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L Space In ZFC

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L Space In ZFC

1. 1. WALKING UP TO AN L-SPACE JUSTIN T MOORE, A SOLUTION TO THE L-SPACE PROBLEM, PREPRINT, 2005. Erik A. Andrejko University of Wisconsin - Madison – SPECIALTY EXAM November 29 2005 ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
2. 2. S AND L-SPACES A space is called hereditarily separable if every subspace has a countable dense subset. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
3. 3. S AND L-SPACES A space is called hereditarily separable if every subspace has a countable dense subset. A space is called hereditarily Lindelöf if every cover of a subspace has a countable subcover. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
4. 4. S AND L-SPACES A space is called hereditarily separable if every subspace has a countable dense subset. A space is called hereditarily Lindelöf if every cover of a subspace has a countable subcover. An S-space is a regular (T3 ) space that is hereditarily separable but not hereditarily Lindelöf. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
5. 5. S AND L-SPACES A space is called hereditarily separable if every subspace has a countable dense subset. A space is called hereditarily Lindelöf if every cover of a subspace has a countable subcover. An S-space is a regular (T3 ) space that is hereditarily separable but not hereditarily Lindelöf. An L-space is a regular space that is hereditarily Lindelöf but not hereditarily separable. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
6. 6. S AND L-SPACES A space is called hereditarily separable if every subspace has a countable dense subset. A space is called hereditarily Lindelöf if every cover of a subspace has a countable subcover. An S-space is a regular (T3 ) space that is hereditarily separable but not hereditarily Lindelöf. An L-space is a regular space that is hereditarily Lindelöf but not hereditarily separable. QUESTION (Juhász and Hajnal 1968) Do there exist S and L spaces? ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
7. 7. CANONICAL S AND L-SPACES A space X is called right-separated if it can be well ordered in type ω1 such that every initial segment is open. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
8. 8. CANONICAL S AND L-SPACES A space X is called right-separated if it can be well ordered in type ω1 such that every initial segment is open. ... ... FIGURE: A right-separated space ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
9. 9. CANONICAL S AND L-SPACES A space X is called right-separated if it can be well ordered in type ω1 such that every initial segment is open. ... ... FIGURE: A right-separated space A space X is called left-separated if it can be well ordered in type ω1 such that every initial segment is open. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
10. 10. CANONICAL S AND L-SPACES A space X is called right-separated if it can be well ordered in type ω1 such that every initial segment is open. ... ... FIGURE: A right-separated space A space X is called left-separated if it can be well ordered in type ω1 such that every initial segment is open. ... FIGURE: A left-separated space ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
11. 11. CANONICAL S AND L-SPACES THEOREM A space is hereditarily separable iff it has no uncountable left separated subspace. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
12. 12. CANONICAL S AND L-SPACES THEOREM A space is hereditarily separable iff it has no uncountable left separated subspace. THEOREM A space is hereditarily Lindelöf iff it has no uncountable right separated subspace. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
13. 13. CANONICAL S AND L-SPACES FACT A regular right separated space of type ω1 is an S-space iff it has no uncountable discrete subspace. A regular left separated space of type ω1 is an L-space iff it has no uncountable discrete subspaces. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
