IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

The 17th topological type of ω∗

(a study in nonhomogeneity)

Jonathan L. Verner

Department of Mathematical SciencesUniversity of North Carolina at Greensboro

April 17th, Thesis Defence

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Outline

1 IntroductionBasic definitionsNonhomogeneity of ω∗, a historical overview

2 ω-uniquely accessible pointsDefinitionVan Mill’s methodConstructing spaces with ω-uniquely accessible points

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Outline

1 IntroductionBasic definitionsNonhomogeneity of ω∗, a historical overview

2 ω-uniquely accessible pointsDefinitionVan Mill’s methodConstructing spaces with ω-uniquely accessible points

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Outline

1 IntroductionBasic definitionsNonhomogeneity of ω∗, a historical overview

2 ω-uniquely accessible pointsDefinitionVan Mill’s methodConstructing spaces with ω-uniquely accessible points

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Homogeneity of topological spaces

DefinitionA topological space is homogeneous if for any two points thereis a homeomorphism taking one to the other.

The real line R is homogeneous.A convergent sequence is not homogeneous.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Homogeneity of topological spaces

DefinitionA topological space is homogeneous if for any two points thereis a homeomorphism taking one to the other.

The real line R is homogeneous.A convergent sequence is not homogeneous.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Homogeneity of topological spaces

DefinitionA topological space is homogeneous if for any two points thereis a homeomorphism taking one to the other.

The real line R is homogeneous.A convergent sequence is not homogeneous.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Topological types

DefinitionA topological type in a topological space X is a subset T ⊆ Xwhich is invariant under homeomorphisms, that is:if f : X → X is a homeomorphism and p ∈ T then f (p) ∈ T .

The set of isolated points of a topological space is a type.The set of points with countable tightness is a type.The set of points of uncountable character is a type.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Topological types

DefinitionA topological type in a topological space X is a subset T ⊆ Xwhich is invariant under homeomorphisms, that is:if f : X → X is a homeomorphism and p ∈ T then f (p) ∈ T .

The set of isolated points of a topological space is a type.The set of points with countable tightness is a type.The set of points of uncountable character is a type.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Topological types

DefinitionA topological type in a topological space X is a subset T ⊆ Xwhich is invariant under homeomorphisms, that is:if f : X → X is a homeomorphism and p ∈ T then f (p) ∈ T .

The set of isolated points of a topological space is a type.The set of points with countable tightness is a type.The set of points of uncountable character is a type.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Topological types

DefinitionA topological type in a topological space X is a subset T ⊆ Xwhich is invariant under homeomorphisms, that is:if f : X → X is a homeomorphism and p ∈ T then f (p) ∈ T .

The set of isolated points of a topological space is a type.The set of points with countable tightness is a type.The set of points of uncountable character is a type.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Ultrafilters (a reminder)

Definition (Hausdorff)A system F of subsets of ω is called an ultrafilter, provided itsatisfies the following conditions(i) ∅ 6∈ F and ω ∈ F(ii) If A ∈ F and A ⊆ B then B ∈ F(iii) If A, B ∈ F then A ∩ B ∈ F(iv) If A is a subset of ω then either A ∈ F or ω \ A ∈ F .

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

The Čech-Stone compactificationof ω

Definition (Stone)The symbol βω stands for the set of all ultrafilters on ω.Together with the topology given by the basic open sets:

A = {p ∈ βω : A ∈ p}, A ⊆ ω

the space is called the Čech-Stone compactification of ω.The space βω \ ω is denoted ω∗ andcalled the Čech-Stone remainder of ω.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Outline

1 IntroductionBasic definitionsNonhomogeneity of ω∗, a historical overview

2 ω-uniquely accessible pointsDefinitionVan Mill’s methodConstructing spaces with ω-uniquely accessible points

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

ω∗ is not homogeneous

QuestionIs ω∗ homogeneous?

1953 Walter Rudin proved that, assuming the continuumhypothesis, there are ultrafilters on ω which are P-points.Obviously, there are non-P-points, so ω∗ is nothomogeneous.

1967 Zdeněk Frolík proved in ZFC that the Čech-Stoneremainder of any nonpseudocompact space X is nothomogeneous.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

ω∗ is not homogeneous

QuestionIs ω∗ homogeneous?

1953 Walter Rudin proved that, assuming the continuumhypothesis, there are ultrafilters on ω which are P-points.Obviously, there are non-P-points, so ω∗ is nothomogeneous.

1967 Zdeněk Frolík proved in ZFC that the Čech-Stoneremainder of any nonpseudocompact space X is nothomogeneous.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

ω∗ is not homogeneous

QuestionIs ω∗ homogeneous?

1953 Walter Rudin proved that, assuming the continuumhypothesis, there are ultrafilters on ω which are P-points.Obviously, there are non-P-points, so ω∗ is nothomogeneous.

1967 Zdeněk Frolík proved in ZFC that the Čech-Stoneremainder of any nonpseudocompact space X is nothomogeneous.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Is there an effective proof?

