Polishness of Some Topologies Related to Automata

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Introduction
This paper is a contribution to the study of the interactions between descriptive set theory and theoretical computer science.These interactions have already been the subject of many studies, see for instance [18,32,21,30,27,25,29,6,28,7,11,9,5].
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 verification of non-terminating systems, see [12,25].The space Σ N of infinite words over a finite alphabet Σ being equipped with the usual Cantor topology, a natural way to study the complexity of ω-languages accepted by various kinds of automata is to study their topological complexity, and particularly to locate them with regard to the Borel and the projective hierarchies.
However, as noticed in [26] by Schwarz and Staiger and in [15] by Hoffmann and Staiger, it turned out that for several purposes some other topologies on a space Σ N are useful, for instance for studying fragments of first-order logic over infinite words or for a topological characterisation of random infinite words (see also [14]).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 iff 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 definable subsets of Polish topological spaces, as well

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Polishness of Some Topologies Related to Automata as the study of hierarchies of complexity (see [16,25] for basic notions).The analytic sets, which are the projections of 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 [24] for 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 machines (see [31]).
In [26], Schwarz and Staiger prove that the Büchi topology, which is generated by the regular ω-languages, is metrizable.It 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.
We now give some more details about the topologies studied by Schwarz and Staiger in [26] that we investigate in this paper.Let Σ be a finite alphabet with at least two symbols.We will consider the topology τ δ on Σ N generated by the set B δ of sets accepted by an unambiguous Büchi Turing machine.Let Σ * be the set of finite sequences of elements of Σ.The following topologies on Σ N , related to automata, are considered in [26].
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 all τ C -closed ω-regular languages are accepted by deterministic Büchi automata), The alphabetic topology τ α , generated by the set B α of sets of the form B w,A := {wσ | σ ∈ A N }, where w ∈ Σ * and A ⊆ Σ (this topology is useful for investigations in restricted first-order theories for infinite words), The strong alphabetic topology τ s , generated by the set B s of sets of the form where w ∈ Σ * and A ⊆ Σ (this topology is derived from τ α , and considered in [4], together with τ α ).In [26], Schwarz and Staiger prove that these topologies are metrizable.We improve this result: From this result, it is already possible to infer many properties of the space Σ N , where Σ is a finite alphabet, equipped with the Büchi topology.In particular, we first 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, and that a set is Borel for the Cantor topology if and only if it is Borel for the Büchi topology, but 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 derive many other properties from the polishness of the Büchi topology.

Background
We first recall the notions required to understand fully the introduction and the sequel (see for example [25,30,16,24]).

Theoretical computer science
, where Σ is the input alphabet, Q is the finite state set, δ is the transition relation, Q i and Q f are the sets of initial and final states.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, that is, q i ∈ Q f for infinitely many i.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.
A Büchi automaton is actually similar to a classical finite automaton.A finite word w of length n is accepted by an 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 us recall that 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 [1]): Theorem 2 (Büchi).Let Σ be a finite alphabet, and L ⊆ Σ N .The following statements are equivalent: The closure properties of the class of ω-regular languages mentioned in the introduction are the following (see [25] and [26]).If Σ is a set and w ∈ Σ * , then w defines the usual basic clopen set N w := {σ ∈ Σ N | w is a prefix ofσ} of the Cantor topology τ C (so Theorem 3 (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.
We now turn to the study of Turing machines (see [3,30]).A Büchi Turing machine is a tuple M = (Σ, Q, Γ, δ, q 0 , Q f ), where Σ and Γ are the input and tape alphabets satisfying Σ ⊆ Γ, Q is the finite state set, δ is the transition relation, q 0 is the initial state and Q f is the set of final states.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 the replaced by symbol b at position j and that all 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.It is complete if the head visits all positions of the tape.This means that for each integer N , the exists an integer i such that C S L 2 0 1 7

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Polishness of Some Topologies Related to Automata j i ≥ N .The run is accepting if it visits infinitely often final states, that is, p i ∈ Q f for infinitely many integers i.The ω-language accepted by M is the set of inputs σ such that there exists an accepting and complete run on σ.
Notice that 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 [3,30].Moreover several types of required behaviour on the input tape have been considered in the literature, see [31,10,8].
A Büchi automaton A is in fact a Büchi Turing machine whose head only move forwards.This means that each of its transition has the form (p, a, q, b, d) where d = 1.Note that the written symbol b does not matter since each position of the tape is just visited once and the 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.

