Topological aspects of Hurewicz tests for the difference hierarchy

We generalize the Baire Category Theorem to the Borel and difference hierarchies, i.e. if Г is any of the classes Σξ⁰, Пξ⁰, Dη(Σξ⁰) or Ďη(Σξ⁰) we find a representative set Pг ∊ Г and a Polish topology τг such that for every A ∊ Ѓ from some assumption on the size of A ∩ Pг we can deduce that A\ Pг is...

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Datum:	2006
1. Verfasser:	Matrai, T.
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Veröffentlicht:	Інститут прикладної математики і механіки НАН України 2006
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spelling	irk-123456789-1245662017-09-30T03:03:59Z Topological aspects of Hurewicz tests for the difference hierarchy Matrai, T. We generalize the Baire Category Theorem to the Borel and difference hierarchies, i.e. if Г is any of the classes Σξ⁰, Пξ⁰, Dη(Σξ⁰) or Ďη(Σξ⁰) we find a representative set Pг ∊ Г and a Polish topology τг such that for every A ∊ Ѓ from some assumption on the size of A ∩ Pг we can deduce that A\ Pг is of second category in the topology τг. This allows us to distinguish the levels of the Borel and difference hierarchies via Baire category. We also present some typical Baire Category Theorem-like applications of the results. 2006 Article Topological aspects of Hurewicz tests for the difference hierarchy / T. Matrai // Український математичний вісник. — 2006. — Т. 3, № 4. — С. 520-546. — Бібліогр.: 6 назв. — англ. 1810-3200 2000 MSC. 03E15, 54H05, 28A05. http://dspace.nbuv.gov.ua/handle/123456789/124566 en Український математичний вісник Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We generalize the Baire Category Theorem to the Borel and difference hierarchies, i.e. if Г is any of the classes Σξ⁰, Пξ⁰, Dη(Σξ⁰) or Ďη(Σξ⁰) we find a representative set Pг ∊ Г and a Polish topology τг such that for every A ∊ Ѓ from some assumption on the size of A ∩ Pг we can deduce that A\ Pг is of second category in the topology τг. This allows us to distinguish the levels of the Borel and difference hierarchies via Baire category. We also present some typical Baire Category Theorem-like applications of the results.
format	Article
author	Matrai, T.
spellingShingle	Matrai, T. Topological aspects of Hurewicz tests for the difference hierarchy Український математичний вісник
author_facet	Matrai, T.
author_sort	Matrai, T.
title	Topological aspects of Hurewicz tests for the difference hierarchy
title_short	Topological aspects of Hurewicz tests for the difference hierarchy
title_full	Topological aspects of Hurewicz tests for the difference hierarchy
title_fullStr	Topological aspects of Hurewicz tests for the difference hierarchy
title_full_unstemmed	Topological aspects of Hurewicz tests for the difference hierarchy
title_sort	topological aspects of hurewicz tests for the difference hierarchy
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2006
url	http://dspace.nbuv.gov.ua/handle/123456789/124566
citation_txt	Topological aspects of Hurewicz tests for the difference hierarchy / T. Matrai // Український математичний вісник. — 2006. — Т. 3, № 4. — С. 520-546. — Бібліогр.: 6 назв. — англ.
series	Український математичний вісник
work_keys_str_mv	AT matrait topologicalaspectsofhurewicztestsforthedifferencehierarchy
first_indexed	2025-07-09T01:38:56Z
last_indexed	2025-07-09T01:38:56Z
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fulltext	Український математичний вiсник Том 3 (2006), № 4, 520 – 546 Topological aspects of Hurewicz tests for the difference hierarchy Tamás Mátrai (Presented by V. Ya. Gutlanskii) Abstract. We generalize the Baire Category Theorem to the Borel and difference hierarchies, i.e. if Γ is any of the classes Σ0 ξ, Π0 ξ, Dη(Σ0 ξ) or Ďη(Σ0 ξ) we find a representative set PΓ ∈ Γ and a Polish topology τΓ such that for every A ∈ Γ̌ from some assumption on the size of A∩PΓ we can deduce that A \PΓ is of second category in the topology τΓ. This allows us to distinguish the levels of the Borel and difference hierarchies via Baire category. We also present some typical Baire Category Theorem- like applications of the results. 2000 MSC. 03E15, 54H05, 28A05. Key words and phrases. Hurewicz test, Baire category, Borel hierar- chy, difference hierarchy. 1. Introduction If an open set of the Polish space (X, τ) is nonempty it is of second category: this is probably the simplest formulation of the Baire Category Theorem. The purpose of this note is to show that this theorem can be extended to the entire Borel and difference hierarchies: if Γ is any of the classes Σ0 ξ , Π0 ξ , Dη(Σ 0 ξ) or Ďη(Σ 0 ξ) in a Polish space (X, τ) we define a representative set PΓ ⊆ X, PΓ ∈ Γ and a fine Polish topology τΓ on X such that if a set A ⊆ X, A ∈ Γ̌ satisfies that A ∩ PΓ is of τΓ-second category then A\PΓ is also of τΓ-second category. This makes possible in particular to distinguish the classes of the Borel and difference hierarchies via Baire category in a suitable topology. We remark that there is a quite classical extension of the dichotomy expressed by the Baire Category Theorem for open sets. More than a half Received 17.05.2006 The research was carried out with the partial support of the OTKA Grants F 43620, T49786 and T 37758, and of the Öveges Project of and . ISSN 1810 – 3200. c© Iнститут математики НАН України T. Mátrai 521 century ago Witold Hurewicz proved the following theorem, also called Hurewicz-dichotomy, about sets failing to be Π0 2. Theorem 1.1 (W. Hurewicz). Let X be a Polish space and A ⊆ X be a coanalytic set. If A is not Π0 2 there is a continuous injection of the Cantor set into X, ϕ : 2ω → X such that ϕ−1(A) is countable dense in 2ω. Again by the Baire Category Theorem in Polish spaces from the weak assumption that a Π0 2 set is dense we can conclude that it is residual; in particular a countable dense set H ⊆ 2ω is never Π0 2. Thus Theorem 1.1 shows via Baire Category that a not Π0 2 set is as far from being Π0 2 as possible on a copy of the Cantor set. This is the reason why the pair (2ω, H) is called the Hurewicz test for Σ0 2 sets. Theorem 1.1 has been strengthened in many successive steps. In some sense the most general existence theorem for Hurewicz tests is the following result (see [3, Theorem 2, p. 27], [2, Corollary 6 and Theorem 7 p. 457] and also [5, Theorem 2, p. 1025]). For the definition of the appearing classes see Section 2; for a class Γ(X) ⊆ 2X , Γ̌(X) = {A ⊆ X : X \ A ∈ Γ(X)} denotes the dual class of Γ. Theorem 1.2 (A. Louveau, J. Saint Raymond). Let Γ be a non- selfdual Borel Wadge class. Then there is a zero topological dimensional compact metric space K and a Γ(K) subset H of K with the following property: for every Borel set B in a nonempty Suslin space E either B ∈ Γ̌(E) or there is a continuous injection ϕ : K → E with ϕ(H) = B. If 3 ≤ ξ < ω1 and 1 ≤ η < ω1 or if ξ = 2 and ω ≤ η < ω1, for the classes Γ = Ďη(Σ 0 ξ) and Γ = Dη(Σ 0 ξ) we can take K = 2ω and H can be an arbitrary set in Γ(2ω) \ Γ̌(2ω). The pair (K, H) is called the Hurewicz test for Γ. Observe that The- orem 1.2 gives in particular that a membership of a Borel set in some Γ depends only on zero dimensional sets and can be witnessed via contin- uous injections ϕ : K → E. On the other hand the aspect of dichotomy which was present in the Baire Category Theorem for open sets and in the Hurewicz-dichotomy Theorem for Π0 2 sets is lacking, e.g. Theorem 1.2 says nothing about how well a set A ∈ Γ can be approximated by sets in Γ̌. Our main result provides a Hurewicz-dichotomy for the classes of the Borel and difference hierarchies. Theorem 1.3. Let 0 < ξ, η < ω1 be fixed. Then there is a Polish space (Cξ,η, τCξ,η ) homeomorphic to (2ω, τ2ω), a Ďη(Σ 0 ξ(τCξ,η )) set Pξ,η ⊆ Cξ,η, a nonempty Π0 ξ(τCξ,η ) set Wξ,η(η) ⊆ Cξ,η and Polish topologies τ< ξ,η, τξ,η on Cξ,η with the following properties: 522 Topological aspects of Hurewicz tests... 