Baire and automata

. In his thesis Baire deﬁned functions of Baire class 1. A function f is of Baire class 1 if it is the pointwise limit of a sequence of continuous functions. Baire proves the following theorem. A function f is not of class 1 if and only if there exists a closed nonempty set F such that the restriction of f to F has no point of continuity. We prove the automaton version of this theorem. An ω -rational function is not of class 1 if and only if there exists a closed nonempty set F recognized by a B¨uchi automaton such that the restriction of f to F has no point of continuity. This gives us the opportunity for a discussion on Hausdorﬀ’s analysis of ∆ 02 , ordinals, transﬁnite induction and some applications of computer science.


Introduction
We would like to dedicate this work to the memory of Pierre Dugac, who was a great historian of Mathematics, and the French specialist of Baire's work.
In his thesis Baire introduced the hierarchy of Baire classes of functions.A function f belongs to class 0 if it is continuous.A function f belongs to class 1 if it is the pointwise limit of a sequence of functions of class 0. A function f belongs to class 2 if it is the pointwise limit of a sequence of functions of class 1, and so on.The present work concerns functions f : A ω → B ω which are ω-rational (A ω and B ω sets of infinite words on finite alphabets A and B).We study these objects from a topological point of view.Let us describe the work done on ω-rational relations.
Acceptance of infinite words by finite automata was first considered in the sixties by Büchi in order to study decidability of the monadic second order of one successor over the integers [14].Since this paper the ω-regular languages have been intensively studied especially because the topological space of infinite words with the usual prefix distance has very interesting properties [69,95,92].
Rational relations on finite words are relations computable by finite automata with two tapes.They were first studied by Rabin and Scott [74].A number of their properties were established by Elgot and Mezei [30].The Decomposition Theorem characterizing functional rational transductions is one of them.A sequential function is a function whose graph is a rational relation with a condition of determinism on the input.A right (resp.left) sequential function reads words from right to left (resp.left to right).A functional rational transduction f satisfying f (ε) = ε is a composition of a left sequential function and of a right sequential function [29,12,78].The extension of rational relations to infinite words, called ω-rational relations, were first studied in 1365-8050 c 2007 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France [7,56,13,40].ω-rational relations are relations computable by a finite automaton with two tapes with a Büchi acceptance condition (and a condition to avoid A * × B * , A ω × B * and A * × B ω ).In [41] Gire shows that functionality is decidable for a ω-rational relation.In [37] Sakarovitch and Frougny show that ω-rational relations of A ω × B ω with bounded delay are exactly the ω-regular languages on the product alphabet A × B. In addition, they prove some undecidability results on ω-rational relations which can be deduced from corresponding undecidability results on rational relations over finite words.The reader should also see [55,24] for other properties and references.
It is only in [31,32] that the topological complexity of ω-rational relations is really investigated.Links between descriptive set theory and automata theory are not new.They go back to Büchi and Landweber's work [54,18,100].Büchi talks very early about analytic set and games [16].In [107] Wagner and Staiger shows that a subset of A ω (A finite) is recognized by a nondeterministic turing machine with Müller conditions if and only if it is an effective analytic set, that is to say a Σ 1 1 set (see Rogers [75] and Moschovakis [65] for a definition of the class Σ 1 1 ).In Staiger papers [88,90,89,91,92] one can have a good overview of the subject.We give here a short account of Finkel's recent work.
Descriptive set theory is the study of definable sets in Polish spaces.A Polish space is a topological space P which is separable (it has a countable dense subset) and have a compatible metric d such that (P, d) is complete.Compact metric spaces are Polish(2 ω the Cantor space, [0, 1]).Complete separable metric spaces are Polish (R, C, C[0, 1]).The most important Polish space is the Baire space ω ω , that is the space of infinite sequence of integers.The family of Borel sets, of a polish space P , is the smallest family of subsets of P which contains open sets and is closed under complements and countable unions.A set E of a polish space P is an analytic set if it is a continuous image of the Baire space ω ω .Another equivalent definition say that E is an analytic set if it is the projection of a Borel set F ⊂ ω ω × P on P .It is easy to construct analytic sets.Let L ⊂ A * , and let L * be the monoid generated by L. Replace star operation * by ω operation, then L ω is an analytic set.If L is finite L ω is compact.If L is not finite L is countable, so we can enumerate elements of L = {u 0 , u 1 , . . ., u n , . ..}. Define an application φ : ω −→ A * by φ(n) = u n .Extend φ in monoid morphism φ : ω * −→ A * .Next extend φ in continuous application φ : ω ω −→ A ω .Since the graph of φ is closed, then L ω is analytic as projection of a closed set.In 1988 Louveau showed that there exists an L such that L ω is not Borel.Unfortunately, he only proved the existence of a such L, he didn't give effectively such a L. His work remains unpublished.An analytic complete set is an analytic set so that any other analytic set can be obtained by continuous inverse image of it.In 2000, Finkel showed that a very simple context free language L is such that L ω is analytic complete [33].Finally in 2001, Finkel showed that one can define an ω-rational relation R such that R is analytic complete (in particular R is not Borel) [31].From this, and using the Post correspondance problem, Finkel discovered new undecidability results about ω-rational relations and gave another proof of the undecidability results of Sakarovitch and Frougny [32].
In this paper an ω-rational function is an (everywhere defined) application f : A ω → B ω whose graph is an ω-rational relation.The ω-rational functions are of Baire class 2. Baire proves the following theorem.
Theorem 1 A function f is not of class 1 if and only if there exists a closed nonempty set F such that the restriction of f to F has no point of continuity.
We prove the automaton version of this theorem.
Theorem 2 An ω-rational function is not of class 1 if and only if there exists a closed nonempty set F recognized by a Büchi automaton such that the restriction of f to F has no point of continuity.
The original proof of Baire uses transfinite induction [5,25].The proof presented in [53,47] is Hausdorff's proof ; we will give a detailed proof of it.The characterization theorem of Baire appears as a corollary of the analysis of ∆ 0 2 sets in an uncountable complete separable metric space.A ∆ 0 2 set is a set which is both F σ (countable union of closed sets) and G δ (countable intersection of open sets).The analysis of ∆ 0 2 sets uses a transfinite derivation over closed sets which is of the same kind of Cantor's derivation.Recall that Cantor discovered countable ordinals iterating in a transfinite way the operation of elimination of the isolated points of a closed set of reals (see Kechris Louveau [48]).
In fact our theorem is just a remark : when we restrict Hausdorff's derivation to ω-regular sets, it stops the derivation at an integer (a greatest fixpoint).This was remarked by the first author in 1986, who, in addition, showed a connection between an old separation theorem and work of Arnold and Nivat [4] about theory of parallelism.
Hausdorff's result is a first step in the study of Wadge's classes of Borel sets [105].Wadge's degrees of Borel sets are essentially well ordered and the type order of the hierarchy is an old uncountable ordinal studied first by Veblen [102].It is usual to present Wadge's degrees with games [103].The restriction of the Wadge's hierarchy to ω-regular sets gives Wagner's hierarchy [106].This is easily seen with Büchi Landweber's result on games such that the winning set is an ω-regular set [18,100].The type order of Wagner's hierarchy is the countable ordinal ω ω .Our proof is of the same type of combinatorial proofs appearing in Wagner's paper [106].
This separation result can be extended to all Wagner classes [83,84], this is easy using well known things from descriptive set theory and Büchi Landweber's result on game [18].On this subject one can also study the work of Barua [6].These results are also automata analogue of effective results of Louveau [60] which give classical results in the plane [76,62].
For recent problems in theory of parallelism one can see [11].Now we return to Elgot Mezei's decomposition theorem.A left sequential machine which reads infinite words is a continuous function.The idea to read from left to right in a right sequential finite state machine suggests non determinism.Moreover, if we work on infinite words rather than finite words, this suggests discontinuity and the Baire hierarchy.If an ω-rational function is not of Baire class 1, one can find a rational tree (tree with a finite number of subtrees) whose set of infinite branches is a Perfect set P (closed set without isolated points [53]) and the restriction of f to P has no point of continuity.This may be interesting, even for finite words.
If the graph of f : A ω → B ω is recognized by a Büchi automaton on the product alphabet A × B we say that f is a synchronous function.Recently we have shown that one can decide if a synchronous function is Baire class 1 [20].Our proof is topological and it is an easy corollary of Sierpinski [81] and Landweber [54].In the present paper we would like to obtain some missing links with works by Beal, Carton, Choffrut, Frougny, Michel, Prieur, Sakarovitch.They have given more algorithmic proofs [23,9,38,39,70,71,8,21]. Talks with Finkel and Carton have given us the impression that for an ω-rational function, being of Baire class 1 is an undecidable property.
This paper is adressed to both computer scientists or set theorists and analysts.We hope that our presentation will be useful for computer scientists.For example, it may help to understand recent results of Duparc [28] and Lecomte [57].On the other hand will expect that a descriptive set theorist will learn automata theory through our examples.He may solve some definability problems about sets recognized by tree automata.
This paper is organized as follows.In sections 2, 3, 4, 5 we present some definitions and properties from automata theory and descriptive set theory.In section 6 we present an example which may be useful to understand the result of Baire.In section 7, we present the difference hierarchy and we give a detailed proof of Hausdorff's result in section 8.In section 9, we give the proof of Baire's result.In section 10, we prove the automaton version of Baire's result.In section 11, we present briefly the Wadge's game and separation games ; we think that this sheds light about results of sections 8 and 10.Finally, we start the discussion about relations between Hausdorff's analysis of ∆ 0 2 sets, ordinals, transfinite induction and applications of computer science.

