Patterns of reflection properties Menachem Magidor Institute of - - PowerPoint PPT Presentation

Compactness and Reflection Classifying reflection properties Corson compacts

Patterns of reflection properties

Menachem Magidor

Institute of Mathematics Hebrew University of Jerusalem

Bagaria60 conference

Compactness and Reflection Classifying reflection properties Corson compacts

• utline

Compactness and Reflection Classifying reflection properties Corson compacts

Compactness and Reflection Classifying reflection properties Corson compacts

This a tribute to Joan Bagaria in appreciation and thanks for his

Compactness and Reflection Classifying reflection properties Corson compacts

This a tribute to Joan Bagaria in appreciation and thanks for his Mathematics

Compactness and Reflection Classifying reflection properties Corson compacts

This a tribute to Joan Bagaria in appreciation and thanks for his Mathematics Professional leadership

Compactness and Reflection Classifying reflection properties Corson compacts

This a tribute to Joan Bagaria in appreciation and thanks for his Mathematics Professional leadership and friendship

Compactness and Reflection Classifying reflection properties Corson compacts

This a tribute to Joan Bagaria in appreciation and thanks for his Mathematics Professional leadership and friendship Do not forget Joan as a political activist and

Compactness and Reflection Classifying reflection properties Corson compacts

This a tribute to Joan Bagaria in appreciation and thanks for his Mathematics Professional leadership and friendship Do not forget Joan as a political activist and mushroom hunter .

Compactness and Reflection Classifying reflection properties Corson compacts

Compactness

We consider properties of mathematical structures.

Compactness and Reflection Classifying reflection properties Corson compacts

Compactness

We consider properties of mathematical structures. Compactness property for a given property is a statement of the form If every "small" substructure has the property then the structure has the property.

Compactness and Reflection Classifying reflection properties Corson compacts

Compactness

We consider properties of mathematical structures. Compactness property for a given property is a statement of the form If every "small" substructure has the property then the structure has the property. Typically "small" means " Having cardinality less than .....

Compactness and Reflection Classifying reflection properties Corson compacts

Combinatorial Examples

The existence of transversals: A transversal for a family of non empty sets is a 1-1 choice function on the family. Suppose that every smaller cardinality subfamily

• f the family F has a transversal. Does F has a

transversal?

Compactness and Reflection Classifying reflection properties Corson compacts

Combinatorial Examples

The existence of transversals: A transversal for a family of non empty sets is a 1-1 choice function on the family. Suppose that every smaller cardinality subfamily

• f the family F has a transversal. Does F has a

transversal? Special case : F is a family of countable sets. Disjointifying F is family of infinite sets. Disjointfying F means picking for every X ∈ F a finite zX ⊆ X such that the family {X − zX|X ∈ F} is a family of mutually disjoint sets.

Compactness and Reflection Classifying reflection properties Corson compacts

Combinatorial Examples

The existence of transversals: A transversal for a family of non empty sets is a 1-1 choice function on the family. Suppose that every smaller cardinality subfamily

• f the family F has a transversal. Does F has a

transversal? Special case : F is a family of countable sets. Disjointifying F is family of infinite sets. Disjointfying F means picking for every X ∈ F a finite zX ⊆ X such that the family {X − zX|X ∈ F} is a family of mutually disjoint sets. Suppose that every smaller cardinality subfamily

• f F can be disjointified . Can F be disjointified ?

Compactness and Reflection Classifying reflection properties Corson compacts

More combinatorics

Chromatic numbers G is a graph such that every (induced) subgroup of G has a chromatic numbers ≤ λ. Does G has chromatic number ≤ λ?

Compactness and Reflection Classifying reflection properties Corson compacts

More combinatorics

Chromatic numbers G is a graph such that every (induced) subgroup of G has a chromatic numbers ≤ λ. Does G has chromatic number ≤ λ? Coloring Number The graph G is said to have a coloring number λ if the nodes of G can be well-ordered such that every node is connected to less than λ nodes appearing before it in the well ordering.

