Journal of Applied Analysis Vol. 6, No. 2 (2000), pp. 159–172

CATEGORY ANALOGUE OF SUP-MEASURABILITY PROBLEM

K. CIESIELSKI AND S. SHELAH

Received September 3, 1999 and, in revised form, April 11, 2000

Abstract. A function F : ℝ^{2} → ℝ is called sup-measurable if F_{f}: ℝ → ℝ given by F_{f} (x) = F (x, f(x)), x ∈ ℝ, is measurable for each measurable function f: ℝ → ℝ. It is known that under different set theoretical assumptions, including CH, there are sup-measurable non-measurable functions, as well as their category analogues. In this paper we will show that the existence of the category analogues of supmeasurable non-measurable functions is independent of ZFC. A similar result for the original measurable case is the subject of a work in prepartion by Rosłanowski and Shelah.

1. Introduction

Our terminology is standard and follows that from [3], [4], [10], or [12]. In particular, pr: X × Y → X will stand for the projection onto the first coordinate. A subset A of a Polish space X is nowhere meager provided A ∩ U is not meager for every non-empty open subset of X.

1991 Mathematics Subject Classification. Primary: 03E35, 26A15; Secondary: 26A30, 26B40.

Key words and phrases. Baire property, composition of functions, sup-measurable functions.

The work of the first author was partially supported by NSF Cooperative Research Grant INT-9600548, with its Polish part financed by Polish Academy of Science PAN. The work of the second author was supported in part by a grant from “Basic Research Foundation” founded by the Israel Academy of Sciences and Humanities. Publication 695.

ISSN 1425-6908 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag.

The ternary Cantor subset of ℝ will be identified with with its homeomorphic copy, 2^{ω}, which stands for the set of all function x: ω → {0, 1} considered with the product topology. In particular, the basic open subsets of 2^{ω} are in the form

[s] =^{def} {x ∈ 2^{ω}: s ⊂ f},

where s ∈ 2^{<ω}. Also, since ℝ \ ℚ is homeomorphic to 2^{ω} \ E for some countable set E (the set of all eventually constant functions in 2^{ω}) in our more technical part of the paper we will be able replace ℝ with 2^{ω}.

The study of sup-measurable functions^{1} comes from the theory of differential equations. More precisely it comes from the question: For which functions F: ℝ^{2} → ℝ does the Cauchy problem

y′ = F (x, y), y(x_{0}) = y_{0}  (1)

have a (unique) a.e.-solution in the class of locally absolutely continuous functions on ℝ in the sense that y(x_{0}) = y_{0} and y′(x) = F (x, y(x)) for almost all x ∈ ℝ? (For more on this motivation see [8] or [2]. Compare also [9].) It is not hard to find measurable functions which are not sup-measurable. (See [13] or [1, Corollary 1.4].) Under the continuum hypothesis CH or some weaker set-theoretical assumptions nonmeasurable sup-measurable functions were constructed in [6], [7], [1], and [8]. The independence from ZFC of the existence of such an example is the subject of a work in prepartion by Rosłanowski and Shelah.

A function F: ℝ^{2} → ℝ is a category analogue of sup-measurable function (or Baire sup-measurable) provided F_{f}: ℝ → ℝ given by F_{f}(x) = F (x, f(x)), x ∈ ℝ, has the Baire property for each function f: ℝ → ℝ with the Baire property. A Baire sup-measurable function without the Baire property has been constructed under CH in [5]. (See also [1] and [2].) The main goal of this paper is to show that the existence of such functions cannot be proved in ZFC. For this we need the following easy fact. (See [1, Proposition 1.5].)

Proposition 1. The following conditions are equivalent.

(i) There is a Baire sup-measurable function F: ℝ^{2} → ℝ without the Baire property.

(ii) There is a function F: ℝ^{2} → ℝ without the Baire property such that F_{f} has the Baire property for every Borel function f: ℝ → ℝ.

(iii) There is a set A ⊂ ℝ^{2} without the Baire property such that the projection pr(A ∩ f) = {x ∈ ℝ: ⟨x, f(x)⟩ ∈ A} has the Baire property for each Borel function f: ℝ → ℝ.

1This is abbreviation from superposition-measurable function.

(iv) There is a Baire sup-measurable function F: ℝ^{2} → {0, 1} without the Baire property.

The equivalence of (i) and (ii) follows from the fact that the function F: ℝ^{2} → ℝ is Baire sup-measurable if and only if F_{f} has a Baire property for every Borel function f: ℝ → ℝ.^{2}

The main theorem of the paper is the following.

Theorem 2. It is consistent with the set theory ZFC that

φ: for every A ⊂ 2^{ω} × 2^{ω} for which the sets A and A^{c} = (2^{ω} × 2^{ω}) \ A are nowhere meager in 2^{ω} × 2^{ω} there exists a homeomorphism f from 2^{ω} onto 2^{ω} such that the set pr(A ∩ f) does not have the Baire property in 2^{ω}.

Before proving this theorem let us notice that it implies easily the following corollary.

Corollary 3. The existence of Baire sup-measurable function F: ℝ^{2} → ℝ without the Baire property is independent from the set theory ZFC.

Proof. As mentioned above under CH there exist Baire sup-measurable functions without the Baire property. So, it is enough to show that the property φ from Theorem 2 implies that there are no such functions.

