Tomek Bartoszynski and Saharon Shelah- Perfectly Meager Sets and Universally Null Sets

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PERFECTLY MEAGER SETS AND UNIVERSALLY NULL SETS

TOMEK BARTOSZYNSKI AND SAHARON SHELAH

Abstract. We will show that there is no ZFC example of a set distinguishingbetween universally null and perfectly meager sets.

1. Introduction

Consider the following three families of sets of reals:

Definition 1. Let X R.

(1) X is perfectly meager if for every perfect set P R, P X is meager in P.(2) X is universally meager if every Borel isomorphic image of X is meager(3) X is universal null if every Borel isomorphic image of X has Lebesgue

measure zero.

Let PM, UM and UN denote these families respectively.

The family UM was studied recently by Zakrzewski [13], and identified as ananalog of UN.

One gets an equivalent definition of UN by replacing Borel isomorphic byhomeomorphic, but this is not the case with UM.

Let M and N denote the -ideals of meager and of measure zero subsets of thereals, respectively.

For a -ideal J P(R) let

non(J) = min{|X| : X R & X J }.

There are many ZFC examples of uncountable sets that are in UM UN. Theseinclude 1

1-gaps, a selector from the constituents of a non-Borel

11 set, etc. (see

[9]) All these sets have size 1, since Miller [8] showed that, consistently, no set ofsize 20 is in UM UN.

Grzegorek found other constructions in ZFC that produce sets of (consistently)different sizes.

Theorem 2 (Grzegorek, [6]).

(1) There exists a set X UN such that |X| = non(N),(2) There exists a set X UM such that |X| = non(M).

The problem whether the equality UM = UN is consistent is open. However,both inclusions are consistent with ZFC; UM UN holds in a model obtained by

1991 Mathematics Subject Classification. 03E17.Key words and phrases. perfectly meager, universally null, consistency.The first author was partially supported by NSF grant DMS 9971282 and Alexander von

Humboldt Foundation.The second author was partially supported by Israel Science Foundation. Publication 732.

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2 TOMEK BARTOSZYNSKI AND SAHARON SHELAH

adding 2 Cohen reals, and UN UM holds in a model obtained by adding 2random reals (side-by-side) (see [4], [9], [8]).

In this paper we investigate the connection between families UN and PM, andshow that both inclusions PM UN and UN PM are consistent with ZFCas well. Observe that trivially UM PM, thus we only need to check thatPM UN is consistent. Recall that PM = UM is consistent ([12]) as well asPM = UM ([2]). We will show that:

Theorem 3. It is consistent with ZFC that PM [R]1 UN.

2. Forcing

Suppose that X 2 is a perfectly meager set in V. Let P be a fixed closedsubset of 2 2 which is universal for perfect sets in 2. In other words, forevery perfect set P 2 there exists an x such that P = (

P)x = {y : (x, y)

P}.

Note that this property is absolute. Since X is perfectly meager, we can find setsQn 2 2 such that for every x 2 and n ,(1) (Qn)x is a closed nowhere dense subset of ( P)x,(2) X ( P)x n Qn

x.

Clearly, the setn

Qn witnesses that X PM sinceX 2 \

x2

( P \ n

Qn)x.Note that the last inclusion makes sense even ifX is not a subset of V. Suppose

that V V and X V is a set of reals. We will say V |= X PM if there

exists a family {

Qn : n } V such that X (

P)x (

n

Qn)x for every real

x V.

The property of being perfectly meager is not absolute so whether X is perfectlymeager in V has no bearing onto whether X is perfectly meager in V. For example,if x V is a Cohen real over V then the set {x} is perfectly meager in V but notin V.

Lemma 4. LetP, Q : < 2} be a countable support iteration of proper forcingnotions over V |= CH. Suppose thatX VP2 R is a perfectly meager set. Thenthere exists an 1-club C 2 such that for every C,

VP |= X PM.

Proof. Let {Qn : n } VP2 be a family witnessing that X is perfectly meager.Let C consist of those ordinals of cofinality 1 that for every n,

Qn

(2 VP)

2 VP . The usual argument involving Skolem-Lowenheim theorem shows thatC has the required property.

Our objective is to find a set of general conditions on a forcing notion P suchthat the countable support iteration of P of length 2 produces a model wherePM [R]1 UN. These conditions are sufficient for the class of forcingnotions defined using norms [10].

