[PPT] - Discrete subspaces of countably compact spaces Istvn Juhsz Alfrd PowerPoint Presentation

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Discrete subspaces of countably compact spaces

István Juhász

Alfréd Rényi Institute of Mathematics

Novi Sad, August, 2014

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 1 / 11

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Introduction

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 2 / 11

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Introduction

FACT. (Folklore??)

If all free sequences in a topological space X have compact closure then X is compact.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 2 / 11

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Introduction

FACT. (Folklore??)

If all free sequences in a topological space X have compact closure then X is compact.

DEFINITION

For a property P of subspaces of X, we say that X is P-bounded iff the closure in X of any subspace with P is compact.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 2 / 11

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Introduction

FACT. (Folklore??)

If all free sequences in a topological space X have compact closure then X is compact.

DEFINITION

For a property P of subspaces of X, we say that X is P-bounded iff the closure in X of any subspace with P is compact. So, F-bounded (and hence D-bounded) spaces are compact.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 2 / 11

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Introduction

FACT. (Folklore??)

If all free sequences in a topological space X have compact closure then X is compact.

DEFINITION

For a property P of subspaces of X, we say that X is P-bounded iff the closure in X of any subspace with P is compact. So, F-bounded (and hence D-bounded) spaces are compact.

COROLLARY

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 2 / 11

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Introduction

FACT. (Folklore??)

If all free sequences in a topological space X have compact closure then X is compact.

DEFINITION

For a property P of subspaces of X, we say that X is P-bounded iff the closure in X of any subspace with P is compact. So, F-bounded (and hence D-bounded) spaces are compact.

COROLLARY

Any non-isolated point of a compact T2 space is discretely touchable,

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 2 / 11

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Introduction

FACT. (Folklore??)

If all free sequences in a topological space X have compact closure then X is compact.

DEFINITION

For a property P of subspaces of X, we say that X is P-bounded iff the closure in X of any subspace with P is compact. So, F-bounded (and hence D-bounded) spaces are compact.

COROLLARY

Any non-isolated point of a compact T2 space is discretely touchable, i.e. the accumulation point of a discrete set.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 2 / 11

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DTTW

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 3 / 11

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DTTW

Dow-Tkachenko-Tkachuk-Wilson, TOPOLOGIES GENERATED BY DISCRETE SUBSPACES, Glasnik Mat., 2002 :

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 3 / 11

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DTTW

Dow-Tkachenko-Tkachuk-Wilson, TOPOLOGIES GENERATED BY DISCRETE SUBSPACES, Glasnik Mat., 2002 :

DEFINITION

A space X is discretely generated iff x ∈ A implies x ∈ D for some D ⊂ A discrete.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 3 / 11

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DTTW

Dow-Tkachenko-Tkachuk-Wilson, TOPOLOGIES GENERATED BY DISCRETE SUBSPACES, Glasnik Mat., 2002 :

DEFINITION

A space X is discretely generated iff x ∈ A implies x ∈ D for some D ⊂ A discrete. X is weakly discretely generated iff A ⊂ X not closed implies D A for some D ⊂ A discrete.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 3 / 11

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DTTW

Dow-Tkachenko-Tkachuk-Wilson, TOPOLOGIES GENERATED BY DISCRETE SUBSPACES, Glasnik Mat., 2002 :

DEFINITION

A space X is discretely generated iff x ∈ A implies x ∈ D for some D ⊂ A discrete. X is weakly discretely generated iff A ⊂ X not closed implies D A for some D ⊂ A discrete. So, compact T2 spaces are weakly discretely generated.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 3 / 11

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DTTW

Dow-Tkachenko-Tkachuk-Wilson, TOPOLOGIES GENERATED BY DISCRETE SUBSPACES, Glasnik Mat., 2002 :

DEFINITION

A space X is discretely generated iff x ∈ A implies x ∈ D for some D ⊂ A discrete. X is weakly discretely generated iff A ⊂ X not closed implies D A for some D ⊂ A discrete. So, compact T2 spaces are weakly discretely generated. Also, countably tight compact T2 spaces are discretely generated.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 3 / 11

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DTTW

Dow-Tkachenko-Tkachuk-Wilson, TOPOLOGIES GENERATED BY DISCRETE SUBSPACES, Glasnik Mat., 2002 :

DEFINITION

A space X is discretely generated iff x ∈ A implies x ∈ D for some D ⊂ A discrete. X is weakly discretely generated iff A ⊂ X not closed implies D A for some D ⊂ A discrete. So, compact T2 spaces are weakly discretely generated. Also, countably tight compact T2 spaces are discretely generated.

