The weak Freeze Nation FN property and the Noetherian types of - - PowerPoint PPT Presentation

The weak Freeze Nation FN property and the Noetherian types of bases

Soukup, L (HAS) RIMS 2010 1 / 23

wFN

Definition (Heindorf and Shapiro)

A poset P = P, ≤ has the weak Freese-Nation (wFN) property iff there is a function f : P →

• P

ω such that if p, q ∈ P and p ≤ q then there is r ∈ f(p) ∩ f(q) with p ≤ r ≤ q. If |P| ≤ ω1, then P has the wfN property. f(pβ) = {pα : α ≤ β}

Definition

Let Q ⊆ P. Write Q ≤σ P if for each p ∈ P the set {q ∈ Q : q ≤ p} has a countable cofinal subset, and the set {q ∈ Q : q ≥ p} has a countable coinitial subset. If Q ⊂ P is f-closed, then Q ≤σ P witnessed by f(p) ∩ Q. If P has wFN, then {Q ∈

• P

ω1 : Q ≤σ P} contains a club. Fuchino-Koppelberg-Shelah: If {Q ∈

• P

ω1 : Q ≤σ P} contains club, then P has wFN.

Soukup, L (HAS) RIMS 2010 2 / 23

wFN

Definition (Heindorf and Shapiro)

A poset P = P, ≤ has the weak Freese-Nation (wFN) property iff there is a function f : P →

• P

ω such that if p, q ∈ P and p ≤ q then there is r ∈ f(p) ∩ f(q) with p ≤ r ≤ q. If |P| ≤ ω1, then P has the wfN property. f(pβ) = {pα : α ≤ β}

Definition

Let Q ⊆ P. Write Q ≤σ P if for each p ∈ P the set {q ∈ Q : q ≤ p} has a countable cofinal subset, and the set {q ∈ Q : q ≥ p} has a countable coinitial subset. If Q ⊂ P is f-closed, then Q ≤σ P witnessed by f(p) ∩ Q. If P has wFN, then {Q ∈

• P

ω1 : Q ≤σ P} contains a club. Fuchino-Koppelberg-Shelah: If {Q ∈

• P

ω1 : Q ≤σ P} contains club, then P has wFN.

Soukup, L (HAS) RIMS 2010 2 / 23

wFN

Definition (Heindorf and Shapiro)

A poset P = P, ≤ has the weak Freese-Nation (wFN) property iff there is a function f : P →

• P

ω such that if p, q ∈ P and p ≤ q then there is r ∈ f(p) ∩ f(q) with p ≤ r ≤ q. If |P| ≤ ω1, then P has the wfN property. f(pβ) = {pα : α ≤ β}

Definition

Let Q ⊆ P. Write Q ≤σ P if for each p ∈ P the set {q ∈ Q : q ≤ p} has a countable cofinal subset, and the set {q ∈ Q : q ≥ p} has a countable coinitial subset. If Q ⊂ P is f-closed, then Q ≤σ P witnessed by f(p) ∩ Q. If P has wFN, then {Q ∈

• P

ω1 : Q ≤σ P} contains a club. Fuchino-Koppelberg-Shelah: If {Q ∈

• P

ω1 : Q ≤σ P} contains club, then P has wFN.

Soukup, L (HAS) RIMS 2010 2 / 23

wFN

Definition (Heindorf and Shapiro)

A poset P = P, ≤ has the weak Freese-Nation (wFN) property iff there is a function f : P →

• P

ω such that if p, q ∈ P and p ≤ q then there is r ∈ f(p) ∩ f(q) with p ≤ r ≤ q. If |P| ≤ ω1, then P has the wfN property. f(pβ) = {pα : α ≤ β}

Definition

Let Q ⊆ P. Write Q ≤σ P if for each p ∈ P the set {q ∈ Q : q ≤ p} has a countable cofinal subset, and the set {q ∈ Q : q ≥ p} has a countable coinitial subset. If Q ⊂ P is f-closed, then Q ≤σ P witnessed by f(p) ∩ Q. If P has wFN, then {Q ∈

• P

ω1 : Q ≤σ P} contains a club. Fuchino-Koppelberg-Shelah: If {Q ∈

• P

ω1 : Q ≤σ P} contains club, then P has wFN.

Soukup, L (HAS) RIMS 2010 2 / 23

wFN

Definition (Heindorf and Shapiro)

A poset P = P, ≤ has the weak Freese-Nation (wFN) property iff there is a function f : P →

• P

ω such that if p, q ∈ P and p ≤ q then there is r ∈ f(p) ∩ f(q) with p ≤ r ≤ q. If |P| ≤ ω1, then P has the wfN property. f(pβ) = {pα : α ≤ β}

Definition

Let Q ⊆ P. Write Q ≤σ P if for each p ∈ P the set {q ∈ Q : q ≤ p} has a countable cofinal subset, and the set {q ∈ Q : q ≥ p} has a countable coinitial subset. If Q ⊂ P is f-closed, then Q ≤σ P witnessed by f(p) ∩ Q. If P has wFN, then {Q ∈

• P

ω1 : Q ≤σ P} contains a club. Fuchino-Koppelberg-Shelah: If {Q ∈

• P

ω1 : Q ≤σ P} contains club, then P has wFN.

Soukup, L (HAS) RIMS 2010 2 / 23

wFN

Definition (Heindorf and Shapiro)

A poset P = P, ≤ has the weak Freese-Nation (wFN) property iff there is a function f : P →

• P

ω such that if p, q ∈ P and p ≤ q then there is r ∈ f(p) ∩ f(q) with p ≤ r ≤ q. If |P| ≤ ω1, then P has the wfN property. f(pβ) = {pα : α ≤ β}

Definition

Let Q ⊆ P. Write Q ≤σ P if for each p ∈ P the set {q ∈ Q : q ≤ p} has a countable cofinal subset, and the set {q ∈ Q : q ≥ p} has a countable coinitial subset. If Q ⊂ P is f-closed, then Q ≤σ P witnessed by f(p) ∩ Q. If P has wFN, then {Q ∈

• P

ω1 : Q ≤σ P} contains a club. Fuchino-Koppelberg-Shelah: If {Q ∈

• P

ω1 : Q ≤σ P} contains club, then P has wFN.

Soukup, L (HAS) RIMS 2010 2 / 23

wFN

Definition (Heindorf and Shapiro)

A poset P = P, ≤ has the weak Freese-Nation (wFN) property iff there is a function f : P →

• P

ω such that if p, q ∈ P and p ≤ q then there is r ∈ f(p) ∩ f(q) with p ≤ r ≤ q. If |P| ≤ ω1, then P has the wfN property. f(pβ) = {pα : α ≤ β}

Definition

Let Q ⊆ P. Write Q ≤σ P if for each p ∈ P the set {q ∈ Q : q ≤ p} has a countable cofinal subset, and the set {q ∈ Q : q ≥ p} has a countable coinitial subset. If Q ⊂ P is f-closed, then Q ≤σ P witnessed by f(p) ∩ Q. If P has wFN, then {Q ∈

• P

ω1 : Q ≤σ P} contains a club. Fuchino-Koppelberg-Shelah: If {Q ∈

• P

ω1 : Q ≤σ P} contains club, then P has wFN.

Soukup, L (HAS) RIMS 2010 2 / 23

wFN

Definition (Heindorf and Shapiro)

A poset P = P, ≤ has the weak Freese-Nation (wFN) property iff there is a function f : P →

• P

ω such that if p, q ∈ P and p ≤ q then there is r ∈ f(p) ∩ f(q) with p ≤ r ≤ q. If |P| ≤ ω1, then P has the wfN property. f(pβ) = {pα : α ≤ β}

Definition

Let Q ⊆ P. Write Q ≤σ P if for each p ∈ P the set {q ∈ Q : q ≤ p} has a countable cofinal subset, and the set {q ∈ Q : q ≥ p} has a countable coinitial subset. If Q ⊂ P is f-closed, then Q ≤σ P witnessed by f(p) ∩ Q. If P has wFN, then {Q ∈

• P

ω1 : Q ≤σ P} contains a club. Fuchino-Koppelberg-Shelah: If {Q ∈

• P

ω1 : Q ≤σ P} contains club, then P has wFN.

Soukup, L (HAS) RIMS 2010 2 / 23

wFN

P has wFN iff ∃f : P →

• P

ω s.t. if p, q ∈ P and p ≤ q then ∃r ∈ f(p) ∩ f(q) with p ≤ r ≤ q. Q ≤σ P iff cf{q ∈ Q : q ≤ p} = ci{q ∈ Q : q ≥ p} = ω. P has wFN iff {Q ∈

• P

ω1 : Q ≤σ P} contains a club

Can we weaken the assumption “{Q ∈

• P

ω1 : Q ≤σ P} contains a club”? Let P be a poset, and χ be a large enough regular cardinal . (Fuchino-Koppelberg-Shelah) TFAE: (1) P has the wFN property (2) P ∩ M ≤σ P for each M ≺ H(χ) with |M| = ω1 and P ∈ M.

Soukup, L (HAS) RIMS 2010 3 / 23

wFN

P has wFN iff ∃f : P →

• P

ω s.t. if p, q ∈ P and p ≤ q then ∃r ∈ f(p) ∩ f(q) with p ≤ r ≤ q. Q ≤σ P iff cf{q ∈ Q : q ≤ p} = ci{q ∈ Q : q ≥ p} = ω. P has wFN iff {Q ∈

• P

ω1 : Q ≤σ P} contains a club

Can we weaken the assumption “{Q ∈

• P

ω1 : Q ≤σ P} contains a club”? Let P be a poset, and χ be a large enough regular cardinal . (Fuchino-Koppelberg-Shelah) TFAE: (1) P has the wFN property (2) P ∩ M ≤σ P for each M ≺ H(χ) with |M| = ω1 and P ∈ M.

Soukup, L (HAS) RIMS 2010 3 / 23

wFN

P has wFN iff ∃f : P →

• P

ω s.t. if p, q ∈ P and p ≤ q then ∃r ∈ f(p) ∩ f(q) with p ≤ r ≤ q. Q ≤σ P iff cf{q ∈ Q : q ≤ p} = ci{q ∈ Q : q ≥ p} = ω. P has wFN iff {Q ∈

• P

ω1 : Q ≤σ P} contains a club

Can we weaken the assumption “{Q ∈

• P

ω1 : Q ≤σ P} contains a club”? Let P be a poset, and χ be a large enough regular cardinal . (Fuchino-Koppelberg-Shelah) TFAE: (1) P has the wFN property (2) P ∩ M ≤σ P for each M ≺ H(χ) with |M| = ω1 and P ∈ M.

