Descriptive complexity of countable unions of Borel rectangles

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Descriptive complexity of countable unions of Borel rectangles

Dominique Lecomte, Miroslav Zeleny

To cite this version:

Dominique Lecomte, Miroslav Zeleny. Descriptive complexity of countable unions of Borel rectangles.

2013. �hal-00852718�

Descriptive complexity of countable unions of Borel rectangles

Dominique LECOMTE and Miroslav ZELENY¹ August 21, 2013

•Universit´e Paris 6, Institut de Math´ematiques de Jussieu, Projet Analyse Fonctionnelle Couloir 16-26, 4`eme ´etage, Case 247, 4, place Jussieu, 75 252 Paris Cedex 05, France

[email protected]

•Universit´e de Picardie, I.U.T. de l’Oise, site de Creil, 13, all´ee de la fa¨ıencerie, 60 107 Creil, France

•¹Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis Sokolovsk´a 83, 186 75 Prague, Czech Republic

[email protected]

Abstract. We give, for each countable ordinalξ≥1, an example of a∆⁰₂ countable union of Borel rectangles that cannot be decomposed into countably manyΠ⁰_ξ rectangles. In fact, we provide a graph of a partial injection with disjoint domain and range, which is a difference of two closed sets, and which has no∆⁰_ξ-measurable countable coloring.

2010 Mathematics Subject Classification.Primary: 03E15, Secondary: 54H05

Keywords and phrases.Borel chromatic number, Borel class, coloring, product, rectangle

Acknowledgements.The results were partly obtained during the second author’s stay at the Universit´e Paris 6 in May 2013. The second author thanks the Universit´e Paris 6 for the hospitality.

The research was supported by the grant GA ˇCR P201/12/0436 for the second author.

1 Introduction

In this paper, we work in products of two Polish spaces. One of our goals is to give an answer to the following simple question. Assume that a countable union of Borel rectangles has low Borel rank. Is there a decomposition of this union into countably many rectangles of low Borel rank? In other words, is there a mapr:ω₁\{0} →ω₁\{0}such thatΠ⁰_ξ∩(∆¹₁×∆¹₁)_σ⊆(Π⁰_r(ξ)×Π⁰_r(ξ))_σ for eachξ∈ω₁\{0}?

By Theorem 3.6 in [Lo], a Borel set with open vertical sections is of the form(∆¹₁×Σ⁰₁)σ. This leads to a similar problem: is there a maps:ω₁\ {0} →ω₁\ {0} such that, for eachξ∈ω₁\ {0}, Π⁰_ξ∩(∆¹₁×Σ⁰₁)_σ⊆(Π⁰_s(ξ)×Σ⁰₁)_σ?

The answer to these questions is negative:

Theorem 1.1 Let1≤ξ < ω₁. Then there exists a partial mapf:ω^ω→ω^ωsuch that the complement

¬Gr(f)of the graph off isΠ⁰₂but not(Σ⁰_ξ×∆¹₁)_σ.

In fact, we prove a result related to∆⁰_ξ-measurable countable colorings. A study of such colorings is made in [L-Z]. It was motivated by the G0-dichotomy (see Theorem 6.3 in [K-S-T]). More pre- cisely, letB be a Borel binary relation having a Borel countable coloring (i.e., a Borel mapc:X→ω such thatc(x)6=c(y)if(x, y)∈B). Is there a relation between the Borel class ofB and that of the coloring? In other words, is there a mapk:ω₁\{0} →ω₁\{0}such that anyΠ⁰_ξbinary relation having a Borel countable coloring has in fact a∆⁰_k(ξ)-measurable countable coloring, for eachξ∈ω₁\{0}?

Here again, the answer is negative:

Theorem 1.2 Let1 ≤ξ < ω₁. Then there exists a partial injection with disjoint domain and range i:ω^ω→ω^ω whose graph is the difference of two closed sets, and has no∆⁰_ξ-measurable countable coloring.

These two results are consequences of Theorem 4 in [M´a] and its proof. This latter can also be used positively, to produce examples of graphs of fixed point free partial injections having reasonable chances to characterize the analytic binary relations without∆⁰_ξ-measurable countable coloring. We will see in Section 4 that such a characterization indeed holds whenξ= 3, and give an example much simpler than the one in [L-Z]. In Section 2, we give a proof of Theorem 4 in [M´a], inω^ωinstead of2^ω, and also prove some additional properties needed for the construction of our partial maps. In Section 3, we prove Theorems 1.1 and 1.2. At the end of Section 4, we show that Theorem 1.2 is optimal in terms of descriptive complexity of the graph, and also give a positive result concerning the first two problems in the case of finite unions of rectangles.

2 M´atrai sets

Before proving our version of Theorem 4 in [M´a], we need some notation, definition, and a few basic facts. The maps with closed graph will be of particular interest for us.

Lemma 2.1 Let(Xi)i∈ω,(Yi)i∈ωbe sequences of metrizable spaces, and, for eachi∈ω,fi:Xi→Yi

be a partial map whose graph is a closed subset of X_i×Y_i. Then the graph of the partial map f:= Π_i∈ωf_i: Π_i∈ωX_i→Π_i∈ωY_iis closed.

