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Strongly unbounded colorings

Strongly unbounded colorings

Kobe Set Theory Workshop 2021

Chris Lambie-Hanson

April 11, 2021
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  1. Strongly unbounded colorings (joint work with Assaf Rinot) Chris Lambie-Hanson

    Department of Mathematics and Applied Mathematics Virginia Commonwealth University Kobe Set Theory Workshop on the occasion of Saka´ e Fuchino’s retirement 11 March 2021
  2. Knaster property Drawing by Leon Jesmanowicz Throughout the talk, κ

    denotes a regular uncountable cardinal. Recall that a poset P is κ-Knaster if, whenever, A ∈ [P]κ, there is B ∈ [A]κ consisting of pairwise compatible conditions.
  3. Knaster property Drawing by Leon Jesmanowicz Throughout the talk, κ

    denotes a regular uncountable cardinal. Recall that a poset P is κ-Knaster if, whenever, A ∈ [P]κ, there is B ∈ [A]κ consisting of pairwise compatible conditions. The κ-Knaster property is a strengthening of the κ-cc.
  4. Knaster property Drawing by Leon Jesmanowicz Throughout the talk, κ

    denotes a regular uncountable cardinal. Recall that a poset P is κ-Knaster if, whenever, A ∈ [P]κ, there is B ∈ [A]κ consisting of pairwise compatible conditions. The κ-Knaster property is a strengthening of the κ-cc. In contrast to the κ-cc, the κ-Knaster property is always productive: if P and Q are κ-Knaster, then P × Q are κ-Knaster.
  5. Infinite productivity For an infinite cardinal θ, we say that

    the κ-Knaster property is θ-productive if, whenever {Pi | i < θ} are all κ-Knaster, the full-support product i<θ Pi is κ-Knaster.
  6. Infinite productivity For an infinite cardinal θ, we say that

    the κ-Knaster property is θ-productive if, whenever {Pi | i < θ} are all κ-Knaster, the full-support product i<θ Pi is κ-Knaster. It’s not hard to show that, if κ is weakly compact, then the κ-Knaster property is <κ-productive.
  7. Infinite productivity For an infinite cardinal θ, we say that

    the κ-Knaster property is θ-productive if, whenever {Pi | i < θ} are all κ-Knaster, the full-support product i<θ Pi is κ-Knaster. It’s not hard to show that, if κ is weakly compact, then the κ-Knaster property is <κ-productive. Theorem (Cox-L¨ ucke, ’17) Assuming the consistency of a weakly compact cardinal, there is consistently an inaccessible cardinal κ that is not weakly compact for which the κ-Knaster property is <κ-productive.
  8. Infinite productivity For an infinite cardinal θ, we say that

    the κ-Knaster property is θ-productive if, whenever {Pi | i < θ} are all κ-Knaster, the full-support product i<θ Pi is κ-Knaster. It’s not hard to show that, if κ is weakly compact, then the κ-Knaster property is <κ-productive. Theorem (Cox-L¨ ucke, ’17) Assuming the consistency of a weakly compact cardinal, there is consistently an inaccessible cardinal κ that is not weakly compact for which the κ-Knaster property is <κ-productive. Theorem (LH-L¨ ucke, ’18) If the κ-Knaster property is ℵ0-productive, then κ is weakly compact in L.
  9. Accessible cardinals The results of the previous slide left open

    the question of whether the κ-Knaster property can consistently be infinitely productive for some accessible cardinal κ, e.g., κ = ℵ2 or κ = ℵω+1.
  10. Accessible cardinals The results of the previous slide left open

    the question of whether the κ-Knaster property can consistently be infinitely productive for some accessible cardinal κ, e.g., κ = ℵ2 or κ = ℵω+1. Theorem (LH-Rinot, [1]) Suppose that κ is a successor cardinal. Then there is a κ-Knaster poset P such that Pℵ0 is not κ-cc.
  11. Accessible cardinals The results of the previous slide left open

    the question of whether the κ-Knaster property can consistently be infinitely productive for some accessible cardinal κ, e.g., κ = ℵ2 or κ = ℵω+1. Theorem (LH-Rinot, [1]) Suppose that κ is a successor cardinal. Then there is a κ-Knaster poset P such that Pℵ0 is not κ-cc. The proof of this theorem involved colorings c : [κ]2 → ω with strong unboundedness properties and initiated a systematic investigation of such colorings, their variations, and their applications.
  12. References [1] Chris Lambie-Hanson and Assaf Rinot, Knaster and friends

    I: Closed colorings and precalibers, Algebra Universalis 79 (2018), no. 4, Art. 90, 39. MR 3878671 [2] , Knaster and friends II: The C-sequence number, J. Math. Log. 21 (2021), no. 01, 2150002. [3] , Knaster and friends III: Subadditive colorings, (2021), In preparation.
  13. U(κ, µ, θ, χ) Definition U(κ, µ, θ, χ) asserts

    the existence of a coloring c : [κ]2 → θ such that,
  14. U(κ, µ, θ, χ) Definition U(κ, µ, θ, χ) asserts

    the existence of a coloring c : [κ]2 → θ such that, for every χ < χ and A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets,
  15. U(κ, µ, θ, χ) Definition U(κ, µ, θ, χ) asserts

    the existence of a coloring c : [κ]2 → θ such that, for every χ < χ and A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets, and for every color i < θ,
  16. U(κ, µ, θ, χ) Definition U(κ, µ, θ, χ) asserts

    the existence of a coloring c : [κ]2 → θ such that, for every χ < χ and A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets, and for every color i < θ, there is B ⊆ A of size µ such that, for all a < b, both from B, we have min(c[a × b]) > i.
  17. U(κ, µ, θ, χ) Definition U(κ, µ, θ, χ) asserts

    the existence of a coloring c : [κ]2 → θ such that, for every χ < χ and A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets, and for every color i < θ, there is B ⊆ A of size µ such that, for all a < b, both from B, we have min(c[a × b]) > i. U(κ, µ, θ, χ) can be seen as asserting a strong failure of Ramsey’s theorem at κ.
  18. U(κ, µ, θ, χ) Definition U(κ, µ, θ, χ) asserts

    the existence of a coloring c : [κ]2 → θ such that, for every χ < χ and A ⊆ [κ]χ consisting of κ-many pairwise disjoint sets, and for every color i < θ, there is B ⊆ A of size µ such that, for all a < b, both from B, we have min(c[a × b]) > i. U(κ, µ, θ, χ) can be seen as asserting a strong failure of Ramsey’s theorem at κ. Note that, for all µ ≤ µ and χ ≤ χ, U(κ, µ, θ, χ) implies U(κ, µ , θ, χ ), but there is no such obvious monotonicity in the third coordinate.
  19. Failure of infinite productivity Lemma Suppose that θ ≤ χ

