Compact covering mappings and cofinal families of compact subsets of a Borel set

by by Gabriel Debs and Jean Saint Raymond

Abstract : Among other results we prove that the topological statement “Any compact covering mapping between two Π_{3}^{0} spaces is inductively perfect ” is equivalent to the set theoretical statement “ ∀α ∈ ω^{ω}, ω_{1}^{L(α)} < ω_{1} ” ; and that the statement “Any compact covering mapping between two coanalytic spaces is inductively perfect ” is equivalent to “Analytic Determinacy ”. We also prove that these statements are connected to some regularity properties of coanalytic cofinal sets in K(X), the hyperspace of all compact subsets of a Borel set X.

This work is a continuation of [2] and [3] which were motivated by the following topological problem :

Problem 1 : Is any compact covering mapping f : X → Y between two Borel spaces, inductively perfect ?

We recall some basic definitions :

The mapping f : X → Y is said to be \underline{compact covering} if any compact subset of Y is the direct image of some compact subset of X.

The mapping f : X → Y is said to be \underline{inductively perfect} if there exists in X a (necessarily closed) subset X′ such that the restriction f′ of f to X′ is a perfect mapping onto Y (i.e. the inverse image by f′ of a compact subset of Y is compact).

The spaces X and Y considered in Problem 1 are supposed to be metrizable and separable ; and in this context, it is shown in [2] that one can reduce the general case to the case where the spaces are zero-dimensional, hence subsets of the Cantor space 2^{ω}. Also in the sequel a topological space will always be viewed as a subset of 2^{ω} endowed with the induced topology. In particular by a “Borel space”, “coanalytic space” ... we simply mean a Borel or a coanalytic subset of 2^{ω}. For descriptive classes we follow standard notations from Effective Set Theory (see Section 1.1 for more details). Notice that since the surrounding space is always assumed to be 2^{ω} the class Π_{2}^{0} = G_{δ} is here the topological class of all zero-dimensional Polish spaces, and the class Σ_{2}^{0} = K_{σ} is the topological class of all zerodimensional σ-compact spaces.

For a detailed discussion on Problem 1 we refer the reader to [3]. We list here some of the basic results :

22/5/2001 – 10:14

1991 Mathematics Subject Classification : Primary 03E15 ; Secondary 03E45, 54H05 a) Any compact covering mapping from a Π_{2}^{0} space onto any space is inductively perfect.

b) Any compact covering mapping from any space onto a Σ_{2}^{0} space is inductively perfect.

c) If we assume Det(Σ_{1}^{1}) ( Σ_{1}^{1} determinacy) then : Any compact covering mapping from a Π_{1}^{1} space onto a Π_{1}^{1} space is inductively perfect.

d) In Gődel’s universe L :

There exists a compact covering mapping f : X → Y between two Borel spaces which is not inductively perfect ; moreover X can be chosen to be the intersection of a Π_{2}^{0} and a Σ_{2}^{0} set, and Y can be chosen to be to be a Π_{2}^{0} set.

Property a) was proved several years ago by Christensen [1] and the second author [10] independently, whereas b) is a much more recent result due to Ostrovsky [9] (also to Just and Wicke [5] in the particular case where the range space is countable) ; c) is from [2] and d) is from [3]. In particular this gives an example of a “natural” topological property of Borel sets which is not decidable in ZFC.

More generally given two classes X and Y of spaces one can ask whether any compact covering mapping from a space in X to a space in Y is inductively perfect. Thus Problem 1 concerns the case X = Y = Δ_{1}^{1} ; but this symmetrical hypothesis on X and Y is misleading, and as we shall see the domain space and the range space play totally different roles in this problem. Also for “reasonable” classes Y we have the following general property :

If any compact covering mapping from a Δ_{1}^{1} space onto a space in Y is inductively perfect, then the same holds for any mapping from a Π_{1}^{1} space onto a space in Y.

In fact this is proved in [3], Theorem 6.5, when Y = Π_{2}^{0}, which was the natural context of that work. However the argument is based on some stability properties of the class Π_{2}^{0}, which are also shared by each of the following classes : Σ_{ξ}^{0} , Π_{ξ}^{0} , Δ_{1}^{1} , Π_{1}^{1}.

In particular it follows from this property that the natural setting for our study is to consider mappings f : X → Y where X is a Π_{1}^{1} space and Y is a Δ_{1}^{1} space.

Another fundamental remark from [3] is that Problem 1 is intimately related to the study of the hyperspace K(Y) of all compact subsets of the range space Y, endowed with the Hausdorff topology. To explain this, let us first fix some terminology :

A subset A of a hyperspace K(Z) will be said \underline{cofinal} if it is cofinal for the inclusion relation ⊂ that is if :

∀ S ∈ K(Z), ∃ T ∈ A such that S ⊂ T

A \underline{domination function} for the cofinal set A is a mapping f : K(Z) → A such that :

∀ S ∈ K(Z) , f(S) ⊃ S

By the Axiom of Choice any cofinal set A admits a domination function ; and we shall say that a A is \underline{continuously cofinal} if it admits a continuous domination function. In [3] we proved the following surprising result :

Any analytic cofinal subset of K(Z) for Z Π_{2}^{0} is continuously cofinal. Notice that if a hyperspace K(Z) contains an analytic cofinal subset A then since we can write K(Z) = {S ∈ K(2^{ω}) : ∃ T ∈ A such that S ⊂ T } it follows that K(Z) is also analytic, hence by a basic result of Christensen and Saint Raymond ([10] or [1]) Z is necessarily Π_{2}^{0} ; this result is the origin of Property a) above.

When Z is a general Borel or coanalytic space the following problem arises then naturally :

Problem 2 : Is any coanalytic cofinal subset of K(Z) for Z Borel, continuously cofinal ? We now explain the connection between these two problems : Given any compact covering mapping f : X → Y one can code (see the proof of [3],

Theorem 6.4) the hyperspace K(X) by a cofinal subset A of K(Y) in such a way that : If A is continuously cofinal then f is inductively perfect. Conversely given any cofinal subset A of K(Z) if we consider the set H = {(S, T) ∈

K(Z) × A : S ⊂ T } then the mapping f : H → K(Z) obtained by restricting to H the canonical projection on the first factor (f(S, T) = S) is compact covering, moreover one can easily proves that :

If f is inductively perfect then f admits a domination function of the first Baire class. but one cannot insure by general arguments the existence of a continuous domination function.

However for mappings valued in a Π_{2}^{0} space we proved in [3] : Theorem A. The following are equivalent :

(i) For all α ∈ ω^{ω} the set ω^{ω} ∩ L(α) is ≤^{⋆} bounded in ω^{ω}. (ii) Any compact covering mapping from a Π_{1}^{1} space onto a Π_{2}^{0} space is inductively perfect. (iii) Any Π_{1}^{1} cofinal subset of K(Z) in a Π_{2}^{0} space Z is continuously cofinal. (iv) Any Π_{1}^{1} cofinal subset of K(ω^{ω}) admits a Borel domination function. where ≤^{⋆} denotes the relation in ω^{ω} of eventual domination of sequences.

In this paper we prove similar equivalences for mappings valued in a Π_{3}^{0} space, or in a Π_{1}^{1} space : Theorem B. The following are equivalent :

(i) For all α ∈ ω^{ω} the set ω^{ω} ∩ L(α) is countable. (ii) Any compact covering mapping from a Π_{1}^{1} space onto a Π_{3}^{0} space is inductively perfect. (iii) Any compact covering mapping from a Π_{3}^{0} space onto a Π_{3}^{0} space is inductively perfect. Theorem C. The following are equivalent :

(i) Det(Σ_{1}^{1}) (ii) Any compact covering mapping from a Π_{1}^{1} space onto a Π_{1}^{1} space is inductively perfect. (iii) Any Π_{1}^{1} cofinal subset of K(Z) in any Π_{1}^{1} space Z is continuously cofinal. (iv) Any Π_{1}^{1} cofinal subset of K(ℚ) admits a Borel domination function. where ℚ denotes a copy of the set of all rational numbers.

In particular, Problem 1 and Problem 2 get positive answers under analytic determinacy.

As one sees the analogy between Theorem C and Theorem A is more complete than between Theorem B and Theorem A. The main reason behind this fact is a closure property of the classes Π_{2}^{0} and Π_{1}^{1} : If a space X is in one of these classes then the hyperspace K(X) is also in the same class. This is not satisfied by the class Π_{3}^{0}, and for this specific reason the proof of Theorem B is much more delicate than the proof of Theorem C.

These results are proved in Section 7. For both theorems the proof decompose in two main parts, that we comment briefly now :

(I) : Proving that (i) is sufficient to insure the other conditions (II) : Proving that (i) is necessary to insure the other conditions

In the case of Theorem C, part (I) follows essentially from previous results of [2] and [3]. In the case of Theorem B we proceed similarly to the proof of Theorem A in [2], by proving first some “continuous lifting property” over Π_{3}^{0} sets ; and for this we introduce a Borel game adapted to the new situation. However the arguments make use of totally new ideas. In fact in both situations (Theorem A and Theorem B) one is reduced to construct, from some strategy σ in the game, a “large” compact set K included in a given Borel set Y. Controlling such a construction when Y is Π_{2}^{0} is quite standard, but totally non trivial when Y is Π_{3}^{0}. In general in such situations one parametrizes the Borel set Y as the projection of some Π_{2}^{0} set Y′, and constructs the compact set K as the projection of a compact subset K′ of Y′. But, for deep reasons, in the present situation this procedure is strictly impossible to realize, and one has to insure “by hand” the condition K ⊂ Y when Y is Π_{3}^{0}. The construction relies on a very technical analysis of the strategy σ. Also to explain partly this analysis, we shall first treat in Section 4 the simpler case when Y is Σ_{2}^{0} and for which the corresponding “lifting property” can be proved in ZFC, without any extra assumption. The case when Y is Π_{3}^{0} is developped in Section 5 which constitutes the heart of this work. This section makes substantial use of Effective Descriptive Set Theory.

Part (II) is non trivial in Theorem B and in Theorem C, but the proof in both cases follows the same scheme, and make use of the fact that in each case condition (i) is equivalent to some regularity property for Π_{1}^{1} sets (see Theorem 1.2 and Theorem 1.3 for more details). The second ingredient of this part is a construction which assigns to any Π_{1}^{1} non Borel set A ⊂ 2^{ω} a cofinal subset A of K(ℚ) which is a kind of replica of A. Then in each case one has to derive from a continuous domination function for A the corresponding regularity proerty for the Π_{1}^{1} set A. The construction of the replica A ⊂ K(ℚ) of A ⊂ 2^{ω} is independent from the rest of the paper and presented in Section 6.

As for Theorem A in [3] one can prove effective (lightface) versions of all these results. In Section 1 we present briefly the main non elementary descriptive results that we use in this work. However we assume the reader to be familiar with Classical and Effective Descriptive Set Theory as well as basic results on the universe L, for example as presented in [8].

1 - DESCRIPTIVE SET THEORY

1.1. Descriptive classes

We follow standard notations from Effective Descriptive Set Theory : Σ_{ξ}^{0} and Π_{ξ}^{0} for the additive and multiplicative Borel classes ; Δ_{1}^{1} for the class of all Borel sets ; Σ_{1}^{1} for the class of all analytic sets ; Π_{1}^{1} for the class of all coanalytic sets ; Σ_{2}^{1} for the class of all projections of coanalytic sets ; and Σ_{ξ}^{0} , Π_{ξ}^{0} , Δ_{1}^{1} , Σ_{1}^{1} , Π_{1}^{1} , Σ_{2}^{1} for the effective versions of the previous classes.

We shall also consider the following two classes :

P_{σ} = {A = B ∩ C : B ∈ Π_{2}^{0} and C ∈ Σ_{2}^{0} }

P̌_{σ} = {A = B ∪ C : B ∈ Π_{2}^{0} and C ∈ Σ_{2}^{0} }

which will play a major role. Obviously P̌_{σ} is the class of all complements of sets in P_{σ}, and both classes lie between the Borel classes Π_{2}^{0} ⊂ Π_{3}^{0}. The class P_{σ} is also denoted sometimes D_{2}(Σ_{2}^{0}). The notation P_{σ} was introduced in [2] where spaces in P_{σ} are called σ-Polish. A typical set in P_{σ} and not in P̌_{σ} is the product space ℚ × ω^{ω} as a subset of 2^{ω} × 2^{ω} (since both ℚ and ω^{ω} embed in 2^{ω}). These classes appear naturally as “limit classes” in several properties of the hyperspace K(X) for X Borel. For example the following two results, that we shall not use in this work, can be found in [11] and [6] :

– The Borel Effros structure of K(X) is standard iff X is in P̌_{σ}

– The σ-ideal K(X) is generated by a Borel subset iff X is in P_{σ}

and we shall see in Section 7 that the classes P_{σ} and P̌_{σ} are also “limit classes” for two instances of Problem 1 above.