Definition 4.
A topological space X is a Polish space if it is 1. separable (there is a countable dense sequence (x n ) in X), 2. completely metrizable (there is a complete distance d on X which is compatible with the topology of X).
Effective descriptive set theory is based on the notion of 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 6.Let (x n ) n∈N , d be a recursive presentation of a Polish space X.

We fix a countable basis of
for some recursive function f .3. A subset S of X is effectively closed (denoted S ∈ Π 0 1 ) if its complement ¬S is semirecursive.

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 [31], see also [8]: Theorem 7. Let Σ be a finite alphabet, and L ⊆ Σ N .The following statements are equivalent: We now recall the 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 x i ∈ U i such that U i ⊆ V i−1 , 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 8.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 Choquet game.
Note that every zero-dimensional space is regular.The following result is 8.18 in [16].

Theorem 9 (Choquet). A nonempty, second countable topological space is Polish if and only if it is T 1 , regular, and strong Choquet.
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Polishness of Some Topologies Related to Automata
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 Σ X .By Theorem 7, 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 [24]).In particular, it is not Polish.

3
Proof of the main result The proof of Theorem 1 is organized as follows.We provide below four properties which ensure that a given topological space is strong Choquet.Then we use Theorem 9 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 := {0, 1} having infinitely many symbols 1.
We will work in the spaces of the form Σ N , where Σ is a finite set with at least two elements.We will fix 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: (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 . This gives α 0 ∈ P ∞ such that (σ 0 , α 0 ) ∈ C 0 .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 so that Player 2 respects the rules of the game if he plays V 0 .Now Player 1 plays ).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 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 that the Gandy-Harrington topology is not Polish in general.However, it is almost Polish as it fulfills the four properties (P1)-(P4).
Note that τ Σ is second countable since there are only countably many Σ 1 1 subsets of X (see 3F.6 in [24]), T 1 since it is finer than the usual topology by the property (P1), and strong Choquet by Theorem 10.
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 [19]).In particular, (Ω X , τ Σ ) is zero-dimensional, and regular.As it is, just like (X, τ Σ ), second countable, T 1 and and strong Choquet, (Ω X , τ Σ ) is a Polish space, by Theorem 9.

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 [26] shows that τ B is metrizable.We now give a distance which is compatible with τ B .This metric was used in [14] (Theorem 2 and Lemma 21 and several corollaries following C S L 2 0 1 7

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Polishness of Some Topologies Related to Automata Lemma 21).A similar argument for subword metrics is in [ [15], Section 4].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 iff The distance δ on Σ N is then defined as follows: for x, y ∈ Σ N , δ(x, y) = 0 if x = y, 2 −n if x = y, 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 complete metric.Proposition 11. 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 definition of δ.Let x, y, z ∈ Σ N , and assume that δ(x, y) + δ(y, z) < δ(x, z) = 2 −n .Then δ(x, y) < 2 −n and δ(y, z) < 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.But this holds for every Büchi automaton with n states and then δ(x, z) < 2 −n .This leads to a contradiction and thus δ(x, z) ≤ δ(x, y) + δ(y, 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 Büchi automata.Let then L(A) be an ω-language accepted by a Büchi automaton A having n states, and x ∈ L(A).We now show that the open ball B(x, 2 −(n+1) ) with center x and δ-radius 2 −(n+1) is a subset of L(A).Indeed, if δ(x, y) < 2 −(n+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) ( 3. Without loss of generality, we set Σ = {0, 1} and we consider, for a natural number n ≥ 1, the ω-word X n = 0 n! • 1 • 0 ω over the alphabet {0, 1} having only one symbol 1 after n! symbols 0, where n! : 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 there is a sequence of symbols 0 of length p ≤ k which can be added several times to the word X n so that the resulting word will still be accepted by A. Formally, any word 0 n!+lp • 1 • 0 ω , for a natural number l ≥ 1, will be accepted by A. In particular

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 ) < 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 would lead to a contradiction, and thus the distance δ is not complete because the sequence (X n ) is a Cauchy sequence which is not convergent.

