Compactness-like Covering Properties Petra Staynova Durham - - PowerPoint PPT Presentation

Compactness-like Covering Properties

Petra Staynova

Durham University

November 7, 2013

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 1 / 31

Something to think about...

[Look at the board]

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 2 / 31

Introduction

One of the main generalisations of compactness is the notion of an H-closed space:

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 3 / 31

Introduction

One of the main generalisations of compactness is the notion of an H-closed space:

Definition (H-closed, general)

A topological space X is said to be H-closed iff it is closed in every Hausdorff space containing it as a subspace.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 3 / 31

Introduction

One of the main generalisations of compactness is the notion of an H-closed space:

Definition (H-closed, general)

A topological space X is said to be H-closed iff it is closed in every Hausdorff space containing it as a subspace. A more useful definition is:

Definition (H-closed)

A topological space X is said to be H-closed iff every open cover has a finite subfamily with dense union.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 3 / 31

Countable generalisations

Another generalisation of compactness is the well-known Lindel¨

• f property:

Definition (Lindel¨

• f)

A topological space X is called Lindel¨

• f iff every open cover has a

countable subcover.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 4 / 31

Countable generalisations

Another generalisation of compactness is the well-known Lindel¨

• f property:

Definition (Lindel¨

• f)

A topological space X is called Lindel¨

• f iff every open cover has a

countable subcover.

Example (Lindel¨

• f spaces)

The following spaces are Lindel¨

• f:

Any countable topological space;

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 4 / 31

Countable generalisations

Another generalisation of compactness is the well-known Lindel¨

• f property:

Definition (Lindel¨

• f)

A topological space X is called Lindel¨

• f iff every open cover has a

countable subcover.

Example (Lindel¨

• f spaces)

The following spaces are Lindel¨

• f:

Any countable topological space; Any space with the co-countable topology;

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 4 / 31

Countable generalisations

Another generalisation of compactness is the well-known Lindel¨

• f property:

Definition (Lindel¨

• f)

A topological space X is called Lindel¨

• f iff every open cover has a

countable subcover.

Example (Lindel¨

• f spaces)

The following spaces are Lindel¨

• f:

Any countable topological space; Any space with the co-countable topology; A countable union of compact spaces;

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 4 / 31

Countable generalisations

Another generalisation of compactness is the well-known Lindel¨

• f property:

Definition (Lindel¨

• f)

A topological space X is called Lindel¨

• f iff every open cover has a

countable subcover.

Example (Lindel¨

• f spaces)

The following spaces are Lindel¨

• f:

Any countable topological space; Any space with the co-countable topology; A countable union of compact spaces; R, with the Euclidean topology.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 4 / 31

Countable generalisations

Another generalisation of compactness is the well-known Lindel¨

• f property:

Definition (Lindel¨

• f)

A topological space X is called Lindel¨

• f iff every open cover has a

countable subcover.

Example (Lindel¨

• f spaces)

The following spaces are Lindel¨

• f:

Any countable topological space; Any space with the co-countable topology; A countable union of compact spaces; R, with the Euclidean topology. The Sorgenfrey line S (R with the topology generated by the base B = {[a, b) : a < b}).

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 4 / 31

Countable generalisations, cont’d

In 1959, Zdenek Frolik introduced a notion that combines the Lindel¨

• f and

H-closed properties:

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 5 / 31

Countable generalisations, cont’d

In 1959, Zdenek Frolik introduced a notion that combines the Lindel¨

• f and

H-closed properties:

Definition (weakly Lindel¨

• f , [Fro59])

A topological space X is weakly Lindel¨

• f if for every open cover U of X

there is a countable subfamily U′ ⊆ U such that X = U′.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 5 / 31

Countable generalisations, cont’d

In 1959, Zdenek Frolik introduced a notion that combines the Lindel¨

• f and

H-closed properties:

Definition (weakly Lindel¨

• f , [Fro59])

A topological space X is weakly Lindel¨

• f if for every open cover U of X

there is a countable subfamily U′ ⊆ U such that X = U′. Later on, while studying cardinal invariants, Dissanayeke and Willard introduced another generalisation of the Lindel¨

• f property, which is

stronger than the weakly Lindel¨

• f one:

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 5 / 31

Countable generalisations, cont’d

In 1959, Zdenek Frolik introduced a notion that combines the Lindel¨

• f and

H-closed properties:

Definition (weakly Lindel¨

• f , [Fro59])

A topological space X is weakly Lindel¨

• f if for every open cover U of X

there is a countable subfamily U′ ⊆ U such that X = U′. Later on, while studying cardinal invariants, Dissanayeke and Willard introduced another generalisation of the Lindel¨

• f property, which is

stronger than the weakly Lindel¨

• f one:

Definition (almost Lindel¨

• f , [WD84])

A topological space X is almost Lindel¨

• f iff for every open cover U of X

there is a countable subfamily U′ ⊆ U such that X = {U : U ∈ U′}.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 5 / 31

Generalisations of compactness: a diagram

keep finiteness keep requirement for cover compact H-closed Lindelof almost Lindelof weakly Lindelof

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 6 / 31

Separation Axioms

= T={[a, infty) for a in R} - topology

• n [0,

infty) Urysohn T0 T1 Hausdorff Regular Functionally Regular Normal Functionally Hausdorff Tychonoff More complex +locallly compact

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 7 / 31

Properties of the generalisations

Proposition

Every regular Hausdorff H-closed space is compact. Every regular Lindel¨

• f space is normal.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 8 / 31

Properties of the generalisations

Proposition

Every regular Hausdorff H-closed space is compact. Every regular Lindel¨

• f space is normal.

Proposition

A regular almost Lindel¨

• f space is Lindel¨
• f.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 8 / 31

Properties of the generalisations

Proposition

Every regular Hausdorff H-closed space is compact. Every regular Lindel¨

• f space is normal.

Proposition

A regular almost Lindel¨

• f space is Lindel¨
• f.

Proposition

A normal weakly Lindel¨

• f space is almost Lindel¨
• f.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 8 / 31

Properties of the generalisations

Proposition

The continuous image of an almost Lindel¨

• f (weakly Lindel¨
• f) space is

almost Lindel¨

• f (weakly Lindel¨
• f).

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 9 / 31

Properties of the generalisations

Proposition

The continuous image of an almost Lindel¨

• f (weakly Lindel¨
• f) space is

almost Lindel¨

• f (weakly Lindel¨
• f).

Proposition

A clopen subset of an almost Lindel¨

• f (weakly Lindel¨
• f) space is almost

Lindel¨

• f (weakly Lindel¨
• f).

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 9 / 31

Properties of the generalisations

Proposition

The continuous image of an almost Lindel¨

• f (weakly Lindel¨
• f) space is

almost Lindel¨

• f (weakly Lindel¨
• f).

Proposition

A clopen subset of an almost Lindel¨

• f (weakly Lindel¨
• f) space is almost

Lindel¨

• f (weakly Lindel¨
• f).

Proposition

If X is almost Lindel¨

• f (weakly Lindel¨
• f) and Y is compact, then X × Y is

almost Lindel¨

• f (weakly Lindel¨
• f).

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 9 / 31

Inheritance

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 10 / 31

A modification

Definition (quasi-Lindel¨

• f , [Arh79])

We call a space X quasi-Lindel¨

• f if for every closed subset Y of X and

every collection U of open in X sets such that Y ⊆ U, there is a countable subfamily U′ ⊆ U such that Y ⊂ U′.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 11 / 31

A modification

Definition (quasi-Lindel¨

• f , [Arh79])

We call a space X quasi-Lindel¨

• f if for every closed subset Y of X and

every collection U of open in X sets such that Y ⊆ U, there is a countable subfamily U′ ⊆ U such that Y ⊂ U′.

Proposition ([Sta12])

Let X be a topological space. The following are equivalent:

1 X is quasi-Lindel¨

• f.

2 Let B be a fixed base for X. Then for any closed subset C ⊂ X and

any cover U of C with U ⊂ B there is a countable subfamily U′ of U such that C ⊂ U′.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 11 / 31

weakly Lindel¨

• f not quasi-Lindel¨
• f example

We modify an example from Song and Zhang [SZ10] and use ideas from Mysior [Mys81] to get:

Example

There exists a Urysohn weakly Lindel¨

• f not quasi-Lindel¨
• f space.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 12 / 31

weakly Lindel¨

• f not quasi-Lindel¨
• f example

We modify an example from Song and Zhang [SZ10] and use ideas from Mysior [Mys81] to get:

Example

There exists a Urysohn weakly Lindel¨

• f not quasi-Lindel¨
• f space.