Proof. Let (x^j)j∈ω be a sequence of elements of Πi∈ω X_i converging to x := (x_i)i∈ω such that f(x^j)

j∈ωconverges toy:= (y_i)_i∈ω∈Π_i∈ω Y_i. Theny_i=f_i(x_i), since Gr(f_i)is closed, for each i∈ω. This implies thaty=f(x)and the proof is finished.

Notation. LetX be a set and F be a family of subsets of X. Then the symbol hFi denotes the smallest topology onXcontainingF.

The next two lemmas can be found in [K] (see Lemmas 13.2 and 13.3).

Lemma 2.2 Let (X, σ) be a Polish space and F be a σ-closed subset of X. Then the topology σ_F:=hσ∪ {F}iis Polish andF isσ_F-clopen.

Lemma 2.3 Let(σ_n)n∈ωbe a sequence of Polish topologies onX. Then the topologyhS

n∈ω σ_niis Polish.

Lemma 2.4 Let(Hn)n∈ω be a disjoint family of sets in a zero-dimensional Polish space(X, σ)and (σ_n)_n∈ω be a sequence of topologies onXsuch that

σ₀ =σ,H₀isσ₀-closed,

σ_n+1=hσn∪ {Hn}i,H_n+1isσ_n+1-closed for everyn∈ω.

Then the topologyσ_∞=hS

n∈ω σ_nisatisfies the following properties:

(a)σ_∞is zero-dimensional Polish, (b)σ_∞|X\^S_n∈ωHn=σ_|X\^S_n∈ωHn, and, for everyn∈ω,

(c)σ_∞|Hn=σ_|Hn, (d)H_nisσ_∞-clopen.

Proof.Using Lemma 2.2 we see that each topologyσ_nis Polish. Then the topologyσ_∞is Polish by Lemma 2.3. Now observe that the following claim holds.

Claim.A setG⊆Xisσ_∞-open if and only ifGcan be written asG=G^′∪(S

n∈ωG_n∩H_n), where G^′, Gnareσ-open.

Note thatH_n∈Σ⁰₁(σ_n+1)⊆Σ⁰₁(σ_∞)andH_n∈Π⁰₁(σ_n)⊆Π⁰₁(σ_∞), thusH_nisσ_∞-clopen. Thus (d) is satisfied. LetBbe a basis forσmade ofσ-clopen sets. Then the family

B ∪ {G∩H_n|G∈ B ∧ n∈ω}

is made ofσ_∞-clopen sets and form a basis forσ_∞by the claim. This gives (a).

LetG∈Σ⁰₁(σ∞). By the claim, we findσ-open setsG^′, G_nsuch thatG=G^′∪(S

n∈ω G_n∩H_n).

ThenG∩(X\S

n∈ω H_n) =G^′∩(X\S

n∈ω H_n). This implies (b). Moreover,G∩H_n=G_n∩H_n,

and (c) holds.

Notation.The symbolτ denotes the product topology onω^ω.

Definition 2.5 We say that a partial mapf:ω^ω→ω^ωisniceif Gr(f)is a(τ×τ)-closed subset of ω^ω×ω^ω.

The construction ofP_ξandτ_ξ, and the verification of the properties (1)_ξ-(3)_ξfrom the next lemma, can be found in [M´a], up to minor modifications.

Lemma 2.6 Let1≤ξ < ω₁. Then there areP_ξ⊆ω^ω, and a topologyτ_ξonω^ωsuch that (1)_ξτ_ξis zero-dimensional perfect Polish andτ⊆τξ⊆Σ⁰_ξ(τ),

(2)ξP_ξis a nonemptyτ_ξ-closed nowhere dense set,

(3)_ξifS∈Σ⁰_ξ(ω^ω, τ)isτ_ξ-nonmeager inP_ξ, thenSisτ_ξ-nonmeager inω^ω, (4)ξ ifU is a nonempty τ_ξ|P

ξ-open subset ofP_ξ, then we can find aτ_ξ-denseG_δ subsetGofU, and a nice(τ_ξ, τ)-homeomorphismϕ_ξ,GfromGontoω^ω,

(5)ξifV is a nonemptyτ_ξ-open subset ofω^ω, then we can find aτ_ξ-denseG_δsubsetHofV, and a nice(τ_ξ, τ)-homeomorphismψ_ξ,HfromHontoω^ω,

(6)_ξ if U is a nonemptyτ_ξ|P

ξ-open subset ofP_ξ and W is a nonempty open subset ofω^ω, then we can find aτ_ξ-dense G_δ subsetGofU, aτ_ξ-dense G_δ subset K ofW\P_ξ, and a nice (τ_ξ, τ_ξ)- homeomorphismϕ_ξ,G,KfromGontoK,

(7)ξ if V, W are nonemptyτ_ξ-open subsets ofω^ω, then we can find a τ_ξ-dense G_δ subsetH of V\P_ξ, aτ_ξ-denseG_δsubsetLofW\P_ξ, and a nice(τ_ξ, τ_ξ)-homeomorphismψ_ξ,H,LfromHontoL.

Proof.We proceed by induction onξ.

The caseξ= 1

We setP₁:={α∈ω^ω | ∀n∈ω α(2n) = 0}and τ₁:=τ. The properties (1)₁-(3)₁ are clearly satisfied.