1. τ< ξ,η refines τCξ,η and τξ,η refines τ< ξ,η; 2. Pξ,η and Cξ,η \ Pξ,η are both τξ,η-open; 3. for every Dη(Σ 0 ξ(τCξ,η )) set A ⊆ Cξ,η and τ< ξ,η-open set G ⊆ Cξ,η, if G ∩ Wξ,η(η) 6= ∅ and A ∩ Pξ,η is τξ,η-residual in G ∩ Pξ,η then A \ Pξ,η is of τξ,η-second category in G \ Pξ,η. Before any further discussion we formulate a corollary of this theorem which is less abundant in notations. Corollary 1.1. Let 0 < ξ, η < ω1 be fixed and let (Cξ,η, τCξ,η ), Pξ,η and τξ,η be as in Theorem 1.3. 1. If A ⊆ Cξ,η is a Dη(Σ 0 ξ(τCξ,η )) set and Pξ,η ⊆ A then A \ Pξ,η is of τξ,η-second category. 2. If A′ ⊆ Cξ,η is a Ďη(Σ 0 ξ(τCξ,η )) set and Cξ,η\Pξ,η ⊆ A′ then A′∩Pξ,η is of τξ,η-second category. The first statement of Corollary 1.1 is the special G = Cξ,η case of Theorem 1.3, while the second follows from Theorem 1.3 applied to A = Cξ,η \ A′ and G = Cξ,η. This corollary has already the feature of dichotomy we are looking for: it allows to derive from the information that a set A is “badly” structured the conclusion that either it does not contain Pξ,η or it is big in category in Cξ,η \ Pξ,η, and vice versa. We remark that for the Borel hierarchy a stronger result can be proved: as we will see in Theorem 4.2, for η = 1 in Theorem 1.3 we can practically conclude that A ∩ Pξ,η is τξ,η-residual in G ∩ Pξ,η. The reason why the result we obtain for the difference hierarchy is weaker than what we have for the Borel hierarchy is that Dη(Σ 0 ξ) (1 < η < ω1) is not closed under taking countable unions. Theorem 1.3 gets its real power when combined with Theorem 1.2, as follows. Corollary 1.2. Let (X, τ) be a nonempty Polish space, 1 < ξ < ω1, 0 < η < ω1 be fixed and let (Cξ,η, τCξ,η ), Pξ,η and τξ,η be as in Theorem 1.3. If a Borel set A ⊆ X is 1. in Dη(Σ 0 ξ(τ)), then for every continuous injection ϕ : (Cξ,η, τCξ,η )→ (X, τ), ϕ(Pξ,η) ⊆ A implies that ϕ−1(A) \ Pξ,η is of τξ,η-second category; 1′. in Ďη(Σ 0 ξ(τ)), then for every continuous injection ϕ : (Cξ,η, τCξ,η )→ (X, τ), ϕ(Cξ,η\Pξ,η) ⊆ A implies that ϕ−1(A)∩Pξ,η is of τξ,η-second category; T. Mátrai 523 2. not in Dη(Σ 0 ξ(τ)) there is a continuous injection ϕ : (Cξ,η, τCξ,η ) → (X, τ) such that ϕ(Pξ,η) = A ∩ ϕ(Cξ,η), hence ϕ(Pξ,η) ⊆ A but ϕ−1(A) \ Pξ,η = ∅; 2′. not in Ďη(Σ 0 ξ(τ)) there is a continuous injection ϕ : (Cξ,η, τCξ,η ) → (X, τ) such that ϕ(Cξ,η \Pξ,η) = A∩ϕ(Cξ,η), hence ϕ(Cξ,η \Pξ,η) ⊆ A but ϕ−1(A) ∩ Pξ,η = ∅. In this note we would like to prove only one more corollary, which represents a typical way to apply the preceding results. Corollary 1.3. Let ℵ0 < λ < 2ℵ0 be a regular cardinal and suppose that in our model the union of λ meager sets is meager in Polish spaces (this assumption holds e.g. under Martin’s Axiom MA(λ)). Let (X, τ) be a nonempty Polish space. Let 1 < ξ < ω1, 0 < η < ω1 be fixed and set Γ = Dη(Σ 0 ξ(τ)) or Γ = Ďη(Σ 0 ξ(τ)). Suppose that A, Aα ⊆ X (α < λ) such that A is Borel, A /∈ Γ̌, Aα ∈ Γ̌ and A ⊆ Aα (α < λ). Then there is a stationary set Λ ⊆ λ such that (X \ A) ∩ ⋂ α∈Λ Aα 6= ∅. The paper is structured as follows. In Section 2 we recall some defini- tions related to the Borel and difference hierarchies and fix some conven- tions for Polish topologies. In Section 3 we define all the Polish spaces and Hurewicz test sets we will use in the sequel. Since for our problem as well, the ξ = 1 case does not fit into the general framework we handle it separately in Section 4.1. In Section 4.2 we prove the theorems concern- ing the Borel hierarchy, and finally in Section 4.3 we extend the results to the difference hierarchy and prove the corollaries. The results presented here were initiated in [4]. A result of similar flavor for the Borel hierar- chy was obtained by S. Solecki [6, Theorem 2.2, p. 526] using effective methods. 2. Preliminaries Our terminology and notation follow [1]. Let (C, τC) denote the Polish space 2ω with its usual product topology. As usual, Π0 ξ(τ) and Σ0 ξ(τ) (0 < ξ < ω) stand for the ξth multiplicative and additive Borel class in the Polish space (X, τ), starting with Π0 1(τ) = closed sets, Σ0 1(τ) = open sets. We derive the difference hierarchy as follows (see also [1, Section 22.E, p. 175]. Every ordinal η can be uniquely written as α+n where α is limit and n < ω. We call η even if n is even and odd if n is odd. 524 Topological aspects of Hurewicz tests... Definition 2.1. Let 0 < ξ, η < ω1 and let (Aα)α<η be a sequence of subsets of a set X, such that Aα ⊆ Aα′ (α ≤ α′ < η). Then Dη((Aα)α<η) ⊆ X is defined by x ∈ Dη((Aα)α<η) ⇐⇒ x ∈ ⋃ α<η Aα and the least α < η with x ∈ Aα has parity opposite to that of η. With this operation in the Polish space (X, τ) we set Dη ( Σ0 ξ(τ) ) = { Dη((Aα)α<η) : Aα ∈ Σ0 ξ(τ), Aα ⊆ Aα′ (α ≤ α′ < η) } . The definition of the Wadge hierarchy can be found e.g. in [1, Section 21.E, p. 156 and Section 22.B, p. 169]. We will use only the Borel and difference hierarchies in the sequel. Let ξ, ϑi (i < ω) be ordinals. We write ϑi → ξ if ξ is successor and ϑi + 1 = ξ (i < ω) or if ξ is limit, ϑi is successor (i < ω), ϑi ≤ ϑj (i ≤ j < ω) and supi<ω ϑi = ξ. For every ordinal ξ < ω1 we fix once and for all a sequence (ϑi)i<ω such that ϑi → ξ. To avoid complicated notations, we do not indicate the dependence of the sequence on ξ, it will be always clear which pair of ordinal and sequence is considered. In this note we will notoriously refine Polish topologies by turning countably many closed sets into open sets. We do this as described in [1], that is the open sets of the ancient topology together with their portion on the members of our collection of closed sets serve as a subbase of the new, finer topology. We will use that the topology obtained in this way is also Polish. Definition 2.2. Let (X, τ) be a Polish space, P = {Pi : i < ω} be a countable collection of Π0 1(τ) sets. Then τ [P] denotes that Polish topology refining τ where each Pi (i < ω) is turned successively into an open set. It is easy to see that the resulting finer topology τ [P] is independent from the enumeration of P. This will be clear shortly when we fix a base of τ [P]. We also use the notation τ [P] when the countable collection of not necessarily Π0 1(τ) sets P can be enumerated on such a way that Pn is Π0 1(τ [{Pi : i < n}]). We need a precise notion of basic open sets in our spaces. Definition 2.3. Let (Xi, τi) (i ∈ I) be Polish spaces; if a basis Gi is fixed in the spaces (Xi, τi) (i ∈ I), which are meant to be the basic open sets in (Xi, τi), then the basic open sets of ( ∏ i∈I Xi, ∏ i∈I τi) are the open sets of the form ∏ i∈J Gi × ∏ i∈I\J Xi, T. Mátrai 525 where J ⊆ I is finite and Gi ∈ Gi for every i ∈ J . If the basic open sets G are fixed in the Polish space (X, τ) and τ [P] makes sense for a countable collection P of subsets of X, then the basic open sets of τ [P] are of the sets of the form G ∩ F0 ∩ · · · ∩ Fn−1 or G with G ∈ G, Fi ∈ P (i < n); a basic τ [P]-open set is said to be proper if it is not τ -open. Observe that the basic open sets defined on this way form a basis of ∏ i∈I τi and τ [P], respectively. From now on whenever a Polish space (X, τ) appears we assume that a countable basis comprised of basic τ - open sets is fixed; and this is done with respect to the convention of Definition 2.3 if it is applicable. We take X to be basic τ -open. The closure of a set A ⊆ (X, τ) is denoted by clτ (A). We recall that a Π0 2(τ) subset G of the Polish space (X, τ) is itself a Polish space with the restricted topology τ \|G (see e.g. [1, (3.11) Theorem]). In particular, the notions related to category in the topology τ make sense relative to G. We will have to return to the topologies on the coordinates in product spaces. If (X, σ), (Y, τ) are arbitrary topological spaces and (X ,S) = (X × Y, σ × τ), then we define PrX(S) = σ. The projection of product sets in product spaces is defined analogously. If GX ⊆ X and GY ⊆ Y , we say that the set of product form G = GX × GY ⊆ X is nontrivial on the X coordinate if GX 6= X. 3. Spaces, sets and topologies In this section we define the Polish spaces and the special sets for which a Polish topology featuring the dichotomy we like can be con- structed. First we handle the Borel hierarchy. Definition 3.1. We set (C1, τC1) = (C2, τC2) = (C, τC), P1 = {x ∈ C1 : x(n) = 1 (n < ω)} and τP1 = τ< P1 = τC1 on C1. Let {U2,n : n < ω} be an enumeration of the set {x ∈ C2 : ∃n < ω (x(i) = 0 (n < i < ω))} and set P2 = C2 \ {U2,n : n < ω}, τ< P2 = τC2 , τP2 = τ< P2 [{U2,n : n < ω}]. Let now 2 < ξ < ω1 and suppose that the spaces (Cϑ, τCϑ ), the Π0 ϑ(τCϑ ) sets Pϑ and the topologies τPϑ , τ< Pϑ are defined for every ϑ < ξ. Then with ϑi → ξ let Cξ = ∏ i<ω Cϑi , τCξ = ∏ i<ω τCϑi , 526 Topological aspects of Hurewicz tests... Pξ = {x ∈ Cξ : x(i, .) ∈ Cϑi \ Pϑi (i < ω)}, (3.1) τ< Pξ = ∏ i<ω τPϑi , and let τPξ = τ< Pξ [{Uξ,n : n < ω}] where Uξ,n = ∏ i<n (Cϑi \ Pϑi ) × Pϑn × ∏ n<i<ω Cϑi ⊆ ∏ i<n Cϑi × Cϑn × ∏ n<i<ω Cϑi = Cξ (n < ω). (3.2) If ξ is a limit ordinal for every m < ω we also define Cm ξ = ∏ m≤i<ω Cϑi , τCm ξ = ∏ m≤i<ω τCϑi , Pm ξ = {x ∈ Cm ξ : x(i, .) ∈ Cϑi \ Pϑi (m ≤ i < ω)}, (3.3) τ< P m ξ = ∏ m≤i<ω τPϑi and let τP m ξ = τ< P m ξ [{Um ξ,n : m ≤ n < ω}] where Um ξ,n = ∏ m≤i<n (Cϑi \ Pϑi ) × Pϑn × ∏ n<i<ω Cϑi ⊆ ∏ m≤i<n Cϑi × Cϑn × ∏ n<i<ω Cϑi = Cξ (m ≤ n < ω). (3.4) If ξ is a limit ordinal and ϑ < ξ let I be minimal such that ϑ ≤ ϑI . Set Hξ(ϑ) = ∏ i<I (Cϑi \ Pϑi ) × ∏ I≤i<ω Cϑi and for every m < ω set Hm ξ (ϑ) = ∏ m≤i<I (Cϑi \ Pϑi ) × ∏ I≤i<ω Cϑi if m < I else let Hm ξ (ϑ) = Cm ξ . If ξ is a successor we set Hξ(ϑ) = Cξ (ϑ < ξ). We prove a lemma on the relation of Pξ, τ< Pξ and τPξ . Lemma 3.1. Let 0 < ξ < ω1 be fixed. 1. (Cξ, τCξ ) is homeomorphic to (C, τC). 2. Pξ is a Π0 ξ(τCξ ) set. 3. Pξ is a τPξ -nowhere dense Π0 1(τPξ ) set. 4. For ξ ≥ 2, Pξ ⊆ Cξ is a τ< Pξ -dense Π0 2(τ < Pξ ) hence τ< Pξ -residual set. T. Mátrai 527 5. For ξ ≥ 2, Cξ \ Pξ is a τ< Pξ -dense Σ0 2(τ < Pξ ) set. 6. If G is basic τPξ -open and G∩Pξ 6= ∅ then G is also basic τ< Pξ -open. 7. If G is proper basic τPξ -open there is a basic τ< Pξ -open set G0 and a unique n < ω satisfying G = G0 ∩ Uξ,n. 8. If G is proper basic τPξ -open the topologies τPξ \|G and τ< Pξ \|G coin- cide. 9. The topologies τPξ \|Pξ and τ< Pξ \|Pξ coincide. If ξ is a limit ordinal then 2–9 hold for Pm ξ and Um ξ,n instead of Pξ and Uξ,n (m ≤ n < ω). Proof. Statement 3.1 follows from the fact that a countable power of (C, τC) is homeomorphic to (C, τC). We prove 2 by induction on ξ. For ξ = 1 we have that P1 is a single point, which is clearly Π0 1(τC1). For ξ = 2, P2 is the complement of a countable set so it is Π0 2(τC2). Let now ξ ≥ 3 and suppose that Pη is Π0 η(τCη) for every η < ξ. With ϑm → ξ we have Pξ = ⋂ m<ω {x ∈ Cξ : x(m, .) ∈ Cϑm \ Pϑm } . (3.5) Since τCξ is the product of the topologies τCϑm and Pϑm is Π0 ϑm (τCϑm ) by the induction hypothesis, Pξ is the intersection of sets of additive class lower than ξ, so the statement follows. We prove 3 and 4 together, by induction on ξ. For ξ = 1, P1 is a single point, which is clearly Π0 1(τP1) and τP1-nowhere dense. For ξ = 2, statements 3 and 4 follow from the fact that {U2,n : n < ω} is a τC2-dense countable set. Let now ξ ≥ 3 and suppose that 3 holds for every η < ξ. We prove 4 for ξ and then 3 for ξ. Let ϑn → ξ. By (3.2) and (3.5), we have Pξ = Cξ \ ( ⋃ n<ω Uξ,n ) . (3.6) By the induction hypothesis Pϑn is τPϑn -nowhere dense and Π0 1(τPϑn ) (n < ω) so since τ< Pξ is the product of the topologies τPϑn (n < ω), Uξ,n is τ< Pξ -nowhere dense (n < ω). Also, Uξ,n is a finite intersection of τ< Pξ -open and Π0 1(τ < Pξ ) sets, thus it is a Σ0 2(τ < Pξ ) set (n < ω). Hence (3.6) shows that Pξ is τ< Pξ -dense and Π0 2(τ < Pξ ). Now we prove 3 for ξ. To obtain τPξ , we made open every Uξ,n on the right hand side of (3.6), so Pξ is Π0 1(τPξ ). Using again that Pϑn is 528 Topological aspects of Hurewicz tests... τPϑn -nowhere dense (n < ω), ⋃ n<ω Uξ,n meets every τ< Pξ -open set, hence it is τPξ -dense τPξ -open. So Pξ is τPξ -nowhere dense. For 5, we have just observed that ⋃ n<ω Uξ,n meets every τ< Pξ -open set, hence it is τ< Pξ -dense, as stated. Statements 6 and 7 follow from the fact that Uξ,n (n < ω) are pairwise disjoint and are disjoint from Pξ. Statements 8 and 9 immediately follow from 7 and 6. Finally if ξ is a limit ordinal and m < ω then by taking ϑ′ i = ϑm+i the sequence obtained satisfies ϑ′ i → ξ so 2–9 hold for Pm ξ and Um ξ,n instead of Pξ and Uξ,n (m ≤ n < ω) and the proof is complete. Next we point out a property of Hξ(ϑ). Lemma 3.2. For every 0 < ξ < ω1, Hξ(ϑ) is a τ< Pξ -dense τ< Pξ -open set in Cξ. We have Pξ ⊆ Hξ(ϑ) ⊆ Hξ(ϑ ′) (ϑ′ ≤ ϑ < ξ). Similarly, Hm ξ (ϑ) is a τ< P m ξ -dense τ< P m ξ -open set in Cm ξ and Pm ξ ⊆ Hm ξ (ϑ) ⊆ Hm ξ (ϑ′) (ϑ′ ≤ ϑ < ξ). Proof. The statements immediately follow from the definition. We continue with the definition of the spaces, the sets and the topolo- gies for the difference hierarchy. Definition 3.2. Let 0 < ξ, η < ω1 be fixed. If ξ is a successor ordinal take sequences (Cξ(α), τCξ(α)), Pξ(α), τ< Pξ(α) and τPξ(α) (α < η) of copies of the Polish space (Cξ, τCξ ), the set Pξ and the topologies τ< Pξ , τPξ defined in Definition 3.1; and set ρξ(α) = 0 (α < η). If ξ is a limit ordinal take an injection ρη : η → ω and let (Cξ(α), τCξ(α)) = (C ρη(α) ξ , τ C ρη(α) ξ ), Pξ(α) = P ρη(α) ξ , τ< Pξ(α) = τ< P ρη(α) ξ and τPξ(α) = τ P ρη(α) ξ (α < η). Let Cξ,η = ∏ α<η Cξ(α), τCξ,η = ∏ α<η τCξ(α), for every α < η set