PERFECT SUBSETS OF GENERALIZED BAIRE SPACES AND LONGGAMES

PHILIPP SCHLICHT

Abstract. We prove that in analogy to Solovay’s theorem for subsets of the Baire space ωω,the perfect set property for all subsets A of κκ definable from elements of Ordκ is consistentrelative to an inaccessible cardinal, where κ is an uncountable regular cardinal with κ

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This is proved in Theorem 2.17. Assuming that there is a proper class of inaccessible cardinals,this result can be extended as follows.

Theorem 1.3. It is consistent relative to a proper class of inaccessible cardinals that for alluncountable regular cardinal κ with κ ω. Then theclub filter Clubκ is not a κ-Baire subset of κ2.

This is generalized to subsets of κκ in [FHK14] as follows. If S is a subset of κ, let ClubSκdenote the set of x ∈ κκ such that there is a club C in κ with x(i) ∈ S for all i ∈ C.

Example 1.6. ([FHK14, Theorem 3.10]) Suppose that κ is an uncountable cardinal with κ

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(1) The perfect set property holds for all subsets A of κκ definable over (Hκ+ , ∈) with parametersin Hκ+ .

(2) The game Gκ(A) is determined for all subsets A of κκ definable over (Hκ+ , ∈) with parametersin Hκ+ .

The topic of this paper is motivated by work of Solovay [Sol70], Mekler and Väänänen [MV93],Donder and Kovachev [Kov09], Friedman, Hyttinen and Kulikov [FHK14], Lücke, Motto Ros andSchlicht [LMRS16], Laguzzi [Lag15] and Friedman, Khomskii and Kulikov [FKK16]. We refer to[FHK14, Lüc12] for definitions in generalized descriptive set theory.

In the remainder of this section, we will collect several additional definitions and facts abouttrees and forcings. In Section 2, we prove the consistency of the perfect set property for defin-able subsets of κκ and related results for definable functions on κκ. In Section 3, we prove theconsistency of the almost Baire property for definable subsets of κκ and related results aboutdefinable functions on κκ. In Section 4, we prove variants of the conclusions of the main resultsfrom Iterated Resurrection Axioms.

1.1. Trees and perfect sets. We always assume that κ is an uncountable regular cardinal withκ

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1.2. Forcings. We state some basic properties of forcings used below. A forcing P is non-trivialif no P-generic filter is in the ground model.

Definition 1.14. (a) Add(κ,1) is the forcing with conditions p∶α → κ for α < κ, ordered byreverse inclusion.

(b) Add∗(κ,1) = {p ∈ Add(κ,1) ∣ dom(p) = ∅ or ∃α < κ dom(p) = α + 1} is ordered by reverseinclusion.

Lemma 1.15. (see [Fuc09, Lemma 2.2]) Suppose that κ is an uncountable cardinal with κ

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Proof. The set

{(p, q̌) ∣ p, q ∈ Add(κ,1), l(p) = l(q), p ⊩Add(κ,1) q̌ ∈ Add∗(κ,1)}is a < κ-closed dense subset of Add(κ,1). �

Remark 1.21. An analogous proof to [Sol70] shows that after forcing with Col(κ,< λ) for a regularcardinal κ and an inaccessible cardinal λ > κ, all Σ11 subsets of κκ have the perfect set property.The proof does not work for Π11 subsets of

κκ because of Lemma 1.20.

2. The perfect set property

2.1. Perfect set games. The perfect set property is characterized by the perfect set game.

Definition 2.1. The perfect set game Gκ(A) of length κ for a subset A of κ2 is defined as follows.The first (even) player, player I, plays in all even rounds. The second (odd) player, player II, playsin all odd rounds. Together, they play a strictly increasing sequence s⃗ = ⟨sα ∣ α < κ⟩ with sα ∈

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Definition 2.6. Let P denote the set of pairs (t, s) such that(a) t ⊆

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there are ζ, η < κ with ζ ≠ η and r⌢⟨ζ⟩, r⌢⟨η⟩ are the unique successors of r in t. Then p ⊩P ḃ(β̌) ∈ ť,contradicting the assumption. �

Let P∗ denote the set of conditions p ∈ P such that ht(t) = l(bp) is a limit.

Lemma 2.11. P∗ is dense in P.

