Open Colorings, Perfect Sets and Games on Generalized Baire Spaces - PowerPoint PPT Presentation

Open Colorings, Perfect Sets and Games on Generalized Baire Spaces Dorottya Sziráki MTA Alfréd Rényi Institute of Mathematics SETTOP 2018 Novi Sad Conference in Set Theory and General Topology July 5, 2018

Generalized Baire spaces Let κ be an uncountable cardinal such that κ <κ = κ . The κ -Baire space κ κ is the set of functions f : κ → κ , with the bounded topology: basic open sets are of the form N s = { f ∈ κ κ : s ⊂ f } , where s ∈ <κ κ .

Generalized Baire spaces Let κ be an uncountable cardinal such that κ <κ = κ . The κ -Baire space κ κ is the set of functions f : κ → κ , with the bounded topology: basic open sets are of the form N s = { f ∈ κ κ : s ⊂ f } , where s ∈ <κ κ . The κ -Cantor space κ 2 is defined similarly.

Generalized Baire spaces Let κ be an uncountable cardinal such that κ <κ = κ . The κ -Baire space κ κ is the set of functions f : κ → κ , with the bounded topology: basic open sets are of the form N s = { f ∈ κ κ : s ⊂ f } , where s ∈ <κ κ . The κ -Cantor space κ 2 is defined similarly. κ -Borel sets: close the family of open subsets under intersections and unions of size ≤ κ and complementation.

Open coloring axioms for subsets of the κ -Baire space Let X ⊆ κ κ . OCA κ ( X ) : Suppose [ X ] 2 = R 0 ∪ R 1 is an open partition (i.e. { ( x, y ) : { x, y } ∈ R 0 } is an open subset of X × X ). Then either X is a union of κ -many R 1 -homogeneous sets, or there exists an R 0 -homogeneous set of size κ + .

Open coloring axioms for subsets of the κ -Baire space Let X ⊆ κ κ . OCA κ ( X ) : Suppose [ X ] 2 = R 0 ∪ R 1 is an open partition (i.e. { ( x, y ) : { x, y } ∈ R 0 } is an open subset of X × X ). Then either X is a union of κ -many R 1 -homogeneous sets, or there exists an R 0 -homogeneous set of size κ + . OCA ∗ κ ( X ) : If [ X ] 2 = R 0 ∪ R 1 is an open partition, then either X is a union of κ -many R 1 -homogeneous sets, or X has a κ -perfect R 0 -homogeneous subset ,

Open coloring axioms for subsets of the κ -Baire space Let X ⊆ κ κ . OCA κ ( X ) : Suppose [ X ] 2 = R 0 ∪ R 1 is an open partition (i.e. { ( x, y ) : { x, y } ∈ R 0 } is an open subset of X × X ). Then either X is a union of κ -many R 1 -homogeneous sets, or there exists an R 0 -homogeneous set of size κ + . OCA ∗ κ ( X ) : If [ X ] 2 = R 0 ∪ R 1 is an open partition, then either X is a union of κ -many R 1 -homogeneous sets, or X has a κ -perfect R 0 -homogeneous subset , i.e., there is a continuous embedding f : κ 2 → X whose image is R 0 -homogeneous.

OCA ∗ κ ( X ) for κ -analytic X κ -analytic or Σ 1 1 ( κ ) sets: continuous images of κ -Borel sets; equivalently: continuous images of closed sets. Theorem (Sz.) If λ > κ is inaccessible and G is Col( κ, <λ ) -generic, then κ ( Σ 1 OCA ∗ 1 ( κ )) holds in V [ G ] .

OCA ∗ κ ( X ) for κ -analytic X κ -analytic or Σ 1 1 ( κ ) sets: continuous images of κ -Borel sets; equivalently: continuous images of closed sets. Theorem (Sz.) If λ > κ is inaccessible and G is Col( κ, <λ ) -generic, then κ ( Σ 1 OCA ∗ 1 ( κ )) holds in V [ G ] . ◮ In the classical setting (when κ = ω ), OCA ∗ ( Σ 1 1 ) holds in ZFC (Feng, 1993).

OCA ∗ κ ( X ) for κ -analytic X κ -analytic or Σ 1 1 ( κ ) sets: continuous images of κ -Borel sets; equivalently: continuous images of closed sets. Theorem (Sz.) If λ > κ is inaccessible and G is Col( κ, <λ ) -generic, then κ ( Σ 1 OCA ∗ 1 ( κ )) holds in V [ G ] . ◮ In the classical setting (when κ = ω ), OCA ∗ ( Σ 1 1 ) holds in ZFC (Feng, 1993). ◮ For uncountable κ = κ <κ , OCA ∗ κ ( Σ 1 1 ( κ )) is equiconsistent with the existence of an inaccessible λ > κ by our result.

OCA ∗ κ ( X ) for definable X ⊆ κ κ Work in progress If λ > κ is inaccessible and G is Col( κ, <λ ) -generic, then in V [ G ] , κ ( X ) holds for all X ⊆ κ κ definable from an element of κ κ . OCA ∗

OCA ∗ κ ( X ) for definable X ⊆ κ κ Work in progress If λ > κ is inaccessible and G is Col( κ, <λ ) -generic, then in V [ G ] , κ ( X ) holds for all X ⊆ κ κ definable from an element of κ κ . OCA ∗ ◮ The classical version of this result is due to Feng (1993).