So, take an arbitrary A ⊂ ℝ^{2} without the Baire property. By (iii) of Proposition 1 it is enough to show there exists a Borel function f: ℝ → ℝ for which the set pr(A ∩ f) does not have the Baire property.

We will first show this under the additional assumption that the sets A and ℝ^{2} \ A are nowhere meager in ℝ^{2}. But then the set A_{0} = A ∩ (ℝ \ ℚ)^{2} and its complement are nowhere meager in (ℝ \ ℚ)^{2}. Moreover, since ℝ \ ℚ is homeomorphic to 2^{ω} \E for some countable set E we can consider A_{0} as a subset of (2^{ω} \E)^{2} ⊂ 2^{ω} ×2^{ω}. Then A_{0} and its complement are still nowhere meager in 2^{ω} × 2^{ω}. Therefore, by φ, there exists an autohomeomorphism f of 2^{ω} such that the set pr(A_{0} ∩ f) = {x ∈ 2^{ω} \ E: ⟨x, f(x)⟩ ∈ A_{0}} does not have the Baire property in 2^{ω}. Now, as before, f ↾ (2^{ω}\E) can be considered as defined on ℝ \ ℚ. So if f̄: ℝ → ℝ is an extension of f ↾ (2^{ω} \ E) (under such identification) to ℝ which is constant on ℚ, then f̄ is Borel and the set pr(A_{0} ∩ f̄) does not have the Baire property in ℝ.

2It is also true that F : ℝ^{2} → ℝ is Baire sup-measurable provided F_{f} has the Baire property for every Baire class one function f: ℝ → ℝ, and that F : ℝ^{2} → ℝ is supmeasurable provided F_{f} is measurable for every continuous function f: ℝ → ℝ. See for example [2, Lemma 1 and Remark 1].

Now, if A is an arbitrary subset of ℝ^{2} without the Baire property we can find non-empty open intervals U and W in ℝ such that A and (U × W) \ A are nowhere meager in U × W. Since U and W are homeomorphic with ℝ the above case implies the existence of Borel function f_{0} : U → W such that pr(A ∩ f_{0}) does not have the Baire property in U. So any Borel extension f: ℝ → ℝ of f_{0} works.

2. Reduction of the proof of Theorem 2 to the main lemma

The theorem will be proved by the method of iterated forcing, a knowledge of which is needed from this point on.

The idea of the proof is quite simple. For every nowhere meager subset A of 2^{ω} ×2^{ω} for which A^{c} = (2^{ω} ×2^{ω})\A is also nowhere meager we will find a natural ccc forcing notion Q_{A} which adds the required homeomorphism f. Then we will start with the constructible universe V = L and iterate with finite support these notions of forcing in such a way that every nowhere meager set A^{∗} ⊂ 2^{ω} × 2^{ω}, with (2^{ω} × 2^{ω}) \ A^{∗} nowhere meager, will be taken care of by some Q_{A} at an appropriate step of iteration.

There are two technical problems with carrying through this idea. First is that we cannot possibly list in our iteration all nowhere meager subsets of 2^{ω} × 2^{ω} with nowhere meager complements since the iteration can be of length at most continuum 𝔠 and there are 2^{𝔠} such sets. This problem will be solved by defining our iteration as P_{ω_{2}} = ⟨⟨P_{α}, Q^{.}_{α}⟩: α < ω_{2}⟩ such that the generic extension V [G] of V with respect to P_{ω_{2}} will satisfy 2^{ω} = 2^{ω_{1}} = ω_{2} and have the property that

(m) every non-Baire subset A^{∗} of 2^{ω} contains a non-Baire subset A of cardinality ω_{1}.

Thus in the iteration we will use only the forcing notions Q_{α} = Q_{A} for the sets A of cardinality ω_{1}, whose number is equal to ω_{2}, the length of iteration. Condition (m) will guarantee that this will give us enough control of all nowhere meager subsets A^{∗} of 2^{ω} × 2^{ω}.

The second problem is that even if at some stage α < ω_{2} of our iteration we will add a homeomorphism f appropriate for a given set A ⊂ 2^{ω} × 2^{ω}, that is such that

V [G_{α}] ⊧ “ pr(A ∩ f) is not Baire in 2^{ω},”

where G_{α} = G ∩ P_{α}, then in general there is no guarantee that the set pr(A∩f) will remain non-Baire in the final model V [G]. The preservation of non-Baireness of each appropriate set pr(A∩f) will be achieved by carefully crafting our iteration following a method known as the oracle-cc forcing iteration.

The theory of the oracle-cc forcings is described in details in [12, Chapter IV] (compare also [11, Chapter IV]) and here we will recall only the fragments that are relevant to our specific situation. In particular if

Γ =^{def} {λ < ω_{1} : λ is a limit ordinal}

then

• an ω_{1}-oracle is any sequence M = ⟨M_{δ}: δ ∈ Γ⟩ where M_{δ} is a countable transitive model of ZFC^{−} that is, ZFC without the power set axiom) with a property that δ + 1 ⊂ M_{δ}, M_{δ} ⊧ “δ is countable,” and the set {δ ∈ Γ: A ∩ δ ∈ M_{δ}} is stationary in ω_{1} for every A ⊂ ω_{1}.

The existence of an ω_{1}-oracle is equivalent to the diamond principle ♢.