Proof. Suppose that p ∈ P. Let p0 = p = (t0, s0). Let pn+1 = (tn+1, sn+1) be a condition withpn+1 ≤ pn and ht(tn) < ht(tn+1) that decides ḃ ↾ ht(tn). Let t = ⋃n∈ω tn and s = ⋃n∈ω sn. Letq = (t, s). Then ht(q) is a limit and ht(q) ≤ l(bq). have l(bq) ≤ ht(q) by Lemma 2.10. �

We expand P as follows to determine the quotient forcing for a branch b in [TG], where G is aP-generic filter over V . Let

Q = {(bp, p) ∣ p ∈ P∗ or p = 1P}.Let (u, p) ≤ (v, q) if v ⊆ u and p ≤P q for (u, p), (v, q) ∈ P. Let Q0 = {(bp,∅) ∣ p ∈ P∗} andQ1 = {(bp, p) ∣ p ∈ P∗}. Then Q = Q0 ∪Q1, Q0 ∩Q1 = {1Q} and Q1 is dense in Q. Moreover themaps f ∶P∗ → Q1, e(p) = (bp, p) and g∶Add(κ,1)→ Q0, g(p) = (p,1P) are isomorphisms.

Hence P, Q are forcing equivalent.

Lemma 2.12. The map π = πQ,Q0 ∶Q→ Q0, π(bp, p) = p is a forcing projection.

Proof. Suppose that u = (bp, p) ∈ Q1 and v = (bq,1P) ∈ Q0 with bp ⊆ bq. We show that u, v arecompatible in Q. This implies that π is a forcing projection. Suppose that p = (t, s). Since p ∈ P∗,l(p) is a limit and bp is cofinal in t. Let t̄ = t ∪ {bq ↾ α ∣ l(bp) ≤ α < l(bq)}, s̄ = s ∪ {bq ↾ α ∣ l(bp) ≤α < l(bq)} and p̄ = (t̄, s̄). Then p̄ ≤ p and bp̄ = bq by Lemma 2.10. Suppose that r ≤ p̄ with r ∈ P∗.Let w = (br, r) ∈ Q1. Since r ≤ p̄ ≤ p and bp ⊆ br, we have w ≤ u. Since bq ⊆ br, we have w ≤ v. �

Lemma 2.13. Q0 is a complete subforcing of Q.

Proof. We claim that every maximal antichain A in Q0 is maximal in Q. Let D0 = {p ∈ Q0 ∣ ∃q ∈A p ≤ q} and D = {p ∈ Q ∣ ∃q ∈ A p ≤ q}. We claim that D is dense in Q. To see this, suppose thatu ∈ Q. If u ∈ Q0, then there is a condition v ≤ u in D0. Suppose that u = (bp, p) ∈ Q1. Since D0 isdense in Q0, there is some v = (bq,1P) ∈ D0 with bp ⊆ bq. There is a common extension w ≤ u, vby Lemma 2.12. Since w ≤ v, we have w ∈ D. Since D is dense in Q, A is a maximal antichain inQ. �

Suppose that G is a P-generic filter over V . Then H = {q ∈ Q ∣ ∃p ∈ G e(p) ≤ q} is a Q-genericfilter over V . Let TH = TG.

Lemma 2.14. Suppose that H is Q-generic over V . Suppose that ċ is a Q-name with ċH = c.Suppose that p0 ∈ Q and p0 ⊩ ċ ∈ [TḢ], where Ḣ is the canonical name for the Q-generic filter.Then the quotient forcing for c is equivalent to Add(κ,1).

Proof. It follows from Lemma 2.12 and Lemma 2.13 that the quotient forcing for c in Q is equalto {(bp, p) ∈ Q ∣ bp ⊆ c}. Since this is < κ-closed, it is forcing equivalent to Add(κ,1) by Lemma1.15. �

Lemma 2.15. Suppose that κ is an uncountable regular cardinal and λ > κ is inaccessible. Supposethat G is Add(κ,1)-generic over V . Then in V [G], there is a perfect subset C of κκ such thateach x ∈ C is Add(κ,1)-generic over V and the quotient forcing for x is equivalent to Add(κ,1).

Proof. By Lemma 2.14. �

Definition 2.16. Suppose that κ is an uncountable regular cardinal and λ is an inaccessiblecardinal with κ < λ. Suppose that X ⊆ λ. Let

Col∗(κ,X) = {p ∈ Col(κ,< λ) ∣ dom(p) ⊆X × κ}and GX = G ∩Col(κ,X) for any Col(κ,< λ)-generic filter G over V .