14. 14. CANONICAL S AND L-SPACES FACT A regular right separated space of type ω1 is an S-space iff it has no uncountable discrete subspace. A regular left separated space of type ω1 is an L-space iff it has no uncountable discrete subspaces. By 2ω1 we denote the Tychanoff product space where 2 is the 2 point discrete space. This has basis elements [σ ] where σ is a ﬁnite function from ω1 to 2 and where [σ ] = {f ∈ 2ω1 : f |dom(σ ) = σ } i.e., the functions which extend σ . ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
15. 15. CANONICAL S AND L-SPACES THEOREM Canonical Form (A) Every S-space contains an uncountable subset which under a possibly weaker topology is homeomorphic to a right separated S-subspace of 2ω1 . ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
16. 16. CANONICAL S AND L-SPACES THEOREM Canonical Form (A) Every S-space contains an uncountable subset which under a possibly weaker topology is homeomorphic to a right separated S-subspace of 2ω1 . (B) Assume ¬CH. Every L-space contains a subset which under a possibly weaker topology is homeomorphic to a left separated L-subspace of 2ω1 . ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
17. 17. CANONICAL S AND L-SPACES THEOREM Canonical Form (A) Every S-space contains an uncountable subset which under a possibly weaker topology is homeomorphic to a right separated S-subspace of 2ω1 . (B) Assume ¬CH. Every L-space contains a subset which under a possibly weaker topology is homeomorphic to a left separated L-subspace of 2ω1 . Thus S and L-spaces, when they exist, exist as subspaces of 2ω1 . ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
18. 18. BUILDING S AND L-SPACES ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
19. 19. BUILDING S AND L-SPACES THEOREM (KUREPA) A Suslin line is an L-space. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
20. 20. BUILDING S AND L-SPACES THEOREM (KUREPA) A Suslin line is an L-space. THEOREM (RUDIN) If there is a Suslin line there is an S-space. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
21. 21. BUILDING S AND L-SPACES THEOREM (KUREPA) A Suslin line is an L-space. THEOREM (RUDIN) If there is a Suslin line there is an S-space. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
22. 22. BUILDING S AND L-SPACES THEOREM (KUREPA) A Suslin line is an L-space. THEOREM (RUDIN) If there is a Suslin line there is an S-space. THEOREM (CH) There exists S and L-spaces. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
23. 23. DESTROYING S AND L-SPACES A strong S-space is a space X such that for all n < ω X n is an S-space. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
24. 24. DESTROYING S AND L-SPACES A strong S-space is a space X such that for all n < ω X n is an S-space. A strong L-space is a space X such that for all n < ω X n is an L-space. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
25. 25. DESTROYING S AND L-SPACES A strong S-space is a space X such that for all n < ω X n is an S-space. A strong L-space is a space X such that for all n < ω X n is an L-space. THEOREM (ROITMAN) Adding a single Cohen real adds a strong S-space and strong L-space. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
26. 26. DESTROYING S AND L-SPACES A strong S-space is a space X such that for all n < ω X n is an S-space. A strong L-space is a space X such that for all n < ω X n is an L-space. THEOREM (ROITMAN) Adding a single Cohen real adds a strong S-space and strong L-space. Under MA + ¬ CH a space is strongly hereditarily Lindelöf iff it is strongly hereditarily separable. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
27. 27. DESTROYING S AND L-SPACES A strong S-space is a space X such that for all n < ω X n is an S-space. A strong L-space is a space X such that for all n < ω X n is an L-space. THEOREM (ROITMAN) Adding a single Cohen real adds a strong S-space and strong L-space. Under MA + ¬ CH a space is strongly hereditarily Lindelöf iff it is strongly hereditarily separable. THEOREM (KUNEN) (MA + ¬ CH) There are no strong S or L-spaces. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
28. 28. A CONJECTURE ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
29. 29. A CONJECTURE ˇ´ THEOREM (TODORCE VI C) (PFA) There are no S-spaces ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