(the original example was of a Urysohn almost Lindel¨

• f space which is not

Lindel¨

• f)

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 12 / 31

Weakly Lindel¨

• f not quasi-Lindel¨
• f example

Construction

Let A = {(aα, −1) : α < ω1} be an ω1-long sequence in the set {(x, −1) : x 0} ⊆ R2. Let Y = {(aα, n) : α < ω1, n ∈ ω}. Let a = (−1, −1). Finally let X = Y ∪ A ∪ {a}. We topologize X as follows:

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 13 / 31

Weakly Lindel¨

• f not quasi-Lindel¨
• f example

Construction

Let A = {(aα, −1) : α < ω1} be an ω1-long sequence in the set {(x, −1) : x 0} ⊆ R2. Let Y = {(aα, n) : α < ω1, n ∈ ω}. Let a = (−1, −1). Finally let X = Y ∪ A ∪ {a}. We topologize X as follows:

• all points in Y are isolated;

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 13 / 31

Weakly Lindel¨

• f not quasi-Lindel¨
• f example

Construction

Let A = {(aα, −1) : α < ω1} be an ω1-long sequence in the set {(x, −1) : x 0} ⊆ R2. Let Y = {(aα, n) : α < ω1, n ∈ ω}. Let a = (−1, −1). Finally let X = Y ∪ A ∪ {a}. We topologize X as follows:

• all points in Y are isolated;
• for α < ω1 the basic neighborhoods of (aα, −1) will be of the form

Un(aα, −1) = {(aα, −1)} ∪ {(aα, m) : m n} for n ∈ ω

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 13 / 31

Weakly Lindel¨

• f not quasi-Lindel¨
• f example

Construction

Let A = {(aα, −1) : α < ω1} be an ω1-long sequence in the set {(x, −1) : x 0} ⊆ R2. Let Y = {(aα, n) : α < ω1, n ∈ ω}. Let a = (−1, −1). Finally let X = Y ∪ A ∪ {a}. We topologize X as follows:

• all points in Y are isolated;
• for α < ω1 the basic neighborhoods of (aα, −1) will be of the form

Un(aα, −1) = {(aα, −1)} ∪ {(aα, m) : m n} for n ∈ ω

• the basic neighborhoods of a = (−1, −1) are of the form

Uα(a) = {a} ∪ {(aβ, n) : β > α, n ∈ ω} for α < ω1.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 13 / 31

Weakly Lindel¨

• f not quasi-Lindel¨
• f example

Let us point out that

Claim

The subset A ⊂ X is closed and discrete in this topology.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 14 / 31

Weakly Lindel¨

• f not quasi-Lindel¨
• f example

Let us point out that

Claim

The subset A ⊂ X is closed and discrete in this topology. Indeed, for any point x ∈ X there is a basic neighborhood U(x) such that A ∩ U(x) contains at most one point and also that X \ A = {a} ∪ Y is

• pen (because Uα(a) ⊂ Y ∪ {a}). Hence X contains an uncontable closed

discrete subset and therefore it cannot be Lindel¨

• f.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 14 / 31

Weakly Lindel¨

• f not quasi-Lindel¨
• f example

Let us point out that

Claim

The subset A ⊂ X is closed and discrete in this topology. Indeed, for any point x ∈ X there is a basic neighborhood U(x) such that A ∩ U(x) contains at most one point and also that X \ A = {a} ∪ Y is

• pen (because Uα(a) ⊂ Y ∪ {a}). Hence X contains an uncontable closed

discrete subset and therefore it cannot be Lindel¨

• f.

Also,

Note

Note that for any open U ∋ a the set X \ U is at most countable.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 14 / 31

Weakly Lindel¨

• f not quasi-Lindel¨
• f example

Let us point out that

Claim

The subset A ⊂ X is closed and discrete in this topology. Indeed, for any point x ∈ X there is a basic neighborhood U(x) such that A ∩ U(x) contains at most one point and also that X \ A = {a} ∪ Y is

• pen (because Uα(a) ⊂ Y ∪ {a}). Hence X contains an uncontable closed

discrete subset and therefore it cannot be Lindel¨

• f.