Suppose that µ is a regular cardinal with κ ≤ µ < λ and µ

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Theorem 2.17. Suppose that κ is an uncountable regular cardinal and λ is an inaccessible cardinalwith κ < λ. Suppose that G is Col(κ,< λ)-generic over V . Then in V [G] every subset of κκ thatis definable from an element of Ordκ has the perfect set property.

Proof. Suppose that ϕ is a formula and y ∈ Ordκ. If M is an inner model of V [G], letAM = AV [G]ϕ ∩M = {x ∈ κκ ∣ ϕ(x, y)}V [G] ∩M.

Suppose that ∣A∣V [G] = (κ+)V [G]. Let Gγ = G ↾ γ for γ < λ. Suppose that γ < λ is such thaty ∈ V [Gγ]. Since ∣A∣V [G] = (κ+)V [G], there is a regular cardinal µ with γ < µ < λ and µ

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Let ψ(x⃗, α, t) denote the formula 1 ⊩V [x⃗]Add(κ,1)

ϕ(x⃗, α, t) in the forcing language with a predicatefor V . Let σ denote the canonical Add(κ,1)n-name for the n-tuple of Add(κ,1)-generic reals withσx⃗ = x⃗ for all x⃗ ∈ [C]n.

Claim. f ↾ [C]n is continuous.

Proof. If x⃗ ∈ [C]n and α < κ, then there is a condition p ∈ Add(κ,1)n with p ⊆ x⃗ and p ⊩VAdd(κ,1)nψ(σ,α, f(x⃗) ↾ α). So f(x⃗) ↾ α = f(y⃗) ↾ α for all y⃗ ∈ C with p ⊆ y⃗. This proves that f ↾ [C]n iscontinuous. �

The proof of the remaining claims is analogous. We force with Add(κ,1)n over an intermediatemodel which contains the parameters and whose quotient forcing is equivalent to Add(κ,λ) or toCol(κ,< λ). �

Theorem 2.19. Suppose there is a proper class of inaccessible cardinals. Then there is a classgeneric extension V [G] of V such that in V [G], for every regular cardinal κ, the perfect setproperty holds for all subsets of κκ that are definable from an element of Ord.

Proof. Let ⟨κα ∣ α ∈ Ord⟩ denote the order-preserving enumeration of the class of all inaccessiblecardinals, their limit points and the additional element ω. We define the following Easton supportiteration ⟨Pα, Ṗα ∣ α ∈ Ord⟩. Let P0 = {1}. If α is a successor or α = 0, let Ṗα denote a Pα-name forCol(κα,< κα+1). If α is a limit and κα is inaccessible, let Ṗα denote a Pα-name for Col(κα,< κα+1).If α is a limit and κα is singular, let ν̇α denote a Pα-name for the least regular cardinal ν withκα ≤ ν < κα+1 that is not singularized by Pα and let Ṗα denote a Pα-name for Col(ν̇α,< κα+1).The support is bounded at regular limits and full at singular limits. Let Ṗ(α) denote the canonicalPα-name for the tail forcing at α. Let P denote the forcing defined by the iteration.

If α is a successor or α = 0, then Pα has the κα-c.c. and 1 ⊩Pα Ṗ(α) is < κα-closed. HenceP preserves the regularity of κα. If α is a limit and κα is inaccessible, then κα = α, Pα has theκα-c.c. and 1 ⊩Pα Ṗ(α) is < κα-closed. Hence P preserves the regularity of κα.

The claim follows from Solovay’s theorem [Sol70], Theorem 2.17 and from the closure of thetail forcings. �

Lemma 2.20. Suppose that κ is an uncountable regular cardinal with κ

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Proof. First suppose that ⟨Uα ∣ α < κ⟩ is a sequence of dense open subsets of κκ and that ⋂α

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The following result implies that Gκ(A) is determined if A has the κ-Baire property. Moreover,it implies that the Baire property for all subsets of the Baire space ωω definable from elements ofωOrd is precisely characterized by the determinacy of the Banach-Mazur game.

