Grothendieck C(K)-Spaces of Small Density

Grothendieck C(K)-spaces of small density

Damian Sobota

Technische Universit¨at Wien, Austria

Winter School, Hejnice 2018 X ∗ — the dual space of X = the Banach space of all continuous linear functionals on X

X ∗∗ — the bidual space of X = the dual space of X ∗ Fact ∗ 1 If x ∈ X , then the evaluation evx : X → R, where ∗ ∗ ∗ ∗ ∗∗ evx (x ) = x (x) for every x ∈ X , is in X . ∗∗ ∗∗ 2 X 3 x 7→ evx ∈ X — isometric embedding of X into X .

Weak topologies

X — an inﬁnite-dimensional Banach space X ∗∗ — the bidual space of X = the dual space of X ∗ Fact ∗ 1 If x ∈ X , then the evaluation evx : X → R, where ∗ ∗ ∗ ∗ ∗∗ evx (x ) = x (x) for every x ∈ X , is in X . ∗∗ ∗∗ 2 X 3 x 7→ evx ∈ X — isometric embedding of X into X .

Weak topologies

X — an inﬁnite-dimensional Banach space

X ∗ — the dual space of X = the Banach space of all continuous linear functionals on X Fact ∗ 1 If x ∈ X , then the evaluation evx : X → R, where ∗ ∗ ∗ ∗ ∗∗ evx (x ) = x (x) for every x ∈ X , is in X . ∗∗ ∗∗ 2 X 3 x 7→ evx ∈ X — isometric embedding of X into X .

Weak topologies

X — an inﬁnite-dimensional Banach space

X ∗ — the dual space of X = the Banach space of all continuous linear functionals on X

X ∗∗ — the bidual space of X = the dual space of X ∗ ∗∗ ∗∗ 2 X 3 x 7→ evx ∈ X — isometric embedding of X into X .

Weak topologies

X — an inﬁnite-dimensional Banach space

X ∗ — the dual space of X = the Banach space of all continuous linear functionals on X

X ∗∗ — the bidual space of X = the dual space of X ∗ Fact ∗ 1 If x ∈ X , then the evaluation evx : X → R, where ∗ ∗ ∗ ∗ ∗∗ evx (x ) = x (x) for every x ∈ X , is in X . Weak topologies on X ∗ weak topology on X ∗ = the smallest topology such that every x∗∗ ∈ X ∗∗ is continuous; weak* topology on X ∗ = the smallest topology such that every evx is continuous;

Weak topologies

X — an inﬁnite-dimensional Banach space

X ∗ — the dual space of X = the Banach space of all continuous linear functionals on X

Weak topologies

X — an inﬁnite-dimensional Banach space

X ∗ — the dual space of X = the Banach space of all continuous linear functionals on X

Weak topologies on X ∗ weak topology on X ∗ = the smallest topology such that every x∗∗ ∈ X ∗∗ is continuous; Weak topologies

X — an inﬁnite-dimensional Banach space

X ∗ — the dual space of X = the Banach space of all continuous linear functionals on X

Weak topologies on X ∗ weak topology on X ∗ = the smallest topology such that every x∗∗ ∈ X ∗∗ is continuous; weak* topology on X ∗ = the smallest topology such that every evx is continuous; Deﬁnition An inﬁnite dimensional Banach space X is Grothendieck if every weak* convergent sequence in the dual X ∗ is weakly convergent.

Grothendieck spaces

Inclusions of the weak topologies on X ∗

weak∗ ⊆ weak ⊆ norm

Question ∗ ∗ Is always a weak* convergent sequence xn ∈ X : n ∈ ω weakly convergent? Grothendieck spaces

Inclusions of the weak topologies on X ∗

weak∗ ⊆ weak ⊆ norm

Question ∗ ∗ Is always a weak* convergent sequence xn ∈ X : n ∈ ω weakly convergent?