We shall use the following basic and classical result of Solovay ([13]) :

Theorem 1.2. (Solovay) For any α ∈ ω^{ω} the following are equivalent :

(i) ω^{ω} ∩ L(α) is countable. (ii) Any uncountable Π_{1}^{1}(α) set contains a perfect set.

We shall commonly write condition (i) with the equivalent form “ω_{1}^{L(α)} < ω_{1}”. Notice that these statements are equiconsistent with ZF C whereas it is not the case for those in the following theorem.

The other basic result we need is a combination of two deep Theorems due to Martin and Harrington. Before we state them we recall some standard terminology. A \underline{reduction} of a set A ⊂ 2^{ω} to a set B ⊂ 2^{ω} is a function f : 2^{ω} → 2^{ω} such that A = f^{−1}(B) ; we also say that A is reducible to B by f . If F is a family of functions from 2^{ω} to 2^{ω} and Γ is a class of subsets of 2^{ω}, let’s say that a set A_{0} in Γ is Γ-\underline{complete by F-reductions} if any A in Γ is reducible to A_{0} by a function f ∈ F ; when F is the set of all continuous functions, we simply say Γ-complete.

Theorem 1.3. (Martin–Harrington) For any α ∈ ω^{ω} the following are equivalent :

(i) Det(Σ_{1}^{1}(α)) (ii) Any Π_{1}^{1}(α) non Borel set is Π_{1}^{1}(α)-complete (by continuous reductions). (iii) Any Π_{1}^{1}(α) non Borel set is Π_{1}^{1}(α)-complete by Borel reductions.

The only known proof of this Theorem is through the following scheme

(i) ⇒ (ii) ⇒ (iii) ⇒ (i)

Aside the middle implication which is trivial both other implications are very deep results. The first implication is based on a result of Martin insuring that analytic determinacy implies the determinacy of games which are difference of analytic sets, hence of Wadge games. The last implication is a Theorem of Harrington ([4]) for which we mention a recent “simple” proof by R. Labib Sami ([7]) avoiding the use of Steel’s forcing as in Harrington’s initial proof.

In this work we shall make use of the last implication, that is Harrington’s Theorem. However we shall also use other results and remarks from [2] and [3] which already used Martin’s Theorem which is the core of the proof of the first implication.

2 - SEQUENTIAL SPACES

2.1. General notations

For any set X we denote by Seq (X) = ⋃_{n} X^{n} the set of all finite sequences in X, and by ≺ the extension relation. If s ∈ X^{n} we denote as usual by

|s| : the length of the sequence s

s_{|m} : the restriction of s to m ≤ |s|

and if s is nonempty by

s^{∗} : the sequence obtained by deleting from s the last element, so s^{∗} = s_{||s|−1}.

When working in product spaces we shall very often make obvious identifications such as (X × Y) × Z = X × Y × Z or Seq (X × Y) ⊂ Seq (X) × Seq (Y). For example if u = (s, t) ∈ X^{n} × Y^{n} then we would write u_{|m} for (s_{|m}, t_{|m})

2.2. Lexicographical ordering on Seq (X)

If (X , <) is a wellordered space, denote for each n ∈ ω by <^{n} the lexicographical ordering of the product space X^{n}. One can extend these orderings to a unique ordering that we shall also denote by < in the following way : for any s and t ∈ Seq (X) set :

s < t ⟺ \{ \begin{matrix}|s| < |t| \\ or \\ |s| = |t| = n and s <^{n} t \end{matrix}

It is clear that Seq (X) is then also wellordered by < . In particular one can iterate the previous procedure inductively to endow the sets Seq (Seq (X)), Seq (Seq (Seq (X))) … with canonical wellorderings that we all denote by < .

In this paper we shall consider the wellorder thus obtained on the space Seq (Seq (κ)) where κ is some ordinal, and to which we sall refer as the canonical wellordering of Seq (Seq (κ)).

2.3. Convergence in Seq (X)

If (s_{n})_{n∈ω} is an infinite sequence in Seq (X) (that is s_{n} ∈ Seq (X) for each n) ; we shall say that the sequence (s_{n})_{n∈ω} is converging if there exists some α ∈ X^{ω} such that :

(1) lim_{n} |s_{n}| = ∞
(2) s_{n}(k) = α(k) for all k ∈ ω and all n large enough.

It is clear that this α is unique ; we shall say that (s_{n}) converges to α and write α = lim_{n} s_{n}. Notice that from condition (1) it follows that if k ∈ ω is fixed then s_{n}(k) is defined for n large enough.

One can easily interprete this convergence notion as a real topological convergence in some product space. We shall not do this but list simply some elementary properties that we will use later.

2.4. Remarks :

a) If there exists some increasing sequence (k_{n})_{n∈ω} in ω such that s_{n} _{|k_{n}} ≺ α for all n, then (s_{n}) converges to α. Conversely if (s_{n}) converges to α then for any increasing sequence (k_{n})_{n∈ω} in ω, we can extract from (s_{n}) a subsequence (s′_{n}) (converging to α) satisfying s_{n} _{|k_{n}} ≺ α for all n.

b) Suppose that α = lim_{n} s_{n}; if all the s_{n}’s are in some tree S ⊂ Seq (X) then α ∈ [S].

c) Suppose that X is wellordered by <. If the sequence (s_{n}) is such that for all m < n :

(1) |s_{m}| < |s_{n}|

(2) the restriction of s_{n} to |s_{m}| is smaller than s_{m} (lexicographically in X^{|s_{m}|})

then for all k the sequence (s_{n}(k))_{n} is eventually nonincreasing hence stationary ; and (s_{n}) is converging.

Finally we shall make large use of the following compactness property :

Lemma 2.5. Let (s_{n}) be an infinite sequence in Seq (X) such that |s_{n}| → ∞; and suppose that for any k ∈ ω the set {s_{n}(k) : n such that |s_{n}| > k} is finite.

Then one can extract from (s_{n}) an infinite converging subsequence.

\underline{Proof} : Starting from the initial sequence σ^{(0)} = (s_{n}^{(0)}) = (s_{n}) one constructs inductively subsequences σ^{(j)} = (s_{n}^{(j)}) such that for all j : σ^{(j+1)} is a subsequence of σ^{(j)} and the sequence (s_{n}^{(j)}(j)) is constant for variable n. Then it is clear that the diagonal subsequence (s′_{n}) = (s_{n}^{(n)}) is converging. ■

3 - THE GAMES OF TYPE Γ(T)

For the proofs of the main results of next two sections we shall introduce two different games each of them adapted to a particular hypothesis, but which are both of the same type. Also to avoid repetitions we shall describe in this section the general form of these games and fix some notations and conventions that we will respect in all the sequel.

Set Σ = Seq (ω_{1}) and let T be a tree on {0, 1} × Σ. By a game of type Γ(T) we shall mean a game in which at each of his moves :

– Player I chooses some element in {0, 1}

– Player II chooses an element in {0, 1} × Σ

We shall identify an infinite run in such a game with a triple (y, z, θ) ∈ 2^{ω} × 2^{ω} × Σ^{ω} ; and we shall say that y and z are the \underline{reals constructed} in the run by Player I by Player II , and we shall refer to the θ(n)’s as the \underline{ordinal moves} of Player II in the run.

\underline{Rules} : To be legal in the game the run (y, z, θ) should satisfy for all n ≥ 0 the following two rules :

(R_{1}) : |θ(n)| ≤ n + 1

(R_{2}) : If |θ(n)| = m then (z_{|m}, θ(n)) ∈ T

Notice that both rules concern Player II ; in particular any move is legal for Player I . In fact in the games that we shall consider later we shall add more rules but which again will concern only Player II . We do not require any particular form for the win condition which will be specifed in each particular game. Notice that if (r_{n}) is a sequence of legal finite runs in the game which is converging (in the sense of 2.3) then by Remark 2.4 b) the limit lim_{n} r_{n} is an infinite run in which the same rules are satisfied.

\underline{Finite runs} : We shall always identify a finite run in the game with a triple

(s, t, θ) ∈ Seq (2) × Seq (2) × Seq (Σ)

where s represents the moves of Player I in the run, and (t, θ) represents the moves of Player II in the run. We always have |t| = |θ|, and |s| = |t| + 1 or |t| depending whether the last move in the run is played by Player I or by Player II .

\underline{Positions} : By a position for Player II in the game we shall mean a couple :

(t, θ) ∈ Seq (2) × Seq (Σ)

which can be realized as the moves of Player II in some run :

(s, t, θ) ∈ Seq (2) × Seq (2) × Seq (Σ)

where all rules are satisfied (not only (R_{1}) and (R_{2})).

One can also define similarly a position for Player I ; but since we shall never deal with this notion in the sequel, by “position” we shall always mean “position for Player II”. The empty set is a position corresponding to the empty run, or to a run with one move made by Player I .

If σ is some fixed strategy for Player I in the game, we shall say that the position u = (t, θ) is compatible with σ if we can find a run r = (s, t, θ) compatible with σ. By convention we shall also impose that in this run Player I made the last move, so that r is uniquely determined by u, and the correspondence between u and r is one-to-one. The position ∅ is always compatible with σ and corresponds to the run with one move σ(∅).

\underline{Notations} : For any position u = (t, θ) we set :

π(u) = t
λ̄(u) = θ

If moreover u is compatible with some strategy σ and is realized by the run (s, t, θ) with |s| = |t| + 1 then we write

σ(u) = s

Notice that |σ(∅)| = 1 and σ(u) is never empty.

4 - CONTINUOUS LIFTINGS OVER Σ_{2}^{0} SPACES

Theorem 4.1. Let R and Y × Z be two subsets of 2^{ω} × 2^{ω} satisfying :

a) Y is Σ_{2}^{0} , Z is Σ_{2}^{1} , R is Δ_{1}^{1}

b) ∀ K compact ⊂ Y , ∃ z ∈ Z such that : K × {z} ⊂ R

Then there exists a continuous mapping f : 2^{ω} → 2^{ω} such that f_{|Y} ⊂ R ∩ (Y × Z) where f_{|Y} denotes the graph of the restriction of f to Y.

The proof of this theorem is easy if Z is assumed to be Borel, but in the general case the problem is to keep the game Borel.

To prove this result we shall introduce a game of type Γ(T) for some tree T such that :

(1) If Player II has a winning strategy then the conclusion of the Theorem holds.

(2) If Player I has a winning strategy then there is a compact set K negating hypothesis b) of the Theorem.

Insuring (1) will be straightforward from the win condition of the game, and the main point will be to prove (2). In fact given an arbitrary strategy σ for Player I we shall define a compact subset K of 2^{ω} such that if σ is winning then K negates hypothesis b).

4.2. Index of an ω–sequence of trees Let S = (S_{k})_{k∈ω} be an ω–sequence of trees on {0, 1}. The index of S is the mapping

k : Seq (2) → ω

defined by

k(s) = \{ \begin{matrix}min\{k < |s| : s ∈ S_{k}\} & if s ∈ ⋃_{k<|s|} S_{k} \\ |s| & if not \end{matrix}

Notice that if s ≺ s′ then k(s) ≤ k(s′). In particular if s ≠ ∅ then k(s^{∗}) ≤ k(s) ; and we shall say that s is \underline{critical} if k(s^{∗}) < k(s).

It is clear then that for any x ∈ 2^{ω} we have :

x ∉ ⋃_{k∈ω}[S_{k}] ⟺ lim_{n} k(x_{|n}) = ∞ ⟺ lim sup_{n} k(x_{|n}) = ∞

4.3. The game G′

Since Z is Σ_{2}^{1} we can fix a tree T on 2 × ω_{1} such Z is the projection of [T] on the first factor (see [8] p. 84). We also fix a sequence (S_{k})_{k∈ω} of trees on 2 such that Y = ⋃_{k}[S_{k}] and denote by k the associated index.

We now consider the game G′ of type Γ(T) in which rule (R_{2}) is unchanged whereas rule (R_{1}) is replaced by the following rule :

(R′_{1}) : If (s, t, θ) is a run with |s| = |t| = |θ| = n + 1 then :

θ(n) ≻ θ(n − 1) and |θ(n)| = |θ(n − 1)| + 1 if k(s) = k(s^{∗})
θ(n) = ∅  if k(s) > k(s^{∗})

It is easy to see check that this rule is stronger than (R_{1}).

\underline{Win condition} : Player II wins the infinite run (y, z, θ) in the game G′ if :

y ∉ Y or (y, z) ∈ R

All game theoretical notions considered in this section are relative to the game G′.

Lemma 4.4. If Player II has a winning strategy then there exists a continuous mapping f : 2^{ω} → 2^{ω} such that f_{|Y} ⊂ R ∩ (Y × Z) .

\underline{Proof} : A winning strategy τ for Player II defines a Lipschitz mapping y ↦ (f(y), g(y)) from 2^{ω} into 2^{ω} × Σ^{ω} where each (y, f(y), g(y)) = (y, z, θ) is an infinite run compatible with τ hence won by Player II .