infinite words
For the concepts introduced in this section we refer the reader to [7,29,69].Let ω be the set of natural numbers (the first infinite ordinal).Complement of a set E will be noted Ě.Let A be a finite alphabet or countable alphabet (A = ω).All alphabets that we consider will have at least two letters.A finite word u over the alphabet A is a finite sequence of elements of A. The set of finite words on A will be called A * .The length (number of letters) of a word u will be noted |u|.A particular word is the empty word ǫ, |ǫ| = 0.As usual A + = A * − {ǫ}.With concatenation, A * is a monoïd with unit element ǫ.An infinite word α over alphabet A is an infinite sequence of elements of A : α = α(0)α(1)...α(n)....The set of infinite words on the alphabet A will be noted A ω .We note α[n] the finite word formed with the n first letters of the infinite word α, α[0] = ǫ, α[1] = α(0).The set A ω , viewed as a product of infinitely many copies of A with the discrete topology, is a metrizable space : ) is a complete metric space, known as the Baire space, which is not compact.The prefix ordering is called <.A finite word u ∈ A * is a prefix of the finite word v ∈ A * (resp infinite word α ∈ A ω ) if there exists a finite word w ∈ A * (resp infinite word β ∈ A ω ) so that v = u.w(resp α = u.β).

Automata on infinite words
Definition 1 A Büchi automaton A is a 5-tuple : A = A, Q, I, T, F , where A is a finite alphabet, Q is a finite set of states, I ⊂ Q is the set of initial states, T ⊂ Q × A × Q is the set of transitions and F ⊂ Q the set of final states.An infinite word α ∈ A ω is recognized by A if there is β ∈ Q ω such that : β(0) ∈ I, ∀ n ∈ ω, (β(n), α(n), β(n + 1)) ∈ T and β(n) ∈ F for infinitely many n.The set of words recognized by a Büchi automaton A is noted L ω (A).
Remark 1 Instead of Büchi automaton one can say automaton with Büchi's acceptation.
T can be viewed as a partial function δ : where u is a finite word and a a letter and δ(p, ǫ) = p.
).This will be denoted by the following graphical notation of path.
The infinite word α ∈ A ω , α = α(0)α(1) . . .α(n) . .., is the label of the path c.Let us note Infinity(c) as the set of states which appears infinitely many times in c.A path c is said to be successful if β(0) ∈ I and Infinity(c) F = ∅.Note that an infinite word α is recognized by A if there is a successful path c in A of label α.
An automaton is called deterministic if it has a unique initial state and for each state p and each letter a there exists at most one transition (p, a, q) ∈ T .Consequently the transition partial function δ can be can be viewed as δ : Q × A → Q. Function δ can be extended to δ : Q × A * → Q by δ(p, ua) = δ(δ(p, u), a), where u is a finite word and a a letter and δ(p, ǫ) = p.Then for all infinite word α there exists at most one path c of label α.
Example 4 Let again B be the deterministic automaton of example 2 and take F = {{0}}.Then this automaton recognizes Q.
2.3 S1S : the monadic second order theory of one successor We now define the terms, atomic formulas, and formulas of S1S the monadic theory of one successor.Let V be a set of variables, its elements noted by x, y, z... , the constant symbol 0 and a unary function symbol s (as successor).We define the set of the terms T by : i) A variable is a term.ii) 0 is a term.iii) if t ∈ T then s(t) ∈ T .Let P be another set of variables.Its variables are noted X , Y, Z... and two predicate symbols =, ∈.Atomic formulas are of the form t = t ′ or t ∈ X where (t, t ′ ) ∈ T 2 and X ∈ P.
Definition 3 A formula of S1S is defined as follows : i) An atomic formula is in S1S.
The interpretation of these formulas is the following : the variables of V are interpreted as natural numbers, symbol 0 as 0 ∈ ω, symbol s as the successor function in ω, the variables of P as subsets of ω and the predicate symbols as equality relation and membership relation in ω.If each integer is assimilated to a singleton and each subset of ω to an infinite word over alphabet {0, 1}, then a S1S formula φ(X 1 , X 2 , ..., X n ), with X 1 , X 2 , ..., X n free variables defines an ω-language L φ ⊂ 2 N × . . . 2 N n .
An ω-language L is said definable in S1S if there exists a formula φ in S1S such that L = L φ .

