（大阪府立大学） 嘉田 勝 Remarks on Scheepers’ Theorem on the cardinality of Lindel¨ of spaces Masaru Kada (Osaka Prefecture University, Japan) RIMS conference “Interplay between large cardinals and small cardinals” Kyoto, Japan / October 25, 2010
Summary Theorem (Scheepers, 2010) CON( ∃ measurable) ⇒ of space has size ≤ 2 ℵ 0 ) CON(every points- G δ indestructibly Lindel¨ This exhibits an interplay between large cardinals and small cardinals ! We will briefly review the role of – a measurable cardinal and – combinatorial (game-theoretic?) properties of precipitous ideals in the proof of the above theorem.
Background: The cardinality of Lindel¨ of spaces Theorem (Arhangel’ski˘ ı, 1969) of spaces have cardinality ≤ 2 ℵ 0 . First-countable T 2 Lindel¨ Question: What about points- G δ T 2 Lindel¨ of spaces? (Remark: points- G δ spaces are T 1 , but not necessarily T 2 )
Background: The cardinality of Lindel¨ of spaces Theorem (Arhangel’ski˘ ı, 1969) of spaces have cardinality ≤ 2 ℵ 0 . First-countable T 2 Lindel¨ Question: What about points- G δ T 2 Lindel¨ of spaces? Consistent negative answer: Theorem (Shelah, 1996)(Gorelic, 1993) ( of space of size 2 ℵ 1 ) ZFC + CH + ∃ 0-dim. points- G δ Lindel¨ CON The consistency of a positive answer is open.
Background: The cardinality of Lindel¨ of spaces Theorem (Arhangel’ski˘ ı, 1969) of spaces have cardinality ≤ 2 ℵ 0 . First-countable T 2 Lindel¨ Question: What about points- G δ T 2 Lindel¨ of spaces? A known ZFC-provable upper bound for T 1 spaces: Theorem (Arhangel’ski˘ ı) Points- G δ Lindel¨ of spaces have cardinality < the least measurable. Theorem (Juh´ asz) There are points- G δ (but not T 2 ) Lindel¨ of spaces of arbitrarily large cardi- nality below the least measurable.
Indestructible Lindel¨ ofness Definition (Tall) A Lindel¨ of space ( X, τ ) is indestructibly Lindel¨ of if its Lindel¨ ofness is preserved by any <ω 1 -closed forcing notion.
Indestructible Lindel¨ ofness Definition (Tall) A Lindel¨ of space ( X, τ ) is indestructibly Lindel¨ of if its Lindel¨ ofness is preserved by any <ω 1 -closed forcing notion. Terminology For a Lindel¨ of space ( X, τ ) , ofness of ( X, τ ) if ( ˇ X, τ P ) is forced to be Lindel¨ of in P preserves Lindel¨ V P , where τ P is a P -name for the topology generated by ˇ τ .
Indestructible Lindel¨ ofness Definition (Tall) A Lindel¨ of space ( X, τ ) is indestructibly Lindel¨ of if its Lindel¨ ofness is preserved by any <ω 1 -closed forcing notion. Example • Hereditarily Lindel¨ of spaces are indestructible. Consequently, any subspace of a Euclidean space R n is indestructibly Lindel¨ of. ofness of 2 ω 1 is destroyed by adding a Cohen subset of • The Lindel¨ ω 1 with countable conditions (forcing with Fn( ω 1 , 2 , ω 1 ) ).
Games on topological spaces of transfinite length Game G <ω 1 ( O , O ) on ( X, τ ) : 1 U 0 U 1 U 2 · · · U ξ · · · One · · · · · · Two O 0 O 1 O 2 O ξ For each ξ < ω 1 , U ξ is an open cover of ( X, τ ) ; O ξ ∈ U ξ . [ ∪ { O ξ : ξ < γ } = X ] Two wins if ∃ γ < ω 1 . Remark ∪ { O ξ : ξ < ω 1 } = X is not enough for Two to win!
