Perfect Set Games and Colorings on Generalized [2 pt] Baire Spaces · 2020. 10. 14. ·...

Transcript of Perfect Set Games and Colorings on Generalized [2 pt] Baire Spaces · 2020. 10. 14. ·...

Perfect Set Games and Colorings on GeneralizedBaire Spaces

Dorottya Sziráki

5th Workshop on Generalized Baire SpacesBristol, 4 February 2020

Dorottya Sziráki Perfect set games and colorings on κκ
Perfectness for the κ-Baire spaceAssume κ
Väänänen’s perfect set game

Let X ⊆ κκ, let x0 ∈ κκ and let ω ≤ γ ≤ κ.

Definition (Väänänen, 1991)

The game Vγ(X,x0) has length γ and is played as follows:

I U1 . . . Uα . . .

II x0 x1 . . . xα . . .

II first plays x0. In each round 0 < α < γ, I plays a basic open subsetUα of X, and then II chooses

xα ∈ Uα with xα 6= xβ for all β < α.

I has to play so that Uβ+1 3 xβ in each successor round β + 1 < γ andUα =

⋂β
Väänänen’s perfect set game

Let X ⊆ κκ, let x0 ∈ κκ and let ω ≤ γ ≤ κ.


The game Vγ(X,x0) has length γ and is played as follows:

I U1 . . . Uα . . .

II x0 x1 . . . xα . . .

II first plays x0. In each round 0 < α < γ, I plays a basic open subsetUα of X, and then II chooses

xα ∈ Uα with xα 6= xβ for all β < α.

I has to play so that Uβ+1 3 xβ in each successor round β + 1 < γ andUα =

⋂β
Perfect and scattered subsets of the κ-Baire space

Let X ⊆ κκ, and suppose ω ≤ γ ≤ κ.


X is a γ-scattered set if I wins Vγ(X,x0) for all x0 ∈ X.X is a γ-perfect set if X is closed and II wins Vγ(X,x0) for all x0 ∈ X.

X is ω-perfect iff X is perfect in the usual sense (i.e., iff X closedand has no isolated points).X is ω-scattered iff X is scattered in the usual sense (i.e., eachnonempty subspace contains an isolated point).Vγ(X,x0) may not be determined when γ > ω.





X is ω-perfect iff X is perfect in the usual sense (i.e., iff X closedand has no isolated points).

X is ω-scattered iff X is scattered in the usual sense (i.e., eachnonempty subspace contains an isolated point).Vγ(X,x0) may not be determined when γ > ω.





X is ω-perfect iff X is perfect in the usual sense (i.e., iff X closedand has no isolated points).X is ω-scattered iff X is scattered in the usual sense (i.e., eachnonempty subspace contains an isolated point).

Vγ(X,x0) may not be determined when γ > ω.





X is ω-perfect iff X is perfect in the usual sense (i.e., iff X closedand has no isolated points).X is ω-scattered iff X is scattered in the usual sense (i.e., eachnonempty subspace contains an isolated point).Vγ(X,x0) may not be determined when γ > ω.

κ-perfect sets vs. strongly κ-perfect sets

Example (Huuskonen)

The following set is κ-perfect but is not strongly κ-perfect:

Yω = {x ∈ κ3 : |{α < κ : x(α) = 2}| < ω}.

PropositionLet X be a closed subset of κκ.

X is κ-perfect ⇐⇒ X =⋃i∈I

Xi for strongly κ-perfect sets Xi.


Example (Huuskonen)


Yω = {x ∈ κ3 : |{α < κ : x(α) = 2}| < ω}.





Example (Huuskonen)


Yω = {x ∈ κ3 : |{α < κ : x(α) = 2}| < ω}.




Väänänen’s generalized Cantor-Bendixson theorem

Theorem (Väänänen, 1991)

The following Cantor-Bendixson theorem for κκ is consistent relative tothe existence of a measurable cardinal λ > κ:

Every closed subset of κκ is the (disjoint) union of

a κ-perfect set and a κ-scattered set, which is of size ≤ κ.

Theorem (Galgon, 2016)

Väänänen’s generalized Cantor-Bendixson theorem is consistent relative tothe existence of an inaccessible cardinal λ > κ.