Lemma 12.
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 the final states.It maps each state q to 1 if q ∈ Q f and to 0 otherwise.The function χ f is extended to Let C be the subset of Σ N × P ∞ defined by Since K is compact as a closed subset of a compact space and Let ∆ be defined by

This allows us to define a Büchi automaton by
Corollary 13.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 the property (P1), and strong Choquet by Theorem 10.Moreover, it is zero-dimensional since the class of ω-regular languages is closed under taking complements (see Theorem 3).It remains to apply Theorem 9.

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Polishness of Some Topologies Related to Automata

The other topologies
Proof of the main result.It is well known that (Σ N , τ C ) is metrizable and compact, and thus Polish.
By Theorem 3.4 in [23], 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 [8], a subset of Σ N is ∆ 1  1 if and only if it is accepted by an unambiguous Büchi Turing machine.Therefore ) is a basis for the zero-dimensional Polish topology τ δ .Corollary 13 gives the result for the Büchi topology.Let (X, τ ) be a Polish space, and (C n ) n∈N be a sequence of closed subsets of (X, τ ).By 13.2 in [16], the topology τ n generated by τ ∪ {C n } is Polish.By 13.3 in [16], 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.This shows that the automatic topology and the alphabetic topology are Polish since they refine the usual product topology on Σ N .Note that S w,A is a G δ subset of Σ N , and thus a Polish subspace of Σ N , by 3.11 in [16].Note also that Σ N = ∅ =A⊆Σ S ∅,A ∪ ∅ =A Σ,w∈Σ * ,b∈Σ\A S wb,A , and that this union is disjoint.As all the sets in this union are open for the strong alphabetic topology, they are also clopen for this topology.This shows that this topology is the countable sum of its restrictions to the sets in this union.As the strong alphabetic topology coincides with the usual topology on each of these sets, it is Polish, by 3.3 in [16].

4
Consequences for our topologies
1.As noted in [26], Ult is the set of isolated points of S B and S A (recall that a point σ ∈ Σ N is isolated if {σ} is an open set).Indeed, each singleton {u•v ω } formed by an ultimately periodic ω-word is an ω-regular language, and thus each ultimately periodic ω-word is an isolated point of S A .Conversely, if {σ} is τ B -open, then it is ω-regular and then the ω-word σ is ultimately periodic (because any countable ω-regular language contains only ultimately periodic ω-words, see [1,25,30]). 2. Every nonempty ω-regular set contains an ultimately periodic ω-word, [1,25,30] [16]).Every subset of X has the Baire property (which is very uncommon, see for exemple 8.24 in [16]).Indeed,

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Polishness of Some Topologies Related to Automata example, U 0 V ω 0 is not countable.This implies that we can find v 0 , v 1 ∈ V 0 which are not powers of the same word.Indeed, we argue by contradiction.Let v 0 ∈ V 0 \ {∅}, and v ∈ Σ * of minimal length such that v 0 is a power of v.We can find w 0 , w 1 , ... ∈ V 0 \ {∅} such that σ := w 0 w 1 ... = v ω .Fix a natural number i. Then w i and v 0 are powers of the same word w.By Corollary 6.2.5 in [22], v and w are powers of the same word u, and v 0 too.By minimality, u = v, and w i is a power of v. Thus σ = v ω , which is absurd.Let u 0 ∈ U 0 , and L := {u 0 }•{v 0 , v 1 } ω .Note that P ∩ L is a τ B -closed subset of K, so that it is τ B -compact.As the identity map from (P ∩ L , τ B ) onto (P ∩ L , τ C ), (where τ C is the Cantor topology), is continuous, P ∩ L is also τ C -compact.But the map α → u 0 v α(0) v α (1) . . . is a homeomorphism from the Cantor space onto L , by Corollaries 6.2.5 and 6.2.6 in [22].Thus P ∩ L = L \Ult is a dense closed subset of L .Thus P ∩ L = L , which is absurd since u 0 • v ω 0 ∈ L \P .(c) By (b) and 7.9 in [16], for each Polish space Y , we can find a τ B -closed set F ⊆ P (an intersection of ω-regular sets) and a continuous bijection from F onto Y .In particular, if Y is not empty, then there is a continuous surjection from P onto Y extending the previous bijection.By 7.15 in [16], if moreover Y is perfect, then there is a continuous bijection from P onto Y .(d) A consequence of (b), Corollary 13, and 7.10 in [16], is that the space S B is not K σ (i.e., countable union of compact sets).