Games on topological spaces of transfinite length Game G <ω 1 ( O , O ) on ( X, τ ) : 1 U 0 U 1 U 2 · · · U ξ · · · One · · · · · · Two O 0 O 1 O 2 O ξ For each ξ < ω 1 , U ξ is an open cover of ( X, τ ) ; O ξ ∈ U ξ . [ ∪ { O ξ : ξ < γ } = X ] Two wins if ∃ γ < ω 1 . Theorem (Scheepers–Tall) ( X, τ ) is indestructibly Lindel¨ of One does not have a winning strategy in G <ω 1 ⇐ ⇒ ( O , O ) on ( X, τ ) 1
Cardinality of indestructibly Lindel¨ of spaces Question Is there a smaller upper bound for the cardinality of points- G δ of spaces than the least measurable? indestructibly Lindel¨ Recall: Theorem (Arhangel’ski˘ ı) Points- G δ Lindel¨ of spaces have cardinality < the least measurable. Theorem (Juh´ asz) There are points- G δ (but not T 2 ) Lindel¨ of spaces of arbitrarily large cardinality below the least measurable.
Cardinality of indestructibly Lindel¨ of spaces Question Is there a smaller upper bound for the cardinality of points- G δ of spaces than the least measurable? indestructibly Lindel¨ Theorem (Tall, 1995) CON ( ∃ supercompact ) ⇒ of space has size ≤ ℵ 1 ) CON ( CH + every points- G δ indestructibly Lindel¨
Cardinality of indestructibly Lindel¨ of spaces Question Is there a smaller upper bound for the cardinality of points- G δ of spaces than the least measurable? indestructibly Lindel¨ Theorem (Tall, 1995) CON ( ∃ supercompact ) ⇒ of space has size ≤ ℵ 1 ) CON ( CH + every points- G δ indestructibly Lindel¨ Theorem (Scheepers, 2010) CON ( ∃ measurable ) ⇒ ( of space has size ≤ 2 ℵ 0 ) CON every points- G δ indestructibly Lindel¨ (Scheepers’ method allows us some flexibility of the value of 2 ℵ 0 .)
Games related to precipitous ideals I : a nonprincipal ℵ 1 -complete (not necessarily κ -complete) ideal on κ Game G ( I ) : · · · · · · One O 0 O 1 O 2 O n · · · · · · Two T 0 T 1 T 2 T n O n ’s, T n ’s are all from I + , O 0 ⊇ T 0 ⊇ O 1 ⊇ · · · ⊇ O n ⊇ T n ⊇ · · · . ⇒ ∩ Two wins ⇐ n<ω T n ̸ = ∅ Definition I is weakly precipitous ⇐ ⇒ One does not have a w.s. in G ( I ) (Note: I is precipitous ⇐ ⇒ I is κ -complete and weakly precipitous)
Games related to precipitous ideals (cont’d) I : a nonprincipal ℵ 1 -complete (not necessarily κ -complete) ideal on κ Game DG ω ( I + , ⊆ ) : · · · · · · One O 0 O 1 O 2 O n · · · · · · Two T 0 T 1 T 2 T n O n ’s, T n ’s are all from I + , O 0 ⊇ T 0 ⊇ O 1 ⊇ · · · ⊇ O n ⊇ T n ⊇ · · · . n<ω T n ∈ I + Two wins ⇐ ⇒ ∩ (A descending chain (Banach–Mazur) game on a poset ( I + , ⊆ ) )
