Remarks on Scheepers Theorem on the cardinality of Lindel of spaces - - PowerPoint PPT Presentation

Remarks on Scheepers’ Theorem

• n the cardinality of Lindel¨
• f spaces

Masaru Kada

(Osaka Prefecture University, Japan)

嘉田 勝

（大阪府立大学）

RIMS conference “Interplay between large cardinals and small cardinals” Kyoto, Japan / October 25, 2010

Summary

Theorem (Scheepers, 2010) CON(∃ measurable) ⇒ CON(every points-Gδ indestructibly Lindel¨

• f space has size ≤ 2ℵ0)

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¨

• f spaces

Theorem (Arhangel’ski˘ ı, 1969) First-countable T2 Lindel¨

• f spaces have cardinality ≤ 2ℵ0.

Question: What about points-Gδ T2 Lindel¨

• f spaces?

(Remark: points-Gδ spaces are T1, but not necessarily T2)

Background: The cardinality of Lindel¨

• f spaces

Theorem (Arhangel’ski˘ ı, 1969) First-countable T2 Lindel¨

• f spaces have cardinality ≤ 2ℵ0.

Question: What about points-Gδ T2 Lindel¨

• f spaces?

Consistent negative answer: Theorem (Shelah, 1996)(Gorelic, 1993) CON ( ZFC + CH + ∃ 0-dim. points-Gδ Lindel¨

• f space of size 2ℵ1)

The consistency of a positive answer is open.

Background: The cardinality of Lindel¨

• f spaces

Theorem (Arhangel’ski˘ ı, 1969) First-countable T2 Lindel¨

• f spaces have cardinality ≤ 2ℵ0.

Question: What about points-Gδ T2 Lindel¨

• f spaces?

A known ZFC-provable upper bound for T1 spaces: Theorem (Arhangel’ski˘ ı) Points-Gδ Lindel¨

• f spaces have cardinality < the least measurable.

Theorem (Juh´ asz) There are points-Gδ (but not T2) Lindel¨

• f spaces of arbitrarily large cardi-

nality below the least measurable.

Indestructible Lindel¨

• fness

Definition (Tall) A Lindel¨

• f space (X, τ) is indestructibly Lindel¨
• f if its Lindel¨
• fness is

preserved by any <ω1-closed forcing notion.

Indestructible Lindel¨

• fness

Definition (Tall) A Lindel¨

• f space (X, τ) is indestructibly Lindel¨
• f if its Lindel¨
• fness is

preserved by any <ω1-closed forcing notion. Terminology For a Lindel¨

• f space (X, τ),

P preserves Lindel¨

• fness of (X, τ) if ( ˇ

X, τ P) is forced to be Lindel¨

• f in

VP, where τ P is a P-name for the topology generated by ˇ τ.

Indestructible Lindel¨

• fness

Definition (Tall) A Lindel¨

• f space (X, τ) is indestructibly Lindel¨
• f if its Lindel¨
• fness is

preserved by any <ω1-closed forcing notion. Example

• Hereditarily Lindel¨
• f spaces are indestructible. Consequently, any

subspace of a Euclidean space Rn is indestructibly Lindel¨

• f.
• The Lindel¨
• fness of 2ω1 is destroyed by adding a Cohen subset of

ω1 with countable conditions (forcing with Fn(ω1, 2, ω1)).

Games on topological spaces of transfinite length

Game G<ω1

1

(O, O) on (X, τ): One U0 U1 U2 · · · Uξ · · · Two O0 O1 O2 · · · Oξ · · · For each ξ < ω1, Uξ is an open cover of (X, τ); Oξ ∈ Uξ. Two wins if ∃γ < ω1 [ ∪{Oξ : ξ < γ} = X ] .

Remark ∪{Oξ : ξ < ω1} = X is not enough for Two to win!

Games on topological spaces of transfinite length

Game G<ω1

1

(O, O) on (X, τ): One U0 U1 U2 · · · Uξ · · · Two O0 O1 O2 · · · Oξ · · · For each ξ < ω1, Uξ is an open cover of (X, τ); Oξ ∈ Uξ. Two wins if ∃γ < ω1 [ ∪{Oξ : ξ < γ} = X ] . Theorem (Scheepers–Tall) (X, τ) is indestructibly Lindel¨

• f

⇐ ⇒ One does not have a winning strategy in G<ω1

1

(O, O) on (X, τ)

Cardinality of indestructibly Lindel¨

• f spaces

Question Is there a smaller upper bound for the cardinality of points-Gδ indestructibly Lindel¨

• f spaces than the least measurable?

Recall: Theorem (Arhangel’ski˘ ı) Points-Gδ Lindel¨

• f spaces have cardinality < the least measurable.

