aa r X i v : . [ m a t h . GN ] M a y On small analytic relations
Dominique LECOMTEMay 28, 2020 •
1) Sorbonne Universit´e, CNRS, Institut de Math´ematiques de Jussieu-Paris Rive Gauche,IMJ-PRG, F-75005 Paris, France2) Universit´e de Paris, IMJ-PRG, F-75013 Paris, [email protected] • Universit´e de Picardie, I.U.T. de l’Oise, site de Creil,13, all´ee de la fa¨ıencerie, 60 100 Creil, France
Abstract.
We study the class of analytic binary relations on Polish spaces, compared with the notionsof continuous reducibility or injective continuous reducibility. In particular, we characterize when alocally countable Borel relation is Σ ξ (or Π ξ ), when ξ ≥ , by providing a concrete finite antichainbasis. We give a similar characterization for arbitrary relations when ξ = 1 . When ξ = 2 , we providea concrete antichain of size continuum made of locally countable Borel relations minimal amongnon- Σ (or non- Π ) relations. The proof of this last result allows us to strengthen a result due toBaumgartner in topological Ramsey theory on the space of rational numbers. We prove that positiveresults hold when ξ = 2 in the acyclic case. We give a general positive result in the non-necessarilylocally countable case, with another suitable acyclicity assumption. We provide a concrete finiteantichain basis for the class of uncountable analytic relations. Finally, we deduce from our positiveresults some antichain basis for graphs, of small cardinality (most of the time 1 or 2). Primary: 03E15, Secondary: 28A05, 54H05
Keywords and phrases. acyclic, analytic relation, antichain basis, Borel class, countable, descriptive complexity, con-tinuous reducibility, graph, Ramsey Introduction
This article presents a continuation of the work in [L5], in which the descriptive complexity ofBorel equivalence relations on Polish spaces was studied (recall that a topological space is
Polish if itis separable and completely metrizable). These relations are compared using the notion of continuousreducibility , which is as follows. Recall that if
X, Y are topological spaces and A ⊆ X , B ⊆ Y , ( X, A ) ≤ c ( Y, B ) ⇔ ∃ f : X → Y continuous with A = ( f × f ) − ( B ) (we say that f reduces A to B ). When the function f can be injective, we write ⊑ c instead of ≤ c . Sometimes, when the space is clear for instance, we will talk about A instead of ( X, A ) . Themotivation for considering these quasi-orders is as follows (recall that a quasi-order is a reflexiveand transitive relation). A standard way of comparing the descriptive complexity of subsets of zero-dimensional Polish spaces is the Wadge quasi-order (see [W]; recall that a topological space is zero-dimensional if there is a basis for its topology made of clopen, i.e., closed and open, sets). If
S, Z are zero-dimensional Polish spaces and C ⊆ S , D ⊆ Z , ( S, C ) ≤ W ( Z, D ) ⇔ ∃ g : S → Z continuous with C = g − ( D ) . However, the pre-image of a graph by an arbitrary continuous map is not in general a graph, for in-stance. Note that the classes of reflexive relations, irreflexive relations, symmetric relations, transitiverelations are closed under pre-images by a square map. Moreover, the class of antisymmetric relationsis closed under pre-images by the square of an injective map. This is the reason why square maps areconsidered to compare graphs, equivalence relations... The most common way of comparing Borelequivalence relations is the notion of Borel reducibility (see, for example, [G], [Ka]). However, veryearly in the theory, injective continuous reducibility was considered, for instance in Silver’s theorem(see [S]).The most classical hierarchy of topological complexity in descriptive set theory is the one givenby the Borel classes. If Γ is a class of subsets of the metrizable spaces, then ˇ Γ := {¬ S | S ∈ Γ } is its dual class , and ∆( Γ ) := Γ ∩ ˇ Γ . Recall that the Borel hierarchy is the inclusion from left to right inthe following picture: Σ = open Σ = F σ Σ ξ =( S η<ξ Π η ) σ ∆ = clopen ∆ = Σ ∩ Π · · · ∆ ξ = Σ ξ ∩ Π ξ · · · Π = closed Π = G δ Π ξ = ˇ Σ ξ This hierarchy is strict in uncountable Polish spaces, in which the non self-dual classes are those ofthe form Σ ξ or Π ξ . In the sequel, by non self-dual Borel class, we mean exactly those classes. A class Γ of subsets of the zero-dimensional Polish spaces is a Wadge class if there is a zero-dimensionalspace Z and a subset D of Z in Γ such that a subset C of a zero-dimensional space S is in Γ exactlywhen ( S, C ) ≤ W ( Z, D ) . The hierarchy of the Wadge classes of Borel sets, compared with theinclusion, refines greatly the hierarchy of the non self-dual Borel classes, and is the finest hierarchyof topological complexity considered in descriptive set theory (see [Lo-SR2]).2e are interested in the descriptive complexity of Borel relations on Polish spaces. In order toapproach this problem, it is useful to consider invariants for the considered quasi-order. A naturalinvariant for Borel reducibility has been studied, the notion of potential complexity (see, for example,[L2], [L3], and [Lo2] for the definition). A Borel relation R on a Polish space X is potentially in aWadge class Γ if we can find a finer Polish topology τ on X such that R is in Γ in the product ( X, τ ) .This is an invariant in the sense that any relation which is Borel reducible to a relation potentially in Γ has also to be potentially in Γ . Along similar lines, any relation which is continuously reducible toa relation in Γ has also to be in Γ .We already mentioned the equivalence relations. A number of other interesting relations can beconsidered on a Polish space X , in the descriptive set theoretic context. Let us mention- the digraphs (which do not meet the diagonal ∆( X ) := { ( x, x ) | x ∈ X } of X ),- the graphs (i.e., the symmetric digraphs),- the oriented graphs (i.e., the antisymmetric digraphs),- the quasi-orders , strict or not,- the partial orders (i.e., the antisymmetric quasi-orders), strict or not.For instance, we refer to [Lo3], [L1], [K-Ma]. For locally countable relations (i.e., relations withcountable horizontal and vertical sections), we refer to [K2] in the case of equivalence relations. Animportant subclass of the class of locally countable Borel equivalence relations is the class of treeablelocally countable Borel equivalence relations, generated by an acyclic locally countable Borel graph.More generally, the locally countable digraphs have been widely considered, not necessarily to studyequivalence relations (see [K-Ma]). All this motivates the work in the present paper, mostly devotedto the study of the descriptive complexity of arbitrary locally countable or/and acyclic Borel relationson Polish spaces.We are looking for characterizations of the relations in a fixed Borel class Γ . So we will considerthe continuous and injective continuous reducibilities. In other words, we want to give answers to thefollowing very simple question: when is a relation Σ ξ (or Π ξ )? We are looking for characterizationsof the following form: a relation is either simple, or more complicated than a typical complex relation.So we need to introduce, for some Borel classes Γ , examples of complex relations. Notation.
Let Γ be a non self-dual Borel class, K be a metrizable compact space, and C ∈ ˇ Γ ( K ) \ Γ ( K ) .If the rank of Γ is one (i.e., if Γ ∈ { Σ , Π } ), then we set K := { } ∪ { − k | k ∈ ω } ⊆ R , C := { } if Γ = Σ , and C := K \{ } if Γ = Π , since we want some injectivity results.If the rank of Γ is at least two, then we set K := 2 ω , and C ∩ N s ∈ ˇ Γ ( N s ) \ Γ ( N s ) for each s ∈ <ω (this is possible, by Lemma 4.5 in [L5]). In particular, C is dense and co-dense in ω . We set C := P ∞ := { α ∈ ω | ∃ ∞ n ∈ ω α ( n ) = 1 } if Γ = Σ , and C := P f := { α ∈ ω | ∀ ∞ n ∈ ω α ( n ) = 0 } if Γ = Π , for injectivity reasons again. Inthe sequel, we will say that ( K , C ) is Γ - good if it satisfies all the properties mentioned here. Examples.
Let Γ be a non self-dual Borel class, and ( K , C ) be Γ -good. We define a relation on D := 2 × K as follows: ( ε, x ) E Γ ( η, y ) ⇔ ( ε, x ) = ( η, y ) ∨ ( x = y ∈ C ) . The graph G Γ m := E Γ \ ∆( D ) will be very important in the sequel. 3 ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣♣♣♣♣♣♣♣♣ (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) C ¬ C C ¬ C C ¬ CC ¬ C E Γ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣♣♣♣♣♣♣♣♣ (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) C ¬ C C ¬ C C ¬ CC ¬ C G Γ m The main result in [L5] is as follows. Most of our results will hold in analytic spaces and not only inPolish spaces. Recall that a separable metrizable space is an analytic space if it is homeomorphic toan analytic subset of a Polish space.
Theorem 1.1
Let Γ be a non self-dual Borel class of rank at least three. Then E Γ is ⊑ c -minimumamong non- Γ locally countable Borel equivalence relations on an analytic space. In fact, this result is also valid for equivalence relations with Σ classes, i.e., Σ sections. Recallthat if ( Q, ≤ ) is a quasi-ordered class, then a basis is a subclass B of Q such that any element of Q is ≤ -above an element of B . We are looking for basis as small as possible, so in fact for antichains(an antichain is a subclass of Q made of pairwise ≤ -incomparable elements). So we want antichainbasis. As we will see, the solution of our problem heavily depends on the rank of the non self-dualBorel class considered. The next result solves our problem for the classes of rank at least three. Theorem 1.2 (1) Let Γ be a non self-dual Borel class of rank at least three. Then there is a concrete34 elements ⊑ c and ≤ c -antichain basis for the class of non- Γ locally countable Borel relations on ananalytic space.(2) G Γ m is ⊑ c -minimum among non- Γ locally countable Borel graphs on an analytic space.(3) These results also hold when Γ has rank two for relations whose sections are in ∆( Γ ) . The next surprising result shows that this complexity assumption on the sections is useful for theclasses of rank two, for which any basis must have size continuum.
Theorem 1.3
Let Γ be a non self-dual Borel class of rank two. Then there is a concrete ≤ c -antichainof size continuum made of locally countable Borel relations on ω which are ≤ c and ⊑ c -minimalamong non- Γ relations on an analytic space. Similar results hold for graphs (see Corollary 3.13 and Theorem 3.15). Our analysis of the ranktwo also provides a basis for the class of non- Σ locally countable Borel relations on an analytic space(see Theorem 2.15). The proof of Theorem 1.3 strengthens Theorem 1.1 in [B] (see also Theorem6.31 in [T], in topological Ramsey theory on the space Q of rational numbers). Theorem 1.4
There is an onto coloring c : Q [2] → ω with the property that, for any H ⊆ Q homeomor-phic to Q , there is h : Q → H , homeomorphism onto its range, for which c ( { x, y } ) = c (cid:0) { h ( x ) , h ( y ) } (cid:1) if x, y ∈ Q . In particular, c takes all the values from ω on H [2] . Question.
For the classes of rank two, we saw that any basis must have size continuum. Is there anantichain basis?The next result solves our problem for the classes of rank one.4 heorem 1.5
Let Γ be a non self-dual Borel class of rank one. Then there is a concrete 7360 elements ≤ c -antichain basis, made of relations on a countable metrizable compact space, for the class of non- Γ relations on a first countable topological space. A similar result holds for ⊑ c , with 2 more elementsin the antichain basis. Note that in Theorem 1.5, the fact of assuming that X is analytic or that R is locally countableBorel does not change the result since the elements of the antichain basis satisfy these stronger as-sumptions. The “first countable” assumption ensures that closures and sequential closures coincide.Here again, similar results hold for graphs, with much smaller antichain basis (of cardinality ≤ c -5, ⊑ c -6 for Π , and 10 for Σ ). We will not give the proof of Theorem 1.5 since it is elementary. Wesimply describe the different antichain basis in Section 5. Remark.
Theorem 1.5 provides a finite antichain basis for the class of non-closed Borel relationson a Polish space, for ⊑ c and ≤ c . This situation is very different for the class C of non-potentiallyclosed Borel relations on a Polish space. Indeed, [L1] provides an antichain of size continuum madeof minimal elements of C , for any of these two quasi-orders. It also follows from [L-M] that in factthere is no antichain basis in C , for any of these two quasi-orders again.The works in [K-S-T], [L-M], [L-Z], [L4] and also [C-L-M] show that an acyclicity assumptioncan give positive dichotomy results (see, for example, Theorem 1.9 in [L4]). This is a way to fix ourproblem with the rank two. If A is a binary relation on a set X , then A − := { ( x, y ) ∈ X | ( y, x ) ∈ A } .The symmetrization of A is s ( A ) := A ∪ A − . An A - path is a finite sequence ( x i ) i ≤ n of points of X such that ( x i , x i +1 ) ∈ A if i < n . We say that A is acyclic if there is no injective A -path ( x i ) i ≤ n with n ≥ and ( x n , x ) ∈ A . In practice, we will consider acyclicity only for symmetric relations. Wewill say that A is s-acyclic if s ( A ) is acyclic. Theorem 1.6 (1) There is a concrete 34 elements ⊑ c and ≤ c -antichain basis for the class of non- Σ s-acyclic locally countable Borel relations on an analytic space.(2) G Σ m is ⊑ c -minimum among non- Σ acyclic locally countable Borel graphs on an analytic space. Notation.
For the class Π , we need some more examples since the complexity of a locally countablerelation can come from the complexity of one of its sections in this case. Let Γ be a non self-dualBorel class of rank at least two, and (2 ω , C ) be Γ -good. We set S := { ∞ } ∪ N (where N is thebasic clopen set { α ∈ ω | α (0) = 1 } ), and G Γ ,am := s ( { (0 ∞ , α ) | α ∈ C } ) . C ¬ C ♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ∞ C ¬ C • • ∞ G Γ ,am Theorem 1.7 (1) There is a concrete 52 elements ⊑ c and ≤ c -antichain basis for the class of non- Π s-acyclic locally countable Borel relations on an analytic space.(2) { ( D , G Π m ) , ( S , G Π ,am ) } is a ⊑ c and ≤ c -antichain basis for the class of non- Π acyclic locallycountable Borel graphs on an analytic space. Theorem 1.8 (1) Let Γ be a non self-dual Borel class of rank two.(1) There is a concrete 76 elements ⊑ c and ≤ c -antichain basis for the class of non- Γ s-acyclic Borelrelations on an analytic space.(2) { ( D , G Γ m ) , ( S , G Γ ,am ) } is a ⊑ c and ≤ c -antichain basis for the class of non- Γ acyclic Borel graphson an analytic space. This result can be extended, with a suitable acyclicity assumption. In [L4], it is shown that thecontainment in a s-acyclic Σ relation allows some positive reducibility results (see, for exampleTheorem 4.1 in [L4]). A natural way to ensure this is to have a s-acyclic closure (recall Theorem 1.9in [L4]). In sequential spaces like the Baire space ω ω , having s-acyclic levels is sufficient to ensurethis (see Proposition 2.7 in [L4]). Moreover, there is a s-acyclic closed relation on ω containing Borelrelations of arbitrarily high complexity (even potential complexity), by Proposition 3.17 in [L4]. Thenext result unifies the classes of rank at least two. Theorem 1.9
Let Γ be a non self-dual Borel class of rank at least two.(1) There is a concrete 76 elements ⊑ c and ≤ c -antichain basis for the class of non- Γ Borel relationson an analytic space contained in a s-acyclic Σ relation.(2) { ( D , G Γ m ) , ( S , G Γ ,am ) } is a ⊑ c and ≤ c -antichain basis for the class of non- Γ Borel graphs on ananalytic space contained in an acyclic Σ graph. Questions.