EXAMPLE 1.

There is a compact T2 space which is not discretely generated

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 3 / 11

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DTTW

Dow-Tkachenko-Tkachuk-Wilson, TOPOLOGIES GENERATED BY DISCRETE SUBSPACES, Glasnik Mat., 2002 :

DEFINITION

A space X is discretely generated iff x ∈ A implies x ∈ D for some D ⊂ A discrete. X is weakly discretely generated iff A ⊂ X not closed implies D A for some D ⊂ A discrete. So, compact T2 spaces are weakly discretely generated. Also, countably tight compact T2 spaces are discretely generated.

EXAMPLE 1.

There is a compact T2 space which is not discretely generated (if there is an L-space).

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 3 / 11

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DTTW

Dow-Tkachenko-Tkachuk-Wilson, TOPOLOGIES GENERATED BY DISCRETE SUBSPACES, Glasnik Mat., 2002 :

DEFINITION

A space X is discretely generated iff x ∈ A implies x ∈ D for some D ⊂ A discrete. X is weakly discretely generated iff A ⊂ X not closed implies D A for some D ⊂ A discrete. So, compact T2 spaces are weakly discretely generated. Also, countably tight compact T2 spaces are discretely generated.

EXAMPLE 1.

There is a compact T2 space which is not discretely generated (if there is an L-space).

EXAMPLE 2.

Consistently, there is an ω-bounded (hence countably compact) regular space with a discretely untouchable point.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 3 / 11

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J-Shelah

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 4 / 11

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J-Shelah

THEOREM (J-Shelah)

For every cardinal κ, there is a κ-bounded 0-dimensional T2 space with a discretely untouchable point.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 4 / 11

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J-Shelah

THEOREM (J-Shelah)

For every cardinal κ, there is a κ-bounded 0-dimensional T2 space with a discretely untouchable point.

DEFINITION. Col(λ, κ) :

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 4 / 11

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J-Shelah

THEOREM (J-Shelah)

For every cardinal κ, there is a κ-bounded 0-dimensional T2 space with a discretely untouchable point.

DEFINITION. Col(λ, κ) : There is c : [λ]2 → 2 s.t.,

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 4 / 11

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J-Shelah

THEOREM (J-Shelah)

For every cardinal κ, there is a κ-bounded 0-dimensional T2 space with a discretely untouchable point.

DEFINITION. Col(λ, κ) : There is c : [λ]2 → 2 s.t., given ξ < κ+ and

h : ξ × ξ → 2,

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 4 / 11

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J-Shelah

THEOREM (J-Shelah)

For every cardinal κ, there is a κ-bounded 0-dimensional T2 space with a discretely untouchable point.

DEFINITION. Col(λ, κ) : There is c : [λ]2 → 2 s.t., given ξ < κ+ and

h : ξ × ξ → 2, for any disjoint {Aα : α < λ} ⊂ [λ]ξ we have α < β < λ s.t.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 4 / 11

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J-Shelah

THEOREM (J-Shelah)

For every cardinal κ, there is a κ-bounded 0-dimensional T2 space with a discretely untouchable point.

DEFINITION. Col(λ, κ) : There is c : [λ]2 → 2 s.t., given ξ < κ+ and

h : ξ × ξ → 2, for any disjoint {Aα : α < λ} ⊂ [λ]ξ we have α < β < λ s.t. c(aα,i, aβ,j) = h(i, j) for any i, j ∈ ξ × ξ.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 4 / 11

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J-Shelah

THEOREM (J-Shelah)

For every cardinal κ, there is a κ-bounded 0-dimensional T2 space with a discretely untouchable point.

DEFINITION. Col(λ, κ) : There is c : [λ]2 → 2 s.t., given ξ < κ+ and

h : ξ × ξ → 2, for any disjoint {Aα : α < λ} ⊂ [λ]ξ we have α < β < λ s.t. c(aα,i, aβ,j) = h(i, j) for any i, j ∈ ξ × ξ.