Theorem (Juh´ asz) There are points-Gδ (but not T2) Lindel¨

• f spaces of arbitrarily large cardinality below the

least measurable.

Cardinality of indestructibly Lindel¨

• f spaces

Question Is there a smaller upper bound for the cardinality of points-Gδ indestructibly Lindel¨

• f spaces than the least measurable?

Theorem (Tall, 1995) CON (∃ supercompact) ⇒ CON (CH + every points-Gδ indestructibly Lindel¨

• f space has size ≤ ℵ1)

Cardinality of indestructibly Lindel¨

• f spaces

Question Is there a smaller upper bound for the cardinality of points-Gδ indestructibly Lindel¨

• f spaces than the least measurable?

Theorem (Tall, 1995) CON (∃ supercompact) ⇒ CON (CH + every points-Gδ indestructibly Lindel¨

• f space has size ≤ ℵ1)

Theorem (Scheepers, 2010) CON (∃ measurable) ⇒ CON ( every points-Gδ indestructibly Lindel¨

• f space has size ≤ 2ℵ0)

(Scheepers’ method allows us some flexibility of the value of 2ℵ0.)

Games related to precipitous ideals

I: a nonprincipal ℵ1-complete (not necessarily κ-complete) ideal on κ Game G(I): One O0 O1 O2 · · · On · · · Two T0 T1 T2 · · · Tn · · · On’s, Tn’s are all from I+, O0 ⊇ T0 ⊇ O1 ⊇ · · · ⊇ On ⊇ Tn ⊇ · · · . Two wins ⇐ ⇒ ∩

n<ω Tn ̸= ∅

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 ℵ1-complete (not necessarily κ-complete) ideal on κ Game DGω(I+, ⊆): One O0 O1 O2 · · · On · · · Two T0 T1 T2 · · · Tn · · · On’s, Tn’s are all from I+, O0 ⊇ T0 ⊇ O1 ⊇ · · · ⊇ On ⊇ Tn ⊇ · · · . Two wins ⇐ ⇒ ∩

n<ω Tn ∈ I+

(A descending chain (Banach–Mazur) game on a poset (I+, ⊆))

Descending chain games of transfinite length

I: a nonprincipal ℵ1-complete (not necessarily κ-complete) ideal on κ Game DG<ω1

Two(I+, ⊆):

One O0 O1 · · ·

(pass)

Oω+1 · · · Oξ · · · Two T0 T1 · · · Tω Tω+1 · · · Tξ · · · Oξ’s, Tξ’s (ξ < ω1) are all from I+. In each limit inning Two has the initiative. O0 ⊇ T0 ⊇ O1 ⊇ · · · ⊇ Tω ⊇ Oω+1 ⊇ Tω+1 ⊇ · · · ⊇ Oξ ⊇ Tξ ⊇ · · · . Two wins ⇐ ⇒ the play is sustained all over ω1 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 DGω(P, ≤) ⇐ ⇒ Two has a w.s. in DG<ω1

Two(P, ≤)

In particular, Two has a w.s. in DGω(I+, ⊆) ⇐ ⇒ Two has a w.s. in DG<ω1

Two(I+, ⊆)

Diagram

For a nonprincipal <ℵ1-complete ideal I on κ: I: dual of measure u.f. − − − − − → I is precipitous ? ? y ? ? y Two ↑tactic DGω(I+, ⊆) ↓ ? ? y ? ? y Two ↑ DG<ω1

Two (I+, ⊆)

Two ↑ DGω(I+, ⊆) ↓ ? ? y ? ? y ? ? y One ̸↑ DG<ω1

Two (I+, ⊆) −

− − − − → One ̸↑ DGω(I+, ⊆) ↓ ? ? y ? ? y One ̸↑ G(I) I is weakly precipitous

Diagram

κ is measurable κ > 2ℵ0 κ may be ℵ1 I: dual of measure u.f. − − − − − → I is precipitous ? ? y ? ? y Two ↑tactic DGω(I+, ⊆) ↓ ? ? y ? ? y Two ↑ DG<ω1

Two (I+, ⊆)

Two ↑ DGω(I+, ⊆) ↓ ? ? y ? ? y ? ? y One ̸↑ DG<ω1

Two (I+, ⊆) −

− − − − → One ̸↑ DGω(I+, ⊆) ↓ ? ? y ? ? y One ̸↑ G(I) I is weakly precipitous

Diagram

After collapsing a measurable κ to ℵ1, I (the dual of the measure u.f.) satisfies: I: dual of measure u.f. − − − − − → I is precipitous ? ? y ? ? y Two ↑tactic DGω(I+, ⊆) ↓ ? ? y ? ? y Two ↑ DG<ω1

