Digital Identity is Under Attack: FIDO Paris Seminar.pptx
Ccc Indestructible S Spaces
1. INDESTRUCTIBLE SPACES WITH STRONG
CCC
SEPARABILITY
AND SEPARABILITY IN 2ω1
MAℵ1
Erik A. Andrejko
University of Wisconsin - Madison
Feb 27 2007
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
2. CH TO MAℵ1 TO PFA
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
3. CH TO MAℵ1 TO PFA
THFD
HFD HFDwω
HFDw
ω
O-space S-space
S-space
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
4. CH TO MAℵ1 TO PFA
THFD
THFD
HFD HFDwω
HFD ω
HFDw
HFDw
HFDw
ω
O-space S-space
ω
O-space S-space
S-space
S-space
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
5. CH TO MAℵ1 TO PFA
?
THFD
THFD
HFD HFDwω
HFD ω
HFDw
HFDw
HFDw
ω
O-space S-space
ω
O-space S-space
S-space
S-space
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
6. SEPARABILITY
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
7. SEPARABILITY
Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular).
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
8. SEPARABILITY
Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular).
DEFINITION
X is separable if X has a countable dense subset.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
9. SEPARABILITY
Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular).
DEFINITION
X is separable if X has a countable dense subset.
X is hereditarily separable if every subspace has a countable dense
subset.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
10. SEPARABILITY
Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular).
DEFINITION
X is separable if X has a countable dense subset.
X is hereditarily separable if every subspace has a countable dense
subset.
If X is hereditarily separable and not Lindelöf then X is called an
S-space.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
11. SEPARABILITY
Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular).
DEFINITION
X is separable if X has a countable dense subset.
X is hereditarily separable if every subspace has a countable dense
subset.
If X is hereditarily separable and not Lindelöf then X is called an
S-space.
DEFINITION
A is finally dense if for some γ < ω1 A is dense in 2ω1 γ .
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
12. SEPARABILITY
Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular).
DEFINITION
X is separable if X has a countable dense subset.
X is hereditarily separable if every subspace has a countable dense
subset.
If X is hereditarily separable and not Lindelöf then X is called an
S-space.
DEFINITION
A is finally dense if for some γ < ω1 A is dense in 2ω1 γ .
X is a weak HFD iff for all Y ∈ [X ]ω1 there is some A ∈ [Y ]ω such that
A is finally dense.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
13. SEPARABILITY
DEFINITION
X is a HFD iff for all A ∈ [X ]ω , A is finally dense.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
14. SEPARABILITY
DEFINITION
X is a HFD iff for all A ∈ [X ]ω , A is finally dense.
LEMMA
If there is an HFD, there is a weak HFD. If there is a weak HFD, there
is an S-space.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
15. SEPARABILITY
DEFINITION
X is a HFD iff for all A ∈ [X ]ω , A is finally dense.
LEMMA
If there is an HFD, there is a weak HFD. If there is a weak HFD, there
is an S-space.
HFD
HFDw
S-space
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
16. O-SPACES
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
17. O-SPACES
DEFINITION
X is an O-space iff every open set is countable or co-countable.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
18. O-SPACES
DEFINITION
X is an O-space iff every open set is countable or co-countable.
LEMMA
If there is a weak HFD there is an O-space. If there is an O-space,
there is S-space.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
19. O-SPACES
DEFINITION
X is an O-space iff every open set is countable or co-countable.
LEMMA
If there is a weak HFD there is an O-space. If there is an O-space,
there is S-space.
HFD
HFDw
S-space
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
20. O-SPACES
DEFINITION
X is an O-space iff every open set is countable or co-countable.
LEMMA
If there is a weak HFD there is an O-space. If there is an O-space,
there is S-space.
HFD
HFDw
O-space
S-space
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
21. STRONG SPACES
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
22. STRONG SPACES
DEFINITION
A Φ space X is called a strong Φ space
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
23. STRONG SPACES
DEFINITION
A Φ space X is called a strong Φ space if every finite power X n is a
Φ space. e.g. HFDw , S-space.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
24. STRONG SPACES
DEFINITION
A Φ space X is called a strong Φ space if every finite power X n is a
Φ space. e.g. HFDw , S-space.
