.1. For A E 0 x Fin let SBA: P(A U loo})/ Fin P(w)/ Fin be given by its lifting,

q1_03) =

fB U (w(2) \ A) U {oo},

if oo E B,

1B,

ifoo¢B

Then 'D O TA is a homomorphism of P(A U {oo})/ Fin into P(w)/ Fin, therefore it has an almost lifting 'l)h for some hA: w -+ A U {oo}. Since OCA implies that the ideal 0 x Fin is l 1-directed under C*, we can uniformize hA (A E 0 x Fin) to get h: w -+ (2) U {oo} such that'Ph is an almost lifting of (D.

It should be clear that the above proof works when 0 x Fin is replaced with any analytic P-ideal, or any ideal which is 111-directed under the inclusion modulo finite.

Following the terminology of §4.4, we say that a function F : a*

ry* is induced

by some continuous h: a -+ ry if F = F1 [ a*, where F1: ,3a -+ Qry is the unique extension of h to la. PROPOSITION 4.10.3. wEP implies that if a is a countable ordinal then every autohomeomorphism of a* is induced by a perfect continuous map h: a -+ a.

Therefore our results reduce the question of describing autohomeomorphism group of a* (and possibly its finite powers) to a question about functions on the integers.

We can consider a version of weak Extension Principle for closed subspaces X of w* and other Cech-Stone remainders. Let weak Extension Principle for X be

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140

wEP(X) If F: X -+ w* is a continuous map, then there is a partition X = Uo U U1 into clopen sets such that F"Uo is nowhere dense and F [ U1 can be continuously extended to F': ,Qw Ow. A similarly-looking extension principle was proved by van Douwen and van Mill ([19]), who extended a classical result of W. Rudin ([104]). They proved that CH implies that every homeomorphism between nowhere dense P-subsets of w* can be extended to an autohomeomorphism of w*. Rudin has proved this in the case when these two sets are singletons. This is incompatible with OCA and MA, because MA implies that there are 2` many P-points in w*, while there are only continuum many autohomeomorphisms of w*. But wEP (and therefore OCA and MA) does imply this principle for many "simply definable" closed nowhere dense P-subsets of w*, i.e., those X C w* such that (by q we denote the closure of the set A)

{ACw : AnX=0} is a non-pathological analytic P-ideal. This follows from and Theorem 3.8.1. Similarly, by Theorem 3.3.6, MA and OCA imply wEP(G). However, the existence of a Baire homomorphism with no completely additive lifting (Theorem 1.9.5) implies the following.

PROPOSITION 4.10.4. There is a continuous surjection F of a closed P-subset of w* onto w* which cannot be continuously extended to w*.

Let us remark that our OCA lifting results of Chapter 3 have been recently extended by A. Dow and K.P. Hart ([24]), who used them to prove that under OCA the Lebesgue measure algebra does not embed into P(w)/ Fin, answering a question of Fremlin ([49]). They have observed the following useful extension of Theorem 3.8.1.

THEOREM 4.10.5. Assume OCA and MA. If qP: P(w) --+ P(w)/Fin is a homomorphism, then 4D has a continuous almost lifting.

It is worth remarking that the Lebesgue measure algebra (moreover any measure algebra of size at most continuum) embeds into P(w)/So, by a result of Fremlin (see [48]). Since such an embedding cannot have a completely additive lifting and Zo is a non-pathological ideal, this result imposes some limitations to possible extensions to our results about liftings.

4.11. Remarks and questions The following theorem is a natural generalization of Corollary 4.5.3; its proof can be found in [41]. THEOREM 4.11.1. wEP implies that the following are equivalent (1) (a*)(d) ^ (-Y*)(1),

(2) a=yandd=l,

for all countable indecomposable ordinals a and -y and all positive integers d, 1.

By a result of van Douwen ([17, Theorem 7.2]) (a*)() .:= (.y*)(I) implies d = 1. By Parovicenko's result, this is as much as can be said without using wEP or some other additional axiom. The following consequence of Theorem 4.11.1 also appears in [41].

THEOREM 4.11.2. wEP implies that the following are equivalent

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141

(1) (a*)(d) _ ('Y*)(1), (2) a > ry and d > 1, for all countable indecomposable ordinals a and -Y and all positive integers d, 1.

Note that wEP is indeed an optimal assumption in this context. By Corollary 4.5.3, wEP implies that (1) is equivalent to its consequence saying that there exists a mapping h: w -* w with certain first-order properties, and the truth of such a statement does not depend on which Set-theoretic axioms we are assuming by Shoenfield's Absoluteness Theorem ([110]). Therefore if it is consistent that (1) and (2) of Theorem 4.11.2 are equivalent, then they are already equivalent under wEP.

Just's original proof that w* does not map onto (w*)(2) ([61]) did not rely on an analysis of liftings of homomorphisms between analytic quotients like our proof from wEP. Instead, it used an axiom (introduced in [63] under then name AT) which states that every map between analytic quotients which merely preserves intersections is "trivial" on a large set. There is still an interest in analyzing liftings of such maps. By employing an idea due to M. Bell ([6]), we have used a version of this axiom which also follows from OCA to prove that the exponential space Exp(w*) is not a continuous image of w* (see [38, §8]). In §4 of [142], Velickovic proved that the Proper Forcing Axiom implies all automorphisms of P(n) /Fin are trivial. His methods suggest that the following (as well as its version for finite powers) should follow from Corollary 4.7.2. CONJECTURE 4.11.3. PFA implies that the following are equivalent

(1) X* -» Y*, (2) there is a perfect h: X -+ Y such that the closure of Y \ h"X is compact, for all locally compact, locally countable spaces X and Y. It would be of an interest to find as large as possible class of spaces for which we

can expect (1) and (2) to be equivalent. By Theorem 4.10.1 all zero-dimensional, locally compact, complete, separable metric spaces belong to this class. The case d = I = 1 of the following question (or rather, its dual version) was already asked in Question 3.14.2. By the main result of [41], one of these questions has a positive answer if and only if the other one does. QUESTION 4.11.4. Do OCA and MA imply the following strong Extension Principle for finite powers of w*:

(sEP) If F: (w*)(d) --> (we)(1) is a continuous map then F can be continuously extended to F': ()3w) (d)

An (expected) positive answer to this question would greatly improve our understanding of the continuous images of w*. In particular, it may be related to the following question asked by M. Bell which is still open:

QUESTION 4.11.5 (Bell). Is it consistent that for every Boolean algebra B, if P(w) embeds into 13/I for some countably generated ideal I, then P(w) embeds into B itself? For more information and some partial results regarding this question (in particular, a negative CH-result) the reader can consult [22]. A related question was asked by A. Dow ([24j):

142

4. WEAK EXTENSION PRINCIPLE

QUESTION 4.11.6 (Dow). Is it consistent that a complete Boolean algebra B embeds into P(w) /Fin if and only if it is or-centered? Equivalently, is it consistent that every extremally disconnected continuous image of w* is separable?

Needless to say, under CH the answer is negative. It is clear that a version of strong extension principle (i.e., a strong lifting theorem) would give a positive answer. The following observation is due to K.P. Hart and the proof is included with his kind permission. PROPOSITION 4.11.7. If there is a nonseparable extremally disconnected continuous image X of w*, then Strong Extension Principle fails.

PROOF. Let f : w* . * X be a continuous surjection. By a result of Simon ([111]), every compact extremally disconnected space can be embedded into w* as

a weak P-set. (Recall that P C w* is a weak P-set if the closure of D is disjoint from P whenever D C w* \ P is countable.) We can assume that X itself is a weak P-subset of w*. We claim that f cannot be continuously extended to /3w. Assume the contrary, that F : ,Ow --* Ow is such an extension, and let D = F"w. Then the closure D of

D includes X. But since X is nonseparable, we have D -r-IX - X. Also, D \ X is disjoint from X, and therefore X \ D is nonempty; a contradiction.

It is believed that PFA implies the positive answer to Question 4.11.6. It should be noted that under PFA every complete Boolean algebra which embeds into P(w)/ Fin has to have ccc (by [125, Theorem 8.8], which is a PFA-reformulation of a result of [81]). We should also note that the assumption that 13 is complete is necessary, since Bell ([7]) has shown that P(w)/Fin has a rich supply of ccc subalgebras with versatile cellularity properties. Results in this Chapter are formulated in terms of the existence of completely additive almost liftings for arbitrary homomorphisms assuming OCA and MA. However, our proofs give analogous results about the existence of completely additive liftings for homomorphisms with Baire-measurable liftings without any set-theoretic assumptions. Therefore we have the following picture for the Baire-embeddability ordering:

P(w('))/Fin xm P(w)/Fin - cl(w2)/(w2) where all the arrows are irreversible. This can be contrasted to a dichotomy result of Kechris and Louveau ([76]) which in particular implies that every quotient of a Borel subalgebra of P(w) over a Borel ideal which is of Borel-cardinality strictly less than P(w('))/Fin xO has to be of Borel-cardinality at most P(w)/Fin.

CHAPTER 5

Gaps and limits in analytic quotients 5.1. Introduction One of the beautiful early discoveries in the study of ?(N)/ Fin was Hausdorff's construction of a gap inside this quotient ([54], [53], see Theorem 5.2.1; all definitions are given at the end of this section). This theorem shows that the degree of saturatedness of P(N)/ Fin is not so large and that this algebra is a saturated structure only in the trivial case when the Continuum Hypothesis holds. A remarkable feature of Hausdorff's original gap is the fact that its construction did not require any additional set-theoretic axioms, a phenomenon which is rarely encountered at this level of complexity in investigating P(N)/ Fin. After the invention of Forcing, Kunen ([81]) proved a rather surprising fact about gaps in P(N)/ Fin: The ordinary axioms of Set Theory are not sufficient to give us any types of gaps other than those discovered by Hausdorff and Rothberger ([54], [53], [103]). Later, Todorcevic (see [125, Theorem 8.6]) deduced the same conclusion from OCA. The question of the existence of gaps in arbitrary analytic quotients was addressed only relatively recently by Just ([105]). Mazur ([95]) constructed (w1,wl)gaps in every quotient over an FF ideal, and, using a weak form of CH, in every quotient over a meager (in particular, analytic) ideal. Then Todorcevic ([130], [128]) proved the following result that gave a new outlook to this kind of problems:

THEOREM 5.1.1. Every monomorphism of ?(N)/Fin into a quotient P(N)/Z over an analytic ideal with a Baire-measurable lifting preserves all Hausdorff gaps. If.T is moreover a P-ideal, then ' preserves all gaps. Since by a theorem of Mathias P(N)/ Fin is Baire-embeddable into every analytic quotient (see §1.2), this implies the existence of Hausdorff gaps in every analytic quotient. Todorcevic ([130]) showed even more, that embeddings with completely additive liftings between analytic quotients (and Mathias' theorem does give an embedding of this form-see §1.2) preserve all gaps, and therefore the gapspectrum of every analytic quotient always includes the gap-spectrum of P(er)/ Fin. It is natural to ask whether analogues of Kunen's or Todorcevic's theorem are true for other analytic quotients (see [130, Problem 2], [128, Problems 3 and 4]). We shall see later (§5.7, §5.8, §5.10) that they both fail in certain analytic quotients. Many authors consider only linear gaps, i.e., those gaps whose both sides are linearly ordered by the inclusion modulo the ideal. But some important gap phenomena that cannot be detected by linear gaps occur in the realm of nonlinear gaps (compare Proposition 5.3.2 to Lemmas 5.8.6 and 5.8.7, also see §5.7, §5.10 below). Any attempt on analysis and classification of gaps in analytic quotients requires a

reasonable notion of a type of a gap. In the case when both sides A, B of a gap are linear, the type of this gap is the pair of regular cardinals (see the end of this section). In an attempt to understand nonlinear gaps, in §5.6 we will suggest a 143

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definition of type of such a gap. This notion can be formulated using the language of Tukey reductions between partially ordered sets (see [138], [124], [46]), which we also use to describe the gap-preservation phenomena.

The organization of this chapter. In §5.2 we shall review some facts about gaps in P(N)/Fin relevant to our results in later sections. In §5.3 we prove that the (linear) gaps spectra of Fin and 0 x Fin coincide and that gaps are often preserved by Baire embeddings of P(IN)/0 x Fin, answering [128, Problem 4]. In §5.4 we prove analogous results for Fin x0, now using a consequence of OCA; this partially answers [128, Problem 3]. In §5.5 we prove that Hurewicz-like separation phenomenon occurring in P(N)/Fin discovered by Todorcevic in [126] also occurs in P(IN)/ Fin x0, but it fails in a quotient over any analytic P-ideal other than Fin (§5.7). (This is where the nonlinear gaps enter the scene; their use is necessary by results of §5.3.) In §5.6 we introduce the notions of Tukey reduction and cofinal similarity and suggest how to define the type of a nonlinear gap. In §5.8 we consider the quotients over the ideals Fin x Fin and 0 x Fin as countable (reduced) powers of P(N)/ Fin to obtain information about their gap spectra. The connection between

the preservation of gaps and the (non)existence of analytic gaps, as well as the preservation of gaps under arbitrary embeddings are discussed in §5.9. In §5.10 we define an analytic Hausdorf gap in a quotient over an analytic P-ideal (this was

rather surprising-cf. [130, page 96]). In §5.11 some easy facts about limits are mentioned, and in §5.12 a "multigap" from [29] is embedded into every analytic quotient. This Chapter closes with some remarks and questions given in §5.13.