ω-regular sets
Recall the following result [100,69] : Theorem 3 for all ω-language L, the following assertions are equivalent : i) L = 1≤i≤n A i B ω i where A i , B i are regular sets and n ∈ ω − {0}.ii) L = L ω (A), where A is a non deterministic Büchi automaton.iii) L = L ω (A), where A is a deterministic Muller automaton.iv) L is definable in S1S.
The family of languages which verify the equivalent conditions of the preceding theorem are usually call the ω-regular sets.We denote by Rec(A ω ) the class of ω-regular sets on alphabet A. Following Louveau 1987, we denote by Auto the family of ω-regular sets.ω-regular sets are denoted by ω regular expression [69].
), with u n ∈ A * and v n ∈ B * This will be denoted by the following graphical notation of path : (α, β) is the label of the path c.A path c is said to be successful if q 0 ∈ I and Infinity(c) F = ∅, where Infinity(c) is still the set of states which appears infinitely many times in c.Let α ∈ A * ∪ A ω and β ∈ B * ∪ B ω , (α, β) is recognized by T if there is a successful path c in T of label (α, β).
Remark 2 A path c of label (α, β) is called admissible if α and β are both infinite words.In [37] it is shown that for every finite Büchi transducer T , it is possible to construct another one T ′ so that every successful path in T ′ is admissible and the paths that are both successful and admissible are the same in T and T ′ .In the sequel of this paper all the labels (α, β) will be in A ω × B ω .
An ω-rational relation is a subset of A ω × B ω which is recognizable by a Büchi's transducer.An ω-rational function f : A ω → B ω is a function whose graph is an ω-rational relation.Recall that a left sequential function f : A * → B * is a function that can be realized by a deterministic automaton with output (sequential transducer).A left sequential function can be extended immediately to f : then this is an example of continuous ω-rational function.This is the case when the deterministic automaton with output realizing f output one letter when he read a letter.We call 1-sequential functions these functions and these functions will be used as strategy for player 2 later.

Borel sets in Polish spaces
For all of the topological concepts introduced in this section we refer the reader to [53,47,69,87].

Ordinals
For a short and comprehensive presentation of ordinals we refer the reader to Srivastava [87].We say that two sets E and F have the same cardinal if there is a bijection from E to F .We say that two well-ordered sets E and F have the same ordinal if there is an order-preserving bijection from E to F .To each well-ordered set W we can associate a particular well-ordered set t(W ) called the type of W which is the ordinal associate to W . Results in the theory of ordinals use the axiom of choice and axiom of remplacement.

The Borel hierarchy
Borel subsets of a topological space X are obtained from open sets using complementation and countable unions.When X is metrizable we can define the hierarchy of Borel sets of finite rank : In particular, we have : One can prove that : This gives us the following picture where any class is contained in every class to the right of it : The Borel hierarchy is also defined for transfinite levels ξ < ω 1 , but we shall not need them in the present study.For all n ∈ ω the classes Σ 0 n (X), Π 0 n (X), ∆ 0 n (X) are closed under finite unions and intersections, moreover Σ 0 n (X) is closed under countable unions, Π 0 n (X) closed under countable intersections and ∆ 0 n (X) closed under complement.All these classes are closed by inverse image by continuous functions.
Sets which are recognized by deterministic Büchi automaton are G δ .One can see this easily as a deterministic automaton gives a continuous function f : A ω → Q ω .Replace q / ∈ F by 0 and q ∈ F by 1.The set recognized by a deterministic Büchi automaton is the inverse image of Q by a continuous function.
Sets which are recognized by deterministic Müller automaton are boolean combinations of sets which are recognized by deterministic Büchi automaton, so they are boolean combinations of Π 0 2 sets hence ∆ 0 3 .

Polish spaces
A Polish space P is a separable topological space which admits a compatible metric d such as (P, d) is complete.A closed subset of a Polish space is Polish.An open subset of a Polish space is Polish.A G δ subset of a Polish space is Polish.This is not true for F σ .Recall the Baire theorem : Theorem 4 Let X be a complete space, the intersection of countably many dense open sets in X is dense.This is equivalent to say that in a complete space X, the union of countably many closed sets of empty interior has empty interior.

Lemma 5
The set Q with the relative topology induced by the one of R is not Polish.
Proof: (Saint Raymond) We have : a countable union of closed sets.Suppose Q was Polish then by the preceding theorem there must be an n such that {q n } has an nonempty interior, otherwise Q would have an empty interior, hence will be empty.But every {q n } has an empty interior because Q is dense in itself.Hence Q can't be Polish.
In fact by Baire's theorem, every countable dense subset of a Polish space is not Polish.As a G δ subset of a Polish space is Polish, every countable dense subset of a Polish space is not G δ .
Remark 3 It is well known (for a descriptive set theorist) that every Polish space is homeomorphic to a G δ set in a compact metric space.For example the Baire space ω ω is homeomorphic to When P is an uncountable Polish space , Borel hierarchy is strict.In the sequel P will be A ω or A ω × B ω or [a, b] with a and b reals.