Compactness and Reflection Classifying reflection properties Corson compacts

More combinatorics

Chromatic numbers G is a graph such that every (induced) subgroup of G has a chromatic numbers ≤ λ. Does G has chromatic number ≤ λ? Coloring Number The graph G is said to have a coloring number λ if the nodes of G can be well-ordered such that every node is connected to less than λ nodes appearing before it in the well ordering. Suppose that every smaller cardinality (induced) subgraph of G has coloring number λ . Does G has coloring number λ?

Compactness and Reflection Classifying reflection properties Corson compacts

Algebraic examples

Freeness of Abelian groups An Abelian group H is free if it can be represented as the direct sum of copies of Z. H =

I Z. Suppose that very smaller cardinality

subgroup of H is free . Is H free?

Compactness and Reflection Classifying reflection properties Corson compacts

Algebraic examples

Freeness of Abelian groups An Abelian group H is free if it can be represented as the direct sum of copies of Z. H =

I Z. Suppose that very smaller cardinality

subgroup of H is free . Is H free? Free* The Abelian group H is said to be free* if it is a subgroup of a direct product of copies of Z.H ⊆

I Z. Suppose that every smaller

cardinality subgroup of H is free*. Is H free*?

Compactness and Reflection Classifying reflection properties Corson compacts

Topological Examples

Metrizability X is a topological space such that every subspace of smaller cardinality is metric. Is X metric?

Compactness and Reflection Classifying reflection properties Corson compacts

Topological Examples

Metrizability X is a topological space such that every subspace of smaller cardinality is metric. Is X metric? In order to avoid simple counter examples assume that X is first countable .

Compactness and Reflection Classifying reflection properties Corson compacts

Topological Examples

Metrizability X is a topological space such that every subspace of smaller cardinality is metric. Is X metric? In order to avoid simple counter examples assume that X is first countable . Collection-wise Hausdorff X is a topological space. Y ⊆ X is a discrete closed set. We say that Y can be separated if there is a family of mutually disjoint

• pen sets {Uy|y ∈ Y} such that for y ∈ Y y ∈ Uy.

Suppose that every smaller cardinality subset of Y can be separated. Can Y be separated?

Compactness and Reflection Classifying reflection properties Corson compacts

Topological Examples

Metrizability X is a topological space such that every subspace of smaller cardinality is metric. Is X metric? In order to avoid simple counter examples assume that X is first countable . Collection-wise Hausdorff X is a topological space. Y ⊆ X is a discrete closed set. We say that Y can be separated if there is a family of mutually disjoint

• pen sets {Uy|y ∈ Y} such that for y ∈ Y y ∈ Uy.

Suppose that every smaller cardinality subset of Y can be separated. Can Y be separated? Here again in order to avoid trivial counter examples we have to assume something about the space, like it is being locally small or having a small character.

Compactness and Reflection Classifying reflection properties Corson compacts

Reflection Principles

A reflection principle is a statement of the form " Suppose that the structure A has a certain property then A has a small substructure with the same property ".

Compactness and Reflection Classifying reflection properties Corson compacts

Reflection Principles

A reflection principle is a statement of the form " Suppose that the structure A has a certain property then A has a small substructure with the same property ". Reflection principles are dual to compactness principles . Namely a reflection principle for a given property is equivalent to a compactness principle for the negation of the property and vice versa.

Compactness and Reflection Classifying reflection properties Corson compacts

Other examples from Set Theory

Stationary set reflection κ a regular cardinal. S ⊆ κ is a stationary subset of κ. Is there α < κ such that S ∩ α is a stationary subset of α?

Compactness and Reflection Classifying reflection properties Corson compacts

Other examples from Set Theory

Stationary set reflection κ a regular cardinal. S ⊆ κ is a stationary subset of κ. Is there α < κ such that S ∩ α is a stationary subset of α? In order to make the problem more concrete assume that the stationary set is made up of points of cofinality ω. Reflection of stationary subsets of Pω1(A) Suppose that S is a stationary subset of the countable subsets of A . Is there B ⊂ A |B| < |A| such that S ∩ Pω1(B) is a stationary subset of Pω1(B)?