    < κ are infinite, regular cardinals, κ is (<χ)-inaccessible, and U(κ, κ, θ, χ) holds. Then there is a poset P such that
  20. Failure of infinite productivity Lemma Suppose that θ ≤ χ

    < κ are infinite, regular cardinals, κ is (<χ)-inaccessible, and U(κ, κ, θ, χ) holds. Then there is a poset P such that • P is χ-directed closed;
  21. Failure of infinite productivity Lemma Suppose that θ ≤ χ

    < κ are infinite, regular cardinals, κ is (<χ)-inaccessible, and U(κ, κ, θ, χ) holds. Then there is a poset P such that • P is χ-directed closed; • Pτ is κ-Knaster for all τ < θ;
  22. Failure of infinite productivity Lemma Suppose that θ ≤ χ

    < κ are infinite, regular cardinals, κ is (<χ)-inaccessible, and U(κ, κ, θ, χ) holds. Then there is a poset P such that • P is χ-directed closed; • Pτ is κ-Knaster for all τ < θ; • Pθ is not κ-c.c.
  23. Failure of infinite productivity Lemma Suppose that θ ≤ χ

    < κ are infinite, regular cardinals, κ is (<χ)-inaccessible, and U(κ, κ, θ, χ) holds. Then there is a poset P such that • P is χ-directed closed; • Pτ is κ-Knaster for all τ < θ; • Pθ is not κ-c.c. Sketch of proof. Let c : [κ]2 → θ witness U(κ, κ, θ, χ).
  24. Failure of infinite productivity Lemma Suppose that θ ≤ χ

    < κ are infinite, regular cardinals, κ is (<χ)-inaccessible, and U(κ, κ, θ, χ) holds. Then there is a poset P such that • P is χ-directed closed; • Pτ is κ-Knaster for all τ < θ; • Pθ is not κ-c.c. Sketch of proof. Let c : [κ]2 → θ witness U(κ, κ, θ, χ). For all i < θ, let Pi be the poset whose conditions are all sets x ∈ [κ]<χ such that min(c“[x]2) > i, ordered by reverse inclusion.
  25. Failure of infinite productivity Lemma Suppose that θ ≤ χ

    < κ are infinite, regular cardinals, κ is (<χ)-inaccessible, and U(κ, κ, θ, χ) holds. Then there is a poset P such that • P is χ-directed closed; • Pτ is κ-Knaster for all τ < θ; • Pθ is not κ-c.c. Sketch of proof. Let c : [κ]2 → θ witness U(κ, κ, θ, χ). For all i < θ, let Pi be the poset whose conditions are all sets x ∈ [κ]<χ such that min(c“[x]2) > i, ordered by reverse inclusion. Let P be the lottery sum i<θ Pi .
  26. Failure of infinite productivity Lemma Suppose that θ ≤ χ

    < κ are infinite, regular cardinals, κ is (<χ)-inaccessible, and U(κ, κ, θ, χ) holds. Then there is a poset P such that • P is χ-directed closed; • Pτ is κ-Knaster for all τ < θ; • Pθ is not κ-c.c. Sketch of proof. Let c : [κ]2 → θ witness U(κ, κ, θ, χ). For all i < θ, let Pi be the poset whose conditions are all sets x ∈ [κ]<χ such that min(c“[x]2) > i, ordered by reverse inclusion. Let P be the lottery sum i<θ Pi . Now check that P works.
  27. Productivity at successor cardinals The principle U(· · · )

    has been implicit in a variety of previous work. Particularly notable for us is the following result of Todorcevic.
  28. Productivity at successor cardinals The principle U(· · · )

    has been implicit in a variety of previous work. Particularly notable for us is the following result of Todorcevic. Theorem (Todorcevic) For every infinite cardinal λ, U(λ+, λ+, ℵ0, cf(λ)) holds.
  29. Productivity at successor cardinals The principle U(· · · )

    has been implicit in a variety of previous work. Particularly notable for us is the following result of Todorcevic. Theorem (Todorcevic) For every infinite cardinal λ, U(λ+, λ+, ℵ0, cf(λ)) holds. Corollary For every successor cardinal κ, the κ-Knaster property fails to be ℵ0-productive.
  30. Provable instances of U(κ, µ, θ, χ) Given a coloring

    c : [κ]2 → θ, an ordinal β < κ, and a color i < θ, let Dc ≤i (β) := {α < β | c(α, β) ≤ i}.
  31. Provable instances of U(κ, µ, θ, χ) Given a coloring

    c : [κ]2 → θ, an ordinal β < κ, and a color i < θ, let Dc ≤i (β) := {α < β | c(α, β) ≤ i}. We say that c is closed if Dc ≤i (β) is a closed subset of β for all β < κ and i < θ.
  32. Provable instances of U(κ, µ, θ, χ) Given a coloring

    c : [κ]2 → θ, an ordinal β < κ, and a color i < θ, let Dc ≤i (β) := {α < β | c(α, β) ≤ i}. We say that c is closed if Dc ≤i (β) is a closed subset of β for all β < κ and i < θ. Theorem (LH-Rinot, [1]) Suppose that θ, χ < κ are regular cardinals. Any one of the following entails U(κ, κ, θ, χ).
  33. Provable instances of U(κ, µ, θ, χ) Given a coloring

    c : [κ]2 → θ, an ordinal β < κ, and a color i < θ, let Dc ≤i (β) := {α < β | c(α, β) ≤ i}. We say that c is closed if Dc ≤i (β) is a closed subset of β for all β < κ and i < θ. Theorem (LH-Rinot, [1]) Suppose that θ, χ < κ are regular cardinals. Any one of the following entails U(κ, κ, θ, χ). 1 There is a non-reflecting stationary subset of κ ∩ cof(≥ χ).
  34. Provable instances of U(κ, µ, θ, χ) Given a coloring

    c : [κ]2 → θ, an ordinal β < κ, and a color i < θ, let Dc ≤i (β) := {α < β | c(α, β) ≤ i}. We say that c is closed if Dc ≤i (β) is a closed subset of β for all β < κ and i < θ. Theorem (LH-Rinot, [1]) Suppose that θ, χ < κ are regular cardinals. Any one of the following entails U(κ, κ, θ, χ). 1 There is a non-reflecting stationary subset of κ ∩ cof(≥ χ). 2 κ = λ+ and λ is regular.
  35. Provable instances of U(κ, µ, θ, χ) Given a coloring