Analytic sets and coanalytic sets
There exists another hierarchy beyond the Borel one, called the projective hierarchy, which is obtained from the Borel hierarchy by successive applications of operations of projection and complementation.We need just the first level of this hierarchy.Let B ⊆ P × ω ω , we will call proj P (B) the projection of B onto P , that is, proj P (B) = {α ∈ P/∃β ∈ ω ω (α, β) ∈ B}.
A set C ⊆ P is called analytic if there is a Borel set B ⊆ P × ω ω such that C = proj P (B).A set C ⊆ P is coanalytic if its complement is analytic.The class of analytic sets in P (resp.coanalytic) is called Σ 1 1 (P ) (resp.Π 1 1 (P )).Borel sets are analytic and coanalytic sets.The famous theorem of Suslin says that in Polish space P , if B ⊆ P is analytic and coanalytic then B is Borel.
Existence of analytic sets which are not Borel is a kind of myth for descriptive set theorists.In 1905 Lebesgue said that the projection of Borel set in the plane was a Borel set.This was false as Suslin discovered in 1917.He called a projection of Borel set an analytic set.Here is the French evidence of Sierpinski [82] : "Par hasard j'étais présent au moment où Michel Souslin communiqua à M. Lusin sa remarque et lui donna le manuscrit de son premier travail".Büchi commented the equivalence of theorem 3 :"What looks like an analytic set (set recognized by a nondeterministic Büchi automaton) is in fact Borel set( a set recognized by a deterministic Müller automaton is a ∆ 0 3 set)" [16].An ω-rational relation is an analytic set of A ω × B ω .

Complete sets
Recall the notion of completeness with regard to reduction by continuous functions.Let Γ be a class of sets in P Polish We call C ⊆ P Γ-complete if C ∈ Γ and for any B ∈ Γ there exists a continuous function f : Finding some simple examples of complete sets is an old tradition in descriptive set theory which goes back to Hurewicz [45] ( see Louveau and Saint Raymond [62], Kechris [47]).It turns out that some simple combinatorial examples of complete sets are recognized by automata.
Example 8 We will see that In fact a countable dense subset of [0, 1] is Σ 0 2 -complete Hurewicz [45], and this true in all uncountable Polish space.Here is some well known examples of coanalytic-complete sets : The set WO, as Well Order, that is the set of E ⊂ ω × ω such that E is the graph of Well ordered linear order, is . The set WF of well founded trees, that is trees on ω which have no infinite branches is Π 1 1complete.
The set NDIFF of continuous functions on [0, 1] which are nowhere differentiable functions in . Finkel showed in [31] that there exists an ω-rational relation which is The following lemma is classical( see [54], [56], [90]).
with L a regular set denoted by the regular expression (0 + 1) * 1.
This lemma is equivalent to the following.The set {α ∈ complete set [83], [93] .In fact we have more : this set is strategically complete [62].We will see it in the game section.

Baire's classes
Definition 7 Definition of Baire's classes for functions is recursive.Let X, Y be metrizable spaces and f : X → Y be a function.
ii) f is of Baire class (n + 1) if f is the pointwise limit of a sequence of Baire class n functions for each integer n ≥ 0.
The Lebesgue, Hausdorff, Banach theorem makes the connexion with the Borel hierarchy : . Remark 4 Note that this result holds for n = 1 if in addition X is separable and either X = A ω or Y = R.
Remark 5 If Y = B ω , as {uB ω |u ∈ B * } is a countable basis of clopen sets, it is equivalent to prove that f : X → Y is of Baire class n iff for all finite word u, f −1 (uB ω ) is in ∆ 0 n+1 (X).Lemma 8 An ω-rational function is of Baire class 2.
Proof: We have to shows that for all finite word u, We have seen that an ω-regular set is a boolean combination of Π 0 2 sets, hence is a ∆ 0 3 set.
(i) Jean Saint Raymond has proved, that this valid for any uncountable Polish space.
(ii) See [47] page 205 for a discussion of the statement : Example 10 The characteristic function of Q ⊂ R, 1 Q is a classical example of Baire class two function which is not of Baire class one [5,25].The function 1 Q is the pointwise limit of the sequence (f m ) m∈N where f m (x) = lim n→∞ cos 2m (n!πx).So 1 Q is of Baire class two.If 1 Q was a Baire class one function, then the inverse image of an open set by 1 Q will be a Σ 0 2 (R) set, hence the inverse image of a closed set by 1 Q will be a Π 0 2 (R) set.But as Let X and Y be metrizable spaces with Y compact, and f : X → Y , it is well known that f is continuous if and only if its graph is closed.
As f is of Baire class n, for any word When A and B are finite alphabets we have : Let cont(f ) denote the set of points of continuity of a function f .Proposition 11 Let X and Y be separable metric spaces and f : X → Y .Then cont(f ) is Π 0 2 (X).Proof: We define the oscillation of f at α by :

An example
In this section we give an example of a function f such that graph(f ) is definable in S1S hence is of Baire class 2. One can easily see that graph(f ) ∈ Π 0 2 and the set of points of continuity of f is a dense open set (hence dense Π 0 2 ).However, f is not of Baire class 1.This example was constructed in 1996 by Tison and the first author and was unpublished.The idea of A. Louveau (1996), is the following : take the characteristic function of the Cantor set on the interval [0, 1].The function is continuous on the complement of the triadic Cantor set, which is a dense open set.Since the graph of f is ∆ 0 2 set, it follows from proposition 10 that f is of Baire class 1.So we have to modify our function on the Cantor set to succeed.Now we will work on space First define g : {0, 1} ω −→ {0, 1} ω by : g(α) = α, if α ∈ (0 * 1) ω (α has infinitely many 1's).
Fig. 6: a dense open set of points of continuity It is easy to see that f has no point of continuity on (0 + 2) ω .The proof is similar to that concerning g, we just have to replace 1 by 2.
In his thesis (1899) Baire has proved that a function is of Baire class 1 if and only if for every non empty closed set F the restriction of this function to F has a point of continuity.Our example is ω-rational function which is not of Baire class 1 and we have found an ω-regular closed set F = (0 + 2) ω such that restriction of f to F has no point of continuity.We will see that our example is generic.If f is an ω-rational function which is not Baire class 1, then there exists a closed F , recognized by a Büchi automaton, such that restriction of f to F has no point of continuity.
The next sections will be devoted to the classical proof of Baire's result.

