Polishness of some topologies related to word or tree automata

We prove that the B\"uchi topology and the automatic topology are Polish. We also show that this cannot be fully extended to the case of a space of infinite labelled binary trees; in particular the B\"uchi and the Muller topologies are not Polish in this case.

In particular, the theory of automata reading infinite words, which is closely related to infinite games, is now a rich theory which is used for the specification and the verification of non-terminating systems, see [15,34].The space Σ N of infinite words over a finite alphabet Σ, equipped with the usual Cantor topology τ C , is a natural place to study the topological complexity of the ω-languages accepted by various kinds of automata.In particular, it is interesting to locate them with respect to the Borel and the projective hierarchies.
However, as noticed in [35] by Schwarz and Staiger and in [18] by Hoffmann and Staiger, it turns out that for several purposes some other topologies on the space Σ N are useful, for instance for studying fragments of the first-order logic over infinite words, or for a topological characterization of the random infinite words (see also [17]).In particular, Schwarz and Staiger studied four topologies on the space Σ N of infinite words over a finite alphabet Σ, which are all related to automata, and refine the Cantor topology on Σ N : the Büchi topology, the automatic topology, the alphabetic topology, and the strong alphabetic topology.
Recall that a topological space is Polish if and only if it is separable, i.e., contains a countable dense subset, and its topology is induced by a complete metric.Classical descriptive set theory is about the topological complexity of the definable subsets of the Polish topological spaces, as well as the study of some hierarchies of topological complexity (see [19,34] for the basic notions).The analytic sets, which are the projections of the Borel sets, are of particular importance.Similar hierarchies of complexity are studied in effective descriptive set theory, which is based on the theory of recursive functions (see [30] for the basic notions).The effective analytic subsets of the Cantor space (2 N , τ C ) are highly related to theoretical computer science, in the sense that they coincide with the sets recognized by some special kind of Turing machine (see [43]).
We now give details about some topologies that we investigate in this paper.Let Σ be a finite alphabet with at least two symbols.We consider the following topologies on Σ N .
• the Büchi topology τ B , generated by the set B B of ω-regular languages, • the automatic topology τ A , generated by the set B A of τ C -closed ω-regular languages (this topology is remarkable because any τ C -closed (or even τ C -Π 0 2 ) ω-regular language is accepted by some deterministic Büchi automaton, [34]), • the topology τ δ , generated by the set B δ of languages accepted by some unambiguous Büchi Turing machine, • the Gandy-Harrington topology τ GH , generated by the set B GH of languages accepted by some Büchi Turing machine.
In [35], Schwarz and Staiger prove that τ B and τ A are metrizable.The topology τ GH is second countable, T 1 and strong Choquet, but it is not regular and thus not metrizable and not Polish.However, there is a dense basic open set on which τ GH is Polish and zero dimensional, which is sufficient in many applications.The topology τ B is separable, by definition, because there are only countably many regular ω-languages.It remains to see that it is completely metrizable to see that it is Polish.This is one of the main results proved in this paper.
From this result, it is already possible to infer many properties of the space Σ N , equipped with the Büchi topology (see [3] for an extended list).In particular, we get some results about the σ-algebra generated by the ω-regular languages.It is stratified in a hierarchy of length ω 1 (the first uncountable ordinal) and there are universal sets at each level of this hierarchy.Notice that this σ-algebra coincides with the σ-algebra of Borel sets for the Cantor topology.However, the levels of the Borel hierarchy differ for the two topologies.For instance, an ω-regular set which is non-Π 0 2 for the Cantor topology is clopen (i.e., ∆ 0 1 ) for the Büchi topology.Therefore the results about the existence of universal sets at each level of the σ-algebra generated by the ω-regular languages are really new and interesting.
We also investigate, following a suggestion of H. Michalewski, whether it is possible to extend these results to the case of a space T ω Σ of infinite binary trees labelled with letters of the alphabet Σ.On the one hand, the automatic topology can be proved to be Polish in a similar way.On the other hand, we show that the Büchi topology (generated by the set of regular tree languages accepted by some Büchi tree automaton) and the Muller topology (generated by the set of regular tree languages accepted by some Muller tree automaton) are both non-Polish.However we prove that these two topologies have quite different properties: the first one is strong Choquet but not metrizable while the second one is metrizable but not strong Choquet.

Background
We first recall the notions required to understand fully the introduction and the sequel (see for example [34,42,19,30]).

Theoretical computer science.
A Büchi automaton is a tuple , where Σ is the input alphabet, Q is the finite set of states, Q i and Q f are the sets of initial and final states, and δ is the transition relation.The transition relation δ is a subset of Q × Σ × Q.