Compactness and Reflection Classifying reflection properties Corson compacts

Other examples from Set Theory

Stationary set reflection κ a regular cardinal. S ⊆ κ is a stationary subset of κ. Is there α < κ such that S ∩ α is a stationary subset of α? In order to make the problem more concrete assume that the stationary set is made up of points of cofinality ω. Reflection of stationary subsets of Pω1(A) Suppose that S is a stationary subset of the countable subsets of A . Is there B ⊂ A |B| < |A| such that S ∩ Pω1(B) is a stationary subset of Pω1(B)? The tree property T is a tree with λ many levels such that every level has cardinality less than λ. Does T has a branch of length of length λ?

Compactness and Reflection Classifying reflection properties Corson compacts

Other examples from Set Theory

Stationary set reflection κ a regular cardinal. S ⊆ κ is a stationary subset of κ. Is there α < κ such that S ∩ α is a stationary subset of α? In order to make the problem more concrete assume that the stationary set is made up of points of cofinality ω. Reflection of stationary subsets of Pω1(A) Suppose that S is a stationary subset of the countable subsets of A . Is there B ⊂ A |B| < |A| such that S ∩ Pω1(B) is a stationary subset of Pω1(B)? The tree property T is a tree with λ many levels such that every level has cardinality less than λ. Does T has a branch of length of length λ?

Compactness and Reflection Classifying reflection properties Corson compacts

Kurepa Hypothesis is a reflection principle

Kurepa hypothesis for the cardinal κ is the statement that if T is a tree of height κ such that the α’s level of T has cardinality ≤ |α| then T has at most κ many branches.

Compactness and Reflection Classifying reflection properties Corson compacts

Kurepa Hypothesis is a reflection principle

Kurepa hypothesis for the cardinal κ is the statement that if T is a tree of height κ such that the α’s level of T has cardinality ≤ |α| then T has at most κ many branches. It is a reflection principle because it means that if T is a tree of height κ with more than κ many branches, then there is α < κ such that the size of the α’s level is more than |α|.

Compactness and Reflection Classifying reflection properties Corson compacts

Kurepa Hypothesis is a reflection principle

Kurepa hypothesis for the cardinal κ is the statement that if T is a tree of height κ such that the α’s level of T has cardinality ≤ |α| then T has at most κ many branches. It is a reflection principle because it means that if T is a tree of height κ with more than κ many branches, then there is α < κ such that the size of the α’s level is more than |α|. The weak Kurepa hypothesis for the cardinal κ is the statement that if T is a tree of height κ with more than κ many branches, then there is α < κ such that Tα has more than |α| many

• branches. (Tα is the subtree made up of the first α many levels
• f the tree.)

Compactness and Reflection Classifying reflection properties Corson compacts

Kurepa Hypothesis is a reflection principle

Kurepa hypothesis for the cardinal κ is the statement that if T is a tree of height κ such that the α’s level of T has cardinality ≤ |α| then T has at most κ many branches. It is a reflection principle because it means that if T is a tree of height κ with more than κ many branches, then there is α < κ such that the size of the α’s level is more than |α|. The weak Kurepa hypothesis for the cardinal κ is the statement that if T is a tree of height κ with more than κ many branches, then there is α < κ such that Tα has more than |α| many

• branches. (Tα is the subtree made up of the first α many levels
• f the tree.)

The weak Kurepa hypothesis is true for ω1.

Compactness and Reflection Classifying reflection properties Corson compacts

Reflection cardinals

Definition

• 1. Given a property of mathematical structure, we say that a

cardinal κ is a reflection cardinal for this property if every structure of cardinality κ has a substructure of cardinality less than κ having the given property.

Compactness and Reflection Classifying reflection properties Corson compacts

Reflection cardinals

Definition

• 1. Given a property of mathematical structure, we say that a

cardinal κ is a reflection cardinal for this property if every structure of cardinality κ has a substructure of cardinality less than κ having the given property.

• 2. κ is a strong reflection cardinal if the every structure (no

restriction on the cardinality ) having this property has a substructure of cardinality less than κ having the given property.