(4)₁ Note that (P₁, τ₁) is homeomorphic to (ω^ω, τ). As any nonempty open subset of (ω^ω, τ) is homeomorphic to(ω^ω, τ),(U, τ1)is homeomorphic to(ω^ω, τ). This givesϕ_ξ,U, which is nice since ω^ωis closed in itself. This shows that we can takeG:=U.

(5)₁As in (4)₁we see that(V, τ₁)is homeomorphic to(ω^ω, τ), and we can takeH:=V.

(6)₁Note thatU is the disjoint union of a sequence(C_n)_n∈ω of nonempty clopen subsets of(P₁, τ₁).

Let(U_1,n)n∈ωbe a partition ofW\P₁into clopen subsets of(ω^ω, τ₁). As any nonempty open subset of(P₁, τ₁)or(ω^ω, τ₁)is homeomorphic to(ω^ω, τ), we can find homeomorphisms

ϕ₀: (C₀, τ₁)→([

n>0

U_1,n, τ₁) andϕ₁: (S

n>0 C_n, τ₁)→(U_1,0, τ₁). AsC₀ andU_1,0 areτ-closed,ϕ₀ andϕ₁ are nice. This shows that the gluing ofϕ₀ andϕ₁ is a nice homeomorphism from(U, τ₁)onto(W\P1, τ₁). Thus we can takeG:=U andK:=W\P₁.

(7)₁As in (6)₁ we writeV as the disjoint union of a sequence(D_n)_n∈ω of nonempty clopen subsets of(ω^ω, τ₁). As the(D_n, τ₁)’s are homeomorphic to(ω^ω, τ₁), we can takeH:=V\P1andL:=W\P1.

The induction step

We assume that1< ξ < ω₁ and that the assertion holds for each ordinalθ < ξ. We fix a sequence of ordinals(ξn)n∈ωcontaining each ordinal inξ\{0}infinitely many times. We set

P_ξ=ω^ω×(Πi∈ω¬Pξi), τ_ξ^<=τ×(Πi∈ωτ_ξi),

U_ξ,n=ω^ω×(Πi<n¬Pξ_i)×Pξn×(ω^ω)^ω (n∈ω).

The family{Uξ,n |n∈ω}is disjoint. We setσ₀=τ_ξ^<andσ_n+1=hσn∪ {Uξ,n}i. It is easy to check thatU_ξ,n∈Π⁰₁(σ_n). Applying Lemma 2.4 we get a topologyτ_ξ:=σ∞such that

(a)τ_ξis zero-dimensional Polish, (b)τ_ξ|P

ξ=τ_ξ^<

|P_ξ, and, for everyn∈ω,

(c)τ_ξ|U

ξ,n=τ_ξ^<

|Uξ,n, (d)U_ξ,nisτ_ξ-clopen.

We defined the topology τ_ξ on (ω^ω)^ω instead of ω^ω. However, since the spaces (ω^ω)^ω, τ^ω) and (ω^ω, τ)are homeomorphic we can replace the latter space by the former one in the proof. Since there is no danger of confusion we will writeτ instead ofτ^ωto simplify the notation.

(1)_ξ Clearly,τ⊆τ_ξ. Note thatU_ξ,n∈Σ⁰_ξ(τ)for everyn∈ω and τ_ξ^<⊆Σ⁰_ξ(τ), so that τ_ξ⊆Σ⁰_ξ(τ).

Moreover,(ω^ω, τ_ξ)is clearly perfect.

(2)_ξAsU_ξ,nisτ_ξ-clopen,P_ξisτ_ξ-closed. Note thatτ_ξ|P

ξ=τ_ξ^<

|P_ξ andP_ξcontains no nonempty basic τ_ξ^<-open set. This implies thatP_ξisτ_ξ-nowhere dense.

(3)_ξ LetS∈Σ⁰_ξ(τ) be τ_ξ-nonmeager inP_ξ. We may assume that S∈Π⁰_θ(τ)for someθ < ξ. As τ_ξ|P

ξ=τ_ξ^<

|P_ξ andShas the Baire property with respect to the topologyτ_ξ^<there exists aτ_ξ^<-open set V such thatSisτ_ξ^<-comeager inP_ξ∩V. Moreover, we may assume thatV has the following form:

V= ˜V×(Πi≤kVi)×(ω^ω)^ω,

whereV˜∈τ,V_i∈τ_ξi andV_i⊆ ¬P_ξifor eachi≤k. The setV^∗= ˜V×(Π_i≤kV_i)×(Π_i>k¬P_ξi)isτ_ξ^<- comeager inV since¬Pξiisτ_ξi-comeager inω^ωfor everyi∈ω. AsP_ξ∩V=V^∗,Sisτ_ξ^<-comeager inV^∗. Letp∈ωbe such thatp > kandξ_p≥θ. Define

τ^∗=τ×(Πi6=pτ_ξi),

Z= ˜V×V0×· · ·×Vk׬Pξk+1×· · ·×¬Pξp−1×(ω^ω)^ω, τ^♯=τ×(Πi<pτ_ξi)×τ×(Πi>pτ_ξi).