Terminology and notation. Let us now define the basic objects of study in this Chapter. Families A, B of sets of integers are I-orthogonal (A 11 B) if

Af1BEIfor all AEAandBEB. (IfI=Fin,thenwesaythatAandBare

orthogonal and write A 113.) A pair of families A, B which are orthogonal modulo

some I is also called a pregap in P(N)/Z. We say that A and B are separated if

thereisCCNsuchthat ACZCandCr1BEIforallAEAandBEB. Sucha set C separates A from B. If a pair of I-orthogonal families A, B is not separated,

then we say that A, B is a gap in P(N)/Z. If there is a sequence C. (n E N) of subsets on N such that for every pair A E A, B E B there is n such that A C' C" and C. n B E 1, then we say that the pregap A, B is countably separated. In order to define the type of a gap, we need some auxiliary definitions about the linearly ordered sets. Recall that cf(P) denotes the cofinality of a partially ordered set P, namely the smallest size of a cofinal subset X of P, i.e., such that every element of P is below some element of X. For A C P(N) by All we denote the poset (A/Z, CZ) (where CT stands for the inclusion modulo 1). If A, B is a gap and both All and B/Z are linear orderings of cofinality a and A, respectively, then we say that a gap (A, B) is linear and that its type is the pair of cardinals (r., A), or that it is a (r,, A)-gap. A linear gap spectrum of a quotient P(N)/Z is the family of all types (n., A) of its linear gaps. We analogously define a Hausdorff gap spectrum and a gap spectrum of a quotient.

5.2. Gaps in the quotient over Fin The study of saturatedness properties of P(N)/Fin with the ordering C* of inclusion modulo finite started as early as in the second half of the last century, when du Bois-Reymond proved there are no (w,1)-limits in P(N)/Fin, namely

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145

whenever A. is a strictly increasing sequence in P(N)/Fin and B D* An or all n, then there is C C* B such that B\C is infinite and it almost includes all An. Later Hadamard ([51]) proved there are no (w,w)-gaps in P(N)/Fin. The next step was made by Hausdorff.

THEOREM 5.2.1 (Hausdorff, [53]). There is an (wl,wl)-gap in P(INY)/Fin. We shall call any pair of orthogonal, a-directed families in some quotient which cannot be separated a Hausdorff gap. Compare Theorem 5.2.1 with another result of Hausdorff ([54]), rediscovered by Rothberger ([103]) (recall that b is the minimal length of an unbounded sequence in (N, <*)). THEOREM 5.2.2 (Hausdorff, Rothberger). There is an unbounded rc-chain of in

(NN, <*) if and only if there is an (w, ,c)-gap in P(N)/Fin. In particular, there is an (w, b) -gap in P(N) / Fin.

Any gap of type (w, ic) will be called a Rothberger gap, while any gap of type (I£, A) where both rc and A are uncountable and regular will be called a Hausdorff gap. (For the definitions of nonlinear Rothberger and Hausdorff gaps, see §5.6.) The cardinal b equals w1 under CH, but it can assume different values in different

models of Set theory (e.g., MA implies b = c, while OCA implies b = U)2; see Theorem 2.2.6). In particular, the existence of (w,wl)-gaps is, unlike the existence of (wj,wl)-gaps, not decided by the usual axioms of Set theory. Are there any types of gaps in P(N)/Fin other than Hausdorff's and Rothberger's? Under CH the answer is trivially negative, while there are situations in which gap-spectrum of P(N)/Fin is rather rich (see e.g., Proposition 5.2.6 below). In ([81]), Kunen proved

that it is impossible to construct any types of gaps other than Hausdorff's and Rothberger's without using some additional assumptions. Then Todorcevic ([125, §8]) formulated Theorem 5.2.4 below which not only gave an explanation to Kunen's theorem, but it raised considerably the confidence about the right formulation of OCA which motivated much of the further research on this axiom. Todorcevic's result is best explained if we leave the domain of linear gaps. The following definition is extracted from [89] in [126]. Two orthogonal families A, 13 form a Luzin gap if there is an uncountable X C 2N and two surjections fo : X A and Ii: X -* B such that for all x, y E X we have if and only if x#y. fo(x)nf1(y):A0 LEMMA 5.2.3. Every Luzin gap is a gap.

PROOF. Assume A,5 is a Luzin gap, and X, fo, fl are as above. Assume the contrary, that C C N separates A from B. By a counting argument, we can find an uncountable X' C X, n E N and s, t C n such that for all x E X we have fo(x)\C = s and fI (x)nC = t. This implies that fo(x)n fl(x) = snt = fo(x)n fl(y) for all x, y E X', contradicting the definition of the Luzin gap. The following result shows that the Luzin gaps are in some sense canonical (see also [126, Theorem 2) for a definable version). THEOREM 5.2.4 (Todorcevic). OCA implies that for every two orthogonal families A and B of subsets of N exactly one of the following applies: (1) A and B can be countably separated from each other,

(2) the restriction of A and B to an end-segment of the integers contains an uncountable Luzin subgap.

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146

In particular, every linear gap in P(N) / Fin is either an (w1i w1)-gap or a Rothberger gap-

PROOF. Recall that A stands for the hereditary closure of A, {A : A C A' for some A' E Al. Let

X = {(A,B) E Ax f3 :An B = 01, and define a partition [X]2 = Ko U K1 by letting {(A, B), (A', B')} in Ko if

An B'¢0or A'nB#0. This partition is open (see the beginning of §2.2). We claim that X is u-K1-homogeneous if and only if A is countably separated from B. For C C N define

7-Lc={(A,B)EX : ACCandBnC=0}. The set 7-1c is clearly always K1-homogeneous. Also, if Cn (n E N) is a countable separation of A from B, then the K1-homogeneous sets Hc,, (n c N, s E Fin) cover X. On the other hand, if 'H is KI-homogeneous, the set

Cx= U A (A,B)E7{

has the property that A \ C7{ C A U B and B n C% C A U B, whenever (A, B) E H. Therefore if X is u-K1-homogeneous, A and B are countably separated, and this concludes the proof that the two properties are equivalent. Now assume that A is not countably separated from B. Since the partition Ko is open, OCA implies that there is an uncountable Ko-homogeneous subset of X. Let X' be the set of all (A, B) such that for som (A', B') E X we have A C A' and

B C B'. Then for some n E N and for uncountably many (A, B) E X' we have A n B C n. Identify 2N x 2N with 2N, let X = X', and let fo, f1 be the projection maps of X to A and B respectively. Then the restriction of the image of X to In, oo) is the required Luzin gap. It remains to prove that a linear gap can be either an (w1, w1)-gap or a Rothberger gap. But every subgap of a (k, A)-gap, where both /c and A are uncountable, has to be of type (,c, A). But a Luzin gap is of size 111, and therefore the conclusion follows.

Note that by identifying a pair (A, B) from X with a partial function fAB : A U

B - {0, 1} such that f-1(0) = A one can see that the above proof is identical to the proof of Theorem 2.2.1. Moreover, it is not difficult to see that the conclusion of Theorem 5.2.4 is equivalent to a slight strengthening of the conclusion of Theorem 2.2.1, saying that every coherent family of partial functions indexed by some ideal is either u-trivial or contains an uncountable subset consisting of pairwise incompatible functions. The following result which appears in [125, Theorem 8.6] shows that the gapexistence results of Theorem 5.2.2 and Theorem 5.2.1 are sharp. COROLLARY 5.2.5 (Todorcevic). Assume OCA and that c = l 2. Then the only types of linear gaps in P(N)/Fin are the following: (L), L02), (w1, w1), and (w2,w). PROOF. By Theorem 2.2.6, b = w2i and by Theorem 5.2.4 and c = 112, the only gaps of type (w, n) or (,c, w) are those for is = w2. Let A, B form a linear Hausdorff

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147

gap. By Theorem 5.2.4 it has a subgap of size 1Z1, and therefore it has to be an (w1, wl)-gap. The existence of such gaps is guaranteed by 5.2.1.

Let us note that Hausdorff gap spectrum of P(N)/Fin can be almost anything. The following (probably folklore) result follows immediately from [30, §6]. PROPOSITION 5.2.6. Assume CH and let 9 be a set of regular uncountable cardinals including w1. Then there is a cardinal-preserving forcing extension in which there is a (rc, n)-gap in P(N)/Fin if and only if r, E 9.

What is the minimal complexity of a gap in P(N)/Fin, considered as a set of reals? It is known (see [126]) that in Godel's constructible universe there are coanalytic Hausdorff gaps, and this is the best possible since by Kunen-Martin theorem (see [74]) no analytic set can have order type w1 under C*. This, however, does not exclude the existence of two analytic families, a-directed under C*, inside

which an (wl,w1)-gap occurs. For example, a Rothberger gap is hidden inside a pair of orthogonal analytic ideals Fin x O and 0 x Fin, and there exists a Luzin gap which is a perfect subset of reals (see e.g., [126, page 57]). In [126], Todorcevic extended the classical Hurewicz separation property (see [74], §5.5, also [77]) and proved that it implies the following. THEOREM 5.2.7. If A and B are orthogonal a-directed families of sets of integers such that A is analytic, then they can be separated.

This implies not only the nonexistence of analytic Hausdorf gaps, but also Theorem 5.1.1 (see Proposition 5.5.3). For more information about this exciting subject and some of its applications the reader can consult [105], [126], [130] and we shall now turn our attention to gaps in analytic quotients other than P(INY)/Fin.

5.3. Gaps in the quotient over 0 x Fin A question of determining the (linear) gap spectra of 0 x Fin and the preservation of gaps via Baire-embeddings of its quotient into other analytic quotients was raised in [128, Problem 4]. In Proposition 5.3.2 and Proposition 5.3.3 below we give a complete answer to this question by using Lemma 5.3.1 below and some results of [130] and [128]. For a family A of sets of integers its restriction to some C C ICY is A f C = {A f1 C : A E Al. Recall that the restriction of 0 x Fin to any [1, n] x N is isomorphic to Fin.

LEMMA 5.3.1. If A, B is a gap in p(N2) /0 x Fin, then there is an n such that A,B is still a gap when restricted to [1, n] x N. PROOF. Assume that there is C. which separates An from Bn for all n. Then C = Un Cn separates A from B, contradicting the assumption that this was a gap and therefore completing the proof. PROPOSITION 5.3.2. The linear gap spectra of 0 x Fin and Fin coincide.

PROOF. By Todorcevic's Theorem 5.1.1, it will suffice to prove that if there is

a gap of certain linear or a-directed type in P(IN)/0 x Fin, then there is such a gap in P(N)/Fin. The restriction of 0 x Fin to any {n} x N is isomorphic to Fin, and we shall use this to reflect gaps to a set of this form.

CLAIM 1. If A, B is a gap in P(IN)/0 x Fin, then there is a natural number n such that A r {n} x N (B [ {n} x N, respectively) is unbounded in A (13, respectively).

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148

PROOF. Assume otherwise. Then families A and B are separated when restricted to {n} x N for every n, contradicting Lemma 5.3.1. To prove (a), assume A, B is a linear gap. Let n1 and n2 be integers guaranteed by Claim 1 and Claim 2 respectively, and let n = max(n1i n2). Then A, B when restricted to {n} x N is a gap in P({ n} x N)/Fin of the same type as the original gap. This completes the proof of theorem.

We should note that it is not necessarily true that spectra of nonlinear gaps in quotients over Fin and 0 x Fin coincide. In §5.7 we will construct an analytic gap in a quotient over 0 x Fin of type which, by Theorem 5.2.7, cannot occur in the quotient over Fin.

PROPOSITION 5.3.3. If : P(N2)/0 x Fin - P(N)/1 is a monomorphism with a Baire-measurable lifting, then 4 preserves all Hausdorff gaps. If I is moreover a P-ideal, then (D preserves all gaps. PROOF. If A, 13 is a gap in P(IN)/0 x Fin, then by Lemma 5.3.1 there is n such that the restriction An, B. of A, B to [1, n] x N is still a gap. Let C = D. ([l, n] x N) is some lifting of '). Then D1 : P([l, n] xN)/ Fin P(C)/I is a homomorphism with a Baire-measurable lifting, so by Todorcevic's Theorem 5.1.1, D1 carries A,, B,

into a gap of P(C)/I, which is clearly also a gap in P(N)/I.

5.4. Gaps in the quotient over Fin xO This section is devoted to study of gap-spectra and preservation of gaps for quotient over the ideal Fin x0, analogous to that of previous section. This question was raised in [128, Problem 3]. The quotient P(IN)/Fin x 0 can be considered as an inverse limit

2(N2)/ Fin x0 = ]im(P(r f)/ Fin, 7rg where r f = {(m, n) : n < f (m)}, and where 7r9 : rf/ Fin -- I'9/ Fin (for f > g) is the restriction mapping. It is clear that the quotient over 0 x Fin can be considered as an inverse limit of quotients P([1, n] x N)/ Fin, and this fact was implicitly used in previous section. Let us say that Hausdorff gaps in the quotient over Fin x0 reflect if for every Hausdorff gap in the quotient over Fin xO there is f E NN such that the restriction of this gap to r f remains a gap. Using an analogous terminology, Lemma 5.3.1 states that all gaps in the quotient over 0 x Fin reflect. However, the reflection of gaps in the quotient over Fin x 0 does not follow from the usual axioms of Set theory since it contradicts CH (by Lemma 5.4.1 below and [90] or [25]). Although the restriction to Hausdorff gaps perhaps looks unnatural, it appears that the Rothberger gaps do not reflect (see Example 5.4.5).

LEMMA 5.4.1. If Hausdorff gaps in the quotient over Fin x 0 reflect, then every coherent family of partial functions indexed by NN is trivial.

PROOF. Let gf : r f .- {0, 1} (f E NN) be a family of coherent functions, and

A= {gf 1({1})

f E N },

B = {gf1({0}) : f E

NN}.

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Then A, B is a pair of orthogonal modulo Fin x0 families. Both A and B are adirected (under the inclusion modulo 0 x Fin) since the ideal 0 x Fin is. Moreover, restrictions of A and B to any rf are separated by Cf = 9f 1({1}). By the assumed reflection of Hausdorff gaps in the quotient over Fin x O, A and B are separated by some C C N2, and taking h to be the characteristic function of C witnesses that the coherent family is trivial. We should note that the converse to Lemma 5.4.1 is not true. This follows from the proof of the main result of [68].