Differences hierarchy
In this section, we introduce the class of differences.Let ξ be an ordinal.Any ξ can be written in a unique way ξ = λ + n with λ a limit ordinal or 0 and n ∈ ω.Parity of ξ is by definition that of n.Definition 8 Let X be a set, ξ an ordinal and < B η : η < ξ > an increasing sequence of subsets of X : the parity of ξ and of η 0 are differents } Then : Let Γ be a family of subsets of X, D ξ (Γ) will be the family of all D ξ (< B η : η < ξ >) where < B η : η < ξ > is an increasing sequence of length ξ of elements of Γ.In the sequel we will be particularly interested in the classes D ξ (Σ 0 1 ) and their dual classes Ďξ (Σ 0 1 ) = {B | B ∈ D ξ (Σ 0 1 )}, where ξ is a countable ordinal.7 gives a deterministic Büchi automaton which recognizes Example 12 Let G n be the following sequence of decreasing dense G δ sets : Taking the complement of these sets we obtain an increasing sequence 2 )-complete set. Figure 8 gives a deterministic Müller automaton which recognizes The loops accessible from the initial state are L = {{0}, {0, 1}, {0, 1, 2}, {0, 1, 2, 3}, {3}}, these are essential loops of the automaton [106].Classify these essential loops in It is well known that in an uncountable Polish space hierarchy of D ξ (Σ 0 η ), ξ < ω 1 , η < ω 1 is strict (see [47] ).

Haussdorff's derivation
Recall that A ω has a countable basis for the topology.Thus we can extract countable covering from each open covering.As a consequence, if ω 1 is the first non countable ordinal, and if (F ξ ) ξ<ω1 is a decreasing sequence of closed sets, it is stationary from a certain rank on, i.e., ∃η < ω 1 such that F ξ = F η , ∀ξ ≥ η.For more details we refer to [47].
Definition 9 Let M and N be two subsets of A ω .Sequence of closed sets (F ξ ) ξ<ω1 is defined by transfinite induction.
The sequence (F ξ ) ξ<ω1 is a decreasing sequence of closed sets, so we know that there exists a smaller η < ω 1 such that We define the following sequences (M ξ ) ξ<ω1 and (N ξ ) ξ<ω1 : Proof: We argue again by a transfinite induction.
For η = 0 we have For λ limit, if M ξ = M ∩ F ξ for ξ < λ, then : Lemma 16 Let M and N be two Π 0 2 subsets of A ω and F η the largest closed set such that 2 sets.So, by lemma 6, there exists Since every open ball containing α 1 meets N η , we can find β 1 ∈ N η so that u 1 is a prefix of β 1 .But N η = Lim(V ), thus we can find v 1 ∈ V such that v 1 is a prefix of β 1 and u 1 is a strict prefix of v 1 .Finally since every ball containing β 1 meet M η we can find α 2 ∈ M η so that v 1 is a prefix of α 2 , and since M η = Lim(U ) we can find u 2 ∈ U prefix of α 2 so that v 1 is a strict prefix of u 2 .Then u 1 < v 1 < u 2 .Iterating this process, we construct two sequences (u i ), (v i ) such that Theorem 17 (Hausdorff ) Let M and N be two disjoint Π 0 2 subsets of A ω .Then : (i) There exists ζ < ω 1 and a set in We have : Let η be the smallest ordinal such that F η = F η+1 .We have : By lemma 16, as M and N are two disjoint Π 0 2 , F η = ∅, thus M and N are disjoint Π 0 2 sets and there exists ζ < ω 1 and a set in ) and the opposite inclusion has been proved in lemma 12.
Corollary 18 Let M and N be two subsets of A ω and let F η be the biggest closed set so that Then M and N can be separated by a ∆ 0 2 set iff F η = ∅.Proof: In the proof of theorem 17 we showed that if Hausdorff's derivation stops to the empty set then M and N are separated by a ∆ 0 2 .Conversely, if M and N are separated by a D ζ (Σ 0 1 ) set C for some ζ < ω 1 , we can operate the Hausdorff derivation on C and Č.As C and Č are disjoint Π 0 2 sets this derivation goes on to the empty set, as the derivation on M and N .Our proof is directly extracted from Kuratowski [53] and documents of Louveau.The result is true in uncountable Polish spaces.Our originality comes from lemma 16 ; this is in such a form that the first author discovered this problem [4].One can see how automata can be used to analyse problems of parallelism in [11].Instead of lemma 16, a descriptive set theorist would use there Baire's theorem : a countable intersection of dense open set is dense.If M and N are two disjoints Π 0 2 and if F is a non empty closed set such that F = F ∩ M ∩ F ∩ N , then F is complete as a closed set in complete space A ω .Sets F ∩ M and F ∩ N are G δ in F , so they cannot be both dense since they are disjoint.9 Baire's theorem Definition 10 A set E is called nowhere dense if its closure E has an empty interior.A set E is called meager if it is included in a countable union of nowhere dense sets.
Baire's theorem asserts that in a polish space [69] a countable intersection of dense open sets is still dense or equivalently that a countable union of nowhere dense closed sets has empty interior.We have seen that for any function f , the set of discontinuity points of f is in Σ 0 2 (X) set.In the case of a Baire class 1 function, we have more.
Proposition 19 Let X and Y be two separable metric spaces and f : X → Y a Baire class 1 function.The set of discontinuity points of f is a meager set in Σ 0 2 (X).Proof: . Thus the set of discontinuity points of f is As f is Baire class 1, all these sets are Σ 0 2 (X) and have empty interior : they are all meager and a countable union of meager sets is still meager.
Theorem 20 Let P be a Polish space, Y a separable metric space and f : P → Y .The following statements are equivalent : (ii) For all nonempty closed set F ⊂ P , the restriction f |F of f to F has a point of continuity.
Proof: (i) ⇒ (ii) Set F is a closed set of a Polish space so F is Polish too.Since f is Baire class 1, so is f |F , thus, by proposition 19, its discontinuity points form a meager subset of F , hence by Baire category theorem, cannot be equal to As Y is a metric space, U can be writen as a countable union of closed sets U = ∪ n≥0 H n .Let H be the complement of U in Y .Suppose that for each n there exists A n ∈ ∆ 0 2 which separates f −1 (H n ) from f −1 (H).Then : and f −1 (U ) will be Σ 0 2 .So there exists an n such that f −1 (H n ) and f −1 (H) can't be separated by a ∆ 0 2 set.We now apply Hausdorff's derivation on f −1 (H n ) and f −1 (H).Let F be the biggest closed set such that . By corollary 18, F is not empty and we will show that f has no point of continuity in Example 13 We have seen that the characteristic function of Using the previous theorem we can see that 1 Q is not Baire class one since it's nowhere continuous.