Soukup, L (HAS) RIMS 2010 3 / 23

wFN

P has wFN iff ∃f : P →

• P

ω s.t. if p, q ∈ P and p ≤ q then ∃r ∈ f(p) ∩ f(q) with p ≤ r ≤ q. Q ≤σ P iff cf{q ∈ Q : q ≤ p} = ci{q ∈ Q : q ≥ p} = ω. P has wFN iff {Q ∈

• P

ω1 : Q ≤σ P} contains a club

Can we weaken the assumption “{Q ∈

• P

ω1 : Q ≤σ P} contains a club”? Let P be a poset, and χ be a large enough regular cardinal . (Fuchino-Koppelberg-Shelah) TFAE: (1) P has the wFN property (2) P ∩ M ≤σ P for each M ≺ H(χ) with |M| = ω1 and P ∈ M.

Soukup, L (HAS) RIMS 2010 3 / 23

wFN

P has wFN iff ∃f : P →

• P

ω s.t. if p, q ∈ P and p ≤ q then ∃r ∈ f(p) ∩ f(q) with p ≤ r ≤ q. Q ≤σ P iff cf{q ∈ Q : q ≤ p} = ci{q ∈ Q : q ≥ p} = ω. P has wFN iff {Q ∈

• P

ω1 : Q ≤σ P} contains a club

Can we weaken the assumption “{Q ∈

• P

ω1 : Q ≤σ P} contains a club”? Let P be a poset, and χ be a large enough regular cardinal . (Fuchino-Koppelberg-Shelah) TFAE: (1) P has the wFN property (2) P ∩ M ≤σ P for each M ≺ H(χ) with |M| = ω1 and P ∈ M.

Soukup, L (HAS) RIMS 2010 3 / 23

wFN

P has wFN iff ∃f : P →

• P

ω s.t. if p, q ∈ P and p ≤ q then ∃r ∈ f(p) ∩ f(q) with p ≤ r ≤ q. Q ≤σ P iff cf{q ∈ Q : q ≤ p} = ci{q ∈ Q : q ≥ p} = ω. P has wFN iff {Q ∈

• P

ω1 : Q ≤σ P} contains a club

Can we weaken the assumption “{Q ∈

• P

ω1 : Q ≤σ P} contains a club”? Let P be a poset, and χ be a large enough regular cardinal . (Fuchino-Koppelberg-Shelah) TFAE: (1) P has the wFN property (2) P ∩ M ≤σ P for each M ≺ H(χ) with |M| = ω1 and P ∈ M.

Soukup, L (HAS) RIMS 2010 3 / 23

A possible genaralization

Let χ = cf(χ) be a large enough. TFAE for a poset P: (1) P has the wFN property (2) P ∩ M ≤σ P for each M ≺ H(χ) with |M| = ω1 and P ∈ M.

Hard to verify P ∩ M ≤σ P for an arbitrary submodel M. Consider nicer submodels

Definition

M ≺ H(χ) is Vω1-like iff M is the union of an ω1 chain of countable elementary submodels of H(χ). (CH) If P =

• κ

ω, ⊆

• , then P ∩ M ≤σ P for each Vω1-like M.

Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤σ P for each Vω1-like M ≺ H(χ) with P ∈ M.

Soukup, L (HAS) RIMS 2010 4 / 23

A possible genaralization

Let χ = cf(χ) be a large enough. TFAE for a poset P: (1) P has the wFN property (2) P ∩ M ≤σ P for each M ≺ H(χ) with |M| = ω1 and P ∈ M.

Hard to verify P ∩ M ≤σ P for an arbitrary submodel M. Consider nicer submodels

Definition

M ≺ H(χ) is Vω1-like iff M is the union of an ω1 chain of countable elementary submodels of H(χ). (CH) If P =

• κ

ω, ⊆

• , then P ∩ M ≤σ P for each Vω1-like M.

Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤σ P for each Vω1-like M ≺ H(χ) with P ∈ M.

Soukup, L (HAS) RIMS 2010 4 / 23

A possible genaralization

Let χ = cf(χ) be a large enough. TFAE for a poset P: (1) P has the wFN property (2) P ∩ M ≤σ P for each M ≺ H(χ) with |M| = ω1 and P ∈ M.

Hard to verify P ∩ M ≤σ P for an arbitrary submodel M. Consider nicer submodels

Definition

M ≺ H(χ) is Vω1-like iff M is the union of an ω1 chain of countable elementary submodels of H(χ). (CH) If P =

• κ

ω, ⊆

• , then P ∩ M ≤σ P for each Vω1-like M.

Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤σ P for each Vω1-like M ≺ H(χ) with P ∈ M.

Soukup, L (HAS) RIMS 2010 4 / 23

A possible genaralization

Let χ = cf(χ) be a large enough. TFAE for a poset P: (1) P has the wFN property (2) P ∩ M ≤σ P for each M ≺ H(χ) with |M| = ω1 and P ∈ M.

Hard to verify P ∩ M ≤σ P for an arbitrary submodel M. Consider nicer submodels

Definition

M ≺ H(χ) is Vω1-like iff M is the union of an ω1 chain of countable elementary submodels of H(χ). (CH) If P =

• κ

ω, ⊆

• , then P ∩ M ≤σ P for each Vω1-like M.

Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤σ P for each Vω1-like M ≺ H(χ) with P ∈ M.

Soukup, L (HAS) RIMS 2010 4 / 23

A possible genaralization

Let χ = cf(χ) be a large enough. TFAE for a poset P: (1) P has the wFN property (2) P ∩ M ≤σ P for each M ≺ H(χ) with |M| = ω1 and P ∈ M.

Hard to verify P ∩ M ≤σ P for an arbitrary submodel M. Consider nicer submodels

Definition

M ≺ H(χ) is Vω1-like iff M is the union of an ω1 chain of countable elementary submodels of H(χ). (CH) If P =

• κ

ω, ⊆

• , then P ∩ M ≤σ P for each Vω1-like M.

Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤σ P for each Vω1-like M ≺ H(χ) with P ∈ M.

Soukup, L (HAS) RIMS 2010 4 / 23

A possible genaralization

Let χ = cf(χ) be a large enough. TFAE for a poset P: (1) P has the wFN property (2) P ∩ M ≤σ P for each M ≺ H(χ) with |M| = ω1 and P ∈ M.

Hard to verify P ∩ M ≤σ P for an arbitrary submodel M. Consider nicer submodels

Definition

M ≺ H(χ) is Vω1-like iff M is the union of an ω1 chain of countable elementary submodels of H(χ). (CH) If P =

• κ

ω, ⊆

• , then P ∩ M ≤σ P for each Vω1-like M.

Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤σ P for each Vω1-like M ≺ H(χ) with P ∈ M.

Soukup, L (HAS) RIMS 2010 4 / 23

A possible genaralization

Let χ = cf(χ) be a large enough. TFAE for a poset P: (1) P has the wFN property (2) P ∩ M ≤σ P for each M ≺ H(χ) with |M| = ω1 and P ∈ M.

Hard to verify P ∩ M ≤σ P for an arbitrary submodel M. Consider nicer submodels

Definition

M ≺ H(χ) is Vω1-like iff M is the union of an ω1 chain of countable elementary submodels of H(χ). (CH) If P =

• κ

ω, ⊆

• , then P ∩ M ≤σ P for each Vω1-like M.

Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤σ P for each Vω1-like M ≺ H(χ) with P ∈ M.

Soukup, L (HAS) RIMS 2010 4 / 23

A possible genaralization

Let χ = cf(χ) be a large enough. TFAE for a poset P: (1) P has the wFN property (2) P ∩ M ≤σ P for each M ≺ H(χ) with |M| = ω1 and P ∈ M.

Hard to verify P ∩ M ≤σ P for an arbitrary submodel M. Consider nicer submodels

Definition

M ≺ H(χ) is Vω1-like iff M is the union of an ω1 chain of countable elementary submodels of H(χ). (CH) If P =

• κ

ω, ⊆

• , then P ∩ M ≤σ P for each Vω1-like M.

Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤σ P for each Vω1-like M ≺ H(χ) with P ∈ M.

Soukup, L (HAS) RIMS 2010 4 / 23

Consistent counterexample

TFAE: (1) P has the wFN property, (2) P ∩ M ≤σ P for each M ≺ H(χ) with M| = ω1 and P ∈ M. Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤σ P for each Vω1-like M ≺ H(χ) with P ∈ M. If CH holds then for each cardinal κ, the poset

• κ

ω ⊆

• satisfies (w2).

Theorem

If GCH holds and (ℵω+1, ℵω) ։ (ℵ1, ℵ0) then the poset

• ℵω

ω, ⊆

• does not have the wFN property.

Levinsky-Magidor-Shelah: The assumption is consistent modulo a huge cardinal.

Soukup, L (HAS) RIMS 2010 5 / 23

Consistent counterexample

TFAE: (1) P has the wFN property, (2) P ∩ M ≤σ P for each M ≺ H(χ) with M| = ω1 and P ∈ M. Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤σ P for each Vω1-like M ≺ H(χ) with P ∈ M. If CH holds then for each cardinal κ, the poset

• κ

ω ⊆

• satisfies (w2).

Theorem

If GCH holds and (ℵω+1, ℵω) ։ (ℵ1, ℵ0) then the poset

• ℵω

ω, ⊆

• does not have the wFN property.

Levinsky-Magidor-Shelah: The assumption is consistent modulo a huge cardinal.

Soukup, L (HAS) RIMS 2010 5 / 23

Consistent counterexample

TFAE: (1) P has the wFN property, (2) P ∩ M ≤σ P for each M ≺ H(χ) with M| = ω1 and P ∈ M. Are the following equivalent? (w1) P has the wFN property, (w2) P ∩ M ≤σ P for each Vω1-like M ≺ H(χ) with P ∈ M. If CH holds then for each cardinal κ, the poset

• κ

ω ⊆

• satisfies (w2).

Theorem

If GCH holds and (ℵω+1, ℵω) ։ (ℵ1, ℵ0) then the poset

• ℵω

ω, ⊆

• does not have the wFN property.