The following Cantor-Bendixson theorem for κκ is consistent relative tothe existence of a measurable cardinal λ > κ:

Every closed subset of κκ is the (disjoint) union of

a κ-perfect set and a κ-scattered set, which is of size ≤ κ.

Theorem (Galgon, 2016)

Väänänen’s generalized Cantor-Bendixson theorem is consistent relative tothe existence of an inaccessible cardinal λ > κ.


Proposition (Sz)

Väänänen’s generalized Cantor-Bendixson theorem is equivalent to theκ-perfect set property for closed subsets of κκ (i.e, the statement thatevery closed subset of κκ of size > κ has a κ-perfect subset).

Remark: The κ-PSP for closed subsets of κκ is equiconsistent with theexistence of an inaccessible cardinal λ > κ.

Sketch of the proof.

Let X be a closed subset of κκ. Its set of κ-condensation points is defined to be

CPκ(X) = {x ∈ X : |X ∩Nx�α| > κ for all α < κ}.

If the κ-PSP holds for closed subsets of κκ, then CPκ(X) is a κ-perfect set andX − CPκ(X) is a κ-scattered set of size ≤ κ.


Proposition (Sz)

Väänänen’s generalized Cantor-Bendixson theorem is equivalent to theκ-perfect set property for closed subsets of κκ (i.e, the statement thatevery closed subset of κκ of size > κ has a κ-perfect subset).

Remark: The κ-PSP for closed subsets of κκ is equiconsistent with theexistence of an inaccessible cardinal λ > κ.

Sketch of the proof.

Let X be a closed subset of κκ. Its set of κ-condensation points is defined to be

CPκ(X) = {x ∈ X : |X ∩Nx�α| > κ for all α < κ}.

If the κ-PSP holds for closed subsets of κκ, then CPκ(X) is a κ-perfect set andX − CPκ(X) is a κ-scattered set of size ≤ κ.

Perfect and scattered trees

Let T be a subtree of

Definition (Galgon, 2016)

T is a γ-scattered tree if player I wins Gγ(T, t) for all t ∈ T .T is a γ-perfect tree if player II wins Gγ(T, t) for all t ∈ T .

Proposition


Definition (Galgon, 2016)

T is a γ-scattered tree if player I wins Gγ(T, t) for all t ∈ T .T is a γ-perfect tree if player II wins Gγ(T, t) for all t ∈ T .

Proposition

γ-perfect sets and trees when γ ≤ κ

Theorem (Sz)


Theorem (Sz)


Theorem (Sz)


Theorem (Sz)


Theorem (Sz)


Theorem (Sz)

Generalizing the Cantor-Bendixson hierarchy

LetX ⊆ κκ, let x0 ∈ κκ, and let S be a tree without branches of length ≥ κ.

Definition (Hyttinen; Väänänen)

The S-approximation VS(X,x0) of Vκ(X,x0) is the following game.

I s1, U1 . . . sα, Uα . . .

II x0 x1 . . . xα . . .

In each round α > 0, I first plays sα ∈ S such that sα >S sβ for all0 < β < α. Then I plays Uα and II plays xα according to the same rulesas in Vκ(X,x0).

The first player who can not move loses, and the other player wins.

Let

ScS(X) = {x ∈ X : I wins VS(X,x)};KerS(X) = {x ∈ κκ : II wins VS(X,x)}.


LetX ⊆ κκ, let x0 ∈ κκ, and let S be a tree without branches of length ≥ κ.

Definition (Hyttinen; Väänänen)

The S-approximation VS(X,x0) of Vκ(X,x0) is the following game.

I s1, U1 . . . sα, Uα . . .

II x0 x1 . . . xα . . .

In each round α > 0, I first plays sα ∈ S such that sα >S sβ for all0 < β < α. Then I plays Uα and II plays xα according to the same rulesas in Vκ(X,x0).

The first player who can not move loses, and the other player wins. Let

ScS(X) = {x ∈ X : I wins VS(X,x)};KerS(X) = {x ∈ κκ : II wins VS(X,x)}.