Frolík’s proof is amazing and very strong. But it iscombinatorial in nature. This has prompted the question:

QuestionIs there an “effective” proof which would givea purely“topological” reason for the nonhomogeneity of ω∗?W. Rudin’s proof is effective, but assumes CH, is there a similarproof in ZFC?

1978 Kenneth Kunen has proved in ZFC that ω∗ containsweak P-points

1982 Saharon Shelah proved that it is consistent with ZFC thatthere are no P-points in ω∗.

1982 Jan van Mill found, in ZFC, 16 topological types of ω∗.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Is there an effective proof?

Frolík’s proof is amazing and very strong. But it iscombinatorial in nature. This has prompted the question:

QuestionIs there an “effective” proof which would givea purely“topological” reason for the nonhomogeneity of ω∗?W. Rudin’s proof is effective, but assumes CH, is there a similarproof in ZFC?

1978 Kenneth Kunen has proved in ZFC that ω∗ containsweak P-points

1982 Saharon Shelah proved that it is consistent with ZFC thatthere are no P-points in ω∗.

1982 Jan van Mill found, in ZFC, 16 topological types of ω∗.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Is there an effective proof?

Frolík’s proof is amazing and very strong. But it iscombinatorial in nature. This has prompted the question:

QuestionIs there an “effective” proof which would givea purely“topological” reason for the nonhomogeneity of ω∗?W. Rudin’s proof is effective, but assumes CH, is there a similarproof in ZFC?

1978 Kenneth Kunen has proved in ZFC that ω∗ containsweak P-points

1982 Saharon Shelah proved that it is consistent with ZFC thatthere are no P-points in ω∗.

1982 Jan van Mill found, in ZFC, 16 topological types of ω∗.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Is there an effective proof?

Frolík’s proof is amazing and very strong. But it iscombinatorial in nature. This has prompted the question:

QuestionIs there an “effective” proof which would givea purely“topological” reason for the nonhomogeneity of ω∗?W. Rudin’s proof is effective, but assumes CH, is there a similarproof in ZFC?

1978 Kenneth Kunen has proved in ZFC that ω∗ containsweak P-points

1982 Saharon Shelah proved that it is consistent with ZFC thatthere are no P-points in ω∗.

1982 Jan van Mill found, in ZFC, 16 topological types of ω∗.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Outline

1 IntroductionBasic definitionsNonhomogeneity of ω∗, a historical overview

2 ω-uniquely accessible pointsDefinitionVan Mill’s methodConstructing spaces with ω-uniquely accessible points

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

17th type

I was interested in another topological type of ω∗. Here is thedefinition:

DefinitionA point p of a topological space is called ω-uniquely accessibleif(i) It is in the closure of a countable set.(ii) It is not in the closure of a discrete set.(iii) If p is in the closure of two countable sets D1, D2, then

D1 ∩ D2 6= ∅.

The definition intuitively says that there is only “one”countable set which “reaches” to the point.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

17th type

I was interested in another topological type of ω∗. Here is thedefinition:

DefinitionA point p of a topological space is called ω-uniquely accessibleif(i) It is in the closure of a countable set.(ii) It is not in the closure of a discrete set.(iii) If p is in the closure of two countable sets D1, D2, then

D1 ∩ D2 6= ∅.

The definition intuitively says that there is only “one”countable set which “reaches” to the point.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

17th type

I was interested in another topological type of ω∗. Here is thedefinition:

DefinitionA point p of a topological space is called ω-uniquely accessibleif(i) It is in the closure of a countable set.(ii) It is not in the closure of a discrete set.(iii) If p is in the closure of two countable sets D1, D2, then

D1 ∩ D2 6= ∅.

The definition intuitively says that there is only “one”countable set which “reaches” to the point.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

17th type

I was interested in another topological type of ω∗. Here is thedefinition:

DefinitionA point p of a topological space is called ω-uniquely accessibleif(i) It is in the closure of a countable set.(ii) It is not in the closure of a discrete set.(iii) If p is in the closure of two countable sets D1, D2, then

D1 ∩ D2 6= ∅.

The definition intuitively says that there is only “one”countable set which “reaches” to the point.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

17th type

I was interested in another topological type of ω∗. Here is thedefinition:

DefinitionA point p of a topological space is called ω-uniquely accessibleif(i) It is in the closure of a countable set.(ii) It is not in the closure of a discrete set.(iii) If p is in the closure of two countable sets D1, D2, then

D1 ∩ D2 6= ∅.

The definition intuitively says that there is only “one”countable set which “reaches” to the point.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Do these points exist?

QuestionDoes there exist an ω-uniquely accessible point?

If we drop condition (i), then any weak P-point is sucha point trivially.If we drop condition (ii), then van Mill has shown thatsuch a point exists.Kunen proved that such points exist, assuming MA.