Lemma 3.8. (see [Kov09, Lemma 7.3.2]) Suppose that A is a subset of κκ and t ∈

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Definition 3.12. (i) An almost strategy for player I in Gκ is a partial strategy σ such thatdom(σ) is dense in the following sense. Suppose that s⃗ = ⟨sα ∣ α < γ⟩ is a strictly increasingsequence in

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(v) if (s, t), (u, v) ∈ p, l(s) = l(u) = γ is even, ⋃(s ↾ γ) = ⋃(u ↾ γ) and ⋃(t ↾ γ) = ⋃(v ↾ γ), thens(γ) = u(γ) and t(γ) = v(γ).

The pairs (s, t) with the properties (i), (ii) and (iii) are called S-nodes.

Remark 3.17. It is possible to replace the conditions ⋃ s ↾ γ = ⋃u ↾ γ and ⋃ t ↾ γ = ⋃ v ↾ γ withthe conditions s ↾ γ = u ↾ γ and t ↾ γ = v ↾ γ in Definition 3.16 (v) and prove analogous results toall results in this section.

The following fullness condition is used to define an almost strategy for player I in Gκ from anS-tree.

Definition 3.18. Suppose that S is a level subset of (

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with the properties2

(a) proj(S) = {s ∈ Add(κ,1) ∣ ∃t (s, t) ∈ S} = Add(κ,1),(b) the map πS,Add(κ,1)∶S → Add(κ,1), πS,Add(κ,1)(s, t) = s is a forcing projection and(c) 1Q ⊩ Q̇ ∼ (S/Add(κ,1))πS,Add(κ,1) .(d) 1Add∗(κ,1) ⊩ Q̇µ ∼ (S/Add(κ,1))πS,Add(κ,1) .

In particular, S ordered by reverse inclusion and Q ∗ Q̇ have the same generic extensions.Proof. There is a dense subset of Q that is isomorphic to Add∗(κ,1). Hence we can assume thatQ is equal to Add∗(κ,1). We can further assume that there is a Q-name 1̇Q̇ for 1Q̇Ġ by adding itto dom(Q̇), where Ġ is a Q-name for the generic filter. Let π∶Q ∗ Q̇ → Q, π(q, q̇) = q denote theprojection.

Since Q∗ Q̇ is equivalent to Add(κ,1), there is a dense subforcing R of Q∗ Q̇ that is isomorphicto Add∗(κ,1) via an isomorphism ν∶Add∗(κ,1) → R. Then π ↾ R∶R → Q is a forcing projection.Hence πAdd∗(κ,1),Q = ν ○ π∶Add∗(κ,1) → Q = Add∗(κ,1) is a forcing projection. Therefore 1Qforces that the quotient forcing (Add∗(κ,1)/Q)πAdd∗(κ,1),Q is equivalent to Q̇.

Let Q̄ denote a Q-name for the set of the elements of (Add∗(κ,1)/Q)πAdd∗(κ,1),Q that are supremaof weakly increasing sequences in (Add∗(κ,1)/Q)πAdd∗(κ,1),Q of length < κ with an upper bound(i.e. a common extension). Let

S = {(s, t) ∈ Add(κ,1) ×Add(κ,1) ∣ l(s) = l(t), ∀α < l(s) ∃β < l(s) s ↾ β ⊩Q ť ∈ Q̄}.and S = {(s, ť) ∣ (s, t) ∈ S}.Claim. The set S is dense in Add(κ,1) ∗ Q̄.Proof. For any condition (p, q̇) ∈ Add(κ,1), we can extend p to decide q̇. Hence the set of(s, ť) ∈ Add(κ,1) ∗ Q̄ with s ⊩Add(κ,1) ť ∈ Q̄ is dense in Add(κ,1) ∗ Q̄. We need to find s, t withthe same length. Suppose that (s, ť) ∈ Add(κ,1) ∗ Q̄ and s ⊩Add(κ,1) ť ∈ Q̄. Since Add(κ,1) ∗(Add∗(κ,1)/Q)πAdd∗(κ,1),Q is dense in Add(κ,1)∗Q̄, we can form a sequence ⟨(sn, ťn) ∈ Add(κ,1)∗(Add∗(κ,1)/Q)πAdd∗(κ,1),Q such that (s0, t0) = (s, t), ⟨l(sn) ∣ n ∈ ω⟩ is strictly increasing andl(sn) ≤ l(tn+1) for all n. Let u = ⋃n∈ω sn and v = ⋃n∈ω tn. Then (u, v̌) ∈ Add(κ,1) ∗ Q̄ by thedefinition of Q̄. The construction implies that (u, v̌) ≤Add(κ,1)∗Q̄ (s, ť) and (u, v) ∈ S. �

Note that S is perfect and closed. Then S and µ have the required properties. �

Suppose that Q ∗ Q̇ is equivalent to Add(κ,1) and that S is a level set as in Lemma 3.21, inparticular S is perfect and S = {(s, ť) ∣ (s, t) ∈ S} is equivalent to Q ∗ Q̇ by Lemma 3.21.Definition 3.22. The conditions in P = PS are S-trees p of size < κ, ordered by reverse inclusion.