Forα∈ω^ωdefine a set(¬S)αby

(¬S)α:={(˜y, y₀, y₁, . . . , y_p−1, y_p+1, . . .)∈ω^ω |(˜y, y₀, y₁, . . . , y_p−1, α, y_p+1, . . .)∈ ¬S}.

DenoteS^∗:={α∈ω^ω |(¬S)αisτ^∗-nonmeager inZ}. Note that¬S∈Σ⁰_θ(τ)⊆Σ⁰_θ(τ^♯). By the Montgomery theorem (see 22.D in [K]),S^∗∈Σ⁰_θ(τ)⊆Σ⁰_ξp(τ). By the Kuratowski-Ulam theorem,S^∗ isτ_ξp-meager in¬Pξp. Using the induction hypothesis, Condition (3)_ξpimplies thatS^∗isτ_ξp-meager inP_ξp. Using the Kuratowski-Ulam theorem again, we see thatSisτ_ξ^<-comeager in theτ_ξ-open set

W= ˜V×V0×· · ·×Vk׬Pξ_k+1×· · ·×¬Pξp−1×Pξp×(ω^ω)^ω. AsW⊆U_ξ,p,τ_ξ|W=τ_ξ^<

|W by (c), and consequentlySisτ_ξ-comeager inW. ThusSisτ_ξ-nonmeager in(ω^ω)^ωsinceW isτ_ξ-open.

(4)_ξWe first construct aτ_ξ-dense open subset ofU, which is the disjoint union of sets of the form Uⁿ:= Wⁿ×(Πi<knW_iⁿ)×(ω^ω)^ω

∩P_ξ=Wⁿ×(Πi<knW_iⁿ\Pξi)×(Πi≥kn ¬Pξi),

whereWⁿis a nonemptyτ-clopen set andW_iⁿis a nonemptyτ_ξi-clopen set. In order to do this, we fix an injectiveτ_ξ-dense sequence (x_n)_n∈ω ofU, which is possible since(P_ξ, τ_ξ) is nonempty and perfect. We first chooseW⁰ and theW_i⁰’s in such a way thatU⁰is a properτ_ξ-clopen neighborhood ofx₀inU, which is possible sinceτ_ξ|P

ξ=τ_ξ^<

|P_ξ. For the induction step, we choosep_nminimal such thatx_pn∈/S

q≤n U^q. Then we choose Wⁿ⁺¹ and theW_iⁿ⁺¹’s in such a way thatUⁿ⁺¹ is a proper τ_ξ-clopen neighborhood ofx_pn inU\(S

q≤n U^q).

There is a nice homeomorphism ψ_n fromWⁿontoN_n:={α∈ω^ω | α(0) =n}. The induction assumption gives,

- fori < k_n, aτ_ξi-denseG_δsubsetGⁿ_i ofW_iⁿ\P_ξi, and a nice(τ_ξi, τ)-homeomorphismψ_ξi,Gⁿ_i of Gⁿ_i ontoω^ω,

- fori≥kn, aτ_ξi-denseG_δsubsetGⁿ_i of¬Pξi, and a nice(τ_ξi, τ)-homeomorphismψ_ξi,Gⁿ_i ofGⁿ_i ontoω^ω.

By Lemma 2.1, the mapψ_n×(Πi∈ω ψ_ξi,Gⁿ_i)is a nice(τ_ξ^<, τ)-homeomorphism from Wⁿ×(Πi∈ωGⁿ_i)

ontoNn×(ω^ω)^ω. If we setG:=S

n∈ω Wⁿ×(Πi∈ωGⁿ_i)

, then we get a nice(τ_ξ^<, τ)-homeomorphism fromGontoω^ω. We are done sinceτ_ξ|P

ξ=τ_ξ^<

|P_ξ.

(5)_ξWe essentially argue as in (4)_ξ. AsP_ξisτ_ξ-closed nowhere dense, we may assume that V⊆ ¬Pξ=[

n∈ω

U_ξ,n.

We first construct aτ_ξ-dense open subset ofV ∩U_ξ,n, which is the disjoint union of sets of the form V^n,p:=W^n,p×(Π_i<nW_i^n,p\P_ξi)×(W_n^n,p∩Pξn)×(Πn<i<k^pnW_i^n,p)×(ω^ω)^ω, whereW^n,pis a nonempty τ-clopen set andW_i^n,pis a nonemptyτ_ξi-clopen set. This is possible sinceτ_ξ|U

ξ,n=τ_ξ^<

|U_ξ,n. We are done sinceU_ξ,nisτ_ξ-clopen.

(6)_ξ As in (4)_ξ we construct aτ_ξ-dense open subset ofU, which is the disjoint union of sets of the formUⁿ:= Wⁿ×(Πi<kn W_iⁿ)×(ω^ω)^ω

∩P_ξ=Wⁿ×(Πi<kn W_iⁿ\Pξi)×(Πi≥kn¬Pξi), whereWⁿ is a nonemptyτ-clopen set andW_iⁿis a nonemptyτ_ξi-clopen set. Recall also that

U_ξ,n=ω^ω×(Πi<n¬Pξi)×Pξn×(ω^ω)^ω.

