Pinning Down versus Density Lajos Soukup Alfrd Rnyi Institute of - - PowerPoint PPT Presentation

Pinning Down versus Density

Lajos Soukup

Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences http://www.renyi.hu/∼soukup

Novi Sad Conference in Set Theory and General Topology

joint work with I. Juhász, J. van Mill and Z. Szentmiklóssy

Cardinal functions

Cardinal functions X → F(X) ∈ Card X ≈ Y = ⇒ F(X) = F(Y)

Cardinal functions X → F(X) ∈ Card X ≈ Y = ⇒ F(X) = F(Y)

• |X|

Cardinal functions X → F(X) ∈ Card X ≈ Y = ⇒ F(X) = F(Y)

• |X|
• w(X)= min{|B| : B is a base}

Cardinal functions X → F(X) ∈ Card X ≈ Y = ⇒ F(X) = F(Y)

• |X|
• w(X)= min{|B| : B is a base}
• d(X)= min{|D| : D ⊂dense X}

Basic inequalities

• |X|
• w(X) = min{|B| : B is a base}
• d(X) = min{|D| : D ⊂dense X}

Basic inequalities

• |X|
• w(X) = min{|B| : B is a base}
• d(X) = min{|D| : D ⊂dense X}
• d(X) ≤ w(X) ≤ 2|X|.

Basic inequalities

• |X|
• w(X) = min{|B| : B is a base}
• d(X) = min{|D| : D ⊂dense X}
• d(X) ≤ w(X) ≤ 2|X|.
• (Posposil) X Hausdorff: |X| ≤ 22d(X). Sharp: βω

Basic inequalities

• |X|
• w(X) = min{|B| : B is a base}
• d(X) = min{|D| : D ⊂dense X}
• d(X) ≤ w(X) ≤ 2|X|.
• (Posposil) X Hausdorff: |X| ≤ 22d(X). Sharp: βω
• X regular: w(X) ≤ 2d(X).

Basic inequalities

• |X|
• w(X) = min{|B| : B is a base}
• d(X) = min{|D| : D ⊂dense X}
• d(X) ≤ w(X) ≤ 2|X|.
• (Posposil) X Hausdorff: |X| ≤ 22d(X). Sharp: βω
• X regular: w(X) ≤ 2d(X).
• X Hausdorff: w(X) ≤ 222d(X)

. Sharp (Juhász)

Pinning down number

Pinning down number U : X → τX is a neighborhood assignment on X iff p ∈ U(p) for all p ∈ X

Pinning down number U : X → τX is a neighborhood assignment on X iff p ∈ U(p) for all p ∈ X A ⊂ X pins down a neighborhood assignment U iff A ∩ U(p) = ∅ for all p ∈ X

Pinning down number U : X → τX is a neighborhood assignment on X iff p ∈ U(p) for all p ∈ X A ⊂ X pins down a neighborhood assignment U iff A ∩ U(p) = ∅ for all p ∈ X ∗ A dense set pins down every neighborhood assignment

Pinning down number U : X → τX is a neighborhood assignment on X iff p ∈ U(p) for all p ∈ X A ⊂ X pins down a neighborhood assignment U iff A ∩ U(p) = ∅ for all p ∈ X ∗ A dense set pins down every neighborhood assignment Definition (T. Banakh, A. Ravsky) pinning down number of a space X: pd(X)=min{κ : ∀U ∈ NEA(X) ∃A ∈

• X

κ (A pins down U)}

Pinning down number U : X → τX is a neighborhood assignment on X iff p ∈ U(p) for all p ∈ X A ⊂ X pins down a neighborhood assignment U iff A ∩ U(p) = ∅ for all p ∈ X ∗ A dense set pins down every neighborhood assignment Definition (T. Banakh, A. Ravsky) pinning down number of a space X: pd(X)=min{κ : ∀U ∈ NEA(X) ∃A ∈

• X

κ (A pins down U)} ⋆ pd(X) ≤ d(X).

Pinning down number U : X → τX is a neighborhood assignment on X iff p ∈ U(p) for all p ∈ X A ⊂ X pins down a neighborhood assignment U iff A ∩ U(p) = ∅ for all p ∈ X ∗ A dense set pins down every neighborhood assignment Definition (T. Banakh, A. Ravsky) pinning down number of a space X: pd(X)=min{κ : ∀U ∈ NEA(X) ∃A ∈

• X

κ (A pins down U)} ⋆ pd(X) ≤ d(X).

