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```Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
Cardinal invariants and Tukey
reduction
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oAŒÆêÆÆ
2016-05-21
E 2016c IênÜ6ï?¬
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
nŽ
½Â
¡ I ⊆ P(ω) ´g
g, ê þ
nŽ , e I ÷vµ
(1). ∅ ∈ I, ω 6∈ I.
(2). e I, J ∈ I, KI ∪ J ∈ I.
(3). e I ∈ I ¿… J ⊆ I, KJ ∈ I.
½Â
¡ F ⊆ P(ω) ´g
g, ê þ
´g,êþ nŽ.
Èf, e F ∗ = {A ∈ P(ω) : Ac ∈ F}
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
nŽ ~f
~f
F in = {A ⊆ ω : |A| < ω}
n
n+1 )|
Z0 = {A ⊆ ω : limn→∞ |A∩[2 2n,2
P
1
I 1 = {A ⊆ ω : n∈A n+1
< ∞}.
= 0}.
n
∅ × F in = {A ⊆ ω × ω : ∀n ∈ ω(|An | < ω)}
KM = {A ∈ K(2ω ) : A ´1˜j8}.
KN = {A ∈ K(2ω ) : A ´"ÿ8}.
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
Borel8Ü
½Â
X ´˜‡
Borel(X) ⊆ P(X) L«•¹m8 • σ-“ê.
Σ11 (X) = {f (A) ⊆ X : f : X → XëY, A ∈ Borel(X)}.
?¿ξ < ω1 :
Σ01 (X) = {U ⊆ X : U ´m8}
Π0ξ (X) = {A ⊆ X : Ac ∈ Σ0ξ (X)}
S
Σ0ξ (X) = { n∈ω An : An ∈ Π0ξn (X), ξn < ξ, n ∈ ω}
∆0ξ (X) = Σ0ξ (X) ∩ Π0ξ (X)
S
S
S
Borel(X) = ξ<ω1 Σ0ξ (X) = ξ<ω1 Π0ξ (X) = ξ<ω1 ∆0ξ (X)
Σ01
∆01
Σ02 (Fσ )
∆02
Π01
···
Π02 (Gδ )
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∆0ξ
Σ0ξ
···
Π0ξ
Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
BorelnŽ
ÏLA ←→ χA ò P(ω) ←→ 2ω À• d. 2ω Dƒ¦ÈÿÀ.
½Â
¡ I ⊆ P(ω) ´Borel, eI ∈ Borel(2ω ).
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
|Ü
|ÜµA½ S ( : Tukey'X, ' Æ‰.
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
Tukey'X
(X, ≤X ), (Y, ≤Y ) ´½•
S8.
N f : X → Y ¡• Tukey N
,e
A ⊆ X Ã.=⇒ f (A) ⊆ Y Ã..
e•3 X
Y
Tukey N , ¡ X Tukey 8
Y . P•
X ≤T Y
e X ≤T Y … Y ≤T X, ¡ X Tukey
du Y . P•
X ≡T Y
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
Tukey'X 5Ÿ1
½n (Tukey 1940)
(X, ≤X ), (Y, ≤Y ) ´½•
S8. X ≡T Y …= •3½
• S8 (Z, ≤Z ) ¦ (X, ≤X ), (Y, ≤Y ) ÑU —i\ (Z, ≤Z ).
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
Tukey'X 5Ÿ2
éó
* :: N g : E −→ D ¡•
X⊆E
—
,e
—=⇒ g(X)
—.
·K
∃f : D −→ E ´TukeyN
⇐⇒
∃g : E −→ D ´ —N
¿…, f, gŒ±
÷vXe'Xµ
∀d ∈ D∀e ∈ E(f (d) ≤E e =⇒ d ≤D g(e))
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
Tukey'X 5Ÿ3
†ÄêØCþ
éX:
add(D) = min{|A| : A ⊆ DÃ.}
cof (D) = min{|A| : A ⊆ D
—}
·K (Schmidt 1955)
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
{¤1
½n (S. Todorčević 1985)
b½ PFA. X ´½•
S8… |X| = ω1 , K X Tukey
u: 1, ω, ω1 , ω × ω1 , [ω1 ]<ω ƒ˜.
d
½n (S. Todorčević 1985)
•3 2ω1 ‡Œ • c …üü Tukey Ø d ½•
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S8.
Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
{¤2
½n (D. Fremlin 1991)
ω ω ≤T KN ≤T Z0 ≤T l1+ .