Compactness and Reflection Classifying reflection properties Corson compacts

Reflection cardinals

Definition

• 1. Given a property of mathematical structure, we say that a

cardinal κ is a reflection cardinal for this property if every structure of cardinality κ has a substructure of cardinality less than κ having the given property.

• 2. κ is a strong reflection cardinal if the every structure (no

restriction on the cardinality ) having this property has a substructure of cardinality less than κ having the given property.

Compactness and Reflection Classifying reflection properties Corson compacts

Compactness cardinals

Definition

• 1. A cardinal κ is a (weakly) compact for the given property if

a structure of cardinality κ has the given property , given that every substructure of cardinality less than κ has the property.

Compactness and Reflection Classifying reflection properties Corson compacts

Compactness cardinals

Definition

• 1. A cardinal κ is a (weakly) compact for the given property if

a structure of cardinality κ has the given property , given that every substructure of cardinality less than κ has the property.

• 2. κ is a strongly compact cardinal for the property if every

structure (no restriction on the cardinality ) has the property given that every substructure of cardinality less than κ has the property.

Compactness and Reflection Classifying reflection properties Corson compacts

Compactness cardinals

Definition

• 1. A cardinal κ is a (weakly) compact for the given property if

a structure of cardinality κ has the given property , given that every substructure of cardinality less than κ has the property.

• 2. κ is a strongly compact cardinal for the property if every

structure (no restriction on the cardinality ) has the property given that every substructure of cardinality less than κ has the property. The duality of reflection and compactness: κ is a reflection cardinal for a certain property iff it is a compactness cardinal for the negation of the property

Compactness and Reflection Classifying reflection properties Corson compacts

Compactness cardinals

Definition

• 1. A cardinal κ is a (weakly) compact for the given property if

a structure of cardinality κ has the given property , given that every substructure of cardinality less than κ has the property.

• 2. κ is a strongly compact cardinal for the property if every

structure (no restriction on the cardinality ) has the property given that every substructure of cardinality less than κ has the property. The duality of reflection and compactness: κ is a reflection cardinal for a certain property iff it is a compactness cardinal for the negation of the property

Compactness and Reflection Classifying reflection properties Corson compacts

Reflection properties form clusters. Some of the parameters defining the cluster to which a given reflection property belong are related to the reflection and strong reflection cardinals for this property.

Compactness and Reflection Classifying reflection properties Corson compacts

Reflection properties form clusters. Some of the parameters defining the cluster to which a given reflection property belong are related to the reflection and strong reflection cardinals for this property.

Fact

If λ is weakly compact then λ is compact for any of the properties considered above.

Compactness and Reflection Classifying reflection properties Corson compacts

Reflection properties form clusters. Some of the parameters defining the cluster to which a given reflection property belong are related to the reflection and strong reflection cardinals for this property.

Fact

If λ is weakly compact then λ is compact for any of the properties considered above.

Fact

Suppose that κ is a regular cardinal κ , S ⊆ κ is a stationary set

• f κ of such that for α ∈ S cof(α) = ω. Suppose that S does not
• reflect. (S ∩ α is not stationary in α for α < κ.). Then for most of

the properties the compactness property fails.

Compactness and Reflection Classifying reflection properties Corson compacts

Reflection properties form clusters. Some of the parameters defining the cluster to which a given reflection property belong are related to the reflection and strong reflection cardinals for this property.

Fact

If λ is weakly compact then λ is compact for any of the properties considered above.

Fact

Suppose that κ is a regular cardinal κ , S ⊆ κ is a stationary set

• f κ of such that for α ∈ S cof(α) = ω. Suppose that S does not
• reflect. (S ∩ α is not stationary in α for α < κ.). Then for most of

the properties the compactness property fails. For almost all the above properties compactness fails for ℵ1.

Compactness and Reflection Classifying reflection properties Corson compacts

Corollary (V = L)

• 1. For each of the above reflection properties a cardinal is a

reflection cardinal for the given property iff it is weakly compact.

Compactness and Reflection Classifying reflection properties Corson compacts

Corollary (V = L)

• 1. For each of the above reflection properties a cardinal is a

reflection cardinal for the given property iff it is weakly compact.

