I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions Baire property and the Ellentuck-Prikry topology Vincenzo Dimonte February 18, 2020 Joint work with Xianghui Shi 1 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions Inspiration (Woodin) I0 is a large cardinal similar to AD. Motivation • Proving theorems that reinforce such statement • Understanding the deep reasons behind such similarity Definition (Woodin, 1980) We say that I0( λ ) holds iff there is an elementary embedding j : L ( V λ +1 ) ≺ L ( V λ +1 ) such that j ↾ V λ +1 is not the identity. It is a large cardinal: if I0( λ ) holds, then λ is a strong limit cardinal of cofinality ω , limit of cardinals that are n -huge for every n ∈ ω . 2 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions These are some similiarities with AD: L ( R ) under AD L ( V λ +1 ) under I0( λ ) DC DC λ Θ L ( V λ +1 ) is regular Θ is regular λ + is measurable ω 1 is measurable the Coding Lemma holds the Coding Lemma holds Theorem (Laver) Let � κ n : n ∈ ω � be a cofinal sequence in λ . For every A ⊆ V λ : • A is Σ 1 1 -definable in ( V λ , V λ +1 ) iff there is a tree T ⊆ � n ∈ ω V κ n × � n ∈ ω V κ n whose projection is A ; • A is Σ 1 2 -definable in ( V λ , V λ +1 ) iff there is a tree n ∈ ω V κ n × λ + whose projection is A . T ⊆ � Let us go a bit deeper. 3 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions We define a topology on V λ +1 : Since V λ +1 = P ( V λ ), the basic open sets of the topology are, for any α < λ and a ⊆ V α , O ( a ,α ) = { b ∈ V λ +1 : b ∩ V α = a } . Theorem (Cramer, 2015) Suppose I0( λ ). Then for every X ⊆ V λ +1 , X ∈ L ( V λ +1 ), either | X | ≤ λ or ω λ can be continuously embedded inside X ( ω λ with the bounded topology). This is similar to AD: in fact, under AD every subset of the reals has the Perfect Set Property. But the proof is completely different: Cramer uses heavily elementary embeddings (inverse limit reflection), while in the classical case involves games. 4 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions In recent work, with Motto Ros we clarified the similarity. The classical case is: • Large cardinals ⇒ every set of reals in L ( R ) is homogeneously Suslin • Every homogeneously Suslin set is determined (so L ( R ) � AD) • Every determined set has the Perfect Set Property 5 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions But there is a shortcut for regularity properties: • Infinite Woodin cardinals ⇒ every set of reals in L ( R ) is weakly homogeneously Suslin • Every weakly homogeneously Suslin set has the Perfect Set Property 6 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions D.-Motto Ros Let λ be a strong limit cardinal of cofinality ω , and let � κ n : n ∈ ω � be a increasing cofinal sequence in λ . Then the following spaces are isomorphic: • λ 2, with the bounded topology; • ω λ , with the bounded topology, and the discrete topology in every copy of λ ; • � n ∈ ω κ n , with the bounded topology and the discrete topology in every κ n ; • if | V λ | = λ , V λ +1 , with the previously defined topology. Moreover, they are λ -Polish, i.e., completely metrizable and with a dense subset of cardinality λ . 7 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions So, for example, we can rewrite Cramer’s result as: Theorem (Cramer, 2015) Suppose I0( λ ). Then L ( λ 2) � ∀ X X ⊆ λ 2 has the λ -PSP. For any λ strong limit of cofinality ω , we defined representable subsets of ω λ , a generalization of weakly homogeneously Suslin sets. D.-Motto Ros Let λ strong limit of cofinality ω . Then every representable subset of ω λ has the λ -PSP. Cramer’s analysis of I0 finalizes the similarity with AD: Theorem (Cramer, to appear) Suppose I0( λ ). Then every X ⊆ V λ +1 , X ∈ L ( V λ +1 ) is representa- ble. 8 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions • Infinite Woodin cardinals ⇒ every set of reals in L ( R ) is weakly homogeneously Suslin • Every weakly homogeneously Suslin set has the Perfect Set Property 9 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions • I0 ⇒ every subset of V λ +1 in L ( V λ +1 ) is representable • Every representable set has the λ -Perfect Set Property 10 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions This approach is more scalable, as it works even in λ -Polish spaces such that λ does not satisfy I0: D.