Also,

Note

Note that for any open U ∋ a the set X \ U is at most countable. Indeed, for any α < ω1, Uα(a) = Uα(a) ∪ {(aβ, −1) : β > α}. Hence X \ Uα(a) is at most countable.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 14 / 31

Weakly Lindel¨

• f not quasi-Lindel¨
• f example

Let us point out that

Claim

The subset A ⊂ X is closed and discrete in this topology. Indeed, for any point x ∈ X there is a basic neighborhood U(x) such that A ∩ U(x) contains at most one point and also that X \ A = {a} ∪ Y is

• pen (because Uα(a) ⊂ Y ∪ {a}). Hence X contains an uncontable closed

discrete subset and therefore it cannot be Lindel¨

• f.

Also,

Note

Note that for any open U ∋ a the set X \ U is at most countable. Indeed, for any α < ω1, Uα(a) = Uα(a) ∪ {(aβ, −1) : β > α}. Hence X \ Uα(a) is at most countable. It is easily seen that X is Hausdorff. With a bit more effort, it can also be proven that X is Urysohn.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 14 / 31

Weakly Lindel¨

• f not quasi-Lindel¨
• f example

Claim

The space X is weakly Lindel¨

• f.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 15 / 31

Weakly Lindel¨

• f not quasi-Lindel¨
• f example

Claim

The space X is weakly Lindel¨

• f.

Let U be an open cover of X. Then there exists a U(a) ∈ U such that a ∈ U(a). We can find a basic neighborhood Uβ(a) ⊂ U(a). Then Uβ(a) ⊂ U(a) and hence X \ U(a) will also be at most countable.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 15 / 31

Weakly Lindel¨

• f not quasi-Lindel¨
• f example

Claim

The space X is weakly Lindel¨

• f.

Let U be an open cover of X. Then there exists a U(a) ∈ U such that a ∈ U(a). We can find a basic neighborhood Uβ(a) ⊂ U(a). Then Uβ(a) ⊂ U(a) and hence X \ U(a) will also be at most countable. Hence X \ U(a) can be covered by (at most) countably many elements of U, say U∗. Set U′ = U∗ ∪ {U(a)}. Then, X ⊆

U∈U′ U ⊆ U∈U′ U. Therefore,

X is weakly Lindel¨

• f.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 15 / 31

Weakly Lindel¨

• f not quasi-Lindel¨
• f example

Claim

The space X is weakly Lindel¨

• f.

Let U be an open cover of X. Then there exists a U(a) ∈ U such that a ∈ U(a). We can find a basic neighborhood Uβ(a) ⊂ U(a). Then Uβ(a) ⊂ U(a) and hence X \ U(a) will also be at most countable. Hence X \ U(a) can be covered by (at most) countably many elements of U, say U∗. Set U′ = U∗ ∪ {U(a)}. Then, X ⊆

U∈U′ U ⊆ U∈U′ U. Therefore,

X is weakly Lindel¨

• f.

Note

In fact, this shows that X is even almost Lindel¨

• f.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 15 / 31

Weakly Lindel¨

• f not quasi-Lindel¨
• f example

Claim

The space X is not quasi-Lindel¨

• f.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 16 / 31

Weakly Lindel¨

• f not quasi-Lindel¨
• f example

Claim

The space X is not quasi-Lindel¨

• f.

Consider the 1-neighborhood of a: U1(a) = {a} ∪ {(aβ, n) : ω1 > β > 1, n ∈ ω}. We have that C = X \ U1(a) is closed.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 16 / 31

Weakly Lindel¨

• f not quasi-Lindel¨
• f example

Claim

The space X is not quasi-Lindel¨

• f.

Consider the 1-neighborhood of a: U1(a) = {a} ∪ {(aβ, n) : ω1 > β > 1, n ∈ ω}. We have that C = X \ U1(a) is closed. We show the uncountable family of basic open sets U = {U0(aα, −1) : α < ω1} forms an open cover of C which has no countable subcover with dense union.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 16 / 31

Weakly Lindel¨

• f not quasi-Lindel¨
• f example

Claim

The space X is not quasi-Lindel¨

• f.