KN ≤T KM ≤T l1+ .
KN 6≤T ω ω .
Z0 6≤T KM .
½n (D. Fremlin 1991)
(a) K(X) ≡T {0}
(b) K(X) ≡T ω
(c) K(X) ≡T
ωω
…=
…=
…=
Π11
X ´;˜m.
X ´šÛÜ;
š Polish
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Polish ˜m.
˜m, K K(X) ≡T K(Q).
Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
{¤3
½n (A. Louveau, B. Velickovic 1999)
?¿nŽ I ≤T ω ω , K I ≤T ω ½ö I ≡T ω ω .
?¿š Fσ
nŽ I Ñk ω ω ≤T I.
?¿Σ11 P-nŽ I 6≤T ω Ñk ω ω ≤T I ≤T I 1 .
n
(P(ω), ⊆∗ ) Œ±i\ Σ11 P-nŽŠ‘ Tukey 'X.
½Â
P-nŽµ?¿{An : n ∈ ω} ⊆ I, •3A ∈ I ¦
∀n ∈ ω(|A \ An | < ω).
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
{¤4
½Â
Ä
X ´˜‡Œ©Ýþ˜m, ≤ ´ X þ ˜‡ S. ¡ (X, ≤) ´
XJ÷vXe^‡:
(1) ?¿ x, y ∈ X, •3˜‡• þ(. ∨(x, y), =,
∨(x, y) ≥ x, y … ∀z ∈ X(x, y ≤ z ⇒ ∨(x, y) ≤ z). ¿…
∨ : X × X → X ´ëYN .
(2) ?¿k.S
k˜‡Âñ fS .
(3) ?¿ÂñS
k˜‡k. fS .
½n ( S. Solecki, S. Todorčević 2004)
X Ú Y ´ Σ11
—N .
Ä
S. e X ≤T Y , K•3˜‡ Borel Œÿ
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
{¤5
½Â
I ´g,êþ Σ11 P-nŽ:
D(I) = {K ∈ MK(2ω ) : ∃x ∈ I∀n ∈ ω(x\n 6∈ K)}.
½n (S. Solecki, S. Todorčević 2004)
I Ú J ´ ω þ Σ11 P-nŽ¿… J 6= F in.
e I ≤T J , K D(I) ≤T D(J ).
I ´Σ11 P-nŽ. e ω ω <T I, K D(I) <T I.
e I ´ ω þ Σ11 P-nŽ, K D(I) ≤T KM .
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
(Ø1
¯K (D.Milovich 2008)
´Ä•3‡È Tukey
duω ω ?
½n
1
F ´ ω þ ˜‡Èf. e F ≤T ω ω , K•3ëY üN
—N g : ω ω → F. AO/, F ´Σ11 .
2 •3˜‡Σ11 3 •3?¿p
nŽ I ≡T ω ω .
Borel E,Ý nŽ Tukey
4 Ø•3 Fσ nŽ Tukey
du ω ω .
du ω ω .
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
(Ø2
¯K (S. Solecki, S. Todorčević 2004)
KM ≡T D(I 1 )? KN ≡T D(I 1 )?
n
n
½n
KM 6≤T D(I 1 ).
n
I1
n
KM
Z0
D(I 1 )
n
KN
D(Z0 )
ωω
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
(Ø3
½n
F ´ ω þ ˜‡1 j P-Èf. e ω ω ≤T F, K:
(1) Ø•3 σ(Σ11 )-Œÿ
(2) •3ëYüN
Tukey N .
—N .
íØ
U ´˜‡‡È. U ´ P-:
„y ω ω ≤T U.
…= Ø•3 Borel Œÿ Tukey N
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
(Ø4
½n
I´ωþ
Fσ nŽ. e (I, ⊆) ≤T KM , K I ≤T ω.