FACT. (Shelah) For any κ, if λ = (2κ)++ + ω4 then Col(λ, κ).

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 4 / 11

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J-Shelah

THEOREM (J-Shelah)

For every cardinal κ, there is a κ-bounded 0-dimensional T2 space with a discretely untouchable point.

DEFINITION. Col(λ, κ) : There is c : [λ]2 → 2 s.t., given ξ < κ+ and

h : ξ × ξ → 2, for any disjoint {Aα : α < λ} ⊂ [λ]ξ we have α < β < λ s.t. c(aα,i, aβ,j) = h(i, j) for any i, j ∈ ξ × ξ.

FACT. (Shelah) For any κ, if λ = (2κ)++ + ω4 then Col(λ, κ).

THEOREM (J-Shelah)

If λ = cf(λ) > κ+ and Col(λ, κ) holds

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 4 / 11

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J-Shelah

THEOREM (J-Shelah)

For every cardinal κ, there is a κ-bounded 0-dimensional T2 space with a discretely untouchable point.

DEFINITION. Col(λ, κ) : There is c : [λ]2 → 2 s.t., given ξ < κ+ and

h : ξ × ξ → 2, for any disjoint {Aα : α < λ} ⊂ [λ]ξ we have α < β < λ s.t. c(aα,i, aβ,j) = h(i, j) for any i, j ∈ ξ × ξ.

FACT. (Shelah) For any κ, if λ = (2κ)++ + ω4 then Col(λ, κ).

THEOREM (J-Shelah)

If λ = cf(λ) > κ+ and Col(λ, κ) holds then the Cantor cube Cλ has a dense κ-bounded subspace with a discretely untouchable point.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 4 / 11

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J-Shelah

THEOREM (J-Shelah)

For every cardinal κ, there is a κ-bounded 0-dimensional T2 space with a discretely untouchable point.

DEFINITION. Col(λ, κ) : There is c : [λ]2 → 2 s.t., given ξ < κ+ and

h : ξ × ξ → 2, for any disjoint {Aα : α < λ} ⊂ [λ]ξ we have α < β < λ s.t. c(aα,i, aβ,j) = h(i, j) for any i, j ∈ ξ × ξ.

FACT. (Shelah) For any κ, if λ = (2κ)++ + ω4 then Col(λ, κ).

THEOREM (J-Shelah)

If λ = cf(λ) > κ+ and Col(λ, κ) holds then the Cantor cube Cλ has a dense κ-bounded subspace with a discretely untouchable point.

FACT. (van Douwen) There is a countable, crowded, regular space in

which every point is discretely untouchable.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 4 / 11

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A PROBLEM

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 5 / 11

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A PROBLEM

TRIVIAL FACT.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 5 / 11

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A PROBLEM

TRIVIAL FACT. If λ = cf(λ) then the (ordered) space λ is (< λ)-bounded but not λ-bounded.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 5 / 11

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A PROBLEM

TRIVIAL FACT. If λ = cf(λ) then the (ordered) space λ is (< λ)-bounded but not λ-bounded.

PROBLEM

What if λ is singular?

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 5 / 11

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A PROBLEM

TRIVIAL FACT. If λ = cf(λ) then the (ordered) space λ is (< λ)-bounded but not λ-bounded.

PROBLEM

What if λ is singular?

THEOREM

If λ is singular and

sup{22µ : µ < λ}

cf(λ) < 22λ

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 5 / 11

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A PROBLEM

TRIVIAL FACT. If λ = cf(λ) then the (ordered) space λ is (< λ)-bounded but not λ-bounded.

PROBLEM

What if λ is singular?

THEOREM

If λ is singular and

sup{22µ : µ < λ}

cf(λ) < 22λ then the Cantor cube C2λ has a dense subspace that is (< λ)-bounded but not λ-bounded.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 5 / 11

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A PROBLEM

TRIVIAL FACT. If λ = cf(λ) then the (ordered) space λ is (< λ)-bounded but not λ-bounded.

PROBLEM

What if λ is singular?

THEOREM

If λ is singular and

sup{22µ : µ < λ}

cf(λ) < 22λ then the Cantor cube C2λ has a dense subspace that is (< λ)-bounded but not λ-bounded. In particular, this is so if λ is strong limit.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 5 / 11

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A PROBLEM

TRIVIAL FACT. If λ = cf(λ) then the (ordered) space λ is (< λ)-bounded but not λ-bounded.