30. 30. A CONJECTURE ˇ´ THEOREM (TODORCE VI C) (PFA) There are no S-spaces CONJECTURE (PFA) There are no L-spaces. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
31. 31. PARTITIONS The existence of S and L-spaces are equivalent to the existence of certain colorings of [ω1 ]2 . ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
32. 32. PARTITIONS The existence of S and L-spaces are equivalent to the existence of certain colorings of [ω1 ]2 . THEOREM (ROITMAN) There does not exist an L-space iff for every partition [ω1 ]2 = K0 ∪ K1 there is an increasing (separated) sequence {aξ : ξ < ω1 } of n elements subsets of ω1 and a k < n such that for all ξ < η either (I) There is an i > k such that {aξ i , aηk } ∈ K1 , (II) or for some i ≤ k {aξ i , aηk } ∈ K1 iff {aξ i , aξ k } ∈ K0 ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
33. 33. UPPER TRACE Let Cδ ⊆ δ be unbounded in δ of order type ω for each δ < ω1 . The sequence Cδ : δ < ω1 is called a ladder system which allows on to quot;walkquot; from β to α. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
34. 34. UPPER TRACE Let Cδ ⊆ δ be unbounded in δ of order type ω for each δ < ω1 . The sequence Cδ : δ < ω1 is called a ladder system which allows on to quot;walkquot; from β to α. Deﬁne the upper trace Tr(α, β ) Tr(α, α) = 0, / Tr(α, β ) = Tr(α, min(Cβ α)) ∪ {β } ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
35. 35. UPPER TRACE Let Cδ ⊆ δ be unbounded in δ of order type ω for each δ < ω1 . The sequence Cδ : δ < ω1 is called a ladder system which allows on to quot;walkquot; from β to α. Deﬁne the upper trace Tr(α, β ) Tr(α, α) = 0, / Tr(α, β ) = Tr(α, min(Cβ α)) ∪ {β } ... ... FIGURE: Tr(α, β ) ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
36. 36. UPPER TRACE Let Cδ ⊆ δ be unbounded in δ of order type ω for each δ < ω1 . The sequence Cδ : δ < ω1 is called a ladder system which allows on to quot;walkquot; from β to α. Deﬁne the upper trace Tr(α, β ) Tr(α, α) = 0, / Tr(α, β ) = Tr(α, min(Cβ α)) ∪ {β } ... ... FIGURE: Tr(α, β ) ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
37. 37. UPPER TRACE Let Cδ ⊆ δ be unbounded in δ of order type ω for each δ < ω1 . The sequence Cδ : δ < ω1 is called a ladder system which allows on to quot;walkquot; from β to α. Deﬁne the upper trace Tr(α, β ) Tr(α, α) = 0, / Tr(α, β ) = Tr(α, min(Cβ α)) ∪ {β } ... ... ... FIGURE: Tr(α, β ) ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
38. 38. UPPER TRACE Let Cδ ⊆ δ be unbounded in δ of order type ω for each δ < ω1 . The sequence Cδ : δ < ω1 is called a ladder system which allows on to quot;walkquot; from β to α. Deﬁne the upper trace Tr(α, β ) Tr(α, α) = 0, / Tr(α, β ) = Tr(α, min(Cβ α)) ∪ {β } ... ... ... ... FIGURE: Tr(α, β ) ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
39. 39. UPPER TRACE Let Cδ ⊆ δ be unbounded in δ of order type ω for each δ < ω1 . The sequence Cδ : δ < ω1 is called a ladder system which allows on to quot;walkquot; from β to α. Deﬁne the upper trace Tr(α, β ) Tr(α, α) = 0, / Tr(α, β ) = Tr(α, min(Cβ α)) ∪ {β } ... ... ... ... ... FIGURE: Tr(α, β ) ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
40. 40. UPPER TRACE Let Cδ ⊆ δ be unbounded in δ of order type ω for each δ < ω1 . The sequence Cδ : δ < ω1 is called a ladder system which allows on to quot;walkquot; from β to α. Deﬁne the upper trace Tr(α, β ) Tr(α, α) = 0, / Tr(α, β ) = Tr(α, min(Cβ α)) ∪ {β } ... ... ... ... ... FIGURE: Tr(α, β ) ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
41. 41. LOWER TRACE Deﬁne the lower trace L(α, β ) by L(α, α) = 0, / L(α, β ) = L(α, min(Cβ α)) ∪ {max(Cβ ∩ α)} max(Cβ ∩ α) ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
42. 42. LOWER TRACE Deﬁne the lower trace L(α, β ) by L(α, α) = 0, / L(α, β ) = L(α, min(Cβ α)) ∪ {max(Cβ ∩ α)} max(Cβ ∩ α) ... ... FIGURE: L(α, β ) ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
43. 43. LOWER TRACE Deﬁne the lower trace L(α, β ) by L(α, α) = 0, / L(α, β ) = L(α, min(Cβ α)) ∪ {max(Cβ ∩ α)} max(Cβ ∩ α) ... ... FIGURE: L(α, β ) ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
44. 44. LOWER TRACE Deﬁne the lower trace L(α, β ) by L(α, α) = 0, / L(α, β ) = L(α, min(Cβ α)) ∪ {max(Cβ ∩ α)} max(Cβ ∩ α) ... ... ... FIGURE: L(α, β ) ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