A run on some sequence σ ∈ Σ N is a sequence (q n ) i∈N ∈ Q N of states such that q 0 is initial (q 0 ∈ Q i ) and q i , σ(i), q i+1 is a transition in δ for each i ≥ 0. It is accepting if it visits infinitely often final states, i.e., q i ∈ Q f for infinitely many i's.An input sequence σ is accepted if there exists an accepting run on α.The set of accepted inputs is denoted L(A).A set of infinite words is called ω-regular if it is equal to L(A) for some automaton A (see [34] for the basic notions about regular ω-languages, which are the ω-languages accepted by some Büchi or Muller automaton).
A Büchi automaton is actually similar to a classical finite automaton.A finite word w of length n is accepted by some automaton A if there is sequence (q i ) i≤n of n + 1 states such that q 0 is initial (q 0 ∈ Q i ), q n is final (q n ∈ Q f ) and (q i , σ(i), q i+1 ) is a transition in δ for each 0 ≤ i < n.The set of accepted finite words is denoted by U (A).A set of finite words is called regular if it is equal to U (A) for some automaton A.
Let U be a set of finite words, and V be a set of finite or infinite words.We recall that An infinite word is ultimately periodic if it is of the form u • v ω , where u, v are finite words.
The ω-power of a set U of finite words is defined by The ω-powers play a crucial role in the characterization of ω-regular languages (see [2]).
Theorem 2.1 (Büchi).Let Σ be a finite alphabet, and L ⊆ Σ N .The following are equivalent: (1) L is ω-regular, (2) there are 2n regular languages In particular, each singleton {uv ω } formed by an ultimately periodic ω-word is an ω-regular language.We now recall some important properties of the class of ω-regular languages (see [34] and [35]).Let Σ be a set, and Σ * be the set of finite sequences of elements of Σ.If w ∈ Σ * , then w defines the usual basic clopen (i.e., closed and open) set Theorem 2.2 (Büchi).The class of ω-regular languages contains the usual basic clopen sets and is closed under finite unions and intersections, taking complements, and projections (from a product alphabet onto one of its coordinates).
We now turn to the study of Turing machines (see [6,42]).A Büchi Turing machine is a tuple M = (Σ, Γ, Q, q 0 , Q f , δ), where Σ and Γ are the input and tape alphabets satisfying Σ ⊆ Γ, Q is the finite set of states, q 0 is the initial state, Q f is the set of final states, and δ is the transition relation.The relation δ is a subset of ( A configuration of M is a triple (q, γ, j) where q ∈ Q is the current state, γ ∈ Γ N is the content of the tape and the non-negative integer j ∈ N is the position of the head on the tape.
Two configurations (q, γ, j) and (q , γ , j ) of M are consecutive if there exists a transition (q, a, q , b, d) ∈ δ such that the following conditions are met.
(1) γ(j) = a, γ (j) = b and γ(i) = γ (i) for each i = j.This means that the symbol a is replaced by the symbol b at the position j and that all the other symbols on the tape remain unchanged.(2) The two positions j and j satisfy the equality j = j + d.
A run of the machine M on some input σ ∈ Σ N is a sequence (p i , γ i , j i ) i∈N of consecutive configurations such that p 0 = q 0 , γ 0 = σ and j 0 = 0.The run is accepting if it visits infinitely often the final states, i.e., p i ∈ Q f for infinitely many i's.The ω-language accepted by M is the set of inputs σ such that there exists an accepting run on σ.
Notice that some other accepting conditions have been considered for the acceptance of infinite words by Turing machines, like the 1' or Muller ones (the latter one was firstly called 3-acceptance), see [6,42].Moreover, several types of required behaviour on the input tape have been considered in the literature, see [43,12,10].
A Büchi automaton A is in fact a Büchi Turing machine whose head only moves forwards.This means that each of its transitions has the form (p, a, q, b, d) where d = 1.Note that the written symbol b is never read.

Descriptive set theory.
Classical descriptive set theory takes place in Polish topological spaces.We first recall that if d is a distance on a set X, and (x n ) n∈N is a sequence of elements of X, then the sequence (x n ) n∈N is called a Cauchy sequence if In a topological space X whose topology is induced by a distance d, the distance d and the metric space (X, d) are said to be complete if every Cauchy sequence in X is convergent.
completely metrizable (there is a complete distance d on X which is compatible with the topology of X).
The most classical hierarchy of topological complexity in descriptive set theory is the one given by the Borel classes.If Γ is a class of sets in metrizable spaces, then Γ := {¬S | S ∈ Γ}, and (Γ) σ is the class of countable unions of sets in Γ. Recall that the Borel hierarchy is the inclusion from left to right in the following picture.
Above the Borel hierarchy sits the projective hierarchy, which is the inclusion from left to right in the following picture.