ScS(X) = {x ∈ X : I wins VS(X,x)}; KerS(X) = {x ∈ κκ : II wins VS(X,x)}.

Given an ordinal α, let Bα = the tree of descending sequences of elements of α.X(α) denotes the αth Cantor-Bendixson derivative of X.

Observation 1 (Väänänen)

X(α) = X ∩KerBα(X) = X − ScBα(X).

Corollary

Kerω(X) =⋂{KerS(X) : S is a tree without infinite branches};

Scω(X) =⋃{ScS(X) : S is a tree without infinite branches}.

Theorem 2 (Hyttinen (1990); Väänänen (1991))

Kerκ(X) =⋂{KerS(X) : S is a tree without branches of length ≥ κ};

Scκ(X) =⋃{ScS(X) : S is a tree without branches of length ≥ κ}.

The sets X ∩ KerS(X) (resp. X − ScS(X)) can be seen as the “levels of ageneralized Cantor-Bendixson hierarchy” for the set X associated to II (resp. I).






Corollary












Corollary












Corollary







Generalizing the Cantor-Bendixson hierarchy for trees

Theorem (Sz, part 1)

There exists a family {G′γ(T, t) : T is a subtree of
Comparing the Cantor-Bendixson hierarchies

ScS(T ) = {t ∈ T : I wins G′S(T, t)}; KerS(T ) = {t ∈ T : II wins G′S(T, t)}.ScS(X) = {x ∈ X : I wins VS(X,x)}; KerS(X) = {x ∈ κκ : II wins VS(X,x)}.


Let S be a tree without branches of length ≥ κ. Then1 KerS([T ]) ⊆ [KerS(T )]

(i.e., if II wins VS([T ], x) then II wins G′S(T, t) when t ( x ∈ κκ.).

2 [T ]− ScS([T ]) ⊆ [T − ScS(T )](i.e., if I wins G′S(T, t) then I wins VS([T ], x) when t ( x ∈ κκ).

3 If κ has the tree property and T is a κ-tree, then

KerS([T ]) = [KerS(T )]

(i.e., VS([T ], x) and G′S(T, t) are equivalent for II when t ( x ∈ κκ).

Density in itself for the κ-Baire space

Definition

A subset X ⊆ κκ is κ-dense in itself if X is a κ-perfect set.A subset X ⊆ κκ is strongly κ-dense in itself if X is a strongly κ-perfectset.

Proposition (Sz)

The following are equivalent for any X ⊆ κκ.X is κ-dense in itself.X =

⋃i∈I Xi where each Xi is strongly κ-dense in itself.

X ⊆ Kerκ(X) (i.e., player II wins Vκ(X,x) for all x ∈ X.)


Definition

A subset X ⊆ κκ is κ-dense in itself if X is a κ-perfect set.A subset X ⊆ κκ is strongly κ-dense in itself if X is a strongly κ-perfectset.

Proposition (Sz)

The following are equivalent for any X ⊆ κκ.X is κ-dense in itself.X =

⋃i∈I Xi where each Xi is strongly κ-dense in itself.

X ⊆ Kerκ(X) (i.e., player II wins Vκ(X,x) for all x ∈ X.)



If λ > κ is measurable and G is Col(κ,
A Cantor-Bendixson theorem for Gδ relations

R is a collection of finitary relations on a set X.Y ⊆ X is R-homogeneous if for all 1 ≤ k < ω and k-ary R ∈ R we have:

(x1, . . . , xk) ∈ R for all pairwise distinct x1, . . . , xk ∈ Y .

Theorem (Kubiś, 2003; Doležal, Kubiś 2015)

Let R be a countable set of Gδ relations on a Polish space X(i.e., every R ∈ R is an Gδ subset of kX for some 1 ≤ k < ω).

1 Either there exists a perfect R-homogeneous set, or there exists α < ω1such that every R-homogeneous set Y has Cantor-Bendixson rank < α(i.e., Y (α) = ∅).

2 If there exists an uncountable R-homogeneous set, then there exists aperfect R-homogeneous set. This also holds for analytic spaces X.

Recall that Y (α) = Y ∩KerBα(Y ) = Y − ScBα(Y ).