Consider the 1-neighborhood of a: U1(a) = {a} ∪ {(aβ, n) : ω1 > β > 1, n ∈ ω}. We have that C = X \ U1(a) is closed. We show the uncountable family of basic open sets U = {U0(aα, −1) : α < ω1} forms an open cover of C which has no countable subcover with dense union. Note that the sets U0(aα, −1) are closed and open. Indeed, X \ U0(aα, −1) = {U0(aβ, −1) : ω1 > β = α} ∪ Uα+1(a). Hence, if we remove even one of the U0(aα, −1), the point (aα, −1) would remain

• uncovered. Therefore, X is not quasi-Lindel¨
• f.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 16 / 31

Another weakly Lindel¨

• f not quasi-Lindel¨
• f example

A shorter example shows that even regularity is not strong enough to make a weakly Lindel¨

• f space quasi-Lindel¨
• f :

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 17 / 31

Another weakly Lindel¨

• f not quasi-Lindel¨
• f example

A shorter example shows that even regularity is not strong enough to make a weakly Lindel¨

• f space quasi-Lindel¨
• f :

Example

The Sorgenfrey plane is weakly Lindel¨

• f but it is not quasi-Lindel¨
• f.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 17 / 31

The quasi-Lindelof property

Theorem

Every separable topological space X is quasi-Lindel¨

• f.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 18 / 31

The quasi-Lindelof property

Theorem

Every separable topological space X is quasi-Lindel¨

• f.

Definition (ccc)

A topological space X satisfies the countable chain condition if every family of non-empty disjoint open subsets of X is countable.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 18 / 31

The quasi-Lindelof property

Theorem

Every separable topological space X is quasi-Lindel¨

• f.

Definition (ccc)

A topological space X satisfies the countable chain condition if every family of non-empty disjoint open subsets of X is countable. Arhangelskii stated without proof that:

Theorem ([Arh79])

Every ccc space is weakly-Lindel¨

• f.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 18 / 31

The quasi-Lindelof property

Theorem

Every separable topological space X is quasi-Lindel¨

• f.

Definition (ccc)

A topological space X satisfies the countable chain condition if every family of non-empty disjoint open subsets of X is countable. Arhangelskii stated without proof that:

Theorem ([Arh79])

Every ccc space is weakly-Lindel¨

• f.

In my 3d year project, I proved that:

Theorem ([Sta11])

Every ccc space is quasi-Lindel¨

• f.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 18 / 31

ccc implies quasi-Lindel¨

• f - the proof

Suppose that X is CCC but not quasi-Lindel¨

• f. Then there exists a closed

nonempty set F ⊂ X and an uncountable family Γ = {Uα : α < β} (β ω1) of non-empty open in X sets such that F ⊂

• α<β

Uα, but for any countable Γ′ ⊂ Γ we have that F \ Γ′ = ∅.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 19 / 31

ccc implies quasi-Lindel¨

• f - the proof

Suppose that X is CCC but not quasi-Lindel¨

• f. Then there exists a closed

nonempty set F ⊂ X and an uncountable family Γ = {Uα : α < β} (β ω1) of non-empty open in X sets such that F ⊂

• α<β

Uα, but for any countable Γ′ ⊂ Γ we have that F \ Γ′ = ∅. We will construct an uncountable collection of nonempty disjoint open in X sets {Vγ : γ < ω1}, thus contradicting CCC.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 19 / 31

ccc implies quasi-Lindel¨

• f - the proof

Suppose that X is CCC but not quasi-Lindel¨

• f. Then there exists a closed

nonempty set F ⊂ X and an uncountable family Γ = {Uα : α < β} (β ω1) of non-empty open in X sets such that F ⊂

• α<β

Uα, but for any countable Γ′ ⊂ Γ we have that F \ Γ′ = ∅. We will construct an uncountable collection of nonempty disjoint open in X sets {Vγ : γ < ω1}, thus contradicting CCC. Let V0 = U0. Then F \ U0 = ∅ (and X \ U0 = ∅). Hence ∅ = F \ U0 ⊂