Effective descriptive set theory is based on the notion of a recursive function.A function from N k to N l is said to be recursive if it is total and computable.By extension, a relation is called recursive if its characteristic function is recursive.
d is a compatible complete distance on X such that the following relations P and Q are recursive: A Polish space X is recursively presented if there is a recursive presentation of it.
Note that the formula (p, q) → 2 p (2q + 1) − 1 defines a recursive bijection N 2 → N. One can check that the coordinates of the inverse map are also recursive.They will be denoted n → (n) 0 and n → (n) 1 in the sequel.These maps will help us to define some of the basic effective classes.Definition 2.5.Let (x n ) n∈N , d be a recursive presentation of a Polish space X.
(1) We fix a countable basis of (4) One can check that a product of two recursively presented Polish spaces has a recursive presentation, and that the Baire space N N has a recursive presentation.A subset S of X is effectively analytic ). ( 6) We will also use the following relativized classes: if X, Y are recursively presented Polish spaces and y ∈ Y , then we say that The crucial link between the effective classes and the classical corresponding classes is as follows: the class of analytic (resp., co-analytic, Borel) subsets of Y is equal to . This allows to use effective descriptive set theory to prove results of classical type.In the sequel, when we consider an effective class in some Σ N with Σ finite, we will always use a fixed recursive presentation associated with the Cantor topology.The following result is proved in [43], see also [10].
Theorem 2.6.Let Σ be a finite alphabet, and L ⊆ Σ N .The following statements are equivalent: (1) L = L(M) for some Büchi Turing machine M, (2) L ∈ Σ 1  1 .We now recall the strong Choquet game played by two players on a topological space X.Players 1 and 2 play alternatively.At each turn i, Player 1 plays by choosing an open subset U i and a point , where V i−1 has been chosen by Player 2 at the previous turn.Player 2, plays by choosing an open subset V i such that x i ∈ V i and V i ⊆ U i .Player 2 wins the game if i∈N V i = ∅.We now recall some classical notions of topology.
Definition 2.7.A topological space X is said to be • T 1 if every singleton of X is closed, • regular if for every point of X and every open neighborhood U of x, there is an open neighborhood V of x with V ⊆ U , • second countable if its topology has a countable basis, • zero-dimensional if there is a basis made of clopen sets, • strong Choquet if X is not empty and Player 2 has a winning strategy in the strong Choquet game.
Note that every zero-dimensional space is regular.The following result is Theorem 8.18 in [19].
Theorem 2.8 (Choquet).A nonempty, second countable topological space is Polish if and only if it is T 1 , regular, and strong Choquet.
Let X be a nonempty recursively presented Polish space.The Gandy-Harrington topology on X is generated by the Σ 1 1 subsets of X, and denoted τ X GH .By Theorem 2.6, this topology is also related to automata and Turing machines.As there are some effectively analytic sets whose complement is not analytic, the Gandy-Harrington topology is not metrizable (in fact not regular) in general (see 3E.9 in [30]).In particular, it is not Polish.
Let Γ be a class of sets in Polish spaces.If Y is a Polish space, then we say that A ∈ Γ(Y ) is Γ-complete if, for each zero-dimensional Polish space X and each B ∈ Γ(X), there is f : X → Y continuous such that B = f −1 (A).By Section 22.B in [19], if Γ is of the form Σ or Π in the Borel or the projective hierarchy, and if A is Γ-complete, then A is not in Γ. Theorem 22.10 in [19] gives a converse in the Borel hierarchy.

Proof of Theorem 1.1
The proof of Theorem 1.1 is organized as follows.We provide below four properties which ensure that a given topological space is strong Choquet.Then we use Theorem 2.8 to prove that the considered spaces are indeed Polish.
Let Σ be a countable alphabet.The set Σ N is equipped with the product topology of the discrete topology on Σ, unless another topology is specified.This topology is induced by a natural metric, called the prefix metric which is defined as follows.For σ = σ ∈ Σ N , the distance d is given by When Σ is finite this topology is the classical Cantor topology.When Σ is countably infinite the topological space is homeomorphic to the Baire space N N .
If Σ is a set, σ ∈ Σ N and l ∈ N, then σ|l is the prefix of σ of length l.We set 2 := {0, 1} and . This latter set is simply the set of infinite words over the alphabet 2 having infinitely many 1's.