QuestionDoes there exist an ω-uniquely accessible point in ZFC?

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Do these points exist?

QuestionDoes there exist an ω-uniquely accessible point?

If we drop condition (i), then any weak P-point is sucha point trivially.If we drop condition (ii), then van Mill has shown thatsuch a point exists.Kunen proved that such points exist, assuming MA.

QuestionDoes there exist an ω-uniquely accessible point in ZFC?

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Do these points exist?

QuestionDoes there exist an ω-uniquely accessible point?

If we drop condition (i), then any weak P-point is sucha point trivially.If we drop condition (ii), then van Mill has shown thatsuch a point exists.Kunen proved that such points exist, assuming MA.

QuestionDoes there exist an ω-uniquely accessible point in ZFC?

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Do these points exist?

QuestionDoes there exist an ω-uniquely accessible point?

If we drop condition (i), then any weak P-point is sucha point trivially.If we drop condition (ii), then van Mill has shown thatsuch a point exists.Kunen proved that such points exist, assuming MA.

QuestionDoes there exist an ω-uniquely accessible point in ZFC?

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Do these points exist?

QuestionDoes there exist an ω-uniquely accessible point?

If we drop condition (i), then any weak P-point is sucha point trivially.If we drop condition (ii), then van Mill has shown thatsuch a point exists.Kunen proved that such points exist, assuming MA.

QuestionDoes there exist an ω-uniquely accessible point in ZFC?

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Outline

1 IntroductionBasic definitionsNonhomogeneity of ω∗, a historical overview

2 ω-uniquely accessible pointsDefinitionVan Mill’s methodConstructing spaces with ω-uniquely accessible points

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Van Mill’s method of constructing points in ω∗

In his 1982 article 16 topological types of βω − ω, van Mill hasintroduced a convenient way to construct a special point in ω∗.

A two step process1. One first finds a space X which contains his point.2. Once we have our space X , we embed it into ω∗ in such

a way, that this point is not “lost”.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Van Mill’s method of constructing points in ω∗

In his 1982 article 16 topological types of βω − ω, van Mill hasintroduced a convenient way to construct a special point in ω∗.

A two step process1. One first finds a space X which contains his point.2. Once we have our space X , we embed it into ω∗ in such

a way, that this point is not “lost”.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Van Mill’s method of constructing points in ω∗

In his 1982 article 16 topological types of βω − ω, van Mill hasintroduced a convenient way to construct a special point in ω∗.

A two step process1. One first finds a space X which contains his point.2. Once we have our space X , we embed it into ω∗ in such

a way, that this point is not “lost”.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Embedding spaces into ω∗

DefinitionA closed set Y ⊆ X is a P-set of X if the closure of any countableset disjoint from Y is also disjoint from Y .

Theorem (Kunen)If X is an extremally disconnected space of weight ≤ c, then X canbe embedded into ω∗ as a weak P-set.

van Mill had a similar theorem in his paper. In his proof he usedtechniques developed by Kunen to show the existence ofweak P-points.Kunen’s theorem is actually much more general and is not limitedto ω.The general version requires the use of some independent matrix.Kunen shows the existence of these matrices in some special cases.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Embedding spaces into ω∗

DefinitionA closed set Y ⊆ X is a P-set of X if the closure of any countableset disjoint from Y is also disjoint from Y .

Theorem (Kunen)If X is an extremally disconnected space of weight ≤ c, then X canbe embedded into ω∗ as a weak P-set.

van Mill had a similar theorem in his paper. In his proof he usedtechniques developed by Kunen to show the existence ofweak P-points.Kunen’s theorem is actually much more general and is not limitedto ω.The general version requires the use of some independent matrix.Kunen shows the existence of these matrices in some special cases.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Embedding spaces into ω∗

DefinitionA closed set Y ⊆ X is a P-set of X if the closure of any countableset disjoint from Y is also disjoint from Y .

Theorem (Kunen)If X is an extremally disconnected space of weight ≤ c, then X canbe embedded into ω∗ as a weak P-set.

van Mill had a similar theorem in his paper. In his proof he usedtechniques developed by Kunen to show the existence ofweak P-points.Kunen’s theorem is actually much more general and is not limitedto ω.The general version requires the use of some independent matrix.Kunen shows the existence of these matrices in some special cases.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

P-sets preserve unique accessibility

ObservationIf Y ⊆ X is a P-set of X and p ∈ Y is an ω-uniquely accessiblepoint of Y , then p is an ω-uniquely accessible point of X .

Proof.Suppose D0, D1 are two countable sets which contain p in theirclosure. Split each of them into two parts Ai , Bi , so thatAi ⊆ X \ Y and Bi ⊆ X . Because Y is a weak P-set, theclosures Ai are disjoint from Y .So p ∈ Di ∩ Y = Ai ∪ Bi ∩ Y = (Ai ∪ Bi ) ∩ Y = Bi .Now, since Bi are countable subsets of Y andp is ω-uniquely accessible in Y , B0 ∩ B1 6= ∅.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

P-sets preserve unique accessibility

ObservationIf Y ⊆ X is a P-set of X and p ∈ Y is an ω-uniquely accessiblepoint of Y , then p is an ω-uniquely accessible point of X .