We also construct aτ_ξ-dense open subset ofW, which is the disjoint union of sets of the form πⁿ:=Zⁿ×(Π_i<lnZ_iⁿ\P_ξi)×(Z_lⁿ_n∩P_ξln)×(Π_ln<i<mnZ_iⁿ)×(ω^ω)^ω⊆U_ξ,ln,

whereZⁿis a nonemptyτ-clopen set andZ_iⁿis a nonemptyτ_ξi-clopen set. Let(W^0,p)_p∈ω(respec- tively, (Z^0,p)p∈ω) be a partition of W⁰ (respectively, Z⁰) into nonempty τ-clopen sets. Using the facts thatτ_ξ|P

ξ=τ_ξ^<

|P_ξ andτ_ξ|U

ξ,n=τ_ξ^<

|U_ξ,n, we will build

- a nice(τ_ξ, τ_ξ)-homeomorphism from a denseG_δsubsetG^0,pof U^0,p:=W^0,p×(Πi<k0 W_i⁰\Pξi)×(Πi≥k0 ¬Pξi)

onto a denseG_δsubsetK^0,pofπ^p+1. Then, using the fact that theW^0,p’s areτ-clopen, the gluing of these homeomorphisms will be a nice(τ_ξ, τ_ξ)-homeomorphismϕ₀fromG⁰:=S

p∈ω G^0,p⊆U⁰onto K⁰:=S

p∈ω K^0,p⊆S

p>0 π^p.

- a nice homeomorphism from a denseG_δsubsetG^1,p ofU^p+1onto a denseG_δ subsetK^1,pof Z^0,p×(Π_i<l0 Z_i⁰\P_ξi)×(Z_l⁰

0 ∩P_ξl

0)×(Π_l0<i<m0 Z_i⁰)×(ω^ω)^ω. Then the gluing of these home- omorphisms will be a nice(τ_ξ, τ_ξ)-homeomorphism ϕ₁ from G¹ :=S

p∈ω G^1,p ⊆S

p>0 U^p onto K¹:=S

p∈ω K^1,p⊆π⁰.

The gluing of these two homeomorphisms will be a nice homeomorphism fromG:=G⁰∪G¹onto K:=K⁰∪K¹. The setG^0,p(respectively,K^0,p) will be of the formW^0,p×(Πi∈ωG^p_i)(respectively, Z^p+1×(Πi∈ω K_i^p)). Note first that there is a homeomorphism ψ_p fromW^0,p ontoZ^p+1. Then we build a permutationi7→ji of the coordinates (with inverseq7→Jq). This permutation is constructed in such a way that ξ_ji =ξ_i, which will be possible since (ξ_n)_n∈ω contains each ordinal in ξ\ {0}

infinitely many times. If i < m_p+1 (respectively, q < k₀), then we choose j_i ≥k₀ (respectively, J_q≥m_p+1), ensuring injectivity. For a remaining coordinateq /∈ {0, ..., k₀−1} ∪ {j_l |l < m_p+1}, we chooseJ_q∈ {0, ..., m/ p+1−1} ∪ {Jl |l < k₀}, ensuring that the mapq7→J_q is a bijection from

¬({0, ..., k0−1} ∪ {jl | l < m_p+1}) onto¬ {0, ..., mp+1−1} ∪ {Jl | l < k₀}

. Then, using the induction assumption, we build our homeomorphism coordinate by coordinate, which means thatG^p_ji will be homeomorphic toK_i^p. The induction assumption gives

- fori < l_p+1, aτ_ξji-denseG_δsubsetG^p_ji of¬Pξ_ji, aτ_ξi-denseG_δsubsetK_i^p ofZ_i^p+1\Pξi, and a nice(τ_ξi, τ_ξi)-homeomorphismψ_ξi,G^p

ji,K_i^pfromG^p_jiontoK_i^p. - aτ_ξ

jlp+1-dense G_δ subsetG^p_j

lp+1 of¬P_ξ

jlp+1, a τ_ξlp+1-denseG_δ subsetK_l^p_p+1 ofP_ξlp+1, and a nice(τ_ξlp+1, τ_ξlp+1)-homeomorphismϕ⁻¹_ξ

lp+1,K_lp+1^p ,G^p

jlp+1 fromG^p_j

lp+1 ontoK_l^p_p+1.

- forlp+1< i < mp+1, aτ_ξji-denseG_δsubsetG^p_jiof¬P_ξji, aτ_ξi-denseG_δsubsetK_i^pofZ_i^p+1\P_ξi, and a nice(τ_ξi, τ_ξi)-homeomorphismψ_ξi,G^p

ji,K_i^pfromG^p_ji ontoK_i^p.

- forq < k₀, aτ_ξq-denseG_δsubsetG^pqofW_q⁰\P_ξq, aτ_ξJq-dense G_δ subsetK_J^p_q of¬P_ξJq, and a nice(τ_ξq, τ_ξq)-homeomorphismψ_ξq,G^p_q,K^p

Jq fromG^pqontoK_J^p_q.

- for a remaining coordinate q /∈ {0, ..., k0−1} ∪ {jl | l < m_p+1}, aτ_ξq-dense G_δsubsetG^p_q of

¬Pξq, aτ_ξJq-dense G_δ subsetK_J^p

q of¬Pξ_Jq, and a nice (τ_ξq, τ_ξq)-homeomorphismψ_ξq,G^p_q,K^p

Jq from G^pqontoK_J^p_q.