PROBLEM

What if λ is singular?

THEOREM

If λ is singular and

sup{22µ : µ < λ}

cf(λ) < 22λ then the Cantor cube C2λ has a dense subspace that is (< λ)-bounded but not λ-bounded. In particular, this is so if λ is strong limit.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 5 / 11

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ωD-bounded 1.

From now on, all spaces are T1.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 6 / 11

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ωD-bounded 1.

From now on, all spaces are T1. ωD ≡ "countable discrete"

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 6 / 11

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ωD-bounded 1.

From now on, all spaces are T1. ωD ≡ "countable discrete" ω-bounded ⇒ ωD-bounded ⇒ countably compact

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 6 / 11

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ωD-bounded 1.

From now on, all spaces are T1. ωD ≡ "countable discrete" ω-bounded ⇒ ωD-bounded ⇒ countably compact SEPARATION 1.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 6 / 11

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ωD-bounded 1.

From now on, all spaces are T1. ωD ≡ "countable discrete" ω-bounded ⇒ ωD-bounded ⇒ countably compact SEPARATION 1. (i) (J. van Mill, 1982) There is a point p ∈ ω∗ that is ω-touchable but not ωD-touchable.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 6 / 11

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ωD-bounded 1.

From now on, all spaces are T1. ωD ≡ "countable discrete" ω-bounded ⇒ ωD-bounded ⇒ countably compact SEPARATION 1. (i) (J. van Mill, 1982) There is a point p ∈ ω∗ that is ω-touchable but not ωD-touchable. So, ω∗ \ {p} is ωD-bounded but not ω-bounded.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 6 / 11

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ωD-bounded 1.

From now on, all spaces are T1. ωD ≡ "countable discrete" ω-bounded ⇒ ωD-bounded ⇒ countably compact SEPARATION 1. (i) (J. van Mill, 1982) There is a point p ∈ ω∗ that is ω-touchable but not ωD-touchable. So, ω∗ \ {p} is ωD-bounded but not ω-bounded. (ii) (R. Hernandez-Gutierrez, 2013) A first countable, locally compact T2 example exists under CH.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 6 / 11

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ωD-bounded 1.

From now on, all spaces are T1. ωD ≡ "countable discrete" ω-bounded ⇒ ωD-bounded ⇒ countably compact SEPARATION 1. (i) (J. van Mill, 1982) There is a point p ∈ ω∗ that is ω-touchable but not ωD-touchable. So, ω∗ \ {p} is ωD-bounded but not ω-bounded. (ii) (R. Hernandez-Gutierrez, 2013) A first countable, locally compact T2 example exists under CH. SEPARATION 2.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 6 / 11

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ωD-bounded 1.

From now on, all spaces are T1. ωD ≡ "countable discrete" ω-bounded ⇒ ωD-bounded ⇒ countably compact SEPARATION 1. (i) (J. van Mill, 1982) There is a point p ∈ ω∗ that is ω-touchable but not ωD-touchable. So, ω∗ \ {p} is ωD-bounded but not ω-bounded. (ii) (R. Hernandez-Gutierrez, 2013) A first countable, locally compact T2 example exists under CH. SEPARATION 2. (i) Any countably compact infinite space in which all compact subspaces are finite.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 6 / 11

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ωD-bounded 1.

From now on, all spaces are T1. ωD ≡ "countable discrete" ω-bounded ⇒ ωD-bounded ⇒ countably compact SEPARATION 1. (i) (J. van Mill, 1982) There is a point p ∈ ω∗ that is ω-touchable but not ωD-touchable. So, ω∗ \ {p} is ωD-bounded but not ω-bounded. (ii) (R. Hernandez-Gutierrez, 2013) A first countable, locally compact T2 example exists under CH. SEPARATION 2. (i) Any countably compact infinite space in which all compact subspaces are finite. (ii) The Franklin-Rajagopalan space is locally compact T2 and sequentially compact but not ωD-bounded.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 6 / 11

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ωD-bounded 1.