By a proof analogous to that of Proposition 5.3.3, if Hausdorff gaps in the quotient over Fin x0 reflect, then every monomorphism of P(IN)/Fin x0 into a quotient P(N)/1 over an analytic ideal with a Baire-measurable lifting preserves all Hausdorff gaps. We shall later see (Corollary 5.5.7) that this conclusion does not require any additional assumptions. Also, if we assume OCA and MA instead, then the assumption of the existence of a Baire-measurable lifting is unnecessary, and

every monomorphism of the quotient over Fin x0 into another analytic quotient preserves all Hausdorff gaps (see Corollary 5.9.2). The following is essentially a reformulation of Theorem 8.7 of [125], which says that under OCA every family of functions coherent modulo Fin is either trivialized by countably many functions or it has a subfamily of size 111 of pairwise inconsistent functions (see §§2.2-2.3 for more on similar results).

PROPOSITION 5.4.2. OCA implies that Hausdorff gaps in the quotient over Fin x0 reflect.

PROOF. Assume OCA, and let A, B be a Hausdorff gap in the quotient over Fin x 0. Recall that for f : N -+ N we have r f = { (m, n) : n < f (m) }, and define

A'={rfnA : AEA} and l3'={rfnB : BEB}. Assume A', B' has a subgap of size 111i and let F C NN be the set of all f appearing

in the definition of A' and B. Since OCA implies that b > w1 (Theorem 2.2.6), we can find f : N -* N which eventually dominates F. Therefore the restriction of A', B' to r f is a gap. Otherwise, by Theorem 5.2.4 the families A' and B' can be countably separated. But since A',13' is a Hausdorff gap and NN is o'-directed under Lemma 2.2.2 implies that A' and 13' can be separated, by a set C C N2, say. We claim that C separates A and B. Assume the contrary, then either there is an A E A such that

A\ C V Fin x O, or there is a B E B such that C n B 0 Fin x O. Then there is an f E NN such that rf n (A \ C) (respectively, r f n (C n B)) is infinite; but this contradicts the assumption on C. PROPOSITION 5.4.3. OCA implies that the linear gap-spectra of quotients over Fin x0 and Fin coincide.

PROOF. Assume A,B is a linear gap in P(N2)/Fin x0. If this is a Hausdorff gap, let f E NN be such that A, B is still a gap when restricted to r f . Such an f exists by Proposition 5.4.2. Then none of the restrictions of A and B to r f has the minimal element. Since both A and B are linear, these restrictions give us a gap in P(F1)/Fin of the same type as A, B. Now assume A, B is a Rothberger gap, i.e., that at least one of A and B (say, A)

is countable. Let A = {Am : m E N} and B = {Be : e < k}, where k is a regular

150

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cardinal. We define an increasing unbounded rs-sequence {ft} in NN by the formula

f£ (n) = min{i : Be fl A. C [1, i] x N}.

Then Be \ B, E Fin x0 implies ft >* f,,. If some g eventually bounds all ft, then the set M

C = U (Am \ [1,g(m)] x N) m=1

separates A and CB, and therefore the sequence { f£} is unbounded. By a classical result ([53], [103]), this implies the existence of an (w, r.) gap in P(N)/Fin. As we have mentioned earlier, the fact that every monomorphism of the quotient over Fin x O into any analytic quotient preserves all Hausdorff gaps follows already from the standard axioms of set theory (Corollary 5.5.7). We do not know whether the assumption of the reflection of Hausdorff gaps can be completely removed from

Proposition 5.4.3, but let us now show that it is not needed in a special case. The case when rc = A of the following lemma is an immediate consequence of [25, Theorem 4.1].

LEMMA 5.4.4. Assume there are an unbounded r-chain and an unbounded Achain in (NN, <*). Then there is a (n, A)-gap in a quotient over Fin x0 if and only if there is a gap of this type in a quotient over Fin. PROOF. Let f£ (1' < rc) and g, (77 < A) be unbounded chains in (NN, <*) and

let A£, By ( < rc, r? < A) be a gap in the quotient over Fin x0. Then the sets A£ = A£ n I fc and B;, = By n r9n form a (ic, A)-gap in P(IN )/ Fin. (See the proof of Proposition 5.4.2.)

Let us now show that Rothberger gaps in the quotient over Fin x O need not reflect even under OCA. Recall that a strictly <*-increasing sequence ft in NN is a scale if it is cofinal in (NN, _<*). Since a scale exists whenever b = c (c is the size of the continuum), and since OCA implies that b > w1 (Theorem 2.2.6), in a model in which OCA is true and there are no scales the size of the continuum would have to be at least ZZ3i and at present no such model is known (see Question 2.5.4). EXAMPLE 5.4.5. Assuming there is a scale { ft l; < k}, there is an (w, c)-gap in P(N2)/Fin x0 such that its restriction to any I'f is not a gap. Our working copy of Fin x 0 will be an ideal on IN generated by sets [1, n] x N2. This ideal is clearly isomorphic to Fin x 0, so we shall call it Fin x 0. For < k define he E NN by he(n) = min{j : ft(j) >_ n}. Let A be generated by sets A. = N2 x [1, n], and for f c NN let Et = {(n, f£(n),he(n)) : n E N}. Let B be the family of all :

B. = U E£ n {(n,i, j) : j ? h£(n)}. Then Be n A, c [1, k] x N2 where k is such that ft(k) > m, therefore A and B are orthogonal. Also, if < 77 then Be \ B,, C [1, k] x N° where k is such that f,, >k ft, therefore Be is (strictly) increasing. We claim that A, B form a gap. Assume some C C Nt3 includes (modulo Fin x0) all Af E A define g: N -+ N by g(j) = min{k : A3 \ C 9 [1, k] x N2}.

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151

` g then Bt n C has an infinite intersection with the set {(n, fe(n), ht(n)) : n E N}, therefore C is not orthogonal to B and A, B is indeed a gap. It remains only to show that the restrictions of A and B to any r f can be separated. In a "standard" interpretation of Fin x O, the family F f (f E NN) generates the ideal 0 x Fin = (Fin x0)1. Therefore analogues of I'f's in our case are sets orthogonal to Fin x 0, and every such set is included in some A£ = {(n, i,j,) i,j : fe(n)} since {f4} is a scale. Then on At families A an B are separated by But if f f

:

Ct = {(n,i,j) E Ae : j > he(n)},

for(A;nAt)\CtC[1,fe(j)]xN2and(BnAe)nCt9[1,k]xN2iff,,>k fe. This ends the proof.

5.5. The Todorcevic separation property, TSP The ideal Fin can be considered as a tree under the ordering C of end-extension. Recall that a tree is superperfect if above every node it has an infinitely branching node.

DEFINITION 5.5.1. A set E C Fin is a superperfect B-tree if it is a superperfect subtree of (Fin, C) such that for every infinitely branching node s the set

U{t C [max s + 1, oo)

:

s U t is an infinitely branching node of E}

is included in some set from B.

An ideal I has the Todorcevic separation property, TSP, if for every analytic hereditary A and an arbitrary set B which is I-orthogonal to A one of the following two conditions applies: (T1) A and B can be countably separated over Z, or (T2) there is a superperfect S-tree all of whose branches are in A and I-positive.

Clearly, if I has the TSP and families A, B are I-orthogonal and such that A is analytic and B is or-directed, then A and 13 can be countably separated. In particular, TSP of I implies there are no analytic Hausdorff gaps in P(N)/I. THEOREM 5.5.2 (Todorcevic, [126]). The ideal Fin has the TSP.

The interest in TSP comes from the following result.

PROPOSITION 5.5.3 (Todorcevic, [126]). Assume I has the TSP and ,7 is an analytic ideal. Then every Baire monomorphism 4): P(N)/I -* P(N)/,7 preserves all Hausdorff gaps. Equivalently, there are no analytic Hausdorff gaps in P(N)/Z.

PROOF. Let 4), be a continuous lifting of 4) and let A, B be a Hausdorff gap in P(N)/Z. If families E'A and V'S are split by some C, let

A'_{A: This is an analytic family which includes A and is I-orthogonal to B. Since A and B cannot be countably separated (for Hausdorff pregaps "countably separated" implies "separated"), TSP implies there is a superperfect B-tree E all of whose branches are in A'. But B is or-directed, so there is B E B which almost includes

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152

(modulo I) all branchings of E. Since E is superperfect, such B includes many of its infinite branches, contradicting the fact that A' and B are orthogonal.

We shall use the language of games to reformulate TSP and prove that ideal Fin x 0 possesses this desirable property. In a game G(A, B) for A, B C P(N) two players, I and II, play according to the following rules: .. I 82 E Fin, B2 E B sl E Fin, B1 E B

II

k1 E N

k2 E N

It is additionally required that sn+1 C Bn n [kn, oo) for all n E N. If both players obey the rules, player I wins if and only if Un sn is in A. LEMMA 5.5.4. Assume A and B are families of sets of integers. There is a superperfect B-tree all of whose branches lie in A if and only if I has a winning strategy in G(A, B).

PROOF. Assume first E is a superperfect B-tree all of whose infinite branches lie in A, and for an infinitely branching node u E E fix B. E B including the set U{s C [max u + 1, oo) : u U s is an infinitely branching node of E}.

A winning strategy T for I is defined as follows: In his first move, I plays some infinitely branching node s1 of E and B1 = B9. In his i-th move, I finds si Bin[ki, oo) such that ti = U.,=1 sj is an infinitely branching node of E and Bi = B. This is clearly a winning strategy. Now assume I has a winning strategy r for G(A, B). By playing against T, II can easily construct a tree satisfying (T2). Lemma 5.5.4 immediately implies the following.

COROLLARY 5.5.5. A Borel ideal I has the TSP if and only if player I has a winning strategy in G(A\I, B) whenever A and B are I-orthogonal, not countably separated families such that A is analytic. THEOREM 5.5.6. The ideal Fin x0 has the TSP.

PROOF. For technical convenience, let us assume Fin x0 is an ideal on N de-

termined by g: N -* N as Fin x0 = {A : g"A is bounded}. Let A and B be two Fin x 0-orthogonal sets which cannot be countably separated and such that A is analytic. Let T be a subtree of Fin x Fin such that A is equal to its projection. By Corollary 5.5.5, it will suffice to describe a winning strategy T for player I in G(A, B) for A, B as above. First pick (sI, ti) E T so that A(sl, t1) = p[T(si, ti)] is not countably separated from B over Fin x g and that set XI = {n : A(sl U {n}, tn) is not countably separated from B for some to extending tn} is unbounded, and let B1 E B be such that B n U Al is positive. In latter moves, I chooses (sn, tn) and Bn so that Xn is unbounded, Bn n U A(sn, tn) is positive, and max(g"(sn \ Sn_I)) > n. It is clear that this is possible and U. sn is positive. Finally, Proposition 5.5.3 and Theorem 5.5.6 imply the following.

COROLLARY 5.5.7. Every Baire monomorphism of P(N2)/Fin x0 into some analytic quotient preserves all Hausdorff gaps.

5.6. TUKEY REDUCTIONS OF NONLINEAR GAPS

153

5.6. Tukey reductions of nonlinear gaps In this section we suggest a natural definition of a type of a nonlinear gap. Let us first review some important definitions. In this section P, Q will always denote directed partially ordered sets. A map f : P - Q is a Takey map if it sends unbounded sets into unbounded sets. If such an f exists, then we say that Q is

cofinally finer than P and write P < Q. We write P1 _ P2 if both P < Q and Q < P. Two partially ordered sets P1 and P2 are said to be cofinally similar if there

is a third partially ordered set Q such that both P1 and P2 are order-isomorphic to some cofinal subset of Q. The following result is due to J.W. Tukey ([138]). THEOREM 5.6.1. Two directed sets P and Q are cofinally similar if and only

ifP
(For more details on these notions the reader can consult [124] and [46].) For example, note that (1) (Na, <*) (0 x Fin, C*), (2) (NN, <) (0 x Fin, C), (3) (N, <) _ (Fin, C). A much deeper result proved recently by Todorcevic is (we consider ideals as partially ordered by the inclusion; in Definition 1.2.7 it was defined when two ideals are isomorphic):

THEOREM 5.6.2. If I is an analytic P-ideal, then either I is isomorphic to Fin or NN < I < Ii/n. This implies that NN / Fin < I/ Fin < I1/n/ Fin. The proofs can be found in [126] and [2]. Since I1/n/ Fin _ N, where N stands for the ideal of Lebesgue measure zero sets ordered by the inclusion, this can be seen as a supplement to the well-known relation between the measure and category investigated by Tomek Bartoszynski and others (see [46], [2]). Recall that add(N) is the additivity of the Lebesgue measure.

LEMMA 5.6.3. Every subset of an analytic P-ideal ,7 of size less than add(.) is C*-bounded in J. In particular, MAN, implies that every subset of size l1 is bounded.

PROOF. By the second part of Todorcevic's Theorem 5.6.2, (,7, C*) is Tukey-

Therefore if a subset of 3 of size < add(N) is C*unbounded, then a subset of I1/n of size < add(.) is also C*-unbounded. It reducible to (I1/n, C*).

is well-known that (I1,n, C*) is Tukey-equivalent to N (see [46], [2]), therefore the first part of the statement follows. Since MAN, implies that the additivity of measure is strictly bigger than 111i the lemma follows.

If A C P(N) and I is an ideal on N, then by All we shall denote the partially ordered set (A/I, CZ).

DEFINITION 5.6.4. If A, B is a gap in P(N)/l it is of type (P, Q) (or, it is a (P, Q)-gap) if

(a) P, Q are directed sets such that All =_ P, 13/I

Q, and

(b) if X C A is unbounded and Y C B is unbounded, then X cannot be separated from B and Y cannot be separated from A. The families A and B are the sides of this gap.

5. GAPS AND LIMITS IN ANALYTIC QUOTIENTS

154

We note that requiring both sides of the gap to be directed under the inclusion modulo I is not a loss of generality, since we can always close A and B under the finite union of their elements and still have a pair of families orthogonal modulo I. Definition 5.6.4 clearly generalizes the linear gaps, i.e., (r,, A)-gaps when is and A

are regular cardinals. We now update our terminology and say that a (nonlinear) gap is Hausdorff if both of its sides are a-directed, and that it is Rothberger if one of its sides is cofinally similarly to N. Let us write 131/1 for an orthogonal of B modulo 1, i.e.,

B1/1= {A: An BE I for all BE 13}. LEMMA 5.6.5. A pair A, B of 1-orthogonal families satisfies the condition (b) above if and only if the identity map from A into B1/1 is a Tukey map.