By Lemma 2.1, the product ϕ⁰_p of ψ_p with these nice homeomorphisms is a nice (τ_ξ^<, τ_ξ^<)- homeomorphism from G^0,p := W^0,p×(Πi∈ω G^p_i) onto K^0,p := Z^p+1×(Πi∈ω K_i^p), as well as a (τ_ξ, τ_ξ)-homeomorphism since τ_ξ|P

ξ =τ_ξ^<

|Pξ and τ_ξ|U

ξ,lp+1 =τ_ξ^<

|U_ξ,lp+1. AsG⁰ is the sum of the G^0,p’s,Gis aτ_ξ-denseG_δ subset ofU⁰. Similarly, K⁰is aτ_ξ-denseG_δsubset ofS

p>0 π^p. More- over, the gluingϕ⁰of theϕ⁰_p’s is a(τ_ξ, τ_ξ)-homeomorphism fromG⁰ontoK⁰.

The construction ofϕ¹is similar.

(7)_ξWe argue as in (6)_ξ.

Lemma 2.7 Let1≤ξ < ω₁. Then there are disjoint familiesF_ξ,G_ξof subsets ofω^ωand a topology T_ξonω^ω such that

(a)_ξT_ξis zero-dimensional perfect Polish andτ⊆T_ξ⊆Σ⁰_ξ(τ),

(b)_ξFξisT_ξ-dense, i.e., for any nonemptyT_ξ-open setV, there isF∈ FξwithF⊆V, and, for everyF∈ F_ξ,

(c)_ξF is nonempty,T_ξ-nowhere dense, and inΠ⁰₂(T_ξ),

(d)_ξifS∈Σ⁰_ξ(τ)isT_ξ-nonmeager inF, thenSisT_ξ-nonmeager inω^ω, (e)_ξthere is a nice(T_ξ, τ)-homeomorphismϕ_F fromF ontoω^ω,

(f)_ξ for any nonemptyT_ξ-open setsV, V^′, there are disjointG, G^′∈ G_ξ withG⊆V,G^′⊆V^′, and there is a nice(T_ξ, T_ξ)-homeomorphismϕ_G,G^′ fromGontoG^′,

and, for everyG∈ G_ξ,

(g)_ξGis nonempty,T_ξ-nowhere dense, and inΠ⁰₂(T_ξ),

(h)_ξifS∈Σ⁰_ξ(τ)isT_ξ-nonmeager inG, thenSisT_ξ-nonmeager inω^ω.

Proof.LetP_ξandτ_ξbe as in Lemma 2.6. We setT_ξ= (τ_ξ)^ω. Let(Un)n∈ωbe a basis for the topology T_ξmade of nonempty sets. For eachn∈ω, there is a finite sequence(V_iⁿ)_i<knof nonemptyτ_ξ-open sets such that(Π_i<kn V_iⁿ)×(ω^ω)^ω⊆U_n. Moreover, the sequence(k_n)_n∈ω is chosen to be strictly increasing. Lemma 2.6 provides

- fori < k_n, aτ_ξ-denseG_δsubsetH_iⁿofV_iⁿ\P_ξand a nice(τ_ξ, τ)-homeomorphismψ_ξ,Hiⁿ, - aτ_ξ-denseG_δsubsetGⁿ_kn ofP_ξand a nice(τ_ξ, τ)-homeomorphismϕ_ξ,Gⁿ

kn,

- fori > kn, aτ_ξ-denseG_δsubsetH_iⁿofω^ωand a nice(τ_ξ, τ)-homeomorphismψ_ξ,Hiⁿ.

We then putFn:= (Π_i<kn H_iⁿ)×Gⁿ_kn×(Π_i>kn H_iⁿ), so thatFn⊆Un. We setF_ξ={Fn|n∈ω}.

ThenFξis clearly a disjoint family and the properties (a)_ξand (b)_ξare obviously satisfied.

(c)_ξAsP_ξisτ_ξ-nowhere dense, eachFnisT_ξ-nowhere dense. EachFnis obviously also inΠ⁰₂(T_ξ).

(d)_ξLetn∈ωandS∈Σ⁰_ξ(τ)beT_ξ-nonmeager inF_n. We define Z= Πi6=kn H_iⁿ,

T_ξ^∗= Π_i6=knτ_ξ|Hn i , T˜_ξ= (Π_i<knτ_ξ|Hn

i )×τ×(Π_i>kn τ_ξ|Hn i).

Ifα∈ω^ω, then we denote

S_α:={(y0, . . . , y_kn−1, y_kn+1, . . .)∈ω^ω|(y₀, . . . , y_kn−1, α, y_kn+1, . . .)∈S}.

We set S^∗ ={α ∈ω^ω | S_αisT_ξ^∗-nonmeager}. By the Montgomery theorem, S^∗ ∈Σ⁰_ξ(τ) since S ∈Σ⁰_ξ( ˜T_ξ). The set S^∗ is τ_ξ-nonmeager in Gⁿ_kn by the Kuratowski-Ulam theorem, in P_ξ also, and thus S^∗ isτ_ξ-nonmeager in ω^ω. Using the Kuratowski-Ulam theorem again, we see thatS is T_ξ-nonmeager in(Π_i<knH_iⁿ)×ω^ω×(Π_i>kn H_iⁿ), and thus in(ω^ω)^ω.