2 If there exists an uncountable R-homogeneous set, then there exists aperfect R-homogeneous set.

This also holds for analytic spaces X.


A dichotomy for infinitely many Π02(κ) relationsR is a Π02(κ) relation on a topological space X iff

R is an intersection of ≤ κ many open subsets of kX for some 1 ≤ k < ω.

Theorem (Sz)

Assume ♦κ or κ is inaccessible.Let R be a collection of ≤ κ many Π02(κ) relations on a closed subset X of κκ.Then either

X has a κ-perfect R-homogeneous subset, or

there exists a tree T without κ-branches, |T | ≤ 2κ, such that for allR-homogeneous Y ⊆ X, we have Y ∩KerT (Y ) = ∅(that is, player II does not win VT (Y, y) for any y ∈ Y ).

If κ is inaccessible, then there exists a tree T of size κ witnessing this.

Sketch of the proof

First, show that if X has a κ-dense in itself R-homogeneous subset, then X has aκ-perfect R-homogeneous subset.



Theorem (Sz)



there exists a tree T without κ-branches, |T | ≤ 2κ, such that for allR-homogeneous Y ⊆ X, we have Y ∩KerT (Y ) = ∅(that is, player II does not win VT (Y, y) for any y ∈ Y ).If κ is inaccessible, then there exists a tree T of size κ witnessing this.

Sketch of the proof




Theorem (Sz)



there exists a tree T without κ-branches, |T | ≤ 2κ, such that for allR-homogeneous Y ⊆ X, we have Y ∩KerT (Y ) = ∅(that is, player II does not win VT (Y, y) for any y ∈ Y ).If κ is inaccessible, then there exists a tree T of size κ witnessing this.

Sketch of the proof


Step 2

Let R be an arbitrary set of finitary relations on κκ.

LemmaIf X does not have a κ-dense in itself R-homogeneous subset, thenthere exists a tree T without κ-branches, |T | ≤ 2κ, such that Y ∩KerT (Y ) = ∅holds for all R-homogeneous Y ⊆ X.

Proof.

The assumption holds iff II does not win Vκ(Y, x) for any R-homogeneousY ⊆ κκ and x ∈ Y .T0 = the tree of winning strategies τ of II in short games Vδ(X,x) (where δ < κand x ∈ X) such that the set of all possible τ -moves of II R-homogeneous.T = σT0, the tree of ascending chains in T0.

Remark

If all R-homogeneous sets Y ⊆ X are κ-scattered (i.e., I wins Vκ(Y, y) for ally ∈ Y ), then there exists a tree S without κ-branches, |S| ≤ 2κ such thatScS(Y ) = Y (i.e., I wins VS(Y, y) for all y ∈ Y ) for all R-homogeneous Y ⊆ κκ.

Step 2



Proof.

The assumption holds iff II does not win Vκ(Y, x) for any R-homogeneousY ⊆ κκ and x ∈ Y .

T0 = the tree of winning strategies τ of II in short games Vδ(X,x) (where δ < κand x ∈ X) such that the set of all possible τ -moves of II R-homogeneous.T = σT0, the tree of ascending chains in T0.

Remark


Step 2



Proof.

The assumption holds iff II does not win Vκ(Y, x) for any R-homogeneousY ⊆ κκ and x ∈ Y .T0 = the tree of winning strategies τ of II in short games Vδ(X,x) (where δ < κand x ∈ X) such that the set of all possible τ -moves of II R-homogeneous.

T = σT0, the tree of ascending chains in T0.

Remark


Step 2



Proof.


Remark


Step 2



Proof.


Remark


A corollary

Corollary

If λ > κ is weakly compact, and G is Col(κ, κ, then X has a κ-perfectR-homogeneous subset. (2)

This was known for measurable λ > κ (Sz, Väänänen).

Question

What is the consistency strength of (2)?

Thank you!

Perfect Set Games and Colorings on Generalized [2 pt] Baire Spaces · 2020. 10. 14. ·...

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Transcript of Perfect Set Games and Colorings on Generalized [2 pt] Baire Spaces · 2020. 10. 14. ·...