    c : [κ]2 → θ, an ordinal β < κ, and a color i < θ, let Dc ≤i (β) := {α < β | c(α, β) ≤ i}. We say that c is closed if Dc ≤i (β) is a closed subset of β for all β < κ and i < θ. Theorem (LH-Rinot, [1]) Suppose that θ, χ < κ are regular cardinals. Any one of the following entails U(κ, κ, θ, χ). 1 There is a non-reflecting stationary subset of κ ∩ cof(≥ χ). 2 κ = λ+ and λ is regular. 3 κ = λ+ and cf(λ) = θ.
  36. Provable instances of U(κ, µ, θ, χ) Given a coloring

    c : [κ]2 → θ, an ordinal β < κ, and a color i < θ, let Dc ≤i (β) := {α < β | c(α, β) ≤ i}. We say that c is closed if Dc ≤i (β) is a closed subset of β for all β < κ and i < θ. Theorem (LH-Rinot, [1]) Suppose that θ, χ < κ are regular cardinals. Any one of the following entails U(κ, κ, θ, χ). 1 There is a non-reflecting stationary subset of κ ∩ cof(≥ χ). 2 κ = λ+ and λ is regular. 3 κ = λ+ and cf(λ) = θ. 4 (κ) holds.
  37. Provable instances of U(κ, µ, θ, χ) Given a coloring

    c : [κ]2 → θ, an ordinal β < κ, and a color i < θ, let Dc ≤i (β) := {α < β | c(α, β) ≤ i}. We say that c is closed if Dc ≤i (β) is a closed subset of β for all β < κ and i < θ. Theorem (LH-Rinot, [1]) Suppose that θ, χ < κ are regular cardinals. Any one of the following entails U(κ, κ, θ, χ). 1 There is a non-reflecting stationary subset of κ ∩ cof(≥ χ). 2 κ = λ+ and λ is regular. 3 κ = λ+ and cf(λ) = θ. 4 (κ) holds. Moreover, in all instances, U(κ, κ, θ, χ) is witnessed by a closed coloring.
  38. Consistent failures of U(κ, µ, θ, χ) Proposition (LH-Rinot, [1])

    1 If κ is weakly compact, then U(κ, 2, θ, 2) fails for all θ < κ.
  39. Consistent failures of U(κ, µ, θ, χ) Proposition (LH-Rinot, [1])

    1 If κ is weakly compact, then U(κ, 2, θ, 2) fails for all θ < κ. 2 If λ is a singular limit of strongly compact cardinals, then U(λ+, 2, θ, cf(λ)+) fails for all regular θ ∈ λ+ \ {cf(λ)}
  40. Consistent failures of U(κ, µ, θ, χ) Proposition (LH-Rinot, [1])

    1 If κ is weakly compact, then U(κ, 2, θ, 2) fails for all θ < κ. 2 If λ is a singular limit of strongly compact cardinals, then U(λ+, 2, θ, cf(λ)+) fails for all regular θ ∈ λ+ \ {cf(λ)} Theorem (LH-Rinot, [2]) 1 Suppose that κ is weakly compact.
  41. Consistent failures of U(κ, µ, θ, χ) Proposition (LH-Rinot, [1])

    1 If κ is weakly compact, then U(κ, 2, θ, 2) fails for all θ < κ. 2 If λ is a singular limit of strongly compact cardinals, then U(λ+, 2, θ, cf(λ)+) fails for all regular θ ∈ λ+ \ {cf(λ)} Theorem (LH-Rinot, [2]) 1 Suppose that κ is weakly compact. • There is a forcing extension in which U(κ, κ, ω, ω) fails but κ is not weakly compact.
  42. Consistent failures of U(κ, µ, θ, χ) Proposition (LH-Rinot, [1])

    1 If κ is weakly compact, then U(κ, 2, θ, 2) fails for all θ < κ. 2 If λ is a singular limit of strongly compact cardinals, then U(λ+, 2, θ, cf(λ)+) fails for all regular θ ∈ λ+ \ {cf(λ)} Theorem (LH-Rinot, [2]) 1 Suppose that κ is weakly compact. • There is a forcing extension in which U(κ, κ, ω, ω) fails but κ is not weakly compact. • For every infinite regular θ < κ, there is a forcing extension in which U(κ, κ, θ, χ) holds for all χ < κ but U(κ, κ, θ , θ+) fails for all regular θ = θ.
  43. Consistent failures of U(κ, µ, θ, χ) Proposition (LH-Rinot, [1])

    1 If κ is weakly compact, then U(κ, 2, θ, 2) fails for all θ < κ. 2 If λ is a singular limit of strongly compact cardinals, then U(λ+, 2, θ, cf(λ)+) fails for all regular θ ∈ λ+ \ {cf(λ)} Theorem (LH-Rinot, [2]) 1 Suppose that κ is weakly compact. • There is a forcing extension in which U(κ, κ, ω, ω) fails but κ is not weakly compact. • For every infinite regular θ < κ, there is a forcing extension in which U(κ, κ, θ, χ) holds for all χ < κ but U(κ, κ, θ , θ+) fails for all regular θ = θ. 2 If there is a supercompact cardinal, then there is a forcing extension in which U(ℵω+1, 2, ℵk , ℵ1) fails for all 1 ≤ k < ω.
  44. C-sequences and weak compactness Definition A C-sequence over κ is

    a sequence Cα | α < κ such that • Cα is a club in α for all α ∈ lim(κ);
  45. C-sequences and weak compactness Definition A C-sequence over κ is

    a sequence Cα | α < κ such that • Cα is a club in α for all α ∈ lim(κ); • Cα+1 = {α} for all α < κ.
  46. C-sequences and weak compactness Definition A C-sequence over κ is

    a sequence Cα | α < κ such that • Cα is a club in α for all α ∈ lim(κ); • Cα+1 = {α} for all α < κ. Theorem (Todorcevic) For every regular, uncountable cardinal κ, the following are equivalent.
  47. C-sequences and weak compactness Definition A C-sequence over κ is

    a sequence Cα | α < κ such that • Cα is a club in α for all α ∈ lim(κ); • Cα+1 = {α} for all α < κ. Theorem (Todorcevic) For every regular, uncountable cardinal κ, the following are equivalent. 1 κ is weakly compact.
  48. C-sequences and weak compactness Definition A C-sequence over κ is