We will work in the spaces of the form Σ N , where Σ is a finite set with at least two elements.We consider a topology τ Σ on Σ N , and a basis B Σ for τ Σ .We consider the following properties of the family (τ Σ , B Σ ) Σ , using the previous identification of Σ N × Γ N and (Σ × Γ) N : (P1) B Σ contains the usual basic clopen sets N w , (P2) B Σ is closed under finite unions and intersections, (P3) B Σ is closed under projections, in the sense that if Γ is a finite set with at least two elements and Proof.We first describe a strategy τ for Player 2. Player 1 first plays We choose l 0 0 ∈ N big enough to ensure that if s 0 0 := α 0 |l 0 0 , then s 0 0 has at least a coordinate equal to 1.We set w 0 := σ 0 |1 and ).We choose l 0 1 > l 0 0 big enough to ensure that if s 0 1 := α 0 |l 0 1 , then s 0 1 has at least two coordinates equal to 1.We set w 1 := σ 1 |2 and Next, Player 1 plays We choose l 2 0 ∈ N big enough to ensure that if s 2 0 := α 2 |l 2 0 , then s 2 0 has at least one coordinate equal to 1.As ).We choose l 0 2 > l 0 1 big enough to ensure that if s 0 2 := α 0 |l 0 2 , then s 0 2 has at least three coordinates equal to 1.We set w 2 := σ 2 |3 and If we go on like this, we build w l ∈ Σ l+1 and s n l ∈ 2 * such that w 0 ⊆ w 1 ⊆ ... and s n 0 s n 1 ... This allows us to define σ := lim l→∞ w l ∈ Σ N and, for each n ∈ N, α n := lim l→∞ s n l ∈ 2 N .Note that α n ∈ P ∞ since s n l has at least l + 1 coordinates equal to 1.As (σ, α n ) is the limit of (w l , s n l ) as l goes to infinity and so that τ is winning for Player 2.

The Gandy-Harrington topology.
We have already mentioned the fact that the Gandy-Harrington topology is not Polish in general.However, it is almost Polish since it fulfills Properties (P1)-(P4).
Let Σ be a finite alphabet with at least two elements and X be the space Σ N equipped with the topology τ Σ := τ X GH generated by the family B Σ of Σ 1 1 subsets of X.Note that the assumption of Theorem 3.1 are satisfied.Indeed, (P1)-(P3) come from 3E.2 in [30].For (P4), let F be a Π 0 1 subset of X × N N such that L = π 0 [F ].Let ϕ be the function from N N to 2 N defined by ϕ(β) = 0 β(0) 10 β(1) 1 . . .Note that ϕ is a homeomorphism from N N onto P ∞ , and recursive (which means that the relation ϕ [30]).
Note that τ Σ is second countable since there are only countably many Σ 1 1 subsets of X (see 3F.6 in [30]), T 1 since it is finer than the usual topology by the property (P1), and strong Choquet by Theorem 3.1.
One can show that there is a dense basic open subset Ω X of (X, τ Σ ) such that S ∩ Ω X is a clopen subset of (Ω X , τ Σ ) for each Σ 1  1 subset S of X (see [22]).In particular, (Ω X , τ Σ ) is zero-dimensional, and regular.As it is, just like (X, τ Σ ), second countable, T 1 and strong Choquet, (Ω X , τ Σ ) is a Polish space, by Theorem 2.8.

The Büchi topology.
Let Σ be a finite alphabet with at least two symbols, and X be the space Σ N equipped with the Büchi topology τ B generated by the family B B of ω-regular languages in X. Theorem 29 in [35] shows that τ B is metrizable.We now give a distance which is compatible with τ B .This metric was used in [17] (Theorem 2 and Lemma 21 and several corollaries following Lemma 21).A similar argument for subword metrics is in Section 4 in [18].If A is a Büchi automaton, then we denote |A| the number of states of A. We say that a Büchi automaton separates x and y if and only if The distance δ on Σ N is then defined as follows: where n := min{|A| | A is a Büchi automaton which separates x and y}.We now describe some properties of the map δ.This is the occasion to illustrate the notion of a complete metric.
Proposition 3.2.The following properties of δ hold: (1) the map δ defines a distance on Σ N , (2) the distance δ is compatible with τ B , (3) the distance δ is not complete.
Proof. 1.If x, y ∈ Σ N , then δ(x, y) = δ(y, x), by the definition of δ.Let x, y, z ∈ Σ N , and assume that δ(x, y) + δ(y, z) < δ(x, z) = 1 2 n .Then δ(x, y) < 1 2 n and δ(y, z) < 1 2 n hold.In particular, if A is a Büchi automaton with n states then it does not separate x and y and similarly it does not separate y and z.Thus either x, y, z ∈ L(A) or x, y, z / ∈ L(A).This implies that the Büchi automaton A does not separate x and z.As this holds for every Büchi automaton with n states, δ(x, z) < 1  2 n .This leads to a contradiction and thus δ(x, z) ≤ δ(x, y) + δ(y, z) for all x, y, z ∈ Σ N .This shows that δ is a distance on Σ N .