Kξ,η(α) = ⋃ β≤α (( ∏ γ<β Cξ(γ) ) × (Cξ(β) \ Pξ(β)) × ∏ β<γ<η Cξ(γ) ) , Vξ,η(α) = ( ∏ γ<α Pξ(γ) ) × (Cξ(α) \ Pξ(α)) × ∏ α<γ<η Cξ(γ), T. Mátrai 529 Vξ,η(η) = ∏ γ<η Pξ(γ) and for 0 < α ≤ η let Wξ,η(α) = ∏ γ<α Pξ(γ) × ∏ α≤γ<η Cξ(γ). Now set Qξ,η = Dη ( (Kξ,η(α)) α<η ) = ⋃ {Vξ,η(α) : α ≤ η, α is odd ↔ η is even}, Pξ,η = Cξ,η \ Qξ,η = ⋃ {Vξ,η(α) : α ≤ η, α is odd ↔ η is odd} and define the topologies τ< ξ,η = ∏ α<η τ< Pξ(α), τ< ξ,η(α) = ∏ γ<α τ< Pξ(γ) × ∏ α≤γ<η τPξ(γ) (α ≤ η), τξ,η = τ< ξ,η(0)[{Wξ,η(α) : 0 < α ≤ η}]. If ξ is a limit ordinal and ϑ < ξ set Hξ,η(ϑ) = ∏ α<η H ρη(α) ξ (ϑ). If ξ is successor we set Hξ,η(ϑ) = Cξ,η Remark 3.1. The notation introduced above is negligent because for limit ξ we do not indicate the dependence of say Cξ(α) from the particular η and ρη we are working with. Since η and ξ will mostly be fixed this will not cause any confusion. On the other hand ρη will vary for the following reason. If ξ is limit and η = η0 + (η \ η0) we identify Cξ,η with Cξ,η0 × Cξ,η\η0 by fixing the injection ρη : η → ω first and taking ρη0 = ρη\|η0 , ρη\η0 = ρη\|η\η0 . In the sequel this convention applies for the terms of product spaces. We will state in advance where this happens. We summarize the basic properties of our new sets and topologies. Notice in advance that Vξ,η(η) = Wξ,η(η), τ< ξ,η = τ< ξ,η(η) (0 < η < ω1) and that the sets Vξ,η(α) (α ≤ η) are pairwise disjoint. Lemma 3.3. Let 0 < ξ, η < ω1 be fixed. 1. (Cξ,η, τCξ,η ) is homeomorphic to (C, τC). 2. Wξ,η(η) is a Π0 ξ(τCξ,η ) set, Kξ,η(α) is a Σ0 ξ(τCξ,η ) set (α < η) hence Qξ,η is a Dη(Σ 0 ξ(τCξ,η )) set and Pξ,η is a Ďη(Σ 0 ξ(τCξ,η )) set. 3. Qξ,η and Pξ,η are τξ,η-open. 530 Topological aspects of Hurewicz tests... Proof. Statements 1 and 2 follows from Lemma 3.1.1 and Lemma 3.1.2. For 3 we have Vξ,η(η) = Wξ,η(η) and Vξ,η(α) = Wξ,η(α) \ (( ∏ γ<α Cξ(γ) ) × Pξ(α) × ∏ α<γ<η Cξ(γ) ) (α < η) hence Vξ,η(α) is τξ,η-open (α ≤ η). Since Qξ,η and Pξ,η are the unions of some Vξ,η(α) (α ≤ η) they are τξ,η-open, as stated. Next we analyze the restriction of the topology τξ,η to our special sets. Lemma 3.4. Let 0 < ξ, η < ω1, α ≤ η be fixed and let (Y, σ) be a nonempty Polish space. Then 1. if G ⊆ Cξ,η ×Y is basic τ< ξ,η(0)×σ-open and G∩ (Vξ,η(α) × Y ) 6= ∅ then G is basic τ< ξ,η(γ) × σ-open (γ ≤ α); 2. if G ⊆ Cξ,η×Y is basic τξ,η×σ-open and G∩(Vξ,η(α) × Y ) 6= ∅ then there is a basic τ< ξ,η(α)×σ-open set G′ such that G∩(Vξ,η(α) × Y ) = G′ ∩ (Vξ,η(α) × Y ); 3. the topologies (τξ,η×σ)\|Vξ,η(α)×Y and (τ< ξ,η(γ)×σ)\|Vξ,η(α)×Y (γ ≤ α) coincide; 4. if 0 < α, G is basic τ< ξ,η(0) × σ-open and G ∩ (Wξ,η(α) × Y ) 6= ∅ then G is basic τ< ξ,η(γ) × σ-open (γ ≤ α); 5. if G is basic τξ,η × σ-open and G ∩ (Wξ,η(η) × Y ) 6= ∅ then there is a basic τ< ξ,η × σ-open set G′ such that G′ ∩ (Wξ,η(η) × Y ) = G ∩ (Wξ,η(η) × Y ); 6. the topologies (τξ,η×σ)\|Wξ,η(η)×Y and (τ< ξ,η(γ)×σ)\|Wξ,η(η)×Y (γ ≤ η) coincide. Proof. For 1, let G be basic a τ< ξ,η(0)×σ-open set satisfying G∩(Vξ,η(α)× Y ) 6= ∅. If for some γ < α, PrCξ(γ)(G) is proper basic τPξ(γ)-open then PrCξ(γ)(G) ∩ Pξ(γ) = ∅ hence G ∩ (Vξ,η(α) × Y ) = ∅, which is not the case. So PrCξ(γ)(G) is τ< Pξ(γ)-open (γ < α), thus 1 holds. For 2, let G be basic τξ,η ×σ-open, say G = G′∩ (Wξ,η(β) × Y ) where G′ is basic τ< ξ,η(0)-open and β ≤ η, such that G ∩ (Vξ,η(α) × Y ) 6= ∅. Since Vξ,η(α) ⊆ Wξ,η(γ) (γ ≤ α) and Vξ,η(α) ∩ Wξ,η(γ) = ∅ (α < γ ≤ η), we have G ∩ (Vξ,η(α) × Y ) = G′ ∩ (Vξ,η(α) × Y ) 6= ∅. Thus by 1 with γ = α, G′ is basic τ< ξ,η(α) × σ-open so 2 holds. Since τ< ξ,η(γ) (γ ≤ α) is finer than τ< ξ,η(α) and is coarser than τξ,η, 2 immediately gives 3. T. Mátrai 531 For 4, observe that Wξ,η(α) = ⋃ α≤β≤η Vξ,η(β). Thus we have G ∩ (Vξ,η(β) × Y ) 6= ∅ for some α ≤ β ≤ η. So by 1, G is basic τ< ξ,η(γ) × σ- open (γ ≤ β), as required. Finally 5 and 6 are the special α = η case of 2 and 3, so the proof is complete. To close this section we prove three lemmata, the first specially for ξ = 1, the second for 1 < ξ < ω1 and the third for Hξ,η(ϑ). Lemma 3.5. Let 0 < η < ω1 be fixed and let (Y, σ) be a nonempty Polish space. If H is τ< 1,η × σ-open and H ∩ (W1,η(η) × Y ) 6= ∅ then H ∩ (V1,η(γ) × Y ) 6= ∅ (γ ≤ η). Proof. By passing to a subset we can assume that H is basic τ< 1,η × σ- open. For γ = η the statement follows from V1,η(η) = W1,η(η) so let γ < η. Observe that τ< 1,η = τC1,η ; thus H ∩ (W1,η(η) × Y ) 6= ∅ implies P1(α) ∈ PrC1(α)(H) (α < η) and PrY (H) 6= ∅. Since PrC1(α)(V1,η(γ)) = P1(α) (α < γ) and PrC1(α)(V1,η(γ)) is a τC1(α)-dense τC1(α)-open set (γ ≤ α < η) we conclude H ∩ (V1,η(γ) × Y ) 6= ∅. Lemma 3.6. Let 1 < ξ, η < ω1, α ≤ γ ≤ η be fixed. Then 1. Vξ,η(α) is a τ< ξ,η(γ)-dense set; 2. Vξ,η(α) is a τ< ξ,η(α)-residual set; 3. Wξ,η(α) is a τ< ξ,η(γ)-residual Π0 2(τ < ξ,η) set. Proof. By Lemma 3.1.4 and Lemma 3.1.5, both Pξ(β) and Cξ(β) \Pξ(β) are τ< Pξ(β)-dense sets (β ≤ α), so 1 follows. By Lemma 3.1.4, Pξ(β) is a τ< Pξ(β)-residual Π0 2(τ < Pξ (β)) set (β < α) while by Lemma 3.1.3, Pξ(α) is a τPξ(α)-nowhere dense, so we have 2 and 3. In the final lemma we apply the convention of Remark 3.1 for the first time. Lemma 3.7. For every 0 < ξ, η < ω1, Hξ,η(ϑ) is a τ< ξ,η-dense τ< ξ,η-open set in Cξ,η. We have Wξ,η(η) ⊆ Hξ,η(ϑ) ⊆ Hξ,η(ϑ ′) (ϑ′ ≤ ϑ < ξ) and for α < η, η̃ = η \ α, Hξ,η(ϑ) = Hξ,α(ϑ) × Hξ,η̃(ϑ) (ϑ < ξ). Proof. The statements immediately follows from Lemma 3.2 and the definition using that ρη̃ = ρη\|η̃ and ρα = ρη\|α by the convention of Remark 3.1. 532 Topological aspects of Hurewicz tests... 4. Testing the difference hierarchy From now on we work to prove that (Cξ,η, τCξ,η ), Wξ,η(η), Pξ,η, τ< ξ,η and τξ,η of Definition 3.2 have the properties required in Theorem 1.3. We start with the easy observations. Proposition 4.1. Let 0 < ξ, η < ω1 be fixed. Then (Cξ,η, τCξ,η ) is home- omorphic to (2ω, τ2ω), Wξ,η(η) is a Π0 ξ(τCξ,η ) set, Pξ,η is a Ďη(Σ 0 ξ(τCξ,η )) set. The topology τξ,η is Polish and refines τ< ξ,η which in turn refines τCξ,η . The sets Pξ,η and Cξ,η \ Pξ,η are both τξ,η-open. Proof. The statements follow from Lemma 3.3.1, Lemma 3.3.2, Defini- tion 3.2 and Lemma 3.3.3. Thus Theorem 1.3.1 and Theorem 1.3.2 hold. Observe that for 1 < ξ < ω1 if G is basic τ< ξ,η-open then G ∩ Wξ,η(η) 6= ∅ follows from Lem- ma 3.6.3 with α = γ = η. So this condition in Theorem 1.3.3 is restrictive only for ξ = 1. In the sequel we use these properties without further refer- ence. Theorem 1.3.3 will be proved through Theorem 4.1, Corollary 4.1.1 and Theorem 4.3. 