Fix y ∈ Y ; it follows from the win condition that (y, f(y)) ∈ R and all we have to prove is that f(y) ∈ Z. Set

k_{0} = min{k ∈ ω : y ∈ [S_{k}]}

It follows from the definition of the index k that there exists some n such that k(y_{|m}) = k_{0} for all m ≥ n. Let n_{0} be the least such n; then by rule (R′_{1}) there exists a unique γ ∈ ω_{1}^{ω} such that γ_{|k} = θ(n_{0} + k) for all k, and since by rule (R_{2}) we have (z_{|k}, γ_{|k}) ∈ T for all k, then (z, γ) ∈ [T] and z = f(y) ∈ Z. ■

We fix now a strategy σ for Player I in the game. All runs and positions considered from now on will implicitly be assumed to be compatible with σ.

Our next goal is to define from σ a compact set K with the properties announced above. The definition of K will rely on some analysis of the strategy σ for which we need to introduce the following notions.

\underline{Minimal positions} : We shall say that a position u is minimal if for any position v the following implication holds :

( σ(v) = σ(u) and π(v) = π(u) ) ⟹ λ̄(u) ≤ λ̄(v)

where as always ≤ refers to the canonical lexicographical ordering on Seq (Σ).

Given any position v it is clear that there exists a unique minimal position u with σ(v) = σ(u) and π(v) = π(u), and we shall denote by μ(v) this unique position u.

Lemma 4.5. If u is a minimal position and v is a position extending u then μ(v) is also an extension of u.

\underline{Proof} : Let u′ denote the position obtained by the restriction of μ(v) to |u|. If we had λ̄(u) < λ̄(u′) then we would also have λ̄(v) < λ̄(μ(v)) which is impossible by the minimality of μ(v). Hence λ̄(u) = λ̄(u′) and since π(u) = π(u′) then u = u′ and so μ(v) ≻ u′ = u. ■

\underline{Good positions} : We shall say that a position u is good if for any subposition v ⪯ u the following implication holds :

σ(v) is critical ⟹ v is minimal

where critical refers to the index k associated to the fixed sequence (S_{k})_{k∈ω}. Set

S = {σ(u) : u good position } ∪ {∅}

Since any subposition of a good position is clearly a good position then S is a tree on {0, 1} and we can define the compact set

K = [S]

We shall say that the strategy σ \underline{takes its values in Y} if in any infinite run (y, z, θ) compatible with σ the real y constructed by Player I belongs to Y. It follows from the win condition of the game that any winning strategy takes its values in Y.

Lemma 4.6. If σ takes its values in Y then K ⊂ Y.

\underline{Proof} : Fix y ∈ K = [S]. For each n the sequence s_{n} = y_{|n+1} is in S, so we can find a good position u_{n} = (t_{n}, θ_{n}) such that |u_{n}| = n and s_{n} = σ(u_{n}).

Fix some n such that s_{n} is critical. We can find some t ∈ 2^{n} such that t_{m} _{|n} = t for infinitely many m’s. Now if we fix such an m, since u_{m} is a good position and σ(u_{m} _{|n}) = s_{n} is critical then the position u_{m} _{|n} is minimal hence there exists θ ∈ Σ^{n} uniquely determined by s_{n} and t such that θ_{m} _{|n} = θ. Thus if s_{n} is critical then there exists a position v = (t, θ) such that σ(v) = s_{n} and u_{m} _{|n} = v for infinitely many m’s.

Suppose now by contradiction that y ∉ Y. Then the set N = {n ∈ ω : s_{n} is critical } is infinite, and applying inductively the previous observations one can construct a family (v_{n})_{n∈N} of increasing positions ( v_{n} ≺ v_{n′} if n < n′) such that σ(v_{n}) = s_{n} ≺ y for all n ∈ N. Then clearly this defines a unique infinite run compatible with σ in which the real constructed by Player I is precisely y; and since y ∉ Y this contradicts the hypothesis of the Lemma. ■

Lemma 4.7. If σ takes its values in Y then for any z ∈ Z there exists an infinite run (y, z, θ) (compatible with σ) such that y ∈ K.

\underline{Proof} : We shall say that a position u is \underline{critical} if σ(u) ∈ Seq (2) is critical with respect to the index k.

Fix z in Z and let U = U_{z} denote the set of all positions u satisfying π(u) ≺ z. We shall construct a finite sequence (u_{k})_{k≤n} in U satisfying for all k ≤ n :

(1) u_{k−1} ≺ u_{k}

(2) u_{k} is a good position

(3) u_{k} is critical

(4) Any extension of u_{n} in U is non critical.

Set u_{0} = ∅. Assume that we have constructed in U a sequence (u_{0}, …, u_{m}) satisfying (1), (2), (3) for all k ≤ m. We distinguish the following two alternatives :

(i) : Any extension of u_{m} in U is non critical.

In this case we define n = m so that condition (4) is realized, and the construction is finished.

(ii) : There exists a critical extension of u_{m} in U.

In this case we fix such an extension v of minimal length, and define u_{m+1} = μ(v). Thus σ(u_{m+1}) = σ(v) is critical and by Lemma 4.5 we have u_{m} ≺ u_{m+1}. Finally since v was chosen of minimal length there is no critical subposition w between u_{m} and u_{m+1}, and since u_{m} is a good position then it is clear that u_{m+1} is also a good position. Hence u_{m+1} satisfies (1), (2), (3).

Notice that in the second alternative (ii) one can repeat again the previous arguments starting from (u_{0}, …, u_{m}, u_{m+1}). If applying this procedure inductively alternative (i) never occurs then we would obtain an infinite sequence (u_{k})_{k∈ω} satisfying (3) for all k, in particular we would have k(σ(u_{k−1})) < k(σ(u_{k})) for all k. This clearly defines an infinite run in the game in which the real y constructed by Player I satisfies lim k(y_{|n}) = ∞, hence not in Y and this contradicts the assumption of the Lemma. Thus after finitely many steps alternative (i) occurs, and this finishes the construction.

Set u = u_{n} and p = |u|, and let r denote the finite run determined by u. Fix some γ ∈ ω_{1}^{ω} such that (z, γ) ∈ [T] and consider the unique infinite run (y, z, θ) compatible with σ, extending r with θ(p + n) = γ_{|n} for all n ≥ 0. It follows easily from (3) and (4) that rule (R′_{1}) is satisfied, and since (z, γ) ∈ [T] then rule (R′_{2}) is also satisfied. Finally notice that it follows from (2) and (4) that any extension of u in U is again a good position ; in particular for all n we have y_{|n+1} = σ(z_{|n}, θ_{|n}) ∈ S, and so y ∈ [S] = K. ■

\underline{Proof of Theorem 4.1} : By Lemma 4.4 it is enough to prove that Player II has a winning strategy in the game G′. So suppose by contradiction that Player II has no winning strategy in this game. Since the game is clearly Borel and so determined, Player I has a winning strategy σ. Let K be the compact set defined above ; then by Lemma 4.6 K is a compact subset of Y, and by hypothesis b) of the Theorem applied to K we can find some z ∈ Z such that K × {z} ⊂ R, and then applying Lemma 4.7 to this z we obtain an infinite run (y, z, θ) compatible with σ and such that y ∈ K ; hence (y, z) ∈ R and Player II wins this run, and this is impossible since σ is winning.

5 - CONTINUOUS LIFTINGS OVER Π_{3}^{0} SPACES

Theorem 5.1. Assume that “ ∀ α ∈ ω^{ω}, ω_{1}^{L(α)} < ω_{1}”. Let R and Y × Z be two subsets of 2^{ω} × 2^{ω} satisfying :

a) Y is Π_{3}^{0} , Z is Σ_{2}^{1} , R is Δ_{1}^{1}

b) ∀ K compact ⊂ Y , ∃ z ∈ Z such that : K × {z} ⊂ R

Then there exists a continuous mapping f : 2^{ω} → 2^{ω} such that f_{|Y} ⊂ R ∩ (Y × Z) where f_{|Y} denotes the graph of the restriction of f to Y.

As in Theorem 4.1 the proof is easy if Z is Borel. But in the general case we have to keep the game Borel.

Precisely we shall prove the following :

Theorem 5.2. Fix α ∈ ω^{ω}, and assume that “ ω_{1}^{L(α)} < ω_{1}”. Let R and Y × Z be two subsets of 2^{ω} × 2^{ω} satisfying :

a) Y is Π_{3}^{0}(α) , Z is Σ_{2}^{1}(α) , R is Δ_{1}^{1}(α)

b) ∀ K compact ⊂ Y , ∃ z ∈ Z such that : K × {z} ⊂ R

Then there exists a continuous mapping f : 2^{ω} → 2^{ω} such that f_{|Y} ⊂ R ∩ (Y × Z) where f_{|Y} denotes the graph of the restriction of f to Y.

\underline{Warning} : All the rest of this section is devoted to the proof of Theorem 5.2. For simplicity we shall give the proof for α = 0 ; the reader can easily check that all the arguments are uniform. The proof will follow a scheme completely similar to the proof of Theorem 4.1 : We shall introduce some adapted game G′′, and as in the Σ_{2}^{0} case we shall derive from some given strategy σ for Player I in the game a compact set K with the desired properties. Here also the definition of K will be based on a particular analysis of the strategy σ; and for this we shall need to consider some particular classes of positions in the game G′′.

We shall make use of the same terminology than in 4.1 and define “minimal positions”, “good positions”, … But we warn the reader that the content of these notions will be here different from the previous case. In fact since the rules of the games are different the comparison of two such notions is meaningless. The reason for this choice is first a matter of simplicity, but also to emphasize the common structure of the proofs in Theorem 4.1 and Theorem 5.1. We shall also introduce a new notion of “very good positions” which has no analog in the previous case, and which will heavily rely on the assumption “ω_{1}^{L} < ω_{1}”.

5.3. Index of an ω^{2}–sequence of trees

Let S = (S_{k}^{j})_{(j,k)∈ω×ω} be an ω^{2}–sequence of trees on {0, 1}. As in 4.2 we denote for each j by k^{(j)} the index of the ω–sequence (S_{k}^{j})_{k∈ω} ; thus k^{(j)} : Seq (2) → ω with

k^{(j)}(s) = \{ \begin{matrix}min\{k < |s| : s ∈ S_{k}^{j} \} & if ∃k < |s| such that s ∈ S_{k}^{j} \\ |s| & if not \end{matrix}

The index of S is the mapping

k̄ : Seq (2) → Seq (ω)

defined inductively as follows :

k̄(∅) = ∅

and for s ≠ ∅ if k̄(s^{∗}) is defined with p = |k̄(s^{∗})|, set m = min{j ≤ p : k^{(j)}(s) > k^{(j)}(s^{∗}) } if k^{(j)}(s) > k^{(j)}(s^{∗}) for some j ≤ p and m = p if not, then define :

k̄(s) = \{ \begin{matrix}k̄(s^{∗})_{|m} & if m < p \\ k̄(s^{∗})^{⌢}k^{(p)}(s) & if m ≥ p \end{matrix}

Notice that we always have k̄(s) ≺ k̄(s^{∗}) or k̄(s) ≻ k̄(s^{∗}) ; moreover in the last case we also have |k̄(s)| = |k̄(s^{∗})| + 1. Since k̄(∅) = ∅ it follows then that for all s :

|k̄(s)| ≤ |s|

Lemma 5.4. For any x ∈ 2^{ω} we have :

x ∈ ⋂_{j} ⋃_{k}[S_{k}^{j}] ⟺ lim_{n} |k̄(x_{|n})| = ∞

\underline{Proof} : Set X = ⋂_{j} ⋃_{k}[S_{k}^{j}] and assume x ∈ X ; then there exists some α ∈ ω^{ω} such that x ∈ [S_{α(j)}^{j}] for all j. We shall construct inductively some β ∈ ω^{ω} and an increasing sequence n_{j} of integers such that

∀n > n_{j}, β_{|j} ⪯ k̄(x_{|n})

which will show that lim_{n→∞} |k̄(x_{|n})| = ∞.

Set n_{0} = 0 and notice that β_{|0} = ∅. Assume n_{j} and β_{|j} are already defined. Since

x ∈ [S_{α(j)}^{j}], the sequence (k^{(j)}(x_{|n}))_{n} is non-decreasing and bounded by α(j), so is eventually constant. Hence there exist n_{j+1} > n_{j} and β(j) ≤ α(j) such that k^{(j)}(x_{|n}) = β(j) for n ≥ n_{j+1}. Then for any n > n_{j+1} and s = x_{|n} we have

\{ \begin{matrix}β_{|j} ⪯ k̄(s) \\ β_{|j} ⪯ k̄(s^{∗}) \\ k^{(j)}(s) = k^{(j)}(s^{∗}) = β(j) \end{matrix}

It follows then from the definition of k̄, that β_{|j} ⪯ k̄(s) which completes the inductive construction.