• 2. There is no strong reflection cardinal for the property.

Compactness and Reflection Classifying reflection properties Corson compacts

Corollary (V = L)

• 1. For each of the above reflection properties a cardinal is a

reflection cardinal for the given property iff it is weakly compact.

• 2. There is no strong reflection cardinal for the property.

Theorem

• 1. A supercompact compact cardinal is a reflection cardinal

for each of the reflection properties considered above.

Compactness and Reflection Classifying reflection properties Corson compacts

Corollary (V = L)

• 1. For each of the above reflection properties a cardinal is a

reflection cardinal for the given property iff it is weakly compact.

• 2. There is no strong reflection cardinal for the property.

Theorem

• 1. A supercompact compact cardinal is a reflection cardinal

for each of the reflection properties considered above.

• 2. It is consistent (Assuming the consistency of the existence
• f a supercompact cardinal) that for each of the reflection

properties above the minimal strong reflection cardinal is the first supercompact.

Compactness and Reflection Classifying reflection properties Corson compacts

Parameters for classifying reflection properties

Reflection cardinal Which is the smallest cardinal that can consistently be a reflection cardinal for this property?

Compactness and Reflection Classifying reflection properties Corson compacts

Parameters for classifying reflection properties

Reflection cardinal Which is the smallest cardinal that can consistently be a reflection cardinal for this property? Since for some of these properties we have that every singular cardinal is a reflection cardinal for the given property ("Shelah singular cardinals compactness") the more pertinent question is "which is the smallest regular cardinal that can consistently be a reflection cardinal for this property ?"

Compactness and Reflection Classifying reflection properties Corson compacts

Parameters for classifying reflection properties

Reflection cardinal Which is the smallest cardinal that can consistently be a reflection cardinal for this property? Since for some of these properties we have that every singular cardinal is a reflection cardinal for the given property ("Shelah singular cardinals compactness") the more pertinent question is "which is the smallest regular cardinal that can consistently be a reflection cardinal for this property ?" The consistency strength What is the consistency strength of the existence of a (regular) reflection cardinal for the property ?

Compactness and Reflection Classifying reflection properties Corson compacts

Parameters for classifying reflection properties

Reflection cardinal Which is the smallest cardinal that can consistently be a reflection cardinal for this property? Since for some of these properties we have that every singular cardinal is a reflection cardinal for the given property ("Shelah singular cardinals compactness") the more pertinent question is "which is the smallest regular cardinal that can consistently be a reflection cardinal for this property ?" The consistency strength What is the consistency strength of the existence of a (regular) reflection cardinal for the property ? Strong reflection cardinal What is the smallest cardinal which consistently be a strong reflection cardinal for the given property ?

Compactness and Reflection Classifying reflection properties Corson compacts

Parameters for classifying reflection properties

Reflection cardinal Which is the smallest cardinal that can consistently be a reflection cardinal for this property? Since for some of these properties we have that every singular cardinal is a reflection cardinal for the given property ("Shelah singular cardinals compactness") the more pertinent question is "which is the smallest regular cardinal that can consistently be a reflection cardinal for this property ?" The consistency strength What is the consistency strength of the existence of a (regular) reflection cardinal for the property ? Strong reflection cardinal What is the smallest cardinal which consistently be a strong reflection cardinal for the given property ? Consistency strength strong reflection What is the consistency

Compactness and Reflection Classifying reflection properties Corson compacts

The ℵ2 cluster

Mitchel, weakly compact ℵ2 can have the tree property. Harington-Shelah, Mahlo cardinal ℵ2 can be a reflection cardinal for the property "S is a stationary set of points of cofinality ω.".

Compactness and Reflection Classifying reflection properties Corson compacts

The ℵ2 cluster

Mitchel, weakly compact ℵ2 can have the tree property. Harington-Shelah, Mahlo cardinal ℵ2 can be a reflection cardinal for the property "S is a stationary set of points of cofinality ω.". Baumgartner, weakly compact cardinal ℵ2 can be a reflection cardinal for the property "S is a stationary subset

• f countable sets".