• T. Banakh, A. Ravsky: e−(X), foredensity ;
• Spadarro: dNA(X),

First results

• U is a NEA on X iff U : X → τX s.t. a ∈ U(a) for all a ∈ X
• P ⊂ X pins down a nea U iff P ∩ U(a) = ∅ for all a ∈ X
• pd(X)= min{κ : ∀U ∈ NEA(X) ∃A ∈
• X

κ (A pins down U)}

• pd(X) ≤ d(X).

First results

• U is a NEA on X iff U : X → τX s.t. a ∈ U(a) for all a ∈ X
• P ⊂ X pins down a nea U iff P ∩ U(a) = ∅ for all a ∈ X
• pd(X)= min{κ : ∀U ∈ NEA(X) ∃A ∈
• X

κ (A pins down U)}

• pd(X) ≤ d(X).

Theorem (T. Banakh, A. Ravsky)

• If X is T2, |X| < ℵω, then pd(X) = d(X).
• If 22cf(κ) > κ > cf(κ), then there is a T2 space X with

pd(X) < d(X).

First results

• U is a NEA on X iff U : X → τX s.t. a ∈ U(a) for all a ∈ X
• P ⊂ X pins down a nea U iff P ∩ U(a) = ∅ for all a ∈ X
• pd(X)= min{κ : ∀U ∈ NEA(X) ∃A ∈
• X

κ (A pins down U)}

• pd(X) ≤ d(X).

Theorem (T. Banakh, A. Ravsky)

• If X is T2, |X| < ℵω, then pd(X) = d(X).
• If 22cf(κ) > κ > cf(κ), then there is a T2 space X with

pd(X) < d(X). A topological space X is an example iff pd(X) < d(X).

First results

• U is a NEA on X iff U : X → τX s.t. a ∈ U(a) for all a ∈ X
• P ⊂ X pins down a nea U iff P ∩ U(a) = ∅ for all a ∈ X
• pd(X)= min{κ : ∀U ∈ NEA(X) ∃A ∈
• X

κ (A pins down U)}

• pd(X) ≤ d(X).

Theorem (T. Banakh, A. Ravsky)

• If X is T2, |X| < ℵω, then pd(X) = d(X).
• If 22cf(κ) > κ > cf(κ), then there is a T2 space X with

pd(X) < d(X). A topological space X is an example iff pd(X) < d(X). Questions

• Regular example?
• ZFC example?

First equivalence

• U is a NEA on X iff U : X → τX s.t. a ∈ U(a) for all a ∈ X
• P ⊂ X pins down a nea U iff P ∩ U(a) = ∅ for all a ∈ X
• pd(X) = min{κ : ∀U ∈ NEA(X) ∃A ∈
• X

κ (A pins down U)}

First equivalence

• U is a NEA on X iff U : X → τX s.t. a ∈ U(a) for all a ∈ X
• P ⊂ X pins down a nea U iff P ∩ U(a) = ∅ for all a ∈ X
• pd(X) = min{κ : ∀U ∈ NEA(X) ∃A ∈
• X

κ (A pins down U)} Theorem (I. Juhász, L.S., Z. Szentmiklóssy) T.F.A.E: (1) 2κ < κ+ω for each cardinal κ, (2) pd(X) = d(X) for each T2 space X, (3) pd(X) = d(X) for each 0-dimensional T2 space X.

A special case

A special case

• dispersion character

∆(X) = min{|U| : ∅ = U ⊂open X}.

A special case

• dispersion character

∆(X) = min{|U| : ∅ = U ⊂open X}.

• X is neat: |X| = ∆(X)

A special case

• dispersion character

∆(X) = min{|U| : ∅ = U ⊂open X}.

• X is neat: |X| = ∆(X)

We prove: If 2ω > ωω then there is a 0-dimensional space X with pd(X) = ω and |X| = ∆(X) = d(X) = ωω.

X0 ω0 Xn ωn Xm ωm

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

X0 ω0 Xn ωn Xm ωm

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A
• .

X0 ω0 Xn ωn Xm ωm f(n) f(m)

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A
• .

X0 ω0 Xn ωn Xm ωm f(n) f(m) A ωn \ f(n)

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A
• .

X0 ω0 Xn ωn Xm ωm f(n) f(m) A ωn \ f(n)

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A
• .

X0 ω0 Xn ωn Xm ωm f(n) f(m) A ωn \ f(n) A ωm \ f(m)

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A
• .

X0 ω0 Xn ωn Xm ωm f(n) f(m) A ωn \ f(n) A ωm \ f(m)

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A
• .