Definition An infinite dimensional Banach space X is Grothendieck if every weak* convergent sequence in the dual X ∗ is weakly convergent. `∞ (Grothendieck ’53) the space H∞ of bounded analytic functions on the unit disc (Bourgain ’83) von Neumann algebras (Pfitzner ’94) C(St(A)) where A is a σ-complete Boolean algebras (Grothendieck ’53)

Examples of non-Grothendieck spaces

`1 and c0 separable C(K)-spaces, e.g. C([0, 1]) or C(2ω) C(K) provided that K has a non-trivial convergent sequence

C(K) such that C(K) = c0 ⊕ Y for some closed subspace Y

Grothendieck spaces

Examples of Grothendieck spaces

reﬂexive spaces, so all `p for 1 < p < ∞ the space H∞ of bounded analytic functions on the unit disc (Bourgain ’83) von Neumann algebras (Pﬁtzner ’94) C(St(A)) where A is a σ-complete Boolean algebras (Grothendieck ’53)

Examples of non-Grothendieck spaces

`1 and c0 separable C(K)-spaces, e.g. C([0, 1]) or C(2ω) C(K) provided that K has a non-trivial convergent sequence

C(K) such that C(K) = c0 ⊕ Y for some closed subspace Y

Grothendieck spaces

Examples of Grothendieck spaces

reﬂexive spaces, so all `p for 1 < p < ∞

`∞ (Grothendieck ’53) von Neumann algebras (Pﬁtzner ’94) C(St(A)) where A is a σ-complete Boolean algebras (Grothendieck ’53)

Examples of non-Grothendieck spaces

`1 and c0 separable C(K)-spaces, e.g. C([0, 1]) or C(2ω) C(K) provided that K has a non-trivial convergent sequence

C(K) such that C(K) = c0 ⊕ Y for some closed subspace Y

Grothendieck spaces

Examples of Grothendieck spaces

reﬂexive spaces, so all `p for 1 < p < ∞

`∞ (Grothendieck ’53) the space H∞ of bounded analytic functions on the unit disc (Bourgain ’83) C(St(A)) where A is a σ-complete Boolean algebras (Grothendieck ’53)

Examples of non-Grothendieck spaces

`1 and c0 separable C(K)-spaces, e.g. C([0, 1]) or C(2ω) C(K) provided that K has a non-trivial convergent sequence

C(K) such that C(K) = c0 ⊕ Y for some closed subspace Y

Grothendieck spaces

Examples of Grothendieck spaces

reﬂexive spaces, so all `p for 1 < p < ∞

`∞ (Grothendieck ’53) the space H∞ of bounded analytic functions on the unit disc (Bourgain ’83) von Neumann algebras (Pﬁtzner ’94) Examples of non-Grothendieck spaces

`1 and c0 separable C(K)-spaces, e.g. C([0, 1]) or C(2ω) C(K) provided that K has a non-trivial convergent sequence

C(K) such that C(K) = c0 ⊕ Y for some closed subspace Y

Grothendieck spaces

Examples of Grothendieck spaces

reﬂexive spaces, so all `p for 1 < p < ∞

`∞ (Grothendieck ’53) the space H∞ of bounded analytic functions on the unit disc (Bourgain ’83) von Neumann algebras (Pﬁtzner ’94) C(St(A)) where A is a σ-complete Boolean algebras (Grothendieck ’53) separable C(K)-spaces, e.g. C([0, 1]) or C(2ω) C(K) provided that K has a non-trivial convergent sequence

C(K) such that C(K) = c0 ⊕ Y for some closed subspace Y

Grothendieck spaces

Examples of Grothendieck spaces

reﬂexive spaces, so all `p for 1 < p < ∞

Examples of non-Grothendieck spaces

`1 and c0 C(K) provided that K has a non-trivial convergent sequence

C(K) such that C(K) = c0 ⊕ Y for some closed subspace Y

Grothendieck spaces

Examples of Grothendieck spaces

reﬂexive spaces, so all `p for 1 < p < ∞

Examples of non-Grothendieck spaces

`1 and c0 separable C(K)-spaces, e.g. C([0, 1]) or C(2ω) C(K) such that C(K) = c0 ⊕ Y for some closed subspace Y