2. Recall that an open set for this topology is a union of ω-languages accepted by some Büchi automaton.Let then L(A) be an ω-language accepted by some Büchi automaton A having n states, and x ∈ L(A).We now show that the open ball B(x, 1  2 n+1 ) with center x and δ-radius 1  2 n+1 is a subset of L(A).Indeed, if δ(x, y) < 1 2 n+1 < 1 2 n , then x and y cannot be separated by any Büchi automaton with n states, and thus y ∈ L(A).This shows that L(A) (and therefore any open set for τ B ) is open for the topology induced by the distance δ.Conversely, let B(x, r) be an open ball for the distance δ, where r > 0 is a positive real.It is clear from the definition of the distance δ that we may only consider the case r = 1  2 n for some natural number n.Then y ∈ B(x, 1  2 n ) if and only if x and y cannot be separated by any Büchi automaton with p ≤ n states.Therefore the open ball B(x, 1  2 n ) is the intersection of the regular ω-languages L(A i ) for some Büchi automata A i having p ≤ n states and such that x ∈ L(A i ), and of the regular ω-languages Σ N \L(B i ) for some Büchi automata B i having p ≤ n states and such that x / ∈ L(B i ).The class of regular ω-languages being closed under taking complements and finite intersections, the open ball B(x, 1  2 n ) is actually a regular ω-language and thus an open set for τ B .
3. Without loss of generality, we set Σ = 2 and we consider, for a natural number n ≥ 1, the ω-word X n = 0 n! • 1 • 0 ω over the alphabet 2 having only one symbol 1 after n! symbols 0, where n! := n × (n − 1) × • • • × 2 × 1.Let now m > n > k and A be a Büchi automaton with k states.Using a classical pumping argument, we can see that the automaton A cannot separate X n and X m .Indeed, assume first that X n ∈ L(A).Then, when reading the first k symbols 0 of X n , the automaton enters at least twice in a same state q.This implies that: of this form and thus X m ∈ L(A).A very similar pumping argument shows that if X m ∈ L(A), then X n ∈ L(A).This shows that δ(X n , X m ) < 1  2 k and finally that the sequence (X n ) is a Cauchy sequence for the distance δ.On the other hand if this sequence was converging to an ω-word x then x should be the word 0 ω because τ B is finer than τ C .But 0 ω is an ultimately periodic word and thus it is an isolated point for τ B .This leads to a contradiction, and thus the distance δ is not complete because the sequence (X n ) is a Cauchy sequence which is not convergent.Proposition 3.2 gives a motivation for deriving Theorem 1.1 from Theorem 3.1.Note that the assumption of Theorem 3.1 are satisfied.Indeed, (P1)-(P3) come from Theorem 2.2.We now check (P4).Lemma 3.3.Let Σ be a finite set with at least two elements, and L ⊆ Σ N be an ω-regular language.Then there is a closed subset C of Σ N × P ∞ , which is ω-regular as a subset of (Σ × 2) N identified with Σ N × 2 N , and such that be a Büchi automaton and let L = L(A) be its set of accepted words.Let χ f be the characteristic function of Q f .It maps the state q to 1 if q ∈ Q f , and to 0 otherwise.The function χ f is extended to Q N by setting α = χ f ((q n ) n∈N ) where α(n) = χ f (q n ).Note that a run ρ of A is accepting if and only if χ f (ρ) ∈ P ∞ .Let C be the subset of Σ N × P ∞ defined by

By the definition of
As K is compact as a closed subset of a compact space and

This allows us to define a Büchi automaton by
Corollary 3.4.Let Σ be a finite set with at least two elements.Then the Büchi topology τ B is zero dimensional and Polish.
Proof.As there are only countably many possible automata (up to identifications), B B is countable.This shows that τ B is second countable.It is T 1 since it is finer than the usual topology by Property (P1), and strong Choquet by Theorem 3.1.Moreover, it is zero-dimensional since the class of ω-regular languages is closed under taking complements (see Theorem 2.2).It remains to apply Theorem 2.8.

3.3.
The other topologies.Lemma 3.5.Let (X, τ ) be a Polish space, and (C n ) n∈N be a sequence of closed subsets of (X, τ ).Then the topology generated by τ ∪ {C n | n ∈ N} is Polish.
Proof.By Lemma 13.2 in [19], the topology τ n generated by τ ∪ {C n } is Polish.By Lemma 13.3 in [19], the topology τ ∞ generated by n∈N τ n is Polish.Thus the topology generated by τ ∪ {C n | n ∈ N}, which is τ ∞ , is Polish.
Proof of Theorem 1.1.It is well known that (Σ N , τ C ) is metrizable and compact, and thus Polish, and zero-dimensional.