Compactness and Reflection Classifying reflection properties Corson compacts

The ℵ2 cluster

Mitchel, weakly compact ℵ2 can have the tree property. Harington-Shelah, Mahlo cardinal ℵ2 can be a reflection cardinal for the property "S is a stationary set of points of cofinality ω.". Baumgartner, weakly compact cardinal ℵ2 can be a reflection cardinal for the property "S is a stationary subset

• f countable sets".

weakly compact ℵ2 can be a reflection cardinal for the property " F is a family of sets that can not be disjointified.

Compactness and Reflection Classifying reflection properties Corson compacts

The ℵ2 cluster

Mitchel, weakly compact ℵ2 can have the tree property. Harington-Shelah, Mahlo cardinal ℵ2 can be a reflection cardinal for the property "S is a stationary set of points of cofinality ω.". Baumgartner, weakly compact cardinal ℵ2 can be a reflection cardinal for the property "S is a stationary subset

• f countable sets".

weakly compact ℵ2 can be a reflection cardinal for the property " F is a family of sets that can not be disjointified. Fleissner-Shelah ,weakly compact ℵ2 can be a reflection cardinal for the property " X is a Hausdorff first countable topological space which is locally of cardinality ≤ ℵ1 . Y ⊂ X is a discrete subset that can not be separated.

Compactness and Reflection Classifying reflection properties Corson compacts

The ℵω+1 cluster

Theorem (M.-Shelah)

Assuming the consistency of ω many supercompacts. ℵω+1 can be a reflection cardinal fpor the following properties:

• 1. An family of countable sets has no transversal.

Compactness and Reflection Classifying reflection properties Corson compacts

The ℵω+1 cluster

Theorem (M.-Shelah)

Assuming the consistency of ω many supercompacts. ℵω+1 can be a reflection cardinal fpor the following properties:

• 1. An family of countable sets has no transversal.
• 2. An abelian group is not free.

Compactness and Reflection Classifying reflection properties Corson compacts

The ℵω+1 cluster

Theorem (M.-Shelah)

Assuming the consistency of ω many supercompacts. ℵω+1 can be a reflection cardinal fpor the following properties:

• 1. An family of countable sets has no transversal.
• 2. An abelian group is not free.
• 3. A group is not free .

Compactness and Reflection Classifying reflection properties Corson compacts

The ℵω+1 cluster

Theorem (M.-Shelah)

Assuming the consistency of ω many supercompacts. ℵω+1 can be a reflection cardinal fpor the following properties:

• 1. An family of countable sets has no transversal.
• 2. An abelian group is not free.
• 3. A group is not free .
• 4. A graph has a coloring number ℵ1

Compactness and Reflection Classifying reflection properties Corson compacts

The ℵω+1 cluster

Theorem (M.-Shelah)

Assuming the consistency of ω many supercompacts. ℵω+1 can be a reflection cardinal fpor the following properties:

• 1. An family of countable sets has no transversal.
• 2. An abelian group is not free.
• 3. A group is not free .
• 4. A graph has a coloring number ℵ1

For items 1 − 3 ℵω+1 is the smallest regular cardinal that can be a reflection cardinal for these properties.

Compactness and Reflection Classifying reflection properties Corson compacts

The ω1-strongly compact family?

Definition

A cardinal κ is ω1 strongly compact if every κ complete filter can be extended to a countably complete ultrafilter.

Compactness and Reflection Classifying reflection properties Corson compacts

The ω1-strongly compact family?

Definition

A cardinal κ is ω1 strongly compact if every κ complete filter can be extended to a countably complete ultrafilter.

Theorem (Bagaria-M.)

κ is ω1 strongly compact if it is a strong reflection cardinal for the property of an Abelian group being not free*.

Compactness and Reflection Classifying reflection properties Corson compacts

The ω1-strongly compact family?

Definition

A cardinal κ is ω1 strongly compact if every κ complete filter can be extended to a countably complete ultrafilter.

Theorem (Bagaria-M.)