For the classes of rank at least three, we gave finite antichain basis for small relations. Isthere an antichain basis if we do not assume smallness? If yes, is it finite? Countable? Is it true thatany basis must have size continuum? The graph (cid:0) ω , C \ ∆(2 ω ) (cid:1) shows that we cannot simply erasethe acyclicity assumptions in Theorems 1.8.(2) and 1.9.(2).Theorems 1.2, 1.3-1.4, 1.6-1.9 are proved in Sections 2, 3, 4 respectively. In Section 6, we closethis study of ⊑ c by providing an antichain basis for the class of uncountable analytic relations on aHausdorff topological space, which gives a perfect set theorem for binary relations. We extend thenotation G Γ m , G Γ ,am to the class Γ = {∅} . Theorem 1.10 (1) There is a concrete 13 elements ⊑ c -antichain basis for the class of uncountableanalytic relations on a Hausdorff topological space.(2) { ( D , G {∅} m ) , ( S , G {∅} ,am ) , (2 ω , =) } is a ⊑ c -antichain basis for the class of uncountable analyticgraphs on a Hausdorff topological space. Note that (2 ω , =) is not acyclic, so that we recover the basis met in Theorems 1.8.(2) and 1.9.(2)in the acyclic case. Also, ( S , G {∅} ,am ) and (2 ω , =) are not locally countable. In conclusion, [L5]and the present study show that, when our finite antichain basis exist, they are small in the cases ofequivalence relations and graphs.We saw that there is no antichain basis in the class of non-potentially closed Borel relations ona Polish space. However, it follows from the main results in [L2] and [L3] that this problem can befixed if we allow partial reductions, on a closed relation (which in fact is suitable for any non self-dualBorel class). 6his solution involves the following quasi-order, less considered than ⊑ c and ≤ c . Let X , Y betopological spaces, and A , A ⊆ X (resp., B , B ⊆ Y ) be disjoint. Then we set ( X, A , A ) ≤ ( Y, B , B ) ⇔ ∃ f : X → Y continuous with ∀ ε ∈ A ε ⊆ ( f × f ) − ( B ε ) . A similar result holds here, for the Borel classes instead of the potential Borel classes. We define O Γ m := (cid:8)(cid:0) (0 , x ) , (1 , x ) (cid:1) | x ∈ C (cid:9) , so that G Γ m = s ( O Γ m ) . Theorem 1.11
Let Γ be a non self-dual Borel class.(1) Let X be an analytic space, and R be a Borel relation on X . Exactly one of the following holds:(a) the relation R is a Γ subset of X ,(b) ( D , O Γ m , O {∅} m \ O Γ m ) ≤ ( X, R, X \ R ) .(2) A similar statement holds for graphs, with ( G Γ m , G {∅} m \ G Γ m ) instead of ( O Γ m , O {∅} m \ O Γ m ) . This last result can be extended to any non self-dual Wadge class of Borel sets. Note that there isno injectivity in Theorem 1.11, because of the examples G Γ m and G Γ , a m for instance. We first extend Lemma 4.1 in [L5].
Lemma 2.1
Let Γ be a class of sets closed under continuous pre-images, Y, Z be topological spaces,and
R, S be a relation on
Y, Z respectively.(a) If R is in Γ , then the sections of R are also in Γ .(b) If S has vertical (resp., horizontal) sections in Γ and ( Y, R ) ≤ c ( Z, S ) , then the vertical(resp., horizontal) sections of R are also in Γ .(c) If S has countable vertical (resp., horizontal) sections and ( Y, R ) ⊑ c ( Z, S ) , then the vertical(resp., horizontal) sections of R are also countable. Proof. (a) comes from the fact that if y ∈ Y , then the maps i y : y ′ ( y, y ′ ) , j y : y ′ ( y ′ , y ) arecontinuous and satisfy R y = i − y ( R ) , R y = j − y ( R ) . The statements (b), (c) come from the facts that R y = f − ( S f ( y ) ) , R y = f − ( S f ( y ) ) . (cid:3) We now extend Theorem 4.3 in [L5].
Notation.
Let R be a relation on D . We set, for ε, η ∈ , R ε,η := (cid:8) ( α, β ) ∈ K | (cid:0) ( ε, α ) , ( η, β ) (cid:1) ∈ R (cid:9) . Theorem 2.2
Let Γ be a non self-dual Borel class of rank at least two, (2 ω , C ) be Γ -good, X be ananalytic space, and R be a Borel relation on X . Exactly one of the following holds:(a) the relation R is a Γ subset of X ,(b) one of the following holds:(1) the relation R has at least one section not in Γ ,(2) there is a relation R on ω such that R ∩ ∆(2 ω ) = ∆( C ) and (2 ω , R ) ⊑ c ( X, R ) ,(3) there is a relation R on D such that R , ∩ ∆(2 ω ) = ∆( C ) and ( D , R ) ⊑ c ( X, R ) . roof. We first note that (a) and (b) cannot hold simultaneously. Indeed, we argue by contradiction,so that R has sections in Γ by Lemma 2.1.(a), R ∈ Γ (cid:0) (2 ω ) (cid:1) , R ∈ Γ ( D ) , and R ∩ ∆(2 ω ) ∈ Γ (cid:0) ∆(2 ω ) (cid:1) , R , ∩ ∆(2 ω ) ∈ Γ (cid:0) ∆(2 ω ) (cid:1) respectively. This implies that C ∈ Γ (2 ω ) , which is absurd.Assume now that (a) and (b).(1) do not hold. Theorem 1.9 in [L5] gives f := ( f , f ) : 2 ω → X continuous with injective coordinates such that C = f − ( R ) . Case 1. f [ ¬ C ] ⊆ ∆( X ) .Note that f [2 ω ] ⊆ ∆( X ) , by the choice of C , so that f = f . We set R := ( f × f ) − ( R ) , so that R ∩ ∆(2 ω ) = ∆( C ) . Note that (2 ω , R ) ⊑ c ( X, R ) , with witness f . Case 2. f [ ¬ C ] ∆( X ) .We may assume that f and f have disjoint ranges, by the choice of C . We define g : D → X by g ( ε, α ) := f ε ( α ) . Note that g is injective continuous, (cid:8)(cid:0) (0 , α ) , (1 , α ) (cid:1) | α ∈ C (cid:9) ⊆ ( g × g ) − ( R ) and (cid:8)(cid:0) (0 , α ) , (1 , α ) (cid:1) | α / ∈ C (cid:9) ⊆ ( g × g ) − ( ¬ R ) . It remains to set R := ( g × g ) − ( R ) . (cid:3) The following property is crucial in the sequel, as well as in [L5].
Definition 2.3
Let f : K → K be a function, and C ⊆ K . We say that f preserves C if C = f − ( C ) . It is strongly related to condition (2) in Theorem 2.2.
Lemma 2.4
Let f : K → K be a function, C ⊆ K , and R be a relation on K with R ∩ ∆( K ) = ∆( C ) .Then the following are equivalent:(1) f preserves C .(2) ( f × f ) − ( R ) ∩ ∆( K ) = ∆( C ) . Proof.
We just apply the definitions. (cid:3)
Finally, we will make a strong use of the following result (see page 433 in [Lo-SR1]).
Theorem 2.5 (Louveau-Saint Raymond) Let ξ ≥ be a countable ordinal, ( K , C ) be Σ ξ -good (orsimply C ∈ Π ξ (2 ω ) \ Σ ξ (2 ω ) if ξ ≥ ), X be a Polish space, and A, B be disjoint analytic subsets of X . Then exactly one of the following holds:(a) the set A is separable from B by a Σ ξ set,(b) we can find f : K → X injective continuous such that C ⊆ f − ( A ) and ¬ C ⊆ f − ( B ) . A first consequence of this is Theorem 1.11 (these two results extend to the non self-dual Wadgeclasses of Borel sets, see Theorem 5.2 in [Lo-SR2]).
Proof of Theorem 1.11. (1) As C / ∈ Γ , O Γ m is not separable from O {∅} m \ O Γ m by a set in Γ , and (a),(b) cannot hold simultaneously. So assume that (a) does not hold. As X is separable metrizable, wemay assume that X is a subset of the Polish space [0 , ω (see 4.14 in [K1]). Note that the analyticset R is not separable from the analytic set X \ R by a Γ subset of ([0 , ω ) . Theorem 2.5 provides h : K → ([0 , ω ) continuous such that C ⊆ h − ( R ) and ¬ C ⊆ h − ( X \ R ) . Note that h takes valuesin X . We define f : D → X by f ( ε, x ) := h ε ( x ) , so that f is continuous. Moreover, (cid:0) (0 , x ) , (1 , x ) (cid:1) ∈ O Γ m ⇔ x ∈ C ⇔ h ( x ) = (cid:0) h ( x ) , h ( x ) (cid:1) ∈ R ⇔ (cid:0) f (0 , x ) , f (1 , x ) (cid:1) ∈ R ,so that (b) holds.(2) We just have to consider the symmetrizations of the relations appearing in (1). (cid:3) .2 Simplifications for the rank two We will see that, for classes of rank two, the basic examples are contained in ∆(2 ω ) ∪ P f . Thenext proof is the first of our two proofs using effective descriptive set theory (see also Theorem 4.6). Lemma 2.6
Let R be a Borel relation on P ∞ whose sections are separable from P f by a Σ set. Thenwe can find a sequence ( R n ) n ∈ ω of relations closed in P ∞ × ω and ω × P ∞ , as well as f : 2 ω → ω injective continuous preserving P f such that ( f × f ) − ( R ) ⊆ S n ∈ ω R n . Proof.
In order to simplify the notation, we assume that R is a ∆ relation on ω . Recall that we canfind Π sets W ⊆ ω × ω and C ⊆ ω × ω × ω such that ∆ ( α )(2 ω ) = { C α,n | ( α, n ) ∈ W } for each α ∈ ω and { ( α, n, β ) ∈ ω × ω × ω | ( α, n ) ∈ W ∧ ( α, n, β ) / ∈ C } is a Π subset of ω × ω × ω (see Section 2 in [Lo1]). Intuitively, W α is the set of codes for the ∆ ( α ) subsets of ω . We set W := { ( α, n ) ∈ W | C α,n is a Π ∩ ∆ ( α ) subset of ω } . Intuitively, ( W ) α is the set of codes forthe Π ∩ ∆ ( α ) subsets of ω . By Section 2 in [Lo1], the set W is Π . We set P := { ( α, n ) ∈ ω × ω | ( α, n ) ∈ W ∧ R α ⊆ ¬ C α,n ⊆ P ∞ } . Note that P is Π . Moreover, for each α ∈ P ∞ , there is n ∈ ω such that ( α, n ) ∈ P , by Theorem 2.B’in [Lo1]. The ∆ -selection principle provides a ∆ -recursive map f : 2 ω → ω such that (cid:0) α, f ( α ) (cid:1) ∈ P if α ∈ P ∞ (see 4B.5 in [Mo]). We set B := (cid:8) ( α, β ) ∈ P ∞ × ω | (cid:0) α, f ( α ) , β (cid:1) / ∈ C (cid:9) . Note that B isa ∆ set with vertical sections in Σ , and R α ⊆ B α ⊆ P ∞ for each α ∈ P ∞ . Theorem 3.6 in [Lo1]provides a finer Polish topology τ on ω such that B ∈ Σ (cid:0) (2 ω , τ ) × ω (cid:1) . Note that the identity mapfrom (2 ω , τ ) into ω is a continuous bijection. By 15.2 in [K1], it is a Borel isomorphism. By 11.5 in[K1], its inverse is Baire measurable. By 8.38 in [K1], there is a dense G δ subset G of ω on which τ coincides with the usual topology on ω . In particular, B ∩ ( G × ω ) ∈ Σ ( G × ω ) and we may assumethat G ⊆ P ∞ . Note that G is not separable from P f by a set in Σ , by Baire’s theorem. Theorem2.5 provides g : 2 ω → ω injective continuous such that P ∞ ⊆ g − ( G ) and P f ⊆ g − ( P f ) . Note that ( g × g ) − ( B ) is Σ in P ∞ × ω and contained in P ∞ . So, replacing B with ( g × g ) − ( B ) if necessary,we may assume that R is contained in a Borel set B which is Σ in P ∞ × ω and contained in P ∞ .Similarly, we may assume that R is contained in a Borel set D which is Σ in ω × P ∞ and contained in P ∞ . Let ( B p ) p ∈ ω , ( D q ) q ∈ ω be sequences of closed relations on ω with B = S p ∈ ω B p ∩ ( P ∞ × ω ) , D = S q ∈ ω D q ∩ (2 ω × P ∞ ) respectively. We set R p,q := B p ∩ D q ∩ P ∞ , so that R p,q is closed in P ∞ × ω and ω × P ∞ since, for example, R p,q := B p ∩ D q ∩ ( P ∞ × ω ) = B p ∩ D q ∩ (2 ω × P ∞ ) , and R ⊆ S p,q ∈ ω R p,q . (cid:3) Notation.
We define a well-order ≤ l of order type ω on <ω by s ≤ l t ⇔ | s | < | t | ∨ ( | s | = | t | ∧ s ≤ lex t ) and, as usual for linear orders, set s < l t ⇔ s ≤ l t ∧ s = t . Let b : ( ω, ≤ ) → (2 <ω , ≤ l ) bethe increasing bijection, α n +1 := b ( n )10 ∞ , α := 0 ∞ , so that P f = { α n | n ∈ ω } . We then set Q := {∅} ∪ { u | u ∈ <ω } , so that P f = { t ∞ | t ∈ Q } . Definition 2.7 A finitely dense Cantor set is a copy C of ω in ω such that P f ∩ C is dense in C . Note that if C is a finitely dense Cantor set, then P f ∩ C is countable dense, and also co-dense, in C , which implies that P f ∩ C is Σ and not Π in C , by Baire’s theorem.9 onventions. In the rest of Sections 2 and 3, we will perform a number of Cantor-like constructions.The following will always hold. We fix a finitely dense Cantor set C , and we want to construct f : 2 ω → C injective continuous preserving P f . We inductively construct a sequence ( n t ) t ∈ <ω of positive natural numbers, and a sequence ( U t ) t ∈ <ω of basic clopen subsets of C , satisfying thefollowing conditions. (1) U tε ⊆ U t (2) α n t ∈ U t (3) diam ( U t ) ≤ −| t | (4) U t ∩ U t = ∅ (5) n t = n t (6) U t ∩ { α n | n ≤ | t |} = ∅ Assume that this is done. Using (1)-(3), we define f : 2 ω → C by { f ( β ) } := T n ∈ ω U β | n , and f isinjective continuous by (4). If t ∈ Q and α = t ∞ , then f ( α ) = α n t by (5), so that P f ⊆ f − ( P f ) .Condition (6) ensures that P ∞ ⊆ f − ( P ∞ ) , so that f preserves P f . For the first step of the induction,we choose n ∅ ≥ in such a way that α n ∅ ∈ C , and a basic clopen neighbourhood U ∅ of α n ∅ . Condition(5) defines n t . It will also be convenient to set s t := b ( n t − . Lemma 2.8
Let ( R n ) n ∈ ω be a sequence of relations on P ∞ which are closed in P ∞ × ω and in ω × P ∞ . Then there is f : 2 ω → ω injective continuous preserving P f with the property that (cid:0) f ( α ) , f ( β ) (cid:1) / ∈ S n ∈ ω R n if α = β ∈ P ∞ . Proof.
We ensure (1)-(6) with C = 2 ω and (7) ( U s × U tε ) ∩ ( [ n ≤ l R n ) = ∅ if s = t ∈ l Assume that this is done. If α = β ∈ P ∞ , then we can find l with α | l = β | l , and a strictly increasingsequence ( l k ) k ∈ ω of natural numbers bigger than l such that α ( l k ) = 1 for each k . Condition (7)ensures that U α | ( l k +1) × U β | ( l k +1) does not meet S n ≤ l k R n , so that (cid:0) f ( α ) , f ( β ) (cid:1) / ∈ S n ∈ ω R n .So it is enough to prove that the construction is possible. We first set n ∅ := 1 and U ∅ := 2 ω .We choose n ≥ such that α n = α n , and U , U disjoint with diameter at most − such that α n ε ∈ U ε ⊆ ω \{ α } . Assume that ( n t ) | t |≤ l and ( U t ) | t |≤ l satisfying (1)-(7) have been constructed forsome l ≥ , which is the case for l = 1 . We set F := { α n t | t ∈ l } ∪ { α n | n ≤ l } and L := S n ≤ l R n . Claim.
Let
U, U , · · · , U m be nonempty open subsets of ω , γ i ∈ P f ∩ U i , for each i ≤ m , and F be a finite subset of P f containing { γ i | i ≤ m } . Then we can find γ ∈ P f ∩ U and clopen subsets V, V , · · · , V m of ω with diameter at most − l − such that γ ∈ V ⊆ U \ ( F ∪ V ) , γ i ∈ V i ⊆ U i and (cid:0) ( V × V i ) ∪ ( V i × V ) (cid:1) ∩ L = ∅ for each i ≤ m . Indeed, fix i ≤ m . Note that P ∞ ×{ γ i } ⊆ P ∞ × P f and L ⊆ P ∞ are disjoint closed subsets of thezero-dimensional space P ∞ × ω . This gives a clopen subset i C of P ∞ × ω with P ∞ ×{ γ i } ⊆ i C ⊆ ¬ L (see 22.16 in [K1]). So for each β ∈ P ∞ we can find a clopen subset i,β O of P ∞ and a clopen subset i,β D of ω such that ( β, γ i ) ∈ i,β O × i,β D ⊆ i C and i,β D ⊆ U i .10s P ∞ = S β ∈ P ∞ i,β O , we can find a sequence ( β p ) p ∈ ω of points of P ∞ with the property that P ∞ = S p ∈ ω i,β p O . We set i,p O := i,β p O \ ( S q
Let R be a locally countable Borel relation on ω . Then we can find f : 2 ω → ω injective continuous preserving P f such that R ′ := ( f × f ) − ( R ) ⊆ ∆(2 ω ) ∪ P f . Proof. As P f is countable and R is locally countable, the set C := S α ∈ P f ( R α ∪ R α ) is countable.We set G := P ∞ \ C , so that G ⊆ P ∞ is a non-meager subset of ω having the Baire property. Lemma7.2 in [L5] provides f : 2 ω → ω injective continuous such that f [ P ∞ ] ⊆ G and f [ P f ] ⊆ P f . Thisproves that we may assume that R ∩ (cid:0) ( P ∞ × P f ) ∪ ( P f × P ∞ ) (cid:1) = ∅ . By Lemma 2.6 applied to R ∩ P ∞ ,we may assume that there is a sequence ( R n ) n ∈ ω of relations closed in P ∞ × ω and ω × P ∞ suchthat R ∩ P ∞ ⊆ S n ∈ ω R n . It remains to apply Lemma 2.8. (cid:3) Corollary 2.9 leads to the following.