PROOF. Both conditions are equivalent to: A subset of A is unbounded in All if and only if it is unbounded in B1/1. The condition (b) is not necessarily satisfied by all gaps and therefore not every gap has a type. However for any gap A, B we can find, by using Zorn's lemma, a

gap A', B' such that A' J A, 13' C B and (b) is satisfied. The purpose of (b) is to avoid having substantially different gaps having the same type. For example, if we did not require (b) then there would always be gaps of type (w1 x w, w1) in P(N) /Fin, but the existence of a gap of type (w x w1iw1) in the sense of Definition 5.6.4 is equivalent to the existence of an (w,w1)-gap in P(N)/Fin. We can even strengthen (b) to require

(b') if X C A is unbounded and Y C B is unbounded, then X cannot be separated from Y.

We can say that A, B has strong type P, Q if (a) and (b') are satisfied. But the condition (b') seems to be too strong, in particular it is unclear whether every gap can be extended to a gap satisfying it. This is the reason why we use (b) instead. We note here that understanding of the relationship between (b) and (b') is tightly connected to the question of whether a gap constructed in §5.10 includes a linear

gap. An important point in favor of (b) in comparison to (b') is that if a pair (A, B) satisfies (b'), then the families A and B cannot be hereditary (i.e., closed under taking subsets of their elements). On the other hand, if A, B satisfy (b), then so do their hereditary closures A, B. A characterization of preservation of gaps by a homomorphism related to the above is given in Proposition 5.6.6 below. The condition (3) has been first isolated in [130, Theorem 12], where it is used to show that the natural homomorphisms of P(N)/Fin into the quotients over Mazur ideals (a large class introduced in [95]) preserve all gaps, i.e., where it is shown that it implies the condition (1). ing

PROPOSITION 5.6.6. For a homomorphism (D: P(N)/1 -> P(N)/3 3 the followare equivalent: (1)

'

preserves gaps,

(2) for every B C P(N) the restriction of

'

to B1/1 is a Tukey map into

(V.13) 1/J.

(3) There is a map F: P(N) -> P(N) such that (i) (A) n B E 3 implies A n F(B) E 1, (ii) (A) \ B E 9 implies A \ F(B) E I.

5.7. TSP IN QUOTIENTS OVER ANALYTIC P-IDEALS

155

PROOF. Note that (2) says that every gap of the form A, B for some A is preserved, therefore (1) and (2) are equivalent. To see that (2) and (3) are equivalent, note that F is a witness that 4? is a Tukey map. An evidence that Definition 5.6.4 is natural is given in: COROLLARY 5.6.7. If a monomorphism 4i: P(N)/I --* P(N)/,7 preserves gaps, then it also preserves their types.

PROOF. We shall check the condition (2) from Proposition 5.6.6. We have to prove that the identity map

id: (

(4> x)1/ 7

is a Tukey map. Note that

81/I : 41(81/ J) idl

(4) 8)1/.T

and that the map 4i is an order-isomorphism between 8'/I and fore the map

There-

ido)-1: 4)"(8 /I) . (4?" 5)1/J is well-defined and equal to a composition of two Tukey maps, and thus a Tukey map itself.

Although the nonlinear gaps have been considered very early (see e.g., [89]) and they do have some use (see e.g., [83]), many authors (see [105] for a survey)

consider only linear gaps. One of the reasons for doing so is that any attempt to describe spectra of nonlinear gaps would very likely require a classification of directed sets of size continuum, a task which we know to be impossible (see [124]). (Even the restriction of this problem to Borel directed sets seems to be quite difficult (see [88]).) Our reason for going into this generality is to point out the new gap phenomena in quotients beyond P(N)/Fin which witnesses that results about gaps in P(N) /Fin cannot be transferred to arbitrary quotients over analytic ideals (see §5.7).

5.7. TSP in quotients over analytic P-ideals We now prove that Theorem 5.5.2 has a converse. Recall that an ideal I is isomorphic to Fin (see Definition 1.2.7) if and only if I is equal to Fin or of the form Fin ®A for some A C N.

THEOREM 5.7.1. If I is an analytic P-ideal, then I has the Todorcevic Separation Property if and only if I is isomorphic to Fin. This theorem will be proved by using the following proposition, due to Todorcevic.

PROPOSITION 5.7.2. Assume that the ideal I has the TSP and (A, 8) is a gap in P(N)/I such that 8 is or-directed and A is analytic. Then A can be countably separated from B.

5. GAPS AND LIMITS IN ANALYTIC QUOTIENTS

156

PROOF. Since for every B-tree there exists a B E B which includes all of its branching nodes (and therefore many of the branches) modulo Z, TSP implies that such a gap is countably separated by some sequence {C}. Recall that [R]<" is a poset of all finite sets of reals ordered by the inclusion. This is a Tukey-maximal directed set of size at most c. COROLLARY 5.7.3. If there is a gap (A, B) in P(N)/I such that A=_ [R]<", A is analytic, B is a-directed and no unbounded subset of All is separated from B, then I does not have the TSP.

PROOF. If I has the TSP, then A and B can be a-separated by Proposition 5.7.2. Therefore an infinite subset of A can be separated from B, contradicting the assumptions.

PROOF OF THEOREM 5.7.1. We shall prove that in every quotient over an analytic P-ideal other than Fin there is an analytic gap (A, B) satisfying the assumptions of Corollary 5.7.3. This is done for Fo P-ideals in Lemma 5.7.4 and for other P-ideals in Lemma 5.8.7 below. The conclusion will then follow by Proposition 5.7.2.

LEMMA 5.7.4. If Z is an F, P-ideal other than Fin, than there is an analytic ([R]<", Q) -gap in its quotient for some a-directed poset Q.

PROOF. First we give an example of a summable quotient with this kind of a gap, and then use this example as a template in proving the general case. Let f : {0,1}
sup

µf (A. fl B),

where

Ay = {x (n : n E N}.

xE{0,1}N

Let B be the analytic P-ideal defined by

B=Exh(cp) _ {B C N : lirnp(B\{0,1}fi') =0}, (this ideal was introduced in Example 1.13.17 as an ideal rather different from both the summable and the density ideals). Now let

A= {Ax:xE{0,1}N}. If (A) is an ideal generated by A, then the triple If, (A), B is, in some sense, analogous to the triple Fin, Fin x0, 0 x Fin. It is clear from the definitions that A/Zf = [R]<", and since B is an analytic P-ideal, B/Zf is a-directed. CLAIM 1. No infinite subset of A can be separated from B.

PROOF. Assume {An} is a subset of A, and let C be such that C D'1 An for all is. We can assume that An forms a converging sequence in {0,1}N with limn An = A,,, so that A(An, Ate) = A(An, An+1) < A(An+l, Ate) for all n. Let un C An n C be disjoint from Ac (and therefore all A,, rn # n) and such that

1/n < p f(un) < 2/n-it exists since µf(An fl C) is infinite. Then B = Un un is in B, but µf (B n C) = oo and therefore C does not separate {An} from B. Therefore the pair A, B satisfies the assumptions of Corollary 5.7.3. This completes the proof in the case of If.

5.8. QUOTIENTS AS REDUCED PRODUCTS

157

Now we prove Lemma 5.7.4 for an arbitrary Fo P-ideal I by mimicking the above construction. Let su (u E {0,1}
for x E {0,1}' and let B = Exh(cp), where is defined by (let p.T be such that I = Exh(wz)) co(B) = sup pz(B n Ax). aE{0,1}N

This is clearly a lower semicontinuous submeasure, therefore B is an analytic Pideal,in particular, B/I is o-directed. Like in the case of If , we have All = [R] < W .

CLAIM 2. No infinite subset of A can be separated from B.

PROOF. Assume An = Axn E A are pairwise distinct elements of A and let C be such that An \ C E I for all n. Pick U. E xn \ Ui#n xi so that wx(su, \ C) < 2 n

(this is possible since limn cpz(su" \C) = 0). We claim there is B C D = Un s," \C such that limn o1(B n sun) = 0 but B ¢ I. Otherwise we have

I I D = {BCD : limcpr(B n s, ,) = 0} = Exh(supcpr(suj), n

n

and the ideal I [ D (and therefore I) is not Fv (by Proposition 1.13.14), contradicting our assumption. Let B be as above. Then B \ C E I, B E B, but B I, and therefore C does not separate {An} from B. This completes the proof.

Therefore A and B satisfy the assumptions of Corollary 5.7.3, and this completes the proof of Lemma 5.7.4. We continue the proof of Theorem 5.7.1 in the following section.

5.8. Quotients as reduced products To complete the proof of Theorem 5.7.1 it remains to construct a required gap in a quotient over every analytic P-ideal which is not F. We shall do this first in the case when l = 0 x Fin and then "embed" this gap into every other quotient over an analytic P-ideal which is not F. Before we continue the proof of Theorem 5.7.1, we shall construct gaps in the quotient over the ideal

FinxFin={ACN 2:{i:{j:(i,j)EA}VFin}EFin}. Recall that if (Pn,
LEMMA 5.8.1. The quotient P(N2)/ Fin x Fin is isomorphic to the reduced pro-

duct (fn P(IN)/ Fin. LEMMA 5.8.2. If for every n there is an (analytic) gap of type (P,, Q,) in the quotient over Fin, then there is an (analytic) gap of type Mn P, /Fin, r In Qn/ Fin) in the quotient over Fin x Fin.

5. GAPS AND LIMITS IN ANALYTIC QUOTIENTS

158

PROOF. Let (An, B,,,) be a gap of type (Pn, Qn) in P(N)/ Fin such that any unbounded subset of A. cannot be separated from Bn, and that any unbounded subset of Bn cannot be separated from An. We can assume that An, tan is a gap in P({n} x N)/ Fin. Define A, B by (let A(n) = A n {n} x N):

A = {ACN2:A(n)EAn} B = {BCN2:B(n)EB,,}. It is clear that A, B are analytic if all An, Bn are. We shall prove that A, B is a gap of type (nn Pn/ Fin, jjn Qn/ Fin). Clearly A/(Fin x Fin) fJn Pn/ Fin, B/(Fin x Fin) fin Qn/ Fin, and A and B are Fin x Fin-, and even 0 x Fin-, orthogonal. Assume A' S A can be separated from B by some C C N2. Then the set

in: (B \ C) (n) E Fin for all B E B} is cofinite (otherwise we can find a B E B such that B \ C V Fin x Fin). Therefore we can assume, possibly by adding a set in Fin x Fin to C, that (B \ C)(n) E Fin for all n. Let

A°={AEAn:AfCEFin}. Then C(n) separates A° from Bn, and since on (An, Bn) satisfies (b) of Definition 5.6.4, the set A° is bounded in An/ Fin by some An. Then A, = Un An is in A, and for every A E A' we have A(n) E Ao for all but finitely many n, therefore the set

{n : (A(n)) \ A, V Fin}

is finite and finally A \ A E Fin x Fin. This means that A' is bounded in A/(Fin x Fin) by A,,,,. We similarly prove that B' C B can be separated from A if and only if it is bounded in B/ Fin x Fin, and therefore (A, B) is a gap of the desired type. PROPOSITION 5.8.3. There is an analytic gap of type (N1' /Fin, NN/ Fin) in the

quotient P(IN)/(Fin x Fin). In particular, Fin x Fin does not have the TSP. PROOF. The ideals Fin x0 and 0 x Fin form a gap of type (N, NN/ Fin) in the quotient over Fin. By Lemma 5.8.2, in the quotient over Fin xO there is a gap of El type (flnN/Fin, (flnNN/Fin)N/Fin) =_ (NN/ Fin, NN/ Fin).

and (w,A)-gaps in P(N)/Fin, then PROPOSITION 5.8.4. If there are there is a (,c, A) -gap in P(IN)/ Fin x Fin. In particular, there is a (b, b) -gap in this quotient.

PROOF. By Lemma 5.8.2, there is a gap of type (icN/ Fin, NN/ Fin) _ (ic,NN/ Fin)

in this quotient. By Rothberger's result (Theorem 5.2.2), there is an unbounded A-chain in NN/ Fin and this gives us the desired (K, A)-gap.

In §26 of [32] we have constructed a model of set theory in which there is an (w2i w2)-gap in the quotient over Fin x Fin yet there are no such gaps in P(INY)/ Fin

(an interested reader familiar with the methods of [30], in particular the model of §2 and Theorems 6.1 and 9.2, should be able to reproduce this result). The above proposition, and its consequence Theorem 5.8.5 below, show that the same conclusion even follows from OCA.

5.8. QUOTIENTS AS REDUCED PRODUCTS

159

THEOREM 5.8.5. Assume OCA. Then there is an (w2iw2)-gap in the quotient

P(IN)/ Fin x Fin. PROOF. By Proposition 5.8.4 there is a (b, b)-gap in this quotient, and by Theorem 2.2.6 we have b = w2.

This result sheds some light on Problem 2 of [130], asking to determine the

gap spectrum of P(N)/I for every analytic ideal I on N. It also shatters the hope expressed in the discussion following the problem that under OCA or PFA {()2iw), (w1, w1), (w,w2)} might be the gap spectrum of every analytic quotient (see [130, page 95]). However, it is still possible that OCA (or PFA) is sufficient for determining the gap spectrum of any analytic quotient which could have been the primary idea behind the discussion in [130]. (See also Corollary 5.10.3.) Note that a (k, A)-gap given by Proposition 5.8.4 is a "subgap" of the gap constructed in Proposition 5.8.3. If this latter gap satisfied the condition (b') instead of (b) (see §5.6), then we would be able to extract a (n, A)-gap from it directly. However, it is easy to see that (b') fails for the gap constructed in the proof of Proposition 5.8.3. (Choose families A' C A n P(A x N) and 13' C_ B x P(B x N) unbounded in A, B respectively for some disjoint infinite sets A, B C N.) A gap witnessing Proposition 5.8.3 and satisfying (b') is not difficult to construct, but at present we do not have an application of such a gap. Let us now go back to the proof of Theorem 5.7.1 and the quotient P(IN)/0 X Fin. In the following lemma the product fi,, Pn is taken with its coordinatewise ordering, f < g if f (n)
PROOF. Assume A'(n) is bounded in An/ Fin by some An, for every n, and that A' is bounded in Al Fin x Fin by some C. Then C U Un An bounds A' in A/0 x Fin. Therefore A' cannot be separated from B over 0 x Fin, by the assumptions on An, Bn and Lemma 5.8.2. We similarly prove that B' C B is separated from A if and only if it is bounded in 8/0 x Fin, and therefore the gap A, B is as required.