These conditions are the following:

(1) VP |= V 2 N,(2) VP |= V 2 M,


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PERFECTLY MEAGER SETS AND UNIVERSALLY NULL SETS 3

(3) P is -bounding, that is V is a dominating family in VP,(4) P adds a real xP 2 such that V |= {xP} PM.

(5) P generic real is minimal, that is, if g is P-generic over V and x V[g] 2

then x V or g V[x].

Condition (1) is necessary to make all sets of size 1 universally null, and condi-tion (2) is necessary to avoid making all 1 sets perfectly meager. Recall that (2)and (3) together are essentially equivalent to VP |= V M is cofinal in M.

For the forcing notions P that we have in mind the following property holds: forevery real x VP there exists a continuous function f V such that x = f(xG),where xG is a generic real.

Condition (5), guarantees that in the above context f can be chosen to be ahomeomorphism. In particular, if X is a set of reals of size 2 then X will containa homeomorphic image of a sequence of generic reals.

The following forcing notion appeared in [5], it is similar (but not identical) to

the infinitely equal real forcing from [7].For a tree p and t p, let succp(t) be the set of all immediate successors of tin p, pt = {v p : t v or v t} the subtree of p determined by t, pn the n-thlevel of p, and let [p] be the set of branches of p. By identifying s


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Note that for p EE the set [p] is a compact subset of X =

n f(n). Moreover,there is a canonical isomorphism between [p] and 2 defined as follows:

For every n let {sn0 , . . . , s

nf(n)} be a fixed enumeration of 0-1 sequences of length

f(n) (recall that f(n) = 2f(n)). Define F : [p] 2 as

F(x) = sn0x(n0+1)

sn1x(n1+1)

. . . ,

where n0, n1, . . . is the increasing enumeration of the set {n : xn split(p)}.

Lemma 6. Let p EE and suppose that H [p] is a meager set in [p]. For everyn there exists q n p such that [q] H = . In particular, EE V |= {g} PM.

Proof. Let H [p] be a meager set, and let n . Fix a descending sequence ofopen sets Uk : k such that each Uk is dense in [p] and H

k Uk = . By

induction build a sequence pk : k such that p0 = p, and for every k,

(1) pk+1 n+k+1 pk EE,(2) [pk+1] Uk.

Suppose that pk is given. For every v splitn+k+1(pk) find qv (pk)v such that[qv] Uk. Let pk+1 =

{qv : v splitn+k+1(pk)}. Condition q = limk pk has the

required property.

Suppose that {Qn : n } V is a possible witness that {g} is perfectlymeager, and let p EE. Find x V such that [p] = (P)x and let q p be such

that [q]

nQn

x= . Clearly,

q EE {g} xV

P \

n

Qn

x

.

In particular, q EE V |= {g} PM. Lemma 7. Suppose that p EE and p EE x 2. For every n there existsq n p and a continuous function F : [q] 2 such that q EE x = F(g), whereg is the canonical name for the generic real.

Moreover, we can require that for every v splitn(q) and any x1, x2 [qv],F(x1)n = F(x2)n.

Proof. The first part is a special case of a more general fact. For n let An EEbe a maximal antichain below p such that r An s 2n r EE xn = s. Use5(5) to find q p such that for every n , {r An : r is compatible with q} isfinite. Let An = {r An : r is compatible with q}. Without loss of generality wecan assume that [q]

rAn

[r]. It follows that [r] [q] is clopen in [q] for every

r A

n. Define F : [q] 2

as F(x) = y if for every n there exists r A

nsuch that x [r] and r EE xn = yn. It is easy to see that F is a continuousfunction that has the required properties.

To show the second part we need to build q in such a way that for every v splitn(q), there is r A

n such that qv r.

Lemma 8. Suppose that p EE, n and p EE x 2. Let F : [q] 2 be acontinuous function such that p EE x = F(g).

There exists q p such that F[q] is constant, or there exists q n p such thatF[q] is one-to-one. In particular, the generic real is minimal.