Definition 2.10
Let Γ be a non self-dual Borel class of rank two, and (2 ω , C ) be Γ -good. A relation R on ω is diagonally complex if it it satisfies the following:(1) R ∩ ∆(2 ω ) = ∆( C ) ,(2) R ⊆ ∆( C ) ∪ P f . Note that a diagonally complex relation is not in Γ by (1), and locally countable Borel by (2). Notation.
We set, for any digraph D on Q , R D := ∆( C ) ∪ { ( s ∞ , t ∞ ) | ( s, t ) ∈ D} . Note that anydiagonally complex relation is of the form R D , for some digraph D on Q .In our future Cantor-like constructions, the definition of n t will be by induction on ≤ l , exceptwhere indicated. Corollary 2.9 simplifies the locally countable Borel relations on ω . Some furthersimplification is possible when the sections are nowhere dense. Lemma 2.11
Let R be a relation on ω with nowhere dense sections. Then there is f : 2 ω → ω injective continuous preserving P f such that (cid:0) f ( α ) , f ( β ) (cid:1) / ∈ R if α = β ∈ P f . Proof.
We ensure (1)-(6) with C = 2 ω and (7) ( α n s , α n t ) , ( α n t , α n s ) / ∈ R if t ∈ Q ∧ s ∈ | t | ∧ s < lex t Assume that this is done. If α = β ∈ P f , then for example α < lex β and there are initial segments s, t of α, β respectively satisfying the assumption in (7). Condition (7) ensures that (cid:0) f ( α ) , f ( β ) (cid:1) / ∈ R .12o it is enough to prove that the construction is possible. We first set n ∅ := 1 and U ∅ := 2 ω .Assume that ( n t ) | t |≤ l and ( U t ) | t |≤ l satisfying (1)-(7) have been constructed, which is the case for l = 0 . Fix t ∈ l . We choose n t ≥ such that α n t ∈ U t \ ( { α n t } ∪ { α n | n ≤ l } ∪ [ s ∈ l +1 ,s< lex t R α ns ∪ R α ns ) ,which exists since R has nowhere dense sections. We then choose a clopen neighbourhood with smalldiameter U tε of α n tε contained in U t , ensuring (4) and (6). (cid:3) Corollary 2.12
Let R be a diagonally complex relation.(1) If R has nowhere dense sections, then (2 ω , R ∅ ) ⊑ c (2 ω , R ) .(2) If P f \R has nowhere dense sections, then (2 ω , R = ) ⊑ c (2 ω , R ) . Proof.
Lemma 2.11 gives f : 2 ω → ω injective continuous preserving P f such that (cid:0) f ( α ) , f ( β ) (cid:1) isnot in R (resp., in R ) if α = β ∈ P f . Note that ( f × f ) − ( R ) = R ∅ (resp., ( f × f ) − ( R ) = R = ). (cid:3) Theorem 1.2 provides a basis. Theorem 2.15 to come provides another one, and is a consequenceof Corollary 2.9.
Notation.
Let Γ be a non self-dual Borel class of rank at least two, and (2 ω , C ) be Γ -good. We set S := C , S := ∅ , S := 2 ω \ C , S := 2 ω . The next result motivates the introduction of these sets. Lemma 2.13
Let Γ be a non self-dual Borel class of rank at least two, (2 ω , C ) be Γ -good, and B bea Borel subset of ω . Then we can find j ∈ and f : 2 ω → ω injective continuous preserving C suchthat f − ( B ) = S j . Proof.
Note that since C ∈ ˇ Γ (2 ω ) \ Γ (2 ω ) , either C \ B is not separable from ¬ C by a set in Γ , or C ∩ B is not separable from ¬ C by a set in Γ , because Γ is closed under finite unions. Assume, for example,that the first case occurs. Similarly, either C \ B is not separable from ( ¬ C ) ∩ ( ¬ B ) by a set in Γ , or C \ B is not separable from ( ¬ C ) ∩ B by a set in Γ , because Γ is closed under finite intersections.This shows that one of the following cases occurs:- C \ B is not separable from ( ¬ C ) ∩ ( ¬ B ) by a set in Γ ,- C \ B is not separable from ( ¬ C ) ∩ B by a set in Γ ,- C ∩ B is not separable from ( ¬ C ) ∩ ( ¬ B ) by a set in Γ ,- C ∩ B is not separable from ( ¬ C ) ∩ B by a set in Γ .Assume, for example, that we are in the first of these four cases. By Theorem 2.5, there is f : 2 ω → ω injective continuous such that C ⊆ f − ( C \ B ) and ¬ C ⊆ f − (cid:0) ( ¬ C ) ∩ ( ¬ B ) (cid:1) . In thiscase, f − ( B ) = ∅ = S . In the other cases, f − ( B ) = S , S , S , respectively. So we proved that f − ( B ) = S j for some j ∈ . (cid:3) Corollary 2.14 Γ be a non self-dual Borel class of rank at least two, (2 ω , C ) be Γ -good, and R be a Borel relation on D . Then there is f : 2 ω → ω injective continuous preserving C such that ( f × f ) − ( R ε,η ) ∩ ∆(2 ω ) ∈ { ∆( S j ) | j ≤ } for each ε, η ∈ . roof. We set, for ε, η ∈ , E ε,η := { α ∈ ω | ( α, α ) ∈ R ε,η } , so that E ε,η is a Borel subset of ω and R ε,η ∩ ∆(2 ω ) = ∆( E ε,η ) . Now fix ε, η ∈ . Lemma 2.13 provides j ∈ and g : 2 ω → ω injectivecontinuous preserving C such that ( g × g ) − ( E ε,η ) = S j . We just have to apply this for each ε, η ∈ . (cid:3) Theorem 2.15
Let Γ be a non self-dual Borel class of rank two, (2 ω , C ) be Γ -good, X be an analyticspace, and R be a locally countable Borel relation on X whose sections are in Γ . Exactly one of thefollowing holds.(a) the relation R is a Γ subset of X ,(b) one of the following holds:(1) there is a diagonally complex relation R on ω such that (2 ω , R ) ⊑ c ( X, R ) ,(2) there is a relation R on D such that, for each ε, η ∈ , if R ε,η ∩ ∆(2 ω ) = ∆( E ε,η ) , then(i) E , = C = S , E , ∈ { S j | j ≤ } , and E ε,ε ∈ { S j | ≤ j ≤ } ,(ii) R ε,η ⊆ ∆( E ε,η ) ∪ P f (in particular, R , is diagonally complex),and ( D , R ) ⊑ c ( X, R ) . Proof.
By Theorem 2.2, (a) and (b) cannot hold simutaneously. Assume that (a) does not hold. ByTheorem 2.2 again, one of the following holds.(1) There is a relation R on ω such that R ∩ ∆(2 ω ) = ∆( C ) and (2 ω , R ) ⊑ c ( X, R ) . By Corollary2.9 we may assume that R ⊆ ∆(2 ω ) ∪ P f , so that R is a diagonally complex relation.(2) There is a relation R ′ on D such that R ′ , ∩ ∆(2 ω ) = ∆( C ) and ( D , R ′ ) ⊑ c ( X, R ) . Note that R ′ ε,η is a locally countable Borel relation on ω , for each ε, η ∈ . Corollary 2.9 provides g ′ : 2 ω → ω injective continuous preserving C such that ( g ′ × g ′ ) − ( S ε,η ∈ R ′ ε,η ) ⊆ ∆(2 ω ) ∪ P f . We define h : D → D by h ( ε, α ) := (cid:0) ε, g ′ ( α ) (cid:1) and set R = ( h × h ) − ( R ′ ) , so that R ε,η ⊆ ∆(2 ω ) ∪ P f . By Corollary2.14, we may assume that R ε,η ∩ ∆(2 ω ) ∈ { ∆( S j ) | j ≤ } . We are done since we are reduced toCase (1) if R ε,ε ∩ ∆(2 ω ) = ∆( C ) . (cid:3) We now introduce a first antichain basis.
Notation.
Let Γ be a non self-dual Borel class of rank at least two, and (2 ω , C ) be Γ -good. We set P := (cid:8) t ∈ (2 ) | t (0 , , t (1 , = 0 ∧ t (0 ,
1) = 0 ∧ (cid:0) t (1 ,
0) = 0 ⇒ t (0 , ≤ t (1 , (cid:1)(cid:9) and, for t ∈ P , R Γ t := (cid:8)(cid:0) ( ε, x ) , ( η, x ) (cid:1) ∈ D | x ∈ S t ( ε,η ) (cid:9) . We order lexicographically, so that, forexample, E Γ = R Γ , , , . Note that G Γ m = R Γ , , , . Finally, A Γ := (cid:8)(cid:0) ω , ∆( C ) (cid:1)(cid:9) ∪ { ( D , R Γ t ) | t ∈ P } .Note that the sections of the elements of A Γ have cardinality at most two, and are in particular closed. Lemma 2.16
Let Γ be a non self-dual Borel class of rank at least two. Then A Γ is a 34 elements ≤ c -antichain. Proof.
Let ( X , R ) = ( X ′ , R ′ ) in A Γ . We argue by contradiction, which gives f : X → X ′ continuous.Assume first that ( X , R ) , ( X ′ , R ′ ) are of the form ( D , R Γ t ) , ( D , R Γ t ′ ) respectively, so that f ( ε, α ) is ofthe form (cid:0) f ( ε, α ) , f ( ε, α ) (cid:1) ∈ × ω . 14et us prove that f (0 , α ) = f (1 , α ) if α ∈ C . We argue by contradiction, which gives l ∈ ω such that f (0 , β ) = f (1 , β ) =: ε if β ∈ N α | l , by continuity of f . This also gives continuous maps g η : N α | l → ω such that f ( η, β ) = g η ( β ) if β ∈ N α | l . If β ∈ C ∩ N α | l , then (cid:0) (0 , β ) , (1 , β ) (cid:1) ∈ R , sothat (cid:0) f (0 , β ) , f (1 , β ) (cid:1) = (cid:16)(cid:0) ε, g ( β ) (cid:1) , ( ε, g ( β ) (cid:1)(cid:17) ∈ R ′ , g ( β ) = g ( β ) and g = g =: g by continuityof g , g . Note that there is j ∈ { , , } such that C ∩ N α | l = g − ( S j ) ∩ N α | l , which contradicts thechoice of C .Fix α ∈ C . Note that there is l ∈ ω such that ε := f (0 , β ) = f (1 , β ) if β ∈ N α | l , by continuity of f . There are g η : N α | l → ω continuous such that f ( η, β ) = g η ( β ) if β ∈ N α | l . If β ∈ C ∩ N α | l , then (cid:0) (0 , β ) , (1 , β ) (cid:1) ∈ R , so that (cid:0) f (0 , β ) , f (1 , β ) (cid:1) = (cid:16)(cid:0) ε , g ( β ) (cid:1) , (1 − ε , g ( β ) (cid:1)(cid:17) ∈ R ′ , g ( β ) = g ( β ) and g = g =: g by continuity of g , g . Note that C ∩ N α | l = g − ( S t ′ ( ε , − ε ) ) ∩ N α | l , so that t ′ ( ε , − ε ) = 0 by the choice of C and C ∩ N α | l = g − ( C ) ∩ N α | l . If ε = 0 , then S t ( ε,η ) ∩ N α | l = g − ( S t ′ ( ε,η ) ) ∩ N α | l = S t ′ ( ε,η ) ∩ N α | l for ε, η ∈ , so that t = t ′ by the choice of C . Thus ε = 1 and S t ( ε,η ) ∩ N α | l = g − ( S t ′ (1 − ε, − η ) ) ∩ N α | l = S t ′ (1 − ε, − η ) ∩ N α | l for ε, η ∈ , so that t ( ε, η ) = t ′ (1 − ε, − η ) if ε, η ∈ , by the choice of C . In particular, note that t ′ (1 ,
0) = t (0 ,
1) = 0 = t ′ (0 ,
1) = t (1 , , t ′ (0 , , t ′ (1 , = 0 , t (1 ,
1) = t ′ (0 , ≤ t ′ (1 ,
1) = t (0 , , t (0 , , t (1 , = 0 and t ′ (1 ,
1) = t (0 , ≤ t (1 ,
1) = t ′ (0 , , so that t = t ′ again.We now have to consider (cid:0) ω , ∆( C ) (cid:1) . Assume first that (cid:0) ω , ∆( C ) (cid:1) ≤ c ( D , R Γ t ) , with ( D , R t ) in A Γ and witness f . Let α ∈ C . Then we can find ε ∈ with f ( α ) = ε , and l ∈ ω such that f ( β ) = ε if β ∈ N α | l , by continuity of f . Note that C ∩ N α | l = f − ( S t ( ε ,ε ) ) ∩ N α | l , so that t ( ε , ε ) = 0 bythe choice of C , which contradicts the definition of A Γ .Assume now that ( D , R Γ t ) ≤ c (cid:0) ω , ∆( C ) (cid:1) , with ( D , R t ) in A Γ and witness f . Let α ∈ C . Then (cid:0) (0 , α ) , (1 , α ) (cid:1) ∈ R Γ t , so that f (0 , α ) = f (1 , α ) ∈ C . By the choice of C , g ( β ) := f (0 , β ) = f (1 , β ) foreach β ∈ ω . We set, for ε, η ∈ , R ε,η := (cid:8) ( α, β ) ∈ ω × ω | (cid:0) ( ε, α ) , ( η, β ) (cid:1) ∈ R Γ t (cid:9) . Note that ( α, β ) ∈ R ε,η ⇔ (cid:0) g ( α ) , g ( β ) (cid:1) ∈ ∆( C ) ,so that R , = R , , which contradicts the definition of A Γ . (cid:3) The next result provides the basis part of Theorem 1.2.
Lemma 2.17
Let Γ be a non self-dual Borel class of rank at least two, X be an analytic space, and R be a locally countable Borel relation on X whose sections are in ∆( Γ ) . Exactly one of the followingholds:(a) the relation R is a Γ subset of X ,(b) there is ( X , R ) ∈ A Γ such that ( X , R ) ⊑ c ( X, R ) . roof. By Theorem 2.2, (a) and (b) cannot hold simutaneously. Assume that (a) does not hold. ByTheorem 2.2 again, one of the following holds.(1) There is a relation R on ω such that R ∩ ∆(2 ω ) = ∆( C ) and (2 ω , R ) ⊑ c ( X, R ) . As R islocally countable Borel, so is R by Lemma 2.1, so that we can apply Corollary 3.3 in [L5] if therank of Γ is at least three. This gives g : 2 ω → ω injective continuous such that g preserves C ,and (cid:0) g ( α ) , g ( β ) (cid:1) / ∈ R if α = β . Note that g is a witness for the fact that (cid:0) ω , ∆( C ) (cid:1) ⊑ c (2 ω , R ) ,so that (cid:0) ω , ∆( C ) (cid:1) ⊑ c ( X, R ) . If the rank of Γ is two, then by Corollary 2.9 we may assume that R ⊆ ∆(2 ω ) ∪ P f . As the sections of R are countable and Π , so are those of R . In particular, R has nowhere dense sections. By Corollary 2.12, we may assume that R = ∆( C ) , so that, here again, (cid:0) ω , ∆( C ) (cid:1) ⊑ c ( X, R ) .(2) There is a relation R on D such that R , ∩ ∆(2 ω ) = ∆( C ) and ( D , R ) ⊑ c ( X, R ) . Note that R ε,η is a locally countable Borel relation on ω . If the rank of Γ is at least three, then Corollary 3.3 in [L5]provides g : 2 ω → ω injective continuous such that g preserves C , and (cid:0) g ( α ) , g ( β ) (cid:1) / ∈ S ε,η ∈ R ε,η if α = β . If the rank of Γ is two, then Corollary 2.9 provides g ′′ : 2 ω → ω injective continuouspreserving C such that R ′′ := ( g ′′ × g ′′ ) − ( S ε,η ∈ R ε,η ) ⊆ ∆(2 ω ) ∪ P f . As the sections of R arecountable and Π , so are those of R ′′ . In particular, R ′′ has nowhere dense sections. By Lemma 2.11,we may assume that R ′′ ⊆ ∆(2 ω ) . So we may assume that g exists in both cases.We define h : D → D by h ( ε, α ) := (cid:0) ε, g ( α ) (cid:1) , so that h is injective and continuous. Wethen set R ′ := ( h × h ) − ( R ) . Repeating the notation above, R ′ ε,η ⊆ ∆(2 ω ) by the property of g , and h is a witness for the fact that ( D , R ′ ) ⊑ c ( D , R ) . This means that we may assume that R ε,η ⊆ ∆(2 ω ) , for ε, η ∈ , and that R , = ∆( C ) = ∆( S ) . By Corollary 2.14, we may assume that R ε,η ∈ { ∆( S j ) | j ≤ } . Note that if R ε,ε = ∆( S ) for some ε ∈ , then (cid:0) ω , ∆( C ) (cid:1) ⊑ c ( X, R ) . So wemay assume that R ε,ε = ∆( S j ) for some j ∈ { , , } . Finally, ( D , R Γ i, , ,j ) ⊑ c ( D , R Γ j, , ,i ) if i > j > with witness ( ε, α ) (1 − ε, α ) . So (b) holds. (cid:3) Proof of Theorem 1.2.