Conversely, assume that lim_{n→∞} |k̄(x_{|n})| = ∞. For all j, there exists some n_{j} such that |k̄(x_{|n})| ≥ j + 1 for all n ≥ n_{j}. Then by definition of k̄(s) we have

k̄(x_{|n})_{|j+1} = k̄(x_{|n_{j}})_{|j+1}

for n ≥ n_{j}, hence x_{|n} ∈ S_{k}^{j} for k = k̄(x_{|n_{j}})(j), and thus x ∈ ⋃_{k∈ω}[S_{k}^{j}]. And since this holds for every j, we have x ∈ X. ■

5.5. The game G′′

As for the game G′ we fix a tree T on {0, 1} × ω_{1} such Z is the projection of [T] on the first factor. We also fix an ω^{2}–sequence S = (S_{k}^{j}) of trees on {0, 1} such that Y = ⋂_{j} ⋃_{k}[S_{k}^{j}] and denote by k̄ denote the associated index. Since Z and Y are lightface we can choose S to be recursive and T in L (see [8] p. 537).

Let G′′ be the game of type Γ(T) in which rule (R_{2}) is unchanged and rule (R_{1}) is replaced by the following rule :

(R′′_{1}) : If (s, t, θ) is any finite run with |s| = |t| = |θ| = n + 1 then :

θ(n) = θ(n − 1)_{|m}  if m = |k̄(s)| ≤ |θ(n − 1)|
θ(n) ⪰ θ(n − 1) and |θ(n)| ≤ |θ(n − 1)| + 1 if not

It is easy to check then that in such a run (in which Player II is making the last move) we have |θ(n)| ≤ |k̄(s)|. Formally, the comparison of (R′′_{1}) and (R′_{1}) is meaningless. However one should think intuitively to (R′′_{1}) as being less restrictive than (R′_{1}) : one of the main possibilities offered to Player II by (R′′_{1}) (and totally forbidden by (R′_{1})) is to “pass” for the ordinal move at any time by playing θ(n) = θ(n − 1). This novelty will play a major role in the arguments.

\underline{Win condition} : Player II wins the infinite run (y, z, θ) in G′′ if :

y ∉ Y or \{ \begin{matrix}(y, z) ∈ R \\ and \\ lim_{n} θ(n) exists in ω_{1}^{ω} \end{matrix}

All game theoretical notions considered in this section are relative to the game G′′.

Lemma 5.6. If Player II has a winning strategy then there exists a continuous mapping f : 2^{ω} → 2^{ω} such that f_{|Y} ⊂ R ∩ (Y × Z) .

\underline{Proof} : Notice that if an infinite run (y, z, θ) in the game is won by Player II with y ∈ Y, then γ = lim_{n} θ(n) ∈ ω_{1}^{ω} exists and it follows from rule (R_{2}) that (z, γ) ∈ [T].

The rest of the argument is completely similar to Lemma 4.4 : A winning strategy for Player II in this game defines canonically a continuous function f : 2^{ω} → 2^{ω} such that f_{|Y} ⊂ R ∩ (Y × Z) . ■

Lemma 5.7. For any infinite run (y, z, θ) in the game, the following are equivalent : (i) lim_{n} θ(n) exists in ω_{1}^{ω} (ii) lim_{n} |θ(n)| = ∞ (iii) y ∈ Y and the sequence of integers (|θ(n)|)_{n∈ω} is not stationary.

\underline{Proof} : (iii) ⟹ (ii) : Define α and β ∈ ω^{ω} by : α(n) = |k̄(y_{|n})| and β(n) = |θ(n)| . It follows then from (R′′_{1}) that for all n :

(1) β(n) ≤ α(n)

(2) β(n) < β(n − 1) ⟹ β(n) = α(n)

Set M = {n > 0 : β(n) < β(n − 1)}; we distinguish two cases :

\underline{Case 1} : M is finite

In this case the sequence of integer β is eventually non-decreasing, and since it is not stationary, then lim_{n} β(n) = ∞.

\underline{Case 2} : M is infinite

Since y ∈ Y then by Lemma 5.4 in this case we have lim_{n} α(n) = ∞; and we now show that lim_{n} β(n) = ∞.

Fix an arbitrary p and then fix n_{0} such that α(n) ≥ p for all n ≥ n_{0}. It follows from condition (2) that for any m ∈ M with m ≥ n_{0} we have β(m) = α(m) ≥ p; moreover if m′ denotes the first element of M greater than m then on the interval [m, m′[ the sequence β is non-decreasing hence for all n ∈ [m, m′[ we have β(n) ≥ β(m) ≥ p. It follows from these observations that if we fix any m_{0} ∈ M with m_{0} ≥ n_{0} then β(n) ≥ p for all n ≥ m_{0}. This proves (ii).

(ii) ⟹ (i) : Fix an infinite sequence (n_{k}) such that |θ(n)| ≥ k for all n ≥ n_{k}. It follows then from (R′′_{1}) that for all m, n ≥ n_{k} we necessarily have θ(n)_{|k} = θ(m)_{|k} ; which proves (i).

(i) ⟹ (iii) : If lim_{n} θ(n) exists in ω_{1}^{ω} then necessarily lim_{n} |θ(n)| = ∞, hence by Lemma 5.4 y ∈ Y ; and obviously (|θ(n)|)_{n∈ω} is not stationary. ■

The win condition defines clearly a Borel subset of 2^{ω} × 2^{ω} × Σ^{ω} and so the game G′′is determined. Moreover we have the following absoluteness property :

Lemma 5.8. The winning player in the game G′′ has a winning strategy in L.

\underline{Proof} : Consider the game G̃ in which at each of their moves : Player I chooses as in G′′ an element in {0, 1}, whereas Player II chooses an element in ω × {0, 1} × Σ. Similarly to G′′ identify a run in G̃ with a quadruple (r, s, t, θ). The rules in G̃ are (R′′_{1}) , (R_{2}) and the additional trivial rule :

(R_{3}) r(n) = |θ(n)|

Also we identify an infinite run in Γ̃ with a quadruple (x, y, z, θ) ∈ ω^{ω} × 2^{ω} × 2^{ω} × Σ^{ω} ; and Player II wins this run if :

y ∉ Y or \{ \begin{matrix}(y, z) ∈ R \\ and \\ lim_{n} x(n) = ∞ \end{matrix}

By Lemma 5.7 the games G′′ and G̃ are equivalent. Moreover any winning strategy σ̃ for one of the players in G̃ determines a winning strategy σ for the same player in G′′, in a trivially definable way ; in particular σ ∈ L(σ̃).

Thus it is enough to prove the Lemma for the game G̃ instead of G′′. For this, notice that the set of all infinite runs won by Player II in G̃ is of the form (B × Σ^{ω}) ∩ [T ] with B a Δ_{1}^{1} subset of ω^{ω} × 2^{ω} × 2^{ω} and T a tree in L on ω × 2 × 2 × Σ determined by the rules of the game ; then the conclusion of the Lemma follows from [3] Proposition 4.3. ■

We now fix a strategy σ for Player I in the game and we suppose that σ ∈ L. All runs and positions considered from now on will implicitly be assumed to be compatible with σ.

\underline{Notations} : We recall previous notations and fix new ones. If u = (t, θ) ∈ 2^{n} × Σ^{n} is a position in the game (compatible with σ) and (s, t, θ) is the unique run associated to u with |s| = n + 1 and |t| = |θ| = n, we set :

σ(u) = s ∈ 2^{n+1}
π(u) = t ∈ 2^{n}
λ(u) = θ(n − 1) ∈ Σ
λ̄(u) = ⟨θ(0), …, θ(n − 1)⟩ = θ ∈ Σ^{n}
ℓ̄(u) = ⟨|θ(0)| , …, |θ(n − 1)|⟩ ∈ ω^{n}
ℓ_{m}(u) = |θ(m)| ∈ ω , for all m < n
φ(u) = (σ(u), π(u), ℓ̄(u))

The mapping φ will play a fundamental role in the proof. One should think of the triple φ(u) = (σ(u), π(u), ℓ̄(u)) as a shadow of the run (σ(u), π(u), λ̄(u)). And to analyze the strategy σ we shall in a first step reduce the information contained in such a run to the shadow φ(u) and to the last ordinal move λ(u) (instead of λ̄(u)). The main property of the shadow is to admit only countably many possible values, but it still contains some information from the real run. In fact for technical reasons we will need to deal with a more precise shadow, that we will introduce later, which again admits only countably many possible values but contains more information about the run.

\underline{Special extensions :} : Let u and v two positions with |u| ≤ |v| (but we do not suppose that u ⪯ v). We shall say that v is \underline{λ-compatible} with u if :

|λ(v_{|m})| ≥ |λ(u)| for all m ∈ [|u| , |v|]

If moreover v is an extension of u it follows then from rule (R′′_{1}) that the following stronger property holds :

λ(v_{|m}) ⪰ λ(u) for all m ∈ [ |u| , |v|]

and we shall say that v is a \underline{λ-compatible extension} of u.

We now discuss more particular types of extensions. Let v be an extension of u :

We shall say that v is a \underline{monotone extension} of u if :

λ(v_{|m}) ⪰ λ(v_{|m−1}) for all m ∈] |u| , |v|]

that is if v_{|m} is λ-compatible with v_{|m−1} for all m ∈] |u| , |v|].

We shall say that v is a \underline{trivial extension} of u if :

λ(v_{|m}) = λ(u) for all m ∈] |u| , |v|]

\underline{Minimal positions} : We shall say that a position u is minimal if for any position v the following implication holds :

( φ(v) = φ(u) and λ(v) = λ(u) ) ⟹ λ̄(u) ≤ λ̄(v)

where as always ≤ refers to the canonical lexicographical ordering on Seq (Σ).

Again given any position v it is clear that there exists a unique minimal position u with φ(v) = φ(u) and λ(v) = λ(u), and we shall denote by μ(v) this unique position u.

The μ operator gives a simple way to construct minimal positions ; we shall now give another (more constructive) way to do this :

Lemma 5.9. If u is a minimal position and v is a monotone extension of u then v is minimal.

\underline{Proof} : Let v′ be a position with φ(v′) = φ(v) and λ(v′) = λ(v) and denote by u′ the restriction of v′ to |u|. Obviously φ(u′) = φ(u). Moreover since v is a monotone extension of u then v′ is also a monotone extension of u′; hence λ(u′) = λ(u) and by the minimality of u we have λ̄(u) ≤ λ̄(u′) and a fortiori λ̄(v) ≤ λ̄(v′) ; and this proves that v is minimal. ■

\underline{The equivalence relation ≡ on Σ} : We recall that the shadow of a position u is a triple :

φ(u) ∈ Seq (2) × Seq (2) × Seq (ω)

We now define a more precise shadow which will be a quadruple :

φ̃(u) = (φ(u), ε(u)) ∈ Seq (2) × Seq (2) × Seq (ω) × Seq (2)

where ε(u) is the characteristic function of the set {m ≤ |u| : u_{|m} is a minimal position}. Now for any ρ ∈ Σ set

Φ(ρ) = {φ̃(u) : u a position with λ(u) = ρ }

and define the equivalence relation ≡ on Σ by :

ρ ≡ ρ′ ⟺ Φ(ρ) = Φ(ρ′)

Lemma 5.10. If ω_{1}^{L} < ω_{1} then there exists a exists a mapping N : Σ → ω satisfying : (N_{0}) : N(∅) = 0 (N_{1}) : N(ρ) = N(ρ′) ⟹ ρ ≡ ρ′

(N_{2}) :  ρ ≺ ρ′  ⟹ N(ρ) ≤ N(ρ′)

\underline{Proof} : Set E = Seq (2) × Seq (2) × Seq (ω) × Seq (2). For any ρ ∈ Σ, the set Φ(ρ) is a subset of E ; and this defines a mapping Φ : Σ → P(E).

We recall that the strategy σ and the tree T were chosen in L. Moreover all the parameters S, k̄ … implicitly involved in the definition of φ are also in L since they are recursive ; and since the lexicographical ordering of Σ is in L then the relation “u_{|m} is a minimal position” (with argument (u, m)) is also in L. Hence the function Φ is definable in L, and Φ(λ) ∈ L for all λ ∈ Σ.

On the other hand the set E is countable in L, and if we assume that ω_{1}^{L} < ω_{1} then the set P(E) ∩ L is countable in V. This shows that the range of Φ is countable in V ; hence the equivalence relation ≡ on Σ has countably many classes. The existence of such a mapping N is then straightforward. ■

From now on we assume that ω_{1}^{L} < ω_{1} and fix a mapping N : Σ → ω satisfying properties (N_{0}), (N_{1}), (N_{2}) above.

\underline{Pseudo-extension} :

Let u and v be two minimal positions ; we shall say that v is is a pseudo-extension of u, and write u ◁ v, if there exists some extension v′ of u with v = μ(v′) ; such a position v′ is not unique.

It is fundamental to observe that on the set of all minimal positions the relation ◁ is not transitive. However as we shall see it has enough transitive consequences.

Lemma 5.11. Let u and v be two minimal positions.

a) If u ≺ v then u ◁ v

b) If u ◁ v and u′ ≺ u is minimal, then u′ ◁ v
c) If u ◁ v then φ(u) ≺ φ(v).
d) If u ◁ v and m ≤ |u| then λ̄(v_{|m}) ≤ λ̄(u_{|m}).