With each ω_{1}-oracle M = ⟨M_{δ}: δ ∈ Γ⟩ there is associated a filter D_{M} generated by the sets I_{M}(A) = {δ ∈ Γ: A ∩ δ ∈ M_{δ}} for A ⊂ ω_{1}. It is proved in [12, Claim 1.4] that D_{M} is a proper normal filter containing every closed unbounded subset of Γ.

We will also need the following fact which, for our purposes, can be viewed as a definition of M-cc property.

Fact 4. Let P be a forcing notion of cardinality ≤ ω_{1}, e: P → ω_{1} be oneto-one, and M = ⟨M_{δ}: δ ∈ Γ⟩ be an ω_{1}-oracle. If there exists a C ∈ D_{M} such that for every δ ∈ Γ ∩ C

e^{−1}(E) is predense in P for every set E ∈ M_{δ} ∩ P(δ), for which

e^{−1}(E) is predense in e^{−1}({γ: γ < δ}),

then P has the M-cc property.

This follows immediately from the definition of M-cc property [12, Definition 1.5, p. 150].

Our proof will rely on the following main lemma.

Lemma 5. For every A ⊂ 2^{ω} × 2^{ω} for which A and A^{c} = (2^{ω} × 2^{ω}) \ A are nowhere meager in 2^{ω} × 2^{ω} and for every ω_{1}-oracle M there exists an M-cc forcing notion Q_{A} of cardinality ω_{1} such that Q_{A} forces

there exists an autohomeomorphism f of 2^{ω} such that the sets pr(f ∩ A) and pr(f \ A) are nowhere meager in 2^{ω}.

The proof of Lemma 5 represents the core of our argument and will be presented in the next section. In the remainder of this section we will sketch how Lemma 5 implies Theorem 2. Since this follows the standard path, as described in [12, Chapter IV], the readers familiar with this treatment may proceed directly to the next section.

First of all, to define an appropriate iteration we will treat forcings Q_{A} from Lemma 5 as defined on ω_{1}. More precisely, in the iteration we will always replace Q_{A} with its order isomorphic copy ⟨ω_{1}, ≤_{A} ⟩. So, we can treat any finite support iteration P_{α} =⟨⟨P_{β}, Q^{.}_{β}⟩: β <α⟩ of Q_{A} forcing notions as having an absolute and fixed universe, say U_{α} = {g ∈ (ω_{1})^{ω_{2}} : g^{−1}(ω_{1} \ {0}) ∈ [α]^{<ω}} . This will allow us to treat the ♢_{ω_{2}}-sequence ⟨X_{α}: α < ω_{2}⟩ as a sequence of P_{α}-names of subsets of 2^{ω} × 2^{ω}. (After appropriate coding.)

We will also need the following variant of [12, Example 2.2].

Lemma 6. Assume that ♢_{ω_{1}} holds and that S ⊂ 2^{ω} is such that S and S^{c} are nowhere meager in 2^{ω}. Then there exists an ω_{1}-oracle M such that if P is an arbitrary M-cc forcing then P forces that

S and S^{c} are nowhere meager in 2^{ω}.

Proof. By [12, Example 2.2] for any non-empty basic open set W of 2^{ω} there are oracles M_{W}^{0} and M_{W}^{1} such any M_{W}^{0} -cc forcing forces that S ∩ W is not meager, and any M_{W}^{1} -cc forcing forces that S^{c} ∩ W is not meager. So, by [12, Claim 3.1], there is a single ω_{1}-oracle M which “extends” all oracles M_{W}^{i}, and it clearly does the job.

Now, the iteration P_{ω_{2}} is defined by choosing by induction the sequence

⟨⟨P_{α}, A^{.}_{α}, M^{.}_{α}, Q^{.}_{α}, f^{.}_{α}⟩: α < ω_{2}⟩ such that for every α < ω_{2}

(a) P_{α} = ⟨⟨P_{β}, Q^{.}_{β}⟩: β < α⟩ is a finite support iteration,

(b) A^{.}_{α} is a P_{α}-name for which P_{α} forces that A^{.}_{α} and (A^{.}_{α})^{c} are nowhere meager subsets of 2^{ω} × 2^{ω},

(c) M^{.}_{α} is a P_{α}-name for which P_{α} forces that

M^{.}_{α} is an ω_{1}-oracle and for every Q^{.} satisfying M^{.}_{α}-cc we have (i) for every β < α if P_{α} = P_{β} ∗ P^{.}_{β,α} then

P_{β} ⊩ “P^{.}_{β,α} ∗ Q^{.} is M^{.}_{β}-cc,”

(ii) if α = γ + 1 then

Q^{.} ⊩ “ pr(f^{.}_{γ} ∩ A^{.}_{γ}), pr(f^{.}_{γ} \ A^{.}_{γ}) ⊂ 2^{ω} are nowhere meager in 2^{ω}”,

(d) Q^{.}_{α} is a P_{α}-name for a forcing such that P_{α} forces Q^{.}_{α} is an M^{.}_{α}-cc forcing Q_{A^{.}}_{α} from Lemma 5,

(e) f^{.}_{α} is a P_{α+1}-name for which P_{α+1} forces that f^{.}_{α} is a Q^{.}_{α}-name for the function f from Lemma 5.

