Perfect Set Games and Colorings on Generalized Baire Spaces - PowerPoint PPT Presentation

Perfect Set Games and Colorings on Generalized Baire Spaces Dorottya Sziráki 5 th Workshop on Generalized Baire Spaces Bristol, 4 February 2020 Dorottya Sziráki Perfect set games and colorings on κ κ

Perfectness for the κ -Baire space Assume κ <κ = κ . Dorottya Sziráki Perfect set games and colorings on κ κ

Perfectness for the κ -Baire space Assume κ <κ = κ . A subset of κ κ is closed iff it is the set of branches [ T ] = { x ∈ κ κ : x ↾ α ∈ T for all α < κ } of a subtree T of <κ κ . Dorottya Sziráki Perfect set games and colorings on κ κ

Perfectness for the κ -Baire space Assume κ <κ = κ . A subset of κ κ is closed iff it is the set of branches [ T ] = { x ∈ κ κ : x ↾ α ∈ T for all α < κ } of a subtree T of <κ κ . Example A subset of a topological space is perfect in the usual sense iff it is closed and contains no isolated points. X ω = { x ∈ κ 2 : |{ α < κ : x ( α ) = 0 }| < ω } is perfect in this usual sense, but | X ω | = κ . Dorottya Sziráki Perfect set games and colorings on κ κ

Perfectness for the κ -Baire space Assume κ <κ = κ . A subset of κ κ is closed iff it is the set of branches [ T ] = { x ∈ κ κ : x ↾ α ∈ T for all α < κ } of a subtree T of <κ κ . Example A subset of a topological space is perfect in the usual sense iff it is closed and contains no isolated points. X ω = { x ∈ κ 2 : |{ α < κ : x ( α ) = 0 }| < ω } is perfect in this usual sense, but | X ω | = κ . Definition A subtree T of <κ κ is a strongly 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 strongly κ -perfect set if X = [ T ] for a strongly κ -perfect tree T . Dorottya Sziráki Perfect set games and colorings on κ κ

Väänänen’s perfect set game Let X ⊆ κ κ , let x 0 ∈ κ κ and let ω ≤ γ ≤ κ . Definition (Väänänen, 1991) The game V γ ( X, x 0 ) has length γ and is played as follows: I U 1 . . . U α . . . . . . . . . II x 0 x 1 x α II first plays x 0 . In each round 0 < α < γ , I plays a basic open subset U α of X , and then II chooses x α ∈ U α with x α � = x β for all β < α . I has to play so that U β +1 ∋ x β in each successor round β + 1 < γ and U α = � β<α U β in each limit round α < γ . II wins a given run of the game if II can play legally in all rounds α < γ . Dorottya Sziráki Perfect set games and colorings on κ κ

Väänänen’s perfect set game Let X ⊆ κ κ , let x 0 ∈ κ κ and let ω ≤ γ ≤ κ . Definition (Väänänen, 1991) The game V γ ( X, x 0 ) has length γ and is played as follows: I U 1 . . . U α . . . . . . . . . II x 0 x 1 x α II first plays x 0 . In each round 0 < α < γ , I plays a basic open subset U α of X , and then II chooses x α ∈ U α with x α � = x β for all β < α . I has to play so that U β +1 ∋ x β in each successor round β + 1 < γ and U α = � β<α U β in each limit round α < γ . II wins a given run of the game if II can play legally in all rounds α < γ . Dorottya Sziráki Perfect set games and colorings on κ κ

Perfect and scattered subsets of the κ -Baire space Let X ⊆ κ κ , and suppose ω ≤ γ ≤ κ . Definition (Väänänen, 1991) X is a γ -scattered set if I wins V γ ( X, x 0 ) for all x 0 ∈ X . X is a γ -perfect set if X is closed and II wins V γ ( X, x 0 ) for all x 0 ∈ X . Dorottya Sziráki Perfect set games and colorings on κ κ

Perfect and scattered subsets of the κ -Baire space Let X ⊆ κ κ , and suppose ω ≤ γ ≤ κ . Definition (Väänänen, 1991) X is a γ -scattered set if I wins V γ ( X, x 0 ) for all x 0 ∈ X . X is a γ -perfect set if X is closed and II wins V γ ( X, x 0 ) for all x 0 ∈ X . X is ω -perfect iff X is perfect in the usual sense (i.e., iff X closed and has no isolated points). Dorottya Sziráki Perfect set games and colorings on κ κ

Perfect and scattered subsets of the κ -Baire space Let X ⊆ κ κ , and suppose ω ≤ γ ≤ κ . Definition (Väänänen, 1991) X is a γ -scattered set if I wins V γ ( X, x 0 ) for all x 0 ∈ X . X is a γ -perfect set if X is closed and II wins V γ ( X, x 0 ) for all x 0 ∈ X . X is ω -perfect iff X is perfect in the usual sense (i.e., iff X closed and has no isolated points). X is ω -scattered iff X is scattered in the usual sense (i.e., each nonempty subspace contains an isolated point). Dorottya Sziráki Perfect set games and colorings on κ κ

