Descriptive and combinatorial set theory at singular...

Descriptive andcombinatorial settheory at singularcardinals and their

successors

Mirna Dzamonja

Introduction

Singular cardinals,descriptively

Singular cardinals,combinatorially

Singular cardinals,topologically

Successor of asingular cardinal

Descriptive and combinatorial set theoryat singular cardinals and their successors

Mirna Dzamonja

School of Mathematics, University of East Anglia,associee IHPST, Universite Pantheon-Sorbonne, Paris 1

Torino, September 2017

successors

Mirna Dzamonja

Introduction

Generalising the Baire spaceThe Baire space ωω is identified with the product ωω andis given the usual product topology.

A naturalgeneralisation of the space is a topology on κκ for someκ > ℵ0 and some natural generalisation of the producttopology.

A natural generalisation of the product topology is to fixsome cardinal λ ≤ κ and to take basic open sets of theform N(f ) = {g : g � dom(f ) = f} for f a partial functionfrom κ to κ with | dom(f )| < λ. The most studied case iswhen λ = κ and it gives what people call the generalisedBaire space.Topologists have studied combinatorial properties ofgeneralised products of spaces since 1920s, usually withdiscouraging results, such as that the compactness is notpreserved. For example even the space Rω with the boxtopology is not connected or first countable, hence notmetrisable.

successors

Mirna Dzamonja

Introduction

Generalising the Baire spaceThe Baire space ωω is identified with the product ωω andis given the usual product topology. A naturalgeneralisation of the space is a topology on κκ for someκ > ℵ0 and some natural generalisation of the producttopology.

successors

Mirna Dzamonja

Introduction

A natural generalisation of the product topology is to fixsome cardinal λ ≤ κ and to take basic open sets of theform N(f ) = {g : g � dom(f ) = f} for f a partial functionfrom κ to κ with | dom(f )| < λ.

The most studied case iswhen λ = κ and it gives what people call the generalisedBaire space.Topologists have studied combinatorial properties ofgeneralised products of spaces since 1920s, usually withdiscouraging results, such as that the compactness is notpreserved. For example even the space Rω with the boxtopology is not connected or first countable, hence notmetrisable.

successors

Mirna Dzamonja

Introduction

A natural generalisation of the product topology is to fixsome cardinal λ ≤ κ and to take basic open sets of theform N(f ) = {g : g � dom(f ) = f} for f a partial functionfrom κ to κ with | dom(f )| < λ. The most studied case iswhen λ = κ and it gives what people call the generalisedBaire space.

Topologists have studied combinatorial properties ofgeneralised products of spaces since 1920s, usually withdiscouraging results, such as that the compactness is notpreserved. For example even the space Rω with the boxtopology is not connected or first countable, hence notmetrisable.

successors

Mirna Dzamonja

Introduction

A natural generalisation of the product topology is to fixsome cardinal λ ≤ κ and to take basic open sets of theform N(f ) = {g : g � dom(f ) = f} for f a partial functionfrom κ to κ with | dom(f )| < λ. The most studied case iswhen λ = κ and it gives what people call the generalisedBaire space.Topologists have studied combinatorial properties ofgeneralised products of spaces since 1920s, usually withdiscouraging results, such as that the compactness is notpreserved.

For example even the space Rω with the boxtopology is not connected or first countable, hence notmetrisable.

successors

Mirna Dzamonja

Introduction

successors

Mirna Dzamonja

Introduction

successors

Mirna Dzamonja

Introduction

The generalised Baire space descriptivelyDescriptive set theory of generalised spaces took longerto develop.

The first paper on this subject was Vaananen(FM, 1991). It considered the analogue of theCantor-Bendixon theorem in ω1ω1 and showed that itsdirect analogue (replacing ω by ω1) is consistently truemodulo a measurable cardinal. It also introducedconnections with games.

Today, the descriptive set theory of generalised Bairespaces is well developed and involves many authors,including S. Friedman, Hyttinnen, Khomskii, Kulikov,Laguzzi, Motto Ros and many others.