If for some α < ω_{2} the sequence ⟨⟨P_{β}, A^{.}_{β}, M^{.}_{β}, Q^{.}_{β}, f^{.}_{β}⟩: β < α⟩ has been

defined then we proceed as follows. Forcing P_{α} is already determined by (a). We choose A^{.}_{α} as X_{α} from the ♢_{ω_{2}}-sequence if it satisfies (b) and arbitrarily, still maintaining (b), otherwise. Since steps (d) and (e) are facilitated by Lemma 5, it is enough to construct M^{.}_{α} satisfying (c). For this we will consider two cases.

Case 1: α is a limit ordinal.

For a moment fix a β < α and work in V^{P_{β}}. Let M_{β} and P_{β,α} be the interpretations of M^{.}_{β} and P^{.}_{β,α}, respectively. By the inductive assumption for every β < γ < α forcing P_{β,γ} is M_{β}-cc. So, by [12, Claim 3.2], P_{β,α} is M_{β}-cc. Thus, by [12, Claim 3.3], in (V^{P_{β}})^{P_{β,α}} = V^{P_{α}} there is an ω_{1}-oracle M_{β}^{∗} such that if Q is M_{β}^{∗}-cc then P_{β,α} ∗ Q is M_{β}-cc.

So, in V^{P_{α}}, we have ω_{1}-oracles M_{β}^{∗} for every β < α. Thus, by [12, Claim 3.1], in V^{P_{α}} there exists an ω_{1}-oracle M_{α} which is stronger than all M_{β}^{∗}’s in a sense that if Q is M_{α}-cc then Q is also M_{β}^{∗}-cc. So, there is a

P_{α}-name M^{.}_{α} for M_{α} for which (c) holds.

Case 2: α is a successor ordinal, α = γ + 1. Then P_{α} = P_{γ} ∗ Q^{.}_{γ}.

Since, by (d), P_{γ} forces that Q^{.}_{γ} is M^{.}_{γ}-cc, using (c) for α = γ we conclude that

P_{β} ⊩ “P^{.}_{β,α} is M^{.}_{β}-cc”

for every β < γ. So, proceeding as in Case 1, in V^{P_{α}} we can find ω_{1}-oracles M^{.}_{β}^{∗} such that

P_{β} ⊩ “P^{.}_{β,α} ∗ Q^{.} is M^{.}_{β}-cc”

for every Q which is M^{.}_{β}^{∗}-cc. Let M be an ω_{1}-oracle from Lemma 6 used

with S = pr(f^{.}_{γ} ∩ A^{.}_{γ}). As above we can find, in V^{P_{α}}, an ω_{1}-oracle M_{α}

which is stronger than all M_{β}^{∗}’s and M. Then, there is a P_{α}-name M^{.}_{α} for M_{α} for which (c) holds. This finishes the construction of the iteration.

To finish the argument first note that the interpretations of pr(f^{.}_{α} ∩ A^{.}_{α}) and pr(f^{.}_{α} \A^{.}_{α}) in the final model V [G] remain nowhere meager in 2^{ω}. This is the case since, by (e), P_{α+1} forces that

pr(f^{.}_{α} ∩ A^{.}_{α}) and pr(f^{.}_{α} \ A^{.}_{α}) are nowhere meager in 2^{ω},

and, by (c)(i), that

every P^{.}_{α+1,γ} is M^{.}_{α+1}-cc

while, by condition (c)(ii), every M^{.}_{α+1}-cc forcing preserves nowhere meagerness of pr(f^{.}_{α}∩A^{.}_{α}) and pr(f^{.}_{α}\A^{.}_{α}). To finish this part of the argument it is enough to note that P_{α+1} forces that “P^{.}_{α+1,ω_{2}} is M^{.}_{α+1}-cc” which follows from [12, Claim 3.2].

To complete the argument it is enough to show that each nowhere meager subset A^{∗} of 2^{ω} × 2^{ω} from V [G] with nowhere meager complement contains an interpretation of some A^{.}_{α}. However, P_{ω_{2}} is ccc. So, if A^{.} is a P_{ω_{2}}-name for A^{∗} then the set

{α ∈ Γ: P_{α} ⊩ A^{.} ∩ V^{P_{α}} is nowhere meager in 2^{ω} × 2^{ω}}

contains a closed unbounded subset of Γ. Thus ♢_{ω_{2}} guarantees that A^{∗} contains an interpretation of some A^{.}_{α}.

3. Proof of Lemma 5

Let K be the family of all sequences h̄ = ⟨h_{ξ}: ξ ∈ Γ⟩ such that each h_{ξ} is a function from a countable set D_{ξ} ⊂ 2^{ω} onto R_{ξ} ⊂ 2^{ω} and that

D_{ξ} ∩ D_{η} = R_{ξ} ∩ R_{η} = ∅ for every distinct ξ, η ∈ Γ.

For each h̄ ∈ K we will define a forcing notion Qh̄. Forcing Q_{A} satisfying Lemma 5 will be chosen as Qh̄ for some h̄ ∈ K.