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase of τ: {G(n, f, An,f) : n ∈ ω, f ∈ P.}

X0 ω0 Xn ωn Xm ωm f(n) f(m) A ωn \ f(n) A ωm \ f(m)

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A
• .

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase of τ: {G(n, f, An,f) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , A ∈ A

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: d(X) = ωω.

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: d(X) = ωω.

• Assume |D| < ωω.

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: d(X) = ωω.

• Assume |D| < ωω.
• |D| < ωn for some n

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: d(X) = ωω.

• Assume |D| < ωω.
• |D| < ωn for some n
• there is f ∈ P such that D ∩ Xm ⊂ f(m) × ω for m ≥ n.

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: d(X) = ωω.

• Assume |D| < ωω.
• |D| < ωn for some n
• there is f ∈ P such that D ∩ Xm ⊂ f(m) × ω for m ≥ n.
• Then G(n, f, An,f ) ∩ D = ∅.

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: d(X) = ωω.

• Assume |D| < ωω.
• |D| < ωn for some n
• there is f ∈ P such that D ∩ Xm ⊂ f(m) × ω for m ≥ n.
• Then G(n, f, An,f ) ∩ D = ∅.
• Thus D is not dense.

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: pd(X) = ω.

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: pd(X) = ω.

• U : X → τ be a NEA.

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: pd(X) = ω.

• U : X → τ be a NEA.
• Then U(p) ⊃G(np, fp, Ap) for all p ∈ X

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: pd(X) = ω.

• U : X → τ be a NEA.
• Then U(p) ⊃G(np, fp, Ap) for all p ∈ X
• there is g∈ P s.t. fp <∗ g for all p ∈ X.

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: pd(X) = ω.

• U : X → τ be a NEA.
• Then U(p) ⊃G(np, fp, Ap) for all p ∈ X
• there is g∈ P s.t. fp <∗ g for all p ∈ X.
• R = {g(n) : n ∈ ω} × ω pins down U

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: pd(X) = ω.

• U : X → τ be a NEA.
• Then U(p) ⊃G(np, fp, Ap) for all p ∈ X
• there is g∈ P s.t. fp <∗ g for all p ∈ X.
• R = {g(n) : n ∈ ω} × ω pins down U
• Let p ∈ X. Then U(p) ⊃ G(np, fp, Ap).

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: pd(X) = ω.

• U : X → τ be a NEA.
• Then U(p) ⊃G(np, fp, Ap) for all p ∈ X
• there is g∈ P s.t. fp <∗ g for all p ∈ X.
• R = {g(n) : n ∈ ω} × ω pins down U
• Let p ∈ X. Then U(p) ⊃ G(np, fp, Ap).
• ∃n ≥ np s.t. fp(n) < g(n)

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: pd(X) = ω.

• U : X → τ be a NEA.
• Then U(p) ⊃G(np, fp, Ap) for all p ∈ X
• there is g∈ P s.t. fp <∗ g for all p ∈ X.
• R = {g(n) : n ∈ ω} × ω pins down U
• Let p ∈ X. Then U(p) ⊃ G(np, fp, Ap).
• ∃n ≥ np s.t. fp(n) < g(n)
• Xn

ωn

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: pd(X) = ω.

• U : X → τ be a NEA.
• Then U(p) ⊃G(np, fp, Ap) for all p ∈ X
• there is g∈ P s.t. fp <∗ g for all p ∈ X.
• R = {g(n) : n ∈ ω} × ω pins down U
• Let p ∈ X. Then U(p) ⊃ G(np, fp, Ap).
• ∃n ≥ np s.t. fp(n) < g(n)
• Xn

ωn fp(n) Ap

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: pd(X) = ω.

• U : X → τ be a NEA.
• Then U(p) ⊃G(np, fp, Ap) for all p ∈ X
• there is g∈ P s.t. fp <∗ g for all p ∈ X.
• R = {g(n) : n ∈ ω} × ω pins down U
• Let p ∈ X. Then U(p) ⊃ G(np, fp, Ap).
• ∃n ≥ np s.t. fp(n) < g(n)
• Xn

ωn fp(n) Ap U(p) ⊃

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: pd(X) = ω.

• U : X → τ be a NEA.
• Then U(p) ⊃G(np, fp, Ap) for all p ∈ X
• there is g∈ P s.t. fp <∗ g for all p ∈ X.
• R = {g(n) : n ∈ ω} × ω pins down U
• Let p ∈ X. Then U(p) ⊃ G(np, fp, Ap).
• ∃n ≥ np s.t. fp(n) < g(n)
• Xn

ωn fp(n) Ap U(p) ⊃ g(n)

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: pd(X) = ω.