• {Uα : α < β, α > 0}

and therefore ∅ = F \ U0 =

• {Uα ∩ (F \ U0) : α < β, α > 0}.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 19 / 31

ccc implies quasi-Lindel¨

• f - the proof cont’d

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 20 / 31

ccc implies quasi-Lindel¨

• f - the proof cont’d

Thus we will have α1 1 such that Uα1 ∩ (F \ U0) = ∅ and moreover Uα1 ∩ (X \ U0) = ∅.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 20 / 31

ccc implies quasi-Lindel¨

• f - the proof cont’d

Thus we will have α1 1 such that Uα1 ∩ (F \ U0) = ∅ and moreover Uα1 ∩ (X \ U0) = ∅. Let V1 = Uα1 ∩ (X \ U0). Then V1 = ∅, V1 is open in X and V1 ∩ V0 = ∅. Again we have ∅ = F \ U0 ∪ Uα1 =

• {Uα ∩ (F \ U0 ∪ Uα1); α < β, α /

∈ {0, α1}}. Hence there is Uα2 ∈ Γ, α1 / ∈ {0, α1} with Uα2 ∩ (F \ U0 ∪ Uα1) = ∅ and moreover Uα2 ∩ (X \ U0 ∪ Uα1) = ∅.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 20 / 31

ccc implies quasi-Lindel¨

• f - the proof cont’d

Thus we will have α1 1 such that Uα1 ∩ (F \ U0) = ∅ and moreover Uα1 ∩ (X \ U0) = ∅. Let V1 = Uα1 ∩ (X \ U0). Then V1 = ∅, V1 is open in X and V1 ∩ V0 = ∅. Again we have ∅ = F \ U0 ∪ Uα1 =

• {Uα ∩ (F \ U0 ∪ Uα1); α < β, α /

∈ {0, α1}}. Hence there is Uα2 ∈ Γ, α1 / ∈ {0, α1} with Uα2 ∩ (F \ U0 ∪ Uα1) = ∅ and moreover Uα2 ∩ (X \ U0 ∪ Uα1) = ∅. Define V2 = Uα2 ∩ (X \ U0 ∪ Uα1). Then V2 = ∅, V2 is open in X and V2 is disjoint from V0, V1. Let γ0 < ω1 and suppose that we have already constructed a family {Vδ : δ < γ0} of non-empty, disjoint open in X sets with Vδ = Uαδ ∩ (X \ ∪{Uασ : σ < δ}), where αδ / ∈ {ασ : σ < δ}.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 20 / 31

ccc implies quasi-Lindel¨

• f - the proof cont’d

Since γ0 is a countable ordinal and F is not weakly Lindel¨

• f in X, we have

that ∅ = F \ ∪{Uαδ : δ < γ0} =

• {Uα ∩ (F \ ∪{Uαδ : δ < γ0}) : α < δ, α /

∈ {αδ : δ < γ0}}.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 21 / 31

ccc implies quasi-Lindel¨

• f - the proof cont’d

Since γ0 is a countable ordinal and F is not weakly Lindel¨

• f in X, we have

that ∅ = F \ ∪{Uαδ : δ < γ0} =

• {Uα ∩ (F \ ∪{Uαδ : δ < γ0}) : α < δ, α /

∈ {αδ : δ < γ0}}. Hence we can choose αγ0 such that Uαγ0 ∩ (F \ ∪{Uαδ : δ < γ0}) = ∅ and αγ0 / ∈ {αδ : δ < γ0}.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 21 / 31

ccc implies quasi-Lindel¨

• f - the proof cont’d

Since γ0 is a countable ordinal and F is not weakly Lindel¨

• f in X, we have

that ∅ = F \ ∪{Uαδ : δ < γ0} =

• {Uα ∩ (F \ ∪{Uαδ : δ < γ0}) : α < δ, α /

∈ {αδ : δ < γ0}}. Hence we can choose αγ0 such that Uαγ0 ∩ (F \ ∪{Uαδ : δ < γ0}) = ∅ and αγ0 / ∈ {αδ : δ < γ0}. Hence moreover Uαγ0 ∩ (X \ ∪{Uαδ : δ < γ0}) = ∅.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 21 / 31

ccc implies quasi-Lindel¨

• f - the proof cont’d

Define Vγ0 = Uαγ0 ∩ (X \ ∪{Uαδ : δ < γ0}).