The forcing P is < κ-closed and has size κ. Hence it is equivalent to Add(κ,1). If G is P-genericover V , let TG = ⋃G.Lemma 3.23. If G is P -generic over V , then TG is a full S-tree.

Proof. Since S is perfect, every condition p can be extended to some q ≤ p so that any terminalnode in p is extended to incompatible nodes in q. Similarly, we can add an upper bound in q to agiven cofinal branch in p of length < κ. Hence genericity of G implies that TG is perfect.

We claim that TG is full. Suppose that (s, t) ∈ TG. Then there is some p ∈ G with (s, t) ∈ p.Suppose that l(s) = l(t) = γ + 1, γ is even, l(s(γ)) = l(t(γ)) = δ, δ < η, u ∈ ηκ, and s(γ) ⊆ u. Themap πS,Add(κ,1)∶S → Add(κ,1), πS,Add(κ,1)(s, t) = s is a forcing projection by Lemma 3.21 (b).Hence there is some (v,w) ∈ S with u ⊆ v. Since the set of conditions q ≤ p with (s⌢⟨v⟩, t⌢⟨w⟩) ∈ qis dense below p, (s⌢⟨v⟩, t⌢⟨w⟩) ∈ TG. �

Suppose that Ġ is the canonical name for the P-generic filter. We further suppose that ḟ , ġ are P-names with 1 ⊩P (ḟ(α), ġ(α)) ∈ TĠ for all α < κ and that ḃ, ċ are P-names with p0 ⊩P ⋃α

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(i) ht(p) = l(p) = γ ∈ Lim and(ii) p ⊩ ḟ ↾ γ = fp, ġ ↾ γ = gp.

If p ∈ P∗, let sp = ⋃α

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(2) πS,Add(κ,1)∶S → Add(κ,1), πS,Add(κ,1)(s, t) = s is a forcing projection.(3) πP∗,Add(κ,1)∶P∗ → Add(κ,1), πP∗,Add(κ,1)(p) = sp is a forcing projection.

Proof. (1) The map πP∗,S is a forcing projection by Lemma 3.27 (2). The quotient forcing foran S-generic filter G is P∗/G = {p ∈ P∗ ∣ (sp, tp) ∈ G}. The quotient forcing P∗/G is closed underincreasing unions of length < κ and hence it is equivalent to Add(κ,1).

(2) The map πS,Add(κ,1) is a forcing projection by Lemma 3.21 (b).(3) This holds since πP∗,Add(κ,1) = πS,Add(κ,1) ○ πP∗,S or by Lemma 3.27 (3). �

For the following, note that for every P-generic filter G and every branch b ∈ p[TG], there is aPS-name ḃ with 1P ⊩ ḃ ∈ p[T ] and ḃG = b.

Lemma 3.29. Suppose that G is PS-generic over V . Suppose that Ġ is a PS-name for G and thatḃ is a PS-name such that 1P ⊩ ḃ ∈ p[TĠ]. Then ḃG is Add(κ,1)-generic over V and the quotientfor ḃG in PS is equivalent to (S/Add(κ,1))πS,Add(κ,1) ×Add(κ,1).

Proof. The map πP∗,Add(κ,1) is a forcing projection by Lemma 3.27 (3). The quotient forcing foran Add(κ,1)-generic filter G is P∗/G = {p ∈ P∗ ∣ sp ∈ G}. Let πP∗,S , πS,Add(κ,1) and πP∗,Add(κ,1)denote the maps in Lemma 3.28. Then the quotient (P∗/Add(κ,1))πP∗,Add(κ,1) is equivalent to theproduct (S/Add(κ,1))πS,Add(κ,1) ×Add(κ,1). �

Suppose that G is P-generic over V and A = p[TG]V [G]. Then player I has a winning strategyfor Gκ(A) in V [G] by Lemma 3.20 and Lemma 3.23. It remains to prove the determinacy of thegames Gκ(Aϕ,z).