• By Theorem 3.4 in [26], the implication (iii) ⇒ (i), ∆ 1 1 (Σ N ) is a basis for a zero-dimensional Polish topology on Σ N .Recall that a Büchi Turing machine is unambiguous if every ω-word σ ∈ Σ N has at most one accepting run.By Theorem 3.6 in [10], a subset of Σ N is ∆ 1  1 if and only if it is accepted by some unambiguous Büchi Turing machine.Therefore B δ = ∆ 1 1 (Σ N ) is a basis for the zero-dimensional Polish topology τ δ .
• Corollary 3.4 gives the result for the Büchi topology.• Lemma 3.5 shows that the automatic topology is Polish since it refines the usual product topology on Σ N .For this reason also, it is zero-dimensional.

The Büchi and Muller topologies on a space of trees
The notion of a Büchi automaton has been extended to the case of a Büchi tree automaton reading infinite binary trees whose nodes are labelled by letters of a finite alphabet.We now recall this notion and some related ones.
A node of an infinite binary tree is represented by a finite word over the alphabet {l, r} where l means "left" and r means "right".An infinite binary tree whose nodes are labelled in Σ is identified with a function t : {l, r} → Σ.The set of infinite binary trees labelled in Σ will be denoted T ω Σ .A finite binary tree is like an "initial finite subtree" of an infinite binary tree.Thus it can be represented by a function s : S ⊆ {l, r} → Σ, where S is a finite subset of {l, r} which is closed under prefix.If t ∈ T ω Σ is an infinite binary tree, and n ≥ 0 is an integer, then we denote by t|n the initial finite subtree of t whose domain is equal to {l, r} ≤n , where {l, r} ≤n is the set of finite words over the alphabet {l, r} of length smaller than or equal to n.
Let t be an infinite binary tree.A branch B of t is a subset of the set of nodes of t which is linearly ordered by the tree partial order and which is closed under prefix relation (i.e., if x and y are nodes of t such that y ∈ B and x y, then x ∈ B).A branch B of a tree is said to be maximal if and only if there is no other branch of t which strictly contains B. Let t be an infinite binary tree in T ω Σ .If B is a maximal branch of t, then this branch is infinite.Let (u i ) i≥0 be the enumeration of the nodes in B which is strictly increasing for the prefix order.The infinite sequence of the labels of the nodes of such a maximal branch B, i.e., t(u It is an ω-word over the alphabet Σ. Let then L ⊆ Σ ω be an ω-language over Σ.We denote ∃Path(L) the set of infinite trees t in T ω Σ such that t has (at least) one path in L.
We now define the tree automata and the recognizable tree languages.
Definition 4.1.A (non deterministic) tree automaton is a quadruple A = (Σ, Q, q 0 , ∆), where Σ is the finite input alphabet, Q is the finite set of states, The tree automaton A is said to be deterministic if the relation ∆ is a functional one, i.e., if for each (q, a) ∈ Q × Σ there is at most one pair of states (q , q ) such that (q, a, q , q ) ∈ ∆.
A run ρ of the Büchi tree automaton A on an infinite binary tree t ∈ T ω Σ is said to be accepting if for each path of ρ there is some accepting state appearing infinitely often on this path.
The tree language L(A) accepted by the Büchi tree automaton A is the set of infinite binary trees t ∈ T ω Σ such that there is (at least) one accepting run of A on t.Definition 4.3.A Muller (non deterministic) tree automaton is a tuple where (Σ, Q, q 0 , ∆) is a tree automaton and F ⊆ 2 Q is the collection of designated state sets.
A run ρ of the Muller tree automaton A on an infinite binary tree t ∈ T ω Σ is said to be accepting if for each path p of ρ, the set of states appearing infinitely often on this path is in F.
The tree language L(A) accepted by the Muller tree automaton A is the set of infinite binary trees t ∈ T ω Σ such that there is (at least) one accepting run of A on t.The class REG of regular, or recognizable, tree languages is the class of tree languages accepted by some Muller automaton.Remark 4.4.Each tree language accepted by some (deterministic) Büchi automaton is also accepted by some (deterministic) Muller automaton.A tree language is accepted by some Muller tree automaton if and only if it is accepted by some Rabin tree automaton.We refer for instance to [44,34] for the definition of a Rabin tree automaton.
Example 4.5.Let L ⊆ Σ ω be a regular ω-language.Then the set ∃Path(L) ⊆ T ω Σ is accepted by some Büchi tree automaton, hence also by some Muller tree automaton.
The set of infinite binary trees t ∈ T ω Σ having all their paths in L, denoted ∀Path(L), is accepted by some deterministic Muller tree automaton.It is in fact the complement of the set ∃Path(Σ ω − L).