(a)
The σ-algebra A z generated by B z is exactly the σ-algebra of Borel subsets of S C (generated by the open subsets of S C ).Indeed, the identity map from S z onto S C is a continuous bijection.By 15.2 in [16], the inverse map is Borel, so that the two Borel structures coincide.It remains to note that the σ-algebra generated by B z is exactly the σ-algebra of Borel subsets of S z .(b) We can define a natural hierarchy in A z : let Σ z 1 := (B z ) σ be the set of countable unions of elements of B z , and, inductively on 1 ≤ ξ < ω 1 , Π z ξ := {¬L | L ∈ Σ z ξ } and Σ z ξ := ( η<ξ Π z η ) σ .This hierarchy is actually the Borel hierarchy in the family of Borel subsets of S z , in the sense that Σ z ξ = Σ 0 ξ (S z ) and Π z ξ = Π 0 ξ (S z ) (see 11.B in [16]; we will also consider ∆ z ξ := Σ z ξ ∩ Π z ξ = ∆ 0 ξ (S z ) and Π 0 0 (S z ) := B z ).By the main result and 22.4 in [16], this hierarchy is strict and of length ω 1 (the first uncountable ordinal).This comes from the existence of universal sets for these classes (see 22.3 in [16]).Recall that a subset U of 2 N ×Σ N is universal for a class Γ of subsets of Σ N if it is in Γ and the Γ subsets of Σ N coincide with the vertical sections U Note that a set is Borel for τ C if and only if it is Borel for any topology τ z , but the levels of the Borel hierarchy may differ for the two topologies.For instance any singleton associated with an ultimately constant word is not open for the Cantor topology and is actually open for the δ, Büchi, automatic, alphabetic and strong alphabetic topologies.Therefore the existence of universal sets for each level of the Borel hierarchy of S z is different from the existence of universal sets for each level of the Borel hierarchy of S C if z = C, and could lead to other consequences or new applications to other domains of theoretical computer science.(c) The Wadge theorem holds (see 22.10 in [16]): let z ∈ {δ, B}, ξ ≥ 1 be a countable ordinal.A subset L of Σ N is in Σ z ξ \ Π z ξ if and only if it is in Σ z ξ , and for each zero-dimensional Polish space X and any Σ 0 ξ subset B of X there is f : X → S z continuous with B = f −1 (L) (we can exchange Σ z ξ and Π z ξ ).(d) The Saint Raymond theorem holds (see 35.45 in [16]): let Σ, Γ be finite sets with at least two elements, L ⊆ Σ N × Γ N be in A z with Σ z 2 vertical sections.Then L = n∈N L n , where L n is in A z and has Π z 1 vertical sections.(e) By 4.(b) and 13.7 in [16], for each L in A B , we can find a τ B -closed set F ⊆ P (an intersection of ω-regular sets) and a continuous bijection from F onto L. In particular, if L is not empty, then there is a continuous surjection from P onto L extending the previous bijection.