Levinsky-Magidor-Shelah: The assumption is consistent modulo a huge cardinal.

Soukup, L (HAS) RIMS 2010 5 / 23

A positive theorem

Let χ = cf(χ) be a large enough regular cardinal . M ≺ H(χ) is Vω1-like iff M is the union of an ω1 chain of countable elemen- tary submodels of H(χ).

Theorem (Fuchino, Soukup)

Assume that λ is a cardinal with (i) cf([µ]ω, ⊆) = µ if ω1 < µ < λ and cf(µ) ≥ ω1 (ii) ∗∗∗

µ

holds for each singular µ < λ with cofinality ω Then for each poset P of cardinality ≤ λ the following are equivalent: (w1) P has the wFN property, (w2) P ∩ M ≤σ P for each Vω1-like M ≺ H(χ) with P ∈ M.

Soukup, L (HAS) RIMS 2010 6 / 23

A positive theorem

Let χ = cf(χ) be a large enough regular cardinal . M ≺ H(χ) is Vω1-like iff M is the union of an ω1 chain of countable elemen- tary submodels of H(χ).

Theorem (Fuchino, Soukup)

Assume that λ is a cardinal with (i) cf([µ]ω, ⊆) = µ if ω1 < µ < λ and cf(µ) ≥ ω1 (ii) ∗∗∗

µ

holds for each singular µ < λ with cofinality ω Then for each poset P of cardinality ≤ λ the following are equivalent: (w1) P has the wFN property, (w2) P ∩ M ≤σ P for each Vω1-like M ≺ H(χ) with P ∈ M.

Soukup, L (HAS) RIMS 2010 6 / 23

A positive theorem

Let χ = cf(χ) be a large enough regular cardinal . M ≺ H(χ) is Vω1-like iff M is the union of an ω1 chain of countable elemen- tary submodels of H(χ).

Theorem (Fuchino, Soukup)

Assume that λ is a cardinal with (i) cf([µ]ω, ⊆) = µ if ω1 < µ < λ and cf(µ) ≥ ω1 (ii) ∗∗∗

µ

holds for each singular µ < λ with cofinality ω Then for each poset P of cardinality ≤ λ the following are equivalent: (w1) P has the wFN property, (w2) P ∩ M ≤σ P for each Vω1-like M ≺ H(χ) with P ∈ M.

Soukup, L (HAS) RIMS 2010 6 / 23

The wFN property of P(ω)

Theorem (Fuchino, Soukup)

Assume that λ is a cardinal with (i) cf([µ]ω, ⊆) = µ if ω1 < µ < λ and cf(µ) ≥ ω1 (ii) ∗∗∗

µ

holds for each singular µ < λ with cofinality ω Then V Fn(λ,2;ω) | = P(ω) has the wFN property.

Theorem (Fuchino, Geschke, Shelah, Soukup)

If GCH holds and (ℵω+1, ℵω) ։ (ℵ1, ℵ0) then V D∗Fn(ℵω,2;ω) | = P(ω) does not have the wFN property.

Soukup, L (HAS) RIMS 2010 7 / 23

The wFN property of P(ω)

Theorem (Fuchino, Soukup)

Assume that λ is a cardinal with (i) cf([µ]ω, ⊆) = µ if ω1 < µ < λ and cf(µ) ≥ ω1 (ii) ∗∗∗

µ

holds for each singular µ < λ with cofinality ω Then V Fn(λ,2;ω) | = P(ω) has the wFN property.

Theorem (Fuchino, Geschke, Shelah, Soukup)

If GCH holds and (ℵω+1, ℵω) ։ (ℵ1, ℵ0) then V D∗Fn(ℵω,2;ω) | = P(ω) does not have the wFN property.

Soukup, L (HAS) RIMS 2010 7 / 23

The wFN property of P(ω)

Theorem (Fuchino, Soukup)

Assume that λ is a cardinal with (i) cf([µ]ω, ⊆) = µ if ω1 < µ < λ and cf(µ) ≥ ω1 (ii) ∗∗∗

µ

holds for each singular µ < λ with cofinality ω Then V Fn(λ,2;ω) | = P(ω) has the wFN property.

Theorem (Fuchino, Geschke, Shelah, Soukup)

If GCH holds and (ℵω+1, ℵω) ։ (ℵ1, ℵ0) then V D∗Fn(ℵω,2;ω) | = P(ω) does not have the wFN property.

Soukup, L (HAS) RIMS 2010 7 / 23

The wFN property of P(ω)

Theorem (Fuchino, Soukup)

Assume that λ is a cardinal with (i) cf([µ]ω, ⊆) = µ if ω1 < µ < λ and cf(µ) ≥ ω1 (ii) ∗∗∗

µ

holds for each singular µ < λ with cofinality ω Then V Fn(λ,2;ω) | = P(ω) has the wFN property.

Theorem (Fuchino, Geschke, Shelah, Soukup)

If GCH holds and (ℵω+1, ℵω) ։ (ℵ1, ℵ0) then V D∗Fn(ℵω,2;ω) | = P(ω) does not have the wFN property.

Soukup, L (HAS) RIMS 2010 7 / 23

Noetherian type

Basic notions

Peregudov, Malykhin, Shapirovskii

Definition

Let X be a topological space. The Noetherian type of X, Nt(X), is the least cardinal κ such that X has a base B such that |{B′ ∈ B : B ⊂ B′}| < κ for each B ∈ B. Noetherian type ≈ cellularity 2κ has countable Noetherian type and cellularity For a topological space X, let X(δ) denote the space obtained by declaring the Gδ-sets to be open.

Theorem (Spadaro)

(GCH) Let X be a compact space such that Nt(X) has uncountable

• cofinality. Then Nt(X(δ)) ≤ 2Nt(X).

Soukup, L (HAS) RIMS 2010 8 / 23

Noetherian type

Basic notions

Peregudov, Malykhin, Shapirovskii

Definition

Let X be a topological space. The Noetherian type of X, Nt(X), is the least cardinal κ such that X has a base B such that |{B′ ∈ B : B ⊂ B′}| < κ for each B ∈ B. Noetherian type ≈ cellularity 2κ has countable Noetherian type and cellularity For a topological space X, let X(δ) denote the space obtained by declaring the Gδ-sets to be open.

Theorem (Spadaro)

(GCH) Let X be a compact space such that Nt(X) has uncountable

• cofinality. Then Nt(X(δ)) ≤ 2Nt(X).

Soukup, L (HAS) RIMS 2010 8 / 23

Noetherian type

Basic notions

Peregudov, Malykhin, Shapirovskii

Definition

Let X be a topological space. The Noetherian type of X, Nt(X), is the least cardinal κ such that X has a base B such that |{B′ ∈ B : B ⊂ B′}| < κ for each B ∈ B. Noetherian type ≈ cellularity 2κ has countable Noetherian type and cellularity For a topological space X, let X(δ) denote the space obtained by declaring the Gδ-sets to be open.

Theorem (Spadaro)

(GCH) Let X be a compact space such that Nt(X) has uncountable

• cofinality. Then Nt(X(δ)) ≤ 2Nt(X).

Soukup, L (HAS) RIMS 2010 8 / 23

Noetherian type

Basic notions

Peregudov, Malykhin, Shapirovskii

Definition

Let X be a topological space. The Noetherian type of X, Nt(X), is the least cardinal κ such that X has a base B such that |{B′ ∈ B : B ⊂ B′}| < κ for each B ∈ B. Noetherian type ≈ cellularity 2κ has countable Noetherian type and cellularity For a topological space X, let X(δ) denote the space obtained by declaring the Gδ-sets to be open.

Theorem (Spadaro)

(GCH) Let X be a compact space such that Nt(X) has uncountable

• cofinality. Then Nt(X(δ)) ≤ 2Nt(X).

Soukup, L (HAS) RIMS 2010 8 / 23

Noetherian type

Basic notions

Peregudov, Malykhin, Shapirovskii

Definition

Let X be a topological space. The Noetherian type of X, Nt(X), is the least cardinal κ such that X has a base B such that |{B′ ∈ B : B ⊂ B′}| < κ for each B ∈ B. Noetherian type ≈ cellularity 2κ has countable Noetherian type and cellularity For a topological space X, let X(δ) denote the space obtained by declaring the Gδ-sets to be open.

Theorem (Spadaro)

(GCH) Let X be a compact space such that Nt(X) has uncountable

• cofinality. Then Nt(X(δ)) ≤ 2Nt(X).

Soukup, L (HAS) RIMS 2010 8 / 23

Noetherian type

Basic notions

Peregudov, Malykhin, Shapirovskii

Definition

Let X be a topological space. The Noetherian type of X, Nt(X), is the least cardinal κ such that X has a base B such that |{B′ ∈ B : B ⊂ B′}| < κ for each B ∈ B. Noetherian type ≈ cellularity 2κ has countable Noetherian type and cellularity For a topological space X, let X(δ) denote the space obtained by declaring the Gδ-sets to be open.

Theorem (Spadaro)

(GCH) Let X be a compact space such that Nt(X) has uncountable

• cofinality. Then Nt(X(δ)) ≤ 2Nt(X).

Soukup, L (HAS) RIMS 2010 8 / 23

Noetherian type

Basic notions

Peregudov, Malykhin, Shapirovskii

Definition

Let X be a topological space. The Noetherian type of X, Nt(X), is the least cardinal κ such that X has a base B such that |{B′ ∈ B : B ⊂ B′}| < κ for each B ∈ B. Noetherian type ≈ cellularity 2κ has countable Noetherian type and cellularity For a topological space X, let X(δ) denote the space obtained by declaring the Gδ-sets to be open.

Theorem (Spadaro)

(GCH) Let X be a compact space such that Nt(X) has uncountable

• cofinality. Then Nt(X(δ)) ≤ 2Nt(X).

Soukup, L (HAS) RIMS 2010 8 / 23

Question

Def: Nt(X) ≤ κ iff |{B′ ∈ B : B ⊂ B′}| < κ for each B ∈ B Def: X(δ) denotes the Gδ-topology Spadaro: If GCH holds, X is compact, cf(Nt(X)) > ω then Nt(X(δ)) ≤ 2Nt(X)

Spadaro’s question

What happens if Nt(X) has countable cofinality? Can we drop GCH? Nt(2κ) = ω. What about Nt(2κ

(δ))?