There is a natural topology on the set T ω Σ [29,24,19].It is defined by the following distance.Let t and s be two distinct infinite trees in T ω Σ .Then the distance between t and s is 1 2 n , where n is the smallest integer such that t(x) = s(x) for some word x ∈ {l, r} of length n.
Let T 0 be a set of finite labelled trees, and T 0 • T ω Σ be the set of infinite binary trees which extend some finite labelled binary tree t 0 ∈ T 0 .Here, t 0 is here a sort of prefix, an "initial subtree" of a tree in t 0 • T ω Σ .The open sets are then of the form T 0 • T ω Σ .It is well known that the set T ω Σ , equipped with this topology, is homeomorphic to the Cantor set 2 ω , hence also to the topological spaces Σ ω , where Σ is a finite alphabet having at least two letters.
We are going to use some notation similar to the one used in the case of the space Σ ω .First, if t is a finite binary tree labelled in Σ, we shall denote by N t the clopen set t • T ω Σ .Notice that it is easy to see that one can take, as a restricted basis for the Cantor topology on T ω Σ , the clopen sets of the form t 0 • T ω Σ , where t 0 is a finite labelled binary tree whose domain is of the special form {l, r} ≤n .
The Borel hierarchy and the projective hierarchy on T ω Σ are defined in the same manner as in the case of the topological space Σ ω .
The ω-language P ∞ = (0 • 1) ω is a well known example of Π 0 2 -complete subset of 2 ω (see Exercise 23.1 in [19]).It is the set of ω-words over 2 having infinitely many occurrences of the letter 1.Its complement 2 ω − (0 • 1) ω is a Σ 0 2 -complete subset of 2 ω .It follows from the definition of the Büchi acceptance condition for infinite trees that each tree language recognized by some (non deterministic) Büchi tree automaton is an analytic set.
Niwiński showed that some Büchi recognized tree languages are actually Σ 1 1 -complete sets, [32].An example is any tree language T ⊆ T ω Σ of the form ∃Path(L), where L ⊆ Σ ω is a regular ω-language which is a Π 0 2 -complete subset of Σ ω .In particular, for Σ = 2, the tree language L = ∃Path(P ∞ ) is Σ 1  1 -complete and hence non Borel [32,34,40].Notice that its complement L − = ∀Path(2 ω − (0 • 1) ω ) is a Π 1 1 -complete set.It cannot be accepted by some Büchi tree automaton because it is not a Σ 1  1 set.On the other hand, it can be easily seen that it is accepted by some deterministic Muller tree automaton.
We now consider the topology on the space T ω Σ generated by the regular languages of trees accepted by some Büchi tree automaton.
We prove a version of Theorem 3.1 as a first step towards the proof that the Büchi topology on a space T ω Σ is strong Choquet.We set T ∞ := {t ∈ T ω 2 | for every path p of t ∀k ≥ 0 ∃i ≥ k p(i) = 1}.This set is simply the set of infinite trees over the alphabet 2 having infinitely many letters 1 on every (infinite) path.We will work in the spaces of the form T ω Σ , where Σ is a finite alphabet with at least two elements.We consider a topology τ Σ on T ω Σ , and a basis B Σ for τ Σ .We consider the following properties of the family (τ Σ , B Σ ) Σ , using the previous identification: (P1) B Σ contains the usual basic clopen sets N t , (P2) B Σ is closed under finite unions and intersections, (P3) B Σ is closed under projections, in the sense that if Γ is a finite set with at least two elements and L ∈ B Σ×Γ , then π 0 [L] ∈ B Σ , (P4) for each L ∈ B Σ there is a closed subset C of T ω Σ × T ∞ (i.e., C is the intersection of a closed subset of the Cantor space Consider now the set of trees T ∞ .It is easy to see that T ∞ is accepted by some deterministic Büchi tree automaton.On the other hand it is well known that the tree languages accepted by some deterministic Büchi tree automaton are Π 0 2 sets, see [1].Thus the set T ∞ is actually a Π 0 2 set, it is the intersection of a countable sequence (O i ) i∈N of open sets.We may assume, without loss of generality, that the sequence (O i ) i∈N is decreasing with respect to the inclusion relation.Moreover, each open set O i is a countable union of basic clopen sets N t i,j , j ≥ 0, and we may also assume, without loss of generality, that for all integers i ≥ 0, and all j ≥ 0, the finite tree t i,j ⊆ {l, r} has a domain of the form {l, r} ≤n for some integer n greater than i.We now state the following result, which is a version of Theorem 3.1 in the case of trees.Proof.We first describe a strategy τ for Player 2. Player 1 first plays t If we go on like this, we build w k ∈ Σ {l,r} ≤k+1 and s n l ∈ 2 {l,r} such that w 0 ⊆ w 1 ⊆ ... (σ, β n ) is the limit of (w l , s n l ) as l goes to infinity and N w l × N s n l meets C n (which is closed in so that τ is winning for Player 2. We now check that the Büchi topology on a space T ω Σ satisfies Properties (P1)-(P4).(P1) It is very easy to see that for each finite tree t labelled in Σ, there exists a Büchi tree automaton accepting the usual basic clopen set N t .(P2) B Σ is closed under finite unions, because any basic open set in the Büchi topology is accepted by some non-deterministic Büchi tree automaton.Moreover one can easily show, using a classical product construction, that the class of tree languages accepted by some Büchi tree automaton is closed under finite intersections.Thus B Σ is closed under finite intersections.(P3) It follows easily, from the fact that any basic open set in the Büchi topology is accepted by some non-deterministic Büchi tree automaton, that B Σ is closed under projections.(P4) This property follows from the following lemma, which is very similar to Lemma 3.3 above.