4.1. The ξ = 1 case First we prove Theorem 1.3.3 for ξ = 1. As we mentioned in the introduction this case does not fit into the general framework. Since the 1 < ξ < ω1 case is complicated enough in itself we treat ξ = 1 separately. Also, this is a simple but informative introduction to the techniques we use. In the proof the product structure of Definition 3.2 must be exploited thus we prove Theorem 1.3.3 for ξ = 1 in the following more general form. When Y is a single point, we get back Theorem 1.3.3. Theorem 4.1. Let 0 < η < ω1 be fixed. Let (Y, σ) be a nonempty Polish space and consider a Dη(Σ 0 1(τC1,η × σ)) set A ⊆ C1,η × Y . Suppose that G ⊆ C1,η ×Y is τ< 1,η ×σ-open, G∩ (W1,η(η) × Y ) 6= ∅ and A∩ (P1,η × Y ) is τ1,η ×σ-residual in G∩ (P1,η × Y ). Then A∩ (Q1,η × Y ) is of τ1,η ×σ- second category in G ∩ (Q1,η × Y ). Proof. Notice first that by Definition 3.2, τ< 1,η(0) = τ< 1,η = τC1,η ; so G is basic τC1,η × σ-open. We prove the statement by induction on η. Let first η = 1; then A∩G is a nonempty Σ0 1(τC1,1 × σ) set, Q1,1 = V1,1(0) = C1(0) \ P1(0) and P1,1 = W1,1(1) = P1(0). That is P1,1 is one point hence Q1,1 is a τC1,1-dense τC1,1-open set. Thus A ∩ G ∩ (Q1,1 × Y ) is a nonempty τC1,1 × σ-open set so it is of τC1,1 × σ-second category, i.e. A ∩ (Q1,1 × Y ) is of τC1,1 × σ-second category in G ∩ (Q1,1 × Y ). By T. Mátrai 533 Lemma 3.4.3 with γ = 0, (τ1,1 × σ) \|Q1,1×Y = ( τC1,1 × σ ) \|Q1,1×Y so the statement follows. Suppose now that 1 < η < ω1 and the statement holds for every η < η and nonempty Polish space (Y , σ). Let A = Dη((Aα)α<η) with Σ0 1(τC1,η × σ) sets Aα (α < η) satisfying Aβ ⊆ Aα (β ≤ α < η). We have that W1,η(η)×Y ⊆ P1,η×Y is τ1,η×σ-open. Since G∩(W1,η(η) × Y ) 6= ∅, A is τ1,η×σ-residual in G∩(W1,η(η) × Y ). Thus there is a minimal α such that the parity of α and η are different and Aα ∩G∩ (W1,η(η) × Y ) is of τ1,η ×σ-second category. Since Aα and G are τC1,η ×σ-open, Lemma 3.5 with H = Aα∩G gives that Aα∩G∩ (V1,η(α) × Y ) is nonempty hence of τ1,η×σ-second category. Now the parity of α and η differ so V1,η(α)×Y ⊆ Q1,η × Y . That is if Aα \ ⋃ β<α Aβ is also of τ1,η × σ-second category in G ∩ (V1,η(α) × Y ) then we are done. Suppose that this is not the case. Then there is a β < α and a basic τ1,η × σ-open set G0 ⊆ Aα ∩ G such that G0 ∩ (V1,η(α) × Y ) 6= ∅, Aβ is τ1,η × σ-residual in G0 ∩ (V1,η(α) × Y ) and the parity of β and α differ, that is parity of β and η coincide. Since Aα, Aβ and G are Σ0 1(τC1,η × σ) we can assume that G0 is in fact basic τC1,η ×σ-open, G0 ⊆ Aβ ∩Aα ∩G and P1(β) /∈ PrC1(β)(G0). Set η = β, Y = ( ∏ β≤γ<η C1(γ) ) × Y, σ = ( ∏ β≤γ<η τC1(γ) ) × σ, A = Dη((Aγ)γ<η) and G = G0. It is clear that G is a basic τC1,η ×σ-open set hence G is basic τ< 1,η × σ-open, as well. Since β < α implies V1,η(α) × Y ⊆ W1,η(β) × Y ⊆ W1,η(η) × Y , we have G ∩ (W1,η(η) × Y ) 6= ∅. We show that A is τ1,η × σ-residual in G ∩ (P1,η × Y ). For this we have to prove that A is τ1,η × σ-residual in G ∩ (V1,η(γ) × Y ) whenever γ ≤ η has parity equal to the parity of η. Now G ⊆ G implies that A is τ1,η × σ-residual in G ∩ (P1,η × Y ), that is A is τ1,η × σ-residual in G ∩ (V1,η(γ) × Y ) for every γ ≤ η with parity equal to the parity of η. Since the parities of η and η are equal, P1(η) /∈ PrC1(η)(G) implies G ∩ (V1,η(γ) × Y ) = G ∩ (V1,η(γ) × Y ) (γ ≤ η) and by Lemma 3.4.3, (τ1,η × σ)\|G∩(V1,η(γ)×Y ) = (τC1,η × σ)\|G∩(V1,η(γ)×Y ) = 534 Topological aspects of Hurewicz tests... = (τC1,η × σ)\|G∩(V1,η(γ)×Y ) = (τC1,η × σ)\|G∩(V1,η(γ)×Y ) = (τ1,η × σ)\|G∩(V1,η(γ)×Y ) (γ ≤ η) we conclude that A is τ1,η ×σ-residual in G∩ (P1,η ×Y ). Now G ⊆ Aβ so Aβ is also τ1,η × σ-residual in G∩ (P1,η × Y ). Since the parities of β and η are equal this is possible only if A = Dβ((Aγ)γ<β) is τ1,η × σ-residual in G ∩ (P1,η × Y ). So the induction hypothesis can be applied for η < η, the Polish space (Y , σ), the Dη(Σ 0 1(τC1,η × σ)) set A and the basic τ< 1,η × σ-open set G satisfying G ∩ (W1,η(η) × Y ) 6= ∅. We get that A is of τ1,η × σ-second category in G ∩ (Q1,η × Y ). Since A = Dβ((Aγ)γ<β) ⊆ A, this means that A is of τ1,η × σ-second category in Q1,η × Y . We have V1,η(γ) × Y = V1,η(γ) × Y (γ < η) so by Lemma 3.4.3, (τ1,η × σ)\|V1,η(γ)×Y = (τC1,η × σ)\|V1,η(γ)×Y = (τC1,η × σ)\|V1,η(γ)×Y = (τC1,η × σ)\|V1,η(γ)×Y = (τ1,η × σ)\|V1,η(γ)×Y (γ < η). (4.7) Since β and η have the same parity, Q1,η × Y = Q1,β × ∏ β≤γ<η C1(γ) × Y = ⋃ {V1,η(α) : α < β, α is odd ↔ β is even} ⊆ Q1,η × Y. So A is of τ1,η × σ-second category in Q1,η × Y , which completes the proof. 4.2. The η = 1 case Similarly to the ξ = 1 case, for 1 < ξ < ω1 the proof of Theorem 1.3.3 goes by induction on η. In this section we prove the first step of the inductive argument, namely the η = 1 case. Theorem 4.2. Fix 1 ≤ ξ < ω1. Let (Y, σ) be a nonempty Polish space, G ⊆ Cξ × Y be a basic τ< Pξ × σ-open set. 1. If ξ ≥ 2 and ϑ < ξ, A ⊆ Cξ × Y is Π0 ϑ(τCξ × σ) and A ∩ (Pξ × Y ) is ( τPξ \|Pξ ) × σ-residual in G ∩ (Pξ × Y ) then A is τPξ × σ-residual in G ∩ (Hξ(ϑ) × Y ). T. Mátrai 535 2. If for a set W ∈ Σ0 ξ(τCξ ×σ), W ∩ (Pξ × Y ) is ( τPξ \|Pξ ) ×σ-residual in G ∩ (Pξ × Y ), then W is τPξ × σ-residual in a τPξ × σ-open set H ⊆ Cξ × Y satisfying G ∩ (Pξ × Y ) ⊆ clτPξ ×σ(H ∩ (Pξ × Y )). The same result holds if (Cξ, τCξ ), Pξ, τPξ , τ< Pξ and Hξ(ϑ) are replaced by (Cm ξ , τCm ξ ), Pm ξ , τP m ξ , τ< P m ξ and Hm ξ (ϑ) (m < ω). Proof. First observe that if ξ is a limit ordinal and m < ω then by taking ϑ′ i = ϑm+i the sequence obtained satisfies ϑ′ i → ξ so the statements for (Cm ξ , τCm ξ ), Pm ξ , τP m ξ , τ< P m ξ and Hm ξ (ϑ) follow from the special m = 0 case. Note also that for ξ ≥ 2 if G ⊆ Cξ × Y is a nonempty basic τ< Pξ × σ- open set then G ∩ (Pξ × Y ) 6= ∅ follows from Lemma 3.1.4. If we have proved 1 for a 2 ≤ ξ < ω1 then 2 is automatic for ξ, as follows. Let ϑi → ξ and write W = ⋃ i<ω Ai where Ai is Π0 ϑi (τCξ × σ). Suppose that W ∩ (Pξ × Y ) is ( τPξ \|Pξ ) ×σ-residual in G∩ (Pξ × Y ). For every i < ω let Hi denote the maximal τPξ × σ-open set in which Ai is τPξ × σ-residual. Now Pξ ⊆ Hξ(ϑ) (ϑ < ξ) by Lemma 3.2 so by 1 the τPξ × σ-open set H = ⋃ i<ω Hi meets G′ ∩ (Pξ × Y ) for every basic τ< Pξ × σ-open set G′ intersecting G ∩ (Pξ × Y ), which proves 2. So we need only to prove 1. We do this by induction on ξ, namely we prove 2 for ξ = 1 and then we prove 1 for a fixed 1 < ξ < ω1 by assuming that 2 holds for every η < ξ. For 2 if ξ = 1, H = W can be chosen by the Baire Category Theorem. Let now ξ ≥ 2 and suppose that 2 holds for every η < ξ and Polish space (Y, σ), no matter how we have fixed ϑi → η for a limit η < ξ. Consider a Π0 ϑ(τCξ × σ) set A ⊆ Cξ × Y for a ϑ < ξ and suppose that A∩(Pξ × Y ) is ( τPξ \|Pξ ) ×σ-residual in G∩(Pξ × Y ) for the basic τ< Pξ ×σ- open set G. By Lemma 3.1.9 the topologies τPξ \|Pξ and τ< Pξ \|Pξ coincide so A ∩ G ∩ (Pξ × Y ) is also τ< Pξ \|Pξ × σ-residual in G ∩ (Pξ × Y ). But by Lemma 3.1.4, Pξ is a τ< Pξ -residual subset of Cξ, so A ∩ G is τ< Pξ × σ-residual in G. (4.8) Suppose that A is not τPξ ×σ-residual in G∩(Hξ(ϑ) × Y ), that is A∩G̃ is τPξ × σ-meager for some basic τPξ × σ-open set G̃ ⊆ G∩ (Hξ(ϑ) × Y ); by passing to a basic τPξ × σ-open subset we can assume that G̃ is not τ< Pξ × σ-open. By Lemma 3.1.7, there is a basic τ< Pξ × σ-open set G0 and a unique J < ω such that G̃ = G0 ∩ (Uξ,J × Y ). By passing to a basic 536 Topological aspects of Hurewicz tests... τ< Pξ × σ-open subset G̃0 ⊆ G0 we can assume that G̃0 ∩ (Uξ,J × Y ) = G̃0 ∩ (( ∏ i<J Cϑi ) ×PϑJ × ( ∏ J+1≤i<ω Cϑi ) × Y ) 6= ∅. (4.9) Note that G̃ ⊆ G ∩ (Hξ(ϑ) × Y ) implies ϑ ≤ ϑJ and G̃0 ⊆ G. To summarize, we obtained that A ∩ G̃0 ∩ (Uξ,J × Y ) is τPξ × σ-meager in G̃0 ∩ (Uξ,J × Y ) . (4.10) Set ξ = ϑJ , Y = ( ∏ i<J Cϑi × ∏ J+1≤i<ω Cϑi ) × Y, σ = ( ∏ i<J τPϑi × ∏ J+1≤i<ω τPϑi ) × σ W = (Cξ × Y ) \ A ⊆ Cξ × Y , G = G̃0 ⊆ Cξ × Y . The space (Y , σ) is clearly Polish, G is a basic τPξ × σ-open subset of Cξ × Y . Since G ∩ (Pξ × Y ) 6= ∅, Lemma 3.1.6 implies that G is basic τ< Pξ × σ-open. By ϑ ≤ ξ, W is a Σ0 ξ(τCξ × σ) set. By (4.9), G ∩ (Pξ × Y ) = G̃0 ∩ (Uξ,J × Y ) thus by Lemma 3.1.8, (τ< Pξ × σ)\|G∩(Pξ×Y ) = (τPξ × σ)\|G̃0∩(Uξ,J×Y ). So we have (τPξ × σ)\|G∩(Pξ×Y ) = (τ< Pξ × σ)\|G∩(Pξ×Y ) = (τPξ × σ)\|G̃0∩(Uξ,J×Y ). Thus by (4.10), W is (τPξ \|Pξ ) × σ-residual in G ∩ (Pξ × Y ). So by the induction hypothesis W is τPξ × σ-residual in some τPξ × σ-open set H ⊆ Cξ×Y such that G∩(Pξ×Y ) ⊆ clτPξ ×σ(H∩(Pξ×Y )); in particular, H ∩ G 6= ∅. Let H = H ⊆ Cξ × Y . Since τPξ = PrCξ (τ< Pξ ) by definition, we have H ∩ G̃0 6= ∅ and A ∩ H ∩ G̃0 is τ< Pξ × σ-meager in H ∩ G̃0 ⊆ G. This contradicts (4.8) so the proof is complete. Theorem 4.2.2 is the key for the proof of the 1 < ξ < ω1, η = 1 case of Theorem 1.3.3, that we prove in Corollary 4.1.1 in the usual more general form. T. Mátrai 537 Corollary 4.1. Let 1 < ξ < ω1 be fixed and let (Y, σ) be a nonempty Polish space. Consider a D1(Σ 0 ξ(τCξ,1 × σ)) set A ⊆ Cξ,1 × Y . Let G ⊆ Cξ,1 × Y be a basic τ< ξ,1 × σ-open set. 1. If A ∩ (Pξ,1 × Y ) is τξ,1 × σ-residual in G ∩ (Pξ,1 × Y ) then A ∩ (Qξ,1 × Y ) is of τξ,1 × σ-second category in G ∩ (Qξ,1 × Y ). 2. If ϑ < ξ, A is Π0 ϑ(τCξ,1 × σ) and A∩ (Pξ,1 × Y ) is τξ,1 × σ-residual in G∩ (Pξ,1 × Y ) then A is τξ,1 × σ-residual in G∩ (Hξ,1(ϑ) × Y ). Proof. In advance, observe that by Definition 3.2 we have D1(Σ 0 ξ(τCξ,1 ))= Σ0 ξ(τCξ,1 ), (Cξ,1, τC ξ,1 ) = (C ρ1(0) ξ , τ C ρ1(0) ξ ), Qξ,1 = Vξ,1(0) = C ρ1(0) ξ \P ρ1(0) ξ , Pξ,1 = Vξ,1(1) = Wξ,1(1) = P ρ1(0) ξ , τ< ξ,1 = τ< P ρ1(0) ξ , τξ,1 = τ P ρ1(0) ξ [{P ρ1(0) ξ }] and Hξ,1(ϑ) = H ρ1(0) ξ (ϑ) (ϑ < ξ). For 1, if A ∩ (Pξ,1 × Y ) is τξ,1 × σ-residual in G ∩ (Pξ,1 × Y ) then A ∩ (P ρ1(0) ξ × Y ) is (τ P ρ1(0) ξ \| P ρ1(0) ξ ) × σ-residual in G ∩ (P ρ1(0) ξ × Y ). So by Theorem 4.2.2, A is of τ P ρ1(0) ξ ×σ-second category in G. But P ρ1(0) ξ is τ P ρ1(0) ξ -nowhere dense by Lemma 3.1.3 so G \ (P ρ1(0) ξ × Y ) 6= ∅ and A is of τ P ρ1(0) ξ × σ-second category G \ (P ρ1(0) ξ × Y ). Now G \ (P ρ1(0) ξ × Y ) = G ∩ (Qξ,1 × Y ) and τξ,1\|Qξ,1 = τ P ρ1(0) ξ \|Qξ,1 so we obtained that A is of τξ,1 × σ-second category in G ∩ (Qξ,1 × Y ), as stated. For 2, by repeating the previous argument Theorem 4.2.1 implies that A is τ P ρ1(0) ξ ×σ-residual in G∩(H ρ1(0) ξ (ϑ)×Y ). Since τξ,1\|Pξ,1 = τ P ρ1(0) ξ \|Pξ,1 , τξ,1\|Qξ,1 = τ P ρ1(0) ξ \|Qξ,1 and A is τξ,1 × σ-residual in G ∩ (Pξ,1 × Y ) by assumption we conclude that A is τξ,1 × σ-residual in G∩ (Hξ,1(ϑ) × Y ), which completes the proof. In the following corollary we show that Wξ,η(η) with the topology τ< ξ,η(0) exhibits the same feature of dichotomy as Pξ with τPξ . Corollary 4.2. Let 1 < ξ, η < ω1, (Y, σ) be a nonempty Polish space and let G be a nonempty basic τ< ξ,η × σ-open set. 1. If ϑ < ξ, A ⊆ Cξ,η ×Y is a Π0 ϑ(τCξ,η ×σ) set and A∩(Wξ,η(η) × Y ) is (τ< ξ,η(0) × σ)\|Wξ,η(η)×Y -residual in G ∩ (Wξ,η(η) × Y ) then A is τ< ξ,η(0) × σ-residual in G ∩ (Hξ,η(ϑ) × Y ). 2. If B ⊆ Cξ,η × Y is a Σ0 ξ(τCξ,η × σ) set and B ∩ (Wξ,η(η) × Y ) is of (τ< ξ,η(0) × σ)\|Wξ,η(η)×Y -second category in G ∩ (Wξ,η(η) × Y ) 538 Topological aspects of Hurewicz tests... then B is of τ< ξ,η(0) × σ-second category in G. 3. If R ⊆ Cξ,η × Y is a Π0 ξ(τCξ,η × σ) set such that R is τ< ξ,η(0) × σ- residual in G then R is τ< ξ,η × σ-residual in G. Proof. First we show that if ξ′ ≤ ξ and 1 holds for Π0 ϑ(τCξ,η × σ) sets A whenever ϑ < ξ′ then 2 and 3 hold for Σ0 ξ′(τCξ,η ×σ) sets and Π0 ξ′(τCξ,η ×σ) sets, respectively. Fix ϑi → ξ′ and let B ⊆ Cξ,η × Y be a Σ0 ξ′(τCξ,η × σ) set such that B ∩ (Wξ,η(η) × Y ) is of (τ< ξ,η(0) × σ)\|Wξ,η(η)×Y -second category in G∩ (Wξ,η(η) × Y ). Let B = ⋃ i<ω Bi where Bi is Π0 ϑi (τCξ,η × σ) (i < ω). Then for some i < ω and basic τ< ξ,η(0) × σ-open set G0 ⊆ G we have G0∩ (Wξ,η(η) × Y ) 6= ∅ and Bi∩ (Wξ,η(η) × Y ) is (τ< ξ,η(0)×σ)\|Wξ,η(η)×Y - residual in G0∩ (Wξ,η(η) × Y ). By Lemma 3.4.5 there is a basic τ< ξ,η ×σ- open set G′ satisfying G′ ∩ (Wξ,η(η) × Y ) = G0 ∩ (Wξ,η(η) × Y ). Since G0 ⊆ G we can assume that G′ ⊆ G. Then the conditions of 1 are satis- fied and we get that Bi is τ< ξ,η(0)× σ-residual in G′ ∩ (Hξ,η(ϑi) × Y ). By Lemma 3.7, Hξ,η(ϑi) is τ< ξ,η-dense τ< ξ,η-open, in particular G′∩(Hξ,η(ϑi)× Y ) 6= ∅. Hence Hξ,η(ϑi) is τ< ξ,η(0)-open as well and G′∩ (Hξ,η(ϑi) × Y ) ⊆ G, that is B is of τ< ξ,η(0) × σ-second category in G, indeed. For 3 let R ⊆ Cξ,η×Y be a Π0 ξ′(τCξ,η ×σ) set such that R is τ< ξ,η(0)×σ- residual in G. Suppose that R is not τ< ξ,η × σ-residual in G, that is for some nonempty basic τ< ξ,η × σ-open set G′ ⊆ G we have that R is τ< ξ,η × σ-meager in G′. By Lemma 3.6.3, G′ ∩ (Wξ,η(η) × Y ) 6= ∅ and Wξ,η(η) × Y is τ< ξ,η × σ-residual in Cξ,η(η) × Y . Thus B = Cξ,η \ R is a Σ0 ξ′(τCξ,η ×σ) set and B∩(Wξ,η(η) × Y ) is (τ< ξ,η×σ)\|Wξ,η(η)×Y -residual in the nonempty G′∩(Wξ,η(η) × Y ). By Lemma 3.4.6 with γ = 0 and γ = n the topologies (τ< ξ,η × σ)\|Wξ,η(η)×Y and (τ< ξ,η(0)× σ)\|Wξ,η(η)×Y coincide so B∩(Wξ,η(η) × Y ) is (τ< ξ,η(0)×σ)\|Wξ,η(η)×Y -residual in G′∩(Wξ,η(η) × Y ). Hence by 2, Cξ,η \ R = B is of τ< ξ,η(0) × σ-second category in G′, which is a contradiction. For every fixed 1 < ξ < ω1 we prove 1 by induction on ϑ and η: we show 3 for ξ = 1, 0 < η < ω1 and we show that if ϑ < ξ and 3 holds for Π0 ϑ(τCξ,η⋆ ×σ) sets R whenever η⋆ ≤ η then 1 holds for Π0 ϑ(τCξ,η ×σ) sets A with our η. As we have seen above this will complete the proof. For ξ = 1, statement 3 follows from τ< 1,η = τ< 1,η(0). So let ϑ < ξ and A ⊆ Cξ,η × Y be a Π0 ϑ(τCξ,η × σ) set such that A ∩ (Wξ,η(η) × Y ) is (τ< ξ,η(0) × σ)\|Wξ,η(η)×Y -residual in G ∩ (Wξ,η(η) × Y ). Suppose that A is not τ< ξ,η(0) × σ-residual in G ∩ (Hξ,η(ϑ) × Y ) that is for some ϑ⋆ < ϑ, Π0 ϑ⋆(τCξ,η ×σ) set A⋆ ⊆ (Cξ,η × Y )\A and nonempty T. Mátrai 539 basic τ< ξ,η(0)×σ-open set G′ ⊆ G∩(Hξ,η(ϑ) × Y ) we have A⋆ is τ< ξ,η(0)×σ- residual in G′. Let I, I< ⊆ η be disjoint finite sets such that G′ = ( ∏ α∈I G′(α) × ∏ α∈I< G′(α) × ∏ α∈η\(I∪I<) Cξ(α) ) × G′(Y ) where G′(α) is proper basic τPξ(α)-open for α ∈ I, G′(α) is basic τ< Pξ(α)- open for α ∈ I< and G′(Y ) is basic σ-open. Let G⋆ be the basic τ< ξ,η ×σ- open set defined by G⋆ = ( ∏ α<η G⋆(α)) × G⋆(Y ) where G′(α) = G⋆(α) (α ∈ I<), G⋆(α) = Cξ(α) (α ∈ η \ ( I ∪ I< ) ) and G⋆(Y ) = G′(Y ), while for α ∈ I let G′(α) = G⋆(α) ∩ Uξ,n(α) with the unique n(α) < ω of Lemma 3.1.7. Since G′ ⊆ G, we can assume that G⋆(α) ⊆ PrCξ(α)(G) (α ∈ I). Thus G⋆ ⊆ G and so by Lemma 3.6.3 A is τ< ξ,η ×σ-residual in G⋆. Observe that G′ ⊆ Hξ,η(ϑ)×Y implies ϑ ≤ ϑn(α) (α ∈ I) and we have G′ ∩ G⋆ ∩ (Hξ,η(ϑ) × Y ) = G′ 6= ∅. Set Y ⋆ = ( ∏ α∈I Cξ(α) ) × Y, σ⋆ = ( ∏ α∈I τPξ(α) ) × σ, τ⋆ = ∏ α∈η\I τ< Pξ(α). Then G′ and G⋆ are both nonempty basic τ⋆ × σ⋆-open sets. We show that A is τ⋆×σ⋆-residual in G⋆∩(Hξ,η(ϑ) × Y ) and A⋆ is τ⋆×σ⋆-residual in G′; since G′ ∩ G⋆ ∩ (Hξ,η(ϑ) × Y ) 6= ∅ this contradicts A ∩ A⋆ = ∅ so the proof will be complete. First we prove that A is τ⋆ × σ⋆-residual in G⋆ ∩ (Hξ,η(ϑ) × Y ). Let