45. 45. LOWER TRACE Deﬁne the lower trace L(α, β ) by L(α, α) = 0, / L(α, β ) = L(α, min(Cβ α)) ∪ {max(Cβ ∩ α)} max(Cβ ∩ α) ... ... ... ... 0 FIGURE: L(α, β ) ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
46. 46. LOWER TRACE Deﬁne the lower trace L(α, β ) by L(α, α) = 0, / L(α, β ) = L(α, min(Cβ α)) ∪ {max(Cβ ∩ α)} max(Cβ ∩ α) ... ... ... ... ... 01 FIGURE: L(α, β ) ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
47. 47. LOWER TRACE Deﬁne the lower trace L(α, β ) by L(α, α) = 0, / L(α, β ) = L(α, min(Cβ α)) ∪ {max(Cβ ∩ α)} max(Cβ ∩ α) ... ... ... ... ... 01 2 FIGURE: L(α, β ) ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
48. 48. COHERENT SEQUENCES If eβ : β < ω1 is a sequence of ﬁnite to one functions such that eβ → ω for all β < ω1 then it is a coherent sequence if whenever β ≤ β eβ β differs from eβ on a ﬁnite set. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
49. 49. COHERENT SEQUENCES If eβ : β < ω1 is a sequence of ﬁnite to one functions such that eβ → ω for all β < ω1 then it is a coherent sequence if whenever β ≤ β eβ β differs from eβ on a ﬁnite set. For α ≤ β deﬁne ρ1 (α, β ): max |Cξ ∩ α| ρ1 (α, β ) = ξ ∈Tr(α,β ) ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
50. 50. COHERENT SEQUENCES If eβ : β < ω1 is a sequence of ﬁnite to one functions such that eβ → ω for all β < ω1 then it is a coherent sequence if whenever β ≤ β eβ β differs from eβ on a ﬁnite set. For α ≤ β deﬁne ρ1 (α, β ): max |Cξ ∩ α| ρ1 (α, β ) = ξ ∈Tr(α,β ) Deﬁne eβ (α) = ρ1 (α, β ). ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
51. 51. COHERENT SEQUENCES If eβ : β < ω1 is a sequence of ﬁnite to one functions such that eβ → ω for all β < ω1 then it is a coherent sequence if whenever β ≤ β eβ β differs from eβ on a ﬁnite set. For α ≤ β deﬁne ρ1 (α, β ): max |Cξ ∩ α| ρ1 (α, β ) = ξ ∈Tr(α,β ) Deﬁne eβ (α) = ρ1 (α, β ). FACT The sequence eβ : β < ω1 is a coherent sequence. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
52. 52. OSCILLATIONS For ﬁnite functions s and t deﬁned on a common set F let Osc(s, t; F ) be the set of all ξ ∈ F {min(F )} such that s(ξ − ) ≤ t(ξ − ) and s(ξ ) > t(ξ ) where ξ − is greatest element of F less than ξ . ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
53. 53. OSCILLATIONS For ﬁnite functions s and t deﬁned on a common set F let Osc(s, t; F ) be the set of all ξ ∈ F {min(F )} such that s(ξ − ) ≤ t(ξ − ) and s(ξ ) > t(ξ ) where ξ − is greatest element of F less than ξ . Deﬁne osc(α, β ) by: osc(α, β ) = | Osc(α, β ; L(α, β ))| ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
54. 54. OSCILLATIONS THEOREM (MOORE) For any A ⊆ [ω1 ]k and B ⊆ [ω1 ]l which are pairwise disjoint and uncountable and every n < ω there exist a ∈ A and {bm }m<n ⊆ B with a < bm and for all i < k, j < l and m < n osc(a(i), bm (j)) = osc(a(i), b0 (j)) + m ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
55. 55. THE MAIN RESULT THEOREM (MOORE) There exists a function o∗ : ω1 → ω such that if A ⊆ [ω1 ]k and 2 B ⊆ [ω1 ] are pairwise disjoint and uncountable and χ : k → ω is any l function then there exist a ∈ A and b ∈ B with a < b and o∗ (a(i), b(φ (i))) = χ(i) for any φ : k → l. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
56. 56. THE MAIN RESULT CONT... COROLLARY There exists an L-space. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
57. 57. THE MAIN RESULT CONT... COROLLARY There exists an L-space. COROLLARY There exists a weak HFC. ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
58. 58. OPEN QUESTIONS ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
59. 59. OPEN QUESTIONS QUESTION Does there exist an L-space whose square is an L-space? ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
60. 60. OPEN QUESTIONS QUESTION Does there exist an L-space whose square is an L-space? QUESTION Does there exist an HFCk space? w ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
61. 61. OPEN QUESTIONS QUESTION Does there exist an L-space whose square is an L-space? QUESTION Does there exist an HFCk space? w QUESTION Does there exist an L-space that is not a weak HFC? ERIK A. ANDREJKO WALKING UP TO AN L-SPACE
62. 62. REFERENCES ERIK A. ANDREJKO WALKING UP TO AN L-SPACE