Proof.Suppose D0, D1 are two countable sets which contain p in theirclosure. Split each of them into two parts Ai , Bi , so thatAi ⊆ X \ Y and Bi ⊆ X . Because Y is a weak P-set, theclosures Ai are disjoint from Y .So p ∈ Di ∩ Y = Ai ∪ Bi ∩ Y = (Ai ∪ Bi ) ∩ Y = Bi .Now, since Bi are countable subsets of Y andp is ω-uniquely accessible in Y , B0 ∩ B1 6= ∅.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

P-sets preserve unique accessibility

ObservationIf Y ⊆ X is a P-set of X and p ∈ Y is an ω-uniquely accessiblepoint of Y , then p is an ω-uniquely accessible point of X .

Proof.Suppose D0, D1 are two countable sets which contain p in theirclosure. Split each of them into two parts Ai , Bi , so thatAi ⊆ X \ Y and Bi ⊆ X . Because Y is a weak P-set, theclosures Ai are disjoint from Y .So p ∈ Di ∩ Y = Ai ∪ Bi ∩ Y = (Ai ∪ Bi ) ∩ Y = Bi .Now, since Bi are countable subsets of Y andp is ω-uniquely accessible in Y , B0 ∩ B1 6= ∅.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

P-sets preserve unique accessibility

ObservationIf Y ⊆ X is a P-set of X and p ∈ Y is an ω-uniquely accessiblepoint of Y , then p is an ω-uniquely accessible point of X .

Proof.Suppose D0, D1 are two countable sets which contain p in theirclosure. Split each of them into two parts Ai , Bi , so thatAi ⊆ X \ Y and Bi ⊆ X . Because Y is a weak P-set, theclosures Ai are disjoint from Y .So p ∈ Di ∩ Y = Ai ∪ Bi ∩ Y = (Ai ∪ Bi ) ∩ Y = Bi .Now, since Bi are countable subsets of Y andp is ω-uniquely accessible in Y , B0 ∩ B1 6= ∅.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

P-sets preserve unique accessibility

ObservationIf Y ⊆ X is a P-set of X and p ∈ Y is an ω-uniquely accessiblepoint of Y , then p is an ω-uniquely accessible point of X .

Proof.Suppose D0, D1 are two countable sets which contain p in theirclosure. Split each of them into two parts Ai , Bi , so thatAi ⊆ X \ Y and Bi ⊆ X . Because Y is a weak P-set, theclosures Ai are disjoint from Y .So p ∈ Di ∩ Y = Ai ∪ Bi ∩ Y = (Ai ∪ Bi ) ∩ Y = Bi .Now, since Bi are countable subsets of Y andp is ω-uniquely accessible in Y , B0 ∩ B1 6= ∅.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

P-sets preserve unique accessibility

ObservationIf Y ⊆ X is a P-set of X and p ∈ Y is an ω-uniquely accessiblepoint of Y , then p is an ω-uniquely accessible point of X .

Proof.Suppose D0, D1 are two countable sets which contain p in theirclosure. Split each of them into two parts Ai , Bi , so thatAi ⊆ X \ Y and Bi ⊆ X . Because Y is a weak P-set, theclosures Ai are disjoint from Y .So p ∈ Di ∩ Y = Ai ∪ Bi ∩ Y = (Ai ∪ Bi ) ∩ Y = Bi .Now, since Bi are countable subsets of Y andp is ω-uniquely accessible in Y , B0 ∩ B1 6= ∅.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

P-sets preserve unique accessibility

ObservationIf Y ⊆ X is a P-set of X and p ∈ Y is an ω-uniquely accessiblepoint of Y , then p is an ω-uniquely accessible point of X .

Proof.Suppose D0, D1 are two countable sets which contain p in theirclosure. Split each of them into two parts Ai , Bi , so thatAi ⊆ X \ Y and Bi ⊆ X . Because Y is a weak P-set, theclosures Ai are disjoint from Y .So p ∈ Di ∩ Y = Ai ∪ Bi ∩ Y = (Ai ∪ Bi ) ∩ Y = Bi .Now, since Bi are countable subsets of Y andp is ω-uniquely accessible in Y , B0 ∩ B1 6= ∅.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

P-sets preserve unique accessibility

ObservationIf Y ⊆ X is a P-set of X and p ∈ Y is an ω-uniquely accessiblepoint of Y , then p is an ω-uniquely accessible point of X .