• U : X → τ be a NEA.
• Then U(p) ⊃G(np, fp, Ap) for all p ∈ X
• there is g∈ P s.t. fp <∗ g for all p ∈ X.
• R = {g(n) : n ∈ ω} × ω pins down U
• Let p ∈ X. Then U(p) ⊃ G(np, fp, Ap).
• ∃n ≥ np s.t. fp(n) < g(n)
• Xn

ωn fp(n) Ap U(p) ⊃ g(n) ω ⊂ R

• X= ωω × ω, τ • Xn= (ωn \ ωn−1) × ω. • P= (ωn \ ωn−1).

If n ∈ ω, f ∈ P, A ⊂ ω let G(n, f, A)=

m≥n

• ωm \ f(m)
• × A.

Fix an independent family A = {An,f : n ∈ ω, f ∈ P}⊂

• ω

ω. Clopen subbase: {G(n, f, An,f ) : n ∈ ω, f ∈ P.} If ∅ = U ⊂open X then G(n, f, A) ⊂ U for some n ∈ ω, f ∈ , AU ∈ A Claim: pd(X) = ω.

• U : X → τ be a NEA.
• Then U(p) ⊃G(np, fp, Ap) for all p ∈ X
• there is g∈ P s.t. fp <∗ g for all p ∈ X.
• R = {g(n) : n ∈ ω} × ω pins down U
• Let p ∈ X. Then U(p) ⊃ G(np, fp, Ap).
• ∃n ≥ np s.t. fp(n) < g(n)
• Then R ∩ Xn ∩ G(np, fp, Ap) = ∅.
• Xn

ωn fp(n) Ap U(p) ⊃ g(n) ω ⊂ R

Some observations

Some observations If pd(X) < d(X), then ∃Y ⊂open X s.t. pd(Y) < d(Y) and ∆(Y) = |Y|.

Some observations If pd(X) < d(X), then ∃Y ⊂open X s.t. pd(Y) < d(Y) and ∆(Y) = |Y|. First examples: pd(X) = cf(|X|) < d(X) = ∆(X) = |X|.

Some observations If pd(X) < d(X), then ∃Y ⊂open X s.t. pd(Y) < d(Y) and ∆(Y) = |Y|. First examples: pd(X) = cf(|X|) < d(X) = ∆(X) = |X|. Questions

• Can d(X) be a regular cardinal?
• Can |X| be a regular cardinal?

Some observations If pd(X) < d(X), then ∃Y ⊂open X s.t. pd(Y) < d(Y) and ∆(Y) = |Y|. First examples: pd(X) = cf(|X|) < d(X) = ∆(X) = |X|. Questions

• Can d(X) be a regular cardinal?
• Can |X| be a regular cardinal?

Modified construction: pd(X) = cf(|X|) <d(X) = cf(d(X))< ∆(X) = |X|

Shelah’s Strong Hypothesis

Shelah’s Strong Hypothesis

• µ > cf(µ)

Shelah’s Strong Hypothesis

• µ > cf(µ)
• S(µ) = {a ∈ [µ ∩ Reg]cf(µ) : sup a = µ}

Shelah’s Strong Hypothesis

• µ > cf(µ)
• S(µ) = {a ∈ [µ ∩ Reg]cf(µ) : sup a = µ}
• U(a) = {D : D is an ultrafilter on a, D ∩ Jbd[a] = ∅}.

Shelah’s Strong Hypothesis

• µ > cf(µ)
• S(µ) = {a ∈ [µ ∩ Reg]cf(µ) : sup a = µ}
• U(a) = {D : D is an ultrafilter on a, D ∩ Jbd[a] = ∅}.
• pp(µ)= sup{cf( a/D) : a ∈ S(µ), D ∈ U(a))}

Shelah’s Strong Hypothesis

• µ > cf(µ)
• S(µ) = {a ∈ [µ ∩ Reg]cf(µ) : sup a = µ}
• U(a) = {D : D is an ultrafilter on a, D ∩ Jbd[a] = ∅}.
• pp(µ)= sup{cf( a/D) : a ∈ S(µ), D ∈ U(a))}

Shelah’s Strong Hypothesis: pp(µ) = µ+ for all singular cardinal µ.

An equiconsistency result

An equiconsistency result Theorem (I. Juhász, L.S., Z. Szentmiklóssy) The following three statements are equiconsistent: (i) There is a singular cardinal λ with pp(λ) > λ+, i.e. Shelah’s Strong Hypothesis fails; (ii) there is a 0-dimensional Hausdorff space X such that |X| = ∆(X) is a regular cardinal and pd(X) < d(X); (iii) there is a topological space X such that |X| = ∆(X) is a regular cardinal and pd(X) < d(X).