    a sequence Cα | α < κ such that • Cα is a club in α for all α ∈ lim(κ); • Cα+1 = {α} for all α < κ. Theorem (Todorcevic) For every regular, uncountable cardinal κ, the following are equivalent. 1 κ is weakly compact. 2 For every C-sequence Cα | α < κ , there is an unbounded D ⊆ κ such that, for every α < κ, there is β < κ for which D ∩ α = Cβ ∩ α.
  49. The C-sequence number Motivated by Todorcevic’s characterization of weak compactness,

    we introduce the notion of the C-sequence number of a cardinal κ (denoted χ(κ)), which can be seen as measuring how far away κ is from being weakly compact.
  50. The C-sequence number Motivated by Todorcevic’s characterization of weak compactness,

    we introduce the notion of the C-sequence number of a cardinal κ (denoted χ(κ)), which can be seen as measuring how far away κ is from being weakly compact. Definition (The C-sequence number) For every regular, uncountable cardinal κ, let χ(κ) = 0 if κ is weakly compact.
  51. The C-sequence number Motivated by Todorcevic’s characterization of weak compactness,

    we introduce the notion of the C-sequence number of a cardinal κ (denoted χ(κ)), which can be seen as measuring how far away κ is from being weakly compact. Definition (The C-sequence number) For every regular, uncountable cardinal κ, let χ(κ) = 0 if κ is weakly compact. Otherwise, let χ(κ) be the least cardinal χ such that, for every C-sequence Cα | α < κ , there is an unbounded D ⊆ κ such that, for every α < κ, there is b ∈ [κ]χ for which D ∩ α ⊆ β∈b Cβ.
  52. Some basic facts Proposition The C-sequence number satisfies the following

    properties. 1 χ(κ) ≤ κ. 2 For every infinite cardinal λ, cf(λ) ≤ χ(λ+) ≤ λ.
  53. Some basic facts Proposition The C-sequence number satisfies the following

    properties. 1 χ(κ) ≤ κ. 2 For every infinite cardinal λ, cf(λ) ≤ χ(λ+) ≤ λ. In particular, if λ is regular, then χ(λ+) = λ.
  54. Some basic facts Proposition The C-sequence number satisfies the following

    properties. 1 χ(κ) ≤ κ. 2 For every infinite cardinal λ, cf(λ) ≤ χ(λ+) ≤ λ. In particular, if λ is regular, then χ(λ+) = λ. 3 If λ is a singular limit of strongly compact cardinals, then χ(λ+) = cf(λ).
  55. Some basic facts Proposition The C-sequence number satisfies the following

    properties. 1 χ(κ) ≤ κ. 2 For every infinite cardinal λ, cf(λ) ≤ χ(λ+) ≤ λ. In particular, if λ is regular, then χ(λ+) = λ. 3 If λ is a singular limit of strongly compact cardinals, then χ(λ+) = cf(λ). 4 If χ(κ) > 1, then χ(κ) ≥ ℵ0.
  56. Some basic facts Proposition The C-sequence number satisfies the following

    properties. 1 χ(κ) ≤ κ. 2 For every infinite cardinal λ, cf(λ) ≤ χ(λ+) ≤ λ. In particular, if λ is regular, then χ(λ+) = λ. 3 If λ is a singular limit of strongly compact cardinals, then χ(λ+) = cf(λ). 4 If χ(κ) > 1, then χ(κ) ≥ ℵ0. 5 If χ(κ) = 1, then κ is greatly Mahlo.
  57. Some basic facts Proposition The C-sequence number satisfies the following

    properties. 1 χ(κ) ≤ κ. 2 For every infinite cardinal λ, cf(λ) ≤ χ(λ+) ≤ λ. In particular, if λ is regular, then χ(λ+) = λ. 3 If λ is a singular limit of strongly compact cardinals, then χ(λ+) = cf(λ). 4 If χ(κ) > 1, then χ(κ) ≥ ℵ0. 5 If χ(κ) = 1, then κ is greatly Mahlo. 6 Every stationary subset of κ ∩ cof(> χ(κ)) reflects.
  58. Some basic facts Proposition The C-sequence number satisfies the following

    properties. 1 χ(κ) ≤ κ. 2 For every infinite cardinal λ, cf(λ) ≤ χ(λ+) ≤ λ. In particular, if λ is regular, then χ(λ+) = λ. 3 If λ is a singular limit of strongly compact cardinals, then χ(λ+) = cf(λ). 4 If χ(κ) > 1, then χ(κ) ≥ ℵ0. 5 If χ(κ) = 1, then κ is greatly Mahlo. 6 Every stationary subset of κ ∩ cof(> χ(κ)) reflects. 7 If µ < κ and (κ, < µ) holds, then χ(κ) ≥ ℵ0.
  59. Some basic facts Proposition The C-sequence number satisfies the following

    properties. 1 χ(κ) ≤ κ. 2 For every infinite cardinal λ, cf(λ) ≤ χ(λ+) ≤ λ. In particular, if λ is regular, then χ(λ+) = λ. 3 If λ is a singular limit of strongly compact cardinals, then χ(λ+) = cf(λ). 4 If χ(κ) > 1, then χ(κ) ≥ ℵ0. 5 If χ(κ) = 1, then κ is greatly Mahlo. 6 Every stationary subset of κ ∩ cof(> χ(κ)) reflects. 7 If µ < κ and (κ, < µ) holds, then χ(κ) ≥ ℵ0. 8 If (κ, < ω) holds, then χ(κ) is as large as possible.
  60. Some basic facts Proposition The C-sequence number satisfies the following

    properties. 1 χ(κ) ≤ κ. 2 For every infinite cardinal λ, cf(λ) ≤ χ(λ+) ≤ λ. In particular, if λ is regular, then χ(λ+) = λ. 3 If λ is a singular limit of strongly compact cardinals, then χ(λ+) = cf(λ). 4 If χ(κ) > 1, then χ(κ) ≥ ℵ0. 5 If χ(κ) = 1, then κ is greatly Mahlo. 6 Every stationary subset of κ ∩ cof(> χ(κ)) reflects. 7 If µ < κ and (κ, < µ) holds, then χ(κ) ≥ ℵ0. 8 If (κ, < ω) holds, then χ(κ) is as large as possible. I.e., χ(κ) = sup(κ ∩ Reg).
  61. Some consistency results Theorem (LH-Rinot, [2]) 1 Suppose that κ

    is weakly compact. • There is a forcing extension in which χ(κ) = 1.
  62. Some consistency results Theorem (LH-Rinot, [2]) 1 Suppose that κ

