Ccc Indestructible S Spaces

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    Ccc Indestructible S Spaces - Presentation Transcript

    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
    33. EXISTENCE AND NONEXISTENCE 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
    39. EXISTENCE AND NONEXISTENCE 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
    48. INDESTRUCTIBLE SPACES CCC 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
    56. INDESTRUCTIBLE SPACES CCC 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
    66. INDESTRUCTIBLE S-SPACE CCC 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 |= \"G is strongly solid\". 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 |= \"G is strongly solid\". For any m there is a P such that V P |= \"G is ccc-indestructibly m-solid\". ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
    86. INDESTRICTIBLE m-SOLID CCC 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
    99. THFD THFD THFD HFD HFDw HFD ω HFD HFDw ω ω 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
    100. THFD THFD THFD HFD HFDw HFD ω HFD HFDw ω ω HFDw HFDw HFDw HFDw ω ω O-space S-space ω O-space S-space O-space S-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
    101. OPEN QUESTIONS ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
    102. OPEN QUESTIONS Consistency questions: ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
    103. OPEN QUESTIONS Consistency questions: HFD HFDwω HFDw ω O-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
    104. OPEN QUESTIONS Consistency questions: HFD HFDw HFD HFDw ω ω HFDw HFDw ω ω O-space S-space O-space S-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
    105. OPEN QUESTIONS Consistency questions: HFD HFDw HFD HFDw ω ω HFDw HFDw ω ω O-space S-space O-space S-space S-space S-space QUESTION (JUHASZ) Does there exists a (c, →)-HFD? ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
    106. REFERENCES ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC

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