An equiconsistency result Theorem (I. Juhász, L.S., Z. Szentmiklóssy) The following three statements are equiconsistent: (i) There is a singular cardinal λ with pp(λ) > λ+, i.e. Shelah’s Strong Hypothesis fails; (ii) there is a 0-dimensional Hausdorff space X such that |X| = ∆(X) is a regular cardinal and pd(X) < d(X); (iii) there is a topological space X such that |X| = ∆(X) is a regular cardinal and pd(X) < d(X). No equivalence: Con(failure of SSH + the limit cardinals are strong limit)

Connected and locally connected spaces

Connected and locally connected spaces Theorem (I. Juhász,J. van Mill, L.S., Z. Szentmiklóssy ) T:F.A.E: (1) 2κ < κ+ω for each cardinal κ, (2) pd(X) = d(X) for each T2 space X, (3) pd(X) = d(X) for each 0-dimensional T2 space X.

Connected and locally connected spaces Theorem (I. Juhász,J. van Mill, L.S., Z. Szentmiklóssy ) T:F.A.E: (1) 2κ < κ+ω for each cardinal κ, (2) pd(X) = d(X) for each T2 space X, (3) pd(X) = d(X) for each 0-dimensional T2 space X. (4) pd(X) = d(X) for all connected, locally connected regular spaces. (5) pd(X) = d(X) for all Abelian topological groups.

Connected and locally connected spaces Theorem (I. Juhász,J. van Mill, L.S., Z. Szentmiklóssy ) T:F.A.E: (1) 2κ < κ+ω for each cardinal κ, (2) pd(X) = d(X) for each T2 space X, (3) pd(X) = d(X) for each 0-dimensional T2 space X. (4) pd(X) = d(X) for all connected, locally connected regular spaces. (5) pd(X) = d(X) for all Abelian topological groups. What about connected Tychonoff spaces?

Connected and locally connected spaces Theorem (I. Juhász,J. van Mill, L.S., Z. Szentmiklóssy ) T:F.A.E: (1) 2κ < κ+ω for each cardinal κ, (2) pd(X) = d(X) for each T2 space X, (3) pd(X) = d(X) for each 0-dimensional T2 space X. (4) pd(X) = d(X) for all connected, locally connected regular spaces. (5) pd(X) = d(X) for all Abelian topological groups. What about connected Tychonoff spaces? Theorem (JvMSSz) It is consistent that

Connected and locally connected spaces Theorem (I. Juhász,J. van Mill, L.S., Z. Szentmiklóssy ) T:F.A.E: (1) 2κ < κ+ω for each cardinal κ, (2) pd(X) = d(X) for each T2 space X, (3) pd(X) = d(X) for each 0-dimensional T2 space X. (4) pd(X) = d(X) for all connected, locally connected regular spaces. (5) pd(X) = d(X) for all Abelian topological groups. What about connected Tychonoff spaces? Theorem (JvMSSz) It is consistent that

• there is a 0-dimensional space X with pd(X) < d(X)

Connected and locally connected spaces Theorem (I. Juhász,J. van Mill, L.S., Z. Szentmiklóssy ) T:F.A.E: (1) 2κ < κ+ω for each cardinal κ, (2) pd(X) = d(X) for each T2 space X, (3) pd(X) = d(X) for each 0-dimensional T2 space X. (4) pd(X) = d(X) for all connected, locally connected regular spaces. (5) pd(X) = d(X) for all Abelian topological groups. What about connected Tychonoff spaces? Theorem (JvMSSz) It is consistent that

• there is a 0-dimensional space X with pd(X) < d(X)
• pd(X) = d(X) for all connected Tychonoff spaces.

A connected, locally connected Tychonoff example

A connected, locally connected Tychonoff example If X is a connected, Tychonoff space then |X| ≥ 2ω.

A connected, locally connected Tychonoff example If X is a connected, Tychonoff space then |X| ≥ 2ω. Theorem (I. Juhász,J. van Mill, L.S., Z. Szentmiklóssy) T:F.A.E: (1) There is a singular cardinal µ ≥ 2ω which is not a strong limit cardinal. (2) There is a connected, locally connected Tychonoff space X with ∆(X) = |X| and pd(X) < d(X).