κ is ω1 strongly compact if it is a strong reflection cardinal for the property of an Abelian group being not free*.

Conjecture

The properties of a first countable space being metric and the property of a graph having chromatic number ℵ0 "behave" like the property of an Abelian group being free*.

Compactness and Reflection Classifying reflection properties Corson compacts

Reflection for topological space

There are two kinds of reflection for topological spaces. Given a topological space with a certain property we can ask if there is a smaller subspace having the property . "Subspace reflection".

Compactness and Reflection Classifying reflection properties Corson compacts

Reflection for topological space

There are two kinds of reflection for topological spaces. Given a topological space with a certain property we can ask if there is a smaller subspace having the property . "Subspace reflection". Another kind of reflection for topological spaces is continuous image reflection , given the space with a certain property , is there a continuous image with a smaller weight with the same

• property. "Image reflection"

Compactness and Reflection Classifying reflection properties Corson compacts

Reflection for topological space

There are two kinds of reflection for topological spaces. Given a topological space with a certain property we can ask if there is a smaller subspace having the property . "Subspace reflection". Another kind of reflection for topological spaces is continuous image reflection , given the space with a certain property , is there a continuous image with a smaller weight with the same

• property. "Image reflection"

Definition

A cardinal κ is an image reflection cardinal for a property (P) of a topological spaces if every space of weight κ which has property (P) has a continuous image of weight < κ having property (P).

Compactness and Reflection Classifying reflection properties Corson compacts

Reflection for topological space

There are two kinds of reflection for topological spaces. Given a topological space with a certain property we can ask if there is a smaller subspace having the property . "Subspace reflection". Another kind of reflection for topological spaces is continuous image reflection , given the space with a certain property , is there a continuous image with a smaller weight with the same

• property. "Image reflection"

Definition

A cardinal κ is an image reflection cardinal for a property (P) of a topological spaces if every space of weight κ which has property (P) has a continuous image of weight < κ having property (P). similarly we can define when κ is image compactness cardinal for the property (P).

Compactness and Reflection Classifying reflection properties Corson compacts

Corson compacta

Definition

a topological space is Corson compactum if it is homeomorphic to a compact subspace Y of some Tychonov cube [0, 1]κ such that this subspace has the property that for every y ∈ Y the set {ξ < κ|y(ξ) = 0} is countable.

Compactness and Reflection Classifying reflection properties Corson compacts

Corson compacta

Definition

a topological space is Corson compactum if it is homeomorphic to a compact subspace Y of some Tychonov cube [0, 1]κ such that this subspace has the property that for every y ∈ Y the set {ξ < κ|y(ξ) = 0} is countable.

Question

For what cardinals can be image compactness cardinals for the property of a space being Corson compactum . (Or equivalently the cardinal being a reflection cardinal for the property of the space not being Corson compactum.)

Compactness and Reflection Classifying reflection properties Corson compacts

Theorem (M.-Plebanek)

Suppose that κ > ω and there is a stationary subset S ⊆ κ such that cof(α) = ω for every α ∈ S and S does not reflect. Then there is a compact space X of weight κ which is not Corson compactum,but every continuous image of X of weight less than κ is a Corson compactum.

Compactness and Reflection Classifying reflection properties Corson compacts

Theorem (M.-Plebanek)

Suppose that κ > ω and there is a stationary subset S ⊆ κ such that cof(α) = ω for every α ∈ S and S does not reflect. Then there is a compact space X of weight κ which is not Corson compactum,but every continuous image of X of weight less than κ is a Corson compactum.

Question

Is it consistent that ω2 is a image compactness cardinal for the property of a topological space being Corson compactum ?

Compactness and Reflection Classifying reflection properties Corson compacts

Guessing models for trees

Definition

Let κ be a regular cardinal. Let T be a tree on κ+ of height κ+ . Let M ≺ Hθ be an elementary substructure , where θ is large enough such that |M| = κ, T ∈ M, κ ⊆ M. M is a guessing model for the tree T if α = M ∩ κ+ then for every branch b of Tα there is a cofinal branch in T, b∗ ∈ M such that b∗ ↾ α = b.