Grothendieck spaces

Examples of Grothendieck spaces

reﬂexive spaces, so all `p for 1 < p < ∞

Examples of non-Grothendieck spaces

`1 and c0 separable C(K)-spaces, e.g. C([0, 1]) or C(2ω) C(K) provided that K has a non-trivial convergent sequence Grothendieck spaces

Examples of Grothendieck spaces

reﬂexive spaces, so all `p for 1 < p < ∞

Examples of non-Grothendieck spaces

`1 and c0 separable C(K)-spaces, e.g. C([0, 1]) or C(2ω) C(K) provided that K has a non-trivial convergent sequence

C(K) such that C(K) = c0 ⊕ Y for some closed subspace Y Examples σ-complete algebras algebras with Haydon’s Subsequential Completeness Property (SCP) algebras with Schachermayer’s property (E), Seever’s property (I), Moltó’s property (f)...

Non-examples A such that St(A) has a non-trivial convergent sequence so countable Boolean algebras algebra J of Jordan-measurable subsets of [0, 1]

The Grothendieck property

Deﬁnition A Boolean algebra A has the Grothendieck property if the space C(St(A)) is a Grothendieck space. algebras with Haydon’s Subsequential Completeness Property (SCP) algebras with Schachermayer’s property (E), Seever’s property (I), Moltó’s property (f)...

Non-examples A such that St(A) has a non-trivial convergent sequence so countable Boolean algebras algebra J of Jordan-measurable subsets of [0, 1]

The Grothendieck property

Deﬁnition A Boolean algebra A has the Grothendieck property if the space C(St(A)) is a Grothendieck space.

Examples σ-complete algebras algebras with Schachermayer’s property (E), Seever’s property (I), Moltó’s property (f)...

Non-examples A such that St(A) has a non-trivial convergent sequence so countable Boolean algebras algebra J of Jordan-measurable subsets of [0, 1]

The Grothendieck property

Deﬁnition A Boolean algebra A has the Grothendieck property if the space C(St(A)) is a Grothendieck space.

Examples σ-complete algebras algebras with Haydon’s Subsequential Completeness Property (SCP) Non-examples A such that St(A) has a non-trivial convergent sequence so countable Boolean algebras algebra J of Jordan-measurable subsets of [0, 1]

The Grothendieck property

Deﬁnition A Boolean algebra A has the Grothendieck property if the space C(St(A)) is a Grothendieck space.

Examples σ-complete algebras algebras with Haydon’s Subsequential Completeness Property (SCP) algebras with Schachermayer’s property (E), Seever’s property (I), Moltó’s property (f)... so countable Boolean algebras algebra J of Jordan-measurable subsets of [0, 1]

The Grothendieck property

Deﬁnition A Boolean algebra A has the Grothendieck property if the space C(St(A)) is a Grothendieck space.

Examples σ-complete algebras algebras with Haydon’s Subsequential Completeness Property (SCP) algebras with Schachermayer’s property (E), Seever’s property (I), Moltó’s property (f)...

Non-examples A such that St(A) has a non-trivial convergent sequence algebra J of Jordan-measurable subsets of [0, 1]

The Grothendieck property

Deﬁnition A Boolean algebra A has the Grothendieck property if the space C(St(A)) is a Grothendieck space.

Examples σ-complete algebras algebras with Haydon’s Subsequential Completeness Property (SCP) algebras with Schachermayer’s property (E), Seever’s property (I), Moltó’s property (f)...

Non-examples A such that St(A) has a non-trivial convergent sequence so countable Boolean algebras The Grothendieck property

Deﬁnition A Boolean algebra A has the Grothendieck property if the space C(St(A)) is a Grothendieck space.