Proof.Suppose D0, D1 are two countable sets which contain p in theirclosure. Split each of them into two parts Ai , Bi , so thatAi ⊆ X \ Y and Bi ⊆ X . Because Y is a weak P-set, theclosures Ai are disjoint from Y .So p ∈ Di ∩ Y = Ai ∪ Bi ∩ Y = (Ai ∪ Bi ) ∩ Y = Bi .Now, since Bi are countable subsets of Y andp is ω-uniquely accessible in Y , B0 ∩ B1 6= ∅.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

P-sets preserve unique accessibility

ObservationIf Y ⊆ X is a P-set of X and p ∈ Y is an ω-uniquely accessiblepoint of Y , then p is an ω-uniquely accessible point of X .

Proof.Suppose D0, D1 are two countable sets which contain p in theirclosure. Split each of them into two parts Ai , Bi , so thatAi ⊆ X \ Y and Bi ⊆ X . Because Y is a weak P-set, theclosures Ai are disjoint from Y .So p ∈ Di ∩ Y = Ai ∪ Bi ∩ Y = (Ai ∪ Bi ) ∩ Y = Bi .Now, since Bi are countable subsets of Y andp is ω-uniquely accessible in Y , B0 ∩ B1 6= ∅.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Outline

1 IntroductionBasic definitionsNonhomogeneity of ω∗, a historical overview

2 ω-uniquely accessible pointsDefinitionVan Mill’s methodConstructing spaces with ω-uniquely accessible points

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Key ingredients

Here we list the main ideas behind our method of constructingspaces with ω-uniquely accessible points.(A) Build a countable space where dense sets intersect

(hereditarily) and find the ω-uniquely accessible point inits Čech-Stone remainder.

(B) If a point in the remainder is a weak P-point, then it isω-uniquely accessible with respect to the remainder.

(C) If a point in the remainder is a remote point, then it isω-uniquely accessible with respect to the space.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Key ingredients

Here we list the main ideas behind our method of constructingspaces with ω-uniquely accessible points.(A) Build a countable space where dense sets intersect

(hereditarily) and find the ω-uniquely accessible point inits Čech-Stone remainder.

(B) If a point in the remainder is a weak P-point, then it isω-uniquely accessible with respect to the remainder.

(C) If a point in the remainder is a remote point, then it isω-uniquely accessible with respect to the space.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Key ingredients

Here we list the main ideas behind our method of constructingspaces with ω-uniquely accessible points.(A) Build a countable space where dense sets intersect

(hereditarily) and find the ω-uniquely accessible point inits Čech-Stone remainder.

(B) If a point in the remainder is a weak P-point, then it isω-uniquely accessible with respect to the remainder.

(C) If a point in the remainder is a remote point, then it isω-uniquely accessible with respect to the space.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Key ingredients

Here we list the main ideas behind our method of constructingspaces with ω-uniquely accessible points.(A) Build a countable space where dense sets intersect

(hereditarily) and find the ω-uniquely accessible point inits Čech-Stone remainder.

(B) If a point in the remainder is a weak P-point, then it isω-uniquely accessible with respect to the remainder.

(C) If a point in the remainder is a remote point, then it isω-uniquely accessible with respect to the space.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

OHI spaces

DefinitionA topological space X is OHI (or Open Hereditarily Irresolvable),provided that for any open subset U ⊆ X any two sets dense in Uhave nonempty intersection.

TheoremIf p is a remote point of an OHI space X , then p is ω-uniquelyaccessible with respect to X .

Proof.Since p is a remote point then, if p ∈ D, we know that there is an openset G such that D is dense in G and p ∈ G . Now if p ∈ D0 ∩ D1, thenthere are G0, G1 so that p ∈ G0 ∩ G1 and Di is dense in Gi . A theorem ofvan Douwen says that βX is extremally disconnected at any remotepoint. As a consequence p cannot be in the closure of two disjoint opensets. So G0 ∩ G1 is nonempty. Now D0, D1 are both dense in G0 ∩ G1 sothey have nonempty intersection, since X is OHI.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

OHI spaces

DefinitionA topological space X is OHI (or Open Hereditarily Irresolvable),provided that for any open subset U ⊆ X any two sets dense in Uhave nonempty intersection.

TheoremIf p is a remote point of an OHI space X , then p is ω-uniquelyaccessible with respect to X .

Proof.Since p is a remote point then, if p ∈ D, we know that there is an openset G such that D is dense in G and p ∈ G . Now if p ∈ D0 ∩ D1, thenthere are G0, G1 so that p ∈ G0 ∩ G1 and Di is dense in Gi . A theorem ofvan Douwen says that βX is extremally disconnected at any remotepoint. As a consequence p cannot be in the closure of two disjoint opensets. So G0 ∩ G1 is nonempty. Now D0, D1 are both dense in G0 ∩ G1 sothey have nonempty intersection, since X is OHI.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

OHI spaces

DefinitionA topological space X is OHI (or Open Hereditarily Irresolvable),provided that for any open subset U ⊆ X any two sets dense in Uhave nonempty intersection.

TheoremIf p is a remote point of an OHI space X , then p is ω-uniquelyaccessible with respect to X .