(e)_ξWe setϕ_F= (Π_i<kn ψ_ξ,Hⁿ_i)×ϕξ,Gⁿ_kn×(Πi>kn ψ_ξ,Hⁿ_i). The mapϕ_F is clearly a(T_ξ, τ)-homeo- morphism fromF onto(ω^ω)^ω. It is nice by Lemma 2.1.

We now construct Gξ. For each m ∈ ω, there are finite sequences (V_i^m)_i<km, (W_i^m)_i<lm of nonemptyτ_ξ-open sets such that(Π_i<km V_i^m)×(ω^ω)^ω⊆U_(m)0 and(Π_i<lmW_i^m)×(ω^ω)^ω⊆U_(m)1. Moreover, the sequences (k_m)m∈ω and (l_m)m∈ω are chosen to be strictly increasing and disjoint.

Assume for example thatk_m< l_m. Lemma 2.6 provides

- fori < k_m, aτ_ξ-dense G_δ subsetH_i^m ofV_i^m\P_ξ, aτ_ξ-dense G_δsubsetL^m_i ofW_i^m\P_ξ, and a nice(τ_ξ, τ_ξ)-homeomorphismψ_ξ,Hi^m_,L^m_i ,

- aτ_ξ-dense G_δ subsetG^m_km ofP_ξ, aτ_ξ-dense G_δ subsetK_k^m_m ofW_i^m\P_ξ, and a nice (τ_ξ, τ_ξ)- homeomorphismϕ_ξ,G^m

km,K^m_km,

- fork_m< i < l_m, aτ_ξ-denseG_δsubsetH_i^m of¬Pξ, aτ_ξ-denseG_δ subsetL^m_i ofW_i^m\Pξ, and a nice(τ_ξ, τ_ξ)-homeomorphismψ_ξ,Hi^m_,L^m_i ,

- a τ_ξ-dense G_δ subset K_l^m_m of ¬P_ξ, a τ_ξ-dense G_δ subset G^m_lm of P_ξ, and a nice (τ_ξ, τ_ξ)- homeomorphismϕ⁻¹_ξ,Gm

lm,K_lm^m,

- for i > l_m, a τ_ξ-dense G_δ subsetH_i^m of¬Pξ, aτ_ξ-dense G_δ subsetL^m_i of ¬Pξ, and a nice (τ_ξ, τ_ξ)-homeomorphismψ_ξ,H^m

i ,L^m_i . We then put

F_m^′ := (Π_i<km H_i^m)×G^m_km×(Π_km<i<lnH_i^m)×K_l^m_m×(Π_i>lm H_i^m), G_m:= (Π_i<kmL^m_i )×K_k^m_m×(Πkm<i<lm L^m_i )×G^m_lm×(Πi>lmL^m_i ),

so thatF_m^′ ×Gm⊆U_(m)0×U_(m)1. We setG_ξ={F_m^′ |m∈ω} ∪ {Gm |m∈ω}. ThenG_ξis clearly a disjoint family.

(f)_ξThe mapϕ_Fm^′ _,Gmis by definition (Π_i<km ψ_ξ,H^m

i ,L^m_i )×ϕξ,G^m_km,K_km^m ×(Πkm<i<lmψ_ξ,H^m

i ,L^m_i )×ϕ⁻¹_ξ,Gm

lm,K_lm^m ×(Πi>lmψ_ξ,H^m

i ,L^m_i ).

Note thatϕ_Fm^′ _,Gmis clearly a(T_ξ, T_ξ)-homeomorphism fromF_m^′ ontoG_m. It is nice by Lemma 2.1.

(g)_ξWe argue as in (c)_ξ.

(h)_ξWe argue as in (d)_ξ.

3 Negative results

Proof of Theorem 1.1.We apply Lemma 2.7 to the ordinalξ+1, which gives a familyFξ+1 and a topologyT_ξ+1satisfying (a)_ξ+1-(e)_ξ+1. Let(U_n×Vn)_n∈ω be a sequence of nonempty sets such that

-U_n∈T_ξ+1,V_nisτ-clopen,

-{Un×Vn|n∈ω}is a basis for the topologyT_ξ+1×τ.

For eachn∈ωwe findF_n∈ F_ξ+1\{F_q |q < n}withF_n⊆U_n. By the property (e)_ξ+1ofF_ξ+1we find, for eachn∈ω, a nice(T_ξ+1, τ)-homeomorphismf_nfromF_nontoV_n. We definef:S

n∈ω F_n→ω^ω byf(x) :=f_n(x)ifx∈F_n. AsFξ+1is a disjoint family,fis well-defined. The graph offisΣ⁰₂(τ×τ) since each Gr(f_n)is(τ×τ)-closed.

Suppose, towards a contradiction, that there exist, forn∈ω,C_n∈Σ⁰_ξ(τ)andD_n∈∆¹1(τ)such that¬Gr(f) =S

n∈ω C_n×Dn. By the Baire category theorem there isn₀∈ωsuch thatC_n0 isT_ξ+1- nonmeager andD_n0 isτ-nonmeager. AsC_n0 has the Baire property, we find a nonemptyT_ξ+1-open setO₁ such thatC_n0 isT_ξ+1-comeager inO₁. Similarly, we find aτ-open set O₂ such that D_n0 is τ-comeager inO₂.