\underline{Proof} : a) and b) are straightforward. For c) and d) fix some v′ such that v = μ(v′) and u ≺ v′; and notice that φ(u) ≺ φ(v′) = φ(v) and λ̄(v) ≤ λ̄(v′), hence

λ̄(v_{|m}) = λ̄(v)_{|m} ≤ λ̄(v′)_{|m} = λ̄(u)_{|m} = λ̄(u_{|m}).

■

More fundamental is the following converse of property a) above.

Lemma 5.12. Let u and v be two minimal λ-compatible positions. If u ◁ v then u ≺ v.

\underline{Proof} : As above fix v′ such that v = μ(v′) and u ≺ v′, and set m = |u|. Then φ(u) = φ(v_{|m}) ; and since ℓ̄(v) = ℓ̄(v′) then v′ is a λ-compatible extension of u and λ(u) = λ(v′)_{|m} = λ(v)_{|m} = λ(v_{|m}). Then by the minimality of u we have λ̄(u) ≤ λ̄(v_{|m}) and a fortiori λ̄(v′) ≤ λ̄(v) ; and it follows from the minimality of v that v′ = v, hence u ≺ v. ■

\underline{Good and very good chains} :

We shall say that a sequence ū = (u^{(0)}, …, u^{(m)}, … u^{(n)}) is a \underline{good chain} of positions if for all m ≤ n :

(1) u^{(m)} is a minimal position

(2) |u^{(m)}| = |u^{(m−1)}| + 1

(3) u^{(m−1)} ◁ u^{(m)}

and we shall say that ū is a \underline{very good chain} if moreover :

(4) N(λ(u^{(m)})) ≤ |u^{(m)}|

\underline{Good and very good positions} :

We shall say that a position u is good (very good) if there exists a good (very good) chain ū = (u^{(0)}, …, u^{(m)}, … u^{(n)}) with u^{(0)} = ∅ and u^{(n)} = u.

Such a chain ū is not unique. By condition (2) for each m we have |u^{(m)}| = m and obviously each u^{(m)} is then also a good (very good) position. We shall say that u^{(m)} is a \underline{good} (\underline{very good}) \underline{minimal version} for u_{|m}. Notice that by Lemma 5.11 c) φ(u_{|m}) = φ(u^{(m)}), however λ(u_{|m}) and λ(u^{(m)}) might be different, and therefore u^{(m)} is not equal to μ(u_{|m}) in general. Still as we shall see this will be the case in many situations.

5.13. Examples :

a) By property (N_{0}) of Lemma 5.10 the empty sequence ∅ is a very good position.

b) If u is any minimal position and v is a monotone extension of u, then it follows from Lemma 5.9 and Lemma 5.11 a) that the sequence

v̄ = (v_{|m})_{|u|≤m≤|v|}

is a good chain of positions. Hence if u is a good position then v is also a good position.

Moreover if u is a very good position and v is a trivial extension of u then it is clear that condition (4) is fulfilled and v̄ is a very good chain, hence v is a very good position.

We now come to the crucial property of good positions. To evaluate this property, recall that the restriction of a minimal (and even of a good) position is not minimal in general.

Lemma 5.14. Let v be a (very) good position. If v is a λ–compatible extension of u then u is a (very) good position ; in particular u is minimal.

\underline{Proof} : Set m = |u| and fix a good chain (v^{(k)})_{k≤n} with v^{(0)} = ∅ and v^{(n)} = v. We shall prove by induction on k ∈ [m, n] that v^{(m)} ⪯ v^{(k)}.

This is obvious for k = m. Suppose that v^{(m)} ⪯ v^{(k)} ; since v^{(k)} ◁ v^{(k+1)} then by Lemma 5.11 b) and c) v^{(m)} ◁ v^{(k+1)} and φ(v^{(m)}) ≺ φ(v^{(k+1)}). In particular v^{(k+1)} is λ–compatible with v^{(m)}, hence by Lemma 5.12 v^{(m)} ≺ v^{(k+1)}.

In particular v^{(m)} ≺ v^{(n)} = v, and since |v^{(m)}| = m then v^{(m)} = v_{|m} = u and u is a good position. Moreover if v is a very good position then u = v_{|m} is a very good position too. ■

\underline{Definition of the compact set K} : Let u be a very good position. We recall that the restriction u_{|m} of u is not a good position in general. However if u^{(m)} is a very good version of u_{|m} (in some very good chain) then σ(u_{|m}) = σ(u^{(m)}) ⪯ σ(u). It follows that the set

S = { σ(u) : u a very good position } ∪ {∅}

is a tree on {0, 1} and we can define the compact set

K = [S]

Lemma 5.15. If σ takes its values in Y then K ⊂ Y.

\underline{Proof} : Fix y ∈ K. For each n, since s_{n} = y_{|n+1} ∈ S fix a very good position u_{n} = (t_{n}, θ_{n}) such that |u_{n}| = n and s_{n} = σ(u_{n}), and set α(n) = |k̄(s_{n})|.

It follows from rule (R′′_{1}) that for all m < n we have ℓ_{m}(u_{n}) ≤ α(m). Hence by Lemma 2.5 we can extract from (u_{n}) a subsequence (v_{n}) such that the

lim(σ(u_{n}), π(u_{n}), ℓ̄(u_{n})) = (y, z, β) ∈ 2^{ω} × 2^{ω} × ω^{ω}

exists ; and by the previous observations we then have β(n) ≤ α(n) for all n.

Suppose that y ∉ Y then by Lemma 5.4 we would have

p = lim inf_{n} β(n) ≤ lim inf_{n} α(n) < ∞

Set

m_{0} = min{m ∈ ω : β(m − 1) = p and ∀ n ≥ m, β(n) ≥ p}

and let (m_{n})_{n∈ω} be the increasing enumeration of the set

M = {m ≥ m_{0} : β(m − 1) = p}

Replacing (v_{n}) by some subsequence if necessary we can suppose that for all n :

|v_{n}| ≥ m_{n} and φ(v_{n})_{|m_{n}} ≺ (y, z, β)

Let w_{n} be a minimal very good version for v_{n} _{|m_{n}} and set ρ_{n} = λ(w_{n}), then for all n we have ℓ̄(w_{n}) ≺ β. It follows then from the definition of the set M that for all k < n, the position w_{n} is a λ–compatible extension of the position w_{(n,k)} = w_{n} _{|m_{k}} (the restriction of w_{n} to m_{k}), hence by Lemma 5.14 w_{(n,k)} is a very good position and since λ(w_{(n,k)}) = ρ_{n} then N(ρ_{n}) ≤ m_{k}.

In particular for k = 0 we obtain N(ρ_{n}) ≤ m_{0} , and replacing (v_{n}) by some subsequence if necessary we can assume that the sequence (N(ρ_{n})) is constant, hence by property (N_{1}) of N all the ρ_{n}’s are equivalent to some fixed ρ ∈ Σ . Applying then the definition of the equivalence relation ≡ we can find for all n a position w′_{n} with

φ̃(w′_{n}) = φ̃(w_{n}) and λ(w′_{n}) = ρ

Again fix k < n and set as above w′_{(n,k)} = w′_{n} _{|m_{k}}. Since w_{(n,k)} is minimal, it follows then from the equality ε(w′_{n}) = ε(w_{n}) that w′_{(n,k)} is also minimal. It follows also from the equality φ(w′_{n}) = φ(w_{n}) that

φ(w′_{(n,k)}) = φ(w_{(n,k)}) = φ(w′_{k}) and λ(w′_{(n,k)}) = λ(w′_{n}) = λ(w′_{k}) = ρ

Hence by the minimality of both positions w′_{(n,k)} and w′_{k} we can conclude that w′_{k} = w′_{(n,k)} ≺ w′_{n}.

Thus the sequence (w′_{n}) defines a unique infinite run in the game in which the real constructed by Player I is clearly y, and this contradicts the hypothesis of the Lemma since y ∉ Y. ■

Lemma 5.16. If σ takes its values in Y then for any z ∈ Z there exists an infinite run (y, z, θ) (compatible with σ) such that y ∈ K and lim_{n} θ(n) exists.

\underline{Proof} : Fix z in Z and γ ∈ Σ^{ω} such that (z, γ) ∈ [T]. Let V = V_{(z,γ)} denote the set of all positions u such that (π(u), λ(u)) ≺ (z, γ).

We shall first construct an infinite sequence (u_{k})_{k∈ω} in V satisfying the following conditions for all k ∈ ω :

(1) u_{k} is a very good position

(2) u_{k−1} ◁ u_{k}

(3) |λ(u_{k})| ≠ |λ(u_{k−1})|

Set u_{0} = ∅ ∈ V which satisfies obviously (1). Assume that v = u_{k−1} is a given very good position in V we shall define w = u_{k} ∈ V satisfying (1), (2) , (3).

Set m = |v| and p = |λ(v)|. For all n ≥ m let v_{n} denote the unique trivial extension of length n of v such that π(v_{n}) = z_{|n}, which is in V, (in particular v_{m} = v) and set q_{n} = |k̄(σ(v_{n}))|.

Say that an extension v′ of v is \underline{critical} if |k̄(σ(v′))| < |λ(v)|. We shall distinguish the following two alternatives (i) and (ii) :

(i) : Any trivial extension of v in V is non critical.

Since by the definition of k̄ we have q_{j} ≠ q_{j−1} for all j, we can find n ≥ N(γ_{|p+1}) such that q_{n} > p. We fix such an n and then define :

w = v_{n}^{⌢}(z(n) , γ_{|p+1})

which is a legal position by rule (R′′_{2}). Notice that by Lemma 5.9 both v_{n} and w are minimal hence v_{n} ◁ w.

(ii) : There exists some trivial extension of v in V which is critical.

Let n be the least j ≥ m such that q_{j} < p and set q = q_{n}. Then v_{n}^{⌢}(z(n) , γ_{|q}) is a legal position by rule (R′′_{2}) ; then we define :

w = μ( v_{n}^{⌢}(z(n) , γ_{|q} ))

Notice that again by Lemma 5.9 v_{n} is minimal ; moreover w is also minimal and v_{n} ◁ w.

By Example 5.13 b) the sequence v̄ = (v_{m} , v_{m+1}, …, v_{n−1}, v_{n}) is a good chain ; and since v_{n} ◁ w with |w| = |v_{n}| + 1, then w̄ = (v_{m} , v_{m+1}, …, v_{n−1}, v_{n}, w) is also a good chain ; and since v is very good then for all k ∈ [m, n] we have

N(λ(w_{|k})) = N(γ_{|p}) ≤ |v| ≤ k

moreover in case (i) by the choice of the integer n we have

N(λ(w)) = N(γ_{|p+1}) ≤ n ≤ |w|

and in case (ii) by property (N_{2}) of the mapping N (see Lemma 5.10) we have

N(λ(w)) = N(γ_{|q}) ≤ N(γ_{|p}) ≤ |v| ≤ |w|

Hence in both cases w̄ is a very good chain, and since v = v_{m} the first element of the sequence w̄ is a very good position, then w is also very good. Moreover since v ≺ v_{n} ◁ w then by Lemma 5.11 b) we have v ◁ w; and clearly |λ(w)| ≠ |λ(v)|. This finishes the construction of the sequence (u_{k}).

It follows then from Lemma 5.11 d) that for all j < k if n = |u_{j}| then λ̄(u_{j}) ≥ λ̄(u_{k} _{|n}), hence by Remarks 2.4 b) and c) lim_{k} λ̄(u_{k}) = θ ∈ Σ^{ω} exists and (y, z, θ) is an infinite run in the game compatible with σ, and it follows from condition (1) that y ∈ [S] = K.

On the other hand it follows from Lemma 5.11 c) that if j < k and n < |u_{j}| < |u_{k}| then ℓ_{n}(u_{j}) = ℓ_{n}(u_{k}), hence |θ(n)| = lim_{k} ℓ_{n}(u_{k}) = ℓ_{n}(u_{k}). In particular for n_{k} = |u_{k}| − 1 we have |θ(n_{k})| = |λ(u_{k})| and it follows then from condition (3) that the sequence (|θ(n)|)_{n} is not stationary. Finally since by the hypothesis of the Lemma, σ takes its values in Y then y ∈ Y and it follows from Lemma 5.7 that lim_{n} θ(n) exists in (ω_{1})^{ω}. ■

\underline{Proof of Theorem 5.2} : The argument is totally similar to 4.1, and we leave the details to the reader. Notice only that by Lemma 5.8 we can choose σ the supposed winning strategy for Player I in L. ■

6 - COFINAL EMBEDDING OF Π_{1}^{1}-SETS IN K(ℚ)

Throughout this section we denote by ℚ the countable dense-in-itself subset of 2^{ω} defined by

ℚ = {α ∈ 2^{ω} : ∃m ∀n > m α(n) = 0}

We denote by 0 the null sequence (0(p) = 0 for all p), and for all n ∈ ω by e_{n} the characteristic function of {n} (e_{n}(p) = 1 ⇔ p = n) ; obviously 0 and all the e_{n}’s are elements of ℚ, and 0 = lim_{n} e_{n}

It is a well-known and basic fact that the Π_{1}^{1} set K(ℚ) is Π_{1}^{1}-complete, in particular non Borel. Thus for any Π_{1}^{1}-set W ⊂ 2^{ω} there exists a continuous mapping g : 2^{ω} → K(2^{ω}) which reduces W to K(ℚ), that is satisfying W = g^{−1}( K(ℚ)). One can even assume that g is a homeomorphic embedding. In fact if g : 2^{ω} → K(2^{ω}) is any continuous reduction of W to K(ℚ) and if we define h from 2^{ω} into K({0, 1} × 2^{ω}) ≈ K(2^{ω}) by

h(α) = ({1} × g(α)) ∪ {0} ∪ {e_{n+1} : α(n) = 1 }

then one easily checks that h a homeomorphic embedding which also reduces W to K(ℚ). In particular :

Lemma 6.1. Any Borel space is homeomorphic to a relatively closed subset of K(ℚ).