So fix an h̄ ∈ K. Then Qh̄ is defined as the set of all triples p = ⟨n, π, h⟩ for which (A) h is a function from a finite subset D of ⋃_{ξ∈Γ} D_{ξ} into 2^{ω}; (B) n < ω and π is a permutation of 2^{n}; (C) |D ∩ D_{ξ}| ≤ 1 for every ξ ∈ Γ; (D) if x ∈ D ∩ D_{ξ} then h(x) = h_{ξ}(x) and h(x) ↾ n = π(x ↾ n). Forcing Qh̄ is ordered as follows. Condition p′ = ⟨n′, π′, h′⟩ is stronger than p = ⟨n, π, h⟩, p′ ≤ p, provided

n ≤ n′, h ⊂ h′, and π′(s) ↾ n = π(s ↾ n) for every s ∈ 2^{n′}. (2)

Note that the second part of (D) says that for every x ∈ D and s ∈ 2^{n}

x ∈ [s] if and only if h(x) ∈ [π(s)].  (3)

Also, if n < ω we will write [s] ↾ 2^{n} for {x ↾ 2^{n}: x ∈ [s]}. Note that in this notation the part of (2) concerning permutations says that π′ extends π in a sense that π′ maps [t] ↾ 2^{n′} onto [π(t)] ↾ 2^{n′} for every t ∈ 2^{n}.

In what follows we will use the following basic property of Qh̄. (∗) For every q = ⟨n, π, h⟩ ∈ Qh̄ and m < ω there exist an n′ ≥ m and a

permutation π′ of 2^{n′} such that q′ = ⟨n′, π′, h⟩ ∈ Qh̄ and q′ extends q. The choice of such n′ and π′ is easy. First pick n′ ≥ max{m, n} such

that x ↾ n′ ≠ y ↾ n′ for every different x and y from either domain D or range R = h[D] of h. This implies that for every t ∈ 2^{n} the set D_{t} = {x ↾ n′: x ∈ D ∩ [t]} ⊂ [t] ↾ 2^{n′} has the same cardinality as D ∩ [t] and H_{t} = {x ↾ n′: x ∈ h[D] ∩ [π(t)]} ⊂ [π(t)] ↾ 2^{n′} has the same cardinality as h[D] ∩ [π(t)]. Since, by (3), we have also |D ∩ [t]| = |h[D] ∩ [π(t)]| we see that |D_{t}| = |H_{t}|. Define π′ on D_{t} by π′(x ↾ n′) = h(x) ↾ n′ for every x ∈ D_{t}. Then π′ is a bijection from D_{t} onto H_{t} and this definition ensures that an appropriate part of the condition (D) for h and π′ is satisfied. Also, if for each t ∈ 2^{n} we extend π′ onto [t] ↾ 2^{n′} as a bijection from ([t] ↾ 2^{n′}) \ D_{t} onto ([π(t)] ↾ 2^{n′}) \ H_{t}, then the condition (2) will be satisfied. Thus such defined q′ = ⟨n′, π′, h⟩ belongs to Qh̄ and extends q.

Next note that forcing Qh̄ has the following property, described in Fact 7, needed to prove Lemma 5. In what follows we will consider 2^{ω} with the standard distance:

d(r_{0}, r_{1}) = 2^{− min{n<ω: r_{0}(n)≠r_{1}(n)}}

for different r_{0}, r_{1} ∈ 2^{ω}.

Fact 7. Let h̄ = ⟨h_{ξ}: ξ ∈ Γ⟩ ∈ K and f = ⋃{h: ⟨n, π, h⟩ ∈ H}, where H is a V -generic filter over Qh̄. Then f is a uniformly continuous one-to-one function from a subset D of 2^{ω} into 2^{ω}. Moreover, if for every ξ ∈ Γ the graph of h_{ξ} is dense in 2^{ω} × 2^{ω}, then D and f[D] are dense in 2^{ω} and f can be uniquely extended to an autohomeomorphism f̃ of 2^{ω}.

Proof. Clearly f is a one-to-one function from a subset D of 2^{ω} into 2^{ω}. To see that it is uniformly continuous choose an ε > 0. We will find δ > 0 such that r_{0}, r_{1} ∈ D and d(r_{0}, r_{1}) < δ imply d(f(r_{0}), f(r_{1})) < ε. For this note that, by (∗), the set

S = {q = ⟨n, π, h⟩ ∈ Qh̄: 2^{−n} < ε}

is dense in Qh̄. So take a q = ⟨n, π, h⟩ ∈ H ∩ S and put δ = 2^{−n}. We claim that this δ works.

Indeed, take r_{0}, r_{1} ∈ D such that d(r_{0}, r_{1}) < δ. Then there exists a q′ = ⟨n′, π′, h′⟩ ∈ H stronger than q such that r_{0} and r_{1} are in the domain of h′. Therefore, n ≤ n′ and for j < 2

f(r_{j}) ↾ n = h′(r_{j}) ↾ n = (h′(r_{j}) ↾ n′) ↾ n = π′(r_{j} ↾ n′) ↾ n = π(r_{j} ↾ n)

by the conditions (D) and (2). Since d(r_{0}, r_{1}) < δ = 2^{−n} implies that r_{0} ↾ n = r_{1} ↾ n we obtain

f(r_{0}) ↾ n = π(r_{0} ↾ n) = π(r_{1} ↾ n) = f(r_{1}) ↾ n

that is, d(f(r_{0}), f(r_{1})) ≤ 2^{−n} < ε. So f is uniformly continuous.

Essentially the same argument (with the same values of ε and δ) shows that f^{−1} : f[D] → D is uniformly continuous. Thus, if f̃ is the unique continuous extension of f into cl(D), then f̃ is a homeomorphism from cl(D) onto cl(f[D]).