From now on, all spaces are T1. ωD ≡ "countable discrete" ω-bounded ⇒ ωD-bounded ⇒ countably compact SEPARATION 1. (i) (J. van Mill, 1982) There is a point p ∈ ω∗ that is ω-touchable but not ωD-touchable. So, ω∗ \ {p} is ωD-bounded but not ω-bounded. (ii) (R. Hernandez-Gutierrez, 2013) A first countable, locally compact T2 example exists under CH. SEPARATION 2. (i) Any countably compact infinite space in which all compact subspaces are finite. (ii) The Franklin-Rajagopalan space is locally compact T2 and sequentially compact but not ωD-bounded. Under CH, it is first countable.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 6 / 11

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ωD-bounded 2.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 7 / 11

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ωD-bounded 2.

THEOREM (folklore ??)

If X is regular, countably compact, and L(X) < p then X is ω-bounded.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 7 / 11

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ωD-bounded 2.

THEOREM (folklore ??)

If X is regular, countably compact, and L(X) < p then X is ω-bounded.

THEOREM (J-Soukup-Szentmiklóssy)

If X is regular, ωD-bounded, and L(X) < cov(M) then X is ω-bounded.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 7 / 11

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ωD-bounded 2.

THEOREM (folklore ??)

If X is regular, countably compact, and L(X) < p then X is ω-bounded.

THEOREM (J-Soukup-Szentmiklóssy)

If X is regular, ωD-bounded, and L(X) < cov(M) then X is ω-bounded. κ < p ⇔ MAκ(σ − centered) ,

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 7 / 11

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ωD-bounded 2.

THEOREM (folklore ??)

If X is regular, countably compact, and L(X) < p then X is ω-bounded.

THEOREM (J-Soukup-Szentmiklóssy)

If X is regular, ωD-bounded, and L(X) < cov(M) then X is ω-bounded. κ < p ⇔ MAκ(σ − centered) , κ < cov(M) ⇔ MAκ(countable) .

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 7 / 11

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ωD-bounded 2.

THEOREM (folklore ??)

If X is regular, countably compact, and L(X) < p then X is ω-bounded.

THEOREM (J-Soukup-Szentmiklóssy)

If X is regular, ωD-bounded, and L(X) < cov(M) then X is ω-bounded. κ < p ⇔ MAκ(σ − centered) , κ < cov(M) ⇔ MAκ(countable) . So, p ≤ cov(M)

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 7 / 11

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ωD-bounded 2.

THEOREM (folklore ??)

If X is regular, countably compact, and L(X) < p then X is ω-bounded.

THEOREM (J-Soukup-Szentmiklóssy)

If X is regular, ωD-bounded, and L(X) < cov(M) then X is ω-bounded. κ < p ⇔ MAκ(σ − centered) , κ < cov(M) ⇔ MAκ(countable) . So, p ≤ cov(M) and p < cov(M) is consistent.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 7 / 11

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ωD-bounded 2.

THEOREM (folklore ??)

If X is regular, countably compact, and L(X) < p then X is ω-bounded.

THEOREM (J-Soukup-Szentmiklóssy)

If X is regular, ωD-bounded, and L(X) < cov(M) then X is ω-bounded. κ < p ⇔ MAκ(σ − centered) , κ < cov(M) ⇔ MAκ(countable) . So, p ≤ cov(M) and p < cov(M) is consistent.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 7 / 11

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ωD-bounded and countably tight

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 8 / 11

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ωD-bounded and countably tight

BIG OPEN PROBLEM:

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 8 / 11

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ωD-bounded and countably tight

BIG OPEN PROBLEM: Is it consistent that countably compact, first countable

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 8 / 11

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ωD-bounded and countably tight

BIG OPEN PROBLEM: Is it consistent that countably compact, first countable (or even countably tight), regular spaces are ω-bounded?

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 8 / 11

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ωD-bounded and countably tight

BIG OPEN PROBLEM: Is it consistent that countably compact, first countable (or even countably tight), regular spaces are ω-bounded?

THEOREM (J-Soukup-Szentmiklóssy)

(i) ωD-bounded and countably tight regular spaces are ωN-bounded.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 8 / 11

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ωD-bounded and countably tight

BIG OPEN PROBLEM: Is it consistent that countably compact, first countable (or even countably tight), regular spaces are ω-bounded?