For (1) and (3), we apply Lemmas 2.16 and 2.17. For (2), we use the closureproperties of ⊑ c and the fact that G Γ m is the only graph in A Γ . (cid:3) In this section, Γ is a non self-dual Borel class of rank two. Up to restrictions to Cantor sets, two comparable diagonally complex relations are bi-reducible.
Lemma 3.1
Let R , R ∗ be diagonally complex, and K be a finitely dense Cantor set with the propertythat ( K, R ∗ ∩ K ) ≤ c (2 ω , R ) . Then ( C, R ∩ C ) ⊑ c ( K, R ∗ ∩ K ) for some C finitely dense. Proof.
Let f be a witness for the fact that ( K, R ∗ ∩ K ) ≤ c (2 ω , R ) . Note that f preserves C since R ∗ , R agree with ∆( C ) on ∆(2 ω ) . This implies that f is nowhere dense-to-one. Let us prove thatthere is g : 2 ω → K continuous such that g preserves C and f (cid:0) g ( α ) (cid:1) < lex f (cid:0) g ( β ) (cid:1) if α < lex β . Weensure (1)-(6) with C = K and (4) f ( α ) < lex f ( β ) if α ∈ U t and β ∈ U t α < lex β , then Condition (4) ensures that f (cid:0) g ( α ) (cid:1) < lex f (cid:0) g ( β ) (cid:1) .So it is enough to prove that the construction is possible. We set U ∅ := K . Assume that ( n t ) | t |≤ l and ( U t ) | t |≤ l satisfying (1)-(6) have been constructed, which is the case for l = 0 . Fix t ∈ l . Note that α n t is in U t ∩ P f , so that f ( α n t ) ∈ P f . This gives s ∈ <ω such that f ( α n t ) = s ∞ . In particular, thereis a clopen neighbourhood N ⊆ U t of α n t such that f ( β ) ∈ N s if β ∈ N . We choose n t ≥ such that α n t ∈ N \ (cid:16) { α n | n ≤ l } ∪ f − (cid:0) { f ( α n t ) } (cid:1)(cid:17) , which is possible since f is nowhere dense-to-one. Notethat f ( α n t ) < lex f ( α n t ) . It remains to choose a small enough clopen neighbourhood U tε of α n tε tofinish the construction, using the continuity of f .Note then that f | g [2 ω ] is a homeomorphism onto ˜ C := f (cid:2) g [2 ω ] (cid:3) . Moreover, (cid:0) g ( α ) , g ( β ) (cid:1) ∈ R ∗ ⇔ (cid:16) f (cid:0) g ( α ) (cid:1) , f (cid:0) g ( β ) (cid:1)(cid:17) ∈ R ,so that f − | g [2 ω ] is a witness for the fact that ( ˜ C, R ∩ ˜ C ) ⊑ c ( K, R ∗ ∩ K ) , and R ∩ ˜ C is not in Γ since the map α (cid:16) f (cid:0) g ( α ) (cid:1) , f (cid:0) g ( α ) (cid:1)(cid:17) reduces C to R ∩ ˜ C . This implies that ˜ C ∩ C is notseparable from ˜ C \ C by a Γ set. Using Theorem 2.5, we get m : 2 ω → ˜ C injective continuous suchthat C = m − ( C ) . It remains to set C := m [2 ω ] . (cid:3) The minimality of diagonally complex relations can be seen on restrictions to Cantor sets.
Lemma 3.2
Let R be a diagonally complex relation, X be an analytic space, and R be a non- Π relation on X such that ( X, R ) ≤ c (2 ω , R ) , with witness f . Then there is g : 2 ω → X injectivecontinuous such that P f = g − (cid:0) f − ( P f ) (cid:1) . Proof.
Note that R = ∆( S ) ∪ ( R ∩ P f ) , where S = P ∞ if Γ = Σ , and S = ∅ if Γ = Π . As P f \R iscountable, it is a Σ subset of P f , and R ∩ P f is a Π subset of P f , which provides G ∈ Π (2 ω × ω ) such that R ∩ P f = G ∩ P f . As R = ( f × f ) − ( R ) = ( f × f ) − (cid:0) ∆( S ) (cid:1) ∪ (cid:0) ( f × f ) − ( G ) ∩ f − ( P f ) (cid:1) , f − ( P f ) is not Π . Theorem 2.5 provides g as desired. (cid:3) Corollary 3.3
Let R be a diagonally complex relation, X be an analytic space, and R be a non- Γ relation on X with ( X, R ) ≤ c (2 ω , R ) . Then there is a finitely dense Cantor set C such that ( C, R ∩ C ) ⊑ c ( X, R ) . Proof.
Assume first that Γ = Π , and let f be a witness for the fact that ( X, R ) ≤ c (2 ω , R ) . Lemma3.2 provides g : 2 ω → X injective continuous such that P f = g − (cid:0) f − ( P f ) (cid:1) . We set R ′ := ( g × g ) − ( R ) .Note that, for each α ∈ ω , ( α, α ) ∈ R ′ ⇔ (cid:0) g ( α ) , g ( α ) (cid:1) ∈ R ⇔ (cid:16) f (cid:0) g ( α ) (cid:1) , f (cid:0) g ( α ) (cid:1)(cid:17) ∈ R ⇔ f (cid:0) g ( α ) (cid:1) ∈ P f ⇔ α ∈ P f . In other words, R ′ ∩ ∆(2 ω ) = ∆( P f ) . This argument also shows that R ′ ⊆ P f . In other words, R ′ is diagonally complex, (2 ω , R ′ ) ⊑ c ( X, R ) and (2 ω , R ′ ) ≤ c (2 ω , R ) . Lemma 3.1 provides a finitelydense Cantor set C such that ( C, R ∩ C ) ⊑ c (2 ω , R ′ ) .17ssume now that Γ = Σ . By Theorem 2.2, one of the following holds:(1) there is a relation R ∗ on ω such that R ∗ ∩ ∆(2 ω ) = ∆( P ∞ ) and (2 ω , R ∗ ) ⊑ c ( X, R ) .(2) there is a relation R ∗ on D such that D ∞ := (cid:8)(cid:0) (0 , α ) , (1 , α ) (cid:1) | α ∈ P ∞ (cid:9) ⊆ R ∗ , D f := (cid:8)(cid:0) (0 , α ) , (1 , α ) (cid:1) | α ∈ P f (cid:9) ⊆ ¬ R ∗ and ( D , R ∗ ) ⊑ c ( X, R ) . We set D := D ∞ ∪ D f .Assume that (2) holds, which gives f : D → ω continuous such that R ∗ = ( f × f ) − ( R ) . Note that- ( f × f )[ D ∞ ] ⊆ R is not separable from ( f × f )[ D f ] by a Σ set,- ( f × f )[ D ∞ ] ∩ ∆(2 ω ) is not separable from ( f × f )[ D f ] by a Σ set since R\ ∆(2 ω ) ⊆ P f is Σ ,- I ∞ := ( f × f )[ D ∞ ] ∩ ∆(2 ω ) is not separable from I f := ( f × f )[ D f ] ∩ ∆(2 ω ) by a Σ set,- R ∞ := D ∩ ( f × f ) − ( I ∞ ) is not separable from R f := D ∩ ( f × f ) − ( I f ) by a Σ set (otherwise R ∞ ⊆ S ⊆ ¬ R f and I ∞ is separable from I f by the K σ set ( f × f )[ S ∩ D ] ),- C ∞ := (cid:8) α ∈ ω | (cid:0) (0 , α ) , (1 , α ) (cid:1) ∈ R ∞ (cid:9) is not separable from C f := (cid:8) α ∈ ω | (cid:0) (0 , α ) , (1 , α ) (cid:1) ∈ R f (cid:9) by a Σ set.Theorem 2.5 provides g : 2 ω → ω injective continuous with the properties that P ∞ ⊆ g − ( C ∞ ) and P f ⊆ g − ( C f ) . We define a map h : 2 ω → { } × ω by h ( α ) := (0 , α ) , and set c := h ◦ g , c ′ := f ◦ c , R ′ := ( c ′ × c ′ ) − ( R ) = ( c × c ) − ( R ∗ ) . As c is injective continuous, R ′ is a Borel rela-tion on ω and (2 ω , R ′ ) ⊑ c ( D , R ∗ ) , ( X, R ) . If α ∈ P ∞ , then (cid:16)(cid:0) , g ( α ) (cid:1) , (cid:0) , g ( α ) (cid:1)(cid:17) ∈ R ∞ , and (cid:16) f (cid:0) , g ( α ) (cid:1) , f (cid:0) , g ( α ) (cid:1)(cid:17) ∈ I ∞ ⊆ ∆(2 ω ) . This implies that (cid:16) f (cid:0) , g ( α ) (cid:1) , f (cid:0) , g ( α ) (cid:1)(cid:17) ∈ R and ( α, α ) ∈ R ′ . Similarly, if α ∈ P f , then ( α, α ) / ∈ R ′ . Thus R ′ ∩ ∆(2 ω ) = ∆( P ∞ ) , and R ′ is a witness forthe fact that (1) also holds. So (1) holds in any case.This implies that (2 ω , R ∗ ) ≤ c (2 ω , R ) , with witness f ′ . By Lemma 2.4, f ′ preserves P f . Inparticular, R ∗ ∩ (cid:0) ( P f × P ∞ ) ∪ ( P ∞ × P f ) (cid:1) = ∅ since R is diagonally complex. Note that R ∗ has Σ sections since R has. This implies that R ∗ ∩ P ∞ is a Borel relation on P ∞ whose sections are separablefrom P f by a Σ set. Lemmas 2.6 and 2.8 provide g ′ : 2 ω → ω injective continuous preserving P f such that ( g ′ × g ′ ) − ( R ∗ ∩ P ∞ ) ⊆ ∆( P ∞ ) . Thus we may assume that R ∗ is diagonally complex. Wenow apply Lemma 3.1. (cid:3) A consequence of Lemma 3.1 is a characterisation of the minimality of diagonally complex rela-tions.
Corollary 3.4
Let R be a diagonally complex relation. The following are equivalent:(a) R is ≤ c and ⊑ c -minimal among non- Γ relations on an analytic space,(b) (2 ω , R ) ⊑ c ( C, R ∩ C ) if C is a finitely dense Cantor set. Proof. (a) ⇒ (b) As P f ∩ C is dense in C , R ∩ ∆( C ) and R ∩ C are not in Γ . As ( C, R ∩ C ) ⊑ c (2 ω , R ) and (a) holds, (b) holds. 18b) ⇒ (a) Let X be an analytic space and R be a non- Γ relation on X with ( X, R ) ≤ c (2 ω , R ) .Corollary 3.3 provides a finitely dense Cantor set C with the property that ( C, R ∩ C ) ⊑ c ( X, R ) .By (b), (2 ω , R ) ⊑ c ( X, R ) . (cid:3) Another consequence of Lemma 3.1 is about the comparison of minimal diagonally complexrelations.
Corollary 3.5
Let R , R ′ be diagonally complex relations, ≤ c and ⊑ c -minimal among among non- Γ relations on an analytic space, with (2 ω , R ) ≤ c (2 ω , R ′ ) . Then (2 ω , R ′ ) ⊑ c (2 ω , R ) and (2 ω , R ) ⊑ c (2 ω , R ′ ) . Proof.
Lemma 3.1 provides a finitely dense Cantor set C such that ( C, R ′ ∩ C ) ⊑ c (2 ω , R ) . ByCorollary 3.4, (2 ω , R ′ ) ⊑ c (2 ω , R ) . An application of this fact implies that (2 ω , R ) ⊑ c (2 ω , R ′ ) . (cid:3) We now introduce our antichain of size continuum.
Notation.
We define i : Q → ω as follows. We want to ensure that i ( z, t ) = i ( s z , s t ) , where s t isdefined before Lemma 2.8. The definition of i is partly inspired by the oscillation map osc definedin [T] after Theorem 6.33 as follows. The elements of <ω are identified with finite subsets of ω ,through the characteristic function. The oscillation between z and t describes the behavious of thesymmetric difference z ∆ t . The equivalence relation ∼ zt is defined on z ∆ t by j ∼ zt k ⇔ [ min ( j, k ) , max ( j, k )] ∩ ( z \ t ) = ∅ ∨ [ min ( j, k ) , max ( j, k )] ∩ ( t \ z ) = ∅ . Then osc ( z, t ) := | z ∆ t/ ∼ zt | . For i , we work on z ∪ t instead of z ∆ t , and the definition depends moreheavily on the initial segments of z and t , in particular on their lexicographic ordering. If z ∈ <ω \{∅} ,then we set z − := z | max { l < | z | | z | l ∈ Q } . We also set ⊥ < := { ( z, t ) ∈ Q | ∃ i < min ( | z | , | t | ) z | i = t | i ∧ z ( i ) < t ( i ) } and, similarly, ⊥ > := { ( z, t ) ∈ Q | ∃ i < min ( | z | , | t | ) z | i = t | i ∧ z ( i ) > t ( i ) } . The definition of i ( z, t ) is by induction on max ( | z | , | t | ) . We set i ( z, t ) := if z = t , i ( z, t − ) if | z | < | t − | ∨ (cid:0) | z | = | t − | ∧ ( z, t − ) ∈⊥ < (cid:1) , i ( z, t − )+1 if ( | z | < | t | ∧ | z | > | t − | ) ∨ (cid:0) | z | = | t − | ∧ ( z, t − ) / ∈⊥ < (cid:1) , i ( z − , t ) if | t | < | z − | ∨ (cid:0) | t | = | z − | ∧ ( t, z − ) ∈⊥ < (cid:1) , i ( z − , t )+1 if ( | t | < | z | ∧ | t | > | z − | ) ∨ (cid:0) | t | = | z − | ∧ ( t, z − ) / ∈⊥ < (cid:1) , i ( z − , t − )+1 if | z | = | t |∧ (cid:0) ( | z − | < | t − |∧ t − < lex z − ) ∨ ( | t − | < | z − |∧ z − < lex t − ) (cid:1) , i ( z − , t − )+2 if | z | = | t |∧ (cid:0) ( | z − | < | t − |∧ z − < lex t − ) ∨ ( | t − | < | z − |∧ t − < lex z − ) ∨ ( | z − | = | t − |∧ z − = t − ) (cid:1) . Note that i ( z, t ) = i ( t, z ) if z, t ∈ Q . 19 emma 3.6 Let ( s t ) t ∈ <ω be a sequence of elements of Q with ( a ) | s z | < | s t | if z < l t are in Q ( b ) s t $ s t if t is in Q Then i ( z, t ) = i ( s z , s t ) if z, t ∈ Q . Proof.