Consequences related to analytic sets
6. Recall that the analytic sets are the projections of the elements of A z .We saw in (5).(a) that τ z and τ C have the same Borel sets.This implies that τ z and τ C have the same analytic sets.Then the following items (a)-(d) are not new, but we cite them as interesting results about the topology τ z .
(a) The class of analytic sets contains strictly A z because of the existence of universal sets (see 14.2 in [16]).(b) The Lusin separation theorem holds (see 14.7 in [16]): assume that L, M ⊆ Σ N are disjoint analytic sets.Then there is S in A z such that L ⊆ S ⊆ ¬M (we say that S separates L from M ).(c) The Souslin theorem holds (see 14.11 in [16]): the elements of A z are exactly the analytic sets whose complement is also analytic.(d) By 14.13 in [16], an analytic set L has the perfect set property : either L is countable, or L contains a copy of S C (this copy has size continuum and is compact).(e) The parametrization theorem for Borel sets holds (see 35.5 in [16]).We say that a set is co-analytic if its complement is analytic.We can find a co-analytic subset D of 2 N , an analytic subset S of 2 N × Σ N , and a co-analytic subset P of 2 N × Σ N such that the vertical sections S α and P α of S and P at any point α of D are equal, and the elements of A z are exactly the sets S α for some α in D.

Consequences related to uniformization problems 7.
Let X, Y be sets, L ⊆ X × Y , and f : X → Y be a partial map.We say that f uniformizes L if the domain of f is π 0 [L] and the graph of f is contained in L. In the sequel, Σ and Γ will be finite sets with at least two elements.
(a) Recall that if L ⊆ Σ N ×Γ N is an infinitary rational relation, then L can be uniformized by a function whose graph is ω-rational (see Theorem 5 in [2]).Moreover, the inverse image of an ω-regular language by such a function is itself ω-regular, and thus the function is continuous for τ B .(b) The Arsenin-Kunugui theorem holds (see 18.18 in [16]): if L ⊆ Σ N × Γ N is in A z and has K σ vertical sections, then L can be uniformized by a function whose graph is in A z (a K σ set is a countable union of compact sets).(c) Let ξ ≥ 1 be a countable ordinal.The class Σ 0 ξ has the number uniformization property : if X is a zero-dimensional Polish space and L ⊆ X × N is Σ 0 ξ , then L can be uniformized by a function whose graph is Σ 0 ξ .Consequently, for z ∈ {δ, B}, Σ z ξ has the generalized reduction property : if (L n ) n∈N is a sequence of Σ z ξ subsets of Σ N , then there is a sequence (B n ) n∈N of pairwise disjoint Σ z ξ subsets of Σ N with n∈N B n = n∈N L n and B n ⊆ L n , C S L 2 0 1 7

22:14
Polishness of Some Topologies Related to Automata Π z ξ has the generalized separation property : if (L n ) n∈N is a sequence of Π z ξ subsets of Σ N with empty intersection, then there is a sequence (B n ) n∈N of ∆ z ξ subsets of Σ N with empty intersection and B n ⊇ L n (see 22.16 in [16]).Moreover, the class of co-analytic sets has the number uniformization property and the generalized reduction property, the class of analytic sets has the generalized separation property (see 35.1 in [16]).(d) The Novikov-Kondo theorem holds (see 36.14 in [16]): if L ⊆ Σ N × Γ N is co-analytic, then L can be uniformized by a function whose graph is co-analytic.

Concluding Remarks
We have obtained in this paper new links and interactions between descriptive set theory and theoretical computer science, showing that four topologies considered in [26] are Polish, and providing many consequences of these results.
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 [13,17,19,20].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.
From the main result, 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 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 = 2 −n for some natural number n.Then y ∈ B(x, 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, 2 −n ) is the intersection of the regular ω-languages L(A i ) for 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 Büchi automata B i having p ≤ n states and such that x / ∈ L(B i ).The class of regular ω-languages being closed under complementation and finite intersection, the open ball B(x, 2 −n ) is actually a regular ω-language and thus an open set for τ B .
. In particular, the set Ult of isolated points of S B and S A is dense , and a subset of S B or S A is dense if and only if it contains Ult. 3. Let X be a topological space in which the set I of isolated points is dense, for example S B by (2).Recall that a subset L of a topological space Y has the Baire property if there is an open subset O of Y such that the symmetric difference L∆O := (L\O) ∪ (O\L) is meager.This is equivalent to say that L = G ∪ M , where G is G δ (i.e. a countable intersection of open sets) and M is meager (see 8.23 in