Examples σ-complete algebras algebras with Haydon’s Subsequential Completeness Property (SCP) algebras with Schachermayer’s property (E), Seever’s property (I), Moltó’s property (f)...

Non-examples A such that St(A) has a non-trivial convergent sequence so countable Boolean algebras algebra J of Jordan-measurable subsets of [0, 1] Problem Describe gr in terms of classical cardinal characteristics of the continuum.

The Grothendieck number

The Grothendieck number gr gr = min |A|: inﬁnite A has the Grothendieck property

ω1 ¬ gr ¬ c The Grothendieck number

The Grothendieck number gr gr = min |A|: inﬁnite A has the Grothendieck property

ω1 ¬ gr ¬ c

Problem Describe gr in terms of classical cardinal characteristics of the continuum. Theorem (Booth ’74) If w(K) < s, then K is sequentially compact.

ZFC lower bounds

If St(A) has a non-trivial convergent sequence, then A does not have the Grothendieck property. Theorem (Geschke ’06) If w(K) < cov(M), then either K is scattered or there exists a perfect subset L ⊆ K with a Gδ-point x ∈ L. Recall that: |A| = w(St(A)). Corollary If |A| < max(s, cov(M)), then A does not have the Grothendieck property. Hence, gr max(s, cov(M)).

ZFC lower bounds

If St(A) has a non-trivial convergent sequence, then A does not have the Grothendieck property. Theorem (Booth ’74) If w(K) < s, then K is sequentially compact. Recall that: |A| = w(St(A)). Corollary If |A| < max(s, cov(M)), then A does not have the Grothendieck property. Hence, gr max(s, cov(M)).

ZFC lower bounds

If St(A) has a non-trivial convergent sequence, then A does not have the Grothendieck property. Theorem (Booth ’74) If w(K) < s, then K is sequentially compact.

Theorem (Geschke ’06) If w(K) < cov(M), then either K is scattered or there exists a perfect subset L ⊆ K with a Gδ-point x ∈ L. ZFC lower bounds

If St(A) has a non-trivial convergent sequence, then A does not have the Grothendieck property. Theorem (Booth ’74) If w(K) < s, then K is sequentially compact.

Theorem (Geschke ’06) If w(K) < cov(M), then either K is scattered or there exists a perfect subset L ⊆ K with a Gδ-point x ∈ L. Recall that: |A| = w(St(A)). Corollary If |A| < max(s, cov(M)), then A does not have the Grothendieck property. Hence, gr max(s, cov(M)). Then, if A is a σ-complete Boolean algebra in a κ ground model V and G is a S -generic ﬁlter over V , then A has the Grothendieck property in V [G].

Corollary κ If CH holds in V and G is a S -generic ﬁlter over V , then gr = ω1 < κ = c holds in V [G].

Con(gr < c)

Theorem (Brech ’06) κ Let κ be a regular cardinal number and S denote the side-by-side Sacks forcing. Corollary κ If CH holds in V and G is a S -generic ﬁlter over V , then gr = ω1 < κ = c holds in V [G].

Con(gr < c)

Theorem (Brech ’06) κ Let κ be a regular cardinal number and S denote the side-by-side Sacks forcing. Then, if A is a σ-complete Boolean algebra in a κ ground model V and G is a S -generic ﬁlter over V , then A has the Grothendieck property in V [G]. Con(gr < c)

Corollary κ If CH holds in V and G is a S -generic filter over V , then gr = ω1 < κ = c holds in V [G]. Examples: Sacks, side-by-side Sacks, Laver, Mathias, Miller, Silver(-like) Definition A forcing P ∈ V preserves the ground model reals non-meager if R ∩ V is a non-meager subset of R ∩ V [G] for any P-generic filter G. Examples: Sacks, side-by-side Sacks, Miller, Silver(-like)