Application to automata theory
Lemma 21 Let M and N be two languages in Rec(A ω ).Then Hausdorff 's derivation on M and N stops in a finite number of steps, i.e.
Proof: Let A (Resp.B) be a deterministic Müller automaton which recognizes M (Resp.N).Construct the cartesian product A × B, this is also a deterministic automaton.Compute the essential loops of the product and classify them in L A , L B , where L A (Resp.L B ) is the set of essential loops such that projection on states of A (Resp.B) is a positive essential loop of A (Resp.B), see example 12.Note that L A , L B are disjoint if and only if M and N are disjoint.Let α in M , then there exists a essential loop F which recognizes α, and one can see that α in M is in the closure of N if and only if a loop of L B is accessible from the loop F in L A .Process as follows : eliminate from L A (Resp.L B ) loops from which every loop in L B (Resp.L A ) is inaccessible, and iterate the work.As there is a finite number of loops, the process will stop in a finite number of steps.At the end, if L A and L B are not empty, then from every loop in L A (Resp.L B ) you can access to some loop in L B (Resp.L A ).
Example 14 Let Q 0 the subset of 2 ω of infinite words with finite even number of 1 and Q 1 the subset of 2 ω of infinite words with finite odd number of 1.This two sets are dense, Σ 0 2 -complete sets.The Hausdorff 's derivation stops after one iteration as Rec(A ω ) are separated by a ∆ 0 2 set.Moreover if they are separated by a ∆ 0 2 set, they are separated by an ω-regular ∆ 0 2 set, i.e., a finite difference of ω-regular open sets.

Corollary 22 One can decide if two langages in
Proof: Let M and N two languages in Rec(A ω ).Using corollary 18, M and N are separated by a ∆ 0 2 set iff Hausdorff's derivation stop to the empty set, and by lemma 21 it comes in a finite number of steps.By closure property of ω-regular sets, the F n which appear in Hausdorff's derivation are ω-regular and emptyness problem is decidable for ω-regular sets.
Corollary 23 Let f : A ω → B ω be an ω-rational function.If f is not Baire class 1 then there exists a nonempty closed set F which is recognizable by a Büchi automaton such that f restricted to F has no point of continuity.
Proof: If f is not Baire class 1 then there exists u ∈ B * such that f −1 (uB ω ) / ∈ ∆ 0 2 (X).So there exists v ∈ B * with |u| = |v| such that f −1 (uB ω ) and f −1 (vB ω ) cannot be separated by a ∆ 0 2 (X) set.Thus, as in the proof of theorem 20, the Hausdorff derivation on f −1 (uB ω ) and f −1 (vB ω ) produces, in finite time, a closed set F such that f |F has no point of continuity.As f −1 (uB ω ) and f −1 (vB ω ) are recognizable by a Büchi automaton so is F by closure properties of the family Rec(A ω ).

Büchi Landweber and Martin
Games are useful in descriptive set theory.They allow to give alternative proofs of some theorems like Cantor Bendixon theorem and Wadge theorem : Definition 11 A game in A ω × B ω between two players I and II can be defined as follows : Player I plays α(0) ∈ A, then player II plays β(0) ∈ B, I plays α(1) ∈ A, and so on.The result of the game is the couple of infinite words (α, β) of A ω × B ω .
Let G be a subset of A ω × B ω .Player II wins the game if (α, β) is in G.
A strategy φ for player I is a winning strategy if for any β in B ω (φ(β), β) is not in G.A strategy ψ for player II is a winning strategy if for any α in A ω (α, ψ(α)) is in G.
Definition 13 A game is called determined if one of the two players has a winning strategy.
It is well known that Borel's games are determined : Martin theorem [47].Sometimes the proofs using games can be adapted to automata theory thanks to Büchi Landweber's theorem [7,69,84,85] Theorem 24 (Büchi Landweber 1969) If the set G is an ω − regular subset of A ω × B ω , one of the players have a winning automaton strategy : either player I has a winning strategy φ such that the tree φ is a rational tree, or either player II has a winning strategy ψ which is a 1-sequential function.There is an algorithm which, given an ω − regular set G, (1) determines which player has a winning strategy, and (2) constructs a winning automaton strategy.In particular if one of the players has a winning strategy, he also has a winning automaton strategy [100,18].

Wadge Game
Let X ⊂ A ω , Y ⊂ A ω , in the Wadge game G(X, Y ) player II wins iff (α ∈ X ⇔ β ∈ Y ), that is to say the winning set for II is G = (X × Y ) ( X × Y ).A winning strategy for II gives a continuous function ( a Lipschitz map) ϕ such that ϕ −1 (Y ) = X.A winning strategy for I gives a continuous function ( a Lipschitz map) ψ such that ψ −1 ( X) = Y .If X and Y are Borel, Wadge game is determined so we have the dichotomy : either there exists ϕ continuous such that ϕ −1 (Y ) = X either there exists ψ continuous such that ψ −1 ( X) = Y .

Wadge's hierarchy, Wagner's Hierarchy, Louveau's hierarchy
Let X ⊂ A ω , Wadge has defined the class of Wadge of X by : The notation W on the right gives in French : W A Droite.WADge has given a complete description of all Wadge classes of Borel set.We have Let X ⊂ Rec(A ω ), Wagner has defined the class of Wagner of X by : And the same definition for Σ 0 n (∆ 1 1 ) with a very simple set in 2 ω n recognized by a finite automaton which reads words of length ω n (see Büchi [15], Shelah [80], Bedon [10], Choffrut Grigoriev [24] for finite automata reading transfinite words).From this effective results Louveau deduces classical results in the plane : If X ⊂ ω ω ×ω ω is a Borel set with Σ 0 n+1 sections then X is a countable union of Borel sets in ω ω × ω ω with Π 0 n sections.The first example of this kind of results is the case of Borel sets in the plane with countable sections studied by Lusin [64].This result was extended by Novikov Arsenin and Kunugui to the case of Borel sets in the plane with compact sections or K σ sections(see Sierpinski [82], Saint Raymond [77], Louveau Saint Raymond [62], Kechris [47], Srivastava [87]).

O and Q
For all Wadge classes of Borel sets Γ, Wadge gives an example of a Γ-complete set in ω ω .As remarked by Professor Jean Saint Raymond "il suffit de le faire pour les ouverts."We will do it for open sets and F σ sets.
Let T be a tree of A * , we denote by [T ] the set of the infinite branches of T : Proposition 25 Let F be a subset of A ω .The set F is closed iff there exists a tree T of A * so that Proof: Let U be in Σ 0 1 (A ω ).The complement of U is closed, so by proposition 25, there exists a tree T so that T = Ǔ .The winning strategy φ : A + → {0, 1} for II is the following : In fact using Martin theorem and Wadge game every open non closed set is Σ 0 1 (A ω )-strategically complete.And using Büchi Landweber theorem we have Σ 0 1 ∩ Auto = Σ 0 1 (Auto).