Letn∈ω andF_n⊆O₁. Suppose thatC_n0 is notT_ξ+1-comeager inF_n. ThenO₁\C_n0 isT_ξ+1- nonmeager inF_n. Note thatO₁∈Σ⁰_ξ+1(τ) andC_n0∈Σ⁰_ξ(τ). ThereforeO₁\C_n0∈Σ⁰_ξ+1(τ). Thus O1\Cn0 isT_ξ+1-nonmeager inω^ω by (d)_ξ+1. Consequently, O1\Cn0 isT_ξ+1-nonmeager inO1, a contradiction. ThusC_n0 isT_ξ+1-comeager inF_nfor anyn∈ωwithF_n⊆O₁.

Findn∈ωsuch that Gr(fn)⊆O1×O2. ThenCn0 isT_ξ+1-comeager inFnandDn0isτ-comeager inV_n. Asf_nis a homeomorphism,f_n⁻¹(V_n∩D_n0)isT_ξ+1-comeager inF_n. AsF_n∈Π⁰₂(T_ξ+1)there existsα∈f_n⁻¹(V_n∩D_n0)∩F_n∩C_n0. This implies that α, f_n(α)

∈C_n0×Dn0, a contradiction.

Proof of Theorem 1.2. Apply Lemma 2.7 to the ordinal ξ + 1, which gives a family Gξ+1 and a topologyT_ξ+1 satisfying (a)_ξ+1-(h)_ξ+1. LetU={Un | n∈ω}be a basis for the space (ω^ω, T_ξ+1) made of nonempty sets. For eachn∈ωwe findT_ξ+1-open setsV_n,W_nsuch that

V_n×Wn⊆B_τ×τ ∆(ω^ω),2⁻ⁿ

∩(U_n×U_n)\∆(ω^ω) (we use the standard metric on(ω^ω, τ)).

By the properties (f)_ξ+1 and (g)_ξ+1 ofG_ξ+1 we find, for eachn∈ω, setsFn andHnfromG_ξ+1 such that

(∗) F_n⊆V_n\([

j<n

F_j∪H_j) ∧ H_n⊆W_n\ F_n∪([

j<n

F_j ∪H_j) .

Moreover, there is a nice(T_ξ+1, T_ξ+1)-homeomorphismf_nfromF_nontoH_n. We set G=[

{Gr(fn)|n∈ω}.

Now we check the desired properties.

Asτ⊆Tξ+1,G^τ×τ=G ∪∆(ω^ω), by construction. ThusGis a difference of two(τ×τ)-closed sets.

As eachf_n is a homeomorphism, the property(∗)implies thatf is a partial injection with disjoint domain and range. In order to see thatGhas no∆⁰_ξ-measurable countable coloring, we proceed by contradiction. Suppose that there areG-discrete setsC_n∈∆⁰_ξ(τ)(a setCisG-discreteifC²∩ G=∅), forn∈ω, such that ∆(ω^ω)⊆S

n∈ω C_n². By the Baire theorem there existsn₀∈ω such that C_n0 isT_ξ+1-nonmeager. AsC_n0 has the Baire property, we find a nonempty T_ξ+1-open set Osuch that C_n0 ∩OisT_ξ+1-comeager inO.

Let F ∈ Gξ+1 with F ⊆O. Suppose that C_n0 is not T_ξ+1-comeager in F. Then O\C_n0 is T_ξ+1-nonmeager in F. Note thatO∈Σ⁰_ξ+1(τ) and C_n0 ∈∆⁰_ξ(τ). Therefore O\C_n0 ∈Σ⁰_ξ+1(τ).

ThusO\Cn0 isT_ξ+1-nonmeager inω^ωby (h)_ξ+1. Consequently,O\Cn0 isT_ξ+1-nonmeager inO, a contradiction. ThusC_n0 isT_ξ+1-comeager inF for anyF∈ G_ξ+1withF⊆O.

Find n∈ω such that Gr(f_n)⊆O². ThenC_n0 isT_ξ+1-comeager in F_n and inH_n. As f_nis a homeomorphism,f_n⁻¹(Hn∩Cn0)isT_ξ+1-comeager inFn∈Π⁰₂(T_ξ+1). Thus there exists

α∈f_n⁻¹(H_n∩C_n0)∩F_n∩C_n0. This implies that α, f_n(α)

∈C_n²₀, a contradiction.

4 Positive results

(A)∆⁰_ξ-measurable countable colorings In [L-Z], the following conjecture is made.

ConjectureLet1≤ξ < ω₁. Then there are - a0-dimensional Polish spaceXξ, - an analytic relationAξonXξ

such that for any (0-dimensional if ξ= 1) Polish space X, and for any analytic relation A on X, exactly one of the following holds:

(a) there is a∆⁰_ξ-measurable countable coloring ofA(i.e., a∆⁰_ξ-measurable mapc:X→ωsuch thatA⊆(c×c)⁻¹(6=)),

(b) there is a continuous mapf:Xξ→Xsuch thatAξ⊆(f×f)⁻¹(A).