Lemma 3.30. Suppose that ϕ(x, y) is a formula and z is a set. Suppose that R, S, G, H, Isatisfy the following conditions.(a) R is a < κ-distributive forcing.(b) S = Add(κ,1) ×Add(κ,1) ×R.(c) G ×H × I is S-generic over V .(d) There is an Add(κ,1)-generic element of κκ over V in AV [G×H×I]ϕ,z ∩ V [G].

Then there is a perfect level subset S of (

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Proof. Let J̇ denote the canonical PS-name for the PS-generic filter over V [I]. Suppose that ḋ isa PS-name in V [I] with 1 ⊩V [I]PS ḋ ∈ p[TJ̇]. Let

Bḋ = ⟨{Jḋ(α) = βK ∣ α,β < κ}⟩B(PS).Then Bḋ is equivalent to Add(κ,1) by Lemma 3.28 (3). Hence there is an isomorphism ν∶Bċ →d Bċ0between dense subsets, a Bċ-name c̄ and a Bḋ-name d̄ such that 1 ⊩Bċ ċ = c̄, 1 ⊩Bḋ ḋ = d̄ and νmaps c̄ to d̄.

We have 1 ⊩Bċ Q̇ ∼ (S/Add(κ,1))πS,Add(κ,1) by Lemma 3.21 (c). Hence1 ⊩Bċ Q̇ ×Add(κ,1) ∼ (S/Add(κ,1))πS,Add(κ,1) ×Add(κ,1).

Moreover1 ⊩Bḋ (PS/Bḋ) ∼ (S/Add(κ,1))

πS,Add(κ,1) ×Add(κ,1)by Lemma 3.29. These two properties and the choice of ḃ imply that 1 ⊩PS ḋ ∈ Aϕ,z �

Player I has a winning strategy for Gκ(p[TJ]V [J∗K]) in V [J ∗K] by Lemma 3.20. This is awinning strategy for Gκ(AV [J∗K]ϕ ) by the previous claim. This shows (1).

Since PS is equivalent to Add(κ,1) × Add(κ,1), it follows from (1) that there is a winningstrategy for player I in Gκ(AV [G×H×I])ϕ,z ) in V [G ×H × I]). This shows (2). �

Theorem 3.31. Suppose that κ is an uncountable regular cardinal and µ is a cardinal. LetP = Col(κ,< λ) × Add(κ,µ). Then in every P-generic extension of V , Gκ(A) is determined forevery subset A of κκ that is definable from an element of Ordκ.

Proof. Suppose that G × H is Col(κ,< λ) × Add(κ,µ)-generic over V . Suppose that ϕ(x, y) isa formula and z ∈ Ordκ is an element of V [G × H]. Let Gγ = G ∩ Col(κ,< γ) for γ ≤ λ. LetAdd(κ,S) = {p∶κ × S → 2 ∣ ∣p∣ < κ} and HS = H ∩ Add(κ,S) for S ⊆ µ. Since Col(κ,< λ) is κ-ccand Add(κ,µ) is κ-cc, there is some γ < λ and some S ∈ [µ]

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(2) Suppose that H is Add(κ,λ)-generic over V and λ ≥ κ+. Then in V [H], for every functionf ∶ [κκ]n ↦ κκ definable from elements of Ordκ, there is a subset C of κκ with the followingproperties.(a) f ↾ [C]n is continuous.(b) Player I has a winning strategy in Gκ(C).(c) There is an Add(κ,1)-generic extension V [G] of V with the following properties.

(i) V [H] is a Add(κ,λ)-generic extension of V [G].(ii) C contains every Add(κ,1)-generic element x of κκ over V [G] whose quotient

forcing in V [H] is equivalent to Q̇x ×Add(κ,λ).(3) Suppose that H is Col(κ,< λ)-generic over V and λ > κ is inaccessible. Then in V [H], for

every function f ∶ [κκ]n ↦ κκ definable from elements of Ordκ, there is a subset C of κκ withthe following properties.(a) f ↾ [C]n is continuous.(b) Player I has a winning strategy in Gκ(C).(c) There is some µ < λ with µ

PERFECT SUBSETS OF GENERALIZED BAIRE SPACES 19

The set S of pairs ⟨t, u⟩ in κκ × κκ with l(t) = l(u) such that t ⊆ ts and u ⊆ us for some s ∈ κ in every generic extension of V and Aϕ does not have a perfectsubset.