THEOREM (J-Soukup-Szentmiklóssy)

(i) ωD-bounded and countably tight regular spaces are ωN-bounded. (ii) If b > ω1 then ωD-bounded and countably tight regular spaces are ω-bounded.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 8 / 11

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ωD-bounded and countably tight

BIG OPEN PROBLEM: Is it consistent that countably compact, first countable (or even countably tight), regular spaces are ω-bounded?

THEOREM (J-Soukup-Szentmiklóssy)

(i) ωD-bounded and countably tight regular spaces are ωN-bounded. (ii) If b > ω1 then ωD-bounded and countably tight regular spaces are ω-bounded.

NOTE. (i) There is an ωD-bounded but not ωN-bounded Tychonov

space.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 8 / 11

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ωD-bounded and countably tight

BIG OPEN PROBLEM: Is it consistent that countably compact, first countable (or even countably tight), regular spaces are ω-bounded?

THEOREM (J-Soukup-Szentmiklóssy)

(i) ωD-bounded and countably tight regular spaces are ωN-bounded. (ii) If b > ω1 then ωD-bounded and countably tight regular spaces are ω-bounded.

NOTE. (i) There is an ωD-bounded but not ωN-bounded Tychonov

space. (ii) If c = ω1 then ( by Hernandez-Gutierrez) there is a first countable, ωD-bounded but not ω-bounded, locally compact T2 space.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 8 / 11

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ωD-bounded and countably tight

BIG OPEN PROBLEM: Is it consistent that countably compact, first countable (or even countably tight), regular spaces are ω-bounded?

THEOREM (J-Soukup-Szentmiklóssy)

(i) ωD-bounded and countably tight regular spaces are ωN-bounded. (ii) If b > ω1 then ωD-bounded and countably tight regular spaces are ω-bounded.

NOTE. (i) There is an ωD-bounded but not ωN-bounded Tychonov

space. (ii) If c = ω1 then ( by Hernandez-Gutierrez) there is a first countable, ωD-bounded but not ω-bounded, locally compact T2 space.

PROBLEM

Does b = ω1 imply the existence of an ωD-bounded and countably tight (or first countable) regular space which is not ω-bounded?

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 8 / 11

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ωD-bounded and countably tight

BIG OPEN PROBLEM: Is it consistent that countably compact, first countable (or even countably tight), regular spaces are ω-bounded?

THEOREM (J-Soukup-Szentmiklóssy)

(i) ωD-bounded and countably tight regular spaces are ωN-bounded. (ii) If b > ω1 then ωD-bounded and countably tight regular spaces are ω-bounded.

NOTE. (i) There is an ωD-bounded but not ωN-bounded Tychonov

space. (ii) If c = ω1 then ( by Hernandez-Gutierrez) there is a first countable, ωD-bounded but not ω-bounded, locally compact T2 space.

PROBLEM

Does b = ω1 imply the existence of an ωD-bounded and countably tight (or first countable) regular space which is not ω-bounded?

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 8 / 11

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On the proof

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 9 / 11

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On the proof

DEFINITION. Let X be any space, U ⊂ τ(X) disjoint, S ⊂ ∪U dense.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 9 / 11

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On the proof

DEFINITION. Let X be any space, U ⊂ τ(X) disjoint, S ⊂ ∪U dense.

I(S, U) = {D ∈ [S]≤ω : ∀ U ∈ U

|D ∩ U| < ω
}

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 9 / 11

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On the proof

DEFINITION. Let X be any space, U ⊂ τ(X) disjoint, S ⊂ ∪U dense.

I(S, U) = {D ∈ [S]≤ω : ∀ U ∈ U

|D ∩ U| < ω
}

LEMMA

If X is regular then I(S, U) is a P-ideal.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 9 / 11

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On the proof

DEFINITION. Let X be any space, U ⊂ τ(X) disjoint, S ⊂ ∪U dense.

I(S, U) = {D ∈ [S]≤ω : ∀ U ∈ U

|D ∩ U| < ω
}

LEMMA

If X is regular then I(S, U) is a P-ideal.

THEOREM (J-Soukup-Szentmiklóssy)

Let X be regular, countably compact, and countably tight. Then for any countable A ⊂ ∪U \ ∪U there is D ∈ I(S, U) s.t. A ⊂ D.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 9 / 11

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On the proof

DEFINITION. Let X be any space, U ⊂ τ(X) disjoint, S ⊂ ∪U dense.