But for our purpose we need more : we would like the set A = h(W) to be cofinal in K(ℚ). As mentioned in the introduction this is impossible if W is Borel, since K(ℚ) is not analytic. But even if W is non Borel one cannot insure in general the cofinality condition on the set A. However we shall prove the following substitute :

Theorem 6.2. For any Π_{1}^{1} subset W of 2^{ω} there exists a homeomorphism h : G → G from a G_{δ} subset G of 2^{ω} × ω^{ω} onto a G_{δ} subset G of K(2^{ω}) satisfying :

(1) W × ω^{ω} ⊂ G ⊂ 2^{ω} × ω^{ω}

(2) W × ω^{ω} = h^{−1}( K(ℚ))

(3) If W is non Borel then h(W × ω^{ω}) is cofinal in K(ℚ).

We start by some general lemmas, some of which might be well-known ; we give the proofs for completeness.

Lemma 6.3. Let E be a perfect subset of 2^{ω}, D a countable dense subset of E and a ∈ D. Then there exists a homeomorphism φ from E onto 2^{ω} such that φ(D) = ℚ and φ(a) = 0.

\underline{Proof} : Let (a_{n}) be an enumeration of D with a_{0} = a. We construct inductively, for every s ∈ 2^{<ω}, a non-empty clopen subset E_{s} of E and a point x_{s} ∈ E_{s} ∩ D such that :

i) E_{∅} = E

ii) E_{s} = E_{s^{⌢}0} ∪ E_{s^{⌢}1} iii) E_{s^{⌢}0} ∩ E_{s^{⌢}1} = ∅ iv) ∀x, y ∈ E_{s} ∀n < |s| x(n) = y(n)

v) x_{∅} = a vi) x_{s^{⌢}0} = x_{s} vii) x_{s^{⌢}1} = a_{n} for n = min{m : a_{m} ∈ E_{s^{⌢}1}} If E_{s} is chosen for |s| = k − 1, it is possible since it is perfect to partition E_{s} into two

clopen nonempty subsets E′_{s} and E′′_{s} such that the k^{th} coordinate is constant on each of them. Since x_{s} ∈ E_{s} we can, up to exchange of E′_{s} and E′′_{s}, assume that x_{s} ∈ E′_{s}. We then put E_{s^{⌢}0} = E′_{s}, E_{s^{⌢}1} = E′′_{s}, x_{s^{⌢}0} = x_{s} and x_{s^{⌢}1} = a_{m} for the least m such that a_{m} ∈ E′′_{s}.

Then for every x ∈ E there exists a unique α = φ(x) ∈ 2^{ω} such that

∀s ∈ 2^{<ω} x ∈ E_{s} ⟺ s ≺ α

It follows from condition vi) that for every s ∈ 2^{<ω} φ(x_{s}) = s^{⌢}0. The function φ is continuous since the E_{s}’s are clopen, bijective since ii) and iii) hold. Thus φ is a homeomorphism because E is compact. Moreover φ(x_{s}) = s^{⌢}0, hence φ(D) ⊃ ℚ.

Finally, let d denote the standard metric on 2^{ω} : d(α, β) = sup{2^{−n} : α(n) ≠ β(n)}. If φ(a_{n}) ∉ ℚ, and 2^{−k} < min_{j<n} d(a_{n}, a_{j}) there exists s ∈ 2^{<ω} such that |s| ≥ k and s^{⌢}1 ≺ φ(a_{n}). Then by condition iv) none of the a_{j}’s for j < n belongs to E_{s^{⌢}1}. But since a_{n} ∈ E_{s^{⌢}1}, x_{s^{⌢}1} = a_{n} and φ(a_{n}) = s^{⌢}1^{⌢}0 ∈ ℚ. This contradiction completes the proof. ■

Lemma 6.4. Let S and T be two compact subsets of ℚ. If f is a homeomorphism from S onto T, then there exists some homeomorphism f̃: 2^{ω} → 2^{ω} such that f̃(ℚ) = ℚ and which extends f.

\underline{Proof} : Let d denote as before the standard metric on 2^{ω}. If S and T are finite then there exists a finite partition of 2^{ω} into clopen sets containing each exactly one point of S (resp. T) ; and it follows from the previous lemma that if U and V are clopen neighborhoods of a ∈ S and b ∈ T there exists a homeomorphism f_{U} from U to V such that f_{U}(a) = b and f_{U}(U ∩ ℚ) = V ∩ ℚ.

If S and T are infinite, denote by S′ and T′ their Cantor derivatives. Let (a_{n}) be an enumeration of the set S \ S′ of isolated points of S. Since f is a homeomorphism and d(a_{n}, S′) → 0, we have T′ = f(S′) and d(f(a_{n}), T′) → 0. Define then for n ∈ ω :

B_{n} = {x ∈ 2^{ω} \ S′ : inf_{p∈ω} d(x, a_{p}) = d(x, a_{n}) < inf_{j<n} d(x, a_{j})}

It is easy to see that the B_{n}’s are clopen in 2^{ω} and pairwise disjoint, that a_{n} ∈ B_{n}, and that ⋃_{n} B_{n} = 2^{ω} \ S′. Moreover diam(B_{n}) → 0 : if not there would exist some δ > 0, some increasing sequence (n_{i}) of integers and some sequence (x_{i}) in 2^{ω} such that x_{i} ∈ B_{n_{i}} and d(x_{i}, a_{n_{i}}) ≥ δ. Without loss of generality, one could assume that the sequence (x_{i}) converges to some x ∈ 2^{ω}. If x ∉ S′, there would be some m such that x ∈ B_{m}, and we would have x_{i} ∈ B_{m}, hence n_{i} = m, for i large enough. Thus x ∈ S′, and there is some p such that d(x, a_{p}) < δ/2 ; hence for large i, d(x_{i}, a_{p}) < δ/2 and n_{i} > p. This would imply that

δ ≤ d(x_{i}, a_{n_{i}}) < inf_{j<n_{i}} d(x_{i}, a_{j}) ≤ d(x_{i}, a_{p}) < δ/2,

a contradiction. Since B_{n} ∩ ℚ is countable and dense in the perfect set B_{n}, there exists by Lemma 6.3 a homeomorphism φ_{n} : B_{n} → 2^{ω} such that φ_{n}(B_{n} ∩ ℚ) = ℚ and φ_{n}(a_{n}) = 0.

Similarly, the sets

C_{n} = {x ∈ 2^{ω} \ T′ : inf_{p∈ω} d(x, f(a_{p})) = d(x, f(a_{n})) < inf_{j<n} d(x, f(a_{j}))}

are clopen in 2^{ω}, pairwise disjoint, cover 2^{ω} \ T′ and diam(C_{n}) → 0. Thus there exists a homeomorphism ψ_{n} : C_{n} → 2^{ω} such that ψ_{n}(C_{n} ∩ ℚ) = ℚ and ψ_{n}(f(a_{n})) = 0. Then f_{n} = ψ_{n}^{−1}∘φ_{n} is a homeomorphism from B_{n} onto C_{n} such that f_{n}(B_{n} ∩ ℚ) = C_{n} ∩ ℚ and f_{n}(a_{n}) = f(a_{n}).

Then if we define f̃ by

f̃(y) = \{ \begin{matrix}f(y) & if y ∈ S′ \\ f_{n}(y) & if y ∈ B_{n} \end{matrix}

it is not hard to check that f̃ is a homeomorphism from 2^{ω} to itself extending f, and that f̃(ℚ) = ℚ. ■

Lemma 6.5. Let W be a coanalytic subset of 2^{ω}. Then there exists a continuous function φ : 2^{ω} → K(2^{ω}) such that

w ∈ W ⟺ φ(w) ⊂ ℚ ⟺ φ(w) is countable

\underline{Proof} : Let T be the set of trees on ω, endowed with the topology induced by 2^{(ω^{<ω})}. It is well-known that the set WF = {T ∈ T : T is well founded} is Π_{1}^{1}-complete in T . Thus there exists a continuous mapping w ↦ T_{w} from 2^{ω} to T such that w ∈ W ⟺ T_{w} ∈ WF.

Let us consider for any s = (n_{0}, n_{1}, …, n_{k}) ∈ ω^{<ω} the point q_{s} ∈ ℚ all coordinates of which are 0 except those of indices n_{0}, n_{0} + n_{1} + 1, …, n_{0} + n_{1} + ⋯ + n_{k} + k. Then we put :

φ_{0}(w) = \overline{\{q_{s} : s ∈ T_{w}\}}

If w ∈ W, then φ_{0}(w) ⊂ ℚ. Indeed, if β ∈ φ_{0}(w) \ ℚ, there is a sequence (s_{i}) in T_{w} such that q_{s_{i}} → β and there is an infinite sequence α = (n_{k}) such that the non-zero coordinates of β are those of indices n_{0}, n_{0} + n_{1} + 1, …, n_{0} + n_{1} + ⋯ + n_{k} + k,. . .. For each integer k we have α_{|k} ≺ s_{i} for i large enough , hence α_{|k} ∈ T_{w} ; this means that α ∈ [T], so T ∉ WF and this is a contradiction with w ∈ W.

Conversely, if w ∉ W, T_{w} ∉ WFand there exists some α ∈ [T]. If q^{(k)} = q_{α_{|k}}, then q^{(k)} ∈ φ_{0}(w) and the sequence (q^{(k)}) converges to some point β ∈ 2^{ω} \ℚ. Then β ∈ φ_{0}(w)\ℚ and φ_{0}(w) ⊄ ℚ.

For every open subset U of 2^{ω}, if φ_{0}(w)∩U ≠ ∅, there is some s ∈ T_{w} such that q_{s} ∈ U. Then N = {y : s ∈ T_{y}} is a clopen neighborhood of w, and φ_{0}(y) ∩ U ≠ ∅ for every y ∈ N ; hence {w ∈ 2^{ω} : φ_{0}(w) ∩ U ≠ ∅} is open.

If F is a closed subset of 2^{ω}, we now prove that Z = {w : φ_{0}(w) ∩ F ≠ ∅} is closed. Assume (w_{i}) is a sequence in Z which converges to some w. For each i we can choose some β_{i} ∈ F ∩φ_{0}(w_{i}), and extracting if necessary some subsequence, we assume that β_{i} → β ∈ F. And there are s_{i} ∈ T_{w_{i}} such that q_{s_{i}} → β.

Then,

- either β ∈ ℚ : in this case β = q_{s} and s ≺ s_{i} for large enough i. This implies that s ∈ T_{w_{i}}, thus that s ∈ T_{w} = lim_{i→∞} T_{w_{i}}, that means β = q_{s} ∈ φ_{0}(w) ∩ F, hence w ∈ Z.

- or β ∉ ℚ : in this case there is an infinite sequence α = (n_{k}) such that the nonzero coordinates of β are those of indices n_{0}, n_{0} + n_{1} + 1, …, n_{0} + n_{1} + ⋯ + n_{k} + k,. . .. For each integer k we have α_{|k} ≺ s_{i}, hence α_{|k} ∈ T_{w_{i}} for large enough i, that means α_{|k} ∈ T_{w} = lim_{i→∞} T_{w_{i}} and β = lim_{k→∞} q_{α_{|k}} ∈ F. Thus w ∈ Z.

This completes the proof of the continuity of φ_{0} : 2^{ω} → K(2^{ω}).

We endow 2^{ω} with the order ≤ defined by β ≤ β′ ⟺ ∀n β(n) ≤ β′(n) and define the mapping φ : 2^{ω} → K(2^{ω}) by

β ∈ φ(w) ⟺ ∃β′ ∈ φ_{0}(w) β ≤ β′

Clearly φ(w) is a compact subset of 2^{ω} for every w ∈ 2^{ω}. Moreover if w ∈ W and β ∈ φ(w) there is some β′ ∈ φ_{0}(w) such that β ≤ β′. Since β′ ∈ ℚ, there is some n such that β′(p) = 0 for p > n; and thus β(p) ≤ β′(p) = 0 for p > n. Hence β ∈ ℚ. This shows that φ(w) ⊂ ℚ for w ∈ W.