We argue by induction on max ( | z | , | t | ) . As i ( z, t ) = i ( t, z ) if z, t ∈ Q , we may assume that z < l t . We go through the cases of the definition of i .If | z | < | t − | , then | s z | < | s t − | ≤ (cid:12)(cid:12) ( s t ) − (cid:12)(cid:12) since s t − $ s t , and i ( z, t ) = i ( z, t − ) = i ( s z , s t − ) = · · · = i (cid:0) s z , ( s t ) − (cid:1) = i ( s z , s t ) . If | z | = | t − | and ( z, t − ) ∈⊥ < , then | s z | < | s t − | ≤ | ( s t ) − | again and we conclude as above.If | z | < | t | and | z | > | t − | , then | s z | < | s t | and | s z | > | s t − | . Let s ∈ Q with s ⊆ s t and | s z | < | s | be ofminimal length. Note that i ( z, t ) = i ( z, t − )+1 = i ( s z , s t − )+1 = i ( s z , s ) = i ( s z , s t ) .If | z | = | t − | and ( z, t − ) / ∈⊥ < , then either z = t − , or ( z, t − ) ∈⊥ > , so that i ( z, t ) = i ( z, t − )+1 = i ( s z , s t − )+1 = · · · = i (cid:0) s z , ( s t ) − (cid:1) = i ( s z , s t ) . If | z | = | t | , | t − | < | z − | and z − < lex t − , then | s t − | < | s z − | < | s z | < | s t | , i ( z, t ) = i ( z − , t − )+1 = i ( s z − , s t − )+1 = i ( s z , s t − )+1 = i ( s z , s t ) . If | z | = | t | , | z − | < | t − | and z − < lex t − , then | s z − | < | s t − | < | s z | < | s t | , i ( z, t ) = i ( z − , t − )+2 = i ( s z − , s t − )+2 = i ( s z , s t − )+1 = i ( s z , s t ) . If | z | = | t | , | z − | = | t − | and z − = t − , then ( z − , t − ) ∈⊥ < , | s z − | < | s t − | < | s z | < | s t | , and we conclude asabove. Thus i ( z, t ) = i ( s z , s t ) in any case, as desired. (cid:3) The next result is the key lemma to prove Theorems 1.3 and 1.4.
Lemma 3.7
Let H ⊆ P f be homeomorphic to P f . Then there is f : 2 ω → ω injective continuoussatisfying the following properties:(i) f [ P f ] ⊆ H ; in particular, for each t ∈ Q there is n t ≥ with f ( t ∞ ) = s t ∞ ,(ii) f [ P ∞ ] ⊆ P ∞ ,(iii) i ( z, t ) = i ( s z , s t ) if z, t ∈ Q . Proof.
We ensure (1)-(6) with C := H , that U t is of the form N z t ∩ C , and (2) α n t ∈ N z t ∩ H if t is in <ω (4) z t < lex z t and | z t | = | z t | if t is in <ω (7) | s z | < | s t | if z < l t are in Q (8) s t $ s t if t is in Q (9) i ( z, t ) = i ( s z , s t ) if z, t ∈ Q
20t is enough to prove that the construction is possible. We choose n ∅ ≥ with s ∅ ∞ ∈ H , andset z ∅ := s ∅ . Assume that ( n t ) | t |≤ l and ( z t ) | t |≤ l satisfying (1)-(9) have been constructed, which is thecase for l = 0 .Fix t ∈ l . As α n t ∈ U t = N z t ∩ C , z t ⊆ s t ∞ . We choose s ⊆ s t ∞ extending z t and s t , insuch a way that N s ∩ ( { α n | n ≤ l } \ { α n t } ) = ∅ and | s z | < | s | if z < l t . Let n t ≥ such that α n t ∈ N s ∩ H \{ α n t } , which is possible by density of H in the perfect space C . We set z t = s t and z t = α n t || z t | . Note that (1) holds. (2) holds by definition. By our extensions, (3) holds. (4) holdsbecause of the choice of s and n t . (5)-(8) hold by construction. By Lemma 3.6, (9) holds. (cid:3) Proof of Theorem 1.4.
We may replace Q with its topological copy P f . We set, for z, t ∈ Q , c ( { z ∞ , t ∞ } ) := i ( z, t ) , which is well-defined by symmetry of i . As i (cid:0) (10) k ∞ , (01) k ∞ (cid:1) = 2 k and i (cid:0) (01) k ∞ , (10) k +1 ∞ (cid:1) = 2 k +1 , c is onto. If H ⊆ P f is homeomorphic to P f , then Lemma 3.7provides f : 2 ω → ω . We set h := f | P f . If z, t ∈ Q , then c (cid:0) { h ( z ∞ ) , h ( t ∞ ) } (cid:1) = c ( { s z ∞ , s t ∞ } ) = i ( s z , s t ) = i ( z, t ) = c ( { z ∞ , t ∞ } ) . In particular, c takes all the values from ω on H [2] . (cid:3) Notation.
We now set, when (2 ω , C ) is Γ -good and β ∈ ω , R β := ∆( C ) ∪ [ β ( p )=1 { ( z ∞ , t ∞ ) | z = t ∈ Q ∧ i ( z, t ) = p } ,so that R β is symmetric. Corollary 3.8
Let β ∈ ω . Then R β is ≤ c and ⊑ c -minimal among non- Γ relations on an analyticspace. Proof.
Note that R β is a diagonally complex relation, and therefore not in Γ . Let C be a finitelydense Cantor set. By Corollary 3.4, it is enough to see that the inequality (2 ω , R β ) ⊑ c ( C, R β ∩ C ) holds. We apply Lemma 3.7 to H := C ∩ P f . As C is finitely dense, H is nonempty, countable,metrizable, perfect, and therefore homeomorphic to P f (see 7.12 in [K]). Lemma 3.7 provides awitness f : 2 ω → ω for the fact that (2 ω , R β ) ⊑ c ( C, R β ∩ C ) . (cid:3) Lemma 3.9
The family ( R β ) β ∈ N is a ≤ c -antichain. Proof.
Assume that β, β ′ ∈ N and (2 ω , R β ) ≤ c (2 ω , R β ′ ) , with witness f . By Lemma 2.4, f preserves C . Claim.
Let u ∈ <ω . Then we can find u ′ ∈ Q and u , u ′ ∈ Q \{∅} such that f ( u ∞ ) = u ′ ∞ and f ( u u ∞ ) = u ′ u ′ ∞ . Indeed, let u ′ ∈ Q with f ( u ∞ ) = u ′ ∞ . We can find a sequence ( β k ) k ∈ ω of points of P ∞ converging to u ∞ . As f preserves C , f ( u ∞ ) = f ( β k ) , and we can find ( n k ) k ∈ ω stricly increasingsuch that α n k | k = β k | k and f ( u ∞ ) = f ( α n k ) , so that ( α n k ) k ∈ ω is a sequence of points differentfrom u ∞ converging to u ∞ . Let v k , v ′ k ∈ Q with α n k = v k ∞ and f ( α n k ) = v ′ k ∞ . We mayassume that u $ v k . As f is continuous, we may assume that f ( α n k ) ∈ N u ′ for each k , so that u ′ $ v ′ k . It remains to choose u , u ′ ∈ Q \{∅} such that v = u u and v ′ = u ′ u ′ . ⋄ t , · · · , t k +2 ∈ <ω are not initial segments of ∞ . Then ( t t | t | · · · t k − | t k | t k +1 ∞ , t | t | t · · · | t k +1 | t k +2 ∞ ) is in { ( z ∞ , t ∞ ) | z = t ∈ Q ∧ i ( z, t ) = 2 k +2 } . Similarly, ( t | t | t · · · | t k − | t k ∞ , t t | t | · · · t k − | t k | t k +1 ∞ ) is in { ( z ∞ , t ∞ ) | z = t ∈ Q ∧ i ( z, t ) = 2 k + 1 } . We first apply the claim to u := t ∈ Q , whichgives t ′ ∈ Q and t , t ′ ∈ Q \{∅} with the properties that f ( t ∞ ) = t ′ ∞ and f ( t t ∞ ) = t ′ t ′ ∞ .The continuity of f provides k ≥ | t | such that f [ N t k ] ⊆ N t ′ | t ′ | . We next apply the claim to u := t k , which gives ˜ t , ˜ t ′ in Q \{∅} such that f ( t k ˜ t ∞ ) = t ′ ˜ t ′ ∞ . Note that | t ′ | ⊆ ˜ t ′ . Weset t := 0 k −| t | ˜ t and t ′ := ˜ t ′ − | t ′ | , so that t , t ′ are not initial segments of ∞ and f ( t | t | t ∞ ) = t ′ | t ′ | t ′ ∞ . This argument shows that we can find sequences of finite binary sequences ( t j ) j ∈ ω and ( t ′ j ) j ∈ ω whichare not initial segments of ∞ and satisfy f ( t | t | t · · · | t k − | t k ∞ ) = t ′ | t ′ | t ′ · · · | t ′ k − | t ′ k ∞ and f ( t t | t | · · · t k − | t k | t k +1 ∞ ) = t ′ t ′ | t ′ | · · · t ′ k − | t ′ k | t ′ k +1 ∞ for each k ∈ ω . By theremark after the claim, β = β ′ . (cid:3) Proof of Theorem 1.3.
We apply Corollary 3.8 and Lemma 3.9. (cid:3)
Theorem 1.3 shows that if ≤ is in {≤ c , ⊑ c } , then the class of non- Π countable relations onanalytic spaces, equipped with ≤ , contains antichains of size continuum made of minimal relations. Theorem 1.3 shows that, for the classes of rank two, any basis must have size continuum. It isnatural to ask whether it is also the case for graphs. We will see that it is indeed the case.
Notation.
We define, for β ∈ ω , G β := s (cid:16)(cid:8)(cid:0) (0 , α ) , (1 , γ ) (cid:1) ∈ D | ( α, γ ) ∈ R β (cid:9)(cid:17) . Note that G β is alocally countable graph. Lemma 3.10
Let Γ := Σ , (2 ω , C ) be Γ -good, β ∈ ω , X be an analytic space, R be a relation on X ,and R be a relation on D such that R , ∩ ∆(2 ω ) = ∆( C ) and ( D , R ) ⊑ c ( X, R ) ⊑ c ( D , G β ) . Then (2 ω , R β ) ⊑ c (2 ω , R , ) with witness h having the property that ( R , ∪ R , ) ∩ h [2 ω ] = ∅ . Proof.
Let g, f be witnesses for the fact that ( D , R ) ⊑ c ( X, R ) , ( X, R ) ⊑ c ( D , G β ) respectively.We set, for ε ∈ , X ε := { x ∈ X | f ( x ) = ε } , which defines a partition of X into clopen sets. Thedefinition of G β shows that R ⊆ ( X × X ) ∪ ( X × X ) . If α ∈ C , then (cid:0) g (0 , α ) , g (1 , α ) (cid:1) ∈ R ,which gives ε ∈ and l ∈ ω such that g (0 , α ) ∈ X ε and g (1 , α ) ∈ X − ε if α ∈ N α | l . This showsthat R ∩ (2 × N α | l ) ⊆ s (cid:16)(cid:8)(cid:0) ( ε , α ) , (1 − ε , γ ) (cid:1) | α, γ ∈ ω (cid:9)(cid:17) . Note that R ε,η is a relation on ω . As R , ∩ ∆(2 ω ) = ∆( C ) , R , is not in Γ . Moreover, by symmetry of R β , ( α, γ ) ∈ R , ⇔ (cid:0) (0 , α ) , (1 , γ ) (cid:1) ∈ R ⇔ (cid:0) g (0 , α ) , g (1 , γ ) (cid:1) ∈ R ⇔ (cid:16) f (cid:0) g (0 , α ) (cid:1) , f (cid:0) g (1 , γ ) (cid:1)(cid:17) ∈ G β ⇔ (cid:16) f (cid:0) g (0 , α ) (cid:1) , f (cid:0) g (1 , γ ) (cid:1)(cid:17) ∈ R β if α ∈ N α | l . 22 laim. f (cid:0) g (0 , α ) (cid:1) = f (cid:0) g (1 , α ) (cid:1) if α ∈ C ∩ N α | l . Indeed, note that α ∈ C ⇔ (cid:0) (0 , α ) , (1 , α ) (cid:1) ∈ R ⇔ (cid:16) f (cid:0) g (0 , α ) (cid:1) , f (cid:0) g (1 , α ) (cid:1)(cid:17) ∈ R β if α ∈ N α | l .We argue by contradiction, which gives α ∈ C ∩ N α | l , t ∈ <ω , η ∈ and l ′ ≥ l with the propertiesthat f (cid:0) g (0 , α ) (cid:1) ∈ N tη and f (cid:0) g (1 , α ) (cid:1) ∈ N t (1 − η ) if α ∈ N α | l ′ . In particular, C ∩ N α | l ′ = n α ∈ N α | l ′ | (cid:16) f (cid:0) g (0 , α ) (cid:1) , f (cid:0) g (1 , α ) (cid:1)(cid:17) ∈ R β ∩ ( N tη × N t (1 − η ) ) o ,which shows that C ∩ N α | l ′ ∈ Γ = Σ since R β \ ∆(2 ω ) is countable, contradicting the choice of C . ⋄ By the claim, continuity of f, g and density of C , f (cid:0) g (0 , α ) (cid:1) = f (cid:0) g (1 , α ) (cid:1) if α ∈ N α | l . Thisshows that ( N α | l , R , ∩ N α | l ) ≤ c (2 ω , R β ) . By the proof of Corollary 3.4 and Corollary 3.8, (2 ω , R β ) ⊑ c ( N α | l , R , ∩ N α | l ) ,and we are done. (cid:3) Lemma 3.11
Let Γ := Σ , (2 ω , C ) be Γ -good, and β ∈ ω . Then G β is ⊑ c -minimal among non- Γ relations on an analytic space. Proof.
Let X be an analytic space, and R be a non- Γ relation on X with ( X, R ) ⊑ c ( D , G β ) . As G β has sections in Γ = Σ , R too. By Theorem 2.2, one of the following holds:(1) there is a relation R on ω such that R ∩ ∆(2 ω ) = ∆( C ) and (2 ω , R ) ⊑ c ( X, R ) ,(2) there is a relation R on D such that R , ∩ ∆(2 ω ) = ∆( C ) and ( D , R ) ⊑ c ( X, R ) .As G β is a digraph, R and R are digraphs too. So (1) cannot hold. By Lemma 3.10, (2 ω , R β ) ⊑ c (2 ω , R , ) ,with witness h having the property that ( R , ∪ R , ) ∩ h [2 ω ] = ∅ . We define k : D → D by k ( ε, α ) := (cid:0) ε, h ( α ) (cid:1) . Note that k is injective, continuous, and (cid:0) (0 , α ) , (1 , γ ) (cid:1) ∈ G β ⇔ (cid:0) h ( α ) , h ( γ ) (cid:1) ∈ R , ⇔ (cid:0) k (0 , α ) , k (1 , γ ) (cid:1) ∈ R ,showing that ( D , G β ) ⊑ c ( X, R ) as desired. (cid:3) Lemma 3.12
Let Γ := Σ , and (2 ω , C ) be Γ -good. Then ( G β ) β ∈ N is a ⊑ c -antichain. Proof.
Let β, β ′ ∈ N . Assume that ( D , G β ) ⊑ c ( D , G β ′ ) . Lemma 3.10 shows that (2 ω , R β ′ ) ⊑ c (cid:0) ω , ( G β ) , (cid:1) = (2 ω , R β ) ,so that β = β ′ by Lemma 3.9. (cid:3) Corollary 3.13
There is a concrete ⊑ c -antichain of size continuum made of locally countable Borelgraphs on D which are ⊑ c -minimal among non- Σ graphs on an analytic space. Proof.
We apply Lemmas 3.11 and 3.12. (cid:3)
We now turn to the study of the class Π . 23 emma 3.14 Assume that Γ = Π and β ∈ N \{ ∞ } . Then R β has a non- Π section. Proof.
We argue by contradiction, so that R β has nowhere dense sections since it is locally countable.By Corollary 2.12, (cid:0) ω , ∆( C ) (cid:1) ⊑ c (2 ω , R β ) . Note that (2 ω , R β ) ⊑ c (cid:0) ω , ∆( C ) (cid:1) by Theorem 1.2 andCorollary 3.5, which contradicts the injectivity since β ∈ N \{ ∞ } . (cid:3) Remark.
Lemma 3.14 shows that a locally countable relation can have a non- Π section, and there-fore be non- Π because of that. This is not the case for the class Σ . This is the reason why wecannot argue for the class Π as we did in Corollary 3.13 for the class Σ . More precisely, notethat G Π ,am is a (locally) countable graph with a non- Π section. Moreover, ( S , G Π ,am ) ⊑ c ( D , G β ) for any β ∈ N \{ ∞ } . Indeed, Lemma 3.14 gives α ∈ ω such that ( R β ) α is not Π since R β issymmetric. By Theorem 2.5, there is g : 2 ω → ω injective continuous such that P f = g − (cid:0) ( R β ) α (cid:1) .We define f : S → D by f (0 ∞ ) := (0 , α ) and f (1 α ) := (cid:0) , g ( α ) (cid:1) , so that f is injective continuous and ( εα, ηγ ) ∈ G Π ,am ⇔ ( ε = 0 ∧ α = 0 ∞ ∧ η = 1 ∧ γ ∈ P f ) ∨ ( ε = 1 ∧ α ∈ P f ∧ η = 0 ∧ γ = 0 ∞ ) ⇔ (cid:18)(cid:0) f ( εα ) , f ( ηγ ) (cid:1) = (cid:16) (0 , α ) , (cid:0) , g ( γ ) (cid:1)(cid:17) ∧ (cid:0) α , g ( γ ) (cid:1) ∈ R β (cid:19) ∨ (cid:18)(cid:0) f ( εα ) , f ( ηγ ) (cid:1) = (cid:16)(cid:0) , g ( α ) (cid:1) , (0 , α ) (cid:17) ∧ (cid:0) α , g ( α ) (cid:1) ∈ R β (cid:19) ⇔ (cid:0) f ( εα ) , f ( ηγ ) (cid:1) ∈ G β . So we need to find other examples for the class Π . Theorem 3.15
There is a concrete ⊑ c -antichain of size continuum made of locally countable Borelgraphs on ω which are ⊑ c -minimal among non- Π graphs on an analytic space. Proof.