THEOREM
(CH) There exists a strong HFDw ,
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
25. STRONG SPACES
DEFINITION
A Φ space X is called a strong Φ space if every finite power X n is a
Φ space. e.g. HFDw , S-space.
THEOREM
(CH) There exists a strong HFDw , and hence a strong S-space.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
26. STRONG SPACES
DEFINITION
A Φ space X is called a strong Φ space if every finite power X n is a
Φ space. e.g. HFDw , S-space.
THEOREM
(CH) There exists a strong HFDw , and hence a strong S-space.
COROLLARY
(CH) There exists a HFDn space and S-spacen for all n < ω.
w
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
27. STRONG SPACES
DEFINITION
A Φ space X is called a strong Φ space if every finite power X n is a
Φ space. e.g. HFDw , S-space.
THEOREM
(CH) There exists a strong HFDw , and hence a strong S-space.
COROLLARY
(CH) There exists a HFDn space and S-spacen for all n < ω.
w
THEOREM
(CH) There exists an HFD.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
28. SPACES UNDER CH
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
29. SPACES UNDER CH
HFD
HFDw
O-space
S-space
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
30. SPACES UNDER CH
HFD HFDwω
HFDw
ω
O-space S-space
S-space
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
31. EXISTENCE AND NONEXISTENCE
THEOREM (ROITMAN)
Let r be a Cohen real.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
32. EXISTENCE AND NONEXISTENCE
THEOREM (ROITMAN)
Let r be a Cohen real.
V [r ] |= ∃a strong HFD
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
34. EXISTENCE AND NONEXISTENCE
THEOREM
(MAℵ1 ) There are no HFDs.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
35. EXISTENCE AND NONEXISTENCE
THEOREM
(MAℵ1 ) There are no HFDs.
LEMMA (SILVER’S LEMMA)
Assume MAℵ1 (or p > ω1 ).
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
36. EXISTENCE AND NONEXISTENCE
THEOREM
(MAℵ1 ) There are no HFDs.
LEMMA (SILVER’S LEMMA)
Assume MAℵ1 (or p > ω1 ). Assume that {An : n < ω} are subsets of
ω1 .
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
37. EXISTENCE AND NONEXISTENCE
THEOREM
(MAℵ1 ) There are no HFDs.
LEMMA (SILVER’S LEMMA)
Assume MAℵ1 (or p > ω1 ). Assume that {An : n < ω} are subsets of
ω1 . Then there is an infinite E ⊆ ω such that either
An
n∈E
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
38. EXISTENCE AND NONEXISTENCE
THEOREM
(MAℵ1 ) There are no HFDs.
LEMMA (SILVER’S LEMMA)
Assume MAℵ1 (or p > ω1 ). Assume that {An : n < ω} are subsets of
ω1 . Then there is an infinite E ⊆ ω such that either
(ω1 An )
An or
n∈E n∈E
is uncountable.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
40. EXISTENCE AND NONEXISTENCE
COROLLARY
If V |= MAℵ1 , and r is a Cohen real, then V [r ] |= MAℵ1 .
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
41. EXISTENCE AND NONEXISTENCE
COROLLARY
If V |= MAℵ1 , and r is a Cohen real, then V [r ] |= MAℵ1 .
THEOREM (KUNEN)
(MAℵ1 ) There are no strong S-spaces.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
42. UNDER MAℵ1
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
43. UNDER MAℵ1
THFD
THFD
HFD HFDwω
HFD ω
HFDw
HFDw
HFDw
ω
O-space S-space
ω
O-space S-space
S-space
S-space
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
44. UNDER MAℵ1
?
THFD
THFD
HFD HFDwω
HFD ω
HFDw
HFDw
HFDw
ω
O-space S-space
ω
O-space S-space
S-space
S-space
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
45. UNDER MAℵ1
THFD
THFD
THFD
HFD HFDw
HFD ω
HFD ω
ω
HFDw
HFDw
HFDw
HFDw
ω
S-spaceω O-space S-space
ω
O-space S-space
S-space
S-space
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
46. UNDER MAℵ1
THFD
THFD
THFD
HFD HFDw
HFD ω
HFD ω
ω
HFDw
HFDw
?
HFDw
HFDw
ω
S-spaceω O-space S-space
ω
O-space S-space
S-space
S-space
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
47. UNDER MAℵ1
THFD
THFD
THFD
HFD HFDw
HFD ω
HFD ω
ω
HFDw
HFDw
?