And if w ∉ W, there is some β′ ∈ φ_{0}(w) \ ℚ. Since β′ has infinitely many non-zero coordinates, {β : β ≤ β′} is uncountable, and so is φ(w).

We conclude that

w ∈ W ⟺ φ(w) ⊂ ℚ ⟺ φ(w) is countable

We now show that φ is continuous. Let F be a closed subset of 2^{ω} ; we want to prove that Z = {w : φ(w) ∩ F ≠ ∅} is closed. But, if F′ is the compact set {β′ : ∃β ∈ F β ≤ β′}, we have

Z = {w : φ_{0}(w) ∩ F′ ≠ ∅}

which is closed since φ_{0} is continuous.

Similarly, if U is an open subset of 2^{ω}, it is easily checked that

U′ = {β′ : ∃β ∈ U β ≤ β′} = ⋃_{β∈U∩ℚ}{β′ : β ≤ β′ }

is open since {β′ : β ≤ β′ } is open for β ∈ ℚ. And we deduce as above that {w : φ(w) ∩ U ≠ ∅} = {w : φ_{0}(w) ∩ U′ ≠ ∅} is open. ■

Lemma 6.6. There exists a homeomorphic embedding Φ : K(ℚ) × 2^{ω} → K(ℚ) such that S ⊂ Φ(S, ε) for all ε ∈ 2^{ω} and all S ∈ K(ℚ).

\underline{Proof} : This is exactly the same mapping than the one constructed in [3], Fact 6.3.1, although the statements are not the same. We give the simple argument for completeness.

Let (U_{n}) be an increasing sequence of clopen subsets of ℚ which covers ℚ (one can assume U_{0} = ∅). For all n, choose in V_{n} = U_{n+1} \ U_{n} a point a_{n} , and a sequence (a_{n,p})_{p} converging to a_{n}.

Define then for n ∈ ω and β ∈ 2^{ω} the compact set

K_{n}(β) = {a_{n}} ∪ {a_{n,p} : β(p) = 1}

It is clear that the function K_{n} is continuous from 2^{ω} into K(ℚ) and one-to-one ; so it is a homeomorphism onto its range. Now define

Φ(S, β) = S ∪ K_{n}(β) if S ⊂ U_{n} and S ∩ V_{n−1} ≠ ∅

Let S_{n} = {S : S ⊂ U_{n} and S ∩ V_{n−1} ≠ ∅}. Since (S_{n})_{n} is a partition of K(ℚ) into clopen subsets, we see that Φ is continuous, one-to-one and satisfies Φ(S, β) ⊃ S.

Furthermore, if S′ = Φ(S, β) and n = min{k : S′ ⊂ U_{k+1}}, we have S′ ∈ S_{n+1}, S = S′ ∩ U_{n} and K_{n}(β) = S′ ∩ V_{n}; and this shows that n, S and β depend continuously of S′, that is Φ^{−1} is continuous on its domain. ■

\underline{Proof of Theorem 6.2} : Let C denote the space of all continuous functions from 2^{ω} into itself equipped with the metric of uniform convergence, and let H′ be the group of homeomorphisms of 2^{ω} preserving ℚ. By the mapping f ↦ (f, f^{−1}) we identify H′ to the set

H = {(f, g) ∈ C × C : f ∘g = g∘f = Id and f(ℚ) = g(ℚ) = ℚ}

One easily checks that H is a Π_{3}^{0} subset of the Polish space C× C. Moreover H is a topological group acting on K(ℚ) and it follows from Lemma 6.4 that H is uncountable since so is the set of images by H of any infinite compact subset of ℚ. Thus the uncountable topological group H has only condensation points, and by a classical result of Sierpinski ([12]), it is a one-to-one continuous image of ω^{ω}.

Fix θ : ω^{ω} → H and ρ : 2^{ω} × 2^{ω} → 2^{ω} with θ a continuous bijection and ρ a homeomorphic embedding. For any (w, α) ∈ W × ω^{ω} with θ(α) = (f, g) ∈ H set :

h(w, α) = Φ(g(φ(w)), ρ(w, α))

where Φ is as in Lemma 6.6 and φ as in Lemma 6.5.

Then h defines clearly a one-to-one and continuous mapping from W × ω^{ω} into K(ℚ). Moreover if h(w, α) = S′ then (g(φ(w)), ρ(w, α)) = (S, β) = Φ^{−1}(S′) and (w, α) = ρ^{−1}(β), which shows that h^{−1} is continuous on its domain.

Since h is a homeomorphism from W ×ω^{ω} onto its range, it extends to a homeomorphim (that we also denote by h) from a G_{δ} subset G of 2^{ω} × ω^{ω} onto some G_{δ} subset G of K(2^{ω}).

From the definition of h we have immediately h(W × ω^{ω}) ⊂ K(ℚ). On the other hand notice that for any (w, α) ∈ W × ω^{ω} with θ(α) = (f, g) ∈ H we have by Lemma 6.6 that h(w, α) ⊃ g(φ(w)), hence by the continuity of h this also holds for any (w, α) ∈ G. Thus for any (w, α) ∈ G with θ(α) = (f, g) ∈ H, if w ∉ W then by Lemma 6.5 φ(w) ⊄ ℚ, and since ℚ is invariant by f then g(φ(w)) ⊄ ℚ, hence by the previous observations we also have h(w, α) ⊄ ℚ. Thus we also have the converse inclusion h(G \ W × ω^{ω}) ⊂ K(2^{ω}) \ K(ℚ), and this proves (2).

To prove (3) notice first that since the Cantor rank γ is a Π_{1}^{1}-norm on the set K_{ω}(2^{ω}) of countable compact subsets of 2^{ω}, and since φ reduces W to K_{ω}(2^{ω}) by Lemma 6.5, then γ∘φ is also a Π_{1}^{1}-norm on W.

Thus if W is not Borel, this Π_{1}^{1}-norm is not bounded on W :

∀ξ < ω_{1}, ∃w ∈ W γ∘φ(w) > ξ

In particular for any S ∈ K(ℚ) there exists some w ∈ W such that γ(φ(w)) > γ(S). Hence S is homeomorphic to some compact subset T of φ(w), and by Lemma 6.4, there exists some homeomorphism f from 2^{ω} to itself such that f(ℚ) = ℚ and T = f(S). If α is such that θ(α) = (f, f^{−1}) = (f, g) ∈ H then

S = g(T) ⊂ g(φ(w)) ⊂ h(w, α) = S′

which proves (3). ■

7 - APPLICATIONS

We first state a more complete version of Theorem A which follows from the results proved in [3] :

Theorem 7.1. The following are equivalent :

(i) For all α ∈ ω^{ω} the set ω^{ω} ∩ L(α) is ≤^{⋆} bounded in ω^{ω}. (ii) Any compact covering mapping from a Π_{1}^{1} space onto a Π_{2}^{0} space is inductively perfect. (iii) Any compact covering mapping from a Π_{1}^{1} space onto a P_{σ} space is inductively perfect. (iv) In any Π_{2}^{0} space X, any Π_{1}^{1} cofinal subset of K(X) is continuously cofinal.

(v) Any Π_{1}^{1} cofinal subset of K(ω^{ω}) admits a Borel domination function.

and we now prove our first main result :

Theorem 7.2. The following are equivalent :

(i) For all α ∈ ω^{ω} the set ω^{ω} ∩ L(α) is countable.

(ii) Any compact covering mapping from a Π_{3}^{0} space (onto any space) is inductively perfect.

(iii) Any compact covering mapping from a Π_{3}^{0} space onto a Π_{3}^{0} space is inductively perfect.

(iv) Any compact covering mapping from a Π_{1}^{1} space onto a Π_{3}^{0} space is inductively perfect.

(v) Any compact covering mapping from a Π_{1}^{1} space onto a P̌_{σ} space is inductively perfect.

\underline{Proof} : (ii) ⟹ (iii) and (iv) ⟹ (v) are obvious.

(i) ⟹ (ii) : Let f : X → Y be a compact covering mapping defined on a Π_{3}^{0} space X. By standard arguments (see Theorem 3.1 of [2]) one can reduce the general case to the case where X ⊂ 2^{ω} × 2^{ω} and Y = π(X) ⊂ 2^{ω}, where π denotes the canonical projection mapping from 2^{ω} × 2^{ω} onto the first factor, and f = π_{X} is the restriction of π to X.

We recall that by [2], Theorem 8.8, under assumption (i), the image by a compact covering mapping of a Borel space is also Borel and of the same additive or multiplicative class ; hence Y is also Π_{3}^{0}. Set

C = K(2^{ω} × 2^{ω}) , Z = K(X) ⊂ C , R = {(y, H) ∈ 2^{ω} × C : y ∈ π(H)}

then clearly R is a closed subset of 2^{ω} × C, and Z is Π_{1}^{1}. Identifying then C to 2^{ω} one can easily check that the hypotheses of Theorem 5.1 are satisfied. Hence there is a continuous mapping F : 2^{ω} → C such that F (y) ⊂ X and y ∈ π(F (y)) for all y ∈ Y.

Then the set H = {(y, z) ∈ 2^{ω} × 2^{ω} : (y, z) ∈ F (y)} is compact and π(H) ⊃ Y ; since for any y ∈ Y we have H ∩ ({y} × 2^{ω}) ⊂ X, the restriction of π to X′ = H ∩ (Y × 2^{ω}) ⊂ X is a perfect mapping from X′ onto Y.

(iii) ⟹ (iv) : This implication follows from the proof of Theorem 6.5 in [3] where given any mapping f : X → Y defined on a Π_{1}^{1} space X we construct a mapping f̃: X̃ → Ỹ with the following properties :

– X̃ is a Boolean combination of Π_{2}^{0} sets and sets homeomorphic to Y.

– Ỹ is the union of Y with some discrete countable set.

– If f is compact covering then f̃ is compact covering.

– If f̃ is inductively perfect then f is inductively perfect.

In particular if Y is Π_{3}^{0} then it is clear that both spaces X̃ and Ỹ are also Π_{3}^{0}, and one can easily derive (iv) from (iii).

(v) ⟹ (i) : Fix W an arbitrary Π_{1}^{1} subset in 2^{ω}. Let h : G → G be the homeomorphism given by Theorem 6.2 and set Z^{∗} = h(W × ω^{ω}) ⊂ K(ℚ). Fix in K(ℚ) a relatively closed subset Y^{∗} which is P̌_{σ} but not P_{σ} ; this is possible by Lemma 6.1. Then the set X^{∗} = {(S, T) ∈ Y^{∗} × Z^{∗} : S ⊂ T } is also Π_{1}^{1}. For simplicity we shall work in 2^{ω} rather than in K(2^{ω}). For this fix some homeomorphim ψ : K(2^{ω}) → 2^{ω} and set :

Y = ψ(Y^{∗}) ⊂ 2^{ω} , Z = ψ(Z^{∗}) ⊂ 2^{ω} , X = (ψ × ψ)(X^{∗}) ⊂ Y × Z

and let π_{X} denote the canonical projection from X onto Y.

Fact 1. If W is non Borel then π_{X} is compact covering.

\underline{Proof} : Since W is non Borel then by Theorem 6.2 Z^{∗} is a cofinal subset of K(ℚ). Let K be some compact subset of Y. Then H = ψ^{−1}(K) is a compact subset of Y^{∗} ⊂ K(ℚ), hence S = ⋃ H is a compact subset of ℚ, and we can find T ∈ Z^{∗} such that S ⊂ T. Set z = ψ(T) ∈ Z ; if y ∈ K then ψ^{−1}(y) ∈ H, hence ψ^{−1}(y) ⊂ S ⊂ T, and so (y, z) ∈ X. This shows that K × {z} ⊂ X and obviously π(K × {z}) = K. ■

Recall that a set is said to be thin if it contains no perfect non-empty subset.

Fact 2. If W is thin then π_{X} is not inductively perfect.

\underline{Proof} : Suppose π_{X} is inductively perfect and let H be some closed subset of 2^{ω} × 2^{ω} such that π(X ∩ H) = Y and that H ∩ ({y} × 2^{ω}) ⊂ X for every y ∈ Y. Replacing if necessary H by \overline{H ∩ X}, we can assume that H ∩ X is dense in H. Since ψ^{−1}(y) ⊂ ψ^{−1}(z) for all (y, z) ∈ X, this holds for all (y, z) ∈ H.

Recall that, for two topological spaces Y and X, a mapping Ψ : Y → K(X) is said to be upper semi-continuous iff for any open subset O of X, the set

Ψ^{−}(O) := {y ∈ Y : Ψ(y) ⊂ O }

is open. In particular, if ψ is a function : Y → X, the mapping Ψ : y ↦ {ψ(y)} is upper semi-continuous if and only if ψ is continuous.