We set, for β ∈ N \{ ∞ } , G β := R β \ ∆(2 ω ) . As R β is symmetric, G β is a graph. In particular,we can speak of proj [ G β ] . By Lemma 3.14, R β has a non- Π section. A consequence of this is that G β is not Π . Let X be an analytic space and R be a non- Π relation on X with ( X, R ) ⊑ c (2 ω , G β ) ,with witness f . By injectivity of f , we get (cid:16) X, R ∪ ( f × f ) − (cid:0) ∆( C ) (cid:1)(cid:17) ⊑ c (2 ω , R β ) . As ∆( C ) ∈ Σ is disjoint from G β , R ∪ ( f × f ) − (cid:0) ∆( C ) (cid:1) is not Π . By Corollary 3.8 and injectivity of f , (2 ω , R β ) ⊑ c (cid:16) X, R ∪ ( f × f ) − (cid:0) ∆( C ) (cid:1)(cid:17) . By injectivity of f again, (2 ω , G β ) ⊑ c ( X, R ) , showingthe minimality of G β .Assume now that β, β ′ ∈ N \{ ∞ } and (2 ω , G β ) ⊑ c (2 ω , G β ′ ) , with witness f . Claim.
Let u ∈ <ω with u ∞ ∈ proj [ G β ] . Then we can find u ′ ∈ Q and u , u ′ ∈ Q \{∅} such that f ( u ∞ ) = u ′ ∞ , u u ∞ ∈ proj [ G β ] and f ( u u ∞ ) = u ′ u ′ ∞ . Indeed, as u ∞ ∈ proj [ G β ] , there is γ such that ( u ∞ , γ ) is in G β , and (cid:0) f ( u ∞ ) , f ( γ ) (cid:1) istherefore in G β ′ . In particular, f ( u ∞ ) ∈ P f and there is u ′ ∈ Q with f ( u ∞ ) = u ′ ∞ . As f iscontinuous, there is k such that f [ N u k ] ⊆ N u ′ .Let us prove that proj [ G β ] is dense in ω . Let s ∈ <ω , and p := 2 k + ε ≥ with β ( p ) = 1 . We set ( z, t ) := (cid:0) s k + ε , s k (cid:1) , so that z = t ∈ Q , i ( z, t ) = p , ( z ∞ , t ∞ ) ∈ G β ∩ N s and proj [ G β ] meets N s as desired. 24et α ∈ proj [ G β ] ∩ N u k , and δ such that ( α, δ ) is in G β . Then (cid:0) f ( α ) , f ( δ ) (cid:1) is in G β ′ , and f ( α ) ∈ P f ∩ N u ′ \ { u ′ ∞ } by injectivity of f . This gives u , u ′ ∈ Q \ {∅} with α = u u ∞ and f ( α ) = u ′ u ′ ∞ . ⋄ The end of the proof is now similar to that of Lemma 3.9. Let us indicate the differences. We firstapply the claim to u := t ∈ Q such that t ∞ ∈ proj [ G β ] , which gives t ′ ∈ Q and t , t ′ ∈ Q \ {∅} such that f ( t ∞ ) = t ′ ∞ , t t ∞ ∈ proj [ G β ] and f ( t t ∞ ) = t ′ t ′ ∞ . The continuity of f provides k ≥ | t | such that f [ N t k ] ⊆ N t ′ | t ′ | . We next apply the claim to u := t k , which gives ˜ t , ˜ t ′ in Q \{∅} such that t k ˜ t ∞ ∈ proj [ G β ] and f ( t k ˜ t ∞ ) = t ′ ˜ t ′ ∞ . This provides sequences ( t j ) j ∈ ω and ( t ′ j ) j ∈ ω as in the proof of Lemma 3.9. By the remark after the claim in the proof of Lemma 3.9, β = β ′ . (cid:3) Remark.
Assume that β ∈ N \ { ∞ } . Then R β is not s-acyclic. Indeed, (0 ∞ , ∞ , ∞ ) is a s ( R β ) -cycle if β (1) = 1 , (10 ∞ , ∞ , ∞ ) is a s ( R β ) -cycle if β (2) = 1 , (101 k +1 ∞ , k +3 ∞ , k ∞ ) is a s ( R β ) -cycle if β (2 k + 3) = 1 , and (0101 k ∞ , (10 ) k ∞ , (101) k ∞ ) is a s ( R β ) -cycle if β (2 k + 4) = 1 . We will see that Theorem 1.2 can be extended, under a suitableacyclicity assumption. We need to introduce new examples. Notation.
Let Γ be a non self-dual Borel class of rank at least two, and (2 ω , C ) be Γ -good. We set A := (cid:8) t ∈ (2 ) | t (0 , ∈ ∧ (cid:0) t (0 ,
1) = 0 ∨ t (1 ,
0) = 0 (cid:1) ∧ t (1 , = 0 (cid:9) . We then set, for t ∈ A , R Γ ,at := { (0 ∞ , ∞ ) | t (0 ,
0) = 1 } ∪ { (0 ∞ , α ) | α ∈ S t (0 , } ∪ { (1 α, ∞ ) | α ∈ S t (1 , } ∪{ (1 α, α ) | α ∈ S t (1 , } . Note that G Γ ,am = R Γ ,a , , , . Finally B Γ := (cid:8) ( S , R Γ ,at ) | t ∈ A (cid:9) . Lemma 4.1
Let Γ be a non self-dual Borel class of rank at least two. Then A Γ ∪ B Γ is a 76 elements ≤ c -antichain. Proof.
Assume that t, t ′ ∈ A and R Γ ,at is ≤ c -below R Γ ,at ′ with witness f : S → S . If R Γ ,at ′ is symmetric,then R Γ ,at is too, and t ′ is of the form ( ε, , , ε ′ ) . As R Γ ,at , R Γ ,at ′ have only one vertical sectionnot in Γ , f (0 ∞ ) = 0 ∞ , f (1 , α )(0) = 1 by the choice of C , and the function f (1 , . ) defined by f (1 , α ) = f (1 α ) preserves C . Thus t = t ′ . So we may assume that R Γ ,at ′ is not symmetric, i.e., t ′ is of the form ( ε, ε ′ , ε ′′ , ε ′′′ ) with ε ′ = ε ′′ , and ε ′ = 0 or ε ′′ = 0 . If, for example, ε ′ = 0 , then R Γ ,at ′ hasvertical sections in Γ , as well as R Γ ,at , so that t is of the form ( η, η ′ , , η ′′′ ) with η ′ = 0 . Note that R Γ ,at , R Γ ,at ′ have only one horizontal section not in Γ , f (0 ∞ ) = 0 ∞ , f (1 , α )(0) = 1 by the choice of C , and f (1 , . ) preserves C . Thus t = t ′ , even if ε ′′ = 0 .As the elements of B Γ have a section not in Γ and the elements of A Γ have closed sections, anelement of B Γ is not ≤ c -below an element of A Γ . By Theorem 1.2, it remains to see that an element R of A Γ is not ≤ c -below an element R Γ ,at of B Γ . We argue by contradiction, which provides f : 2 ω → S or f : D → S . 25n the first case, note first that f ( α )(0) = 1 , since otherwise f ( β )(0) = 0 if β is in a clopenneighbourhood C of α . If β = γ ∈ C ∩ C , then f ( β ) = f ( γ ) = 0 ∞ , so that (0 ∞ , ∞ ) ∈ R Γ ,at and ( β, γ ) ∈ ∆( C ) , which is absurd. This shows that ∆( C ) ∈ Γ since ( R Γ ,at ) , ∈ Γ , which isabsurd. In the second case, note first that f (0 , α )(0) = 1 , since otherwise f (0 , β )(0) = 0 if β isin a clopen neighbourhood C of α . We may assume that there is ε ∈ with the property that f (1 , β )(0) = ε if β ∈ C . If β = γ ∈ C ∩ C , then either ε = 0 , (cid:0) f (0 , β ) , f (1 , γ ) (cid:1) = (0 ∞ , ∞ ) ∈ R Γ ,at and (cid:0) (0 , β ) , (1 , γ ) (cid:1) ∈ R , or ε = 1 , (cid:0) f (0 , β ) , f (1 , β ) (cid:1) , (cid:0) f (0 , γ ) , f (1 , γ ) (cid:1) ∈ R Γ ,at , f (1 , β ) , f (1 , γ ) arein { α | α ∈ C } , (cid:0) f (0 , β ) , f (1 , γ ) (cid:1) ∈ R Γ ,at and (cid:0) (0 , β ) , (1 , γ ) (cid:1) ∈ R , which is absurd. Similarly, f (1 , α )(0) = 1 . This shows that R , ∈ Γ since ( R Γ t ) , ∈ Γ , which is absurd. (cid:3) We now study the rank two.
Lemma 4.2
Let R be a s-acyclic Borel relation on P ∞ . Then we can find a sequence ( R n ) n ∈ ω ofrelations closed in P ∞ × ω and ω × P ∞ , as well as f : 2 ω → ω injective continuous preserving P f such that ( f × f ) − ( R ) ⊆ S n ∈ ω R n . Proof.
Assume first that R α is not separable from P f by a set in Σ , for some α ∈ P ∞ . Theorem 2.5provides h : 2 ω → ω \{ α } injective continuous such that P ∞ ⊆ h − ( R α ) and P f ⊆ h − ( P f ) . We set R ′ := ( h × h ) − ( R ) . If α, β, γ ∈ P ∞ are pairwise distinct and β, γ ∈ R ′ α , then (cid:0) h ( β ) , h ( α ) , h ( γ ) , α (cid:1) isa s ( R ) -cycle, which is absurd. Thus R ′ has vertical sections of cardinality at most two. So, replacing R with R ′ if necessary, we may assume that R α is separable from P f by a set in Σ for each α ∈ ω .Similarly, we may assume that R α is separable from P f by a set in Σ for each α ∈ ω . It remains toapply Lemma 2.6. (cid:3) Lemma 4.3 (a) Let R be a s-acyclic subrelation of P f . Then there is f : 2 ω → ω injective continuouspreserving P f such that ( f × f ) − ( R ) ⊆ ∆( P f ) .(b) Let R be a s-acyclic subrelation of (2 × P f ) . Then there is f : 2 ω → ω injective continuouspreserving P f such that ( f × f ) − ( S ε,η ∈ R ε,η ) ⊆ ∆( P f ) . Proof. (a) Assume that R α is not nowhere dense for some α ∈ P f . This gives s ∈ <ω with the prop-erty that N s ⊆ R α . Note that N s ∩ R α \{ α } is not separable from N s ∩ P ∞ by a Π set, by Baire’stheorem. Theorem 2.5 provides h : 2 ω → N s \{ α } injective continuous such that P f ⊆ h − ( R α ) and P ∞ ⊆ h − ( P ∞ ) . If R ′ := ( h × h ) − ( R ) , α, β, γ ∈ P f are pairwise distinct and β, γ ∈ R ′ α , then (cid:0) h ( β ) , h ( α ) , h ( γ ) , α (cid:1) is a s ( R ) -cycle, which is absurd. This shows that, replacing R with R ′ if nec-essary, we may assume that R has vertical sections of cardinality at most two. In any case, we mayassume that R has nowhere dense vertical sections. Similarly, we may assume that R has nowheredense horizontal sections. It remains to apply Lemma 2.11.(b) We argue similarly. Fix ε, η ∈ . We replace R with R ε,η . We just have to note that the cycle (cid:0) h ( β ) , h ( α ) , h ( γ ) , α (cid:1) becomes (cid:16)(cid:0) η, h ( β ) (cid:1) , (cid:0) ε, h ( α ) (cid:1) , (cid:0) η, h ( γ ) (cid:1) , (cid:0) ε, α (cid:1)(cid:17) . (cid:3) Lemma 4.4 (a) Let R be a s-acyclic Borel subrelation of ( P f × P ∞ ) ∪ ( P ∞ × P f ) . Then there is f : 2 ω → ω injective continuous preserving P f such that ( f × f ) − ( R ) = ∅ .(b) Let R be a s-acyclic Borel subrelation of (cid:0) (2 × P f ) × (2 × P ∞ ) (cid:1) ∪ (cid:0) (2 × P ∞ ) × (2 × P f ) (cid:1) . Thenthere is f : 2 ω → ω injective continuous preserving P f such that ( f × f ) − ( S ε,η ∈ R ε,η ) = ∅ . roof. (a) By symmetry, we may assume that R ⊆ P f × P ∞ . Assume first that R α is not meagerfor some α ∈ P f . This gives s ∈ <ω with the property that N s ∩ R α is comeager in N s . Inparticular, N s ∩ P f \ { α } is not separable from N s ∩ R α by a Π set, by Baire’s theorem. Theorem2.5 provides h : 2 ω → N s \ { α } injective continuous with P f ⊆ h − ( P f ) and P ∞ ⊆ h − ( R α ) . If R ′ := ( h × h ) − ( R ) , α ∈ P f , β = γ ∈ R ′ α , then (cid:0) h ( β ) , h ( α ) , h ( γ ) , α (cid:1) is a s ( R ) -cycle, which isabsurd. This shows that, replacing R with R ′ if necessary, we may assume that R has meager verticalsections. Let F be a meager Σ subset of ω containing S α ∈ P f R α . Note that P f is not separablefrom P ∞ \ F by a Π set. Theorem 2.5 provides f : 2 ω → ω injective continuous with P f ⊆ f − ( P f ) and P ∞ ⊆ f − ( P ∞ \ F ) . Thus ( f × f ) − ( R ) is empty.(b) We argue as in the proof of Lemma 4.3.(b). (cid:3) Corollary 4.5 (a) Let R be a s-acyclic Borel relation on ω . Then there is f : 2 ω → ω injectivecontinuous preserving P f such that ( f × f ) − ( R ) ⊆ ∆(2 ω ) .(b) Let R be a s-acyclic Borel relation on D . Then there is f : 2 ω → ω injective continuouspreserving P f such that ( f × f ) − ( S ε,η ∈ R ε,η ) ⊆ ∆(2 ω ) . Proof. (a) By Lemma 4.3, we may assume that R ∩ P f ⊆ ∆(2 ω ) . By Lemma 4.4, we may assumethat R ∩ (cid:0) ( P f × P ∞ ) ∪ ( P ∞ × P f ) (cid:1) = ∅ . By Lemma 4.2, we may assume that R \ ∆(2 ω ) is containedin the union of a sequence ( R n ) n ∈ ω of relations on P ∞ which are closed in P ∞ × ω and in ω × P ∞ .By Lemma 2.8 applied to ( R n ) n ∈ ω , we may assume that R ∩ P ∞ ⊆ ∆(2 ω ) .(b) By Lemma 4.3, we may assume that S ε,η ∈ R ε,η ∩ P f ⊆ ∆(2 ω ) . By Lemma 4.4, we may assumethat R ∩ (cid:16)(cid:0) (2 × P f ) × (2 × P ∞ ) (cid:1) ∪ (cid:0) (2 × P ∞ ) × (2 × P f ) (cid:1)(cid:17) = ∅ . By Lemma 4.2, we may assume that ( S ε,η ∈ R ε,η ) \ ∆(2 ω ) is contained in the union of a sequence ( R n ) n ∈ ω of relations on P ∞ whichare closed in P ∞ × ω and in ω × P ∞ . By Lemma 2.8 applied to ( R n ) n ∈ ω , we may assume that S ε,η ∈ R ε,η ∩ P ∞ ⊆ ∆(2 ω ) . (cid:3) Theorem 1.9 for classes of rank at least three is a consequence of Lemma 4.1 and the followingresult since the elements of A Γ are contained in either ∆(2 ω ) , or in (cid:8)(cid:0) ( ε, x ) , ( η, x ) (cid:1) ∈ D | x ∈ ω (cid:9) ,and the elements of B Γ are contained in ∆( S ) ∪ ( { ∞ }× N ) ∪ ( N ×{ ∞ } ) , which are s-acyclic andclosed on the one side, and G Γ m and G Γ ,am are the only graphs in A Γ ∪ B Γ on the other side. The nextproof is the last one using effective descriptive set theory. Notation.