Theorem (Milovich)

Nt(2κ

(δ)) = ω1 for κ < ℵω. If ℵω + (ℵω)ω = ℵω+1 then Nt(2κ (δ)) = ω1

Question: Nt(2ℵω

(δ)) ≤ 2ω in ZFC?

Soukup, L (HAS) RIMS 2010 9 / 23

Question

Def: Nt(X) ≤ κ iff |{B′ ∈ B : B ⊂ B′}| < κ for each B ∈ B Def: X(δ) denotes the Gδ-topology Spadaro: If GCH holds, X is compact, cf(Nt(X)) > ω then Nt(X(δ)) ≤ 2Nt(X)

Spadaro’s question

What happens if Nt(X) has countable cofinality? Can we drop GCH? Nt(2κ) = ω. What about Nt(2κ

(δ))?

Theorem (Milovich)

Nt(2κ

(δ)) = ω1 for κ < ℵω. If ℵω + (ℵω)ω = ℵω+1 then Nt(2κ (δ)) = ω1

Question: Nt(2ℵω

(δ)) ≤ 2ω in ZFC?

Soukup, L (HAS) RIMS 2010 9 / 23

Question

Def: Nt(X) ≤ κ iff |{B′ ∈ B : B ⊂ B′}| < κ for each B ∈ B Def: X(δ) denotes the Gδ-topology Spadaro: If GCH holds, X is compact, cf(Nt(X)) > ω then Nt(X(δ)) ≤ 2Nt(X)

Spadaro’s question

What happens if Nt(X) has countable cofinality? Can we drop GCH? Nt(2κ) = ω. What about Nt(2κ

(δ))?

Theorem (Milovich)

Nt(2κ

(δ)) = ω1 for κ < ℵω. If ℵω + (ℵω)ω = ℵω+1 then Nt(2κ (δ)) = ω1

Question: Nt(2ℵω

(δ)) ≤ 2ω in ZFC?

Soukup, L (HAS) RIMS 2010 9 / 23

Question

Def: Nt(X) ≤ κ iff |{B′ ∈ B : B ⊂ B′}| < κ for each B ∈ B Def: X(δ) denotes the Gδ-topology Spadaro: If GCH holds, X is compact, cf(Nt(X)) > ω then Nt(X(δ)) ≤ 2Nt(X)

Spadaro’s question

What happens if Nt(X) has countable cofinality? Can we drop GCH? Nt(2κ) = ω. What about Nt(2κ

(δ))?

Theorem (Milovich)

Nt(2κ

(δ)) = ω1 for κ < ℵω. If ℵω + (ℵω)ω = ℵω+1 then Nt(2κ (δ)) = ω1

Question: Nt(2ℵω

(δ)) ≤ 2ω in ZFC?

Soukup, L (HAS) RIMS 2010 9 / 23

Question

Def: Nt(X) ≤ κ iff |{B′ ∈ B : B ⊂ B′}| < κ for each B ∈ B Def: X(δ) denotes the Gδ-topology Spadaro: If GCH holds, X is compact, cf(Nt(X)) > ω then Nt(X(δ)) ≤ 2Nt(X)

Spadaro’s question

What happens if Nt(X) has countable cofinality? Can we drop GCH? Nt(2κ) = ω. What about Nt(2κ

(δ))?

Theorem (Milovich)

Nt(2κ

(δ)) = ω1 for κ < ℵω. If ℵω + (ℵω)ω = ℵω+1 then Nt(2κ (δ)) = ω1

Question: Nt(2ℵω

(δ)) ≤ 2ω in ZFC?

Soukup, L (HAS) RIMS 2010 9 / 23

Question

Def: Nt(X) ≤ κ iff |{B′ ∈ B : B ⊂ B′}| < κ for each B ∈ B Def: X(δ) denotes the Gδ-topology Spadaro: If GCH holds, X is compact, cf(Nt(X)) > ω then Nt(X(δ)) ≤ 2Nt(X)

Spadaro’s question

What happens if Nt(X) has countable cofinality? Can we drop GCH? Nt(2κ) = ω. What about Nt(2κ

(δ))?

Theorem (Milovich)

Nt(2κ

(δ)) = ω1 for κ < ℵω. If ℵω + (ℵω)ω = ℵω+1 then Nt(2κ (δ)) = ω1

Question: Nt(2ℵω

(δ)) ≤ 2ω in ZFC?

Soukup, L (HAS) RIMS 2010 9 / 23

Answer

Thm: Nt(2κ

(δ)) = ω1 for κ < ℵω. If ℵω + (ℵω)ω = ℵω+1 then Nt(2κ (δ)) = ω1

Question: Nt(2ℵω

(δ)) ≤ 2ω in ZFC?

Theorem

If GCH holds and (ℵω+1, ℵω) ։ (ℵ1, ℵ0) then Nt(2ℵω

(δ)) ≥ ω2.

Theorem

Assume that (ℵω)ω = ℵω+1 and (ℵω+1, ℵω) ։ (ℵ1, ℵ0). If D is cofinal in

• ℵω

ω, ⊆

• , then there is A ∈
• ℵω

ω such that |D ∩ P(A)| > ω. What is the relationship between these properties and the wFN property of

• ℵω

ω, ⊂

• ?

Soukup, L (HAS) RIMS 2010 10 / 23

Answer

Thm: Nt(2κ

(δ)) = ω1 for κ < ℵω. If ℵω + (ℵω)ω = ℵω+1 then Nt(2κ (δ)) = ω1

Question: Nt(2ℵω

(δ)) ≤ 2ω in ZFC?

Theorem

If GCH holds and (ℵω+1, ℵω) ։ (ℵ1, ℵ0) then Nt(2ℵω

(δ)) ≥ ω2.

Theorem

Assume that (ℵω)ω = ℵω+1 and (ℵω+1, ℵω) ։ (ℵ1, ℵ0). If D is cofinal in

• ℵω

ω, ⊆

• , then there is A ∈
• ℵω

ω such that |D ∩ P(A)| > ω. What is the relationship between these properties and the wFN property of

• ℵω

ω, ⊂

• ?

Soukup, L (HAS) RIMS 2010 10 / 23

Answer

Thm: Nt(2κ

(δ)) = ω1 for κ < ℵω. If ℵω + (ℵω)ω = ℵω+1 then Nt(2κ (δ)) = ω1

Question: Nt(2ℵω

(δ)) ≤ 2ω in ZFC?

Theorem

If GCH holds and (ℵω+1, ℵω) ։ (ℵ1, ℵ0) then Nt(2ℵω

(δ)) ≥ ω2.

Theorem

Assume that (ℵω)ω = ℵω+1 and (ℵω+1, ℵω) ։ (ℵ1, ℵ0). If D is cofinal in

• ℵω

ω, ⊆

• , then there is A ∈
• ℵω

ω such that |D ∩ P(A)| > ω. What is the relationship between these properties and the wFN property of

• ℵω

ω, ⊂

• ?

Soukup, L (HAS) RIMS 2010 10 / 23

Answer

Thm: Nt(2κ

(δ)) = ω1 for κ < ℵω. If ℵω + (ℵω)ω = ℵω+1 then Nt(2κ (δ)) = ω1

Question: Nt(2ℵω

(δ)) ≤ 2ω in ZFC?

Theorem

If GCH holds and (ℵω+1, ℵω) ։ (ℵ1, ℵ0) then Nt(2ℵω

(δ)) ≥ ω2.

Theorem

Assume that (ℵω)ω = ℵω+1 and (ℵω+1, ℵω) ։ (ℵ1, ℵ0). If D is cofinal in

• ℵω

ω, ⊆

• , then there is A ∈
• ℵω

ω such that |D ∩ P(A)| > ω. What is the relationship between these properties and the wFN property of

• ℵω

ω, ⊂

• ?

Soukup, L (HAS) RIMS 2010 10 / 23

Resolvability of monotonically normal spaces

Soukup, L (HAS) RIMS 2010 11 / 23

The beginnings

Basic notions

• E. Hewitt, 1943

Definition

A topological space X is κ-resolvable iff X contains κ disjoint dense subsets. resolvable iff it is 2-resolvable irresolvable it is not resolvable

Soukup, L (HAS) RIMS 2010 12 / 23

The beginnings

Basic notions

• E. Hewitt, 1943

Definition

A topological space X is κ-resolvable iff X contains κ disjoint dense subsets. resolvable iff it is 2-resolvable irresolvable it is not resolvable

Soukup, L (HAS) RIMS 2010 12 / 23

The beginnings

Basic notions

• E. Hewitt, 1943

Definition

A topological space X is κ-resolvable iff X contains κ disjoint dense subsets. resolvable iff it is 2-resolvable irresolvable it is not resolvable

Soukup, L (HAS) RIMS 2010 12 / 23

The beginnings

Basic notions

• E. Hewitt, 1943

Definition

A topological space X is κ-resolvable iff X contains κ disjoint dense subsets. resolvable iff it is 2-resolvable irresolvable it is not resolvable

Soukup, L (HAS) RIMS 2010 12 / 23

The beginnings

Basic notions

• E. Hewitt, 1943

Definition

A topological space X is κ-resolvable iff X contains κ disjoint dense subsets. resolvable iff it is 2-resolvable irresolvable it is not resolvable

Soukup, L (HAS) RIMS 2010 12 / 23

The beginnings.

Basic notions

X is κ-resolvable iff X contains κ disjoint dense subsets.

If D is dense and U is a non-empty open set, then U ∩ D = ∅. So if X is κ-resolvable then κ ≤ min{|U| : U ∈ τX \ {∅}} =∆(X). ∆(X) is the dispersion character of X.

Definition (Ceder, Pearson, 1967)

X is maximally resolvable iff it is ∆(X)-resolvable.

Soukup, L (HAS) RIMS 2010 13 / 23

The beginnings.

Basic notions

X is κ-resolvable iff X contains κ disjoint dense subsets.

If D is dense and U is a non-empty open set, then U ∩ D = ∅. So if X is κ-resolvable then κ ≤ min{|U| : U ∈ τX \ {∅}} =∆(X). ∆(X) is the dispersion character of X.

Definition (Ceder, Pearson, 1967)

X is maximally resolvable iff it is ∆(X)-resolvable.

Soukup, L (HAS) RIMS 2010 13 / 23

The beginnings.