Proof. There is a subtree T of κ by [LMRS16, Proposition7.2]. Suppose that ϕ(x) is a formula which states that x ∈ [T ] or x ∉ L. Suppose that V [G] is ageneric extension of V . If (κ+)L has size < κ+ in V [G], then V [G] ⊧ ∣Aϕ∣ ≥ ∣[T ]∣ > κ. If (κ+)L iscollapsed in V [G], then V [G] ⊧ (κκ)L ≤ κ and hence V [G] ⊧ ∣Aϕ∣ > κ. Since T is a κ-tree, [T ]does not have a perfect subset. �

Lemma 4.1 can be extended to definable sets as follows.

Lemma 4.3. Let ν = (2κ)+. Suppose that ϕ(x, y) is a formula and z is a set. If 1 ⊩Col(κ, κ, then 1Col(κ,

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In particular, the 1-uplifting cardinals are inaccessible. The consistency strength of a cardinalκ that is n-fold uplifting for all n ∈ ω is below a Mahlo cardinal [AV15, Proposition 3.4].Definition 4.8. Suppose that ψ(x, y) is a formula and Γψ = {x ∣ ψ(x, y)}. We say that the classΓψ is absolute to inaccessibles if for all sets x and all inaccessible cardinals κ with x, y ∈Hκ, ψ(x, y)holds if and only if ψ(x, y) holds in Vκ.

The following result allows us to force RAκn(Γ) for all n ∈ ω by an iteration of least counterex-amples.

Lemma 4.9. Suppose that κ is a regular uncountable cardinal and λ > κ is n-fold uplifting.Suppose that Γψ is a definable class of forcings preserving κ that is closed under < κ-supportiterations and lottery sums and is absolute to inaccessibles.(1) Suppose that λ > κ is n-fold uplifting and P,Q ∈ Γ are forcings in Hλ. Then the condition

that P satisfies RAκn(Γ) below Q is absolute between Hλ and V .(2) Suppose that λ > κ is n-fold uplifting, m < n and P is a forcing in Γ. Then there is some

m-fold uplifting cardinal ν > λ with P ∈Hν such that the condition that P is a counterexampleto RAκn(Γ) of minimal rank is absolute between Hν and V .

Proof. (1) We prove this by induction on n. Suppose that P = Q ∗ Ṙ for some forcing Q ∈ Γ andsome Q-name Ṙ for a forcing in Γ.

Suppose that in Hλ, P satisfies RAκn(Γ) and that m < n. Then there is a P-name Ṡ in Hλ suchthat RAκm(Γ) holds below P∗ Ṡ in Hλ and We have H

V [G]λ =Hλ[G] for every (P∗ Ṡ)-generic filter

G over V by [AV15, Lemma 4.4]. Moreover λ is n-fold uplifting in V [G] by a straightforwardinduction. It follows from the inductive hypothesis that RAκm(Γ) holds below P ∗ Ṡ in V . Thisshows that P satisfies RAκn(Γ).

Suppose that P satisfies RAκn(Γ) and that m < n. Then there is a P-name Ṡ for a forcing inΓ such that RAκm(Γ) holds below P ∗ Ṡ. Since λ is n-fold uplifting, there is an m-fold upliftingcardinal ν > λ such that Q̇ ∈Hν and Hλ ≺+ Hν . We have HV [G]ν =Hν[G] for every (P∗ Ṡ)-genericfilter G over V by [AV15, Lemma 4.4]. Then RAκm(Γ) holds below P ∗ Ṡ in Hν by the inductivehypothesis. Since Hλ ≺+ Hν , there is a P-name Ṫ in Hλ for a forcing in Γ such that RAκm(Γ) holdsbelow P ∗ Ṫ in Hλ. This shows that P satisfies RAκn(Γ) in Hλ.