HFDw
HFDw
ω
S-spaceω O-space S-space
ω
O-space S-space
S-space
S-space
QUESTION
Does there exist an S-space, O-space, or weak HFD under MAℵ1 ?
Finite powers?
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
49. INDESTRUCTIBLE SPACES
CCC
DEFINITION
A set X is said to be ccc-indestructibly ϕ
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
50. INDESTRUCTIBLE SPACES
CCC
DEFINITION
A set X is said to be ccc-indestructibly ϕ iff for any ccc poset P and
any P-generic filter G over V ,
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
51. INDESTRUCTIBLE SPACES
CCC
DEFINITION
A set X is said to be ccc-indestructibly ϕ iff for any ccc poset P and
any P-generic filter G over V ,
V |= ϕ(X )
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
52. INDESTRUCTIBLE SPACES
CCC
DEFINITION
A set X is said to be ccc-indestructibly ϕ iff for any ccc poset P and
any P-generic filter G over V ,
V |= ϕ(X ) =⇒ V [G] |= ϕ(X )
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
53. INDESTRUCTIBLE SPACES
CCC
DEFINITION
A set X is said to be ccc-indestructibly ϕ iff for any ccc poset P and
any P-generic filter G over V ,
V |= ϕ(X ) =⇒ V [G] |= ϕ(X )
e.g. If X is an S-space, then X is ccc-indestructible iff X is an S-space
in any ccc extension.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
54. INDESTRUCTIBLE SPACES
CCC
DEFINITION
A set X is said to be ccc-indestructibly ϕ iff for any ccc poset P and
any P-generic filter G over V ,
V |= ϕ(X ) =⇒ V [G] |= ϕ(X )
e.g. If X is an S-space, then X is ccc-indestructible iff X is an S-space
in any ccc extension.
DEFINITION
Let X be an S-space.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
55. INDESTRUCTIBLE SPACES
CCC
DEFINITION
A set X is said to be ccc-indestructibly ϕ iff for any ccc poset P and
any P-generic filter G over V ,
V |= ϕ(X ) =⇒ V [G] |= ϕ(X )
e.g. If X is an S-space, then X is ccc-indestructible iff X is an S-space
in any ccc extension.
DEFINITION
Let X be an S-space. Then let PX be the natural order to add an
uncountable discrete subspace with finite conditions.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
57. INDESTRUCTIBLE SPACES
CCC
THEOREM (SZENTMIKLÓSSY)
If X is a ccc destructible S-space, then some uncountable A ⊆ PX has
ccc.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
58. INDESTRUCTIBLE SPACES
CCC
THEOREM (SZENTMIKLÓSSY)
If X is a ccc destructible S-space, then some uncountable A ⊆ PX has
ccc.
DEFINITION
X ⊆ 2ω1 is a tight HFD iff X is an HFD
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
59. INDESTRUCTIBLE SPACES
CCC
THEOREM (SZENTMIKLÓSSY)
If X is a ccc destructible S-space, then some uncountable A ⊆ PX has
ccc.
DEFINITION
X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit
type,
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
60. INDESTRUCTIBLE SPACES
CCC
THEOREM (SZENTMIKLÓSSY)
If X is a ccc destructible S-space, then some uncountable A ⊆ PX has
ccc.
DEFINITION
X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit
type, and associated β
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
61. INDESTRUCTIBLE SPACES
CCC
THEOREM (SZENTMIKLÓSSY)
If X is a ccc destructible S-space, then some uncountable A ⊆ PX has
ccc.
DEFINITION
X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit
type, and associated β for every neighborhood ε ∈ [2ω1 β ]<ω :
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
62. INDESTRUCTIBLE SPACES
CCC
THEOREM (SZENTMIKLÓSSY)
If X is a ccc destructible S-space, then some uncountable A ⊆ PX has
ccc.
DEFINITION
X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit
type, and associated β for every neighborhood ε ∈ [2ω1 β ]<ω :
[ε] ∩ A is tight in A
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
63. INDESTRUCTIBLE SPACES
CCC
THEOREM (SZENTMIKLÓSSY)
If X is a ccc destructible S-space, then some uncountable A ⊆ PX has
ccc.