Proof.Since p is a remote point then, if p ∈ D, we know that there is an openset G such that D is dense in G and p ∈ G . Now if p ∈ D0 ∩ D1, thenthere are G0, G1 so that p ∈ G0 ∩ G1 and Di is dense in Gi . A theorem ofvan Douwen says that βX is extremally disconnected at any remotepoint. As a consequence p cannot be in the closure of two disjoint opensets. So G0 ∩ G1 is nonempty. Now D0, D1 are both dense in G0 ∩ G1 sothey have nonempty intersection, since X is OHI.

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Van Mill’s method

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OHI spaces

DefinitionA topological space X is OHI (or Open Hereditarily Irresolvable),provided that for any open subset U ⊆ X any two sets dense in Uhave nonempty intersection.

TheoremIf p is a remote point of an OHI space X , then p is ω-uniquelyaccessible with respect to X .

Proof.Since p is a remote point then, if p ∈ D, we know that there is an openset G such that D is dense in G and p ∈ G . Now if p ∈ D0 ∩ D1, thenthere are G0, G1 so that p ∈ G0 ∩ G1 and Di is dense in Gi . A theorem ofvan Douwen says that βX is extremally disconnected at any remotepoint. As a consequence p cannot be in the closure of two disjoint opensets. So G0 ∩ G1 is nonempty. Now D0, D1 are both dense in G0 ∩ G1 sothey have nonempty intersection, since X is OHI.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

OHI spaces

DefinitionA topological space X is OHI (or Open Hereditarily Irresolvable),provided that for any open subset U ⊆ X any two sets dense in Uhave nonempty intersection.

TheoremIf p is a remote point of an OHI space X , then p is ω-uniquelyaccessible with respect to X .

Proof.Since p is a remote point then, if p ∈ D, we know that there is an openset G such that D is dense in G and p ∈ G . Now if p ∈ D0 ∩ D1, thenthere are G0, G1 so that p ∈ G0 ∩ G1 and Di is dense in Gi . A theorem ofvan Douwen says that βX is extremally disconnected at any remotepoint. As a consequence p cannot be in the closure of two disjoint opensets. So G0 ∩ G1 is nonempty. Now D0, D1 are both dense in G0 ∩ G1 sothey have nonempty intersection, since X is OHI.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

OHI spaces

DefinitionA topological space X is OHI (or Open Hereditarily Irresolvable),provided that for any open subset U ⊆ X any two sets dense in Uhave nonempty intersection.

TheoremIf p is a remote point of an OHI space X , then p is ω-uniquelyaccessible with respect to X .

Proof.Since p is a remote point then, if p ∈ D, we know that there is an openset G such that D is dense in G and p ∈ G . Now if p ∈ D0 ∩ D1, thenthere are G0, G1 so that p ∈ G0 ∩ G1 and Di is dense in Gi . A theorem ofvan Douwen says that βX is extremally disconnected at any remotepoint. As a consequence p cannot be in the closure of two disjoint opensets. So G0 ∩ G1 is nonempty. Now D0, D1 are both dense in G0 ∩ G1 sothey have nonempty intersection, since X is OHI.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

OHI spaces

DefinitionA topological space X is OHI (or Open Hereditarily Irresolvable),provided that for any open subset U ⊆ X any two sets dense in Uhave nonempty intersection.

TheoremIf p is a remote point of an OHI space X , then p is ω-uniquelyaccessible with respect to X .

Proof.Since p is a remote point then, if p ∈ D, we know that there is an openset G such that D is dense in G and p ∈ G . Now if p ∈ D0 ∩ D1, thenthere are G0, G1 so that p ∈ G0 ∩ G1 and Di is dense in Gi . A theorem ofvan Douwen says that βX is extremally disconnected at any remotepoint. As a consequence p cannot be in the closure of two disjoint opensets. So G0 ∩ G1 is nonempty. Now D0, D1 are both dense in G0 ∩ G1 sothey have nonempty intersection, since X is OHI.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

OHI spaces

DefinitionA topological space X is OHI (or Open Hereditarily Irresolvable),provided that for any open subset U ⊆ X any two sets dense in Uhave nonempty intersection.

TheoremIf p is a remote point of an OHI space X , then p is ω-uniquelyaccessible with respect to X .

Proof.Since p is a remote point then, if p ∈ D, we know that there is an openset G such that D is dense in G and p ∈ G . Now if p ∈ D0 ∩ D1, thenthere are G0, G1 so that p ∈ G0 ∩ G1 and Di is dense in Gi . A theorem ofvan Douwen says that βX is extremally disconnected at any remotepoint. As a consequence p cannot be in the closure of two disjoint opensets. So G0 ∩ G1 is nonempty. Now D0, D1 are both dense in G0 ∩ G1 sothey have nonempty intersection, since X is OHI.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

OHI spaces

DefinitionA topological space X is OHI (or Open Hereditarily Irresolvable),provided that for any open subset U ⊆ X any two sets dense in Uhave nonempty intersection.

