Characters of countably tightspaces
Toshimichi Usuba (薄葉 季路)
Nagoya University
2012 General Topology シンポジウム
Kobe Univ. December 14, 2012

X: topological space, A ⊆ X!

A: closure of A

Definition 1. A space X is countably tight"def.

⇐⇒ For every A ⊆ X and x ∈ A, there is a countable

B ⊆ A with x ∈ B.

#$

In this talk, we will observe some connections between

characters of countably tight spaces (of size ℵ1) and large
cardinals.

1Topological games!

"Definition 2. X: space. A, B: families of sets.
α: ordinal.

X satisfies Gα(A, B) if Player ONE does not have awinning strategy in the following game of length α:
ONE A0 A1 · · · Aξ · · · (Aξ ∈ A)
(ξ < α)TWO B0 B1 · · · Bξ · · · (Bξ ∈ Aξ )TWO wins if {Bξ : ξ < α} ∈ B, otherwise ONE wins.

#A class of spaces which satisfy Gω (A, B) are studied$

widely, but Gκ(A, B) for κ ≥ ω1 does not.

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A space X is Lindelöfdef.

!

⇐⇒ Every open cover of X has a countable subcover.

Definition 3 (Tall). A space X is indestructibly Lindelöf"

if X is Lindelöf and satisfies Gω1 (U , U ), where U is theset of all open covers of X.
ONE U0 U1 · · · Uξ · · · (Uξ : open cover)TWO O0 O1 · · · Oξ · · · (Oξ ∈ Uξ )TWO wins ⇐⇒ {Oξ : ξ < ω1} is an open cover.
#$!

"Fact 4 (Scheepers-Tall). A space X is indestructiblyLindelöf ⇐⇒ X is Lindelöf and every σ-closed forcing
preserves the Lindelöfness of X.

#$3

Examples➀ Every Lindelöf space of size ≤ ℵ1 is indestructiblyLindelöf.

➁ Every second countable space is indestructibly Lindelöf (and a second countable T2 space has size ≤ 2ℵ0 ).

➂ (Tall) {0, 1}ℵ1 is compact, has size 2ℵ1 , and is not!

indestructibly Lindelöf.

Question 5. Is there an indestructibly Lindelöf space

ℵ0 .
which
has
size>
2
#"

$4

Large cardinals and long game!

Fact 6 (Dias-Tall, Tall-Usuba). The following are"equiconsistent:➀ 2ℵ0 = ℵ1 and there is no indestructibly Lindelöf T3
space of weight ≤ ℵ1 and size > ℵ1.

➁ 2ℵ0 = ℵ1, 2ℵ1 > ℵ2,and there is no Lindelöf T3

space of weight ≤ ℵ1 and size ℵ2.

➂ There is an inaccessible cardinal.

#$5➀ One cannot prove “there is no indestructibly LindelöfT3 space of weight ≤ ℵ1 and size > ℵ1”.

➁ And (assuming the consistency of “∃ inaccessible cardinal”), one cannot disprove “there is no indestructibly
Lindelöf T3 space of weight ≤ ℵ1 and size > ℵ1”.

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Indestructible countable tightness!

Definition 7 (Scheepers). A space X is indestructibly"countably tight if X is countably tight and satisfies
Gω1 (Ωx, Ωx) for every x ∈ X, where Ωx is the set ofall A ⊆ X with x ∈ A.

#$!

"ONE A0 A1 · · · Aξ · · · (x ∈ Aξ )TWO x0 x1 · · · xξ · · · (xξ ∈ Aξ )TWO wins ⇐⇒ x ∈ {xξ : ξ < ω1}.
Fact 8 (Scheepers). A space X is indestructibly Lindelöf⇐⇒

X is countably tight and ever σ-closed forcing

preserves the countable tightness of X.

#$7

Examples➀ If X is countably tight and has character ≤ ℵ1, then
X is indestructibly countably tight.

➁ Every countable space is indestructibly countably tight.
➂ (Scheepers) Cp({0, 1}ℵ1 ) is countably tight, has size
2ℵ1 , character 2ℵ1 , and it is not indestructibly countably
tight.
➃ (Kada, M. Sakai) The sequential fun Sℵ1 is countably

tight, has size ℵ1, character 2ℵ1 , and it is not indestructibly countably tight.

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!

Question 9. Is there a small indestructibly countably
tight space of character > ℵ1?

#"$9Non-CH case!

Lemma 10. For each κ ≤ 2ℵ0 , there is an indestructiblycountably tight T3 1 space of size ℵ1 and character κ.
2#"$Proof. (M. Sakai) Fix a filter F over ω with character κ.
Let X be the space ω + 1 with the topology {{n} : n <ω} ∪ {A ∪ {ω} : A ∈ F }.

Let Y = X⊕D(ℵ1). Then Y is an indestructibly LindelöfT3 1 space of size ℵ1 and character κ.
2Hence if 2ℵ0 > ℵ1, there is an indestructibly countably

tight T3 1 space of size ℵ1 and character ℵ2.
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Main results!

Theorem 11. Let κ be an inaccessible cardinal. Then"the following hold in V Col(ω1,<κ):➀ 2ℵ0 = ℵ1 and 2ℵ1 = ℵ2.

➁ There is no indestructibly countably tight space ofsize ℵ1 and character > ℵ1.

➂ There is no indestructibly countably tight T3 space#of density ≤ ℵ1 and character > ℵ1.