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Proof. Consider the following two cases.Case 1 p EE x V. Let x V and q p be such that q EE x = x. Clearly

F[q] is constant with value x.

Case 2 p EE x V.Build by induction a sequence of conditions pk : k such that p0 = p and

for every k,

(1) pk+1 n+k+1 pk,

(2) sets

F

(pk+1)s

: s splitn+k+1(pk+1)

are pairwise disjoint and have

diameter < 2k

Suppose that pk is given. Note that F

[(pk)s]

is uncountable for every s pk.

For v splitn+k+1(pk) choose pairwise different reals xv F

[(pk)v ]

. It is notimportant now but will be relevant in the sequel, that we can choose these reals

effectively from a fixed countable subset of [pk]. Let > k be such that sequencesxv are also pairwise different. For every v splitn+k+1(pk) let sv split(pk) besuch that for every z [(pk)sv ], F(z) = xv. If F is as in the second partof lemma 7 then we can find sv in split(pk). Define pk+1 =

{(pk)sv : v

splitn+k+1(pk)}. Observe that q = limk pk has the required property.

Note that the above lemma shows that the reals added by EE are minimal.Infinitely equal forcing from [7] or [4] does not have this property.

3. Iteration of EE.

Let 2 be an ordinal and suppose that EE is a countable support iterationofEE of length . In other words, p EE is

(1) p is a function and dom(p) = ,(2) supp(p) = { : p() = } is countable,(3) < pEE p() EE.

For F []


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The countable support iteration P, Q : < sup(w) such that

< sup(w) P Q EE if w if w .It is clear that EEw EEp . Moreover, we can view condition p as a member ofEEw.

For the rest of the section we will consider only the iteration ofEE of countablelength and show that EE has the same properties that EE has.

Let be a countable ordinal and p EE. Define p X as follows:x : < p if for every < ,

x p()

x : <

.

Note that p()[x : < ] is the interpretation of p() using reals x : < so may be undefined if these reals are not sufficiently generic.

For a set G X, u , and x Xu let

(G)x = {y X\u : z G zu = x & z( \ u) = y},

and for let (G) = {x() : x G}.We say that p EE is good if

(1) p is compact,

(2) for every < and x p, p[x] = (p)x and p()[x] = ((p)x) .(3) p is homeomorphic to X via a homeomorphism h such that for every <

and x p, h((p)x) is a homeomorphism between ((p)x) and X.

Lemma 10. {p EE : p is good} is dense inEE.

Proof. Case 1. = + 1.Fix p EE and for n let An be a maximal antichain below p such that

(1) r An r is compact.(2) r An t

j


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(3)n Fn = ,

(4) supn n = .

By induction build a sequence qn : n such that for n ,(1) qn EE,(2) supp(qn) n,(3) qn+1 Fn,n qn,(4) qnn pn,(5) qnn is compact in X

n .

Let q = limn qn. Note that q =n qnn X

\n is as required.

From now on we will always work with conditions p such that p is good. Wenoticed earlier that for every condition p EE, [p] is canonically isomorphic to 2,in exactly the same way we can verify that if p EE and p is good then p isisomorphic to (2).

As in the lemma 7 we show that:Lemma 11. Suppose that p EE and p EE x 2

. Then there exists q pand a continuous function F : p 2 such that q EE x = F(g), where g =g : < is the sequence of generic reals.

Lemma 12. Let p EE and suppose that H p is a meager set in p. For everyF []


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(3)n Fn = ,

(4) supn n = .

By induction build a sequence qn : n such that for n ,(1) qn EE,(2) supp(qn) n,(3) qn+1 Fn,n qn,(4) qnn pn,

(5) qnn Hn = , where Hn =

x qnn : (H)x is not meager in p[x]

.

As before (5) is possible by Kuratowski-Ulam theorem. Let q = limn qn. It is clearthat q H = .

The following lemma is an analog of lemma 8.

Lemma 14. Suppose that p EE, n and p EE x 2. Let F : p 2

be a continuous function such that p EE x = F(g), where g = g : < is thesequence of generic reals. There exists q p such that exactly one of the followingconditions hold:

(1) Fq is constant,(2) there exists < such that Fq is one-to-one and for every x q,

F(q)x is constant,(3) Fq is one-to-one.