    is weakly compact. • There is a forcing extension in which χ(κ) = 1. • For every infinite regular θ < κ, there is a forcing extension in which χ(κ) = θ.
  63. Some consistency results Theorem (LH-Rinot, [2]) 1 Suppose that κ

    is weakly compact. • There is a forcing extension in which χ(κ) = 1. • For every infinite regular θ < κ, there is a forcing extension in which χ(κ) = θ. 2 If there is a supercompact cardinal, then there is a forcing extension in which χ(ℵω+1) = ℵ0.
  64. Some consistency results Theorem (LH-Rinot, [2]) 1 Suppose that κ

    is weakly compact. • There is a forcing extension in which χ(κ) = 1. • For every infinite regular θ < κ, there is a forcing extension in which χ(κ) = θ. 2 If there is a supercompact cardinal, then there is a forcing extension in which χ(ℵω+1) = ℵ0. Note the similarity to the previous consistency result about failures of U(κ, µ, θ, χ).
  65. Some consistency results Theorem (LH-Rinot, [2]) 1 Suppose that κ

    is weakly compact. • There is a forcing extension in which U(κ, κ, ω, ω) fails but κ is not weakly compact. • For every infinite regular θ < κ, there is a forcing extension in which U(κ, κ, θ, χ) holds for all χ < κ but U(κ, κ, θ , θ+) fails for all regular θ = θ. 2 If there is a supercompact cardinal, then there is a forcing extension in which U(ℵω+1, 2, ℵk , ℵ1) fails for all 1 ≤ k < ω.
  66. χ(κ) and closed colorings Theorem (LH-Rinot, [2]) Suppose that ℵ0

    ≤ χ ≤ θ = cf(θ) < κ, and suppose that there is a closed witness to U(κ, 2, θ, χ).
  67. χ(κ) and closed colorings Theorem (LH-Rinot, [2]) Suppose that ℵ0

    ≤ χ ≤ θ = cf(θ) < κ, and suppose that there is a closed witness to U(κ, 2, θ, χ). Then χ(κ) ≥ χ.
  68. χ(κ) and closed colorings Theorem (LH-Rinot, [2]) Suppose that ℵ0

    ≤ χ ≤ θ = cf(θ) < κ, and suppose that there is a closed witness to U(κ, 2, θ, χ). Then χ(κ) ≥ χ. Theorem (LH-Rinot, [2]) There is a closed witness to U(κ, κ, ω, χ(κ)).
  69. χ(κ) and closed colorings Theorem (LH-Rinot, [2]) Suppose that ℵ0

    ≤ χ ≤ θ = cf(θ) < κ, and suppose that there is a closed witness to U(κ, 2, θ, χ). Then χ(κ) ≥ χ. Theorem (LH-Rinot, [2]) There is a closed witness to U(κ, κ, ω, χ(κ)). Theorem (LH-Rinot, [2]) There is a closed witness to U(κ, κ, χ(κ), χ(κ)).
  70. χ(κ) and closed colorings Theorem (LH-Rinot, [2]) Suppose that ℵ0

    ≤ χ ≤ θ = cf(θ) < κ, and suppose that there is a closed witness to U(κ, 2, θ, χ). Then χ(κ) ≥ χ. Theorem (LH-Rinot, [2]) There is a closed witness to U(κ, κ, ω, χ(κ)). Theorem (LH-Rinot, [2]) There is a closed witness to U(κ, κ, χ(κ), χ(κ)). Theorem (LH-Rinot, [2]) For infinite regular θ < κ, TFAE:
  71. χ(κ) and closed colorings Theorem (LH-Rinot, [2]) Suppose that ℵ0

    ≤ χ ≤ θ = cf(θ) < κ, and suppose that there is a closed witness to U(κ, 2, θ, χ). Then χ(κ) ≥ χ. Theorem (LH-Rinot, [2]) There is a closed witness to U(κ, κ, ω, χ(κ)). Theorem (LH-Rinot, [2]) There is a closed witness to U(κ, κ, χ(κ), χ(κ)). Theorem (LH-Rinot, [2]) For infinite regular θ < κ, TFAE: 1 there is a C-sequence C over κ such that χ(C) = θ; 2 there is a closed witness to U(κ, κ, θ, θ).
  72. Some conjectures Conjectures 1 For all regular θ < κ,

    U(κ, κ, θ, χ(κ)) holds. 2 (Slightly less ambitious) For all regular θ ≤ χ(κ), U(κ, κ, θ, χ(κ)) holds.
  73. Some conjectures Conjectures 1 For all regular θ < κ,

    U(κ, κ, θ, χ(κ)) holds. 2 (Slightly less ambitious) For all regular θ ≤ χ(κ), U(κ, κ, θ, χ(κ)) holds. 3 For all regular, uncountable κ, TFAE: • χ(κ) ≤ 1;
  74. Some conjectures Conjectures 1 For all regular θ < κ,

    U(κ, κ, θ, χ(κ)) holds. 2 (Slightly less ambitious) For all regular θ ≤ χ(κ), U(κ, κ, θ, χ(κ)) holds. 3 For all regular, uncountable κ, TFAE: • χ(κ) ≤ 1; • the κ-Knaster property is ℵ0 -productive.
  75. Some conjectures Conjectures 1 For all regular θ < κ,

    U(κ, κ, θ, χ(κ)) holds. 2 (Slightly less ambitious) For all regular θ ≤ χ(κ), U(κ, κ, θ, χ(κ)) holds. 3 For all regular, uncountable κ, TFAE: • χ(κ) ≤ 1; • the κ-Knaster property is ℵ0 -productive. 4 For all regular, uncountable κ, TFAE: • κ is weakly compact;
  76. Some conjectures Conjectures 1 For all regular θ < κ,

    U(κ, κ, θ, χ(κ)) holds. 2 (Slightly less ambitious) For all regular θ ≤ χ(κ), U(κ, κ, θ, χ(κ)) holds. 3 For all regular, uncountable κ, TFAE: • χ(κ) ≤ 1; • the κ-Knaster property is ℵ0 -productive. 4 For all regular, uncountable κ, TFAE: • κ is weakly compact; • U(κ, 2, θ, 2) fails for all infinite regular θ < κ;
  77. Some conjectures Conjectures 1 For all regular θ < κ,