DEFINITION
X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit
type, and associated β for every neighborhood ε ∈ [2ω1 β ]<ω :
[ε] ∩ A is tight in A
Let A, B we well ordered of type α, β < ω1 limit ordinals.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
64. INDESTRUCTIBLE SPACES
CCC
THEOREM (SZENTMIKLÓSSY)
If X is a ccc destructible S-space, then some uncountable A ⊆ PX has
ccc.
DEFINITION
X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit
type, and associated β for every neighborhood ε ∈ [2ω1 β ]<ω :
[ε] ∩ A is tight in A
Let A, B we well ordered of type α, β < ω1 limit ordinals. Then A is
tight in B iff for some n,
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
65. INDESTRUCTIBLE SPACES
CCC
THEOREM (SZENTMIKLÓSSY)
If X is a ccc destructible S-space, then some uncountable A ⊆ PX has
ccc.
DEFINITION
X ⊆ 2ω1 is a tight HFD iff X is an HFD and for every A ∈ [X ]ω of limit
type, and associated β for every neighborhood ε ∈ [2ω1 β ]<ω :
[ε] ∩ A is tight in A
Let A, B we well ordered of type α, β < ω1 limit ordinals. Then A is
tight in B iff for some n, every interval of B of length n meets A.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
67. INDESTRUCTIBLE S-SPACE
CCC
THEOREM (SZENTMIKLÓSSY)
(CH) There exists a tight HFD.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
68. INDESTRUCTIBLE S-SPACE
CCC
THEOREM (SZENTMIKLÓSSY)
(CH) There exists a tight HFD.
THEOREM (SZENTMIKLÓSSY)
If X is a tight HFD, then X is a ccc indestructible S-space.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
69. INDESTRUCTIBLE S-SPACE
CCC
THEOREM (SZENTMIKLÓSSY)
(CH) There exists a tight HFD.
THEOREM (SZENTMIKLÓSSY)
If X is a tight HFD, then X is a ccc indestructible S-space.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
70. INDESTRUCTIBLE S-SPACE
CCC
THEOREM (SZENTMIKLÓSSY)
(CH) There exists a tight HFD.
THEOREM (SZENTMIKLÓSSY)
If X is a tight HFD, then X is a ccc indestructible S-space.
THFD
THFD
THFD
HFD HFDw
HFD HFDw ω
HFD HFDw ω
ω
?
HFDw
HFDw
ω
S-spaceω O-space S-space
ω
O-space S-space
S-space
S-space
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
71. INDESTRUCTIBLE S-SPACE
CCC
THEOREM (SZENTMIKLÓSSY)
(CH) There exists a tight HFD.
THEOREM (SZENTMIKLÓSSY)
If X is a tight HFD, then X is a ccc indestructible S-space.
THFD
THFD
THFD
HFD HFDw
HFD HFDw ω
HFD HFDw ω
ω
HFDw
HFDw
ω
S-spaceω O-space S-space
ω
O-space S-space
S-space
S-space
S-space
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
72. STRONGLY SOLID GRAPHS
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
73. STRONGLY SOLID GRAPHS
DEFINITION
A graph G on ω1 × K and m ∈ ω. Then G is m-solid if given any
domain disjoint sequence sα : α < ω1 ⊆ Fnm (ω1 , K ) there are
α < β < ω1 such that
[sα , sβ ] ⊆ G
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
74. STRONGLY SOLID GRAPHS
DEFINITION
A graph G on ω1 × K and m ∈ ω. Then G is m-solid if given any
domain disjoint sequence sα : α < ω1 ⊆ Fnm (ω1 , K ) there are
α < β < ω1 such that
[sα , sβ ] ⊆ G
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
75. STRONGLY SOLID GRAPHS
DEFINITION
A graph G on ω1 × K and m ∈ ω. Then G is m-solid if given any
domain disjoint sequence sα : α < ω1 ⊆ Fnm (ω1 , K ) there are
α < β < ω1 such that
[sα , sβ ] ⊆ G
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
76. STRONGLY SOLID GRAPHS
DEFINITION
A graph G on ω1 × K and m ∈ ω. Then G is m-solid if given any
domain disjoint sequence sα : α < ω1 ⊆ Fnm (ω1 , K ) there are
α < β < ω1 such that
[sα , sβ ] ⊆ G
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
77. STRONGLY SOLID GRAPHS
DEFINITION
A graph G on ω1 × K and m ∈ ω. Then G is m-solid if given any
domain disjoint sequence sα : α < ω1 ⊆ Fnm (ω1 , K ) there are
α < β < ω1 such that
[sα , sβ ] ⊆ G
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