These authors have developed a rich theory, mostlyconcentrating on the case κ regular, in particularsuccessor of regular or inaccessible. Often, thegeneralised Baire space does not allow directgeneralisations of theorems about the Baire space andnew techniques and expectations have to be made.

successors

Mirna Dzamonja

Introduction

The generalised Baire space descriptivelyDescriptive set theory of generalised spaces took longerto develop. The first paper on this subject was Vaananen(FM, 1991).

It considered the analogue of theCantor-Bendixon theorem in ω1ω1 and showed that itsdirect analogue (replacing ω by ω1) is consistently truemodulo a measurable cardinal. It also introducedconnections with games.

successors

Mirna Dzamonja

Introduction

The generalised Baire space descriptivelyDescriptive set theory of generalised spaces took longerto develop. The first paper on this subject was Vaananen(FM, 1991). It considered the analogue of theCantor-Bendixon theorem in ω1ω1 and showed that itsdirect analogue (replacing ω by ω1) is consistently truemodulo a measurable cardinal.

It also introducedconnections with games.

successors

Mirna Dzamonja

Introduction

The generalised Baire space descriptivelyDescriptive set theory of generalised spaces took longerto develop. The first paper on this subject was Vaananen(FM, 1991). It considered the analogue of theCantor-Bendixon theorem in ω1ω1 and showed that itsdirect analogue (replacing ω by ω1) is consistently truemodulo a measurable cardinal. It also introducedconnections with games.

successors

Mirna Dzamonja

Introduction

successors

Mirna Dzamonja

Introduction

These authors have developed a rich theory, mostlyconcentrating on the case κ regular, in particularsuccessor of regular or inaccessible.

Often, thegeneralised Baire space does not allow directgeneralisations of theorems about the Baire space andnew techniques and expectations have to be made.

successors

Mirna Dzamonja

Introduction

successors

Mirna Dzamonja

Introduction

A related space

I have been interested in the generalised Baire space inthe case that κ is a singular cardinal.

In this case it is alsointeresting to consider the space κcf(κ). For simplicity letus work with κ strong limit singular of countable cofinality.In the space κω there is a dense set of size κ, the

topology is 0-dimensional (ultra)metrizable and eachopen set is the union of κ closed sets.

DefinitionA set A ⊆ κω is Π1

1 if there is an open set B ⊆ κω × κω (inthe product topology) such that for every f ∈ ωκ

f ∈ A ⇐⇒ ∀g ((f ,g)) ∈ B). (1)

A set is Σ11 if its complement is Π1

1 and it is ∆11 if it is both

Π11 and Σ1

successors

Mirna Dzamonja

Introduction

A related space

I have been interested in the generalised Baire space inthe case that κ is a singular cardinal. In this case it is alsointeresting to consider the space κcf(κ).

For simplicity letus work with κ strong limit singular of countable cofinality.In the space κω there is a dense set of size κ, the

f ∈ A ⇐⇒ ∀g ((f ,g)) ∈ B). (1)

Π11 and Σ1

successors

Mirna Dzamonja

Introduction

A related space

I have been interested in the generalised Baire space inthe case that κ is a singular cardinal. In this case it is alsointeresting to consider the space κcf(κ). For simplicity letus work with κ strong limit singular of countable cofinality.In the space κω there is a dense set of size κ, the

f ∈ A ⇐⇒ ∀g ((f ,g)) ∈ B). (1)

Π11 and Σ1

successors

Mirna Dzamonja

Introduction

A related space

f ∈ A ⇐⇒ ∀g ((f ,g)) ∈ B). (1)

Π11 and Σ1

successors

Mirna Dzamonja

Introduction

A related space

f ∈ A ⇐⇒ ∀g ((f ,g)) ∈ B). (1)

Π11 and Σ1

successors

Mirna Dzamonja

Introduction

Covering and boundednessWe shall present two results, from our paper withVaananen (JML, 2011), corresponding to what is knownabout the Baire space.