The mapping y ↦ H(y) = {z : (y, z) ∈ H} is then upper semicontinuous from 2^{ω} to K(2^{ω}) ; furthermore, if y ∈ Y then H(y) ⊂ ψ(Z^{∗}) ⊂ ψ(G). Since the set Γ := ψ(G) is G_{δ} then the set

B = \{β ∈ \overline{Y} : H(β) ⊂ Γ \}

is also G_{δ} and contains Y.

Consider the set-valued mapping M : β ↦ h^{−1}∘ψ^{−1}(H(β)) defined on B. Clearly M is u.s.c. with non-empty compact values in G; moreover for all y ∈ Y since H(y) ⊂ ψ(Z^{∗}) then M(y) ⊂ W × ω^{ω}. It follows that the set {(w, α, y) ∈ 2^{ω} × ω^{ω} × Y : (w, α) ∈ M(y) } is a closed subset of 2^{ω} × ω^{ω} × Y and is contained in W × ω^{ω} × Y. Hence its projection D on the first factor is an analytic subset of W ; and since W is thin, then D is countable. Fix some enumeration (w_{k}) of D, and set for all k :

B_{k} = {β ∈ B : M(β) ∩ ({w_{k}} × 2^{ω}) ≠ ∅ }

Since M is u.s.c. each B_{k} is closed in B. If y ∈ Y then M(y) is non-empty and contained in D; hence Y ⊂ ⋃_{k} B_{k}.

Conversely, if β ∈ B \ Y then since Y^{∗} is a closed subset of K(ℚ) we have ψ^{−1}(β) ∈ \overline{Y^{∗}} \ Y^{∗} ⊂ K(2^{ω}) \ K(ℚ). And for each z such that (β, z) ∈ H we have ψ^{−1}(z) ⊃ ψ^{−1}(β), thus ψ^{−1}(z) ⊄ ℚ, and since W × ω^{ω} = h^{−1}( K(ℚ)), (w, α) = h^{−1}∘ψ^{−1}(z) ∉ W × ω^{ω}. But (w, α) ∈ 2^{ω} × ω^{ω}, hence w ∉ W and w ∉ D. We conclude that β ∉ ⋃ B_{k} if β ∈ B \ Y.

This proves that Y = ⋃_{k} B_{k} is F_{σ} in the Polish space B, hence that Y is P_{σ} ; and this impossible since Y is homeomorphic to Y^{∗} which is not P_{σ}. ■

Thus if W is any Π_{1}^{1} non Borel set, then by Fact 1 π_{X} is a compact covering mapping from the Π_{1}^{1} space X onto the P̌_{σ} space Y, hence by (v) π_{X} is inductively perfect, and from Fact 2 we derive that W is not thin. By Theorem 1.2 this proves (i), and finishes the proof of the Theorem. ■

Theorem 7.3. The following are equivalent :

(i) Det(Σ_{1}^{1}) (ii) Any compact covering mapping from a Π_{1}^{1} space onto a Π_{1}^{1} space is inductively perfect. (iii) In any Π_{1}^{1} space X, any Π_{1}^{1} cofinal subset of K(X) is continuously cofinal. (iv) Any Π_{1}^{1} cofinal subset of K(ℚ) admits a Borel domination function.

\underline{Proof} : The implications (i) ⟹ (iii) ⟹ (ii) follow from the remarks in Section 7 of [3].

(ii) ⟹ (iv) : If A is a Π_{1}^{1} cofinal subset of K(ℚ) consider the set

X^{∗} = {(S, T) ∈ K(ℚ) × A : S ⊂ T }

Then as in the proof of Fact 1 in Theorem 7.2 one easily checks that the canonical projection on the first factor from X onto K(ℚ) is compact covering, hence by assumption (ii) its restriction on some relatively closed subset P of X is perfect and onto. Then by classical results P admits a first Baire class section, that is a function f : K(ℚ) → A of the first Baire class whose graph lies in P, thus satisfying S ⊂ f(S) for all S ∈ K(ℚ) which shows that f is a domination function for A.

(iv) ⟹ (i) : Asuming (iv) we shall prove that any Π_{1}^{1} non Borel set is Π_{1}^{1}-complete by Borel reductions, which will prove (i) by Harrington’s result quoted in Section 1 (see Theorem 1.3).

So fix a Π_{1}^{1} non Borel subset W of 2^{ω} and let h be the homeomorphism given by Theorem 6.2. Since W is not Borel, the set A = h(W × ω^{ω}) is a Π_{1}^{1} cofinal subset of K(ℚ). Hence by (iv) there exists a Borel function f : K(ℚ) → A such that S ⊂ f(S) for all S ∈ K(ℚ). Then h^{−1}∘f defines a Borel mapping from K(ℚ) into W × ω^{ω}, which extends to a Borel mapping g : B → 2^{ω} × ω^{ω} defined on a Borel subset B containing K(ℚ).

Let g_{0} : B → 2^{ω} denote the first coordinate function of g. Since S ⊂ f(S) = h∘g(S) for all S ∈ K(ℚ), the set B′ = {S ∈ B : S ⊂ h∘g(S) } is Borel and contains K(ℚ) ; and for all S ∈ B′ :

- if S ⊂ ℚ , then g(S) ∈ W × ω^{ω} and g_{0}(S) ∈ W.

- if S ⊄ ℚ, then h∘g(S) ⊃ S and h∘g(S) ∉ K(ℚ), hence g(S) ∉ W ×ω^{ω} and g_{0}(S) ∉ W. This proves that K(ℚ) = g_{0}^{−1}(W), hence g_{0} is a Borel reduction of K(ℚ) to W, and since K(ℚ) is Π_{1}^{1} complete, this proves that W is Π_{1}^{1}-complete by Borel reductions. ■

We finish with some variations of the previous results. In the next two results π denotes the canonical projection from 2^{ω} × 2^{ω} on the first factor ; we also identify any mapping f : A → B with its graph and write f ⊂ A × B.

Theorem 7.4. Assume that “ ∀ α ∈ ω^{ω}, ω_{1}^{L(α)} < ω_{1}”.

Let X ⊂ 2^{ω} × 2^{ω} be a Π_{1}^{1} set such that Y = π(X) is Π_{3}^{0}. If for any compact set K ⊂ Y there exists a continuous mapping f_{K} : K → 2^{ω} such that f_{K} ⊂ X then there exists a continuous mapping f : Y → 2^{ω} such that f ⊂ X.

\underline{Proof} : The argument is similar to the proof of (i) ⟹ (ii) in Theorem 7.2 and we sketch it briefly : apply Theorem 5.1 with R as the proof of (i) ⇒ (ii) in Theorem 7.2 above and

Z = {H ∈ K(2^{ω} × 2^{ω}) : H ⊂ X and H is a graph }

Notice that the condition “H is a graph” is Borel (in fact Π_{2}^{0}), hence Z is Π_{1}^{1} ; and one easily checks that hypothesis b) of Theorem 5.1 holds. Then the conclusion of this theorem provides a continuous mapping F : Y → Z with graph contained in R. Then the set H_{0} := {(y, z) : (y, z) ∈ F (y)} is the graph of some function f̃: 2^{ω} → 2^{ω}, which is continuous since H_{0} is compact. Then f = f̃_{|Y} : Y → 2^{ω} is continuous and f ⊂ X. ■

By similar arguments one can prove :

Theorem 7.5. Assume “Det(Σ_{1}^{1})”.

Let X ⊂ 2^{ω} × 2^{ω} be a Π_{1}^{1} set such that Y = π(X) is Π_{1}^{1}. If for any compact set K ⊂ Y there exists a continuous mapping f_{K} : K → 2^{ω} such that f_{K} ⊂ X then there exists a continuous mapping f : Y → 2^{ω} such that f ⊂ X.

7.6. Extensions and remarks : We give now some variations and extensions of the previous results. The proofs that we omit use essentially either the same arguments or some complementary ideas from [2] and [3] .

1) All the results have clearly effective versions. Notice however that in Theorem 5.1 the function f of the conclusion is only continuous and not recursive. Similarly the continuous or Borel domination functions considered in Theorem 7.3 are not effective.

2) Examining the arguments of Theorem 7.2 one easily sees that the converse of Theorem 7.4 holds, that is one can derive the assumption “ ∀ α ∈ ω^{ω}, ω_{1}^{L(α)} < ω_{1}” from the rest of the statement. This is also true for Theorem 7.5.

3) One can add to the list of equivalent propositions in Theorem 7.2 the following analog of proposition (iv) in Theorem 7.1 and of proposition (iii) in Theorem 7.3 :

In any Π_{1}^{1} space X (or in X = ℚ), for any Π_{1}^{1} cofinal subset A of K(X) and any Π_{3}^{0} subset B of K(X) there exists a continuous mapping f : B → A satisfying f(S) ⊃ S for all S ∈ B

But one cannot, as in the other theorems, replace here “continuous” by “Borel”. Also one cannot replace the cofinality hypothesis on A by the more natural but weaker hypothesis “∀ S ∈ B, ∃ T ∈ A such that S ⊂ T ”.

4) For simplicity we focused on Π_{1}^{1} cofinal subset of K(X). In fact all the previous results are still valid for Σ_{2}^{1} cofinal subsets. This can be seen either by slight modifications in the arguments, or by using the homeomorphic embedding Φ : K(X) × 2^{ω} → K(X) constructed in Lemma 6.6 which enables one to reduce the study of Σ_{2}^{1} cofinal subsets to the study of Π_{1}^{1} cofinal subsets (see the proof of Theorem 5.2 in [3]).

But one cannot replace in Theorem 7.4 and Theorem 7.5, the hypothesis “X Π_{1}^{1}” by “X Σ_{2}^{1}”.

5) Most of the classes considered in this work are Wadge classes. We recall that Y is Wadge class if there exists some Y_{0} in Y such that any space Y in Y is of the form Y = φ^{−1}(Y_{0}) for some continuous mapping φ : 2^{ω} → 2^{ω} ; but then the mapping ψ : Y → Y_{0}, obtained by restricting the domain and the range of φ, is clearly perfect. For example one can take :

Y_{0} = ω^{ω} , ℚ × ω^{ω} , (2^{ω} × 2^{ω}) \ (ℚ × ω^{ω}) , ℚ^{ω} , K(ℚ) if Y = Π_{2}^{0} , P_{σ} , P̌_{σ} , Π_{3}^{0} , Π_{1}^{1}. If we consider now any mapping f : X → Y it is straightforward to check the following

properties :

– If f is compact covering then ψ∘f is compact covering.

– If ψ∘f is inductively perfect then f is inductively perfect. It follows then that for a given space X :

If any compact covering mapping from X onto Y_{0} is inductively perfect then the same holds for any mapping from X onto any space in Y. Applying this observation to the particular classes considered in Theorems 7.1, 7.2 and 7.3, one obtains new equivalent propositions in these theorems.

6) It is interesting to compare the following statements :

a) Any compact covering mapping f : X → Y with X Borel and Y Σ_{2}^{0} is inductively perfect A) Any compact covering mapping f : X → Y with X Borel and Y Π_{2}^{0} is inductively perfect b) Any compact covering mapping f : X → Y with X Π_{2}^{0} is inductively perfect B) Any compact covering mapping f : X → Y with X Π_{3}^{0} is inductively perfect

As mentioned in the introduction a) and b) are theorems in ZF C ; whereas by Theorem 7.1, A) is equivalent to “∀ α ∈ ω^{ω}, ω^{ω} ∩ L is ≤^{∗} bounded” ; and by Theorem 7.2, B) is equivalent to “∀ α ∈ ω^{ω}, ω^{ω} ∩ L is countable”

7) The results of this paper give strong evidence for the following Conjecture : Assume that “ ∀α ∈ ω^{ω}, ω_{1}^{L(α)} < ω_{1}”. Then any compact covering mapping f : X → Y between two Borel spaces X and Y is inductively perfect. which would solve completely Problem 1, since by propositions (ii) and (iii) of Theorem 7.2 the assumption “∀α ∈ ω^{ω}, ω_{1}^{L(α)} < ω_{1}” is necessary even if one restricts the conjecture to mappings between Π_{3}^{0} spaces.

Notice that as in the proof of Theorem 7.2 one obtains an equivalent form of this conjecture if the hypothesis “X and Y are Borel” is replaced by one of the following alternative hypothesis :

– “X is Π_{1}^{1} and Y is Borel”

– “X is Borel” (with no assumption on Y).

Other equivalent forms can be obtained in terms of Π_{1}^{1} cofinal subsets of K(X) as in Remark 3 above, where B would now be any Borel (or even analytic) subset of K(X).

8) Finally we recall that inductively perfect mappings preserve Borel (additive or multiplicative) classes (see [2]) ; and as mentioned in the proof of Theorem 7.2 this property extends to compact covering mappings if one assumes that “ ∀ α ∈ ω^{ω}, ω_{1}^{L(α)} < ω_{1}”.

But the following problem, raised by Ostrovsky [9], is still open in ZF C Problem : Is the image of a Borel space by a compact covering mapping also Borel ?

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