Generalization

Definition A forcing P ∈ V has the Laver property if for every P-generic filter G over V , every f ∈ ωω ∩ V and g ∈ ωω ∩ V [G] such that g ¬∗ f , there exists H : ω → [ω]<ω such that g(n) ∈ H(n) and |H(n)| ¬ n + 1 for every n ∈ ω. Definition A forcing P ∈ V preserves the ground model reals non-meager if R ∩ V is a non-meager subset of R ∩ V [G] for any P-generic filter G. Examples: Sacks, side-by-side Sacks, Miller, Silver(-like)

Generalization

Definition A forcing P ∈ V has the Laver property if for every P-generic filter G over V , every f ∈ ωω ∩ V and g ∈ ωω ∩ V [G] such that g ¬∗ f , there exists H : ω → [ω]<ω such that g(n) ∈ H(n) and |H(n)| ¬ n + 1 for every n ∈ ω. Examples: Sacks, side-by-side Sacks, Laver, Mathias, Miller, Silver(-like) Definition A forcing P ∈ V preserves the ground model reals non-meager if R ∩ V is a non-meager subset of R ∩ V [G] for any P-generic filter G. Generalization

Definition A forcing P ∈ V has the Laver property if for every P-generic filter G over V , every f ∈ ωω ∩ V and g ∈ ωω ∩ V [G] such that g ¬∗ f , there exists H : ω → [ω]<ω such that g(n) ∈ H(n) and |H(n)| ¬ n + 1 for every n ∈ ω. Examples: Sacks, side-by-side Sacks, Laver, Mathias, Miller, Silver(-like) Definition A forcing P ∈ V preserves the ground model reals non-meager if R ∩ V is a non-meager subset of R ∩ V [G] for any P-generic filter G. Examples: Sacks, side-by-side Sacks, Miller, Silver(-like) Then, if A is a σ-complete Boolean algebra in V and G is a P-generic filter over V , then A has the Grothendieck property in V [G].

Corollary

1 in the Miller model: gr = ω1 < ω2 = d = g = c

2 in the Silver model: gr = ω1 < ω2 = r = u = c Recall that: Con(r = u < s) and Con(g < cov(M)) Corollary No ZFC inequality between gr and any of the numbers r, u and g.

Question Con(d < gr)?

Generalization

Theorem (S.–Zdomskyy ’17) Let P ∈ V be a notion of proper forcing having the Laver property and preserving the ground model reals non-meager. Corollary

1 in the Miller model: gr = ω1 < ω2 = d = g = c

2 in the Silver model: gr = ω1 < ω2 = r = u = c Recall that: Con(r = u < s) and Con(g < cov(M)) Corollary No ZFC inequality between gr and any of the numbers r, u and g.

Question Con(d < gr)?

Generalization

Theorem (S.–Zdomskyy ’17) Let P ∈ V be a notion of proper forcing having the Laver property and preserving the ground model reals non-meager. Then, if A is a σ-complete Boolean algebra in V and G is a P-generic ﬁlter over V , then A has the Grothendieck property in V [G]. 2 in the Silver model: gr = ω1 < ω2 = r = u = c Recall that: Con(r = u < s) and Con(g < cov(M)) Corollary No ZFC inequality between gr and any of the numbers r, u and g.

Question Con(d < gr)?

Generalization

Corollary

1 in the Miller model: gr = ω1 < ω2 = d = g = c Recall that: Con(r = u < s) and Con(g < cov(M)) Corollary No ZFC inequality between gr and any of the numbers r, u and g.

Question Con(d < gr)?

Generalization

Corollary

1 in the Miller model: gr = ω1 < ω2 = d = g = c

2 in the Silver model: gr = ω1 < ω2 = r = u = c Question Con(d < gr)?

Generalization

Corollary

1 in the Miller model: gr = ω1 < ω2 = d = g = c

2 in the Silver model: gr = ω1 < ω2 = r = u = c Recall that: Con(r = u < s) and Con(g < cov(M)) Corollary No ZFC inequality between gr and any of the numbers r, u and g. Generalization

Corollary

1 in the Miller model: gr = ω1 < ω2 = d = g = c

2 in the Silver model: gr = ω1 < ω2 = r = u = c Recall that: Con(r = u < s) and Con(g < cov(M)) Corollary No ZFC inequality between gr and any of the numbers r, u and g.