LEMMA 5.8.7. If I is an analytic P-ideal which is not F,, then there is an analytic gap of type (NN, NN/ Fin) in the quotient over I.

PROOF. Let us first assume that I = 0 x Fin. By Lemma 5.8.6, since there is a (N, NN/ Fin)-gap in ?(N)/ Fin, there is a gap of type (NN, (N"/ Fin)N) in P(N2)/O x Fin. Note that this gap is analytic. To prove that P = (NN/Fin) N = NN/ Fin, note that if f, g E (NN)N then

f

n) <'

n) for all n if and only if f (m, n) < g(m, n) for all but finitely many pairs m, n

5. GAPS AND LIMITS IN ANALYTIC QUOTIENTS

160

NN>

/ Fin = NN / Fin, this completes the proof of lemma. Now let I be any analytic P-ideal which is not F. By a result of Solecki ([112]) we have 0 x Fin
proof.

5.9. Preservation of gaps The study of gaps in arbitrary analytic quotients was given a new impetus after Todorcevic ([130]) proved that gaps are frequently preserved by embeddings

of P(N)/Fin into other analytic quotients. Namely, it appears that the existence of a certain kind of analytic gap in P(N)/I is equivalent to non-preservation result of gaps by a monomorphism with a completely additive lifting of P(N)/I into a quotient over some analytic ideal (see proofs of Proposition 5.7.2 and Proposition 5.5.3). It appears that gaps can be preserved by arbitrary monomorphisms (as always, this fails under CH, since monomorphisms constructed using saturatedness of P(N)/ Fin clearly need not preserve gaps). We have an analogue of Theorem 3.4.1 in this context. PROPOSITION 5.9.1. Let I, ,7 be analytic P-ideals or Fin x 0. Then the following are equivalent:

(1) Every Baire monomorphism 4i: P(N)/I -+ P(N)/,7 preserves all (Hausdorff) gaps.

(2) OCA and MA imply that every monomorphism 4i: P(N)/I

P(N)/,7

preserves all (Hausdorff) gaps.

PROOF. (2) implies (1) is trivial. Assume (1) and let A, B be a gap in P(N)/I and let 1) be a monomorphism. By Theorem 3.3.5 (or 3.3.6), 4i is an amalgamation of a Baire homomorphism 4i1 and a homomorphism 4?2 with a a large kernel. Since 4) is a monomorphism, 4 # 4?2i so the conclusion follows from the fact that 4?1 preserves the gap A, B.

By Proposition 5.9.1 and Theorem 5.1.1, we have the following. COROLLARY 5.9.2. Under OCA and MA every monomorphism 4) of P(N)/Fin

into a quotient P(N)/I over an analytic ideal preserves all Hausdorff gaps. If I is moreover a P-ideal, then 4i preserves all gaps.

By Shoenfield's Absoluteness theorem and Proposition 5.9.1, we obtain the following (see §2.1 for a discussion of such statements).

PROPOSITION 5.9.3. For every analytic P-ideal I the statement "all gaps are preserved by embeddings of P(N)/I into other quotients over analytic P-ideals" is either false or it follows from OCA and MA.

The following (and an analogous theorem for arbitrary gaps and analytic Pideals) was proved by Todorcevic in [130], but for the convenience of the reader we will sketch the proof.

PROPOSITION 5.9.4. Let I be an analytic ideal. Then the following are equivalent:

5.10. AN ANALYTIC HAUSDORFF GAP

161

(1) Every Baire monomorphism D: P(N)/I -* P(N)/,7 preserves all Hausdorff gaps for every analytic ideal J. (2) There are no analytic Hausdorff gaps in P(N)/Z.

PROOF. Assume that (2) fails, and let A, B be an analytic Hausdorff gap in P(N)/Z. Let J be the ideal N x {0, 1} generated by (IUA) x {0} and (ZUB) x {1}. This is an analytic ideal, and if h: N x {0, 1} - N is the projection h(i,j) = i, then the 4'h-image of A, B is separated by N x {1}, therefore (1) falls. Assume that (1) fails, A, B is a Hausdorff gap whose image under a monomorphism 4': P(N)/I P(N)/,7 is split by some C, and that 4'. is a continuous lifting

of 4'. Then A' = {A:4'.(A)nCE,7}andB'={Bn4'.(B)\C E 7} is an analytic Hausdorff gap in P(N)/Z (for details see [130]).

0

5.10. An analytic Hausdorff gap In [130, page 96] Todorcevic writes that in view of the results from his paper it is natural to expect that OCA (or PFA) imply that there are only (w, w2), (w2i w) and (u)1,w1)-gaps in P(N)/T for any analytic ideal Z. (Recall that this is true for I = Fin, by a result of Todorcevic ([125], see also Theorem 5.2.4)). We have already seen in Corollary 5.8.5 that this fails in case when I is Fin x Fin. Moreover, the "definable" version also fails: There exists an analytic Hausdorff gap in P(N2)/ Fin x Fin (the gap constructed in Lemma 5.8.2 is analytic). On the other hand, in P(N)/Fin there are no analytic Hausdorff gaps ([126]). Velickovic's proof that all automorphisms of P(N)/ Fin are trivial under OCA and MA is related to

the analysis of gap-spectra of this structure under OCA. Therefore there was a hope that some elements of our analysis of homomorphisms between quotients over analytic P-ideals under OCA (Theorem 3.3.5) would help us determine gap spectra of P(N)/Z at least in the case of analytic P-ideals, and perhaps even disprove the

existence of analytic Hausdorff gaps in such quotients. We shall now see that it is not so. Let us concentrate on analytic gaps for a moment. A submeasure co is absolutely continuous with respect to v if cp(A) = 0 implies ss/.(A) = 0 for all A C N. Recall that IIAII, = lim"
(1) There are no analytic Hausdorff gaps in quotients over analytic P-ideals. (2) For all lower semicontinuous submeasures o, ?p on N there is C C N such that II - II, is absolutely continuous with respect to II IIw on C and II IIA is IIw on N \ C. absolutely continuous with respect to II

-

PROOF. Assume (2), and let A, B be or-directed I-orthogonal families, where I is an analytic P-ideal. We can assume that A, B are ideals, and therefore by a result of Solecki (see Theorem 1.2.5) there are lower semicontinuous submeasures cp and 0 such that A = Exh(cp) and B = Exh(zi). Let C be as in (2). We claim that C splits

the pregap A,B: If A E A and A \ C E A, then IIA \ CI[,, = 0, therefore by the absolute continuity IIA \ C110 = 0 as well, and A E A n B C Z. If on the other hand A is such that A n C E B, we similarly get that A n C E A n B C Z. This completes the proof that (2) implies (1). Now assume (1), and let gyp, zG be two lower semicontinuous submeasures on N.

Let (be the pointwise maximum of co, 0, let I = Exh((), A = Exh(V) and B = Exh(7b). Clearly A and B are I-orthogonal, analytic and a-directed. By (1) they can be separated, and there is C such that

5. GAPS AND LIMITS IN ANALYTIC QUOTIENTS

162

forallACN\CifAEAthenAE1,and for allACC,ifAE5then AoI. We claim that C is as required in (2). Note that n), zb(A\ n)) = max(IIAII,, IIAIIu,),

IIAIIs = lim n

o0

and therefore for A C N \ C the condition II A II, = 0 implies I I A I I S= 0 and A E 1.

Similarly if A C C then IIAII, = 0 implies IIAIIS = 0, and this concludes the proof.

Proposition 5.10.1 gives a very good hint on how to find Hausdorff gaps in quotients over analytic P-ideals, and the following result is not very surprising anymore.

THEOREM 5.10.2. There exists an F. Hausdorff gap in a quotient over some Fo

P-ideal. PROOF. Let {un} be a (disjoint) partition of N into consecutive intervals, so that Iu, i = n3 for all n. On un we define three submeasures: min(Isi,n) Wn(s) =

vn(s) =

n2 In3I,

Wn(s) = max(cpn(s), vn(n))

(Therefore vn is simply a uniform probability measure.) These sequences define lower semicontinuous submeasures on N by 00

ao

ao

cpn(A n un),

co(A) _

vn(A n un),

v(A) _

b(A)

?,b, (A n un). n=1

n=1

n=1

and the corresponding Fo-P-ideals (recall that IIAII,p = limncp(A \ [1, n]) and Exh(cp) = {A : IIAII, = 0}): A = Exh(So),

B = Exh(v),

I = Exh(o).

These ideals are Fo, for all three submeasures have the additional property cp(A)
max(cpn(A nun), vn(A n un))

max(cp(A), -(A))< O(A) _ n=1

< cp(A) + v(A).

0

Therefore, if A E A and B E B, then b((An B)) \ [1, n)) < max(cp(A \ [1, n)), v(B \ [1, n)), and this expression approaches 0 as n -> oo.

CLAIM 2. Families A and B form a gap inside the quotient P(N)/S. PROOF. It suffices to prove that for every C C N one of the following applies:

(1) C is too large, i.e., there is B C C such that B E B but B ¢ Z, or (2) C is too small, i.e., there is A C N \ C such that A E A but A 0 Z.

5.10. AN ANALYTIC HAUSDORFF GAP

163

So let us fix CC N, and let I= {n:ICflunl>n}. CASE 1. [1nEI 1/n = no. By shrinking C, we may assume that IC n unI = n for all n E I. Then the set

B=

U(Cnun).

nEI

is in B since v(B) = >nEI vn(C fl un) _< E°°_1 1/n2 < oo. On the other hand we have co(B) = EnEI (P,(C fl un) = ZnEJ 1/n = co, therefore B is not in A and (by Claim 1) not in I. CASE 2.

/--/nEI

1/n < oo. Then we can find an infinite J C N \ I such that 1

n nEJ


Let A = UnEJ(un \ C). We have co(A) = EnEJ Pn(un \ C) < EnEJ 1/n < no, and on the other hand

n3-n n3 = 00

vn(un \ C)

v(A) nEJ

nEJ

therefore A is in A \ B, and (by Claim 1) not in I. This completes the proof of Claim 2 ...

...

and Theorem 5.10.2.

The following Corollary in particular shows that OCA does not imply that the gap-spectrum of P(N)/I for the ideal I defined in Theorem 5.10.2 is equal to (w,w2), (w2,w), (w1, w1) (see the introduction to this section, also Theorem 5.8.5 and the discussion following it).

COROLLARY 5.10.3. Assume MA, and let I be the Fo P-ideal constructed in Theorem 5.10.2. Then there is a (c, c) -gap in P(N)/I.

PROOF. By Lemma 5.6.3, we can recursively construct unbounded c-chains inside the families A and B defined in Theorem 5.10.2, while taking care that the resulting pair forms a gap in the end.

Note that Claim 2 is a statement about families A and B, rather than a statement about the quotient P(N)/I (see also Proposition 5.10.1). The above family B is a summable ideal, since v is a measure, while A is equal to the first known example of a non-summable Fo P-ideal (see §1.11). It is very easy to see that we cannot hope to get both families A and B as above to be summable ideals (if v and µ are the corresponding measures, then the set C = {n : p({n}) > v({n})} separates A from B). We do not know much about the class C of all analytic P-ideals I such that in P(N)/I there are analytic Hausdorff gaps; e.g., the ideals Fin and 0 x Fin are not in this class. Although the ideal I E C constructed above is non-pathological, it seems possible (plausible?) that no summable ideals and no Erdos-Ulam ideals belong to the class C. Theorem 5.10.2 was used in [99] (see also [132, Theorem 8.4]) to construct a topological space with interesting cellularity properties.

164

5. GAPS AND LIMITS IN ANALYTIC QUOTIENTS

5.11. Limits in analytic quotients In [112] Solecki proved the equivalence of conditions (1) and (2) below. We give a simple-minded extension of his result. THEOREM 5.11.1. The following are equivalent for an analytic P-ideal Z on N:

(1) Z is not Fo, (2) 0 x Fin
Z={B : BnAn EI} (4) there is an (w, 1) -limit (Bn, D) in P(N)/I PROOF. The equivalence of (2) and (3) was established by Solecki in [114]. To prove that (2) implies (3), assume h: N -* N2 is a Rudin-Blass reduction of I to 0 x Fin, and let An = h-1({n} x N). To see that this sequence satisfies (3), let B be such that B n An E I for all n. Since I is a P-ideal, there is C E I which includes all B n An modulo finite. Then there is a finite set sn C {n} x N such that

h-1(sn) 2 (BnAn) \ C, so the set B C CUh-1(Un sn) is in I. To see that (3) implies (4), let B,n = UU1 An where {An} is as in (3). Then every C such that CnBm, is in I for all m is, by (3), in Z, so that (B,n, ICY) is the required limit. To see that (4) implies (3), let (B,n, D) be an (w, l)-limit. We can assume D = N by letting B;n = B, U D for all m, and we can also assume that all An = Bn \ Bn_1 (Bo = 0) are I-positive sets. To see

that An satisfy (3), let C be such that C n A. E I for all n. Let D be such that D D* C n An for all n. The set 00

E= U(CnAn)\D n=1

has a finite intersection with every An, so it is in I as well as C C D n E. For (3) implies (2), let hn: An - {n} x N be a Rudin-Blass reduction of I ( An to Fin guaranteed by Mathias' theorem. Then h = Un hn : N2 -+ N is a Rudin-Blass reduction of I to 0 x Fin. This completes the proof.