Perfect and scattered subsets of the κ -Baire space Let X ⊆ κ κ , and suppose ω ≤ γ ≤ κ . Definition (Väänänen, 1991) X is a γ -scattered set if I wins V γ ( X, x 0 ) for all x 0 ∈ X . X is a γ -perfect set if X is closed and II wins V γ ( X, x 0 ) for all x 0 ∈ X . X is ω -perfect iff X is perfect in the usual sense (i.e., iff X closed and has no isolated points). X is ω -scattered iff X is scattered in the usual sense (i.e., each nonempty subspace contains an isolated point). V γ ( X, x 0 ) may not be determined when γ > ω . Dorottya Sziráki Perfect set games and colorings on κ κ

κ -perfect sets vs. strongly κ -perfect sets Dorottya Sziráki Perfect set games and colorings on κ κ

κ -perfect sets vs. strongly κ -perfect sets Example (Huuskonen) The following set is κ -perfect but is not strongly κ -perfect: Y ω = { x ∈ κ 3 : |{ α < κ : x ( α ) = 2 }| < ω } . Dorottya Sziráki Perfect set games and colorings on κ κ

κ -perfect sets vs. strongly κ -perfect sets Example (Huuskonen) The following set is κ -perfect but is not strongly κ -perfect: Y ω = { x ∈ κ 3 : |{ α < κ : x ( α ) = 2 }| < ω } . Proposition Let X be a closed subset of κ κ . � X is κ -perfect ⇐ ⇒ X = X i for strongly κ -perfect sets X i . i ∈ I Dorottya Sziráki Perfect set games and colorings on κ κ

Väänänen’s generalized Cantor-Bendixson theorem Theorem (Väänänen, 1991) The following Cantor-Bendixson theorem for κ κ is consistent relative to the existence of a measurable cardinal λ > κ : Every closed subset of κ κ is the (disjoint) union of a κ -perfect set and a κ -scattered set, which is of size ≤ κ . Dorottya Sziráki Perfect set games and colorings on κ κ

Väänänen’s generalized Cantor-Bendixson theorem Theorem (Väänänen, 1991) The following Cantor-Bendixson theorem for κ κ is consistent relative to the existence of a measurable cardinal λ > κ : Every closed subset of κ κ 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 to the existence of an inaccessible cardinal λ > κ . Dorottya Sziráki Perfect set games and colorings on κ κ

Väänänen’s generalized Cantor-Bendixson theorem Proposition (Sz) Väänänen’s generalized Cantor-Bendixson theorem is equivalent to the κ -perfect set property for closed subsets of κ κ (i.e, the statement that every closed subset of κ κ of size > κ has a κ -perfect subset). Remark: The κ -PSP for closed subsets of κ κ is equiconsistent with the existence of an inaccessible cardinal λ > κ . Dorottya Sziráki Perfect set games and colorings on κ κ

Väänänen’s generalized Cantor-Bendixson theorem Proposition (Sz) Väänänen’s generalized Cantor-Bendixson theorem is equivalent to the κ -perfect set property for closed subsets of κ κ (i.e, the statement that every closed subset of κ κ of size > κ has a κ -perfect subset). Remark: The κ -PSP for closed subsets of κ κ is equiconsistent with the existence of an inaccessible cardinal λ > κ . Sketch of the proof. Let X be a closed subset of κ κ . Its set of κ -condensation points is defined to be CP κ ( X ) = { x ∈ X : | X ∩ N x ↾ α | > κ for all α < κ } . If the κ -PSP holds for closed subsets of κ κ , then CP κ ( X ) is a κ -perfect set and X − CP κ ( X ) is a κ -scattered set of size ≤ κ . Dorottya Sziráki Perfect set games and colorings on κ κ

Perfect and scattered trees Let T be a subtree of <κ 2 , let t ∈ T , and let ω ≤ γ ≤ κ . Definition (Galgon, 2016) The game G γ ( T, t ) has length γ and is played as follows: I δ 0 i 0 . . . δ α i α . . . . . . . . . II t 0 t α In each round α < γ , player I first plays δ α < κ . Then II plays a node t α ∈ T of height ≥ δ α , and I chooses i α < 2 . II has to play so that t ⊆ t 0 , and t β⌢ � i β � ⊆ t α for all β < α < γ . II wins a given run of the game if II can play legally in all rounds α < γ . Dorottya Sziráki Perfect set games and colorings on κ κ

Perfect and scattered trees Let T be a subtree of <κ 2 , let t ∈ T , and let ω ≤ γ ≤ κ . Definition (Galgon, 2016) The game G γ ( T, t ) has length γ and is played as follows: I δ 0 i 0 . . . δ α i α . . . . . . . . . II t 0 t α In each round α < γ , player I first plays δ α < κ . Then II plays a node t α ∈ T of height ≥ δ α , and I chooses i α < 2 . II has to play so that t ⊆ t 0 , and t β⌢ � i β � ⊆ t α for all β < α < γ . II wins a given run of the game if II can play legally in all rounds α < γ . Dorottya Sziráki Perfect set games and colorings on κ κ