Question Con(d < gr)? then there exists a Boolean algebra A with the Grothendieck property and of cardinality κ.

Corollary If cof([cof(N )]ω) = cof(N ), then gr ¬ cof(N ).

Recall that Con(ω2 = cof(N ) < a = ω3) (Brendle ’03). Corollary No ZFC inequality between gr and a.

A ZFC upper bound

Theorem (S. ’18) If κ is a cardinal number such that cof([κ]ω) = κ cof(N ), Corollary If cof([cof(N )]ω) = cof(N ), then gr ¬ cof(N ).

Recall that Con(ω2 = cof(N ) < a = ω3) (Brendle ’03). Corollary No ZFC inequality between gr and a.

A ZFC upper bound

Theorem (S. ’18) If κ is a cardinal number such that cof([κ]ω) = κ cof(N ), then there exists a Boolean algebra A with the Grothendieck property and of cardinality κ. Recall that Con(ω2 = cof(N ) < a = ω3) (Brendle ’03). Corollary No ZFC inequality between gr and a.

A ZFC upper bound

Theorem (S. ’18) If κ is a cardinal number such that cof([κ]ω) = κ cof(N ), then there exists a Boolean algebra A with the Grothendieck property and of cardinality κ.

Corollary If cof([cof(N )]ω) = cof(N ), then gr ¬ cof(N ). A ZFC upper bound

Theorem (S. ’18) If κ is a cardinal number such that cof([κ]ω) = κ cof(N ), then there exists a Boolean algebra A with the Grothendieck property and of cardinality κ.

Corollary If cof([cof(N )]ω) = cof(N ), then gr ¬ cof(N ).

Recall that Con(ω2 = cof(N ) < a = ω3) (Brendle ’03). Corollary No ZFC inequality between gr and a. 1 gr cov(M) and Con(cov(M) > non(M)) 2 gr s and Con(s > cov(M)) 3 Con(gr < d)

Cichoń’s diagram and gr

cov(N ) / non(M) / cof(M) / cof(N ) / c O O O O

b / d O O

ω1 / add(N ) / add(M) / cov(M) / non(N )

What’s known: 2 gr s and Con(s > cov(M)) 3 Con(gr < d)

Cichoń’s diagram and gr

X X X covX(XNX) / non(XMX) / cof(M) / cof(N ) / c O O O O

b¡A / d O O

XX XX ¨ωH¨H1 / add(XNX) / add(XMX) / cov(M) / non(N )

What’s known: 1 gr cov(M) and Con(cov(M) > non(M)) 3 Con(gr < d)

Cichoń’s diagram and gr

X X X covX(XNX) / non(XMX) / cof(M) / cof(N ) / c O O O O

b¡A / d O O

XX XX XX ¨ωH¨H1 / add(XNX) / add(XMX) / cov(XMX) / non(N )

What’s known: 1 gr cov(M) and Con(cov(M) > non(M)) 2 gr s and Con(s > cov(M)) Cichoń’s diagram and gr

X XX X X covX(XNX) / non(XMX) / cofX(MXX) / cofX(NXX) / ¡Ac O O O O

b¡A / d¡A O O

XX XX XX ¨ωH¨H1 / add(XNX) / add(XMX) / cov(XMX) / non(N )

What’s known: 1 gr cov(M) and Con(cov(M) > non(M)) 2 gr s and Con(s > cov(M)) 3 Con(gr < d) 2 b ¬ gr? (the Laver model?) 3 Con(gr < cov(N ))? (the random model?)

Cichoń’s diagram and gr

X XX X X covX(XNX) / non(XMX) / cofX(MXX) / cofX(NXX) / ¡Ac O O O O

b¡A / d¡A O O

XX XX XX ¨ωH¨H1 / add(XNX) / add(XMX) / cov(XMX) / non(N )

Questions: 1 Con(non(N ) < gr)? 3 Con(gr < cov(N ))? (the random model?)