TheoremIf p is a remote point of an OHI space X , then p is ω-uniquelyaccessible with respect to X .

Proof.Since p is a remote point then, if p ∈ D, we know that there is an openset G such that D is dense in G and p ∈ G . Now if p ∈ D0 ∩ D1, thenthere are G0, G1 so that p ∈ G0 ∩ G1 and Di is dense in Gi . A theorem ofvan Douwen says that βX is extremally disconnected at any remotepoint. As a consequence p cannot be in the closure of two disjoint opensets. So G0 ∩ G1 is nonempty. Now D0, D1 are both dense in G0 ∩ G1 sothey have nonempty intersection, since X is OHI.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Building OHI spaces

The following theorem is a tool to guarantee (A).

TheoremAny zerodimensional crowded topology can be refined to acrowded extremally disconnected OHI topology.

The theorem can be proved by methods going back toHewitt.Hewit looked at certain maximal topologies.Our method is similar to Hewitt’s.Our proof of this theorem uses Boolean algebras and givessome additional results for boolean algebras.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Building OHI spaces

The following theorem is a tool to guarantee (A).

TheoremAny zerodimensional crowded topology can be refined to acrowded extremally disconnected OHI topology.

The theorem can be proved by methods going back toHewitt.Hewit looked at certain maximal topologies.Our method is similar to Hewitt’s.Our proof of this theorem uses Boolean algebras and givessome additional results for boolean algebras.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Building OHI spaces

The following theorem is a tool to guarantee (A).

TheoremAny zerodimensional crowded topology can be refined to acrowded extremally disconnected OHI topology.

The theorem can be proved by methods going back toHewitt.Hewit looked at certain maximal topologies.Our method is similar to Hewitt’s.Our proof of this theorem uses Boolean algebras and givessome additional results for boolean algebras.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Building OHI spaces

The following theorem is a tool to guarantee (A).

TheoremAny zerodimensional crowded topology can be refined to acrowded extremally disconnected OHI topology.

The theorem can be proved by methods going back toHewitt.Hewit looked at certain maximal topologies.Our method is similar to Hewitt’s.Our proof of this theorem uses Boolean algebras and givessome additional results for boolean algebras.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Weak P-points

Turning to (B) we get, modifying a proof of a theorem ofvan Mill, the following theorem

Theorem (Mill)If a space X of weight ≤ c contains an independent linkedsystem mod some closed filter, then X ∗ contains a weakP-point in the intersection of the filter.

In the following theorem we show (using our method to buildOHI spaces), that we can have OHI spaces with weak P-points.

TheoremIf M is an i.l.s. on ω, then there is a crowded zerodimensionaltopology on ω, where the members of M are closed.

IntroductionBasic definitions

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Van Mill’s method

Finding the point

Summary

Weak P-points

Turning to (B) we get, modifying a proof of a theorem ofvan Mill, the following theorem

Theorem (Mill)If a space X of weight ≤ c contains an independent linkedsystem mod some closed filter, then X ∗ contains a weakP-point in the intersection of the filter.

In the following theorem we show (using our method to buildOHI spaces), that we can have OHI spaces with weak P-points.

TheoremIf M is an i.l.s. on ω, then there is a crowded zerodimensionaltopology on ω, where the members of M are closed.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Weak P-points

Turning to (B) we get, modifying a proof of a theorem ofvan Mill, the following theorem

Theorem (Mill)If a space X of weight ≤ c contains an independent linkedsystem mod some closed filter, then X ∗ contains a weakP-point in the intersection of the filter.

In the following theorem we show (using our method to buildOHI spaces), that we can have OHI spaces with weak P-points.

TheoremIf M is an i.l.s. on ω, then there is a crowded zerodimensionaltopology on ω, where the members of M are closed.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Weak P-points

Turning to (B) we get, modifying a proof of a theorem ofvan Mill, the following theorem

Theorem (Mill)If a space X of weight ≤ c contains an independent linkedsystem mod some closed filter, then X ∗ contains a weakP-point in the intersection of the filter.

In the following theorem we show (using our method to buildOHI spaces), that we can have OHI spaces with weak P-points.

TheoremIf M is an i.l.s. on ω, then there is a crowded zerodimensionaltopology on ω, where the members of M are closed.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Remote points

To get (C) we proved the following:

TheoremThere is a zerodimensional OHI space with a remote point.

Proof.In c many steps we construct a topology on ω and a remotefilterAt each step we look at a subset of ω and either make ithave nonempty interior or add a set to our filter, whichmisses itWe have to be careful not to get isolated pointsWe use the fact that there is an ideal I on ω such thatP(ω)/I has hereditary independence c to guarantee that theprocess does not stop before ω

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Remote points

To get (C) we proved the following:

TheoremThere is a zerodimensional OHI space with a remote point.