A connected, locally connected Tychonoff example If X is a connected, Tychonoff space then |X| ≥ 2ω. Theorem (I. Juhász,J. van Mill, L.S., Z. Szentmiklóssy) T:F.A.E: (1) There is a singular cardinal µ ≥ 2ω which is not a strong limit cardinal. (2) There is a connected, locally connected Tychonoff space X with ∆(X) = |X| and pd(X) < d(X). (3) There is a pathwise connected, locally pathwise connected Tychonoff Abelian topological group X with ∆(X) = |X| and pd(X) < d(X).

Embedding theorem

Embedding theorem

• µ is a singular cardinal µ ≥ 2ω, not a strong limit cardinal.

Embedding theorem

• µ is a singular cardinal µ ≥ 2ω, not a strong limit cardinal.
• Need pathwise connected, locally pathwise connected Tychonoff

Abelian topological group H with ∆(H) = |H| and pd(H) < d(H).

Embedding theorem

• µ is a singular cardinal µ ≥ 2ω, not a strong limit cardinal.
• Need pathwise connected, locally pathwise connected Tychonoff

Abelian topological group H with ∆(H) = |H| and pd(H) < d(H).

• there is a 0-dimensional neat space X such that pd(X) < d(X)

and |X| ≥ 2ω.

Embedding theorem

• µ is a singular cardinal µ ≥ 2ω, not a strong limit cardinal.
• Need pathwise connected, locally pathwise connected Tychonoff

Abelian topological group H with ∆(H) = |H| and pd(H) < d(H).

• there is a 0-dimensional neat space X such that pd(X) < d(X)

and |X| ≥ 2ω. Theorem Let X be a T3.5 neat space such that |X| ≥ 2ω. Then X has a closed embedding into a T3.5 Abelian topological group H such that

Embedding theorem

• µ is a singular cardinal µ ≥ 2ω, not a strong limit cardinal.
• Need pathwise connected, locally pathwise connected Tychonoff

Abelian topological group H with ∆(H) = |H| and pd(H) < d(H).

• there is a 0-dimensional neat space X such that pd(X) < d(X)

and |X| ≥ 2ω. Theorem Let X be a T3.5 neat space such that |X| ≥ 2ω. Then X has a closed embedding into a T3.5 Abelian topological group H such that

• 1. d(X) = d(H),
• 2. pd(X) = pd(H),
• 3. H is neat,
• 4. H is pathwise connected and locally pathwise connected.

Step 1: embedding into a group

Step 1: embedding into a group If X is a T3.5-space, then F(X) and A(X) denote the free topological group and the free abelian topological group on X.

Step 1: embedding into a group If X is a T3.5-space, then F(X) and A(X) denote the free topological group and the free abelian topological group on X. F(X) is a topological group containing (a homeomorphic copy of) X such that

• 1. X generates F(X) algebraically,
• 2. every continuous function f : X → H, where H is any topological

group, can be extended to a continuous homomorphism ¯ f : F(X) → H.

Step 1: embedding into a group If X is a T3.5-space, then F(X) and A(X) denote the free topological group and the free abelian topological group on X. F(X) is a topological group containing (a homeomorphic copy of) X such that

• 1. X generates F(X) algebraically,
• 2. every continuous function f : X → H, where H is any topological

group, can be extended to a continuous homomorphism ¯ f : F(X) → H. Similarly for A(X).

Step 1: embedding into a group If X is a T3.5-space, then F(X) and A(X) denote the free topological group and the free abelian topological group on X. F(X) is a topological group containing (a homeomorphic copy of) X such that

• 1. X generates F(X) algebraically,
• 2. every continuous function f : X → H, where H is any topological

group, can be extended to a continuous homomorphism ¯ f : F(X) → H. Similarly for A(X). The existence of these groups was proved by Markov.

Step 1: embedding into a group If X is a T3.5-space, then F(X) and A(X) denote the free topological group and the free abelian topological group on X. F(X) is a topological group containing (a homeomorphic copy of) X such that

• 1. X generates F(X) algebraically,
• 2. every continuous function f : X → H, where H is any topological

group, can be extended to a continuous homomorphism ¯ f : F(X) → H. Similarly for A(X). The existence of these groups was proved by Markov. Theorem (JvMSSz) Let X be a T3.5-space. Then d(X) = d(F(X)) = d(A(X)). If X is neat, then so are A(X) and F(X), and pd(X) = pd(A(X)) = pd(F(X)).

Step 2: embedding a group into a pathwise connected one

Step 2: embedding a group into a pathwise connected one

• Hartman Mycielski construction

Step 2: embedding a group into a pathwise connected one

• Hartman Mycielski construction
• Let (G, ·, e) be a Tychonoff topological group.