-Motto Ros Suppose I0( λ ). Then it is consistent that there is κ strong limit of cofinality ω such that all the subsets of ω κ in L ( V κ +1 ) have the κ -PSP, and ¬ I0( κ ). 11 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions The next step would be to analyze the Baire Property. 12 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions The most natural thing is to define nowhere dense sets as usual, λ -meager sets as λ -union of nowhere dense sets and λ -comeager sets as complement of λ -meager sets. 13 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions The most natural thing is to define nowhere dense sets as usual, λ -meagre sets as λ -union of nowhere dense sets and λ -comeagre sets as complement of λ -meagre sets. 14 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions Let f : ω → ω . n ∈ ω κ f ( n ) is nowhere dense in ω λ . Then D f = � But ω λ = � f ∈ ω ω D f , therefore the whole space is λ -meagre (in fact, it is c -meagre), and the Baire property in this setting is just nonsense. 15 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions Or is it? 16 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions From now on, we work with λ strong limit of cofinality ω , � κ n : n ∈ ω � a strictly increasing cofinal sequence of measurable cardinals in λ . The space we work in is � n ∈ ω κ n . Idea • Baire category is closely connected to Cohen forcing • The space κ 2, with κ regular, is κ -Baire (i.e., every nonempty open set is not κ -meagre) because Cohen forcing on κ is < κ -distributive • “Cohen” forcing on λ singular is not < λ -distributive, and this is why λ 2 is not λ -Baire • But there are other forcings on λ that are < λ -distributive, like Prikry forcing • We can try to define Baire category via Prikry forcing instead of Cohen forcing. 17 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions Definition Let λ be strong limit of cofinality ω , � κ n : n ∈ ω � a strictly increasing cofinal sequence of measurable cardinals in λ , and fix µ n a measure for each κ n . The Prikry forcing P � µ on λ respect to � µ has conditions of the form � α 1 , . . . , α n , A n +1 , A n +2 . . . � , where α i ∈ κ i and A i ∈ µ i . � α 1 , . . . , α n � is the stem of the condition. � β 1 , . . . , β m , B m +1 , B m +2 . . . � ≤ � α 1 , . . . , α n , A n +1 , A n +2 . . . � iff m ≥ n and • for i ≤ n β i = α i • for n < i ≤ m β i ∈ A i • for i > m B i ⊆ A i . p ≤ ∗ q if p ≤ q and they have the same stem. 18 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions Definition The Ellentuck-Prikry � µ -topology (in short EP-topology) on � n ∈ ω κ n is the topology generated by the family { O p : p ∈ P � µ } , where if p = � α 1 , . . . , α n , A n +1 , A n +2 . . . � , then � O p = { x ∈ κ n : ∀ i ≤ n x ( i ) = α i , ∀ i > n x ( i ) ∈ A i } . n ∈ ω The EP-topology is a refinement of the bounded topology: if a set is open in the bounded topology, it is open also in the EP-topology, but not viceversa (in fact, many open sets in the EP-topology are nowhere dense in the bounded topology). 19 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions There is a connection between the concepts of “open” and “dense” relative to the forcing and relative to the topology: µ (forcing) � n ∈ ω κ n (EP-topology) P � l O = { x ∈ � O open → n ∈ ω κ n : ∃ p ∈ P � µ x ∈ O p } open ← l U = U open { p ∈ P � µ : O p ⊆ U } open l O open dense O open dense → l U open dense ← U open dense l ( l U ) = U , but not viceversa. 20 / 30
I0 and the prehistory of singular GDST The λ -PSP and I0 λ -Baire Category Questions Definition Let X be a topological space. • a set A ⊆ X is λ -meagre iff it is the λ -union of nowhere dense sets • a set A ⊆ X is λ -comeagre iff it is the complement of a λ -meagre set • a set A ⊆ X has the λ -Baire property iff there is an open set U such that A △ U is λ -meagre • X is a λ -Baire space iff every nonempty open set in X is not λ -meagre, i.e., the intersection of λ -many open dense sets is dense. 21 / 30
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