Basic notions

X is κ-resolvable iff X contains κ disjoint dense subsets.

If D is dense and U is a non-empty open set, then U ∩ D = ∅. So if X is κ-resolvable then κ ≤ min{|U| : U ∈ τX \ {∅}} =∆(X). ∆(X) is the dispersion character of X.

Definition (Ceder, Pearson, 1967)

X is maximally resolvable iff it is ∆(X)-resolvable.

Soukup, L (HAS) RIMS 2010 13 / 23

The beginnings.

Basic notions

X is κ-resolvable iff X contains κ disjoint dense subsets.

If D is dense and U is a non-empty open set, then U ∩ D = ∅. So if X is κ-resolvable then κ ≤ min{|U| : U ∈ τX \ {∅}} =∆(X). ∆(X) is the dispersion character of X.

Definition (Ceder, Pearson, 1967)

X is maximally resolvable iff it is ∆(X)-resolvable.

Soukup, L (HAS) RIMS 2010 13 / 23

The beginnings.

Basic notions

X is κ-resolvable iff X contains κ disjoint dense subsets.

If D is dense and U is a non-empty open set, then U ∩ D = ∅. So if X is κ-resolvable then κ ≤ min{|U| : U ∈ τX \ {∅}} =∆(X). ∆(X) is the dispersion character of X.

Definition (Ceder, Pearson, 1967)

X is maximally resolvable iff it is ∆(X)-resolvable.

Soukup, L (HAS) RIMS 2010 13 / 23

The beginnings

X is κ-resolvable iff X contains κ disjoint dense subsets. X is maximally resolvable iff it is ∆(X)-resolvable. irresolvable ≡ not 2-resolvable

There are countable, regular, dense-in-itself irresolvable spaces. A topological space X is maximally resolvable provided it is (1) metric, or (2) ordered, or (3) compact, or (4) pseudo-radial (Pytkeev). What about MONOTONICALLY NORMAL spaces?

Soukup, L (HAS) RIMS 2010 14 / 23

The beginnings

X is κ-resolvable iff X contains κ disjoint dense subsets. X is maximally resolvable iff it is ∆(X)-resolvable. irresolvable ≡ not 2-resolvable

There are countable, regular, dense-in-itself irresolvable spaces. A topological space X is maximally resolvable provided it is (1) metric, or (2) ordered, or (3) compact, or (4) pseudo-radial (Pytkeev). What about MONOTONICALLY NORMAL spaces?

Soukup, L (HAS) RIMS 2010 14 / 23

The beginnings

X is κ-resolvable iff X contains κ disjoint dense subsets. X is maximally resolvable iff it is ∆(X)-resolvable. irresolvable ≡ not 2-resolvable

There are countable, regular, dense-in-itself irresolvable spaces. A topological space X is maximally resolvable provided it is (1) metric, or (2) ordered, or (3) compact, or (4) pseudo-radial (Pytkeev). What about MONOTONICALLY NORMAL spaces?

Soukup, L (HAS) RIMS 2010 14 / 23

The beginnings

X is κ-resolvable iff X contains κ disjoint dense subsets. X is maximally resolvable iff it is ∆(X)-resolvable. irresolvable ≡ not 2-resolvable

There are countable, regular, dense-in-itself irresolvable spaces. A topological space X is maximally resolvable provided it is (1) metric, or (2) ordered, or (3) compact, or (4) pseudo-radial (Pytkeev). What about MONOTONICALLY NORMAL spaces?

Soukup, L (HAS) RIMS 2010 14 / 23

The beginnings

X is κ-resolvable iff X contains κ disjoint dense subsets. X is maximally resolvable iff it is ∆(X)-resolvable. irresolvable ≡ not 2-resolvable

There are countable, regular, dense-in-itself irresolvable spaces. A topological space X is maximally resolvable provided it is (1) metric, or (2) ordered, or (3) compact, or (4) pseudo-radial (Pytkeev). What about MONOTONICALLY NORMAL spaces?

Soukup, L (HAS) RIMS 2010 14 / 23

The beginnings

X is κ-resolvable iff X contains κ disjoint dense subsets. X is maximally resolvable iff it is ∆(X)-resolvable. irresolvable ≡ not 2-resolvable

There are countable, regular, dense-in-itself irresolvable spaces. A topological space X is maximally resolvable provided it is (1) metric, or (2) ordered, or (3) compact, or (4) pseudo-radial (Pytkeev). What about MONOTONICALLY NORMAL spaces?

Soukup, L (HAS) RIMS 2010 14 / 23

The beginnings

X is κ-resolvable iff X contains κ disjoint dense subsets. X is maximally resolvable iff it is ∆(X)-resolvable. irresolvable ≡ not 2-resolvable

There are countable, regular, dense-in-itself irresolvable spaces. A topological space X is maximally resolvable provided it is (1) metric, or (2) ordered, or (3) compact, or (4) pseudo-radial (Pytkeev). What about MONOTONICALLY NORMAL spaces?

Soukup, L (HAS) RIMS 2010 14 / 23

The beginnings

X is κ-resolvable iff X contains κ disjoint dense subsets. X is maximally resolvable iff it is ∆(X)-resolvable. irresolvable ≡ not 2-resolvable

There are countable, regular, dense-in-itself irresolvable spaces. A topological space X is maximally resolvable provided it is (1) metric, or (2) ordered, or (3) compact, or (4) pseudo-radial (Pytkeev). What about MONOTONICALLY NORMAL spaces?

Soukup, L (HAS) RIMS 2010 14 / 23

Monotonically normal spaces

Given a space X, define the family of marked open sets as follows: M(X) =

• x, U ∈ X × τ(X) : x ∈ U
• X is monotonically normal (MN) if

(1) X is T1, (2) X admits a monotone normality operator i.e. there is function H : M(X) → τ(X) such that

(1) x ∈ H(x, U) ⊂ U for each x, U ∈ M(X), (2) if (x, U), (y, V) ∈ M(X), x / ∈ V and y / ∈ U then H(x, U) ∩ H(y, V) = ∅.

Metric and linearly ordered spaces are MN Are the monotonically normal spaces maximally resolvable?

Soukup, L (HAS) RIMS 2010 15 / 23

Monotonically normal spaces

Given a space X, define the family of marked open sets as follows: M(X) =

• x, U ∈ X × τ(X) : x ∈ U
• X is monotonically normal (MN) if

(1) X is T1, (2) X admits a monotone normality operator i.e. there is function H : M(X) → τ(X) such that

(1) x ∈ H(x, U) ⊂ U for each x, U ∈ M(X), (2) if (x, U), (y, V) ∈ M(X), x / ∈ V and y / ∈ U then H(x, U) ∩ H(y, V) = ∅.

Metric and linearly ordered spaces are MN Are the monotonically normal spaces maximally resolvable?

Soukup, L (HAS) RIMS 2010 15 / 23

Monotonically normal spaces

Given a space X, define the family of marked open sets as follows: M(X) =

• x, U ∈ X × τ(X) : x ∈ U
• X is monotonically normal (MN) if

(1) X is T1, (2) X admits a monotone normality operator i.e. there is function H : M(X) → τ(X) such that

(1) x ∈ H(x, U) ⊂ U for each x, U ∈ M(X), (2) if (x, U), (y, V) ∈ M(X), x / ∈ V and y / ∈ U then H(x, U) ∩ H(y, V) = ∅.

Metric and linearly ordered spaces are MN Are the monotonically normal spaces maximally resolvable?

Soukup, L (HAS) RIMS 2010 15 / 23

Monotonically normal spaces

Given a space X, define the family of marked open sets as follows: M(X) =

• x, U ∈ X × τ(X) : x ∈ U
• X is monotonically normal (MN) if

(1) X is T1, (2) X admits a monotone normality operator i.e. there is function H : M(X) → τ(X) such that

(1) x ∈ H(x, U) ⊂ U for each x, U ∈ M(X), (2) if (x, U), (y, V) ∈ M(X), x / ∈ V and y / ∈ U then H(x, U) ∩ H(y, V) = ∅.

Metric and linearly ordered spaces are MN Are the monotonically normal spaces maximally resolvable?

Soukup, L (HAS) RIMS 2010 15 / 23

Monotonically normal spaces

Given a space X, define the family of marked open sets as follows: M(X) =

• x, U ∈ X × τ(X) : x ∈ U
• X is monotonically normal (MN) if

(1) X is T1, (2) X admits a monotone normality operator i.e. there is function H : M(X) → τ(X) such that

(1) x ∈ H(x, U) ⊂ U for each x, U ∈ M(X), (2) if (x, U), (y, V) ∈ M(X), x / ∈ V and y / ∈ U then H(x, U) ∩ H(y, V) = ∅.

Metric and linearly ordered spaces are MN Are the monotonically normal spaces maximally resolvable?

Soukup, L (HAS) RIMS 2010 15 / 23

Monotonically normal spaces

Given a space X, define the family of marked open sets as follows: M(X) =

• x, U ∈ X × τ(X) : x ∈ U
• X is monotonically normal (MN) if

(1) X is T1, (2) X admits a monotone normality operator i.e. there is function H : M(X) → τ(X) such that

(1) x ∈ H(x, U) ⊂ U for each x, U ∈ M(X), (2) if (x, U), (y, V) ∈ M(X), x / ∈ V and y / ∈ U then H(x, U) ∩ H(y, V) = ∅.

Metric and linearly ordered spaces are MN Are the monotonically normal spaces maximally resolvable?

Soukup, L (HAS) RIMS 2010 15 / 23

Monotonically normal spaces

Given a space X, define the family of marked open sets as follows: M(X) =

• x, U ∈ X × τ(X) : x ∈ U
• X is monotonically normal (MN) if

(1) X is T1, (2) X admits a monotone normality operator i.e. there is function H : M(X) → τ(X) such that

(1) x ∈ H(x, U) ⊂ U for each x, U ∈ M(X), (2) if (x, U), (y, V) ∈ M(X), x / ∈ V and y / ∈ U then H(x, U) ∩ H(y, V) = ∅.

Metric and linearly ordered spaces are MN Are the monotonically normal spaces maximally resolvable?