Descending chain games of transfinite length I : a nonprincipal ℵ 1 -complete (not necessarily κ -complete) ideal on κ Game DG <ω 1 Two ( I + , ⊆ ) : · · · · · · · · · One O 0 O 1 O ω +1 O ξ (pass) · · · · · · · · · Two T 0 T 1 T ω T ω +1 T ξ O ξ ’s, T ξ ’s ( ξ < ω 1 ) are all from I + . In each limit inning Two has the initiative. O 0 ⊇ T 0 ⊇ O 1 ⊇ · · · ⊇ T ω ⊇ O ω +1 ⊇ T ω +1 ⊇ · · · ⊇ O ξ ⊇ T ξ ⊇ · · · . Two wins ⇐ ⇒ the play is sustained all over ω 1 innings (A descending chain (Banach–Mazur) game on a poset ( I + , ⊆ ) of transfinite length )
Descending chain games of transfinite length (cont’d) Theorem (Foreman)(Veliˇ ckovi´ c) For a poset ( P, ≤ ) , ⇒ Two has a w.s. in DG <ω 1 Two has a w.s. in DG ω ( P, ≤ ) ⇐ Two ( P, ≤ ) In particular, Two has a w.s. in DG ω ( I + , ⊆ ) ⇐ ⇒ Two has a w.s. in DG <ω 1 Two ( I + , ⊆ )
Diagram For a nonprincipal < ℵ 1 -complete ideal I on κ : I : dual of measure u.f. − − − − − → I is precipitous ? ? ? ? y y Two ↑ tactic DG ω ( I + , ⊆ ) ↓ ? ? ? ? y y Two ↑ DG <ω 1 Two ( I + , ⊆ ) Two ↑ DG ω ( I + , ⊆ ) ↓ ? ? ? ? ? ? y y y One ̸↑ DG <ω 1 Two ( I + , ⊆ ) − One ̸↑ DG ω ( I + , ⊆ ) − − − − → ↓ ? ? ? ? y y One ̸↑ G ( I ) I is weakly precipitous
Diagram κ > 2 ℵ 0 κ is measurable κ may be ℵ 1 I : dual of measure u.f. − − − − − → I is precipitous ? ? ? ? y y Two ↑ tactic DG ω ( I + , ⊆ ) ↓ ? ? ? ? y y Two ↑ DG <ω 1 Two ( I + , ⊆ ) Two ↑ DG ω ( I + , ⊆ ) ↓ ? ? ? ? ? ? y y y One ̸↑ DG <ω 1 Two ( I + , ⊆ ) − One ̸↑ DG ω ( I + , ⊆ ) − − − − → ↓ ? ? ? ? y y One ̸↑ G ( I ) I is weakly precipitous
Diagram After collapsing a measurable κ to ℵ 1 , I (the dual of the measure u.f.) satisfies: I : dual of measure u.f. − − − − − → I is precipitous ? ? ? ? y y Two ↑ tactic DG ω ( I + , ⊆ ) ↓ ? ? ? ? y y Two ↑ DG <ω 1 Two ( I + , ⊆ ) Two ↑ DG ω ( I + , ⊆ ) ↓ ? ? ? ? ? ? y y y One ̸↑ DG <ω 1 Two ( I + , ⊆ ) − One ̸↑ DG ω ( I + , ⊆ ) − − − − → ↓ ? ? ? ? y y One ̸↑ G ( I ) I is weakly precipitous
Diagram After collapsing a measurable κ to ℵ 2 , I (the dual of the measure u.f.) satisfies: I : dual of measure u.f. − − − − − → I is precipitous ? ? ? ? y y Two ↑ tactic DG ω ( I + , ⊆ ) ↓ ? ? ? ? y y Two ↑ DG <ω 1 Two ( I + , ⊆ ) Two ↑ DG ω ( I + , ⊆ ) ↓ ? ? ? ? ? ? y y y One ̸↑ DG <ω 1 Two ( I + , ⊆ ) − One ̸↑ DG ω ( I + , ⊆ ) − − − − → ↓ ? ? ? ? y y One ̸↑ G ( I ) I is weakly precipitous
The main theorem Theorem (Scheepers) If there is a nonprincipal ℵ 1 -complete ideal I on κ such that Two has a winning tactic in DG ω ( I + , ⊆ ) , then every points- G δ indestructibly Lindel¨ of space has cardinality < κ .
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