We set, for any relation R on S , R , := { ( α, β ) ∈ ω × ω | (1 α, β ) ∈ R } . Theorem 4.6
Let Γ be a non self-dual Borel class of rank at least three, X be an analytic space, and R be a Borel relation on X contained in a s-acyclic Borel relation with Σ vertical sections. Exactlyone of the following holds:(a) the relation R is a Γ subset of X ,(b) there is ( X , R ) ∈ A Γ ∪ B Γ such that ( X , R ) ⊑ c ( X, R ) . Proof.
By Theorem 1.2 and since the elements of B Γ have a section not in Γ , (a) and (b) cannot holdsimultaneously. Assume that (a) does not hold. By Theorem 2.2, one of the following holds:(1) the relation R has at least one section not in Γ ,(2) there is a relation R on ω such that R ∩ ∆(2 ω ) = ∆( C ) and (2 ω , R ) ⊑ c ( X, R ) ,(3) there is a relation R on D such that R , ∩ ∆(2 ω ) = ∆( C ) and ( D , R ) ⊑ c ( X, R ) .271) Let x ∈ X such that, for example, R x is not in Γ , the other case being similar. Note that R x \{ x } is not separable from X \ ( R x ∪ { x } ) by a set in Γ . Theorem 2.5 provides h : 2 ω → X \{ x } injectivecontinuous with C = h − ( R x ) . We define g : S → X by g (0 ∞ ) := x and g (1 α ) := h ( α ) , so that g isinjective continuous. Considering ( g × g ) − ( R ) if necessary, we may assume that X = S , x = 0 ∞ and R x ∩ N = { α | α ∈ C } . Let A be a s-acyclic Borel relation with Σ vertical sections containing R .For the simplicity of the notation, we assume that the rank of Γ is less than ω CK , and C , A are ∆ .Theorem 3.5 in [Lo1] gives a sequence ( C n ) of ∆ relations with closed vertical sections such that A = S n ∈ ω C n . By Lemma 2.2.2 in [L5], ∆ ∩ ω is countable and Π , so that V := 2 ω \ ( ∆ ∩ ω ) is Σ , disjoint from ∆ ∩ ω , and V ∩ C is not separable from V \ C by a set in Γ . We will applyTheorem 3.2 in [L5], where the Gandy-Harrington topology Σ ω on ω generated by Σ (2 ω ) isused. Let us prove that A , ∩ V is ( Σ ω ) -meager in V . It is enough to see that ( C n ) , ∩ V is ( Σ ω ) -nowhere dense in V for each n . By Lemma 3.1 in [L5], ( C n ) , is ( Σ ω ) -closed. Weargue by contradiction, which gives n and nonempty Σ subsets S, T of ω with the property that S × T ⊆ ( C n ) , ∩ V . By the effective perfect set Theorem (see 4.F1 in [Mo]), S, T are uncountable.So pick x, y ∈ S and z, t ∈ T pairwise different. Then (1 x, z, y, t ) is a s ( A ) -cycle, which is absurd.Theorem 3.2 in [L5] provides f : 2 ω → ω injective continuous preserving C with the property that (cid:0) f ( α ) , f ( β ) (cid:1) / ∈ A , if α = β .Considering the set ( f × f ) − ( R ) if necessary, we may assume that R , ⊆ ∆(2 ω ) . We set E , := { α ∈ ω | ( α, α ) ∈ R , } , so that E , is a Borel subset of ω and R , = ∆( E , ) . By Lemma2.13, we may assume that E , = S j for some j ∈ . If j = 0 , then (cid:0) ω , ∆( C ) (cid:1) ⊑ c ( X, R ) . So we mayassume that j = 0 . Similarly, { α ∈ ω | (1 α, ∞ ) ∈ R } = S j for some j ∈ . This provides t ∈ A with ( S , R Γ ,at ) ⊑ c ( X, R ) .(2) We partially argue as in (1). Note that R is Borel and contained in a s-acyclic Borel relation with Σ vertical sections A .This time, C n ∈ ∆ (cid:0) (2 ω ) (cid:1) . Let us prove that C n ∩ V is ( Σ ω ) -nowhere dense in V for each n .We argue by contradiction, which gives n and nonempty Σ subsets S, T of ω with S × T ⊆ C n ∩ V .Note that ( x, z, y, t ) is a s ( A ) -cycle, which is absurd. Theorem 3.2 in [L5] provides f : 2 ω → ω injective continuous preserving C such that (cid:0) f ( α ) , f ( β ) (cid:1) / ∈ A if α = β .Considering ( f × f ) − ( R ) if necessary, we may assume that R ⊆ ∆(2 ω ) , which means that (cid:0) ω , ∆( C ) (cid:1) ⊑ c ( X, R ) .(3) We partially argue as in (2).This time, C n ∈ ∆ ( D ) . Fix ε, η ∈ . Let us prove that ( C n ) ε,η ∩ V is ( Σ ω ) -nowhere densein V for each n . We argue by contradiction, which gives n and nonempty Σ subsets S, T of ω with S × T ⊆ ( C n ) ε,η ∩ V . Note that (cid:0) ( ε, x ) , ( η, z ) , ( ε, y ) , ( η, t ) (cid:1) is a s ( A ) -cycle, which isabsurd. Theorem 3.2 in [L5] provides f : 2 ω → ω injective continuous preserving C such that (cid:0) f ( α ) , f ( β ) (cid:1) / ∈ S ε,η ∈ A ε,η if α = β .Considering ( f × f ) − ( R ) if necessary, we may assume that R ε,η ⊆ ∆(2 ω ) and R , = ∆( C ) .Theorem 1.2 provides ( X , R ) ∈ A Γ such that ( X , R ) ⊑ c ( X, R ) .So (b) holds in any case. (cid:3) Theorem 1.8 is an immediate consequence of the following result.28 heorem 4.7
Let Γ be a non self-dual Borel class of rank two, X be an analytic space, and R be as-acyclic Borel relation on X . Exactly one of the following holds:(a) the relation R is a Γ subset of X ,(b) there is ( X , R ) ∈ A Γ ∪ B Γ such that ( X , R ) ⊑ c ( X, R ) . Proof.
We partially argue as in the proof of Theorem 4.6. For the case (1), recall that we may assumethat X = S , x = 0 ∞ and R x ∩ N = { α | α ∈ C } . Corollary 4.5 provides f : 2 ω → ω injectivecontinuous preserving C such that (cid:0) f ( α ) , f ( β ) (cid:1) / ∈ R , if α = β . For the case (2), R is s-acyclic Borel,and by Corollary 4.5 we may assume that R ⊆ ∆(2 ω ) , which means that (cid:0) ω , ∆( C ) (cid:1) ⊑ c ( X, R ) . Forthe case (3), by Corollary 4.5 we may assume that S ε,η ∈ R ε,η ⊆ ∆(2 ω ) and R , = ∆( C ) . (cid:3) Theorem 1.6 is an immediate consequence of the following result.
Corollary 4.8
Let X be an analytic space, and R be a s-acyclic Borel relation on X whose sectionsare in Σ . Exactly one of the following holds:(a) the relation R is a Σ subset of X ,(b) there is ( X , R ) ∈ A Σ such that ( X , R ) ⊑ c ( X, R ) .In particular, A Σ is a 34 elements ⊑ c and ≤ c -antichain basis. Proof.
We apply Theorem 4.7 and use the fact that the elements of B Γ have a section not in Γ . (cid:3) Notation.
We set C Π := A Π ∪ (cid:8) ( S , R Π ,at ) | t ∈ (2 ) ∧ t (0 , , t (0 , , t (1 , ∈ ∧ (cid:0) t (0 ,
1) = 0 ∨ t (1 ,
0) = 0 (cid:1) ∧ t (1 , = 0 (cid:9) .Theorem 1.7 is an immediate consequence of the following result. Corollary 4.9
Let X be an analytic space, and R be a s-acyclic locally countable Borel relation on X . Exactly one of the following holds:(a) the relation R is a Π subset of X ,(b) there is ( X , R ) ∈ C Π such that ( X , R ) ⊑ c ( X, R ) .Moreover, C Π is a 52 elements ≤ c -antichain (and thus a ⊑ c and a ≤ c -antichain basis). Proof.
We apply Theorem 4.7 and use the fact that { α ∈ ω | (1 α, ∞ ) ∈ R } = S j for some j ∈ since R is locally countable. This provides ( S , R Π ,at ) in C Π below ( X, R ) . (cid:3) Notation.
Let K := { − k | k ∈ ω } ∪ { } , and C := { − k | k ∈ ω } . We first set S := { ( x, y ) ∈ K | x, y ∈ C ∧ x < y } , S := { ( x, y ) ∈ K | x = y ∈ C } , S := { ( x, y ) ∈ K | x, y ∈ C ∧ x > y } , S := { ( x, y ) ∈ K | x ∈ C ∧ y / ∈ C } , S := { ( x, y ) ∈ K | x / ∈ C ∧ y ∈ C } , S := { ( x, y ) ∈ K | x, y / ∈ C } . ( S j ) j< is a partition of K , and the vertical sections of S and the horizontal sections of S are infinite. We set N := (cid:8) t ∈ | (cid:0) t / ∈ ∧ t (5) = 0 (cid:1) ∨ (cid:0) t (2) = 1 ∧ t (3) = 0 (cid:1) ∨ (cid:0) t (0) = 1 ∧ t (4) = 0 (cid:1)(cid:9) .We first consider the class Π and set, for t ∈ , R Π N,t := S j< ,t ( j )=1 S j .We next code relations having just one non-closed vertical section, with just one limit point onthis vertical section, out of the diagonal. We set V := (cid:8) t ∈ (2 ) | t (0 , ∈ ( { } × ∧ t (0 ,
1) = (0 , , , , , ∧ t (1 , ∈ ( { } × ×{ }× ∧ t (1 , / ∈ N (cid:9) , and L := ( ¬ C ) ⊕ K . We set, for t ∈ V , R Π V,t := [ ( ε,η ) ∈ ,j< ,t ( ε,η )( j )=1 (cid:8)(cid:0) ( ε, x ) , ( η, y ) (cid:1) ∈ L | ( x, y ) ∈ S j (cid:9) . Similarly, we code relations having closed vertical sections, and just one non-closed horizontal sec-tion, with just one limit point on this horizontal section, out of the diagonal. We set H := (cid:8) t ∈ (2 ) | t (0 , / ∈ N ∧ t (0 ,
1) = (0 , , , , , ∧ t (1 , ∈ ( { } × ) \{ (0 , , , , , }∧ t (1 , ∈ ( { } × (cid:9) , and M := K ⊕ ( ¬ C ) . We set, for t ∈ H , R Π H,t := [ ( ε,η ) ∈ ,j< ,t ( ε,η )( j )=1 (cid:8)(cid:0) ( ε, x ) , ( η, y ) (cid:1) ∈ M | ( x, y ) ∈ S j (cid:9) . We define a set of codes for relations on K with closed sections as follows: C := { t ∈ | t (0) = 1 ⇒ t (4) = 1 ⇒ t (5) = 1 ∧ t (2) = 1 ⇒ t (3) = 1 ⇒ t (5) = 1 } . We define a set of codes for the missing relations in our antichain basis. We set S := (cid:8) t ∈ (2 ) | t (0 , , t (1 , / ∈ N ∧ t (0 ,
1) = (0 , , , , , ∧ t (1 , ∈ C ∧ t (1 ,
0) = (0 , , , , , ⇒ t (0 , ≤ lex t (1 , (cid:9) . We set, for t ∈ S , R Π S,t := S ( ε,η ) ∈ ,j< ,t ( ε,η )( j )=1 (cid:8)(cid:0) ( ε, x ) , ( η, y ) (cid:1) ∈ D | ( x, y ) ∈ S j (cid:9) . Finally, A Π := { ( K , R Π N,t ) | t ∈ N } ∪ { ( L , R Π V,t ) | t ∈ V } ∪ { ( M , R Π H,t ) | t ∈ H } ∪ { ( D , R Π S,t ) | t ∈ S } is the 7360 elements ≤ c -antichain basis mentioned in the statement of Theorem 1.5.We set, for j ∈ , T j := { (2 − k − j (0) , − k − j (1) ) | k ∈ ω } . We then define relations on K by R Π := T (0 , and R Π := T (0 , ∪ T (1 , . They describe the 2 elements mentioned at the end of thestatement of Theorem 1.5. For the class Σ , we simply pass to complements.For graphs, in the case of the class Π , the 5-elements ≤ c -antichain basis is described by thefollowing codes:- (0 , , , , , (acyclic) , (1 , , , , , , (1 , , , , , ∈ N ,- (cid:0) (0 , , , , , , (0 , , , , , , (0 , , , , , , (0 , , , , , (cid:1) ∈ V (acyclic),- (cid:0) (0 , , , , , , (0 , , , , , , (0 , , , , , , (0 , , , , , (cid:1) ∈ S (acyclic).30n order to get the 6-elements ⊑ c -antichain basis, we just have to add ( K , R Π ) (which is acyclic).In the case of the class Σ , the 10-elements ≤ c and ⊑ c -antichain basis is described as follows: take- for (1 , , , , , ∈ N , ∪ j< ,t ( j )=0 S j (which is acyclic),- for t ∈ (cid:8)(cid:0) (0 , , , , , , (0 , , , , , , (0 , , , , , , ( ε , , ε , ε , ε , (cid:1) ∈ V | ( ε , ε ) = (1 , (cid:9) , ∪ ( ε,η ) ∈ ,j< ,t ( ε,η )( j )=0 (cid:8)(cid:0) ( ε, x ) , ( η, y ) (cid:1) ∈ L | ( x, y ) ∈ S j (cid:9) (which is acyclic when ( ε , ε ) = (1 , ),- for t ∈ (cid:8)(cid:0) ( ε , , ε , ε , ε , , (0 , , , , , , (0 , , , , , , ( ε , , ε , ε , ε , (cid:1) ∈ S | ( ε , ε ) , ( ε , ε ) = (1 , ∧ ( ε , , ε , ε , ε , ≤ lex ( ε , , ε , ε , ε , (cid:9) . ∪ ( ε,η ) ∈ ,j< ,t ( ε,η )( j )=0 (cid:8)(cid:0) ( ε, x ) , ( η, y ) (cid:1) ∈ D | ( x, y ) ∈ S j (cid:9) . Notation.
We set H := N ×{ ∞ } , V := { ∞ } × N , and L := H ∪ V . If A ∈ { H , V , L } , then A + := A ∪ { (0 ∞ , ∞ ) } . Let o : 2 ω → ω be defined by o ( α )( n ) := α ( n ) exactly when n > . Then o isa homeomorphism and an involution. We set A c := (cid:8)(cid:0) ω , (2 ω ) (cid:1) , ( S , H ) , ( S , V ) , ( S , L ) , ( S , H + ) , ( S , V + ) , ( S , L + ) , (2 ω , =) , (2 ω , < lex ) , (2 ω , =) , (2 ω , ≤ lex ) , (cid:0) ω , Graph ( o ) (cid:1) , (cid:0) ω , Graph ( o | N ) (cid:1)(cid:9) . We enumerate A c := {E i | i ≤ } (in the previous order). Lemma 6.1 A c is an antichain. Proof.
Note that if ( X, A ) ⊑ c ( Y, B ) and B is in some Borel class Γ , then A is in Γ too. The secondcoordinate of a member of E - E (resp., E - E , E - E ) is clopen (resp., open not closed, closed notopen). This proves that no member of E - E is reducible to a member of E - E , and that the membersof E - E are incomparable to the members of E - E . Note that the second coordinate of- E and E - E is reflexive.- E - E , E - E and E - E is irreflexive.- E , E , E , E , E and E is symmetric.- E - E , E - E , E - E and E is antisymmetric.- E - E , E - E , E - E and E is transitive.Assume that ( X, A ) ⊑ c ( Y, B ) , and that P is one of the following properties of relations: reflex-ive, irreflexive, symmetric, antisymmetric, transitive. We already noticed that ( X, A ) has P if ( Y, B ) does. Note also that if ( X, A ) has P , then there is a copy C of X in Y such that ( C, B ∩ C ) has P . This implies that the only cases to consider are the following. In all these cases, we will prove bycontradiction a result of the form ( X, A ) c ( Y, B ) , which gives i : X → Y injective continuous with A = ( i × i ) − ( B ) . 31 c E : i (10 ∞ ) = i (1 ∞ ) = 0 ∞ , which contradicts the injectivity of i . Similarly, E c E , E isincomparable with E , E c E , E c E . E c E : i (0 ∞ ) = 0 ∞ , so that (0 ∞ , ∞ ) ∈ H , which is absurd. Similarly, E c E and E c E . E c E : for example i (10 ∞ ) < lex i (1 ∞ ) and (10 ∞ , ∞ ) ∈ H , which is absurd. Similarly, E c E . E c E : (cid:0) i (10 ∞ ) , i (0 ∞ ) (cid:1) = (0 α, α ) , (cid:0) i (1 ∞ ) , i (0 ∞ ) (cid:1) = (0 β, β ) , so that α = β , whichcontradicts the injectivity of i . Similarly, E c E . E c E : i (10 ∞ ) = i (1 ∞ ) , so that (10 ∞ , ∞ ) ∈ L , which is absurd. E c E : (cid:0) i (0 ∞ ) , i (1 ∞ ) (cid:1) , (cid:0) i (0 ∞ ) , i (10 ∞ ) (cid:1) ∈ Graph ( o ) , so that i (1 ∞ ) = i (10 ∞ ) , which contra-dicts the injectivity of i . (cid:3) From now on, Y will be a Hausdorff topological space and B will be an uncountable analyticrelation on Y . Note that B ∩ ∆( Y ) is analytic. Lemma 6.2
Assume that B ∩ ∆( Y ) is uncountable. Then (cid:0) ω , (2 ω ) (cid:1) ⊑ c ( Y, B ) , (2 ω , =) ⊑ c ( Y, B ) or (2 ω , ≤ lex ) ⊑ c ( Y, B ) . Proof.