G• =

• f ∈ [0,1)G :

for some sequence 0 = a0 < a1 < · · · < an = 1 f is constant on [ak, ak+1) for every k = 0, . . . , n−1

• .

Step 2: embedding a group into a pathwise connected one

• Hartman Mycielski construction
• Let (G, ·, e) be a Tychonoff topological group.

G• =

• f ∈ [0,1)G :

for some sequence 0 = a0 < a1 < · · · < an = 1 f is constant on [ak, ak+1) for every k = 0, . . . , n−1

• .
• Define ∗ on G• by (f ∗ g)(x) = f(x) · g(x) for all f, g ∈ G• and

x ∈ [0, 1).

Step 2: embedding a group into a pathwise connected one

• Hartman Mycielski construction
• Let (G, ·, e) be a Tychonoff topological group.

G• =

• f ∈ [0,1)G :

for some sequence 0 = a0 < a1 < · · · < an = 1 f is constant on [ak, ak+1) for every k = 0, . . . , n−1

• .
• Define ∗ on G• by (f ∗ g)(x) = f(x) · g(x) for all f, g ∈ G• and

x ∈ [0, 1).

• (G•, ∗, e•) is a group, where e•(r) = e for each r ∈ [0, 1).

Step 2: embedding a group into a pathwise connected one

• Hartman Mycielski construction
• Let (G, ·, e) be a Tychonoff topological group.

G• =

• f ∈ [0,1)G :

for some sequence 0 = a0 < a1 < · · · < an = 1 f is constant on [ak, ak+1) for every k = 0, . . . , n−1

• .
• Define ∗ on G• by (f ∗ g)(x) = f(x) · g(x) for all f, g ∈ G• and

x ∈ [0, 1).

• (G•, ∗, e•) is a group, where e•(r) = e for each r ∈ [0, 1).
• G embeds into G• via x → x•, where x•(r) = x for every

r ∈ [0, 1).

Step 2: embedding a group into a pathwise connected one

• Hartman Mycielski construction
• Let (G, ·, e) be a Tychonoff topological group.

G• =

• f ∈ [0,1)G :

for some sequence 0 = a0 < a1 < · · · < an = 1 f is constant on [ak, ak+1) for every k = 0, . . . , n−1

• .
• Define ∗ on G• by (f ∗ g)(x) = f(x) · g(x) for all f, g ∈ G• and

x ∈ [0, 1).

• (G•, ∗, e•) is a group, where e•(r) = e for each r ∈ [0, 1).
• G embeds into G• via x → x•, where x•(r) = x for every

r ∈ [0, 1).

• For e ∈ V ∈ τG and ε > 0, put

O(V, ε) = {f ∈ G• : λ({r ∈ [0, 1) : f(r) ∈ V})} < ε}

Step 2: embedding a group into a pathwise connected one

• Hartman Mycielski construction
• Let (G, ·, e) be a Tychonoff topological group.

G• =

• f ∈ [0,1)G :

for some sequence 0 = a0 < a1 < · · · < an = 1 f is constant on [ak, ak+1) for every k = 0, . . . , n−1

• .
• Define ∗ on G• by (f ∗ g)(x) = f(x) · g(x) for all f, g ∈ G• and

x ∈ [0, 1).

• (G•, ∗, e•) is a group, where e•(r) = e for each r ∈ [0, 1).
• G embeds into G• via x → x•, where x•(r) = x for every

r ∈ [0, 1).

• For e ∈ V ∈ τG and ε > 0, put

O(V, ε) = {f ∈ G• : λ({r ∈ [0, 1) : f(r) ∈ V})} < ε}

• The O(V, ε) are the neighborhoods of the element e• of G• that

generate the topology.

Properties of Hartman Mycielski extension G•

Properties of Hartman Mycielski extension G• Theorem G• is a topological group and is pathwise connected and locally pathwise connected. d(G•) ≤ d(G).

Properties of Hartman Mycielski extension G• Theorem G• is a topological group and is pathwise connected and locally pathwise connected. d(G•) ≤ d(G). Theorem (JvMSSz)

• d(G) = d(G•).
• If G is neat, and |G| ≥ 2ω, then G• is neat and pd(G•) = pd(G).

Positive results

Positive results Theorem If X is compact then pd(X) = d(X).

Positive results Theorem If X is compact then pd(X) = d(X). Proof.

Positive results Theorem If X is compact then pd(X) = d(X). Proof.

• Spec. case: ∆(X) = |X|.