(2) We can assume that n = m + 1. Suppose that λ < µ and P ∈ Hµ. Suppose that θ > µ issufficiently large so that for every forcing S ∈ Γ in Hµ which satisfies RAκn(Γ), this condition iswitnessed by some S-name Ṫ in Hθ. We have HV [G]ν =Hν[G] for every forcing S ∈ Γ in Hµ, everyS-name Ṫ for a forcing in Γ in Hθ and every (S∗ Ṫ)-generic filter G over V by [AV15, Lemma 4.4].Moreover λ is n-fold uplifting in V [G] by a straightforward induction. Since λ is n-fold uplifting,there is some m-fold uplifting cardinal ν > θ. It follows from the inductive hypothesis (1) for mand the choice of θ that for every forcing S ∈ Γ in Hµ, the property that S satisfies RAκm(Γ) isabsolute between Hν and V . This implies the claim. �

Lemma 4.10. Suppose that κ is an uncountable regular cardinal and Γ is a definable class offorcings preserving κ that is closed under < κ-support iteration and lottery sums. Suppose thatn ∈ ω, RAn(Γ) holds, P is a forcing in Γ with 1 ⊩P RAn(Γ) and G is P-generic over V . ThenHκ+ ≺Σn (Hκ+)V [G].Proof. The proof is analogous to [AV15, Theorem 2.6]. �

Lemma 4.11. Suppose that κ is a regular uncountable cardinal and λ > κ. Suppose that Γψ is adefinable class of forcings preserving κ that is closed under < κ-support iterations and lottery sums,is absolute to inaccessibles and contains the forcings Col(κ,< ν) for all cardinals ν with ν

PERFECT SUBSETS OF GENERALIZED BAIRE SPACES 21

(b) If m ∈ ω is least such that (a) fails, then Ṗα is a Pα-name for the lottery sum of all forcingsQ ∈ Γ in Hλ of minimal rank such that RAκm(Γ) fails for Q in Hλ, and the forcings Col(κ,< ν)for all cardinals ν < λ with ν λ such that Q ∈ Hν , Hλ ≺+ Hν and in Hν , Qis a counterexample to RAκm(Γ) of minimal rank by Lemma 4.9 (2). Let Pν denote the iterationdefined in (Hν , THν ) as Pλ in (Hλ, THλ). Then the lottery iteration can opt for a Q-generic filterH in step λ. Suppose that I is ṘG∗H -generic over V [G ∗H], where Ṙ denotes the tail forcingfollowing step λ. Since Hλ ≺+ Hν , Hλ[G] ≺ Hν[G ∗H ∗ I] by the forcing theorem as in [AV15,Lemma 4.6] This contradicts the choice of H. �

Lemma 4.11 shows that we can obtain a model of RAκn(Γ) for all n ∈ ω from a cardinal λ > κthat is n-fold uplifting for all n ∈ ω.Theorem 4.12. Suppose that κ is an uncountable regular cardinal. Suppose that Γψ is a definableclass of forcings preserving κ that is closed under < κ-support iterations and lottery sums, isabsolute to inaccessibles and contains the forcings Col(κ,< ν) for all cardinals ν with ν κ suggest the questionwhether this connection extends to other regularity properties.

Question 5.1. Are there analogous results to Theorem 2.17 and Theorem 3.31 for the determinacyof games associated to regularity properties such as the Hurewicz dichotomy?

An inaccessible cardinal is necessary for the perfect set property. We ask if the conclusion ofTheorem 3.31 can be achieved without the inaccessible cardinal as in [She84].

Question 5.2. Is the consistency strength of the almost Baire property for all subsets of κκdefinable from an element of Ordκ for an uncountable regular cardinal κ strictly less than aninaccessible cardinal?

It would be useful to obtain different models in which the conlusions of the main theorems hold.

Question 5.3. Do the conclusions of Theorem 2.17 and Theorem 3.31 hold in the Silver collapse[Cum10, Definition 20.1] and in the Kunen collapse [Cum10, Section 20] of an inaccessible cardinalλ to κ+, where κ is regular?

For the Banach-Mazur game of length ω, the existence of strategies and tactics are equivalentby 3.8. This suggests the following question.

Question 5.4. Is it consistent that there is an uncountable regular cardinal κ with κ

22 PHILIPP SCHLICHT

The following question arose in a discussion with Philipp Lücke.

Question 5.5. Does the almost Baire property for all subsets of κκ definable from an element ofOrdκ imply a version of the Kuratowski-Ulam theorem?

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1. Introduction1.1. Trees and perfect sets1.2. Forcings1.3. A forcing with no strategically closed quotient

2. The perfect set property2.1. Perfect set games2.2. Consistency of the perfect set property

3. The almost Baire property3.1. Banach-Mazur games3.2. Consistency of the almost Baire property

4. Consequences of Iterated Resurrection Axioms5. QuestionsReferences