We should note that the implication from (1) to (4) in the case of EU-ideals (see §1.13) is implicit in [65] (see also the paragraph after Lemma 1.3.2).

5.12. A coherent family of functions Inspired by a question in cohomology theory, D. Talayco Q123]) recently proved the existence of a new kind of gap in P(N)/ Fin:

THEOREM 5.12.1. There are sets Atn (E; < w1i n E N) such that for every surjection g: N --+ {0, 1} the families

B£ = U A£n, g(n)=0

form a Hausdorff gap in P(N)/ Fin.

CC = U At, 9(n)=1

5.13. REMARKS AND QUESTIONS

165

Talayco ([123]) has also proved, using some additional assumptions, that there can be an uncountable coherent family of gaps in P(N) /Fin (see Theorem 5.12.2 below). We have extended Theorem 5.12.1 in [29], and we would now like to point out a corollary of this result and Todorcevic's gap preservation theorem (Theorem

5.1.1). Let I be an ideal of sets of integers. A family P of partial functions on integers is coherent modulo I if for all f, g E F the set of all n E dom(f) fl dom(g) such that f (n) # g(n) is in Z. Obviously, the family of all subfunctions of a given f : N --+ N is always coherent, but the following theorem shows there can be a highly nontrivial coherent family.

THEOREM 5.12.2. Let I be an analytic ideal. Then there is a family of func-

tions f : A,,, - a (A,, C N, a < w1) coherent modulo I and such that for every g: w1 -> {0, 1} which is onto, the families B. = In : g(fC(n)) = 0},

C. = In : g(fC(n)) = 1}

form a Hausdorff gap in P(N)/I. PROOF. In [29] we have proved there is such a family F of coherent functions modulo Fin. Let -(b: P(N)/Fin -> P(N)/I be a monomorphism with a Bairemeasurable lifting. The image of F under 1P is coherent, and it is nontrivial by Theorem 5.1.1.

The existence of a nontrivial coherent family of partial functions was proved in [29] in an indirect way, by using Keisler's completeness theorem for the logic L''(Q) with the additional quantifier "there exist uncountably many" ([78]) and forcing. It would be interesting to see a direct construction of such a family supported by a given arbitrary C'-increasing sequence At (l; < wl), similar to one given in [5, pages 96-98]. In this construction, due to Todorcevic, Hausdorf's gap is not constructed recursively as usual but it is given by an explicit description. QUESTION 5.12.3. Is there a direct construction of a nontrivial coherent family of partial functions as in Theorem 5.12.2? Can such a family be constructed inside an arbitrary C*-increasing w1-sequence {A,,} ?

5.13. Remarks and questions The fact that we were able to analyze gap spectra and preservation of gaps in the case of quotients over ideals 0 x Fin and Fin xm is neither surprising nor new. These two ideals were, besides Fin, the first two ideals for which the RadonNikodym property was known to be true (see §1.6). On the other hand, extending these results to a larger class of ideals (see §1.9) required a much deeper analysis involving a notion of c-approximate homomorphism (see §1.9 or [33]) connecting this area with finite combinatorics and analysis. It is therefore plausible that further analysis of gap-spectra and preservation of gaps in other analytic quotients will lead to similar discoveries and connections (as extension in another direction already did-see [37]). Let us therefore reproduce a problem first asked by Todorcevic in [130]:

PROBLEM 5.13.1. Determine the gap-spectrum of P(N)/I for every analytic ideal I on N. In particular:

5. GAPS AND LIMITS IN ANALYTIC QUOTIENTS

166

QUESTION 5.13.2. Are there analytic ideals other than 0 x Fin whose linear gap-spectra are equal to that of Fin? In particular, does the linear gaps spectrum of Fin x 0 always coincide with that of Fin? By Proposition 5.4.3 the answer to the latter question is positive under OCA. A positive answer to the following question, essentially suggested by Todorcevic in [130, page 95], would give even more support that OCA is the `right' axiom for the study of Hausdorff-completeness. QUESTION 5.13.3. Can the Hausdorff gap spectrum of every analytic quotient be determined under OCA? Another related question is whether some form of Martin's Axiom is necessary in Corollary 5.10.3. We conjecture that it is not, if Corollary 5.10.3 is stated in the right way. Recall that add(.) is the additivity of the Lebesgue measure.

CONJECTURE 5.13.4. There is an (add(.), add(.))-gap in the quotient over an Fo P-ideal defined in Theorem 5.10.2. A confirmation of this Conjecture would give a positive answer to Problem 2

of [130], where it was asked whether some standard cardinal invariants of the continuum other than w, wl and b naturally occur in the (linear) gap spectra of some analytic quotient. Note that, using some form of MA linear gaps can be found inside every analytic gap we have constructed in this Chapter (see Lemma 5.8.2, Proposition 5.8.3, Lemma 5.8.7, and Theorem 5.10.2). PROBLEM 5.13.5. Describe the class of analytic ideals I such that there are no analytic Hausdorff gaps in its quotient.

PROBLEM 5.13.6. Describe the class of analytic ideals I such that OCA (or PFA) implies that there are only (wl,w1), (w,W2), and (w2,w)-gaps in its quotient. QUESTION 5.13.7. Do the classes of ideals defined in Problem 5.13.5 and Prob-

lem 5.13.6 coincide? Do Il/n and Zo belong to either of these two classes? If there is an analytic Hausdorff gap in a quotient over I then Proposition 5.7.2

implies that I fails the TSP, therefore Fin and 0 x Fin belong to this class. But these two conditions on ideals, having the TSP and not having analytic Hausdorff gaps in its quotient, are not equivalent. This is because the ideal 0 x Fin fails the TSP (by Theorem 5.7.1) yet by Proposition 5.3.3 and Proposition 5.9.4 there are no analytic Hausdorff gaps in its quotient. QUESTION 5.13.8. Is there an analytic ideal other than Fin and Fin xO which has the TSP?

QUESTION 5.13.9. Is there some other "natural" separation property for analytic ideals which the summable ideals, or Zo, satisfy? To motivate the above question, let us note that Todorcevic has shown in [130, page 59] that the TSP of Fin has both the classical Hurewicz separation property ([56]) and the one proved by Kechris-Louveau-Woodin ([77]) as its corollaries. Therefore the discovery of some new separation phenomena in analytic quotients over the ideals closely related to a-ideals on reals like Zo (see [112]) may also lead to discoveries of some new phenomena in the realm of classical Descriptive Set Theory.

Bibliography [1] U. Abraham, M. Rubin, and S. Shelah. On the consistency of some partition theorems for continuous colorings, and the structure of lii-dense real order types. Annals of Pure and Applied Logic, 29:123-206, 1985.

[2] T. Bartoszynski. Invariants of measure and category. In M. Foreman and A. Kanamori, editors, Handbook of Set Theory. to appear. [3] J.E. Baumgartner. All R1-dense sets of reals can be isomorphic. Fundamenta Mathematicae, 79:101-106, 1973.

[4] J.E. Baumgartner. Chains and antichains in P(w). The Journal of Symbolic Logic, 45:85-92, 1980.

[5] M. Bekkali. Topics in Set Theory, volume 1476 of Lecture Notes in Mathematics. SpringerVerlag, 1991.

[6] M. Bell. an email of October 1997. [7] M. Bell. Two Boolean algebras with extreme cellular and compactness properties. Canadian Journal of Mathematics, XXXV:824-838, 1983.

[8] A. Blass. Near coherence of filters II: Applications to operator ideals, the Stone-tech remainder of a half-line, order ideals of sequences, and slenderness of groups. Transactions of the American Mathematical Society, 300:557-581, 1987. [9] A. Blass and C. Laflamme. Consistency results about filters and the number of inequivalent growth types. The Journal of Symbolic Logic, 54:50-56, 1989. [10] M. Burke. Liftings for Lebesgue measure. In Set Theory of the reads, volume 6 of Israel Mathematical Conference Proceedings, pages 119-150. 1993. [11] P. Casevitz. Dichotomies pour les espaces de suites reelles. PhD thesis, Universite Paris 6, 1999.

[12] C.C. Chang and H.J. Keisler. Model Theory. North-Holland, 1973. [13] J.P.R. Christensen. Some results with relation to the control measure problem. In R.M. Aron and S. Dineen, editors, Vector space measures and applications II, volume 645 of Lecture Notes in Mathematics, pages 27-34. Springer, 1978. [14] W.W. Comfort and S. Negrepontis. Theory of ultrafilters. Springer, 1974. [15] H.G. Dales and WE. Woodin. An Introduction to Independence for Analysts, volume 115 of London Mathematical Society Lecture Note Series. Cambridge University Press, 1987. [16] J. Diestel and J.J. Uhl, Jr. Vector measures. Amer. Math. Soc. Math. Surveys, Number 15, 1977.

[17] E.K. van Douwen. Prime mappings, number of factors and binary operations, volume 199 of Dissertationes Mathematicae. Warszawa, 1981. [18] E.K. van Douwen. The automorphism group of P(w)/ Fin need not be simple. Topology and its Applications, pages 97-104, 1990. [19] E.K. van Douwen and J. van Mill. The homeomorphism extension theorem for Ow \ w. In S. Andima at al., editors, Papers on general topology and applications, volume 704 of Ann. New York Acad. Sci., pages 345-350. 1993. [20] E.K. van Douwen, D. Monk, and M. Rubin. Some questions about Boolean algebras. Algebra Universalis, 11:220-243, 1980. [21] A. Dow. Extending real-valued functions in On. Fundamenta Mathematicae, 152:21-41, 1997. [22] A. Dow and I. Farah. Is P(w) a subalgebra? in preparation, 2001. [23] A. Dow and K.P. Hart. w' has (almost) no continuous images. Israel Journal of Mathematics, 109:29-39, 1999.

[24] A. Dow and K.P. Hart. The measure algebra does not always embed. Fundamenta Mathematicae, 163:163-176, 2000. 167

BIBLIOGRAPHY

168

[25] A. Dow, P. Simon, and J.E. Vaughan. Strong homology and the Proper Forcing Axiom. Proceedings of the American Mathematical Society, 106 (3):821-828, 1989. [26] A. Ehrenfeucht, J. Kahn, R. Maddux, and J. Mycielski. On the dependence of functions on their variables. Journal of Combinatorial Theory, ser. A, 33:106-108, 1982. [27] R. Engelking. General Topology. Heldermann, Berlin, 1989. [28] P. Erd6s. My Scottish Book "problems". In D. Mauldin, editor, The Scottish Book, pages 35-44. Birkhauser, Boston, 1981. [29] I. Farah. A coherent family of functions. Proceedings of the American Mathematical Society, 124:2845-2852,1996. [30] I. Farah. Embedding partially ordered sets into wW. Fundamenta Mathematicae, 151:53-95, 1996.

[31] I Farah. OCA and towers in P(N)/ Fin. Commentationes Mathematicae Universitatis Carolinas, 37:861-866, 1996. [32] I. Farah. Analytic ideals and their quotients. PhD thesis, University of Toronto, 1997. [33] I. Farah. Approximate homomorphisms. Combinatorica, 18:335-348, 1998. [34] I. Farah. Cauchy nets and open colorings. Publ. Inst. Math. (Beograd) (N.S.), 64(78):146152, 1998. 50th anniversary of the Mathematical Institute, Serbian Academy of Sciences and Arts (Belgrade, 1996). [35] I. Farah. Completely additive liftings. The Bulletin of Symbolic Logic, 4:37-54, 1998. [36] I. Farah. Ideals induced by Tsirelson submeasures. Fundamenta Mathematicae, 159:243-258, 1999.

[37] I. Farah. Approximate homomorphisms II: Group homomorphisms. Combinatorica, 20:4760, 2000.

[38] I. Farah. Liftings of homomorphisms between quotient structures and Ularn stability. In [39]

S. Buss, P. H£.jek, and P. Pudl£k, editors, Logic Colloquium '98, volume 13 of Lecture notes in logic, pages 173-196. A.K. Peters, 2000. I. Farah. Basis problem for turbulent actions I: Tsirelson submeasures. In Proceedings of XI Latin American Symposium in Mathematical Logic, Merida, July 1998, Annals of Pure and Applied Logic. to appear.

[40] I. Farah. Basis problem for turbulent actions II: co-equalities. Proceedings of the London

[41]

Mathematical Society, 81, to appear. I. Farah. Dimension phenomena associated with )311-spaces. Topology and its Applications, to appear.

[42] I. Farah. Functions essentially depending on at most one variable. Journal of Combinatorial

Theory, Ser. A, to appear. [43] Qi Feng. Homogeneity for open partitions of pairs of reals. Transactions of the American Mathematical Society, 339:659-684, 1993. [44] G.L. Forti. Hyers-Ulam stability of functional equations in several variables. Aequationes Math., 50:143-190, 1995. [45] D.H. Fremlin. Measure algebras. In D. Monk and R. Bonnett, editors, Handbook of Boolean algebras, pages 877-980. North-Holland, Amsterdam, 1989. [46] D.H. Fremlin. The partially ordered sets of measure theory and Tukey's ordering. Note di Matematica, Vol. XI:177-214, 1991. [47] D.H. Fremlin. Notes on FARAH P99. preprint, University of Essex, June 1999. [48] D.H. Fremlin. Embedding measure algebras in P(N)/Zo. preprint, University of Essex, May 1999.

[49] D.H. Fremlin. Problems. University of Essex, version of April 21, 1994. [50] F. Galvin. Letter of August 5, 1995. [51] J. Hadamard. Sur les caracteres de convergence des series a termes positifs et sur les fonctions indefiniment croissantes. Acta Mathematicae, 18:319-336, 1884.

[52] K.P. Hart and J. van Mill. Open problems on fiw. In J. van Mill and G.M. Reed, editors, Open problems in topology, pages 97-125. North-Holland, 1990. [53] F. Hausdorff. Die Graduierung nach dem Endverlauf. Abhandlungen der Kdniglich Sdchsischen Gesellschaft der Wissenschaften; Mathematisch-Physiche Masse, 31:296-334, 1909. [54] F. Hausdorff. Summen von b Mengen. Fundamenta Mathematicae, 26:241-255, 1936. [55] G. Hjorth and A.S. Kechris. New dichotomies for Borel equivalence relations. The Bulletin of Symbolic Logic, 3:329-346, 1997.