I(S, U) = {D ∈ [S]≤ω : ∀ U ∈ U

|D ∩ U| < ω
}

LEMMA

If X is regular then I(S, U) is a P-ideal.

THEOREM (J-Soukup-Szentmiklóssy)

Let X be regular, countably compact, and countably tight. Then for any countable A ⊂ ∪U \ ∪U there is D ∈ I(S, U) s.t. A ⊂ D. The proof of ωD-bounded ⇒ ωN-bounded easily follows.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 9 / 11

SLIDE 72

On the proof

DEFINITION. Let X be any space, U ⊂ τ(X) disjoint, S ⊂ ∪U dense.

I(S, U) = {D ∈ [S]≤ω : ∀ U ∈ U

|D ∩ U| < ω
}

LEMMA

If X is regular then I(S, U) is a P-ideal.

THEOREM (J-Soukup-Szentmiklóssy)

Let X be regular, countably compact, and countably tight. Then for any countable A ⊂ ∪U \ ∪U there is D ∈ I(S, U) s.t. A ⊂ D. The proof of ωD-bounded ⇒ ωN-bounded easily follows.

COROLLARY

Regular, countably compact, and countably tight spaces are discretely determined.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 9 / 11

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On the proof

DEFINITION. Let X be any space, U ⊂ τ(X) disjoint, S ⊂ ∪U dense.

I(S, U) = {D ∈ [S]≤ω : ∀ U ∈ U

|D ∩ U| < ω
}

LEMMA

If X is regular then I(S, U) is a P-ideal.

THEOREM (J-Soukup-Szentmiklóssy)

Let X be regular, countably compact, and countably tight. Then for any countable A ⊂ ∪U \ ∪U there is D ∈ I(S, U) s.t. A ⊂ D. The proof of ωD-bounded ⇒ ωN-bounded easily follows.

COROLLARY

Regular, countably compact, and countably tight spaces are discretely determined.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 9 / 11

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PRODUCTS

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 10 / 11

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PRODUCTS

ω-boundedness is fully productive,

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 10 / 11

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PRODUCTS

ω-boundedness is fully productive, countable compactness is not productive at all.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 10 / 11

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PRODUCTS

ω-boundedness is fully productive, countable compactness is not productive at all. From now on, all spaces are T2.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 10 / 11

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PRODUCTS

ω-boundedness is fully productive, countable compactness is not productive at all. From now on, all spaces are T2.

THEOREM (J-S-Sz)

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 10 / 11

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PRODUCTS

ω-boundedness is fully productive, countable compactness is not productive at all. From now on, all spaces are T2.

THEOREM (J-S-Sz)

If Π{Xi : i ∈ I} is ωD-bounded but not ω-bounded, then there is j ∈ I s.t. Xj is not ω-bounded and

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 10 / 11

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PRODUCTS

ω-boundedness is fully productive, countable compactness is not productive at all. From now on, all spaces are T2.

THEOREM (J-S-Sz)

If Π{Xi : i ∈ I} is ωD-bounded but not ω-bounded, then there is j ∈ I s.t. Xj is not ω-bounded and Π{Xi : i ∈ I \ {j}} is finite.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 10 / 11

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PRODUCTS

ω-boundedness is fully productive, countable compactness is not productive at all. From now on, all spaces are T2.

THEOREM (J-S-Sz)

If Π{Xi : i ∈ I} is ωD-bounded but not ω-bounded, then there is j ∈ I s.t. Xj is not ω-bounded and Π{Xi : i ∈ I \ {j}} is finite.

DEFINITION. X is weakly bounded iff for each A ∈ [X]ω there is

B ∈ [A]ω s.t. B is compact.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 10 / 11

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PRODUCTS

ω-boundedness is fully productive, countable compactness is not productive at all. From now on, all spaces are T2.

THEOREM (J-S-Sz)

If Π{Xi : i ∈ I} is ωD-bounded but not ω-bounded, then there is j ∈ I s.t. Xj is not ω-bounded and Π{Xi : i ∈ I \ {j}} is finite.

DEFINITION. X is weakly bounded iff for each A ∈ [X]ω there is

B ∈ [A]ω s.t. B is compact. ωD-bounded

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 10 / 11

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PRODUCTS

ω-boundedness is fully productive, countable compactness is not productive at all. From now on, all spaces are T2.