To finish the argument assume that all functions h_{ξ} have dense graphs, take a t ∈ 2^{m} for some m < ω, and notice that the set

S_{t} = {q = ⟨n, π, h⟩ ∈ Qh̄: the domain D′ of h intersects [t]}

is dense in Qh̄. Indeed, if q = ⟨n, π, h⟩ ∈ Qh̄ then, by (∗), strengthening q if necessary, we can assume that m ≤ n. Then, refining t if necessary, we can also assume that m = n, that is, that t is in the domain of π. Now, if [t] intersects the domain of h, then already q belongs to S_{t}. Otherwise take ξ ∈ Γ with D′ ∩ D_{ξ} = ∅ and pick ⟨x, h_{ξ}(x)⟩ ∈ [t] × [π(t)], which exists by the density of the graph of h_{ξ}. Then ⟨n, π, h ∪ {⟨x, h_{ξ}(x)⟩}⟩ belongs to S_{t} and extends q.

This shows that D ∩ [t] ≠ ∅ for every t ∈ 2^{<ω}, that is, D is dense in 2^{ω}. A similar argument shows that for every t ∈ 2^{<ω} the set

S^{t} = {q = ⟨n, π, h⟩ ∈ Qh̄: the range of h intersects [t]}

is dense in Qh̄, which implies that h[D] is dense in 2^{ω}. Thus f̃ is a homeomorphism from cl(D) = 2^{ω} onto cl(h[D]) = 2^{ω}.

Now take A ⊂ 2^{ω} × 2^{ω} for which A and A^{c} = (2^{ω} × 2^{ω}) \ A are nowhere meager in 2^{ω} × 2^{ω} and fix an ω_{1}-oracle M = ⟨M_{δ}: δ ∈ Γ⟩. By Fact 7 in order to prove Lemma 5 it is enough to find an h̄ = ⟨h_{ξ}: ξ ∈ Γ⟩ ∈ K such that

Q_{A} = Qh̄ is M-cc  (4)

and Qh̄ forces that, in V [H],

the sets pr(f ∩ A) and pr(f \ A) are nowhere meager in 2^{ω}.  (5)

(In (5) function f is defined as in Fact 7.)

To define h̄ we will construct a sequence ⟨⟨x_{α}, y_{α}⟩ ∈ 2^{ω} × 2^{ω}: α < ω_{1}⟩ aiming at h_{ξ} = {⟨x_{ξ+n}, y_{ξ+n}⟩: n < ω}, where ξ ∈ Γ.

Let {⟨s_{n}, t_{n}⟩: n < ω} be an enumeration of 2^{<ω} × 2^{<ω} with each pair ⟨s, t⟩ appearing for an odd n and for an even n. Points ⟨x_{ξ+n}, y_{ξ+n}⟩ are chosen inductively in such a way that

(i) ⟨x_{ξ+n}, y_{ξ+n}⟩ is a Cohen real over M_{δ}[⟨⟨x_{α}, y_{α}⟩: α < ξ + n⟩] for every δ ≤ ξ, δ ∈ Γ, that is, ⟨x_{ξ+n}, y_{ξ+n}⟩ is outside all meager subsets of 2^{ω} × 2^{ω} which are coded in M_{δ}[⟨⟨x_{α}, y_{α}⟩: α < ξ + n⟩];

(ii) ⟨x_{ξ+n}, y_{ξ+n}⟩ ∈ A if n is even, and ⟨x_{ξ+n}, y_{ξ+n}⟩ ∈ A^{c} otherwise. (iii) ⟨x_{ξ+n}, y_{ξ+n}⟩ ∈ [s_{n}] × [t_{n}]. The choice of ⟨x_{ξ+n}, y_{ξ+n}⟩ is possible since both sets A and A^{c} are nowhere meager, and we consider each time only countably many meager sets. Condition (iii) guarantees that the graph of each of h_{ξ} will be dense in 2^{ω} × 2^{ω}.

Note that if Γ ∋ δ ≤ α_{0} < ⋯ < α_{k−1}, where k < ω, then (by the product lemma in M_{δ})

⟨⟨x_{α_{i}}, y_{α_{i}}⟩: i < k⟩ is an M_{δ}-generic Cohen real in (2^{ω} × 2^{ω})^{k}. (6)

For q = ⟨n, π, h⟩ ∈ Qh̄ define

q̂ = ⋃_{⟨s,t⟩∈π}[s] × [t].

Clearly q̂ is an open subset of 2^{ω} × 2^{ω} and condition (2) implies that for every q, r ∈ Qh̄ with r = ⟨n′, π′, h′⟩

if q ≤ r then q̂ ⊂ r̂ and q̂ ∩ ([s] × [t]) ≠ ∅ for every ⟨s, t⟩ ∈ π′. (7)

Also for δ ∈ Γ let (Qh̄)^{δ} = {⟨n, π, h⟩ ∈ Qh̄: h ⊂ ⋃_{ζ<δ} h_{ζ}}. To prove (4) and (5) we will use also the following fact.

Fact 8. Let δ ∈ Γ and let E ∈ M_{δ} be a predense subset of (Qh̄)^{δ}. Then for every k < ω and p = ⟨n, π, h⟩ ∈ (Qh̄)^{δ} the set

B_{p}^{k} = ⋃ {(q̂)^{k}: q extends p and some q_{0} ∈ E}  (8)

is dense in (p̂)^{k} ⊂ (2^{ω} × 2^{ω})^{k}.