$(Assuming the consistency of ∃inaccessible cardinal,)

One cannot prove the existence of an indestructibly countably tight space of size ℵ1 and character > ℵ1.
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Main results!

Theorem 12. Let κ be an inaccessible cardinal. Then"+,2,ω )Col(ω,<κ)×Fn(κ11 :the following hold in V

➀ 2ℵ0 = ℵ1 and 2ℵ1 > ℵ2.

➁ There is no countably tight space of size ℵ1 andcharacter ℵ2.

➂ There is no countably tight T3 space of density ≤ ℵ1#and character ℵ2.

(Assuming the consistency of ∃inaccessible cardinal,) One$

cannot prove the existence of a countably tight space ofsize ℵ1 and character ℵ2.

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Main results!

Theorem 13. Suppose ω2 is not inaccessible in the con-"structible universe L. Then there is an indestructiblycountably tight T3 1 space of size ℵ1 and character ℵ2.

#2$One cannot disprove the existence of a(n indestruc-tibly) countably tight space of size ℵ1 and character ℵ2.

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Main results!

"Corollary 14. The following are equiconsistent:➀ There is no indestructibly countably tight space ofsize ℵ1 and character > ℵ1.

➁ There is no countably tight space of size ℵ1 andcharacter ℵ2.

➂ There is an inaccessible cardinal.

#Hence one cannot prove and disprove the existence of$a(n indestructibly) countably tight space of size ℵ1 andcharacter ℵ2.

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!

In this talk, we will sketch the proof of:

Proposition 15. Suppose 2ℵ0 = ℵ1 and there is aKurepa tree which has just κ cofinal branches. Then"

there is an indestructibly countably tight T3 1 space ofsize ℵ1 and character κ.

#!

2$Theorem 13 follows from combining this with:

Fact 16 (H. Sakai). If ω2 is not inaccessible in the con-"structible universe L, then there is a Kurepa tree which
has just ℵ2 cofinal branches.

#$15

!

"Definition 17. X: space
Cp(X) is the space of all continuous functions from X
to R with pointwise convergence topology.

#!

Note that χ(Cp(X)) = |X| and |Cp(X)| ≤ 2d(X).

Fact 18 (Arkhangelskii-Pytkeev). X: T3 1 .
Then the following are equivalent:$"2➀ Each finite product of X is Lindelöf.
➁ Cp(X) is countably tight.

#$16

!

"Lemma 19. X: T3 1 .
2Then the following are equivalent:➀ Each finite product of X is indestructibly Lindelöf.
➁ Cp(X) is indestructibly countably tight.

#$Proof. Repeat the proof of Arkhangelskii-Pytkeev theorem in generic extension.

17We will find a good indestructibly Lindelöf space Xsuch that:➀ Cp(X) is indestructibly countably tight, and➁ Cp(X) has a subspace which has size ℵ1 and character> ℵ1 .

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A tree T is Kurepa if➀ the height of T is ω1,➁ each level of T is countable,➂ T has strictly more than ℵ1 cofinal branches.

!

Fact 20 (Dias-Tall). Suppose there is a Kurepa tree"

T which has just κ cofinal branches. Then there is aindestructibly Lindelöf compact T2 space of weight ℵ1and size max{2ℵ0 , κ}.

#That space is a branch space of T .

$19!

Lemma 21. Let X be a compact space.

If X is in-"destructibly Lindelöf, then each finite product of X isindestructibly Lindelöf.

#Hence if 2ℵ0 = ℵ1 and there is a Kurepa tree which has$just κ cofinal branches, then there is a compact T2 space
X of weight ℵ1, size κ such that Cp(X) is indestructiblycountably tight.

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Elementary submodel!

Let θ be a large regular cardinal.

Lemma 22. Suppose 2ℵ0 = ℵ1. Suppose that there is"a Kurepa tree which has just κ cofinal branch, and let
X be Dias-Tall space obtained from T .
Let M ≺ H(θ) with |M | = ℵ1 ⊆ M and X, T, Cp(X), . . . ∈M . Then the subspace M ∩ Cp(X) of Cp(X) has size ℵ1and character κ.

#$M ∩ Cp(X) ≈ a subspace of Cp(X) of size ℵ1 which

is closed under all definable functions from X to H(θ)
without parameters.
M ∩ Cp(X) reflects some properties of Cp(X), hence its

21character is κ.

!

"Definition 23. X has countable fan tightnessdef.

⇐⇒ For every x ∈ X and An (n < ω) with x ∈ An, there!
are finite Bn ⊆ An (n < ω) such that x ∈ n<ω Bn.
def.

X is Menger ⇐⇒ For every open covers Un (n < ω),!
!
there are finite Vn ⊆ Un such that X = n<ω Vn.

#countable fan tightness ⇒ countable tightness.

$compact ⇒ Menger ⇒ Lindelöf.

22!

Fact 24 (Arkhangelskii). X: T3 1 .

"2Then the following are equivalent.
➀ Each finite product of X is Menger.
➁ Cp(X) has countable fan tightness.

#

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!

Proposition 25. There is a T3 1 space X of size ℵ1 and"2character 2ℵ1 such that X has countable fan tightness,but is not indestructibly countably tight.

#$Proof. Since {0, 1}ℵ1 is a compact T2 space, Cp({0, 1}ℵ1 )has countable fan tightness.

Take a large regular θ and take M ≺ H(θ) with |M | =ℵ1 ⊆ M . Then M ∩ Cp(X) has size ℵ1, character 2ℵ1 , has
countable fan tightness, but is not indestructibly countably tight.

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