Compactness and Reflection Classifying reflection properties Corson compacts

Guessing models for trees

Definition

Let κ be a regular cardinal. Let T be a tree on κ+ of height κ+ . Let M ≺ Hθ be an elementary substructure , where θ is large enough such that |M| = κ, T ∈ M, κ ⊆ M. M is a guessing model for the tree T if α = M ∩ κ+ then for every branch b of Tα there is a cofinal branch in T, b∗ ∈ M such that b∗ ↾ α = b.

Definition

The combinatorial statement ∗κ is the statement that if T is a tree of height κ+ such that for every α < κ+ cof(α) = κ Tα has at most κ many cofinal branches. Then for θ large enough there is an increasing sequence of elementary submodels of Hθ Mα|α < κ+ which is continuous at every α < κ+, cof(α) = κ such that each Mα is guessing model for T.

Compactness and Reflection Classifying reflection properties Corson compacts

Fact

∗κ implies that κ+ satisfies Kurepa hypothesis.

Compactness and Reflection Classifying reflection properties Corson compacts

Fact

∗κ implies that κ+ satisfies Kurepa hypothesis.

Theorem (Farah-M)

Suppose that κ is a regular cardinal , κ < λ where λ is supercompact past the next inaccessible. Then there is < κ-closed forcing such that in the resulting model λ = κ+ and ∗κ holds.

Compactness and Reflection Classifying reflection properties Corson compacts

Theorem (Farah-M.)

Suppose that CH and ∗ω1. Then ω2 is a image compactness cardinal for the property of a topological space being Corson compactum.

Compactness and Reflection Classifying reflection properties Corson compacts

Theorem (Farah-M.)

Suppose that CH and ∗ω1. Then ω2 is a image compactness cardinal for the property of a topological space being Corson compactum. The proof goes by dualizing the problem. Rather than talking about the space X we talk about the C∗ algebra of continuous complex valued functions on X- C(X). We have a notion of an Abelian C∗ algebra being Corson.

Compactness and Reflection Classifying reflection properties Corson compacts

Theorem (Farah-M.)

Suppose that CH and ∗ω1. Then ω2 is a image compactness cardinal for the property of a topological space being Corson compactum. The proof goes by dualizing the problem. Rather than talking about the space X we talk about the C∗ algebra of continuous complex valued functions on X- C(X). We have a notion of an Abelian C∗ algebra being Corson.

Fact

The compact space X is Corson compactum iff the C∗ algebra C(C) is Corson.

Compactness and Reflection Classifying reflection properties Corson compacts

Theorem (Farah-M.)

Suppose that CH and ∗ω1. Then ω2 is a image compactness cardinal for the property of a topological space being Corson compactum. The proof goes by dualizing the problem. Rather than talking about the space X we talk about the C∗ algebra of continuous complex valued functions on X- C(X). We have a notion of an Abelian C∗ algebra being Corson.

Fact

The compact space X is Corson compactum iff the C∗ algebra C(C) is Corson.

Compactness and Reflection Classifying reflection properties Corson compacts

Since the compact space Y is a continuous image of the compact space X iff C(Y) is a subalgebra of C(X) we get:

Lemma

A cardinal κ is a image compactness cardinal for the property

• f a topological space being Corson iff Every Abelian C∗

algebra which is generated by κ many elements and which is not Corson has a subalgebra generated by less than κ many elements which is not Corson.

Compactness and Reflection Classifying reflection properties Corson compacts

Theorem (Farah-M.)

Assuming the consistency of the existence of a cardinal λ which is supercompact past the next inaccessible. Then it is consistent that ω2 is a image compactness cardinal for the property of a topological space being Corson compactum.

Compactness and Reflection Classifying reflection properties Corson compacts

Theorem (Farah-M.)

Assuming the consistency of the existence of a cardinal λ which is supercompact past the next inaccessible. Then it is consistent that ω2 is a image compactness cardinal for the property of a topological space being Corson compactum. So the image reflection problem for a space not being Corson seems to be a sort of it own.

Compactness and Reflection Classifying reflection properties Corson compacts

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