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 22 / 31

ccc implies quasi-Lindel¨

• f - the proof cont’d

Define Vγ0 = Uαγ0 ∩ (X \ ∪{Uαδ : δ < γ0}). Then Vγ0 = ∅, Vγ0 is open in X and by construction Vγ0 ∩ Vδ = ∅ for every δ < γ0.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 22 / 31

ccc implies quasi-Lindel¨

• f - the proof cont’d

Define Vγ0 = Uαγ0 ∩ (X \ ∪{Uαδ : δ < γ0}). Then Vγ0 = ∅, Vγ0 is open in X and by construction Vγ0 ∩ Vδ = ∅ for every δ < γ0. Thus we have constructed a family {Vγ : γ < ω1} of nonempty disjoint

• pen in X sets, contradicting CCC.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 22 / 31

ccc implies quasi-Lindel¨

• f - the proof cont’d

Define Vγ0 = Uαγ0 ∩ (X \ ∪{Uαδ : δ < γ0}). Then Vγ0 = ∅, Vγ0 is open in X and by construction Vγ0 ∩ Vδ = ∅ for every δ < γ0. Thus we have constructed a family {Vγ : γ < ω1} of nonempty disjoint

• pen in X sets, contradicting CCC.

Hence, X is quasi-Lindel¨

• f.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 22 / 31

What about the converse?

Though we have that ccc implies quasi-Lindel¨

• f, the converse is false:

Example

The lexicographic square is quasi-Lindel¨

• f, but is not ccc.

The quasi-Lindel¨

• f property follows from the fact that the lexicographic

square is compact, that it is not ccc can be found in [SS96].

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 23 / 31

Relations between Lindel¨

• f-type covering properties

+ normal = implies implies + regular = + regular Misra, Kilicman & Fawakhreh Kilicman & Fawakhreh Kilicman & Fawakhreh Kilicman & Fawakhreh Kilicman & Fawakhreh Kilicman & Fawakhreh Lindelof Products Subspaces quotient spaces mappings Almost Lindelof Products Subspaces quotient spaces mappings Weakly Lindelof Products Subspaces quotient spaces mappings quasi- Lindelof Products quotient spaces mappings compact Lindelof continuous Open Closed Clopen regular Subspaces Open? Closed Clopen? regular? continuous continuous continuous ccc compact clopen regularly closed +regular does not = +Hausdorff + 1st ctble not= compact clopen + Urysohn not = normal + regular not = +Hausdorff + 1st ctble not= +regular does not = + Urysohn not = Lindelof P- space nearly compact nearly compact almost Lindelof weak P-space weakly Lindelof weak P-space +Tychonoff +CCC not = regularly closed? weakly Lindelof weak P-space + normal = closed? Lindelof P-space P-space - T1 and countable intersections

• f open sets are open (ie

G-delta sets are open) + Urysohn... combine Mysior and Song&Zhang + regular... Song& Zhang (not proven but can be) +regular does n... Sorgenfrey plane +Hausdorff... BGW - I've proved this +Tychonoff... Song&Zhang Every CCC, non Lindelof Tychonoff space eg - Sigma-product in 2^\kappa +Hausdorff... Bell-Ginsburgh-Woods + Urysohn... combine Mysior and Song&Zhang

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 24 / 31

Open questions - preliminaries

The famous Tychonoff’s Theorem states that an arbitrary product of compact spaces is compact. However, for Lindel¨

• f spaces, this fails even in

the finite case, as exemplified by the Sorgenfrey plane.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 25 / 31

Open questions - preliminaries

The famous Tychonoff’s Theorem states that an arbitrary product of compact spaces is compact. However, for Lindel¨

• f spaces, this fails even in

the finite case, as exemplified by the Sorgenfrey plane. In [TAAJ11] Frank Tall introduced the productively Lindel¨

• f property:

Definition (productively Lindel¨

• f)

A space X is called productively Lindel¨

• f if for every Lindel¨
• f space Y ,

the product X × Y is Lindel¨

• f.