Proposition 27
The set Proof: Let U be in Σ 0 2 (A ω ).Then there exists a family (F n ) n∈ω of closed sets so that U = n∈ω F n .By proposition 25, for any integer n, there exists a tree T n so that F n = [T n ].The winning strategy φ : A + → {0, 1} for II is given by the following induction : Let (α, φ(α)) be a result of the game.
-If α is in U , there exists n so that α is in Then for all m in ω, α[m] is in T n .So there exists m 0 so that for all m ≥ m 0 , φ(α[m]) = 0 and φ(α) is in Q : II wins.-If α is not in U then for all n and m in ω, there exists m ≥ m so that α[ m] is not in T n i.e.
the sequence of finite words (α[m]) m∈ω leaves any tree T n in finite time.Then φ(α) has an infinite number of 1, so φ(α) is not in Q and II wins.
In fact using Martin theorem and Wadge game every F σ set which is not a G δ set is Σ 0 2 (A ω )strategically complete.And using Büchi Landweber theorem we have Σ 0 2 ∩ Auto = Σ 0 2 (Auto).

Separation games
In October 1984 Louveau was presenting joint work of his and Saint Raymond in the seminary of theory of effective borel sets.The title of the talk was "Jeux de Mistigri (Mistigri Games)".This was a sort of Wadge game, a separation game.Let Y and Z be two analytic disjoint subsets of ω ω .
In a first Game Player II wins the game iff (α  Note that these theorems hold for all classes of Wagner's hierarchy which is of type order the ordinal of ω ω .These results where presented to Louveau in 1987 and appear in [84,85].R. Barua solves the case of the D n (Σ 0 2 ) classes with a proof without games [6].Separation games appear in Van Wesep [101].The game where Player II wins if Kechris Louveau Woodin [50,48,47] to give new proof of the old Hurewicz's theorem [45] :

Mistigri Couleur
Van Wesep and Steel Games [101,94] were used to study structural properties of Wadge classes of Borel sets like separation property.A Wadge class Γ has the separation property if for any pair X, Y of disjoints sets in Γ there exists Z in ∆(Γ) which separate X from Y .In fact for each pair of nonselfdual Γ, Γ Wadgeclass of Borel sets then exactly one of the classes has the separation property.And if Γ doesn't have the separation property then one can find a simple pair X, Y of disjoints sets in Γ such that X can't be separated from Y by a set in ∆(Γ).Another very interesting property is the norm property, in [63] Louveau and Saint Raymond study norm property of Borel Wadge classes.The classes Σ 0 ξ , D η (Σ 0 ξ ) , ξ < ω 1 , η < ω 1 , and Π 1 1 have the norm property, and the D n (Σ 0 1 (Auto)), D n (Σ 0 2 (Auto)) have the norm property [83].If a class Γ has a norm property then Γ has the reduction property [63] that is to say for all X, Y in Γ one can find Moreover if Γ has the reduction property then Γ has the separation property.For example we have seen that the classes Π 0 2 , Π 0 2 (Auto) have the separation property ; in fact this is true because the classes Σ 0 2 , Σ 0 2 (Auto) have the reduction property.The reduction property of classes Σ 0 ξ is used to prove the Lebesgue, Hausdorff, Banach's theorem : X). Remark 6 Note another time that this result holds for n = 1 if in addition X is separable and either The proof use finite valued Borel functions, finite Borel partition.We know four descriptions of Borel Wadge classes : The one of Wadge(Descriptive Set Theorist) [104] is useful for a Computer Scientist (DST : Hello this set is Γ-complete and we give you a proof of that.Do you know countable choice and fundamental sequences ?CS : Thank you, I will try to find some device to recognize this Γ-complete set.Is this the simplest one ?What is countable choice ?Do I use it ?Do you think I can recognize this Γ-complete set with a Muller tree automaton ?DST : what is a Muller tree automaton ?).The one of Louveau [60] is useful to study structural properties of classes, Selivanov [79] uses this description and describes the topological invariants of Wagner classes by finite trees.The one of Saint Raymond [77] uses Borel functions of class ξ (ξ < ω 1 ).This description has the advantage to extend immediately to the case of finite Borel coloring of A ω (Wadge case is Black and White, X ⊂ A ω , X 0 = X, X 1 = X, A ω = X 0 ∪ X 1 ).The one of Duparc [27] follows the Cantor normal form of ordinals.This description has been used by Finkel [33,35] to study the order type of the Wadge hierarchy restricted to ω-context-free languages.
In [77] it is quoted that games used by Van Wesep, Steel, Louveau Saint Raymond are particular case of elementary games with winning set G = ∪ n i=0 X i × Y i , where the X i , Y i are Borel subsets of A ω .Let (X 0 , . . ., X n ) and (Y 0 , . . ., Y n ) be two Borel partitions of A ω .Define the game G(X 0 , . . ., X n ; Y 0 , . . ., Y n ) where I plays α, II plays β and where II wins the game if ∀iα ∈ X i ⇒ Y i .This game enable us to compare finite Borel partitions.This gives for all n the Wadge n + 1 colors hierarchy.If the X i , Y i are Auto this gives for all n the Wagner n + 1 colors hierarchy.Then you can use Büchi Landweber and this certainly has to do with the algebra of finite monoïds (see Carton Michel [21], Carton Perrin [22], Perrin Pin [69], Wilk [109]).
In [44] Hertling and Weihrauch study discontinuity of finite valued Borel functions, for understanding degeneracy in computational geometry.Here is their abstract : "We introduce levels of discontinuity and prove that they correspond to the number of tests in "continuous computation trees".We illustrate the concept of level by various simple examples from computational geometry.For a finer comparison of kinds of discontinuity we introduce a continuous reducibility relation for finite valued functions.We show that each of the resulting degrees (of finite level) can be characterized by a finite set of finite trees which describes the type of discontinuity of its functions.The ordering of the degrees is decidable in the tree sets and each level consists only of finitely many degrees".The description of Saint Raymond may have very interesting applications in computational geometry.