    U(κ, κ, θ, χ(κ)) holds. 2 (Slightly less ambitious) For all regular θ ≤ χ(κ), U(κ, κ, θ, χ(κ)) holds. 3 For all regular, uncountable κ, TFAE: • χ(κ) ≤ 1; • the κ-Knaster property is ℵ0 -productive. 4 For all regular, uncountable κ, TFAE: • κ is weakly compact; • U(κ, 2, θ, 2) fails for all infinite regular θ < κ; • U(κ, 2, ℵ0 , 2) fails.
  78. Subadditivity Definition A coloring c : [κ]2 → θ is

    said to be subadditive if, for all α < β < γ < κ,
  79. Subadditivity Definition A coloring c : [κ]2 → θ is

    said to be subadditive if, for all α < β < γ < κ, 1 c(α, γ) ≤ max{c(α, β), c(β, γ)};
  80. Subadditivity Definition A coloring c : [κ]2 → θ is

    said to be subadditive if, for all α < β < γ < κ, 1 c(α, γ) ≤ max{c(α, β), c(β, γ)}; 2 c(α, β) ≤ max{c(α, γ), c(β, γ)}.
  81. Subadditivity Definition A coloring c : [κ]2 → θ is

    said to be subadditive if, for all α < β < γ < κ, 1 c(α, γ) ≤ max{c(α, β), c(β, γ)}; 2 c(α, β) ≤ max{c(α, γ), c(β, γ)}. We say that c is subadditive of the first (resp. second) kind if it satisfies 1 (resp. 2).
  82. Subadditivity Definition A coloring c : [κ]2 → θ is

    said to be subadditive if, for all α < β < γ < κ, 1 c(α, γ) ≤ max{c(α, β), c(β, γ)}; 2 c(α, β) ≤ max{c(α, γ), c(β, γ)}. We say that c is subadditive of the first (resp. second) kind if it satisfies 1 (resp. 2). Subadditive witnesses to U(κ, µ, θ, χ) are particularly useful in applications.
  83. Subadditivity Definition A coloring c : [κ]2 → θ is

    said to be subadditive if, for all α < β < γ < κ, 1 c(α, γ) ≤ max{c(α, β), c(β, γ)}; 2 c(α, β) ≤ max{c(α, γ), c(β, γ)}. We say that c is subadditive of the first (resp. second) kind if it satisfies 1 (resp. 2). Subadditive witnesses to U(κ, µ, θ, χ) are particularly useful in applications. Theorem (LH-Rinot, [3]) Suppose that (κ) holds. Then, for every regular θ < κ, there is a subadditive witness to U(κ, κ, θ, χ) for all χ < κ.
  84. Consistent negative results Subadditive witnesses to U(κ, λ, θ, χ)

    do not necessarily exist, though, even when U(κ, λ, θ, χ) holds.
  85. Consistent negative results Subadditive witnesses to U(κ, λ, θ, χ)

    do not necessarily exist, though, even when U(κ, λ, θ, χ) holds. Theorem (LH-Rinot, [3]) If the P-ideal dichotomy holds, then, for every regular κ > ℵ1, there is no subadditive witness to U(κ, 2, ℵ0, 2).
  86. Consistent negative results Subadditive witnesses to U(κ, λ, θ, χ)

    do not necessarily exist, though, even when U(κ, λ, θ, χ) holds. Theorem (LH-Rinot, [3]) If the P-ideal dichotomy holds, then, for every regular κ > ℵ1, there is no subadditive witness to U(κ, 2, ℵ0, 2). Theorem (Shani, ’16, LH, ’17) Relative to the consistency of large cardinals, there are consistently regular cardinals κ for which (κ, 2) holds but, for every regular θ < κ, there is no subadditive witness to U(κ, 2, θ, 2).
  87. Consistent negative results Subadditive witnesses to U(κ, λ, θ, χ)

    do not necessarily exist, though, even when U(κ, λ, θ, χ) holds. Theorem (LH-Rinot, [3]) If the P-ideal dichotomy holds, then, for every regular κ > ℵ1, there is no subadditive witness to U(κ, 2, ℵ0, 2). Theorem (Shani, ’16, LH, ’17) Relative to the consistency of large cardinals, there are consistently regular cardinals κ for which (κ, 2) holds but, for every regular θ < κ, there is no subadditive witness to U(κ, 2, θ, 2). (Here κ can be arranged to be a successor of a regular cardinal, a successor of a singular cardinal, or an inaccessible cardinal.)
  88. Tightness of Gδ-modifications Definition Let X be a topological space.

    1 The tightness of X, denoted t(X), is the least cardinal κ such that, for every T ⊆ X and every x ∈ cl(T), there is T ∈ [T]≤κ such that x ∈ cl(T ).
  89. Tightness of Gδ-modifications Definition Let X be a topological space.

    1 The tightness of X, denoted t(X), is the least cardinal κ such that, for every T ⊆ X and every x ∈ cl(T), there is T ∈ [T]≤κ such that x ∈ cl(T ). 2 X is Fr´ echet if for every T ⊆ X and every x ∈ cl(T), there is a (countable) sequence of elements of T converging to x.
  90. Tightness of Gδ-modifications Definition Let X be a topological space.

    1 The tightness of X, denoted t(X), is the least cardinal κ such that, for every T ⊆ X and every x ∈ cl(T), there is T ∈ [T]≤κ such that x ∈ cl(T ). 2 X is Fr´ echet if for every T ⊆ X and every x ∈ cl(T), there is a (countable) sequence of elements of T converging to x. Note that, if X is Fr´ echet, then t(X) ≤ ℵ0.
  91. Tightness of Gδ-modifications Definition Let X be a topological space.

    1 The tightness of X, denoted t(X), is the least cardinal κ such that, for every T ⊆ X and every x ∈ cl(T), there is T ∈ [T]≤κ such that x ∈ cl(T ). 2 X is Fr´ echet if for every T ⊆ X and every x ∈ cl(T), there is a (countable) sequence of elements of T converging to x. Note that, if X is Fr´ echet, then t(X) ≤ ℵ0. 3 The Gδ-modification of X, denoted Xδ, is the space with the same underlying set whose topology is generated by the Gδ sets of X.
  92. Tightness of Gδ-modifications Definition Let X be a topological space.