It is well-known that compact spaces are productively Lindel¨

• f; Tall proved

that under certain additional axioms, there exist other productively Lindel¨

• f properties.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 25 / 31

Open questions - preliminaries

Similarly, we can define productively weakly-Lindel¨

• f:

Definition ((PS) productively weakly Lindel¨

• f)

A space X is called productively weakly Lindel¨

• f if for every weakly

Lindel¨

• f space Y , the product X × Y is weakly Lindel¨
• f.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 26 / 31

Open questions - preliminaries

Similarly, we can define productively weakly-Lindel¨

• f:

Definition ((PS) productively weakly Lindel¨

• f)

A space X is called productively weakly Lindel¨

• f if for every weakly

Lindel¨

• f space Y , the product X × Y is weakly Lindel¨
• f.

Proposition

If X is weakly Lindel¨

• f and Y is compact, then X × Y is weakly Lindel¨
• f.

This was stated by Song and Zhang in [SZ10], a proof can be found in [Sta11].

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 26 / 31

Open questions - continued

We can also define:

Definition ((PS) productively quasi-Lindel¨

• f )

A space X is called productively quasi-Lindel¨

• f if for every quasi-Lindel¨
• f

space Y , the product X × Y is quasi-Lindel¨

• f.

Thus, it is natural to ask:

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 27 / 31

Open questions - continued

We can also define:

Definition ((PS) productively quasi-Lindel¨

• f )

A space X is called productively quasi-Lindel¨

• f if for every quasi-Lindel¨
• f

space Y , the product X × Y is quasi-Lindel¨

• f.

Thus, it is natural to ask:

Open Question ([Sta12])

Are compact spaces productively quasi-Lindel¨

• f?

This question is interesting even in the partial case:

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 27 / 31

Open questions - continued

We can also define:

Definition ((PS) productively quasi-Lindel¨

• f )

A space X is called productively quasi-Lindel¨

• f if for every quasi-Lindel¨
• f

space Y , the product X × Y is quasi-Lindel¨

• f.

Thus, it is natural to ask:

Open Question ([Sta12])

Are compact spaces productively quasi-Lindel¨

• f?

This question is interesting even in the partial case:

Open Question ([Sta12])

Is the product of the unit interval [0, 1] with a quasi-Lindel¨

• f space,

quasi-Lindel¨

• f?

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 27 / 31

Open questions, cont’d

We know that there is a regular weakly Lindel¨

• f space that is not

quasi-Lindel¨

• f, and also that every normal weakly Lindel¨
• f space is

quasi-Lindel¨

• f.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 28 / 31

Open questions, cont’d

We know that there is a regular weakly Lindel¨

• f space that is not

quasi-Lindel¨

• f, and also that every normal weakly Lindel¨
• f space is

quasi-Lindel¨

• f. Hence, it makes sense to ask:

Open Question

In completely regular spaces, do the weakly Lindel¨

• f and quasi-Lindel¨
• f

properties coincide?

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 28 / 31

Bibliography I

• A. Arhangelskii.

A theorem on cardinality.

• Russ. Math. Surv., 34(153), 1979.
• Z. Frolik.

Generalizations of compact and Lindel¨

• f spaces.
• Czech. Math. J., 9, 1959.
• A. Mysior.

A regular space which is not completely regular. Proceedings of the American Mathematical Society, 81(4), 1981. Lynn A. Steen and J. Arthur Seebach. Counterexamples in Topology. Dover Publications Inc., new edition, 1996.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 29 / 31

Bibliography II

Petra Staynova. A Comparison of Lindel¨

• f-type Covering Properties of Topological

Spaces. Rose-Hulman Undergraduate Journal of Mathematics, 12(2), 2011. Petra Staynova. A Note on Quasi-Lindel¨

• f spaces.

Proceedings of the 41st Spring Conference of the Union of Bulgarian Mathematicians, 2012.

• Y. Song and Y. Zhang.

Some remarks on almost Lindel¨

• f spaces and weakly Lindel¨
• f spaces.

Matematicheskii Vestnik, 62, 2010. Franklin Tall, O. T. Alas, L. F. Aurichi, and L. R. Junqueira. Non-productively Lindel¨

• f spaces and small cardinals.

Houston Journal of Mathematics, 37:1373–1381, 2011.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 30 / 31

Bibliography III

• S. Willard and U. N. B. Dissanayake.

The almost Lindel¨

• f degree.
• Canad. Math. Bull., 27(4), 1984.

Petra Staynova (Durham University) Compactness-like Covering Properties November 7, 2013 31 / 31