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
ÄêØCþ
½Â (Vojtáš 1993)
˜‡ Vojtáš n |´• A = (A− , A+ , A), Ù¥ A ´l A−
A+
'X. ½Â† A k' ÄêØCþXe:
kAk = min{|D| : D ⊆ A+ , ∀a ∈ A− ∃d ∈ D(aAd)}
ù
ÄêØCþ•):
d =k (ω ω , ω ω , ≤∗ ) k.
b =k (ω ω , ω ω , 6≥∗ ) k.
r =k ([ω]ω , [ω]ω , Ø©•) k.
s =k ([ω]ω , [ω]ω ,
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
•õ ÄêØCþ
½Â (S. Coskey, T. Mátrai , J. Steprāns , 2013)
A Ú B ´ Vojtáš n |, P ´ A+
A k' XeÄêØCþ:
f8 ˜‡5Ÿ. ½Â†
kAkP = min{|D| : D ⊆ A+ k5Ÿ P , ∀a ∈ A− ∃d ∈ D(aAd)}
ù
ÄêØCþ•):
a =k ([ω]ω , [ω]ω , 6⊥) kÃ¡A
u =k ([ω]ω , [ω]ω , Ø
Ø
x,
i =k
p =k
([ω]ω , [ω]ω , 6⊆∗ )
k)¤Õáx ,
k¥% .
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
ÄêØCþBorel
½Â
A = (A− , A+ , A) Ú B = (B− , B+ , B) ´Borel
Vojtáš n
|, P ´ A+ f8 5ŸÚ Q ´ B+
f8 5Ÿ. ¡
(ϕ, ψ) ´l A
B Borel
, e Borel N ϕ : B− −→ A− ,
ψ : A+ −→ B+ ÷vXe'X:
(1) F k5Ÿ P =⇒ ψ(F) k5Ÿ Q.
(2) ∀b− ∈ B− ∀a+ ∈ A+ (ϕ(b− )Aa+ =⇒ b− Bψ(a+ )). eXþ
Borel
•3, ·‚P kAkP ≤BT kBkQ .
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
{¤
Theorem[Coskey, Mátrai, Steprāns, 2013]
ZFC-provable (≤)
d
i
u
?
a
r
Borel Tukey (≤BT )
i
@
r
@
@
@ ?
? R
s
?
d
u
a
@
@ ? ? R
b
?
b
s
@
@
R
@ ?
p
p
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
(Ø
¯K (Coskey, Mátrai, Steprāns, 2013)
t ≤BT b?
½n
(1) t 6≤BT b.
(2) t 6≤BT a.
(3) t 6≤BT d.
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
½Â
I Ú J ´ ω þ nŽ. I Ú J ƒm ' Æ‰, P•
G(I, J ), ´•:
· · · In ∈ I
···
J0 ∈ J · · ·
Jn ∈ J · · ·
S
S
[ II I, XJ n∈ω In ∈ I ⇔ n∈ω Jn ∈ J ¤á. ÄK,
I.
XJ [ II k˜‡-I üÑ, ·‚P I v J .
XJ I v J ¿… J v I, ·‚P I ' J .
I
II
I0 ∈ I
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[I
Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
ÄÅ
½Â
N f : I → J ¡•ü
üN Tukey, e f ´ Tukey N ¿… f ´ü
N . eù
N •3, ·‚¡ I üN Tukey u J , P•:
I ≤MT J .
·K
I ≤MT J =⇒ I v J .
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
{¤
½n (M. Hrušák, D. M. Alcántara 2011)
?¿
Borel nŽ I, J , Æ‰ G(I, J ) ´û½ .
S v ´ûÄ
.
S v ´A ‚5
( ?¿‡ó •ÝØ‡L2 ).
•3ØŒêõ‡ ' - da.
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
¯K
¯K (M. Hrušák, D. M. Alcántara 2011)
S v ´‚5
(ûS )?
´Ø´•kü‡ Fσδ š Fσ -nŽ ' - da?
˜ kõ ‡ Fσδσ -nŽ ' - da?
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
(Ø1
½Â
X ´ 2ω
Borel f8. ½Â X ,nŽ T (X)• <ω 2 þd
{{x|n : n ∈ ω} : x ∈ X} )¤ nŽ.
½n
T ((∅ × F in)+ ) Ú ∅ × F in ´ v -ØƒN .
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
(Ø2
½Â
A, B ⊆ 2ω , ¡A Wadge uB XJ•3ëYN
¦ f −1 (B) = A. P•A ≤W B.
A ≡W B XJA ≤W B ¿…B ≤W A.
f : 2ω → 2ω
½n
I, J ´ ω þ Borel nŽ¿…Ù Wadge Ýpu ∆(Dω (Σ02 )),
K I ≡W J ⇔ T (I) ' T (J ).
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Cardinal invariants an
Ä Vg nŽm Tukey'X ÄêØCþƒm Tukey'X nŽm ' Æ‰
!
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Cardinal invariants an
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