Positive results Theorem If X is compact then pd(X) = d(X). Proof.

• Spec. case: ∆(X) = |X|.
• X compact, so w(X) ≤ |X| = ∆(X)

Positive results Theorem If X is compact then pd(X) = d(X). Proof.

• Spec. case: ∆(X) = |X|.
• X compact, so w(X) ≤ |X| = ∆(X)
• ∃U ∈ NEA(X) s.t. {U(y) : y ∈ X} is a base.

Positive results Theorem If X is compact then pd(X) = d(X). Proof.

• Spec. case: ∆(X) = |X|.
• X compact, so w(X) ≤ |X| = ∆(X)
• ∃U ∈ NEA(X) s.t. {U(y) : y ∈ X} is a base.
• If A pins down U, then A is dense.

Positive results Theorem If X is compact then pd(X) = d(X). Proof.

• Spec. case: ∆(X) = |X|.
• X compact, so w(X) ≤ |X| = ∆(X)
• ∃U ∈ NEA(X) s.t. {U(y) : y ∈ X} is a base.
• If A pins down U, then A is dense.

Question (JSSz)

• What about (regular) Lindelöf spaces?
• What about (regular) countably compact spaces?

Positive results Theorem If X is compact then pd(X) = d(X). Proof.

• Spec. case: ∆(X) = |X|.
• X compact, so w(X) ≤ |X| = ∆(X)
• ∃U ∈ NEA(X) s.t. {U(y) : y ∈ X} is a base.
• If A pins down U, then A is dense.

Question (JSSz)

• What about (regular) Lindelöf spaces?
• What about (regular) countably compact spaces?

Theorem (Juhász,van Mill, S, Szentmiklóssy) It is consistent that pd(X) < d(X) for some hereditarily Lindelöf regular space X.

Estimate d(X) using pd(X)

Estimate d(X) using pd(X) Theorem (JSSz) d(X) ≤ 2pd(X).

Estimate d(X) using pd(X) Theorem (JSSz) d(X) ≤ 2pd(X). Sharp?

Estimate d(X) using pd(X) Theorem (JSSz) d(X) ≤ 2pd(X). Sharp? Theorem (JSSz) d(X) < 2pd(X).

Estimate d(X) using pd(X) Theorem (JSSz) d(X) ≤ 2pd(X). Sharp? Theorem (JSSz) d(X) < 2pd(X). Sharp?

Estimate d(X) using pd(X) Theorem (JSSz) d(X) ≤ 2pd(X). Sharp? Theorem (JSSz) d(X) < 2pd(X). Sharp? Yes. It is consistent that 2pd(X) is as large as you wish and d(X)+ = 2pd(X).

Inequalities

• Pospisil: |X| ≤ 22d(X) for T2 spaces
• w(x) ≤ 2d(x) for T3 spaces

Inequalities

• Pospisil: |X| ≤ 22d(X) for T2 spaces
• w(x) ≤ 2d(x) for T3 spaces

Theorem (JSSz) |X| ≤ 22pd(X) for T2 spaces.

Inequalities

• Pospisil: |X| ≤ 22d(X) for T2 spaces
• w(x) ≤ 2d(x) for T3 spaces

Theorem (JSSz) |X| ≤ 22pd(X) for T2 spaces. Theorem (JSSz) If |X| = ∆(X), then

Inequalities

• Pospisil: |X| ≤ 22d(X) for T2 spaces
• w(x) ≤ 2d(x) for T3 spaces

Theorem (JSSz) |X| ≤ 22pd(X) for T2 spaces. Theorem (JSSz) If |X| = ∆(X), then

• either pd(X) = d(X) and |X| ≤ 22pd(X), or

Inequalities

• Pospisil: |X| ≤ 22d(X) for T2 spaces
• w(x) ≤ 2d(x) for T3 spaces

Theorem (JSSz) |X| ≤ 22pd(X) for T2 spaces. Theorem (JSSz) If |X| = ∆(X), then

• either pd(X) = d(X) and |X| ≤ 22pd(X), or
• pd(X) < d(X) and |X| < 2pd(X).

Inequalities

• Pospisil: |X| ≤ 22d(X) for T2 spaces
• w(x) ≤ 2d(x) for T3 spaces

Theorem (JSSz) |X| ≤ 22pd(X) for T2 spaces. Theorem (JSSz) If |X| = ∆(X), then

• either pd(X) = d(X) and |X| ≤ 22pd(X), or
• pd(X) < d(X) and |X| < 2pd(X).

Problem Does w(x) ≤ 2pd(x) hold for regular spaces?