Cichoń’s diagram and gr

X XX X X covX(XNX) / non(XMX) / cofX(MXX) / cofX(NXX) / ¡Ac O O O O

b¡A / d¡A O O

XX XX XX ¨ωH¨H1 / add(XNX) / add(XMX) / cov(XMX) / non(N )

Questions: 1 Con(non(N ) < gr)? 2 b ¬ gr? (the Laver model?) Cichoń’s diagram and gr

X XX X X covX(XNX) / non(XMX) / cofX(MXX) / cofX(NXX) / ¡Ac O O O O

b¡A / d¡A O O

XX XX XX ¨ωH¨H1 / add(XNX) / add(XMX) / cov(XMX) / non(N )

Questions: 1 Con(non(N ) < gr)? 2 b ¬ gr? (the Laver model?) 3 Con(gr < cov(N ))? (the random model?) 1 No ZFC ineq. between gr and r, u, g and a 2 cov(M) ¬ gr and Con(s < cov(M)) 3 Con(gr < d) 4 s ¬ gr

Van Douwen’s diagram and gr

What’s known: c G O `

u i O > O

a r d ^ O ? O _

b s g _ O ?

p O

ω1 2 cov(M) ¬ gr and Con(s < cov(M)) 3 Con(gr < d) 4 s ¬ gr

Van Douwen’s diagram and gr

What’s known: c 1 No ZFC ineq. between gr G¡AO _ and r, u, g and a

u i ¡AO ? O

a r d ¡A ] ¡AO ? O _

b¡A s g _ O @ ¡A

p¡A O

¨ωH¨H1 3 Con(gr < d) 4 s ¬ gr

Van Douwen’s diagram and gr

What’s known: c 1 No ZFC ineq. between gr G¡AO _ and r, u, g and a 2 cov(M) ¬ gr and u i ¡AO ? O Con(s < cov(M))

a r d ¡A ] ¡AO ? O _

b¡A s g _ ¡AO @ ¡A

p¡A O

¨ωH¨H1 4 s ¬ gr

Van Douwen’s diagram and gr

What’s known: c 1 No ZFC ineq. between gr H¡AO _ and r, u, g and a 2 cov(M) ¬ gr and u i£C ¡AO ? O Con(s < cov(M)) 3 Con(gr < d) a r d¡A ¡A ] ¡AO ? O _

b¡A s g _ ¡AO @ ¡A

p¡A O

¨ωH¨H1 Van Douwen’s diagram and gr

What’s known: c 1 No ZFC ineq. between gr H¡AO _ and r, u, g and a 2 cov(M) ¬ gr and u i£C ¡AO ? O Con(s < cov(M)) 3 Con(gr < d) a r d¡A 4 s ¬ gr ¡A ] ¡AO ? O _

b¡A s g _ ¡AO @ ¡A

p¡A O

¨ωH¨H1 Van Douwen’s diagram and gr

What’s known: c 1 No ZFC ineq. between gr H¡AO _ and r, u, g and a 2 cov(M) ¬ gr and u i£C ¡AO ? O Con(s < cov(M)) 3 Con(gr < d) a r d¡A 4 s ¬ gr ¡A ] ¡AO ? O _ Questions: 1 b ¬ gr? b¡A ¡As g¡A _ O @ 2 gr ¬ d?

p¡A O

¨ωH¨H1 Fact gr may be either regular (CH) or singular (in every model where cov(M) = c > cf(c)).

A word on the coﬁnality of gr

Theorem (Schachermayer ’82) cf(gr) > ω. A word on the coﬁnality of gr

Theorem (Schachermayer ’82) cf(gr) > ω.

Fact gr may be either regular (CH) or singular (in every model where cov(M) = c > cf(c)). The end

Thank you for the attention!