Proof. We have three cases:Case 1. There exists q p such that q EE x V. Without loss of generality

we can assume that for some x V 2 q EE x = x. It follows that Fq isconstant.

Case 2. There exists q p such that q EE < x VEE

. By shrinkingq we can assume that there exists a continuous function G : q 2 such thatq EE x = G(g). In particular, for x [q], F(x) = G(x). If was minimalthen, using the argument below, we can also assume that G is one-to-one.

Suppose that q EE, F []


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of p. Let > k be such that sequences xv are also pairwise different. Defineconditions rj : j k, qj : j k such that for every j k,

(1) rj EE ,(2) rj+1 rj ,(3) rj EE qj (p)vj [, ),(4) z rjqj, F(z) = F(xvj ).

Let q = qk and q[, ) =j


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By thinning out we can assume that ot(w) = , F = F and p = p. Moreover,since V |= CH, we can assume that w w = w for = . Finally, without loss

of generality we can assume that w

= .Let P = F(p). Since F is a homeomorphism, P is perfect. We will show that

X P is not meager in VEE2 (relative to P).Assume otherwise and let H P be a meager set such that for some p EE2 ,

p EE2 X P H. By 9(4) we can assume that H V. Set G = (F)1(H) and

notice that G is a meager subset of p.Find < 2 such that w cl(p) = . By lemma 12 there exists q p,

q EEw EE such that q G = .Since p and q are compatible let r p, q. It follows that

r EE2 x = F(q : w) H,

which finishes the proof.Acknowledgments: The work was done while the first author was spending his

sabbatical year at the Rutgers University and the College of Staten Island, CUNY,and their support is gratefully acknowledged.

References

[1] Tomek Bartoszynski. Measure and Category Invariants. In M. Magidor M.Foreman,A. Kanamori, editor, Handbook of Set Theory. submitted.

[2] Tomek Bartoszynski. On perfectly meager sets. available from front.math.ucdavis.edu.

[3] Tomek Bartoszynski and Haim Judah. Set Theory: on the structure of the real line. A.K.Peters, 1995.

[4] Paul Corazza. The generalized Borel conjecture and strongly proper orders. Transactions ofthe American Mathematical Society, 316(1):115140, 1989.

[5] Martin Goldstern, Haim Judah, and Saharon Shelah. Strong measure zero sets without Cohenreals. The Journal of Symbolic Logic, 58(4):13231341, 1993.

[6] Edward Grzegorek. Solution of a problem of Banach on -fields without continuous measures.

LAcademie Polonaise des Sciences. Bulletin. Serie des Sciences Mathematiques, 28(1-2):710, 1980.

[7] Arnold W. Miller. Some properties of measure and category. Transactions of the AmericanMathematical Society, 266(1):93114, 1981.

[8] Arnold W. Miller. Mapping a set of reals onto the reals. The Journal of Symbolic Logic,48(3):575584, 1983.

[9] Arnold W. Miller. Special subsets of the real line. In K. Kunen and J. E. Vaughan, editors,Handbook of Set Theoretic Topology, pages 201235. North-Holland, Amsterdam, 1984.

[10] Andrzej Roslanowski and Saharon Shelah. Norms on possibilities I: forcing with trees andcreatures. Memoirs of the American Mathematical Society. American Mathematical Society,1999.

[11] Saharon Shelah. Proper and Improper Forcing. Perspectives in Logic. Springer-Verlag, 1998.[12] Waclaw Sierpinski. Sur une probleme de M.Kuratowski concernant la propriete de Baire des

ensambles. Fundamenta Mathematicae, 22:262266, 1934.[13] Piotr Zakrzewski. Universally meager sets. to appear in Proceedings of the American Math-

ematical Society.

Department of Mathematics and Computer Science, Boise State University, Boise,

Idaho 83725 U.S.A.

E-mail address: [email protected], http://math.boisestate.edu/~tomek

Department of Mathematics, Hebrew University, Jerusalem, Israel and Department

of Mathematics, Rutgers University, New Brunswick, NJ

E-mail address: [email protected], http://math.rutgers.edu/~shelah/

Tomek Bartoszynski and Saharon Shelah- Perfectly Meager Sets and Universally Null Sets

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