Soukup, L (HAS) RIMS 2010 15 / 23

Monotonically normal spaces

Given a space X, define the family of marked open sets as follows: M(X) =

• x, U ∈ X × τ(X) : x ∈ U
• X is monotonically normal (MN) if

(1) X is T1, (2) X admits a monotone normality operator i.e. there is function H : M(X) → τ(X) such that

(1) x ∈ H(x, U) ⊂ U for each x, U ∈ M(X), (2) if (x, U), (y, V) ∈ M(X), x / ∈ V and y / ∈ U then H(x, U) ∩ H(y, V) = ∅.

Metric and linearly ordered spaces are MN Are the monotonically normal spaces maximally resolvable?

Soukup, L (HAS) RIMS 2010 15 / 23

Monotonically normal spaces

Given a space X, define the family of marked open sets as follows: M(X) =

• x, U ∈ X × τ(X) : x ∈ U
• X is monotonically normal (MN) if

(1) X is T1, (2) X admits a monotone normality operator i.e. there is function H : M(X) → τ(X) such that

(1) x ∈ H(x, U) ⊂ U for each x, U ∈ M(X), (2) if (x, U), (y, V) ∈ M(X), x / ∈ V and y / ∈ U then H(x, U) ∩ H(y, V) = ∅.

Metric and linearly ordered spaces are MN Are the monotonically normal spaces maximally resolvable?

Soukup, L (HAS) RIMS 2010 15 / 23

Monotonically normal spaces

Are the monotonically normal spaces maximally resolvable?

Theorem (Juhász,S, Szentmiklóssy)

A dense-in-itself monotonically normal space is ω-resolvable

Problem (Ceder, Pearson 1967)

Does ω-resolvable imply maximally resolvable?

Theorem (Juhász, S, Szentmiklóssy)

For each infinite κ there is a 0-dimensional T2 space X = κ, τ, s.t. ∆(X) = κ, X is ω-resolvable, but not ω1-resolvable.

Soukup, L (HAS) RIMS 2010 16 / 23

Monotonically normal spaces

Are the monotonically normal spaces maximally resolvable?

Theorem (Juhász,S, Szentmiklóssy)

A dense-in-itself monotonically normal space is ω-resolvable

Problem (Ceder, Pearson 1967)

Does ω-resolvable imply maximally resolvable?

Theorem (Juhász, S, Szentmiklóssy)

For each infinite κ there is a 0-dimensional T2 space X = κ, τ, s.t. ∆(X) = κ, X is ω-resolvable, but not ω1-resolvable.

Soukup, L (HAS) RIMS 2010 16 / 23

Monotonically normal spaces

Are the monotonically normal spaces maximally resolvable?

Theorem (Juhász,S, Szentmiklóssy)

A dense-in-itself monotonically normal space is ω-resolvable

Problem (Ceder, Pearson 1967)

Does ω-resolvable imply maximally resolvable?

Theorem (Juhász, S, Szentmiklóssy)

For each infinite κ there is a 0-dimensional T2 space X = κ, τ, s.t. ∆(X) = κ, X is ω-resolvable, but not ω1-resolvable.

Soukup, L (HAS) RIMS 2010 16 / 23

Monotonically normal spaces

Are the monotonically normal spaces maximally resolvable?

Theorem (Juhász,S, Szentmiklóssy)

A dense-in-itself monotonically normal space is ω-resolvable

Problem (Ceder, Pearson 1967)

Does ω-resolvable imply maximally resolvable?

Theorem (Juhász, S, Szentmiklóssy)

For each infinite κ there is a 0-dimensional T2 space X = κ, τ, s.t. ∆(X) = κ, X is ω-resolvable, but not ω1-resolvable.

Soukup, L (HAS) RIMS 2010 16 / 23

The basic construction

X dense-in-itself, monotonically normal

?

= ⇒ X maximally resolvable?

If T is a trees, t ∈ T, then succ(t) = the succerrors of t in T T is everywhere infinitely branching iff succT(t) is infinite a filtration on T is a map F s.t. dom(F) = T and F(t) is a filter

• n succT(t).

Define the topological space XF = T, τF: U ⊂ T is open iff for each t ∈ U the set succT(t) ∩ U ∈ F(t). XF is monotonically normal: H(t, V)= {u ∈ V : [t, u] ⊂ V} We conjectured: If T = ω<ω

1

and F(t) is an uniform ultrafilter on “ω1” then XF is not ω1-resolvable

Proposition

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable.

Soukup, L (HAS) RIMS 2010 17 / 23

The basic construction

X dense-in-itself, monotonically normal

?

= ⇒ X maximally resolvable?

If T is a trees, t ∈ T, then succ(t) = the succerrors of t in T T is everywhere infinitely branching iff succT(t) is infinite a filtration on T is a map F s.t. dom(F) = T and F(t) is a filter

• n succT(t).

Define the topological space XF = T, τF: U ⊂ T is open iff for each t ∈ U the set succT(t) ∩ U ∈ F(t). XF is monotonically normal: H(t, V)= {u ∈ V : [t, u] ⊂ V} We conjectured: If T = ω<ω

1

and F(t) is an uniform ultrafilter on “ω1” then XF is not ω1-resolvable

Proposition

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable.

Soukup, L (HAS) RIMS 2010 17 / 23

The basic construction

X dense-in-itself, monotonically normal

?

= ⇒ X maximally resolvable?

If T is a trees, t ∈ T, then succ(t) = the succerrors of t in T T is everywhere infinitely branching iff succT(t) is infinite a filtration on T is a map F s.t. dom(F) = T and F(t) is a filter

• n succT(t).

Define the topological space XF = T, τF: U ⊂ T is open iff for each t ∈ U the set succT(t) ∩ U ∈ F(t). XF is monotonically normal: H(t, V)= {u ∈ V : [t, u] ⊂ V} We conjectured: If T = ω<ω

1

and F(t) is an uniform ultrafilter on “ω1” then XF is not ω1-resolvable

Proposition

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable.

Soukup, L (HAS) RIMS 2010 17 / 23

The basic construction

X dense-in-itself, monotonically normal

?

= ⇒ X maximally resolvable?

If T is a trees, t ∈ T, then succ(t) = the succerrors of t in T T is everywhere infinitely branching iff succT(t) is infinite a filtration on T is a map F s.t. dom(F) = T and F(t) is a filter

• n succT(t).

Define the topological space XF = T, τF: U ⊂ T is open iff for each t ∈ U the set succT(t) ∩ U ∈ F(t). XF is monotonically normal: H(t, V)= {u ∈ V : [t, u] ⊂ V} We conjectured: If T = ω<ω

1

and F(t) is an uniform ultrafilter on “ω1” then XF is not ω1-resolvable

Proposition

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable.

Soukup, L (HAS) RIMS 2010 17 / 23

The basic construction

X dense-in-itself, monotonically normal

?

= ⇒ X maximally resolvable?

If T is a trees, t ∈ T, then succ(t) = the succerrors of t in T T is everywhere infinitely branching iff succT(t) is infinite a filtration on T is a map F s.t. dom(F) = T and F(t) is a filter

• n succT(t).

Define the topological space XF = T, τF: U ⊂ T is open iff for each t ∈ U the set succT(t) ∩ U ∈ F(t). XF is monotonically normal: H(t, V)= {u ∈ V : [t, u] ⊂ V} We conjectured: If T = ω<ω

1

and F(t) is an uniform ultrafilter on “ω1” then XF is not ω1-resolvable

Proposition

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable.

Soukup, L (HAS) RIMS 2010 17 / 23

The basic construction

X dense-in-itself, monotonically normal

?

= ⇒ X maximally resolvable?

If T is a trees, t ∈ T, then succ(t) = the succerrors of t in T T is everywhere infinitely branching iff succT(t) is infinite a filtration on T is a map F s.t. dom(F) = T and F(t) is a filter

• n succT(t).

Define the topological space XF = T, τF: U ⊂ T is open iff for each t ∈ U the set succT(t) ∩ U ∈ F(t). XF is monotonically normal: H(t, V)= {u ∈ V : [t, u] ⊂ V} We conjectured: If T = ω<ω

1

and F(t) is an uniform ultrafilter on “ω1” then XF is not ω1-resolvable

Proposition

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable.

Soukup, L (HAS) RIMS 2010 17 / 23

The basic construction

X dense-in-itself, monotonically normal

?

= ⇒ X maximally resolvable?

If T is a trees, t ∈ T, then succ(t) = the succerrors of t in T T is everywhere infinitely branching iff succT(t) is infinite a filtration on T is a map F s.t. dom(F) = T and F(t) is a filter

• n succT(t).

Define the topological space XF = T, τF: U ⊂ T is open iff for each t ∈ U the set succT(t) ∩ U ∈ F(t). XF is monotonically normal: H(t, V)= {u ∈ V : [t, u] ⊂ V} We conjectured: If T = ω<ω

1

and F(t) is an uniform ultrafilter on “ω1” then XF is not ω1-resolvable

Proposition

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable.

Soukup, L (HAS) RIMS 2010 17 / 23

The basic construction

X dense-in-itself, monotonically normal

?

= ⇒ X maximally resolvable?

If T is a trees, t ∈ T, then succ(t) = the succerrors of t in T T is everywhere infinitely branching iff succT(t) is infinite a filtration on T is a map F s.t. dom(F) = T and F(t) is a filter

• n succT(t).

Define the topological space XF = T, τF: U ⊂ T is open iff for each t ∈ U the set succT(t) ∩ U ∈ F(t). XF is monotonically normal: H(t, V)= {u ∈ V : [t, u] ⊂ V} We conjectured: If T = ω<ω

1

and F(t) is an uniform ultrafilter on “ω1” then XF is not ω1-resolvable

Proposition

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable.

Soukup, L (HAS) RIMS 2010 17 / 23

A consistent counterexample

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable.

An ultrafilter U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅.

Theorem (J-S-Sz)

Let F be an ultrafiltration on T. If every F(t) is ω-decomposable, then XF is ω1-resolvable.

Theorem (J-S-Sz)

If κ is a measurable cardinal, T = κ<ω, F(t) is a measure on succT(t) ≈ κ, then XF is ω1-irresolvable.

Soukup, L (HAS) RIMS 2010 18 / 23

A consistent counterexample

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable.

An ultrafilter U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅.

Theorem (J-S-Sz)

Let F be an ultrafiltration on T. If every F(t) is ω-decomposable, then XF is ω1-resolvable.

Theorem (J-S-Sz)

If κ is a measurable cardinal, T = κ<ω, F(t) is a measure on succT(t) ≈ κ, then XF is ω1-irresolvable.