THEOREM (J-S-Sz)

If Π{Xi : i ∈ I} is ωD-bounded but not ω-bounded, then there is j ∈ I s.t. Xj is not ω-bounded and Π{Xi : i ∈ I \ {j}} is finite.

DEFINITION. X is weakly bounded iff for each A ∈ [X]ω there is

B ∈ [A]ω s.t. B is compact. ωD-bounded and sequentially compact spaces are both weakly bounded.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 10 / 11

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PRODUCTS

ω-boundedness is fully productive, countable compactness is not productive at all. From now on, all spaces are T2.

THEOREM (J-S-Sz)

If Π{Xi : i ∈ I} is ωD-bounded but not ω-bounded, then there is j ∈ I s.t. Xj is not ω-bounded and Π{Xi : i ∈ I \ {j}} is finite.

DEFINITION. X is weakly bounded iff for each A ∈ [X]ω there is

B ∈ [A]ω s.t. B is compact. ωD-bounded and sequentially compact spaces are both weakly

bounded. Weakly bounded spaces are countably compact.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 10 / 11

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PRODUCTS

ω-boundedness is fully productive, countable compactness is not productive at all. From now on, all spaces are T2.

THEOREM (J-S-Sz)

If Π{Xi : i ∈ I} is ωD-bounded but not ω-bounded, then there is j ∈ I s.t. Xj is not ω-bounded and Π{Xi : i ∈ I \ {j}} is finite.

DEFINITION. X is weakly bounded iff for each A ∈ [X]ω there is

B ∈ [A]ω s.t. B is compact. ωD-bounded and sequentially compact spaces are both weakly

bounded. Weakly bounded spaces are countably compact.

THEOREM (J-S-Sz)

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 10 / 11

SLIDE 86

PRODUCTS

ω-boundedness is fully productive, countable compactness is not productive at all. From now on, all spaces are T2.

THEOREM (J-S-Sz)

If Π{Xi : i ∈ I} is ωD-bounded but not ω-bounded, then there is j ∈ I s.t. Xj is not ω-bounded and Π{Xi : i ∈ I \ {j}} is finite.

DEFINITION. X is weakly bounded iff for each A ∈ [X]ω there is

B ∈ [A]ω s.t. B is compact. ωD-bounded and sequentially compact spaces are both weakly

bounded. Weakly bounded spaces are countably compact.

THEOREM (J-S-Sz)

(i) The product of < t weakly bounded spaces is weakly bounded.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 10 / 11

SLIDE 87

PRODUCTS

ω-boundedness is fully productive, countable compactness is not productive at all. From now on, all spaces are T2.

THEOREM (J-S-Sz)

If Π{Xi : i ∈ I} is ωD-bounded but not ω-bounded, then there is j ∈ I s.t. Xj is not ω-bounded and Π{Xi : i ∈ I \ {j}} is finite.

DEFINITION. X is weakly bounded iff for each A ∈ [X]ω there is

B ∈ [A]ω s.t. B is compact. ωD-bounded and sequentially compact spaces are both weakly

bounded. Weakly bounded spaces are countably compact.

THEOREM (J-S-Sz)

(i) The product of < t weakly bounded spaces is weakly bounded. (ii) The product of t weakly bounded spaces is countably compact.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 10 / 11

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PRODUCTS

ω-boundedness is fully productive, countable compactness is not productive at all. From now on, all spaces are T2.

THEOREM (J-S-Sz)

If Π{Xi : i ∈ I} is ωD-bounded but not ω-bounded, then there is j ∈ I s.t. Xj is not ω-bounded and Π{Xi : i ∈ I \ {j}} is finite.

DEFINITION. X is weakly bounded iff for each A ∈ [X]ω there is

B ∈ [A]ω s.t. B is compact. ωD-bounded and sequentially compact spaces are both weakly

bounded. Weakly bounded spaces are countably compact.

THEOREM (J-S-Sz)

(i) The product of < t weakly bounded spaces is weakly bounded. (ii) The product of t weakly bounded spaces is countably compact.

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 10 / 11

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István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 11 / 11

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THANK YOU FOR YOUR ATTENTION !

István Juhász (Rényi Institute) Discrete subspaces Novi Sad 2014 11 / 11