Proof. By way of contradiction assume that B_{p}^{k} is not dense in (p̂)^{k}. Then there exist m < ω and s_{0}, t_{0}, …, s_{k−1}, t_{k−1} ∈ 2^{m} with the property that P = ∏_{i<k}([s_{i}] × [t_{i}]) ⊂ (p̂)^{k} is disjoint from B_{p}^{k}. Increasing m and refining the s_{i}’s and t_{j}’s, if necessary, we may assume that m ≥ n, all s_{i}’s and t_{j}’s are different, ⋃_{i<k}[s_{i}] is disjoint from the domain D of h, and h[D]∩⋃_{i<k}[t_{i}] = ∅. We can also assume that x ↾ m ≠ y ↾ m for every different x and y from D and from h[D]. Now, refining slightly the argument for (∗) we can find r = ⟨m, π′, h⟩ ∈ (Qh̄)^{δ} extending p such that π′(s_{i}) = t_{i} for every i < k. (Note that P ⊂ (p̂)^{k}.) We will obtain a contradiction with the predensity of E in (Qh̄)^{δ} by showing that r is incompatible with every element of E.

Indeed if q were an extension of r ≤ p and an element q_{0} of E, then we would have (q̂)^{k} ⊂ B_{p}^{k}. But then, by (7) and the fact that ⟨s_{i}, t_{i}⟩ ∈ π′ for i < k, we would also have (q̂)^{k} ∩ P ≠ ∅, contradicting P ∩ B_{p}^{k} = ∅. This finishes the proof of Fact 8.

Now we are ready to prove (4), that is, that Qh̄ is M-cc. So, fix a bijection e: Qh̄ → ω_{1} and let

C = {δ ∈ Γ: (Qh̄)^{δ} = e^{−1}(δ) ∈ M_{δ}} .

Then C ∈ D_{M}. (Just use a suitable nice codding or [12, Claim 1.4(4)].) Take a δ ∈ C and fix an E ⊂ δ, E ∈ M_{δ}, for which e^{−1}(E) is predense in (Qh̄)^{δ}. By Fact 4 it is enough to show that

e^{−1}(E) is predense in Qh̄.

Take p_{0} = ⟨n, π, h_{0}⟩ from Qh̄, let h = h_{0} ↾ ⋃_{η<δ} D_{η} and h_{1} = h_{0} \ h, and notice that the condition p = ⟨n, π, h⟩ belongs to (Qh̄)^{δ}. Assume that h_{1} = {⟨x_{i}, y_{i}⟩: i < k}. Since s(h_{1}) = ⟨⟨x_{i}, y_{i}⟩: i < k⟩ ∈ (p̂)^{k}, B_{p}^{k} ∈ M_{δ} (as defined from (Qh̄)^{δ} ∈ M_{δ}) and, by Fact 8, B_{p}^{k} is dense in (p̂)^{k} condition (6) implies that s(h_{1}) ∈ B_{p}^{k}. So there are q = ⟨n_{0}, π_{0}, g⟩ ∈ (Qh̄)^{δ} extending p and some q_{0} ∈ e^{−1}(E) for which s(h_{1}) ∈ q̂^{k}. But then p′ = ⟨n_{0}, π_{0}, g ∪ h_{1}⟩ belongs to Qh̄ and extends q. This finishes the proof of (4).

The proof of (5) is similar. We will prove only that pr(f \A) = pr(f ∩A^{c}) is nowhere meager in 2^{ω}, the argument for pr(f ∩ A) being essentially the same.

By way of contradiction assume that pr(f \ A) is not nowhere meager in 2^{ω}. So there exists an s^{∗} ∈ 2^{<ω} such that pr(f \ A) is meager in [s^{∗}]. Let a condition p^{∗} ∈ Qh̄ and Qh̄-names U^{.}_{m}, for m < ω, be such

p^{∗} ⊩_{Qh̄} each U^{.}_{m} is an open dense subset of [s^{∗}] and pr(f\A)∩ ⋂_{m<ω} U^{.}_{m} = ∅.

For each m < ω, since p^{∗} forces that U^{.}_{m} is an open dense subset of [s^{∗}], for every t ∈ 2^{<ω} extending s^{∗} there is a maximal antichain ⟨p_{s,k}^{m}: k < κ_{s}^{m}⟩ in

Qh̄ forcing that U^{.}_{m} ∩ [t] contains some basic open subset [s].

Note that each of these antichains must be countable, since the forcing notion Qh̄ is M-cc and therefore ccc. Combining all these antichains we get a sequence ⟨p_{s,k}^{m} ∈ Qh̄: m < ω, s ∈ 2^{<ω}, k < κ_{s}^{m}⟩ such that

• κ_{s}^{m} ≤ ω,

• p_{s,k}^{m} ⊩_{Qh̄} [s] ⊆ U^{.}_{m},

• for every q ∈ Qh̄ extending p^{∗} and t ∈ 2^{<ω} extending s^{∗} there are s ∈ 2^{<ω} and k < κ_{s}^{m} such that the conditions q and p_{s,k}^{m} are compatible and t ⊂ s.

Note that for sufficiently large δ ∈ Γ we have p_{s,k}^{m} ∈ (Qh̄)^{δ} for all m < ω, s ∈ 2^{<ω}, and k < κ_{s}^{m}.