Proof.In c many steps we construct a topology on ω and a remotefilterAt each step we look at a subset of ω and either make ithave nonempty interior or add a set to our filter, whichmisses itWe have to be careful not to get isolated pointsWe use the fact that there is an ideal I on ω such thatP(ω)/I has hereditary independence c to guarantee that theprocess does not stop before ω

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Remote points

To get (C) we proved the following:

TheoremThere is a zerodimensional OHI space with a remote point.

Proof.In c many steps we construct a topology on ω and a remotefilterAt each step we look at a subset of ω and either make ithave nonempty interior or add a set to our filter, whichmisses itWe have to be careful not to get isolated pointsWe use the fact that there is an ideal I on ω such thatP(ω)/I has hereditary independence c to guarantee that theprocess does not stop before ω

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Remote points

To get (C) we proved the following:

TheoremThere is a zerodimensional OHI space with a remote point.

Proof.In c many steps we construct a topology on ω and a remotefilterAt each step we look at a subset of ω and either make ithave nonempty interior or add a set to our filter, whichmisses itWe have to be careful not to get isolated pointsWe use the fact that there is an ideal I on ω such thatP(ω)/I has hereditary independence c to guarantee that theprocess does not stop before ω

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Remote points

To get (C) we proved the following:

TheoremThere is a zerodimensional OHI space with a remote point.

Proof.In c many steps we construct a topology on ω and a remotefilterAt each step we look at a subset of ω and either make ithave nonempty interior or add a set to our filter, whichmisses itWe have to be careful not to get isolated pointsWe use the fact that there is an ideal I on ω such thatP(ω)/I has hereditary independence c to guarantee that theprocess does not stop before ω

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Summary

Main resultsA method for building OHI spaces in a Boolean-algebraicsetting.An OHI space with a weak P-point.An OHI space with a remote point.

Unfortunately, the main question still remains open:

QuestionDoes there exist an ω-uniquely accessible point in ZFC?

It may be possible to modify the method for buildingremote points to get a remote weak P-filter.If we are careful enough, then we might be able to obtaina remote filter in our construction of a space with an i.l.s.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Summary

Main resultsA method for building OHI spaces in a Boolean-algebraicsetting.An OHI space with a weak P-point.An OHI space with a remote point.

Unfortunately, the main question still remains open:

QuestionDoes there exist an ω-uniquely accessible point in ZFC?

It may be possible to modify the method for buildingremote points to get a remote weak P-filter.If we are careful enough, then we might be able to obtaina remote filter in our construction of a space with an i.l.s.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Summary

Main resultsA method for building OHI spaces in a Boolean-algebraicsetting.An OHI space with a weak P-point.An OHI space with a remote point.

Unfortunately, the main question still remains open:

QuestionDoes there exist an ω-uniquely accessible point in ZFC?

It may be possible to modify the method for buildingremote points to get a remote weak P-filter.If we are careful enough, then we might be able to obtaina remote filter in our construction of a space with an i.l.s.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Summary

Main resultsA method for building OHI spaces in a Boolean-algebraicsetting.An OHI space with a weak P-point.An OHI space with a remote point.

Unfortunately, the main question still remains open:

QuestionDoes there exist an ω-uniquely accessible point in ZFC?

It may be possible to modify the method for buildingremote points to get a remote weak P-filter.If we are careful enough, then we might be able to obtaina remote filter in our construction of a space with an i.l.s.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Summary

Main resultsA method for building OHI spaces in a Boolean-algebraicsetting.An OHI space with a weak P-point.An OHI space with a remote point.

Unfortunately, the main question still remains open:

QuestionDoes there exist an ω-uniquely accessible point in ZFC?

It may be possible to modify the method for buildingremote points to get a remote weak P-filter.If we are careful enough, then we might be able to obtaina remote filter in our construction of a space with an i.l.s.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Summary

Main resultsA method for building OHI spaces in a Boolean-algebraicsetting.An OHI space with a weak P-point.An OHI space with a remote point.

Unfortunately, the main question still remains open:

QuestionDoes there exist an ω-uniquely accessible point in ZFC?

It may be possible to modify the method for buildingremote points to get a remote weak P-filter.If we are careful enough, then we might be able to obtaina remote filter in our construction of a space with an i.l.s.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Summary

Main resultsA method for building OHI spaces in a Boolean-algebraicsetting.An OHI space with a weak P-point.An OHI space with a remote point.

Unfortunately, the main question still remains open:

QuestionDoes there exist an ω-uniquely accessible point in ZFC?

It may be possible to modify the method for buildingremote points to get a remote weak P-filter.If we are careful enough, then we might be able to obtaina remote filter in our construction of a space with an i.l.s.

IntroductionBasic definitions

History

Special pointsDefinition

Van Mill’s method

Finding the point

Summary

Thank you. . .

I would like to thank Dr. Jerry Vaughan for his guidance,Dr. Jan Rychtář for his comments and his invitation toGreensboro and the comittee for the time they spent readingmy thesis.