Lemma 4.7.Let Σ be a finite set with at least two elements, and L ⊆ T ω Σ be a regular tree language accepted by some Büchi tree automaton.Then there is a closed subset C of T ω Σ × T ∞ , which is accepted by some Büchi tree automaton as a subset of T ω (Σ×2) identified with T ω Σ × T ω 2 , and such that L = π 0 [C].Proof.Let A = (Σ, Q, q 0 , Q f , ∆) be a Büchi tree automaton, and L = L(A) be its set of accepted trees.We call χ f the characteristic function of Q f .It maps the state q to 1 if q ∈ Q f , and to 0 otherwise.The function χ f is extended to T ω Q by setting t = χ f (t) and t (s) = χ f t(s) .Note that a run ρ of A is accepting if and only if χ f (ρ) ∈ T ∞ .Let C be the subset of T ω Σ × T ∞ defined by C := (t, t ) ∈ T ω Σ × T ∞ | ∃ρ run of A on t such that t = χ F (ρ) .By definition of C, L = π 0 [C].Let K be the subset of T ω Σ × T ω 2 × T ω Q defined by K := (t, t , ρ) ∈ T ω Σ × T ω 2 × T ω Q | ρ is a run of A on t such that t = χ F (ρ) .As K is compact as a closed subset of the compact space Moreover, it is easy to construct a Büchi tree automaton accepting the tree language C. Corollary 4.8.Let Σ be a finite set with at least two elements.Then the Büchi topology on T ω Σ is strong Choquet.Proof.This follows from the fact that the Büchi topology on T ω Σ satisfies Properties (P1)-(P4), and from Theorem 4.6.
On the other hand, as in the case of the Büchi topology on Σ ω , the Büchi topology on T ω Σ is second countable since there are only countably many possible Büchi tree automata (up to identifications), and it is T 1 since it is finer than the usual Cantor topology by Property (P1).However, the Büchi topology on T ω Σ is not Polish, by the following result.

Concluding remarks
We obtained in this paper new links and interactions between descriptive set theory and theoretical computer science, showing that two topologies considered in [35] are Polish.
Notice that this paper is also motivated by the fact that the Gandy-Harrington topology, generated by the effective analytic subsets of a recursively presented Polish space, is an extremely powerful tool in descriptive set theory.In particular, this topology is used to prove some results of classical type (without reference to effective descriptive set theory in their statement).Among these results, let us mention the dichotomy theorems in [16,20,22,23].Sometimes, no other proof is known.Part of the power of this technique comes from the nice closure properties of the class Σ 1  1 of effective analytic sets (in particular the closure under projections).
The class of ω-regular languages has even stronger closure properties.So our hope is that the study of the Büchi topology, generated by the ω-regular languages, will help to prove some automatic versions of known descriptive results in the context of theoretical computer science.For instance, more precisely, let Σ, Γ be finite sets with at least two elements, and L be a subset of Σ N ×Γ N which is ω-regular and also a countable union of Borel rectangles.It would be very interesting to know whether L is open for the product topology τ B ×τ B .Indeed, this would give a version of the G 0 -dichotomy for ω-regular languages, and thus a very serious hope to get versions of many difficult dichotomy results of descriptive set theory for ω-regular languages (see [27]).
From Theorem 1.1, we know that there is a complete distance which is compatible with τ z .It would be interesting to have a natural complete distance compatible with τ B .We leave this as an open question for further study.

and s n 0 s n 1 .
.. This allows us to define σ := lim l→∞ w l ∈ T ω Σ and, for each n ∈ N, β n := lim l→∞ s n l ∈ T ω 2 .Note that β n ∈ T ∞ since the basic open set N s n l is included in the open set O l+1 .