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Ccc Indestructible S Spaces

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A presentation on Soukup's work on ccc indestructible spaces with strong separability properties.

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Ccc Indestructible S Spaces

  1. 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. 2. CH TO MAℵ1 TO PFA ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  3. 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. 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. 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. 6. SEPARABILITY ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  7. 7. SEPARABILITY Let X ⊆ 2ω1 be uncountable, zero-dimensional (hence regular). ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  8. 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. 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. 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. 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. 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. 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. 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. 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. 16. O-SPACES ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  17. 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. 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. 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. 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. 21. STRONG SPACES ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  22. 22. STRONG SPACES DEFINITION A Φ space X is called a strong Φ space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  23. 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. 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. 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. 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. 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. 28. SPACES UNDER CH ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  29. 29. SPACES UNDER CH HFD HFDw O-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  30. 30. SPACES UNDER CH HFD HFDwω HFDw ω O-space S-space S-space ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  31. 31. EXISTENCE AND NONEXISTENCE THEOREM (ROITMAN) Let r be a Cohen real. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  32. 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. 33. EXISTENCE AND NONEXISTENCE ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  34. 34. EXISTENCE AND NONEXISTENCE THEOREM (MAℵ1 ) There are no HFDs. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  35. 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. 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. 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. 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. 39. EXISTENCE AND NONEXISTENCE ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  40. 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. 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. 42. UNDER MAℵ1 ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  43. 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. 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. 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. 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. 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. 48. INDESTRUCTIBLE SPACES CCC ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  49. 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. 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. 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. 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. 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. 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. 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. 56. INDESTRUCTIBLE SPACES CCC ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  57. 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. 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. 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. 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. 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. 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. 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. 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. 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. 66. INDESTRUCTIBLE S-SPACE CCC ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  67. 67. INDESTRUCTIBLE S-SPACE CCC THEOREM (SZENTMIKLÓSSY) (CH) There exists a tight HFD. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  68. 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. 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. 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. 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. 72. STRONGLY SOLID GRAPHS ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  73. 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. 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. 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. 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. 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. 78. STRONGLY SOLID GRAPHS ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  79. 79. STRONGLY SOLID GRAPHS THEOREM (SOUKUP) For a space X , ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  80. 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. 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. 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. 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. 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. 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
  86. 86. INDESTRICTIBLE m-SOLID CCC ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  87. 87. INDESTRICTIBLE m-SOLID CCC Assume 2ω1 = ω2 . ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  88. 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. 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. 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. 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. 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. 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. 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. 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. 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. 97. ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  98. 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. 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. 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. 101. OPEN QUESTIONS ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  102. 102. OPEN QUESTIONS Consistency questions: ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC
  103. 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. 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. 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. 106. REFERENCES ERIK A. ANDREJKO INDESTRUCTIBLE SPACES WITH STRONG SEPARABILITY CCC

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