OCA ∗ κ ( X ) for definable X ⊆ κ κ Work in progress If λ > κ is inaccessible and G is Col( κ, <λ ) -generic, then in V [ G ] , κ ( X ) holds for all X ⊆ κ κ definable from an element of κ κ . OCA ∗ ◮ The classical version of this result is due to Feng (1993). ◮ The κ -perfect set property holds for such subsets X (Schlicht, 2017).

Question Let OCA κ say: “ OCA κ ( X ) holds for all X ⊆ κ κ ”. Is OCA κ consistent? If so, how does it influence the structure of the κ -Baire space?

Perfectness for the κ -Baire space A subset of κ κ is closed if and only if it is the set of branches [ T ] = { x ∈ κ κ : x ↾ α ∈ T for all α < κ } of a subtree T of <κ κ .

Perfectness for the κ -Baire space A subset of κ κ is closed if and only if it is the set of branches [ T ] = { x ∈ κ κ : x ↾ α ∈ T for all α < κ } of a subtree T of <κ κ . Definition A subtree T of <κ κ is a strong κ -perfect tree if

Perfectness for the κ -Baire space A subset of κ κ is closed if and only if it is the set of branches [ T ] = { x ∈ κ κ : x ↾ α ∈ T for all α < κ } of a subtree T of <κ κ . Definition A subtree T of <κ κ is a strong κ -perfect tree if T is <κ -closed

Perfectness for the κ -Baire space A subset of κ κ is closed if and only if it is the set of branches [ T ] = { x ∈ κ κ : x ↾ α ∈ T for all α < κ } of a subtree T of <κ κ . Definition A subtree T of <κ κ is a strong κ -perfect tree if T is <κ -closed and every node of T extends to a splitting node.

Perfectness for the κ -Baire space A subset of κ κ is closed if and only if it is the set of branches [ T ] = { x ∈ κ κ : x ↾ α ∈ T for all α < κ } of a subtree T of <κ κ . Definition A subtree T of <κ κ is a strong κ -perfect tree if T is <κ -closed and every node of T extends to a splitting node. A set X ⊆ κ κ is a strong κ -perfect set if X = [ T ] for a strong κ -perfect tree T .

Väänänen’s perfect set game Perfectness was first generalized for the κ -Baire space by Väänänen, based on the following game. Definition (Väänänen) Let X ⊆ κ κ , let x 0 ∈ X and let ω ≤ γ ≤ κ . Then V γ ( X, x 0 ) is the following game. U 1 . . . U α . . . I . . . . . . II x 0 x 1 x α

Väänänen’s perfect set game Perfectness was first generalized for the κ -Baire space by Väänänen, based on the following game. Definition (Väänänen) Let X ⊆ κ κ , let x 0 ∈ X and let ω ≤ γ ≤ κ . Then V γ ( X, x 0 ) is the following game. U 1 . . . U α . . . I . . . . . . II x 0 x 1 x α I plays a basic open sets U α of X such that U α � U β for all β < α , and x β ∈ U β +1 at successor rounds α = β + 1 , and U α = � β<α U β at limit rounds α . II responds with x α ∈ U α such that x α � = x β for all β < α . Player II wins the run if she can make all her γ moves legally.

Väänänen’s perfect set game Let X ⊆ κ κ and let ω ≤ γ ≤ κ . Definition (Väänänen) X is γ -perfect if II wins V γ ( X, x 0 ) for all x 0 ∈ X .

Väänänen’s perfect set game Let X ⊆ κ κ and let ω ≤ γ ≤ κ . Definition (Väänänen) X is γ -perfect if II wins V γ ( X, x 0 ) for all x 0 ∈ X . X is γ -scattered if I wins V γ ( X, x 0 ) for all x 0 ∈ X .

κ -perfect and κ -scattered trees Definition Let T be a subtree of <κ κ , and let t 0 ∈ T . Then G ∗ κ ( T, t 0 ) is the following game. i 1 . . . i α . . . I t 0 1 , t 1 t 0 α , t 1 . . . . . . II 1 α α ⊃ t i β α and t i II plays t 0 α , t 1 α ∈ T such that t 0 α ⊥ t 1 β for all β < α and i < 2 . Then I chooses, by playing i α < 2 . (Thus, II plays a pair of disjoint basic open subsets of [ T ] which are contained in the previously chosen basic open sets).

κ -perfect and κ -scattered trees Definition Let T be a subtree of <κ κ , and let t 0 ∈ T . Then G ∗ κ ( T, t 0 ) is the following game. i 1 . . . i α . . . I t 0 1 , t 1 t 0 α , t 1 . . . . . . II 1 α α ⊃ t i β α and t i II plays t 0 α , t 1 α ∈ T such that t 0 α ⊥ t 1 β for all β < α and i < 2 . Then I chooses, by playing i α < 2 . (Thus, II plays a pair of disjoint basic open subsets of [ T ] which are contained in the previously chosen basic open sets). Player II wins the run if she can make all her κ moves legally.

κ -perfect and κ -scattered trees Definition Let T be a subtree of <κ κ , and let t 0 ∈ T . Then G ∗ κ ( T, t 0 ) is the following game. i 1 . . . i α . . . I t 0 1 , t 1 t 0 α , t 1 . . . . . . II 1 α α ⊃ t i β α and t i II plays t 0 α , t 1 α ∈ T such that t 0 α ⊥ t 1 β for all β < α and i < 2 . Then I chooses, by playing i α < 2 . (Thus, II plays a pair of disjoint basic open subsets of [ T ] which are contained in the previously chosen basic open sets). Player II wins the run if she can make all her κ moves legally. T is a κ -perfect tree if II wins G ∗ κ ( T, t 0 ) for all t 0 ∈ T .