    1 The tightness of X, denoted t(X), is the least cardinal κ such that, for every T ⊆ X and every x ∈ cl(T), there is T ∈ [T]≤κ such that x ∈ cl(T ). 2 X is Fr´ echet if for every T ⊆ X and every x ∈ cl(T), there is a (countable) sequence of elements of T converging to x. Note that, if X is Fr´ echet, then t(X) ≤ ℵ0. 3 The Gδ-modification of X, denoted Xδ, is the space with the same underlying set whose topology is generated by the Gδ sets of X. Some recent work has been done studying the relationship between t(X) and t(Xδ). Of particular interest is whether there is an upper bound on t(Xδ) for countably tight (or stronger) spaces X.
  93. Some results Theorem (Dow-Juh´ asz-Soukup-Szentmikl´ ossy-Weiss, ’19) If there is

    a non-reflecting stationary subset of κ ∩ cof(ω), then there is a Fr´ echet space X such that t(Xδ) = κ.
  94. Some results Theorem (Dow-Juh´ asz-Soukup-Szentmikl´ ossy-Weiss, ’19) If there is

    a non-reflecting stationary subset of κ ∩ cof(ω), then there is a Fr´ echet space X such that t(Xδ) = κ. Theorem (DHSSW, ’19) If λ is strongly compact and t(X) < λ, then t(Xδ) ≤ λ.
  95. Some results Theorem (Dow-Juh´ asz-Soukup-Szentmikl´ ossy-Weiss, ’19) If there is

    a non-reflecting stationary subset of κ ∩ cof(ω), then there is a Fr´ echet space X such that t(Xδ) = κ. Theorem (DHSSW, ’19) If λ is strongly compact and t(X) < λ, then t(Xδ) ≤ λ. Theorem (Chen-Mertens–Szeptycki, ’2X) If (κ) holds, then there is a Fr´ echet α1-space X such that t(Xδ) = κ.
  96. Some results Theorem (Dow-Juh´ asz-Soukup-Szentmikl´ ossy-Weiss, ’19) If there is

    a non-reflecting stationary subset of κ ∩ cof(ω), then there is a Fr´ echet space X such that t(Xδ) = κ. Theorem (DHSSW, ’19) If λ is strongly compact and t(X) < λ, then t(Xδ) ≤ λ. Theorem (Chen-Mertens–Szeptycki, ’2X) If there is a subadditive witness to U(κ, 2, ℵ0, 2), then there is a Fr´ echet α1-space X such that t(Xδ) = κ.
  97. Some results Theorem (Dow-Juh´ asz-Soukup-Szentmikl´ ossy-Weiss, ’19) If there is

    a non-reflecting stationary subset of κ ∩ cof(ω), then there is a Fr´ echet space X such that t(Xδ) = κ. Theorem (DHSSW, ’19) If λ is strongly compact and t(X) < λ, then t(Xδ) ≤ λ. Theorem (Chen-Mertens–Szeptycki, ’2X) If there is a subadditive witness to U(κ, 2, ℵ0, 2), then there is a Fr´ echet α1-space X such that t(Xδ) = κ. Theorem (Chen-Mertens–Szeptycki, ’2X) If PID holds and X is a Fr´ echet α1-space, then t(Xδ) ≤ ℵ1.
  98. An example from a failure of SCH Theorem (LH-Rinot, [3])

    Suppose that µ is a singular cardinal of countable cofinality and SCH fails at µ. Then there is a Fr´ echet α1-space X such that t(Xδ) = µ+.
  99. An example from a failure of SCH Theorem (LH-Rinot, [3])

    Suppose that µ is a singular cardinal of countable cofinality and SCH fails at µ. Then there is a Fr´ echet α1-space X such that t(Xδ) = µ+. We’ll end by sketching a proof of this theorem.
  100. An example from a failure of SCH Theorem (LH-Rinot, [3])

    Suppose that µ is a singular cardinal of countable cofinality and SCH fails at µ. Then there is a Fr´ echet α1-space X such that t(Xδ) = µ+. We’ll end by sketching a proof of this theorem. Lemma (folklore) For every infinite cardinal µ, there is a witness c to U(µ+, 2, cf(µ), 2) such that
  101. An example from a failure of SCH Theorem (LH-Rinot, [3])

    Suppose that µ is a singular cardinal of countable cofinality and SCH fails at µ. Then there is a Fr´ echet α1-space X such that t(Xδ) = µ+. We’ll end by sketching a proof of this theorem. Lemma (folklore) For every infinite cardinal µ, there is a witness c to U(µ+, 2, cf(µ), 2) such that • c is subadditive of the first kind;
  102. An example from a failure of SCH Theorem (LH-Rinot, [3])

    Suppose that µ is a singular cardinal of countable cofinality and SCH fails at µ. Then there is a Fr´ echet α1-space X such that t(Xδ) = µ+. We’ll end by sketching a proof of this theorem. Lemma (folklore) For every infinite cardinal µ, there is a witness c to U(µ+, 2, cf(µ), 2) such that • c is subadditive of the first kind; • c is locally small, i.e., |Dc ≤i (β)| < µ for all i < cf(µ) and all β < µ+.
  103. Proof sketch Fix a coloring c : [µ+]2 → ω

    as in the Lemma. The underlying set of X will be µ+ ∪ {∞}.
  104. Proof sketch Fix a coloring c : [µ+]2 → ω

    as in the Lemma. The underlying set of X will be µ+ ∪ {∞}. Every element of µ+ is isolated in X.
  105. Proof sketch Fix a coloring c : [µ+]2 → ω

    as in the Lemma. The underlying set of X will be µ+ ∪ {∞}. Every element of µ+ is isolated in X. Basic open neighborhoods of ∞ in X are of the form Ni,β := {∞} ∪ (µ+ \ Dc ≤i (β)),
  106. Proof sketch Fix a coloring c : [µ+]2 → ω

    as in the Lemma. The underlying set of X will be µ+ ∪ {∞}. Every element of µ+ is isolated in X. Basic open neighborhoods of ∞ in X are of the form Ni,β := {∞} ∪ (µ+ \ Dc ≤i (β)), i.e., the closed subsets of µ+ are precisely the intersections of the sets Dc ≤i (β), so, if T ⊆ µ+, then ∞ ∈ cl(T) ⇔ ( ∃(i, β) ∈ ω × µ+)(T ⊆ Dc ≤i (β))
  107. X is Fr´ echet Suppose that T ⊆ µ+ and

    ∞ ∈ cl(T). Suppose first that |T| < µ. For each β > sup(T) + 1, define fβ : T → ω by fβ(α) := c(α, β).
  108. X is Fr´ echet Suppose that T ⊆ µ+ and

    ∞ ∈ cl(T). Suppose first that |T| < µ. For each β > sup(T) + 1, define fβ : T → ω by fβ(α) := c(α, β). Since µ is strong limit, there are an unbounded A ⊆ µ+ and a function f : T → ω such that fβ = f for all β ∈ A.
  109. X is Fr´ echet Suppose that T ⊆ µ+ and