BIBLIOGRAPHY

169

[56) W. Hurewicz. Relativ Perfecte Teile von Punktmengen and Mengen (A). Fundamenta Mathematicae, 12:78-109, 1928.

[57] A. lonescu Tulcea and C. Ionescu Tulcea. Topics in the theory of lifting. Ergebnisse der Mathematik and ihrer Grenzgebiete, Band 48. Springer-Verlag, New York, 1969. [58) S.-A. Jalali-Naini. The monotone subsets of Cantor space, filters and descriptive set theory. PhD thesis, Oxford, 1976. [59] B Jonsson and P. Olin. Almost direct products and saturation. Compositio Mathematicae, 20:125-132; 1968. [60] W. Just. Nowhere dense P-subsets of w". Proceedings of the American Mathematical Society, 106:1145-1146, 1989.

[61] W. Just. The space (w")n+l is not always a continuous image of (w")". Pundamenta Mathematicae, 132:59-72, 1989. [62] W. Just. Repercussions on a problem of Erdos and Ulam about density ideals. Canadian Journal of Mathematics, 42:902-914, 1990. [63] W. Just. A modification of Shelah's oracle chain condition with applications. Transactions of the American Mathematical Society, 329:325-341, 1992. [64] W. Just. A weak version of AT from OCA. Mathematical Science Research Institute Publications, 26:281-291, 1992. [65] W. Just and A. Krawczyk. On certain Boolean algebras P(w)/I. Transactions of the American Mathematical Society, 285:411-429, 1984. [66] N.J. Kalton. The Maharani problem. In G. Choquet et al., editors, S6minaire Initiation a d Analyse, 28e Andee, volume 18, pages 1-13. 1988/89. [67] N.J. Kalton and J.W. Roberts. Uniformly exhaustive submeasures and nearly additive set functions. Transactions of the American Mathematical Society, 278:803-816, 1983. [68] S. Kamo. Almost coinciding families and gaps in P(w). J. Math. Soc. Japan, 45:357-368, 1993.

[69] A. Kanamori. The higher infinite. Springer-Verlag, 1995. [70] V. Kanovei and M. Reeken. New Radon-Nikodym ideals. Mathematika, to appear. [71] V. Kanovei and M. Reeken. On Baire measurable homomorphisms of quotients of the additive group of the reals. Mathematical Logic Quarterly, to appear. [72] V. Kanovei and M. Reeken. On Ulam's problem of stability of non-exact homomorphisms. Proceedings of Moscow Steklov Mathematical Institute, to appear. [73] I. Kaplansky. Normed algebras. Duke Math. Journal, 16:399-418, 1949. [74] A.S. Kechris. Classical descriptive set theory, volume 156 of Graduate texts in mathematics. Springer, 1995. [75] AS. Kechris. Rigidity properties of Borel ideals on the integers. In 8th Prague symposium on general topology and its relations to modern analysis and algebra, 1996, volume 85 of Topology and its applications, pages 195-205, 1998. [76] A.S. Kechris and A. Louveau. The structure of hypersmooth Borel equivalence relations. Journal of the American Mathematical Society, 10:215-242, 1997.

[77] A.S. Kechris, A. Louveau, and W.H. Woodin. The structure of a-ideals of compact sets. Transactions of the American Mathematical Society, 301:263-288, 1987. [78) H. J. Keisler. Logic with the quantifier "There exists uncountably many". Annals of Mathematical Logic, 1:1-93, 1970. [79] S. Koppelberg. personal communication. [80] K. Kunen. Some comments on box products. In A. Hajnal et al., editors, Infinite and finite sets, Keszthely (Hungary), 1978, volume 10 of Coll. Math. Soc. Jdnos Bolyai, pages 10111016. North-Holland, Amsterdam, 1975. [81] K. Kunen. (w, A")-gaps under MA. preprint, 1976. [82] K. Kunen. An Introduction to Independence Proofs. North-Holland, 1980. [83] K. Kunen. Where MA first fails. The Journal of Symbolic Logic, 53:429-433, 1988. [84] C. Laflamme. Forcing with filters and complete combinatorics. Annals of pure and applied logic, 42:125-163, 1989.

[85] C. Laflamme. Combinatorial aspects of F filters with an application to N-sets. Proceedings of the American Mathematical Society, 125:3019-3025, 1997. [86] C. Laflamme and J. Zhu. The Rudin-Blass ordering on ultrafilers. Journal of Symbolic Logic, 63:584-592, 1998.

BIBLIOGRAPHY

170

[87] A. Louveau and B. Velickovic. A note on Borel equivalence relations. Proceedings of the

American Mathematical Society, 120:255-259, 1994. [88] A. Louveau and B. Velickovic. Analytic ideals and cofinal types. Annals of Pure and Applied Logic, 99:171-195, 1999. [89] H. JIy3141i. 0 ^tacmrrx xamypaddrtozo psrda. I43B. AH CCCP, cepNA MaT., 11, 105:714-722, 1947.

[90] S. Mardesic and A. Prasolov. Strong homology is not additive. Transactions of the American Mathematical Society, 307:725-744, 1988. [91] D.A. Martin and R.M. Solovay. Internal Cohen extensions. Annals of Mathematical Logic, 2:143-178, 1970. [92] A.R.D. Mathias. Solution of problems of Choquet and Puritz. In Conference in Mathematical Logic-London '70, volume 255 of Springer Lecture Notes in Mathematics, pages 204-210. 1972.

[93] A.R.D. Mathias. A remark on rare filters. In A. Hajnal at al., editors, Infinite and finite sets, Vol. III, volume 10 of Coll. Math. Soc. Jhnos Bolyai, pages 1095-1097. North Holland, 1975.

[94] K. Mazur. A modification of Louveau and Velickovic construction for F ideals. preprint, 199?

[95] K. Mazur. F-ideals and wlwi-gaps in the Boolean algebra P(w)/I. Fundamenta Mathematicae, 138:103-111, 1991. [96] K. Mazur. Towards the dichotomy for Fe-ideals. preprint, 1996.

[97] J. van Mill. An introduction to Ow. In K. Kunen and J. Vaughan, editors, Handbook of Set-theoretic topology, pages 503-560. North-Holland, 1984. [98] J.D. Monk and R.M. Solovay. On the number of complete Boolean algebras. Algebra Universalis, 2:365-368, 1972. [99] J.T. Moore. Linearly fibered Souslinean space under MA. Topology Proceedings, 24, Spring 1999.

[100] MR. Oliver. The density ideal is equireducible with co. preprint, UCLA, 1997. [101] I.I. Parovicenko. A universal bicompact of weight N. Soviet Mathematics Doklady, 4:592-592, 1963.

[102] R.S. Pierce. Countable Boolean algebras. In J.D. Monk and R. Bonnett, editors, Handbook of Boolean algebras, volume 3, pages 775-886. North-Holland, Amsterdam, 1989. [103] F. Rothberger. Sur la families indenombrables de suites de nombres naturels et lea problemes concernant la propriete C. Proc. Cambridge Phil. Soc., 37:109-126, 1941. [104] W. Rudin. Homogeneity problems in the theory of tech compactifications. Duke Mathematics Journal, 23:409-419, 1956.

[105] M. Scheepers. Gaps in "w. In Set theory of the reads, volume 6 of Israel Mathematical Conference Proceedings, pages 439-561. 1993. [106] S. Shelah. Proper Forcing. Lecture Notes in Mathematics 940. Springer, 1982. [107] S. Shelah and J. Steprans. PFA implies all automorphisms are trivial. Proceedings of the American Mathematical Society, 104:1220-1225, 1988. [108] S. Shelah and J. Steprans. Somewhere trivial automorphisms. Journal of the London Mathematical Society, 49:569-580, 1994. [109] S. Shelah and J. Steprans. Martin's axiom is consistent with the existence of nowhere trivial automorphisms. preprint, Rutgers University, 2000. [110] JR. Shoenfield. The problem of predicativity. In Y. Bar-Hillel at al., editors, Essays on the Foundations of Mathematics, pages 132-142. The Magens Press, Jerusalem, 1961. [111] P. Simon. Applications of independent linked families. In Topology, theory and applications (Eger, 1983), pages 561-580. North-Holland, Amsterdam, 1985. [112] S. Solecki. Analytic ideals. The Bulletin of Symbolic Logic, 2:339-348, 1996. [113] S. Solecki. an untitled handwritten note. University of Indiana, Bloomington, 1997. [114] S. Solecki. Analytic ideals and their applications. Annals of Pure and Applied Logic, 99:5172, 1999.

[115] S. Solecki. Cofinal G6 subsets of analytic ideals of compact sets. handwritten note, University of Indiana, Bloomington, 1999. [116] S. Solecki and S. Todorcevic. Borel bases for analytic ideals. preprint, University of Toronto, 1998.

BIBLIOGRAPHY

171

[117] R. Solovay. A model of set theory in which every set of reals is Lebesgue measurable. Annals of Mathematics, 92:1-56, 1970. ' [118] R. Solovay. Discontinuous homomorphisms of Banach algebras. preprint, UC Berkeley, 1976. [119] J. Steprans. The size of the autohomeomorphism group of the Cech-Stone compactification of the integers. preprint, York University, 2000. [120] E. Szemeredi. On sets of integers containing no k elements in arithmetic progression. Acta Arithmetica, 27:199-245, 1975. [121] M. Talagrand. Compacts de fonctions mesurables et filters nonmesurables. Studio Mathematica, 67:13-43, 1980.

[122] M. Talagrand. A simple example of a pathological submeasure. Mathematische Annalen, 252:97-102, 1980.

[123] D. E. Talayco. Applications of Cohomology to Set Theory I: Hausdorff gaps. Annals of Pure and Applied Logic, 71:69-106, 1995. [124] S. Todorcevic. Directed sets and cofinal types. Transactions of the American Mathematical Society, 290:711-723, 1985. [125] S. Todorcevic. Partition Problems in Topology, volume 84 of Contemporary mathematics. American Mathematical Society, Providence, Rhode Island, 1989. [126] S. Todorcevic. Analytic gaps. Fundamenta Mathematicae, 150:55-67, 1996. [127] S. Todorcevic. Comparing the continuum with the first two uncountable cardinals. In M.L. Dalla Chiara et al., editors, Proceedings of the 10th International Congress of Logic, Methodology and Philosophy of Science, Florence 1995, pages 145-155. Kluwer Academic Publishers, 1997.

[128] S. Todorcevic. Definable ideals and gaps in their quotients. In C.A. DiPrisco et al., editors, Set Theory: Techniques and Applications, pages 213-226. Kluwer Academic Press, 1997. [129] S. Todorcevic. The first derived limit and compactly Fo-sets. J. Math. Soc. Japan, 50:831836, 1998.

[130] S. Todorcevic. Gaps in analytic quotients. Fundamenta Mathematicae, 156:85-97, 1998. [131] S. Todorcevic. Compact sets of Baire class-1 functions. Journal of the American Mathematical Society, 12:1179-1212, 1999. [132] S. Todorcevic. Chain condition methods in topology. Topology and its applications, 101:4582, 2000.

[133] S. Todorcevic. The orthogonal of every analytic P-ideal is countably generated. handwritten note, University of Toronto, July 1996. [134] S. Todorcevic. A dichotomy for P-ideals of countable sets. Fundamenta Mathematicae, to appear.

[135] S. Todorcevic and I. Farah. Some Applications of the Method of Forcing. Mathematical Institute, Belgrade and Yenisei, Moscow, 1995. [136] F. Topspe. Some remarks concerning pathological submeasures. Mathematica Scandinavica, 38:159-166, 1976.

[137] B. Tsirelson. Not every Banach space contains an embedding of fp or co. Functional Anal. Appl., 8:138-141, 1974. translated from the Russian. [138] J.W. Tukey. Convergence and uniformity in topology. Princeton University Press, 1940. [139] S.M. Ulam. Problems in modern mathematics. John Wiley & Sons, 1964. [140] S.M. Ulam and D. Mauldin. Mathematical problems and games. Advances in Applied Mathematics, 8:281-344, 1987. [141] B. Velickovic. Definable automorphisms of P(w)/Fin. Proceedings of the American Mathematical Society, 96:130-135, 1986. [142] B. Velickovic. OCA and automorphisms of P(w)/Fin. Topology and its Applications, 49:112, 1992.

[143] P. §t6pdnek and M. Rubin. Homogeneous Boolean algebras. In D. Monk and R. Bonnett, editors, Handbook of Boolean algebras, volume 2, pages 679-715. North-Holland, Amsterdam, 1989.

[144] W. Weiss. Partitioning topological spaces. In J. Nesetfil and V. Rodl, editors, Mathematics of Ramsey Theory, volume 5 of Algorithms and Combinatorics, pages 154-171. SpringerVerlag, Berlin, 1990. [145] W.H. Woodin. E'-absoluteness. Note of May 1985. [146] W.H. Woodin. The Axiom of Determinacy, forcing axioms and the nonstationary ideal, volume 1 of de Gruyter Series in Logic and Its Applications. de Gruyter, 1999.