Now, by the definition of ω_{1}-oracle, the set B_{0} of all δ ∈ Γ for which

⟨p_{s,k}^{m} ∈ Qh̄: m < ω, s ∈ 2^{<ω}, k < κ_{s}^{m}⟩ ∈ M_{δ} and (Qh̄)^{δ} ∈ M_{δ}

is stationary in ω_{1}. (Just use a suitable nice coding, or see [12, Chapter IV, Claim 1.4(4)]). Thus, using clause (iii) of the choice of x_{ξ}’s, we may find a δ ∈ B_{0}, an odd j < ω, and a condition p_{0} = ⟨n_{0}, π_{0}, h_{0}⟩ ∈ Qh̄ such that • p_{0} ≤ p^{∗}, s^{∗} ⊂ x_{δ+j}, and

• x_{δ+j} belongs to the domain of h_{0}.

Then p_{0} ⊩ “x_{δ+j} ∈ [s^{∗}]∩pr(f \A)” (remember j is odd so ⟨x_{δ+j}, y_{δ+j}⟩ ∈ A^{c}). We will show that

p_{0} ⊩ x_{δ+j} ∈ ⋂_{m<ω} U^{.}_{m},

which will finish the proof.

So, assume that this is not the case. Then there exist an i < ω and a p_{1} = ⟨n, π, h_{1}⟩ ∈ Qh̄ stronger than p_{0} such that p_{1} ⊩ “x_{δ+j} ∉ U^{.}_{i}.” Let us define h = h_{1} ↾ {x_{α} : α < δ} and h_{1} \ h = {⟨a_{l}, b_{l}⟩: l < m}. Notice that the condition p = ⟨n, π, h⟩ belongs to (Qh̄)^{δ}. We can also assume that ⟨x_{δ+j}, y_{δ+j}⟩ = ⟨a_{0}, b_{0}⟩.

Now consider the set Z of all ⟨z_{0}, z′_{0}, …, z_{m−1}, z′_{m−1}⟩ ∈ (2^{ω} × 2^{ω})^{m} for which

• there exist s ∈ 2^{<ω}, k < κ_{s}^{i}, and q ∈ (Qh̄)^{δ} such that s ⊂ z_{0}, q extends p and p_{s,k}^{i}, and ⟨z_{0}, z′_{0}, …, z_{m−1}, z′_{m−1}⟩ ∈ (q̂)^{m}.

Claim. The set Z belongs to the model M_{δ} and it is an open dense subset of (p̂)^{m}.

Proof. It should be clear that Z is (coded) in M_{δ}. (Remember the choice of δ.) To show that it is dense in (p̂)^{m} we proceed like in the proof of Fact 8. We choose s_{0}, t_{0}, …, s_{m−1}, t_{m−1} and r exactly as there. Next pick a condition q ∈ Qh̄, a sequence s ∈ 2^{<ω}, and k < κ_{s}^{m} such that

s_{0} ⊂ s and q extends p_{s,k}^{i} and r.

(Remember the choice of the p_{s,k}^{i}’s.) Clearly we can demand that q ∈ (Qh̄)^{δ}. Now note that it is possible to choose a z̄ = ⟨z_{0}, z′_{0}, …, z_{m−1}, z′_{m−1}⟩ ∈ (q̂)^{m} such that s ⊂ z_{0}, s_{i} ⊂ z_{i}, t_{i} ⊂ z′_{i}. Then z̄ ∈ Z ∩ ∏_{i<k}([s_{i}] × [t_{i}]).

Since Z is clearly open, this completes the proof of Claim.

Now, by (6) and the Claim above, ⟨⟨a_{l}, b_{l}⟩: l < m⟩ belongs to Z since ⟨⟨a_{l}, b_{l}⟩: l < m⟩ belongs to (p̂_{1})^{m} = (p̂)^{m}. But this means that there exist q = ⟨n^{q}, π^{q}, h^{q}⟩ ∈ (Qh̄)^{δ} and s ∈ 2^{<ω} such that:

• q ≤ p, q ⊩“[s] ⊆ U_{i}^{.}”, and

• ⟨⟨a_{l}, b_{l}⟩: l < m⟩ ∈ (q̂)^{m}, and x_{δ+j} = a_{0} ∈ [s].

But then p_{2} = ⟨n^{q}, π^{q}, h^{q} ∪ {⟨a_{l}, b_{l}⟩: l < m}⟩ belongs to Qh̄ and extends both q and p_{1}. So, p_{2} forces that x_{δ+j} = a_{0} ∈ [s] ⊆ U^{.}_{i}, contradicting our assumption that p_{1} ⊩ “x_{δ+j} ∉ U^{.}_{i}.”

This finishes the proof of (5) and of Lemma 5.

Acknowledgments. The authors wish to thank Professor Andrzej Rosłanowski for reading earlier versions of this paper and helping in improving its final form.

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Krzysztof Chris Ciesielski Saharon Shelah Department of Mathematics Institute of Mathematics West Virginia University the Hebrew University of Jerusalem Morgantown, WV 26506-6310 91904 Jerusalem USA Israel e-mail: K Cies@math.wvu.edu and

Department of Mathematics

Rutgers University

web page: New Brunswick, NJ 08854 http://www.math.wvu.edu/homepages/kcies USA