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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.
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
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
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
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.