Two (I+, ⊆)

Two ↑ DGω(I+, ⊆) ↓ ? ? y ? ? y ? ? y One ̸↑ DG<ω1

Two (I+, ⊆) −

− − − − → One ̸↑ DGω(I+, ⊆) ↓ ? ? y ? ? y One ̸↑ G(I) I is weakly precipitous

Diagram

After collapsing a measurable κ to ℵ2, I (the dual of the measure u.f.) satisfies: I: dual of measure u.f. − − − − − → I is precipitous ? ? y ? ? y Two ↑tactic DGω(I+, ⊆) ↓ ? ? y ? ? y Two ↑ DG<ω1

Two (I+, ⊆)

Two ↑ DGω(I+, ⊆) ↓ ? ? y ? ? y ? ? y One ̸↑ DG<ω1

Two (I+, ⊆) −

− − − − → One ̸↑ DGω(I+, ⊆) ↓ ? ? y ? ? y One ̸↑ G(I) I is weakly precipitous

The main theorem

Theorem (Scheepers) If there is a nonprincipal ℵ1-complete ideal I on κ such that Two has a winning tactic in DGω(I+, ⊆), then every points-Gδ indestructibly Lindel¨

• f space has cardinality < κ.

The main theorem

Theorem (Scheepers) If there is a nonprincipal ℵ1-complete ideal I on κ such that Two has a winning tactic in DGω(I+, ⊆), then every points-Gδ indestructibly Lindel¨

• f space has cardinality < κ.

Now, by collapsing a measurable cardinal κ to ℵ2, we obtain a model where every points-Gδ indestructibly Lindel¨

• f space

has cardinality ≤ ℵ1 = 2ℵ0.

Raising 2ℵ0 with λ+ random reals and then collapsing κ to λ++ allows us to force “every points-Gδ indestructibly Lindel¨

• f space has cardinality ≤ λ+ = 2ℵ0.”

The main theorem

Theorem (Scheepers) If there is a nonprincipal ℵ1-complete ideal I on κ such that Two has a winning tactic in DGω(I+, ⊆), then every points-Gδ indestructibly Lindel¨

• f space has cardinality < κ.

Actually the above theorem can be slightly improved as follows:

Theorem If there is a nonprincipal ℵ1-complete ideal I on κ such that Two has a winning stragety in DGω(I+, ⊆), then every points-Gδ indestructibly Lindel¨

• f space has cardinality < κ.

How games work

What we actually prove is the following statement:

If there is a nonprincipal ℵ1-complete ideal I on κ such that Two has a winning stragety in DG<ω1

Two(I+, ⊆),

then, for any points-Gδ space (X, τ) with |X| ≥ κ, One has a winning strategy in G<ω1

1

(O, O) on (X, τ). The proof goes along the plays of two games, DG<ω1

Two(I+, ⊆)

and G<ω1

1

(O, O) on (X, τ), which are simultaneously played.

How games work (cont’d)

A (very rough) sketch of the proof: Lemma Let I be a nonprincipal ℵ1-complete ideal on κ, X a points-Gδ space with X ⊇ κ. Then for each x ∈ X, B ∈ I+ and 〈Un : n < ω〉: a sequence of open neighborhoods of x such that S

n<ω Un = {x}, there is a C ∈ I+ such that C ⊆ B and Un ∩ C ∈ I for all n.

Using the above lemma, we find out a winning strategy for One in G<ω1

1

(O, O) on X. For each x ∈ X fix a ⊆-decreasing sequence 〈Ux,n : n < ω〉 of neighborhoods of x whose intersection is {x}. In each α-th inning One will play an open cover of X of the form {Ux,fα(x,n) : x ∈ X} for some fα : X → ω. We shall manipulate fα’s to lead One to win the game. The winning strategy for Two in another game DG<ω1

Two (I+, ⊆) and the above

lemma help us defend an I-positive “off-limits area” against Two’s attempt to complete a cover of X.

Conclusion

We have reviewed how the following theorem is proved: Theorem (Scheepers, 2010) CON(∃ measurable) ⇒ CON(every points-Gδ indestructibly Lindel¨

• f space has size ≤ 2ℵ0)

and ...

Conclusion

We have reviewed how the following theorem is proved: Theorem (Scheepers, 2010) CON(∃ measurable) ⇒ CON(every points-Gδ indestructibly Lindel¨

• f space has size ≤ 2ℵ0)

and (hopefully) examined how large cardinals and small cardinals interplay! Thank you for your attention!

Remarks on Scheepers’ Theorem on the cardinality of Lindel¨

• f spaces

Masaru Kada (Osaka Prefecture University, Japan)