Index of partial functions indexed by A, 60 of partitions indexed by NN, 63 trivial, 64

additive lifting, 1 agree below k, 101 N1-dense, 88

coloring, 55 open, 55 comeager, 91 completely additive lifting, 1 conjecture Kechris-Mazur, 51, 53 Todorcevic, 14

Ni-directed, 60 almost bijection, 123 almost disjoint, 7 almost disjoint sets, 73 almost equal, 7 almost includes, 7 almost lifting, 73, 132 almost refines, 135 amalgamation of homomorphisms, 12, 71 approximate homomorphism, 3 approximation 5-approximation for H, 21 E-approximation on X, 93 asymptotic density zero, 9 asymptotically additive, 16

version for arbitrary connecting maps,

114

weaker form of, 51 van Douwen's, 120, 123

countably saturated, 75 countably separated, 10 families, 77 pregap, 144

AT, 141 (d,1)-perfect, 125

automorphism group, 78

d-bounded subset of a(d), 132 d-compact, 117 5-approximation for H, 21 dense ideal, 41 density ideal, 6 density ideal generated by a sequence of measures, 42 depends on at most one coordinate, 118 depends on at most one variable, 118 directed, 60 directed, 60 a-directed, 60 discrete sequence, 72 downwards closure, 91 dual filter, 16

Baire, see also Baire-measurable Baire-measurable, 12 basic question, 1, 5, 74 /3w-space, 120

Boolean algebra dual Hopfian, 82 homogeneous, 83 Hopfian, 82 weakly homogeneous, 83 Borel cardinality, 12 bounding number, 61 ccc over Fin, 73

ccc over ,7, 132 closed approximation to an ideal, 9

elementary function, 123

cofinal, 144 cofinality, 144 cofinally finer, 153 cofinally similar, 153 coherent family

piecewise, 123 embeddable, 76 e-approximate epimorphism, 52 homomorphism, 21 E-approximation on X, 93 E-tame, 93 eventual dominance, 61 everywhere nonmeager, 94 Extension Principle, 117

v-directed, 66 indexed by A, 60 indexed by IAN, 61

modulo Fin, 85 modulo Z, 66, 165 a nontrivial one, 165 172

INDEX

173

strong, 141

1-positive, 16

weak, 117

ideal analytic, 7

f-density zero, 8 family, see also coherent family neat, 99 a-trivial, 59, 62 tree-like, 99 trivial, 59, 62 finite-to-one reduction, 12 Fr6chet ideal, 7 generalized, 53 Fr6chet property, 11, 83 Fubini product of two ideals, 8 function Baire-measurable, 12 elementary, 123 Erdbs-Ulam function, 42 incompatible, 59 induced by h, 139 monotonic, 20 perfect, 124 R-measurable, 125 a-Borel, 101 subadditive, 20

analytic P-ideals, 5 atomic, 36 co-like, 51

ccc over J, 132 ccc over Fin, 73

countably generated, 74

dense, 11, 41 dense summable, 34 density, 6 density ideal, 42

OxFin, 8 Erdos-Ulam, 8, 42 EUf, see also ideal, Erdds-Ulam EU-ideals, 8 F P-ideal which is not summable, 33, see also [36]

f-ideal, see also ideal, Erdos-Ulam Fin x Fin, 157 Fin xO, 8 Frechet, 7

generalized Fr6chet ideal, 53 homogeneous, 53, 85 2

gap, see also nonlinear gap, 4, 76, 143 analytic ([]R]<", Q)-gap, 156 (b, b)-gap, 158 coherent, 165

F, Hausdorff gap, 162 Hausdorff, 145 reflects, 148

in P(N)/Z, 144 linear, 144, 154 Luzin, 145 of type (P, Q), 153 (P, Q)-gap, 153 Rothberger, 145 gap spectrum, 144 Hausdorff, 144 linear, 144 groupwise density, 92

Hausdorff gaps in the quotient over Fin x 0 reflect, 148 hereditary, 9, 86, 91 closure, 91 homogeneous, 83, 122 Ks-homogeneous, 55, 122 weakly, 83 homomorphism

e-approximate, 21 approximate, 3 Baire, 5 non-exact, 3 Hyers-Ulam stability, 25 .T-orthogonal, 144

11/,<, 1

I, , see also ideal, ordinal If, see also ideal, summable isomorphic, 10

Z®9, 8

Ix,7,8 Louveau-Velickovic, 31 Mazur, 53, 154

non-pathological analytic P-ideal, 25 nontrivial, 10 of d-bounded subsets of a(d), 132 of asymptotic density zero sets, 9 of logarithmic density zero sets, 9 ordinal ideal, 53 topological ordinal ideal, 54 P-ideal, 7 pathological analytic P-ideal, 25 summable, 8 tall, 11 Weiss, 54

Sµ, see also ideal, density Z1,g, 9 EC), 9

incompatible, 59 indexed by A, 62 induced by h, 139 induces, 125 K;-homogeneous, 55, 122

lemma Prime Mapping Lemma, 120 Todorcevic, 11

174

lifting, 1, 11 additive, 1 almost lifting, 73 asymptotically additive, 16 Baire-measurable, 5 completely additive, 1

of 4? on J, 98 logarithmic density zero, 9, 42 lower semicontinuous submeasure, 7

INDEX

K,-, 100 L,-, 107 open, 55 Velickovic, 100

pathological, see also ideal submeasure, 21 perfect mapping, 124 piecewise elementary, 123 Polish space, 135 pregap

meager, 11, 91

non-exact homomorphism, 3 non-pathological Fo ideal, 53 analytic P-ideal, 25 submeasure, 21 nonlinear gap Hausdorff, 154 Luzin, 145 Rothberger, 154 nonmeager, 12 everywhere, 94

countably separated, 144

in 2(N)/Z, 144 separated, 144

Principle of Open Coloring, see also Open Coloring Axiom, 63 Property of Baire, 12 question basic, 1 Bell's, 141 Dow's, 142 Just-Krawczyk, 84 72-measurable, 125

OCA, 55, see also Open Coloring Axiom OCA lifting theorem, 69 (w, 1)-limits, 144 open coloring, 55, 63 Open Coloring Axiom, 55 the reflection component of, 56 the evolution of, 56 open partition, 55 oracle-chain condition, 68 ordering Baire embeddability, 12 Bore! cardinality, 12 dense, 34 finite-to-one reduction, 6 Rudin-Blass, 12 Rudin-Keisler, 12 Tukey, 153

ordinal additively indecomposable, see also indecomposable indecomposable, 53, 124 multiplicatively indecomposable, 54 orthogonal, 10, 36, 144 Z-orthogonal, 144 of an ideal, 76 submeasures, 42 P-ideal, 7 P-set, 77 weak, 142 partial ordering directed, 60

o-directed, 60

partition isomorphic, 55

Radon-Nikodym property, 5, 15 of 0 x Fin, 31 of Fin, 19

of Fin x 0, 19 of density ideals, 42

of Erdds-Ulan ideals, 31 of generalized ideals, 53 of Louveau-Velickovic ideals, 31 of Mazur ideals, 53 of non-pathological F.-ideals, 53 of non-pathological P-ideals, 31 of summable ideals, 31 of Weiss ideals, 54 rectangle, 62 reduced product, 157 reduction, see also ordering

refines, 135

restriction, 147 saturated quotient, 56

scale, 150 Schroder-Bernstein property, 41

sEP, see also strong Extension Principle, 141 separated, 76 separates, 144 sequential fan, 139 set tij-dense, 88 almost disjoint, 73 cofinal, 144 comeager, 91 d-compact, 117 e-tame, 93 everywhere nonmeager, 94 groupwise dense, 92

INDEX

hereditary, 9, 86, 91 homogeneous, 122 1-positive, 16 K;-homogeneous, 55, 122 meager, 11, 91 nonmeager, 12 P-set, 77 weak, 142 a-Ks-homogeneous, 55 2-1-small, 120 sides (of a gap), 153 a-Ks-homogeneous, 55 a-Borel function, 101 a-directed, 60 subset of P(N), 60 a-trivial, 59 space 3w-space, 120 Parovicenko, 3 Polish, 135 Stone, 132 Tsirelson, 34 stability program of Ulam, 3

stabilizer, 17 (n, n')-e-stabilizer, 95 (n, n')-stabilizer, 17 stable, 3 non-exact homomorphisms, 21 Stone space, 132 subadditive, 20 submeasure, 20 absolutely continuous, 161 Erdos-Ulam submeasure, 42 Fhbini, 53 lower semicontinuous, 7 non-pathological, 21 norm of, 21 orthogonal, 42 pathological, 21 supported by, 7 sum of two ideals, 8 summable ideal, 8 superperfect, see also tree theorem Dow-Hart, 117 Jalali-Naini, Talagrand, 92 Jankov, von Neumann, 93 Just, 76 Just-Krawczyk, 115 Kanovei-Reeken, 53 Kechris, 47, 52 Keisler's completeness theorem, 165 lifting theorem for non-pathological analytic P-ideals, 31 Mathias, 12 Mazur, 9 non-lifting theorem, 31 OCA lifting theorem, 74

175

OCA uniformization theorem, 64 Olin, 14 Parovicenko, 117 Shelah, 68 Shoenfield's absoluteness theorem, 57 E2-reflection for coherent families, 65 Solecki, 9, 52 Todorcevic, 10, 32, 76, 143, 145, 153 Tukey, 153

W. Rudin, 140 Woodin, 58 Todorcevic separation property, 151 tree

downwards closed, 102 superperfect, 151 B-tree, 151 trivial autohomeomorphism of (w*)idl, 123 automorphism, 75 coherent family, 64 family, 62 family of functions, 59 isomorphism, 16 trivializes, 59, 62 .F on En., 64

TSP, see also Todorcevic separation property Tukey map, 153 2-1-small, 120 type

of a linear gap, 144 of a nonlinear gap, 153 strong, 154 uncountable rectangle of compatible conditions, 88, 111 of incompatible conditions, 88, 111 uniformization, 55, 67, 136 weak Extension Principle, 117 for X, 139 wEP, 117 wEP(c,ry), 117 wEP(X), 140 working part, 88 zero density, 41

Index of special symbols 11/n .................................1

'Py ................................. 16

CP . .................................. 1

rf ................................. 19

Z1/.. ...............................2

supp(wp) ............................ 21

'Dh .................................. 5

III,II ................................ 21

If ...................................6

cp ...................................21

0 .................................... 7

P(So) ............................... 21

1IAII. ................................7

K,o ................................. 21

Exh(v) .............................. 7

µf ................................ 21

Fin(w) ............................... 7

so ..................................22

A 1 B ............................... 7

A[c, C] ............................. 35

AD* B ..............................7

35

A = B .............................. 7

A(., C] ..............................35

AAB ................................ 7

Z1 ................................. 36

A C_z B ............................. 7

af k .................................. 38

A =z B ............................. 7 A. .................................. 8

Z1 .r .............................. 41 µf ................................ 42

Z e j ............................... 8

(pf ................................. 42

Z x 9 ............................... 8

EUf ................................ 42

Z [ A ................................ 8

at+(W) ............................. 42

V00 .................................. 8

at-(W) .............................. 42

3. ..................................8

pµ ................................. 42

Fin x O ............................... 8

Sµ ................................. 42

0 x Fin .............................. 8

Fµ ................................. 48

N .................................... 8

Gµ6 ................................ 48

EUf ................................. 8

{0,1}
"pf ................................... 8

Jbr ................................. 50

zo ...................................9

[t]

.................................. 50

Zlog .................................9

Za ................................. 53

JVM ...............................10 f"X ............................... 10

P(a) ............................... 54 [X]2 ................................ 55

Z1 .................................10

112,111 ............................. 58

P(N)/Z .............................11

f < g ............................... 61

[A]z ................................ 11

f <. g ............................. 61

4'. ................................. 11

b ...................................61

7rT,7rj ............................. 11

&r(gf,gf') ......................... 62

Z

U ® V .............................. 62 [A]
Z
E2 ................................. 65

Z
4D ® lB ..............................71

Z

Z <_BC 9 ........................... 12

P(N)/Z ti P(M/,7 .................75

4> ®T ..............................12

Z <EM .7 ........................... 76

d, (A, B) ...........................

14

If .................................

76

7T ...................................15

Aut(P(N)/I) ....................... 78

Z. ..................................16

S00 ................................. 79

Z+ ................................. 16

Gf ................................. 79 176

INDEX OF SPECIAL SYMBOLS Hf ................................. 79

A ................................. 140

A ................................... 80

A 1.1 5 ........................... 144

.7cont ............................... 85

A 1 B ............................. 144

hA ................................. 85

cf(P) .............................. 144

.7i ................................. 85

All ............................... 144

9A ................................. 85

C_................................. 144

Lo(D) .............................. 86

b .................................. 145

rf ................................. 87

A I C ............................. 147

k(A) ............................... 87

rf ................................ 148

P .................................. 87

c .................................. 150

P, 1 ................................89

C ................................. 151

X .................................. 91

G(A, B)

[s; m] ............................... 91

P < Q ............................ 153

........................... 152

IAIIw ...............................92

Pi = P2 ...........................153

Exh(w) ............................. 92

N ................................. 153

d, (A, B) ........................... 92

All ............................... 153

.7cont ............................... 99

13-L /I ............................. 154 [II2]<.

A D B ............................ 99 {0,1}
............................. 156

Exh(
A ................................. 100

P,/ Fin ....................... 157

Ao ................................ 100

IL lp* ............................ 159 L' (Q) ............................ 165

Ko , K1 ........................... 100 (a,b) I k = (a', V) I k .............. 101

.70 ................................103 ,7cont ..............................103 ............................106

CI q

.71 ................................ 106 .72 ................................ 106

Ka ................................ 106 x(b) ............................... 108 i(d,a) ............................. 108 k(d,a) ............................. 108

Ta ................................ 108 Y. ................................. 108

P ................................. 110

PW1 ...............................112 D ................................. 117 w .................................. 117

X(d) .............................. 117 Rd . ................................117

Rd ................................117 X -.. Y ............................ 117 (Lfx)r(x) .......................... 118

AA, [A]2, (A)2 ................... 121 [A]3 ............................... 122

{1,m,n}< ......................... 122 G ................................. 131 c1(a) .............................. 131

(a) ................................131 (a) (d) .............................132 cl(X) .............................. 132

K(X) ............................. 132 Stone(S) .......................... 132 ir :.................................134

A -< B ............................ 135

A <` B ...........................135 S(m) .............................. 139

add(N) ........................... 166

177

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*Real and harmonic analysis and geometric partial differential equations to WILLIAM

BECKNER, Department of Mathematics, University of Texas, Austin, TX 78712-1082; email:

[email protected] All other communications to the editors should be addressed to the Managing Editor, WILLIAM BECKNER, Department of Mathematics, University of Texas, Austin, TX 787121082; email: beckner@math. utexas . edu.