    ∞ ∈ cl(T). Suppose first that |T| < µ. For each β > sup(T) + 1, define fβ : T → ω by fβ(α) := c(α, β). Since µ is strong limit, there are an unbounded A ⊆ µ+ and a function f : T → ω such that fβ = f for all β ∈ A. Since ∞ ∈ cl(T), for each i < ω, we can fix αi ∈ T such that f (αi ) ≥ i.
  110. X is Fr´ echet Suppose that T ⊆ µ+ and

    ∞ ∈ cl(T). Suppose first that |T| < µ. For each β > sup(T) + 1, define fβ : T → ω by fβ(α) := c(α, β). Since µ is strong limit, there are an unbounded A ⊆ µ+ and a function f : T → ω such that fβ = f for all β ∈ A. Since ∞ ∈ cl(T), for each i < ω, we can fix αi ∈ T such that f (αi ) ≥ i. Then every Dc ≤i (β) contains only finitely many of the αi ’s, so αi | i < ω converges to ∞.
  111. X is Fr´ echet Suppose that T ⊆ µ+ and

    ∞ ∈ cl(T). Suppose first that |T| < µ. For each β > sup(T) + 1, define fβ : T → ω by fβ(α) := c(α, β). Since µ is strong limit, there are an unbounded A ⊆ µ+ and a function f : T → ω such that fβ = f for all β ∈ A. Since ∞ ∈ cl(T), for each i < ω, we can fix αi ∈ T such that f (αi ) ≥ i. Then every Dc ≤i (β) contains only finitely many of the αi ’s, so αi | i < ω converges to ∞. Suppose now that |T| ≥ µ.
  112. X is Fr´ echet Suppose that T ⊆ µ+ and

    ∞ ∈ cl(T). Suppose first that |T| < µ. For each β > sup(T) + 1, define fβ : T → ω by fβ(α) := c(α, β). Since µ is strong limit, there are an unbounded A ⊆ µ+ and a function f : T → ω such that fβ = f for all β ∈ A. Since ∞ ∈ cl(T), for each i < ω, we can fix αi ∈ T such that f (αi ) ≥ i. Then every Dc ≤i (β) contains only finitely many of the αi ’s, so αi | i < ω converges to ∞. Suppose now that |T| ≥ µ. For each (i, β), we have |Dc ≤i (β)| < µ, so |[Dc ≤i (β)]ℵ0 | < µ.
  113. X is Fr´ echet Suppose that T ⊆ µ+ and

    ∞ ∈ cl(T). Suppose first that |T| < µ. For each β > sup(T) + 1, define fβ : T → ω by fβ(α) := c(α, β). Since µ is strong limit, there are an unbounded A ⊆ µ+ and a function f : T → ω such that fβ = f for all β ∈ A. Since ∞ ∈ cl(T), for each i < ω, we can fix αi ∈ T such that f (αi ) ≥ i. Then every Dc ≤i (β) contains only finitely many of the αi ’s, so αi | i < ω converges to ∞. Suppose now that |T| ≥ µ. For each (i, β), we have |Dc ≤i (β)| < µ, so |[Dc ≤i (β)]ℵ0 | < µ. Therefore, the number of elements of [T]ℵ0 that are contained in some Dc ≤i (β) is at most µ+.
  114. X is Fr´ echet Suppose that T ⊆ µ+ and

    ∞ ∈ cl(T). Suppose first that |T| < µ. For each β > sup(T) + 1, define fβ : T → ω by fβ(α) := c(α, β). Since µ is strong limit, there are an unbounded A ⊆ µ+ and a function f : T → ω such that fβ = f for all β ∈ A. Since ∞ ∈ cl(T), for each i < ω, we can fix αi ∈ T such that f (αi ) ≥ i. Then every Dc ≤i (β) contains only finitely many of the αi ’s, so αi | i < ω converges to ∞. Suppose now that |T| ≥ µ. For each (i, β), we have |Dc ≤i (β)| < µ, so |[Dc ≤i (β)]ℵ0 | < µ. Therefore, the number of elements of [T]ℵ0 that are contained in some Dc ≤i (β) is at most µ+. Since SCH fails at µ, |[T]ℵ0 | > µ+, so we can find a countable subset of T that is not contained in any Dc ≤i (β), and proceed as in the previous case.
  115. t(Xδ ) = µ+ In X, every closed subset of

    µ+ is bounded in µ+. Since every closed subset of µ+ in Xδ is contained in a countable union of closed subsets of µ+ in X, this is also true in Xδ.
  116. t(Xδ ) = µ+ In X, every closed subset of

    µ+ is bounded in µ+. Since every closed subset of µ+ in Xδ is contained in a countable union of closed subsets of µ+ in X, this is also true in Xδ. In particular, in Xδ, ∞ ∈ cl(µ+).
  117. t(Xδ ) = µ+ In X, every closed subset of

    µ+ is bounded in µ+. Since every closed subset of µ+ in Xδ is contained in a countable union of closed subsets of µ+ in X, this is also true in Xδ. In particular, in Xδ, ∞ ∈ cl(µ+). However, for every β < µ+, we have β = i<ω Dc ≤i (β).
  118. t(Xδ ) = µ+ In X, every closed subset of

    µ+ is bounded in µ+. Since every closed subset of µ+ in Xδ is contained in a countable union of closed subsets of µ+ in X, this is also true in Xδ. In particular, in Xδ, ∞ ∈ cl(µ+). However, for every β < µ+, we have β = i<ω Dc ≤i (β). Since Dc ≤i (β) is closed in X, β is closed in Xδ.
  119. t(Xδ ) = µ+ In X, every closed subset of

    µ+ is bounded in µ+. Since every closed subset of µ+ in Xδ is contained in a countable union of closed subsets of µ+ in X, this is also true in Xδ. In particular, in Xδ, ∞ ∈ cl(µ+). However, for every β < µ+, we have β = i<ω Dc ≤i (β). Since Dc ≤i (β) is closed in X, β is closed in Xδ. Therefore, ∞ is not in the closure of any bounded subset of µ+ in Xδ.
  120. t(Xδ ) = µ+ In X, every closed subset of

    µ+ is bounded in µ+. Since every closed subset of µ+ in Xδ is contained in a countable union of closed subsets of µ+ in X, this is also true in Xδ. In particular, in Xδ, ∞ ∈ cl(µ+). However, for every β < µ+, we have β = i<ω Dc ≤i (β). Since Dc ≤i (β) is closed in X, β is closed in Xδ. Therefore, ∞ is not in the closure of any bounded subset of µ+ in Xδ. It follows that t(Xδ) = µ+.