78. STRONGLY SOLID GRAPHS
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
79. STRONGLY SOLID GRAPHS
THEOREM (SOUKUP)
For a space X ,
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
80. STRONGLY SOLID GRAPHS
THEOREM (SOUKUP)
For a space X , there is a graph GX such that
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
81. STRONGLY SOLID GRAPHS
THEOREM (SOUKUP)
For a space X , there is a graph GX such that
GX is m-solid
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
82. STRONGLY SOLID GRAPHS
THEOREM (SOUKUP)
For a space X , there is a graph GX such that
GX is m-solid ⇐⇒ X is HFDm
w
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
83. STRONGLY SOLID GRAPHS
THEOREM (SOUKUP)
For a space X , there is a graph GX such that
GX is m-solid ⇐⇒ X is HFDm
w
DEFINITION
A graph G is strongly solid iff G is m-solid for every m < ω.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
84. STRONGLY SOLID GRAPHS
THEOREM (SOUKUP)
For a space X , there is a graph GX such that
GX is m-solid ⇐⇒ X is HFDm
w
DEFINITION
A graph G is strongly solid iff G is m-solid for every m < ω.
THEOREM (SOUKUP)
Let V |= quot;G is strongly solidquot;. For any m there is a P such that
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
85. STRONGLY SOLID GRAPHS
THEOREM (SOUKUP)
For a space X , there is a graph GX such that
GX is m-solid ⇐⇒ X is HFDm
w
DEFINITION
A graph G is strongly solid iff G is m-solid for every m < ω.
THEOREM (SOUKUP)
Let V |= quot;G is strongly solidquot;. For any m there is a P such that
V P |= quot;G is ccc-indestructibly m-solidquot;.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
87. INDESTRICTIBLE m-SOLID
CCC
Assume 2ω1 = ω2 .
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
88. INDESTRICTIBLE m-SOLID
CCC
◦
Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
89. INDESTRICTIBLE m-SOLID
CCC
◦
Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that
◦
|Qη | = ω1
1Pη
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
90. INDESTRICTIBLE m-SOLID
CCC
◦
Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that
◦
|Qη | = ω1 and so (2ω1 )V
Pω2
= ω2 .
1Pη
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
91. INDESTRICTIBLE m-SOLID
CCC
◦
Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that
◦
1Pη |Qη | = ω1 and so (2ω1 )V
Pω2
= ω2 . Furthermore P satisfies the
previous theorem.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
92. INDESTRICTIBLE m-SOLID
CCC
◦
Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that
◦
1Pη |Qη | = ω1 and so (2ω1 )V
Pω2
= ω2 . Furthermore P satisfies the
previous theorem.
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
93. INDESTRICTIBLE m-SOLID
CCC
◦
Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that
◦
1Pη |Qη | = ω1 and so (2ω1 )V
Pω2
= ω2 . Furthermore P satisfies the
previous theorem.
{
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
94. INDESTRICTIBLE m-SOLID
CCC
◦
Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that
◦
1Pη |Qη | = ω1 and so (2ω1 )V
Pω2
= ω2 . Furthermore P satisfies the
previous theorem.
{
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
95. INDESTRICTIBLE m-SOLID
CCC
◦
Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that
◦
1Pη |Qη | = ω1 and so (2ω1 )V
Pω2
= ω2 . Furthermore P satisfies the
previous theorem.
{
{
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
96. INDESTRICTIBLE m-SOLID
CCC
◦
Assume 2ω1 = ω2 . There is a P = Pω2 = Pη , Qη : η < ω2 such that
◦
1Pη |Qη | = ω1 and so (2ω1 )V
Pω2
= ω2 . Furthermore P satisfies the
previous theorem.
{
{
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
97. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC
98. ?
THFD
THFD
HFD HFDwω
HFD HFDwω
HFDw
HFDw
ω
O-space S-space
ω
O-space S-space
S-space
S-space
ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY
CCC