I = {ηi : i < \|I\|} and for every i < \|I\| set Yi = ( ∏ α∈{ηj : j<i} Cξ(α) × ∏ α∈η\{ηj : j≤i} Cξ(α) ) × Y, σi = ( ∏ α∈{ηj : j<i} τPξ(α) × ∏ α∈η\{ηj : j≤i} τ< Pξ(α) ) × σ, Hi = ( ∏ α∈{ηj : j<i} PrCξ(α)(Hξ,η(ϑ)) × ∏ α∈{ηj : i<j<\|I\|} Cξ(α) × ∏ α∈η\{ηj : j<\|I\|} PrCξ(α)(Hξ,η(ϑ)) ) × Y, 540 Topological aspects of Hurewicz tests... Gi = ∏ α∈η\{ηi} G⋆(α). Then (Yi, σi) is a Polish space, Gi ⊆ Yi is a basic σi-open set so G⋆ is a basic τ< Pξ(ηi) × σi-open set (i < \|I\|). We show by induction on i that A is τPξ(ηi) × σi-residual in G⋆ ∩ (PrCξ(ηi)(Hξ,η(ϑ)) × Hi) ⊆ Cξ(ηi) × Yi = Cξ,η × Y (i < \|I\|). Let first i = 0. Since A is τ< ξ,η×σ residual in G⋆ and τ< ξ,η×σ = τ< Pξ(η0)× σ0, by Lemma 3.1.9 we have that A∩(Pξ(η0) × Y0) is (τPξ(η0)\|Pξ(η0))×σ0- residual in G⋆ ∩ (Pξ(η0) × Y0). We apply Theorem 4.2.1 with the pair Pξ(η0), τPξ(η0), the Polish space (Y0, σ0) for the Π0 ϑ(τCξ(η0)×σ0) set A and basic τ< Pξ(η0) × σ0-open set G⋆. We get that A is τPξ(η0) × σ0-residual in G⋆∩(Hξ(ϑ)×Y0) if ξ is successor and in G⋆∩(H ρη(η0) ξ (ϑ)×Y0) if ξ is limit. So by Lemma 3.7 A is τPξ(η0)×σ0-residual in G⋆∩(PrCξ(η0)(Hξ,η(ϑ))×H0), as well, which proves the i = 0 case. Suppose now that the statement holds for some i < \|I\| − 1; we prove it for i + 1. Observe that τPξ(ηi) × σi = τ< Pξ(ηi+1) × σi+1 and PrCξ(ηi)(Hξ,η(ϑ))×Hi = Cξ(ηi+1)×Hi+1 so by the induction hypothesis we have A is τ< Pξ(ηi+1)×σi+1-residual in G⋆∩(Cξ(ηi+1) × Hi+1); hence by Lemma 3.1.9, A ∩ (Pξ(ηi+1) × Yi+1) is (τPξ(ηi+1)\|Pξ(ηi+1)) × σi+1-residual in G⋆ ∩ (Pξ(ηi+1) × Hi+1). So we can apply Theorem 4.2.1 with the pair Pξ(ηi+1), τPξ(ηi+1), the Polish space (Yi+1, σi+1) for the Π0 ϑ(τCξ(ηi+1) × σi+1) set A and basic τ< Pξ(ηi+1)×σi+1-open set G⋆∩(Cξ(ηi+1) × Hi+1). We get that A is τPξ(ηi+1)×σi+1-residual in G⋆∩(PrCξ(ηi+1)(Hξ,η(ϑ))×Hi+1), as stated. Since τPξ(η\|I\|−1) × σ\|I\|−1 = τ⋆ × σ⋆ and PrCξ(η\|I\|−1)(Hξ,η(ϑ)) × H\|I\|−1 = Hξ,η(ϑ) we conclude that A is τ⋆×σ⋆-residual in G⋆∩(Hξ,η(ϑ)× Y ). It remains to prove that A⋆ is τ⋆ × σ⋆-residual in G′. Let C⋆ = ∏ α∈η\I Cξ(α) and η⋆ = η \ I. Using the convention of Remark 3.1, (C⋆, τ⋆) = (Cξ,η⋆ , τ< ξ,η⋆) and τ< ξ,η⋆(0)× σ⋆ = τ< ξ,η(0)× σ. So by the defini- tion of G′, A⋆ is τ< ξ,η⋆(0)×σ⋆-residual in G′. By our assumption, 3 holds for the Π0 ϑ⋆(τCξ,η⋆ × σ⋆) set A⋆ ⊆ Cξ,η⋆ × Y ⋆ and the nonempty basic τ< ξ,η⋆ × σ⋆-open set G′, and we get that A⋆ is τ< ξ,η⋆ × σ⋆-residual hence τ⋆ × σ⋆-residual in G′. The proof is complete. 4.3. The general case This final section contains the proof of Theorem 1.3.3 for 1 < ξ < ω1. We start with a claim which slightly strengthens Corollary 4.2.2. T. Mátrai 541 Proposition 4.2. Let 1 < ξ, η < ω1, (Y, σ) be a nonempty Polish space and G ⊆ Cξ,η × Y be a nonempty τ< ξ,η × σ-open set. If A ⊆ Cξ,η × Y is Σ0 ξ(τCξ,η × σ) and A ∩ (Wξ,η(η) × Y ) is of (τξ,η × σ) \|Wξ,η(η)×Y -second category in G∩(Wξ,η(η) × Y ) then there is a nonempty basic τ< ξ,η×σ-open set G0 ⊆ G such that A is τξ,η × σ-residual in G0. Proof. Before starting the proof, observe that by Lemma 3.6.3, G ∩ (Wξ,η(η) × Y ) 6= ∅ for every nonempty τ< ξ,η × σ-open set G. Let ϑi → ξ. Since A is Σ0 ξ(τCξ,η ×σ), there is an i < ω, a Π0 ϑi (τCξ,η ×σ) set B ⊆ A and a nonempty basic τξ,η × σ-open set G⋆ ⊆ G such that G⋆ ∩ (Wξ,η(η) × Y ) 6= ∅ and B ∩ (Wξ,η(η) × Y ) is (τξ,η × σ) \|Wξ,η(η)×Y - residual in G⋆ ∩ (Wξ,η(η) × Y ). By Lemma 3.4.5 and Lemma 3.7 there is a basic τ< ξ,η × σ-open set G0 for which G0 ∩ (Wξ,η(η) × Y ) = G⋆ ∩ (Wξ,η(η) × Y ) and G0 ⊆ Hξ,η(ϑi) × Y . Then by Lemma 3.4.6 for γ = η, B ∩ (Wξ,η(η) × Y ) is (τ< ξ,η × σ)\|Wξ,η(η)×Y -residual in G0 ∩ (Wξ,η(η) × Y ). We show that B is τξ,η ×σ-residual in G0; then by B ⊆ A, G0 fulfills the requirements. We have Cξ,η×Y = ⋃ α≤η Vξ,η(α)×Y and Vξ,η(α) is τξ,η×σ-open (α ≤ η) so we have to prove that B is τξ,η × σ-residual in G0 ∩ (Vξ,η(α) × Y ) (α ≤ η). For α = η this follows from Vξ,η(η) = Wξ,η(η); so fix some α < η. Set η̃ = η \ α, Ỹ = ( ∏ γ<α Cξ(γ) ) × Y, σ̃ = ( ∏ γ<α τ< Pξ(γ) ) × σ, G̃ = G0 and B̃ = B. Then (Ỹ , σ̃) is a nonempty Polish space and G̃ is a nonempty basic τ< ξ,η̃ × σ̃-open set. By Lemma 3.6.3, Wξ,η(η) × Y = Wξ,η̃(η̃)×Wξ,α(α)×Y is τ< ξ,η̃× σ̃-residual in Cξ,η̃× Ỹ so B̃∩(Wξ,η̃(η̃)× Ỹ ) is (τ< ξ,η̃ × σ̃)\|Wξ,η̃(η̃)×Ỹ -residual in G̃∩ (Wξ,η̃(η̃)× Ỹ ). By Lemma 3.4.6 the topologies (τξ,η̃ × σ̃)\|Wξ,η̃(η̃)×Ỹ and (τ< ξ,η̃(γ) × σ̃)\|Wξ,η̃(η̃)×Ỹ (γ ≤ η̃) all coincide so B̃ ∩ (Wξ,η̃(η̃) × Ỹ ) is (τ< ξ,η̃(0) × σ̃)\|Wξ,η̃(η̃)×Ỹ -residual and (τξ,η̃ × σ̃) \|Wξ,η̃(η̃)×Ỹ -residual in G̃ ∩ (Wξ,η̃(η̃) × Ỹ ). Thus we can apply Corollary 4.1.2 if η̃ = 1 or Corollary 4.2.1 if 1 < η̃ for B̃ and G̃ in Cξ,η̃ × Ỹ and we get that B̃ is τ< ξ,η̃(0) × σ̃-residual in G̃∩ (Hξ,η̃(ϑi)× Ỹ ). Since G0 ⊆ Hξ,η(ϑi)×Y implies G0 ⊆ (Hξ,η̃(ϑi)× Ỹ ) 542 Topological aspects of Hurewicz tests... by Lemma 3.7 and τ< ξ,η̃(0)× σ̃ = τ< ξ,η(α)×σ we get that B is τ< ξ,η(α)×σ- residual in G0. By Lemma 3.6.2, Vξ,η(α) × Y is τ< ξ,η(α) × σ-residual in Cξ,η × Y so G0 ∩ (Vξ,η(α) × Y ) 6= ∅ and B is (τ< ξ,η(α)× σ)\|Vξ,η(α)×Y -residual in G0 ∩ (Vξ,η(α) × Y ). By Lem- ma 3.4.3 with γ = α the topologies (τξ,η × σ)\|Vξ,η(α)×Y and (τ< ξ,η(α) × σ)\|Vξ,η(α)×Y coincide. So we obtained that B is τξ,η × σ-residual in G0 ∩ (Vξ,η(α) × Y ). The proof is complete. As above, in the proof of the remaining part of Theorem 1.3.3 the product structure of Definition 3.2 must be exploited. So we prove it in the following more general form. When Y is a single point we get back Theorem 1.3.3. Theorem 4.3. Let 1 < ξ, η < ω1 be fixed. Let (Y, σ) be a nonempty Polish space and consider a Dη(Σ 0 ξ(τCξ,η × σ)) set A ⊆ Cξ,η × Y . If G ⊆ Cξ,η ×Y is τ< ξ,η ×σ-open such that A∩ (Pξ,η × Y ) is τξ,η ×σ-residual in G ∩ (Pξ,η × Y ) then A ∩ (Qξ,η × Y ) is of τξ,η × σ-second category in G ∩ (Qξ,η × Y ). Proof. We prove the statement by induction on η. The η = 1 case is Corollary 4.1.1. Suppose now that 1 < η < ω1 and that the statement holds for every η < η and Polish space (Y , σ). Let A = Dη((Aα)α<η) with Σ0 ξ(τCξ,η × σ) sets Aα (α < η) satisfying Aβ ⊆ Aα (β ≤ α < η). We have that Wξ,η(η)× Y ⊆ Pξ,η × Y is τξ,η × σ-open. By Lemma 3.6.3, G ∩ (Wξ,η(η) × Y ) 6= ∅ so A is τξ,η ×σ-residual in G∩ (Wξ,η(η) × Y ). Thus there is a minimal α such that the parity of α and η are different and for some basic τξ,η × σ- open set G⋆ ⊆ G, Aα is of τξ,η × σ-second category in the nonempty G⋆ ∩ (Wξ,η(η) × Y ). Then by Lemma 3.4.5 there is a τ< ξ,η × σ-open set G′ such that G⋆ ∩ (Wξ,η(η) × Y ) = G′ ∩ (Wξ,η(η) × Y ) 6= ∅. We apply Proposition 4.2 for Aα and G′; we obtain that for some non- empty basic τ< ξ,η×σ-open set G0 ⊆ G′, Aα is τξ,η×σ-residual in G0. So in particular, Aα is τξ,η×σ-residual in G0∩(Vξ,η(α) × Y ), which is nonempty by Lemma 3.6.1. But the parity of α and η differ, so Vξ,η(α) × Y ⊆ Qξ,η × Y . That is if Aα \ ⋃ β<α Aβ is also of τξ,η × σ-second category in G0 ∩ (Vξ,η(α) × Y ) then we are done. Suppose that this is not the case. Then there is a β < α and a basic τξ,η × σ-open set G⋆ 0 ⊆ G0 such that G⋆ 0 ∩ (Vξ,η(α) × Y ) 6= ∅ and Aβ is τξ,η × σ-residual in G⋆ 0 ∩ (Vξ,η(α) × Y ), moreover the parity of β and α differ, that is the parity of β and η