Soukup, L (HAS) RIMS 2010 18 / 23

A consistent counterexample

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable.

An ultrafilter U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅.

Theorem (J-S-Sz)

Let F be an ultrafiltration on T. If every F(t) is ω-decomposable, then XF is ω1-resolvable.

Theorem (J-S-Sz)

If κ is a measurable cardinal, T = κ<ω, F(t) is a measure on succT(t) ≈ κ, then XF is ω1-irresolvable.

Soukup, L (HAS) RIMS 2010 18 / 23

A consistent counterexample

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable.

An ultrafilter U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅.

Theorem (J-S-Sz)

Let F be an ultrafiltration on T. If every F(t) is ω-decomposable, then XF is ω1-resolvable.

Theorem (J-S-Sz)

If κ is a measurable cardinal, T = κ<ω, F(t) is a measure on succT(t) ≈ κ, then XF is ω1-irresolvable.

Soukup, L (HAS) RIMS 2010 18 / 23

Smaller counterexamples

An ultrafilter U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅.

An ultrafilter U is λ-descendingly complete iff {Xξ : ξ < λ} ∈ U for each decreasing sequence {Xξ : ξ < λ} ⊂ U . Theorems: (J-S-Sz) If λ = cf(λ), F is an ultrafiltration on T, F(t) is λ-descendingly complete for all t ∈ T, then XF is λ+-irresolvable. (Magidor): It is consistent from a supercompact that there is an ω1-descendingly complete uniform ultrafilter on ℵω Con (∃supercompact) implies Con( there is a monotonically normal space X with |X| = ∆(X) = ℵω that is no ω2-resolvable).

Soukup, L (HAS) RIMS 2010 19 / 23

Smaller counterexamples

An ultrafilter U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅.

An ultrafilter U is λ-descendingly complete iff {Xξ : ξ < λ} ∈ U for each decreasing sequence {Xξ : ξ < λ} ⊂ U . Theorems: (J-S-Sz) If λ = cf(λ), F is an ultrafiltration on T, F(t) is λ-descendingly complete for all t ∈ T, then XF is λ+-irresolvable. (Magidor): It is consistent from a supercompact that there is an ω1-descendingly complete uniform ultrafilter on ℵω Con (∃supercompact) implies Con( there is a monotonically normal space X with |X| = ∆(X) = ℵω that is no ω2-resolvable).

Soukup, L (HAS) RIMS 2010 19 / 23

Smaller counterexamples

An ultrafilter U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅.

An ultrafilter U is λ-descendingly complete iff {Xξ : ξ < λ} ∈ U for each decreasing sequence {Xξ : ξ < λ} ⊂ U . Theorems: (J-S-Sz) If λ = cf(λ), F is an ultrafiltration on T, F(t) is λ-descendingly complete for all t ∈ T, then XF is λ+-irresolvable. (Magidor): It is consistent from a supercompact that there is an ω1-descendingly complete uniform ultrafilter on ℵω Con (∃supercompact) implies Con( there is a monotonically normal space X with |X| = ∆(X) = ℵω that is no ω2-resolvable).

Soukup, L (HAS) RIMS 2010 19 / 23

Smaller counterexamples

An ultrafilter U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅.

An ultrafilter U is λ-descendingly complete iff {Xξ : ξ < λ} ∈ U for each decreasing sequence {Xξ : ξ < λ} ⊂ U . Theorems: (J-S-Sz) If λ = cf(λ), F is an ultrafiltration on T, F(t) is λ-descendingly complete for all t ∈ T, then XF is λ+-irresolvable. (Magidor): It is consistent from a supercompact that there is an ω1-descendingly complete uniform ultrafilter on ℵω Con (∃supercompact) implies Con( there is a monotonically normal space X with |X| = ∆(X) = ℵω that is no ω2-resolvable).

Soukup, L (HAS) RIMS 2010 19 / 23

Smaller counterexamples

An ultrafilter U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅.

An ultrafilter U is λ-descendingly complete iff {Xξ : ξ < λ} ∈ U for each decreasing sequence {Xξ : ξ < λ} ⊂ U . Theorems: (J-S-Sz) If λ = cf(λ), F is an ultrafiltration on T, F(t) is λ-descendingly complete for all t ∈ T, then XF is λ+-irresolvable. (Magidor): It is consistent from a supercompact that there is an ω1-descendingly complete uniform ultrafilter on ℵω Con (∃supercompact) implies Con( there is a monotonically normal space X with |X| = ∆(X) = ℵω that is no ω2-resolvable).

Soukup, L (HAS) RIMS 2010 19 / 23

Smaller counterexamples

An ultrafilter U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅.

An ultrafilter U is λ-descendingly complete iff {Xξ : ξ < λ} ∈ U for each decreasing sequence {Xξ : ξ < λ} ⊂ U . Theorems: (J-S-Sz) If λ = cf(λ), F is an ultrafiltration on T, F(t) is λ-descendingly complete for all t ∈ T, then XF is λ+-irresolvable. (Magidor): It is consistent from a supercompact that there is an ω1-descendingly complete uniform ultrafilter on ℵω Con (∃supercompact) implies Con( there is a monotonically normal space X with |X| = ∆(X) = ℵω that is no ω2-resolvable).

Soukup, L (HAS) RIMS 2010 19 / 23

Filtrations

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable. Con(there is an ultrafiltration F on T = (ωω)<ω s.t. XF is not ω2-resolvable.)

An ultrafilter U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅. Theorems: (J-So-Sz) Let F be a filtration on κ<ω and λ be an infinite cardinal such that F(t) is µ-decomposable whenever t ∈ κ<ω and ω ≤ µ ≤ λ. Then there X(F) is λ-resolvable. (Kunen, Prikry): If λ = cf(λ), then “λ+-decomposable” = ⇒ “λ-decomposable”. For k ≤ n < ω, every uniform ultrafilter on ωn is ωk-decomposable There is no filtration type counterexample of size < ℵω

Soukup, L (HAS) RIMS 2010 20 / 23

Filtrations

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable. Con(there is an ultrafiltration F on T = (ωω)<ω s.t. XF is not ω2-resolvable.)

An ultrafilter U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅. Theorems: (J-So-Sz) Let F be a filtration on κ<ω and λ be an infinite cardinal such that F(t) is µ-decomposable whenever t ∈ κ<ω and ω ≤ µ ≤ λ. Then there X(F) is λ-resolvable. (Kunen, Prikry): If λ = cf(λ), then “λ+-decomposable” = ⇒ “λ-decomposable”. For k ≤ n < ω, every uniform ultrafilter on ωn is ωk-decomposable There is no filtration type counterexample of size < ℵω

Soukup, L (HAS) RIMS 2010 20 / 23

Filtrations

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable. Con(there is an ultrafiltration F on T = (ωω)<ω s.t. XF is not ω2-resolvable.)

An ultrafilter U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅. Theorems: (J-So-Sz) Let F be a filtration on κ<ω and λ be an infinite cardinal such that F(t) is µ-decomposable whenever t ∈ κ<ω and ω ≤ µ ≤ λ. Then there X(F) is λ-resolvable. (Kunen, Prikry): If λ = cf(λ), then “λ+-decomposable” = ⇒ “λ-decomposable”. For k ≤ n < ω, every uniform ultrafilter on ωn is ωk-decomposable There is no filtration type counterexample of size < ℵω

Soukup, L (HAS) RIMS 2010 20 / 23

Filtrations

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable. Con(there is an ultrafiltration F on T = (ωω)<ω s.t. XF is not ω2-resolvable.)

An ultrafilter U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅. Theorems: (J-So-Sz) Let F be a filtration on κ<ω and λ be an infinite cardinal such that F(t) is µ-decomposable whenever t ∈ κ<ω and ω ≤ µ ≤ λ. Then there X(F) is λ-resolvable. (Kunen, Prikry): If λ = cf(λ), then “λ+-decomposable” = ⇒ “λ-decomposable”. For k ≤ n < ω, every uniform ultrafilter on ωn is ωk-decomposable There is no filtration type counterexample of size < ℵω

Soukup, L (HAS) RIMS 2010 20 / 23

Filtrations

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable. Con(there is an ultrafiltration F on T = (ωω)<ω s.t. XF is not ω2-resolvable.)

An ultrafilter U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅. Theorems: (J-So-Sz) Let F be a filtration on κ<ω and λ be an infinite cardinal such that F(t) is µ-decomposable whenever t ∈ κ<ω and ω ≤ µ ≤ λ. Then there X(F) is λ-resolvable. (Kunen, Prikry): If λ = cf(λ), then “λ+-decomposable” = ⇒ “λ-decomposable”. For k ≤ n < ω, every uniform ultrafilter on ωn is ωk-decomposable There is no filtration type counterexample of size < ℵω

Soukup, L (HAS) RIMS 2010 20 / 23

Filtrations

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable. Con(there is an ultrafiltration F on T = (ωω)<ω s.t. XF is not ω2-resolvable.)

An ultrafilter U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅. Theorems: (J-So-Sz) Let F be a filtration on κ<ω and λ be an infinite cardinal such that F(t) is µ-decomposable whenever t ∈ κ<ω and ω ≤ µ ≤ λ. Then there X(F) is λ-resolvable. (Kunen, Prikry): If λ = cf(λ), then “λ+-decomposable” = ⇒ “λ-decomposable”. For k ≤ n < ω, every uniform ultrafilter on ωn is ωk-decomposable There is no filtration type counterexample of size < ℵω

Soukup, L (HAS) RIMS 2010 20 / 23

Filtrations

If F is a uniform ultrafiltration on T = ω<ω

1

then XF is ω1-resolvable. Con(there is an ultrafiltration F on T = (ωω)<ω s.t. XF is not ω2-resolvable.)

An ultrafilter U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅. Theorems: (J-So-Sz) Let F be a filtration on κ<ω and λ be an infinite cardinal such that F(t) is µ-decomposable whenever t ∈ κ<ω and ω ≤ µ ≤ λ. Then there X(F) is λ-resolvable. (Kunen, Prikry): If λ = cf(λ), then “λ+-decomposable” = ⇒ “λ-decomposable”. For k ≤ n < ω, every uniform ultrafilter on ωn is ωk-decomposable There is no filtration type counterexample of size < ℵω

Soukup, L (HAS) RIMS 2010 20 / 23

Universality

There is no filtration type counterexample of size < ℵω

Theorem (J-So-Sz)

Assume that κ = cf(κ) ≥ λ. Then the following are equivalent. (1) If X is a MN space with |X| = ∆(X) = κ then X is λ-resolvable. (2) For every uniform ultrafiltration F on κ<ω, the space X(F) is λ-resolvable.