The perfect set theorem gives j : 2 ω → B ∩ ∆( Y ) injective continuous. Note that π := proj (cid:2) j [2 ω ] (cid:3) = proj (cid:2) j [2 ω ] (cid:3) is a copy of ω , ∆( π ) ⊆ B ∩ π and ( π, B ∩ π ) ⊑ c ( Y, B ) , so that we may assume that Y = 2 ω and ∆(2 ω ) ⊆ B ∈ Σ (cid:0) (2 ω ) (cid:1) . By 19.7 in [K], there is a copy P of ω in ω such that < lex ∩ P ⊆ B or < lex ∩ P ⊆ ¬ B . Similarly, there is a copy Q of ω in P such that > lex ∩ Q ⊆ B or > lex ∩ Q ⊆ ¬ B . Case 1. < lex ∩ Q ⊆ B and > lex ∩ Q ⊆ B .Note that Q = B ∩ Q and (cid:0) ω , (2 ω ) (cid:1) ⊑ c ( Y, B ) . Case 2. < lex ∩ Q ⊆ ¬ B and > lex ∩ Q ⊆ ¬ B .Note that ∆( Q ) = B ∩ Q and (2 ω , =) ⊑ c ( Y, B ) . Case 3. < lex ∩ Q ⊆ B and > lex ∩ Q ⊆ c ¬ B .Note that ≤ lex ∩ Q = B ∩ Q and (2 ω , ≤ lex ) ⊑ c ( Y, B ) . Case 4. < lex ∩ Q ⊆ ¬ B and > lex ∩ Q ⊆ B .Note that ≥ lex ∩ Q = B ∩ Q and (2 ω , ≥ lex ) ⊑ c ( Y, B ) . But (2 ω , ≤ lex ) ⊑ c (2 ω , ≥ lex ) , withwitness i defined by i ( α )( n ) := 1 − α ( n ) . Thus (2 ω , ≤ lex ) ⊑ c ( Y, B ) . (cid:3) Lemma 6.3
Assume that there is C ⊆ Y countable such that B ⊆ ( C × Y ) ∪ ( Y × C ) . Then there is ≤ i ≤ such that E i ⊑ c ( Y, B ) . Proof. As B is uncountable, there is y ∈ C such that B y or B y is uncountable. As B y and B y areanalytic, there is a copy P of ω in Y , disjoint from C , such that P ×{ y } ⊆ B or { y }× P ⊆ B . Case 1. P ×{ y } ⊆ B . Case 1.1. ( { y }× P ) ∩ B is countable.Note that there is a copy Q of ω in P such that { y }× Q ⊆ ¬ B .32 ase 1.1.1. ( y, y ) / ∈ B .Note that Q ×{ y } = B ∩ ( { y } ∪ Q ) and ( S , H ) ⊑ c ( Y, B ) . Case 1.1.2. ( y, y ) ∈ B .Note that Q ×{ y } ∪ { ( y, y ) } = B ∩ ( { y } ∪ Q ) and ( S , H + ) ⊑ c ( Y, B ) . Case 1.2. ( { y }× P ) ∩ B is uncountable.Note that there is a copy Q of ω in P such that { y }× Q ⊆ B . Case 1.2.1. ( y, y ) / ∈ B .Note that Q ×{ y } ∪ { y }× Q = B ∩ ( { y } ∪ Q ) and ( S , L ) ⊑ c ( Y, B ) . Case 1.2.2. ( y, y ) ∈ B .Note that Q ×{ y } ∪ { y }× Q ∪ { ( y, y ) } = B ∩ ( { y } ∪ Q ) and ( S , L + ) ⊑ c ( Y, B ) . Case 2. { y }× P ⊆ B .Similarly, we show that E ⊑ c ( Y, B ) , E ⊑ c ( Y, B ) , E ⊑ c ( Y, B ) or E ⊑ c ( Y, B ) . (cid:3) So from now on we will assume that B ∩ ∆( Y ) is countable, and that there is no countablesubset C of Y such that B ⊆ ( C × Y ) ∪ ( Y × C ) . In particular, we may assume that B is irreflexive.By Theorem 1 and Remark 2 in [P], there are ϕ : 2 ω → Y and h : ϕ [2 ω ] → Y injective continuouswith Graph ( h ) ⊆ B . As B is irreflexive, we may assume that h has disjoint domain and range. Wedefine i : 2 ω → Y by i (0 α ) := ϕ ( α ) and i (1 α ) := h (cid:0) ϕ ( α ) (cid:1) , so that i is injective continuous. We set A := ( i × i ) − ( B ) . Note that A is an analytic digraph on ω , which contains Graph ( o | N ) , and that (2 ω , A ) ⊑ c ( Y, B ) . So from now on we will assume that Y = 2 ω , B ∈ Σ (cid:0) (2 ω ) (cid:1) is a digraph, andGraph ( o | N ) ⊆ B . Lemma 6.4
Assume that B is meager. Then there is ≤ i ≤ such that E i ⊑ c ( Y, B ) . Proof.
It is enough to find a Cantor subset P of N such that B ∩ ( P ∪ o [ P ]) ⊆ Graph ( o ) . In orderto see this, we distinguish two cases. Case 1.
Graph ( o | o [ P ] ) ∩ B is uncountable.There is a copy Q of ω in o [ P ] with Graph ( o | Q ) ⊆ B , so that Graph ( o | Q ∪ o [ Q ] ) = B ∩ ( Q ∪ o [ Q ]) .Let ψ : 2 ω → Q be a homeomorphism. We define j : 2 ω → ω by the formulas j (0 α ) := ψ ( α ) and j (1 α ) := o (cid:0) ψ ( α ) (cid:1) , so that j is injective continuous. Note that Graph ( o ) = ( j × j ) − ( B ) , so that (cid:0) ω , Graph ( o ) (cid:1) ⊑ c ( Y, B ) . Case 2.
Graph ( o | o [ P ] ) ∩ B is countable.There is a copy Q of ω in o [ P ] with Graph ( o | Q ) ⊆ ¬ B , so that Graph ( o | o [ Q ] ) = B ∩ ( Q ∪ o [ Q ]) .Let ψ : 2 ω → o [ Q ] be a homeomorphism. We define j : 2 ω → ω by the formulas j (0 α ) := ψ ( α ) and j (1 α ) := o (cid:0) ψ ( α ) (cid:1) , so that j is injective continuous. Note that Graph ( o | N ) = ( j × j ) − ( B ) , so that (cid:0) ω , Graph ( o | N ) (cid:1) ⊑ ( Y, B ) .As B is meager, there is a sequence ( K m ) m ∈ ω of meager compact subsets of (2 ω ) satisfying theinclusions B \ Graph ( o ) ⊆ S m ∈ ω K m ⊆ ¬ Graph ( o ) .33e build a sequence ( U s ) s ∈ <ω of nonempty clopen subsets of N satisfying the following: (1) U sε ⊆ U s (2) diam ( U s ) ≤ −| s | (3) U s ∩ U s = ∅ (4) ∀ ( f, g ) ∈ { Id ω , o } ∀ ( s, t ) ∈ (2 × <ω ( f [ U s ] × g [ U t ]) ∩ ( S m< | s | K m ) = ∅ Assume that this is done. We define ϕ : 2 ω → N by { ϕ ( α ) } := T n ∈ ω U α | n . Note that ϕ is injectiveand continuous, so that P := ϕ [2 ω ] is a Cantor subset of N . It remains to check the inclusion B ∩ ( P ∪ o [ P ]) ⊆ Graph ( o ) . So let ( α, β ) ∈ B ∩ ( P ∪ o [ P ]) \ Graph ( o ) , and m with ( α, β ) ∈ K m .We get s, t ∈ m +1 such that ( α, β ) ∈ (cid:0) ( U s ∪ o [ U s ]) × ( U t ∪ o [ U t ]) (cid:1) ∩ K m , which is absurd.Let us prove that the construction is possible. We first set U ∅ := N . Note that { ( α , α ) ∈ U ∅ | α = α and ∀ ( f, g ) ∈ { Id ω , o } ∀ ( s, t ) ∈ × (cid:0) f ( α s ) , g ( α t ) (cid:1) / ∈ K } is a dense open subset of U ∅ . In particular, it is not empty, and we can pick ( α , α ) in it. Wechoose a clopen neighbourhood U ε ⊆ U ∅ of α ε with diameter at most − such that U ∩ U = ∅ and S ( s,t ) ∈ × U s × U t ⊆ T ( f,g ) ∈{ Id ω ,o } ( f × g ) − ( ¬ K ) . Assume that ( U s ) | s |≤ l satisfying conditions(1)-(4) have been constructed, which is the case for l = 1 , so that from now on l ≥ . Note that (cid:8) ( α s ) s ∈ l +1 ∈ (2 ω ) l +1 | ∀ s ∈ l +1 α s ∈ U s | l and ∀ u ∈ l α u = α u and ∀ ( f, g ) ∈ { Id ω , o } ∀ ( s, t ) ∈ l +1 × l +1 (cid:0) f ( α s ) , g ( α t ) (cid:1) / ∈ S m ≤ l K m (cid:9) is a dense open subset of Π s ∈ l +1 U s | l . We pick ( α s ) s ∈ l +1 in it, and choose a clopen neighbourhood U s ⊆ U s | l of α s with diameter at most − l − such that U u ∩ U u = ∅ and [ ( s,t ) ∈ l +1 × l +1 U s × U t ⊆ \ ( f,g ) ∈{ Id ω ,o } ,m ≤ l ( f × g ) − ( ¬ K m ) . This finishes the proof. (cid:3)
So we may assume that B is not meager. The Baire property of B and 19.6 in [K] give a productof Cantor sets contained in B . This means that we may assume that N × N ⊆ B ⊆ ¬ ∆(2 ω ) . Proof of Theorem 1.10. (1) We distinguish several cases.
Case 1. B ∩ ( N × N ) is not meager.By 19.6 in [K], B ∩ ( N × N ) contains a product of Cantor sets, so that we may assume that ( N × N ) ∪ ( N × N ) ⊆ B . Case 1.1.
There is a Cantor subset of ω which is B -discrete. Then ( S , L ) ⊑ c ( Y, B ) .Indeed, assume for example that Q ⊆ N is a Cantor B -discrete set. Let h : 2 ω → Q be ahomeomorphism. We define i : S → Y by i (0 ∞ ) := 0 ∞ and i (1 α ) := h ( α ) , so that i is injectivecontinuous. Clearly L ⊆ ( i × i ) − ( B ) , and the converse holds since B is a digraph and Q is B -discrete. 34 ase 1.2. No Cantor subset of ω is B -discrete. Then (2 ω , =) ⊑ c ( Y, B ) or (2 ω , < lex ) ⊑ c ( Y, B ) .Indeed, as in the proof of Lemma 6.2 there is a Cantor subset Q of ω with < lex ∩ Q ⊆ B or < lex ∩ Q ⊆ ¬ B , and > lex ∩ Q ⊆ B or > lex ∩ Q ⊆ ¬ B . As B is irreflexive and no Cantor subset of ω is B -discrete, we cannot have < lex ∩ Q ⊆ ¬ B and > lex ∩ Q ⊆ ¬ B . Case 1.2.1. < lex ∩ Q ⊆ B and > lex ∩ Q ⊆ B .Note that Q \ ∆( Q ) = B ∩ Q and (2 ω , =) ⊑ c ( Y, B ) . Case 1.2.2. < lex ∩ Q ⊆ B and > lex ∩ Q ⊆ ¬ B .Note that < lex ∩ Q = B ∩ Q and (2 ω , < lex ) ⊑ c ( Y, B ) . Case 1.2.3. < lex ∩ Q ⊆ ¬ B and > lex ∩ Q ⊆ B .Note that > lex ∩ Q = B ∩ Q and (2 ω , > lex ) ⊑ c ( Y, B ) . But (2 ω , < lex ) ⊑ c (2 ω , > lex ) , with witness i defined by i ( α )( n ) := 1 − α ( n ) . Thus (2 ω , < lex ) ⊑ c ( Y, B ) . Case 2. B ∩ ( N × N ) is meager.By 19.6 in [K], ( ¬ B ) ∩ ( N × N ) contains a product of Cantor sets, so that we may assume that N × N ⊆ B ⊆ ¬ ( N × N ) . Case 2.1.
There is a B -discrete Cantor subset of ω . Then ( S , H ) ⊑ c ( Y, B ) or ( S , V ) ⊑ c ( Y, B ) .Indeed, assume for example that Q ⊆ N is a Cantor B -discrete set. Then as in Case 1.1 we seethat ( S , H ) ⊑ c ( Y, B ) . Similarly, if Q ⊆ N is a Cantor B -discrete set, then ( S , V ) ⊑ c ( Y, B ) . Case 2.2.
No Cantor subset of ω is B -discrete. Then (2 ω , =) ⊑ c ( Y, B ) or (2 ω , < lex ) ⊑ c ( Y, B ) .Indeed, we argue as in Case 1.2.(2) The indicated elements are the only graphs in A c , up to the isomorphism ( ε, α ) εα . (cid:3) [B] J. E. Baumgartner, Partition relations for countable topological spaces, J. Combin. Theory Ser.A
43 (1986), 178-195[C-L-M] J. D. Clemens, D. Lecomte and B. D. Miller, Essential countability of treeable equivalencerelations,
Adv. Math.
265 (2014), 1-31[G] S. Gao,
Invariant Descriptive Set Theory,
Pure and Applied Mathematics, A Series of Mono-graphs and Textbooks, 293, Taylor and Francis Group, 2009[Ka] V. Kanovei,
Borel equivalence relations,
Amer. Math. Soc., 2008[K1] A. S. Kechris,
Classical Descriptive Set Theory,
Springer-Verlag, 1995[K2] A. S. Kechris, The theory of countable Borel equivalence relations, ∼ kechris/papers/lectures on CBER02.pdf) [K-Ma] A. S. Kechris, and A. S. Marks, Descriptive graph combinatorics, ∼ kechris/papers/combinatorics20.pdf) [K-S-T] A. S. Kechris, S. Solecki and S. Todorˇcevi´c, Borel chromatic numbers, Adv. Math.
Topology Appl. Π ξ subsets of the plane?, J. Math. Log.
9, 1(2009), 39-62[L3] D. Lecomte, Potential Wadge classes,
Mem. Amer. Math. Soc.,
Ann. Pure Appl. Logic
Trans. Amer. Math. Soc.
J. Math. Log.
Math. LogicQuart.
65, 2 (2019), 134-169[Lo1] A. Louveau, A separation theorem for Σ sets, Trans. Amer. Math. Soc.
260 (1980), 363-378[Lo2] A. Louveau, Ensembles analytiques et bor´eliens dans les espaces produit,
Ast´erisque (S. M. F.)
78 (1980)[Lo3] A. Louveau, Two Results on Borel Orders,
J. Symbolic Logic
54, 3 (1989), 865-874[Lo-SR1] A. Louveau and J. Saint Raymond, Borel classes and closed games: Wadge-type andHurewicz-type results,
Trans. Amer. Math. Soc.
304 (1987), 431-467[Lo-SR2] A. Louveau and J. Saint Raymond, The strength of Borel Wadge determinacy,
CabalSeminar 81-85, Lecture Notes in Math.
Descriptive set theory,
North-Holland, 1980[P] T. C. Przymusi´nski, On the notion of n -cardinality, Proc. Amer. Math. Soc.
69 (1978), 333-338[S] J. H. Silver, Counting the number of equivalence classes of Borel and coanalytic equivalencerelations,
Ann. Math. Logic
18, 1 (1980), 1-28[T] S. Todorˇcevi´c,
Introduction to Ramsey spaces
Annals of Mathematics Studies, 174. PrincetonUniversity Press, Princeton, NJ, 2010. viii+287 pp[W] W. W. Wadge,