coincide. By Lemma 3.4.2, T. Mátrai 543 there is a basic τ< ξ,η(α) × σ-open set G′ 0 such that G⋆ 0 ∩ (Vξ,η(α) × Y ) = G′ 0 ∩ (Vξ,η(α) × Y ); and since G0 is basic τ< ξ,η × σ-open, we can assume that G′ 0 ⊆ G0. By Lemma 3.4.3 with γ = α, Aβ is τ< ξ,η(α)×σ-residual in G′ 0 ∩ (Vξ,η(α) × Y ). By passing to a subset if necessary, we assume that PrCξ(α)(G ′ 0) is proper τPξ (α)-open, that is PrCξ(α)(G ′ 0) ⊆ Cξ(α) \ Pξ(α). (4.11) Set η̃ = α, Ỹ = ( ∏ α≤γ<η Cξ(γ) ) × Y, σ̃ = ( ∏ α≤γ<η τPξ(γ) ) × σ and G̃ = G′ 0. With this setting, using (4.11), G̃ ∩ (Wξ,η̃(η̃) × Ỹ ) = G′ 0 ∩ (Vξ,η(α) × Y ) 6= ∅ (4.12) and by Lemma 3.4.6, (τξ,η̃ × σ̃) \| G̃∩(Wξ,η̃(η̃)×Ỹ ) = (τ< ξ,η̃ × σ̃)\| G̃∩(Wξ,η̃(η̃)×Ỹ ) = (τ< ξ,η̃ × σ̃)\| G′ 0∩(Vξ,η(α)×Y ) = (τ< ξ,η(α) × σ)\| G′ 0∩(Vξ,η(α)×Y ). (4.13) By (4.12) and (4.13) we have G̃ ∩ (Wξ,η̃(η̃) × Ỹ ) 6= ∅ and Aβ is (τξ,η̃×σ̃)\|Wξ,η̃(η̃)×Ỹ -residual in G̃∩(Wξ,η̃(η̃)×Ỹ ). So we can apply Propo- sition 4.2 in Cξ,η̃×Ỹ for Aβ and G̃. We get that for some nonempty basic τ< ξ,η̃ × σ̃-open set G̃0 ⊆ G̃, Aβ is τξ,η̃ × σ̃-residual in G̃0. In particular, G̃0∩(Vξ,η̃(γ)× Ỹ ) 6= ∅ (γ ≤ η̃) by Lemma 3.6.1 and Aβ is τξ,η̃× σ̃-residual in G̃0 ∩ (Vξ,η̃(γ) × Ỹ ) (γ ≤ η̃). Now G̃0 ⊆ G is τξ,η × σ-open and Vξ,η̃(γ) × Ỹ is also τξ,η × σ-open (γ ≤ η̃). So A is τξ,η × σ-residual in G̃0 ∩ (Vξ,η̃(γ) × Ỹ ) for every γ ≤ η̃ with parity different from the parity of η̃. We have G̃0 ∩ (Vξ,η(γ) × Y ) = G̃0 ∩ (Vξ,η̃(γ) × Ỹ ) 6= ∅ (γ < η̃). So by Lemma 3.4.3, (τξ,η × σ) \| G̃0∩(Vξ,η(γ)×Y ) = (τ< ξ,η(0) × σ)\| G̃0∩(Vξ,η(γ)×Y ) = (τ< ξ,η̃(0) × σ̃)\| G̃0∩(Vξ,η(γ)×Y ) = (τ< ξ,η̃(0) × σ̃)\| G̃0∩(Vξ,η̃(γ)×Ỹ ) = (τξ,η̃ × σ̃) \| G̃0∩(Vξ,η̃(γ)×Ỹ ) (γ < η̃). (4.14) We get that A is τξ,η̃×σ̃-residual in G̃0∩(Vξ,η̃(γ)×Ỹ ) for every γ < η̃ with parity different from the parity of η̃. Since Aβ is also τξ,η̃ × σ̃-residual in 544 Topological aspects of Hurewicz tests... G̃0∩ (Vξ,η̃(γ)× Ỹ ) for every γ ≤ η̃, this is possible only if Dβ((Aγ) γ<β ) is τξ,η̃× σ̃-residual in G̃0∩(Vξ,η̃(γ)× Ỹ ) for every γ < η̃ with parity different from the parity of η̃. Let H ⊆ Cξ,η be a basic τ< ξ,η(0)-open set which is nontrivial only on the Cξ(β) coordinate and PrCξ(β)(H) is proper basic τPξ(β)-open, i.e. PrCξ(β)(H) ⊆ Cξ(β) \ Pξ(β), and H ∩ G̃0 6= ∅. Set η = β, Y = ( ∏ β≤γ<α Cξ(γ) ) × Ỹ , σ = ( ∏ β≤γ<α τPξ(γ) ) × σ̃, A = Dη((Aγ) γ<η ) and G = G̃0 ∩H. Since G̃0 is basic τ< ξ,η̃ × σ̃-open, it is τ< ξ,η̃(β) × σ̃-open and so G is basic τ< ξ,η × σ-open. As above, we have G0 ∩ (Vξ,η(γ) × Y ) = G0 ∩ (Vξ,η̃(γ) × Ỹ ) 6= ∅ (γ ≤ η) so by Lemma 3.4.3, (τξ,η × σ)\| G0∩(Vξ,η(γ)×Y ) = (τ< ξ,η(0) × σ)\| G0∩(Vξ,η(γ)×Y ) = (τ< ξ,η̃(0) × σ̃)\| G0∩(Vξ,η(γ)×Y ) = (τ< ξ,η̃(0) × σ̃)\| G0∩(Vξ,η̃(γ)×Ỹ ) = (τξ,η̃ × σ̃)\| G0∩(Vξ,η̃(γ)×Ỹ ) (γ ≤ η). (4.15) Since A = Dη((Aγ) γ<η ) is τξ,η̃ × σ̃-residual in G̃0 ∩ (Vξ,η̃(γ) × Ỹ ), we get that A is τξ,η × σ-residual in G ∩ (Vξ,η(γ) × Y ) for every γ ≤ η with parity different from the parity of η̃, that is with parity equal to the parity of η. In particular, A is τξ,η×σ-residual in G∩(Pξ,η×Y ). So by the induction hypothesis A is of τξ,η × σ-second category in G ∩ (Qξ,η × Y ). Since A = Dβ((Aγ) γ<β ) ⊆ A, this means that A is of τξ,η × σ-second category in Qξ,η × Y . We have Vξ,η(γ) × Y = Vξ,η(γ) × Y (γ < η) so by Lemma 3.4.3, (τξ,η × σ)\|Vξ,η(γ)×Y = (τ< ξ,η(0) × σ)\|Vξ,η(γ)×Y = (τ< ξ,η(0) × σ)\|Vξ,η(γ)×Y = (τ< ξ,η(0) × σ)\|Vξ,η(γ)×Y = (τξ,η × σ)\|Vξ,η(γ)×Y (γ < η). (4.16) Since β and η have the same parity, T. Mátrai 545 Qξ,η × Y = Qξ,β × ∏ β≤γ<η Cξ(γ) × Y = ⋃ {Vξ,η(α) : α < β, α is odd ↔ β is even} ⊆ Qξ,η × Y. So A is of τξ,η × σ-second category in Qξ,η × Y, which completes the proof. Before proving Corollary 1.2 and Corollary 1.3 we need to show that Theorem 1.2 can be applied for our pairs (Cξ,η, Pξ,η) and (Cξ,η, Qξ,η) for 1 < ξ < ω1 and 0 < η < ω1. Lemma 4.1 (A. Louveau, J. Saint Raymond). Let 1 < ξ < ω1 and 0 < η < ω1 be fixed. Then (Cξ,η, Pξ,η) is a Hurewicz test pair for Ďη(Σ 0 ξ) and (Cξ,η, Cξ,η \ Pξ,η) is a Hurewicz test pair for Dη(Σ 0 ξ). Proof. Let first 3 ≤ ξ < ω1 and 1 ≤ η < ω1 or ξ = 2 and ω ≤ η < ω1. By Lemma 3.3.1, (Cξ,η, τCξ,η ) is homeomorphic to (C, τC). By Lemma 3.3.2, Pξ,η is Ďη(Σ 0 ξ(τCξ,η )) and Qξ,η is Dη(Σ 0 ξ(τCξ,η )) so it remains to prove that Pξ,η /∈ Dη(Σ 0 ξ(τCξ,η )) and Qξ,η /∈ Ďη(Σ 0 ξ(τCξ,η )). Suppose that Pξ,η is Dη(Σ 0 ξ(τCξ,η )). By Corollary 1.1.1 with A = Pξ,η, Pξ,η \ Pξ,η = ∅ is of τξ,η-second category, a contradiction. The same argument using Corollary 1.1.2 gives that Qξ,η /∈ Ďη(Σ 0 ξ(τCξ,η )). If ξ = 2 and 0 < η < ω the statement follows from [2, Corollary 9, p. 458] and the special choice of P2,η. This completes the proof. Proof of Corollary 1.2. Let first A ⊆ X be a Dη(Σ 0 ξ(τ)) set and sup- pose that the continuous injection ϕ : (Cξ,η, τCξ,η ) → (X, τ) satisfies ϕ(Pξ,η) ⊆ A. Then Pξ,η ⊆ ϕ−1(A) so by Corollary 1.1.1, ϕ−1(A) \Pξ,η is of τξ,η-second category, which proves 1. If A ⊆ X is Ďη(Σ 0 ξ(τ)) the same argument using Corollary 1.1.2 proves 1′. For 2 let A ⊆ X be not in Dη(Σ 0 ξ(τ)). By Lemma 4.1, (Cξ,η, Pξ,η) is a Hurewicz test pair for Ďη(Σ 0 ξ) so by applying Theorem 1.2 we get a continuous injection ϕ : (Cξ,η, τCξ,η ) → (X, τ) which satisfies ϕ(Pξ,η) = A ∩ ϕ(Cξ,η), as stated. The same argument gives 2′ so the proof is complete. Proof of Corollary 1.3. We give the proof for Γ = Dη(Σ 0 ξ), the proof for Γ = Ďη(Σ 0 ξ) is the same. By Lemma 4.1 we can apply Theo- rem 1.2 to have a continuous injection ϕ : (Cξ,η, τCξ,η ) → (X, τ) satisfying ϕ(Cξ,η \ Pξ,η) = A ∩ ϕ(Cξ,η). Fix a countable base B in the Polish space ( Pξ,η, τξ,η\|Pξ,η ) . Set ΛB = { α < λ : ϕ−1(Aα) is τξ,η\|Pξ,η -residual in B } (B ∈ B). 546 Topological aspects of Hurewicz tests... By Corollary 1.1.2, ϕ−1(Aα) is of τξ,η-second category in Pξ,η thus we have λ = ⋃ B∈B ΛB. So Λ = ΛB is stationary for some B ∈ B. Since in our model the union of λ meager sets is meager in Polish spaces, ⋂ α∈ΛB ϕ−1(Aα) is τξ,η-residual in B, in particular B∩ ⋂ α∈ΛB ϕ−1(Aα) 6= ∅. Now ϕ is one-to-one hence this implies (X \ A)∩ ⋂ α∈Λ Aα 6= ∅, so the proof is complete. References [1] A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer-Verlag, 1994. [2] A. Louveau, J. Saint Raymond, Borel Classes and Closed Games: Wadge-type and Hurewicz-type Results // Trans. Amer. Math. Soc., 304 (1987), N 2, 431–467. [3] A. Louveau, J. Saint-Raymond, The strength of Borel Wadge determinacy, Cabal Seminar 81–85, Lecture Notes in Math., 1333, Springer, Berlin, 1988, 1–30. [4] T. Mátrai, Hurewicz tests: separating and reducing analytic sets the conscious way, PhD Thesis, Central European University, 2005. [5] S. Solecki, Covering analytic sets by families of closed sets // J. Symbolic Logic 59 (1994), N 3, 1022–1031. [6] S. Solecki, Decomposing Borel sets and functions and the structure of Baire class 1 functions // J. Amer. Math. Soc. 11 (1998), N 3, 521–550. Contact information Tamás Mátrai Department of Mathematics and its Applications Central European University Nádor u. 6, 1051 Budapest, Hungary E-Mail: matrait@renyi.hu

Topological aspects of Hurewicz tests for the difference hierarchy

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