Theorem

A crowded monotonically normal space X is maximally resolvable provided |X| < ℵω.

Soukup, L (HAS) RIMS 2010 21 / 23

Universality

There is no filtration type counterexample of size < ℵω

Theorem (J-So-Sz)

Assume that κ = cf(κ) ≥ λ. Then the following are equivalent. (1) If X is a MN space with |X| = ∆(X) = κ then X is λ-resolvable. (2) For every uniform ultrafiltration F on κ<ω, the space X(F) is λ-resolvable.

Theorem

A crowded monotonically normal space X is maximally resolvable provided |X| < ℵω.

Soukup, L (HAS) RIMS 2010 21 / 23

Universality

There is no filtration type counterexample of size < ℵω

Theorem (J-So-Sz)

Assume that κ = cf(κ) ≥ λ. Then the following are equivalent. (1) If X is a MN space with |X| = ∆(X) = κ then X is λ-resolvable. (2) For every uniform ultrafiltration F on κ<ω, the space X(F) is λ-resolvable.

Theorem

A crowded monotonically normal space X is maximally resolvable provided |X| < ℵω.

Soukup, L (HAS) RIMS 2010 21 / 23

Universality

There is no filtration type counterexample of size < ℵω

Theorem (J-So-Sz)

Assume that κ = cf(κ) ≥ λ. Then the following are equivalent. (1) If X is a MN space with |X| = ∆(X) = κ then X is λ-resolvable. (2) For every uniform ultrafiltration F on κ<ω, the space X(F) is λ-resolvable.

Theorem

A crowded monotonically normal space X is maximally resolvable provided |X| < ℵω.

Soukup, L (HAS) RIMS 2010 21 / 23

Universality

There is no filtration type counterexample of size < ℵω

Theorem (J-So-Sz)

Assume that κ = cf(κ) ≥ λ. Then the following are equivalent. (1) If X is a MN space with |X| = ∆(X) = κ then X is λ-resolvable. (2) For every uniform ultrafiltration F on κ<ω, the space X(F) is λ-resolvable.

Theorem

A crowded monotonically normal space X is maximally resolvable provided |X| < ℵω.

Soukup, L (HAS) RIMS 2010 21 / 23

Decomposable ultrafilters

U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅.

Let U be a uniform ultrafilter on λ. We say that U is maximally decomposable iff it is µ-decomposable for all ω ≤ µ ≤ λ. Any "measure" is ω-descendingly complete, hence not ω-decomposable. (Kunen - Prikry, 1971) If λ < ℵω then every uniform ultrafilter on λ is maximally decomposable. (Donder, 1988) If there is a not maximally decomposable uniform ultrafilter then there is a measurable cardinal in some inner model.

Soukup, L (HAS) RIMS 2010 22 / 23

Decomposable ultrafilters

U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅.

Let U be a uniform ultrafilter on λ. We say that U is maximally decomposable iff it is µ-decomposable for all ω ≤ µ ≤ λ. Any "measure" is ω-descendingly complete, hence not ω-decomposable. (Kunen - Prikry, 1971) If λ < ℵω then every uniform ultrafilter on λ is maximally decomposable. (Donder, 1988) If there is a not maximally decomposable uniform ultrafilter then there is a measurable cardinal in some inner model.

Soukup, L (HAS) RIMS 2010 22 / 23

Decomposable ultrafilters

U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅.

Let U be a uniform ultrafilter on λ. We say that U is maximally decomposable iff it is µ-decomposable for all ω ≤ µ ≤ λ. Any "measure" is ω-descendingly complete, hence not ω-decomposable. (Kunen - Prikry, 1971) If λ < ℵω then every uniform ultrafilter on λ is maximally decomposable. (Donder, 1988) If there is a not maximally decomposable uniform ultrafilter then there is a measurable cardinal in some inner model.

Soukup, L (HAS) RIMS 2010 22 / 23

Decomposable ultrafilters

U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅.

Let U be a uniform ultrafilter on λ. We say that U is maximally decomposable iff it is µ-decomposable for all ω ≤ µ ≤ λ. Any "measure" is ω-descendingly complete, hence not ω-decomposable. (Kunen - Prikry, 1971) If λ < ℵω then every uniform ultrafilter on λ is maximally decomposable. (Donder, 1988) If there is a not maximally decomposable uniform ultrafilter then there is a measurable cardinal in some inner model.

Soukup, L (HAS) RIMS 2010 22 / 23

Decomposable ultrafilters

U is λ-decomposable iff there is a decreasing sequence {Xξ : ξ < λ} ⊂ U with {Xξ : ξ < λ} = ∅.

Let U be a uniform ultrafilter on λ. We say that U is maximally decomposable iff it is µ-decomposable for all ω ≤ µ ≤ λ. Any "measure" is ω-descendingly complete, hence not ω-decomposable. (Kunen - Prikry, 1971) If λ < ℵω then every uniform ultrafilter on λ is maximally decomposable. (Donder, 1988) If there is a not maximally decomposable uniform ultrafilter then there is a measurable cardinal in some inner model.

Soukup, L (HAS) RIMS 2010 22 / 23

Recent results:

If λ < ℵω then every uniform ultrafilter on λ is maximally decomposable. If there is a not maximally decomposable uniform ultrafilter then there is a measurable cardinal in some inner model.

Theorem (Juhász, Magidor)

The following are equivalent: Every MN space of cardinality < κ is maximally resolvable. Every uniform ultrafilter on a cardinal < κ is maximally decomposable. The following are equiconsistent: There is a measurable cardinal. There is a MN space that is not maximally resolvable. There is a MN space X with |X| = ∆(X) = ℵω that is not ω1-resolvable.

Soukup, L (HAS) RIMS 2010 23 / 23

Recent results:

If λ < ℵω then every uniform ultrafilter on λ is maximally decomposable. If there is a not maximally decomposable uniform ultrafilter then there is a measurable cardinal in some inner model.

Theorem (Juhász, Magidor)

The following are equivalent: Every MN space of cardinality < κ is maximally resolvable. Every uniform ultrafilter on a cardinal < κ is maximally decomposable. The following are equiconsistent: There is a measurable cardinal. There is a MN space that is not maximally resolvable. There is a MN space X with |X| = ∆(X) = ℵω that is not ω1-resolvable.

Soukup, L (HAS) RIMS 2010 23 / 23

Recent results:

If λ < ℵω then every uniform ultrafilter on λ is maximally decomposable. If there is a not maximally decomposable uniform ultrafilter then there is a measurable cardinal in some inner model.

Theorem (Juhász, Magidor)

The following are equivalent: Every MN space of cardinality < κ is maximally resolvable. Every uniform ultrafilter on a cardinal < κ is maximally decomposable. The following are equiconsistent: There is a measurable cardinal. There is a MN space that is not maximally resolvable. There is a MN space X with |X| = ∆(X) = ℵω that is not ω1-resolvable.

Soukup, L (HAS) RIMS 2010 23 / 23

Recent results:

If λ < ℵω then every uniform ultrafilter on λ is maximally decomposable. If there is a not maximally decomposable uniform ultrafilter then there is a measurable cardinal in some inner model.

Theorem (Juhász, Magidor)

The following are equivalent: Every MN space of cardinality < κ is maximally resolvable. Every uniform ultrafilter on a cardinal < κ is maximally decomposable. The following are equiconsistent: There is a measurable cardinal. There is a MN space that is not maximally resolvable. There is a MN space X with |X| = ∆(X) = ℵω that is not ω1-resolvable.

Soukup, L (HAS) RIMS 2010 23 / 23

Recent results:

If λ < ℵω then every uniform ultrafilter on λ is maximally decomposable. If there is a not maximally decomposable uniform ultrafilter then there is a measurable cardinal in some inner model.

Theorem (Juhász, Magidor)

The following are equivalent: Every MN space of cardinality < κ is maximally resolvable. Every uniform ultrafilter on a cardinal < κ is maximally decomposable. The following are equiconsistent: There is a measurable cardinal. There is a MN space that is not maximally resolvable. There is a MN space X with |X| = ∆(X) = ℵω that is not ω1-resolvable.

Soukup, L (HAS) RIMS 2010 23 / 23

Recent results:

If λ < ℵω then every uniform ultrafilter on λ is maximally decomposable. If there is a not maximally decomposable uniform ultrafilter then there is a measurable cardinal in some inner model.

Theorem (Juhász, Magidor)

The following are equivalent: Every MN space of cardinality < κ is maximally resolvable. Every uniform ultrafilter on a cardinal < κ is maximally decomposable. The following are equiconsistent: There is a measurable cardinal. There is a MN space that is not maximally resolvable. There is a MN space X with |X| = ∆(X) = ℵω that is not ω1-resolvable.

Soukup, L (HAS) RIMS 2010 23 / 23

Recent results:

If λ < ℵω then every uniform ultrafilter on λ is maximally decomposable. If there is a not maximally decomposable uniform ultrafilter then there is a measurable cardinal in some inner model.

Theorem (Juhász, Magidor)

The following are equivalent: Every MN space of cardinality < κ is maximally resolvable. Every uniform ultrafilter on a cardinal < κ is maximally decomposable. The following are equiconsistent: There is a measurable cardinal. There is a MN space that is not maximally resolvable. There is a MN space X with |X| = ∆(X) = ℵω that is not ω1-resolvable.

Soukup, L (HAS) RIMS 2010 23 / 23

Recent results:

If λ < ℵω then every uniform ultrafilter on λ is maximally decomposable. If there is a not maximally decomposable uniform ultrafilter then there is a measurable cardinal in some inner model.

Theorem (Juhász, Magidor)

The following are equivalent: Every MN space of cardinality < κ is maximally resolvable. Every uniform ultrafilter on a cardinal < κ is maximally decomposable. The following are equiconsistent: There is a measurable cardinal. There is a MN space that is not maximally resolvable. There is a MN space X with |X| = ∆(X) = ℵω that is not ω1-resolvable.

Soukup, L (HAS) RIMS 2010 23 / 23