Conclusion
12.1 Π 1 1 sets and ω 1 , the boundedness theorem of Lusin There is a lot of to say about the story of transfinite.For example Borel did not believe in ω 1 ("cette totalité illégitime"), Lusin seems to refuse the Third Middle excluded for projective sets.It is well known that Baire, Borel, Lebesgue, did not believe in the axiom of choice.Zermelo has proven with the axiom of choice that you can put a well order on every set.Sierpinski believed in the axiom choice, but has shown which theorems need the axiom of choice.In general, there is no need of axiom of choice when you dispose of a well order.Do you believe in ω 1 ?Do we need countable ordinals in computer science ?It is an old result of Lusin and Sierpinski that WO (as Well Order), the code of countable ordinals, is The Example of a coanalytic set.It is a coanalytic complete set.The first examples of coanalytic-complete sets in analysis are due to Hurewicz [45,47,83].These are the set K(Q) of compacts subsets of the rationals of the interval [0, 1] and the set K ω of countable compacts subsets of the interval [0, 1].Consider the stupid game in 2 ω × 2 ω where Player I wins the game if α ∈ Q and β ∈ 2 ω .Player I has a simple automaton strategy : always play 0. Call, by analogy with K(Q), KQ the set of winning strategies of player I, this set is Π 1 1 and extending ϕ : ω * −→ 2 * of remark 3 in ϕ : 2 ω * −→ 2 2 * one has that ϕ −1 (KQ) = WF, hence KQ is a Π 1 1 -complete set.This was first observed by Niwinski in 1986 [67].This set is recognized by a deterministic Muller infinite tree automaton.In fact a set is recognized by deterministic Muller infinite tree automaton if and only if it is the set of winning strategies of player I in an ω-regular game, and a set is recognized by a nondeterministic Muller infinite tree automaton if it is the projection of a set recognized by a deterministic Muller infinite tree automaton.Infinite tree automaton where used by Rabin [72] to show the decidability of the monadic second order theory of the tree 2 * , S2S.Rabin shows that a set is recognized by a nondeterministic Muller infinite tree automaton if and only if it is definable in S2S.Note that it is quite clear when you read the first pages of Rabin that the set K(Q) of compacts subsets of Q = {α ∈ 2 ω | ∃m ∀n ≥ m α(n) = 0} is definable in S2S and we know since Hurewicz that this set is Π 1 1 -complete (see [45,48,50,47]).Rabin shows by transfinite induction on countable ordinals that if a set is recognized by a nondeterministic Muller infinite tree automaton then its complements is also recognized by a nondeterministic Muller infinite tree automaton.The complement of KQ is recognized by a nondeterministic Büchi infinite tree automaton but KQ is not recognized by a nondeterministic Büchi infinite tree automaton ( otherwise KQ will be Borel, see also Rabin [73] for a combinatorial proof of that, the paper of Rabin [73] is a kind of Suslin Kleene Automata theorem [83]).Finkel codes the complement of KQ to obtain an ω-context free set which is analytic complete.This led to undecidability results : The problem of knowing whether an ω-context free languages is Borel is undecidable, the problem of knowing whether an ω-context free languages is Σ 0 ξ is undecidable.ω-context free languages are Σ 1 1 .And one can play separation games where Player II wins the game iff (α ∈ O ⇒ β ∈ Y and α ∈ Ǒ ⇒ β ∈ Z) (Resp.Player II wins the game iff (α ∈ Q ⇒ β ∈ Y and α ∈ Q ⇒ β ∈ Z)) with Y and Z ω-context free languages.Probably these games are undecidable, probably one can't decide if two ω-context free languages are separated by a Σ 0 ξ set.But Louveau's theorem is true if they are separated by a Σ 0 ξ set ( ξ < ω 1 ), then they are separated by a Σ 0 ξ (∆ 1 1 ) set.
Later on, Finkel, with the same kind of coding, obtained an ω-rational relation which is analytic complete.This gave other undecidability results : The problem of knowing whether an ω-rational relation is Borel is undecidable, the problem of knowing whether an ω-rational relation is Σ 0 ξ is undecidable.Finally we can remark that simple models of asynchronous parallelism on infinite words gives analytic complete sets [36].Note that Kuratowski shows how to eliminate transfinite numbers in mathematical proofs [52].He takes as examples the Cantor Bendixon theorem and the derivation of Felix Hausdorff.For example, in games, ordinals are hidden in the construction of strategies.One can say : don't hide countable ordinals and you will see some true coanalytic sets.This has to do with the boundedness theorem of Lusin [64,65,47].If a set X is a coanalytic set then for all α ∈ X one can associate a countable ordinal ϕ(α).And X is borel if and only if there exists ξ < ω 1 such that for all α ∈ X, ϕ(α) < ξ.Note that Lusin don't think that such a procedure can be effective to decide if a Π 1 1 set is a true (not Borel) Π 1 1 set, and the undecidability results of Finkel shows that he was right.We think that countable ordinals are inherently hidden in models of parallelism, verification and XM L.

Baire class 1 functions
In conclusion, let us say that other properties of ω-rational function which are Baire class 1 can be derived from work by Kechris and Louveau [49].One can find concrete examples of ω-rational Baire class 1 functions in [38,39,51,66] and one can even define Baire class 1 functions on real numbers by using representation of real numbers in Pisot Basis [20].
Finally we remark that Baire has introduced semi continuity, oscillation and the space ω ω .The bibliography is big but still incomplete.We have certainly forgotten some work, especially work of Schupp on alternating automata and work on fixed point theory of Arnold, Niwinski, Kozen, Bradfield, Walukiewicz, Wilke.
A winning strategy for II gives a continuous function ( a Lipschitz map) ϕ such that ϕ −1 (Y ) = O.A winning strategy for I gives a continuous function ( a Lipschitz map) ψ such that : ψ(Y ) ⊂ Ǒ, ψ(Z) ⊂ O.That is to say ψ −1 ( Ǒ) separate Y from Z.It is easy to see that there is a closed set which separates Y from Z if and only if player I has a winning strategy in the first separation game.If Y and Z are ω-regular sets one can deduce from Büchi Landweber's theorem that one can decide if two ω-regular disjoint sets are separated by a closed set.Moreover if they are separated by a closed set, they are separated by a Π 0 1 (Auto).In a second game, Player II wins the game iff (α ∈ Q ⇒ β ∈ Y and α ∈ Q ⇒ β ∈ Z).A winning strategy for II gives a continuous function (a Lipschitz map) ϕ such that ϕ −1 (Y ) = Q.A winning strategy for I gives a continuous function (a Lipschitz map) ψ such that : ψ It is easy to see that there is a Π 0 2 set which separates Y from Z if and only if player I has a winning strategy in the first separation game.If Y and Z are ω-regular sets, one can deduce from Büchi Landweber's theorem that one can decide if two ω-regular disjoint sets are separated by a Π 0 2 set.Moreover if they are separated by a Π 0