To introduce them, we need somenotation. Let T O denote the class of all well-foundedtrees of size κ. Order them by letting T ≤ T ′ if there is a≤-preserving function from T to T ′. If S is a tree of pairs(f ,g) ∈ ωκ ordered by initial segment and f ∈ ωκ, thenS(f ) = {g : (f ,g) ∈ S}.

Theorem (Representation Theorem)A set A is Π1

1 in the space κω iff there exists a tree onωκ× ωκ such that f ∈ A ⇐⇒ T (f ) ∈ T O.

We say A is represented by T .

Theorem (Boundedness Theorem)Suppose that A is Π1

1 in κω and represented by the tree T .Then A is ∆1

1 if and only if there is g ∈ T O such that∀f ∈ A (T (f ) ≤ T (g)).

successors

Mirna Dzamonja

Introduction

Covering and boundednessWe shall present two results, from our paper withVaananen (JML, 2011), corresponding to what is knownabout the Baire space. To introduce them, we need somenotation.

Let T O denote the class of all well-foundedtrees of size κ. Order them by letting T ≤ T ′ if there is a≤-preserving function from T to T ′. If S is a tree of pairs(f ,g) ∈ ωκ ordered by initial segment and f ∈ ωκ, thenS(f ) = {g : (f ,g) ∈ S}.

successors

Mirna Dzamonja

Introduction

Covering and boundednessWe shall present two results, from our paper withVaananen (JML, 2011), corresponding to what is knownabout the Baire space. To introduce them, we need somenotation. Let T O denote the class of all well-foundedtrees of size κ.

Order them by letting T ≤ T ′ if there is a≤-preserving function from T to T ′. If S is a tree of pairs(f ,g) ∈ ωκ ordered by initial segment and f ∈ ωκ, thenS(f ) = {g : (f ,g) ∈ S}.

successors

Mirna Dzamonja

Introduction

Covering and boundednessWe shall present two results, from our paper withVaananen (JML, 2011), corresponding to what is knownabout the Baire space. To introduce them, we need somenotation. Let T O denote the class of all well-foundedtrees of size κ. Order them by letting T ≤ T ′ if there is a≤-preserving function from T to T ′.

If S is a tree of pairs(f ,g) ∈ ωκ ordered by initial segment and f ∈ ωκ, thenS(f ) = {g : (f ,g) ∈ S}.

successors

Mirna Dzamonja

Introduction

Covering and boundednessWe shall present two results, from our paper withVaananen (JML, 2011), corresponding to what is knownabout the Baire space. To introduce them, we need somenotation. Let T O denote the class of all well-foundedtrees of size κ. Order them by letting T ≤ T ′ if there is a≤-preserving function from T to T ′. If S is a tree of pairs(f ,g) ∈ ωκ ordered by initial segment and f ∈ ωκ, thenS(f ) = {g : (f ,g) ∈ S}.

successors

Mirna Dzamonja

Introduction

successors

Mirna Dzamonja

Introduction

successors

Mirna Dzamonja

Introduction

successors

Mirna Dzamonja

Introduction

CofinalitiesOne may feel that the above theorems are easy becausewe deal with countable cofinality, so there is a naturalnotion of well-founded trees.

However, one should not betoo quick to say that this is obviously the case, sincesome similarities that one may naively think should holdtrue between κ of countable cofinality and ω, in fact do nothold. For example, the analogue of Konig’s lemma fails.

Observation There is a κ-Souslin tree.

Proof.Let 〈κn : n < ω〉 be an increasing sequence in κ. Adisjoint rooted union of the ordinals κn (n < ω) providesan example.

In fact our descriptive set theory theorem works for anycofinality in place of ω, with natural replacement ofwell-founded by ”with no branches of length ...”.Mekler and Vaananen (FM 1993) proved that under CHboundedness holds in ω1ω1.

successors

Mirna Dzamonja

Introduction

CofinalitiesOne may feel that the above theorems are easy becausewe deal with countable cofinality, so there is a naturalnotion of well-founded trees. However, one should not betoo quick to say that this is obviously the case, sincesome similarities that one may naively think should holdtrue between κ of countable cofinality and ω, in fact do nothold.

For example, the analogue of Konig’s lemma fails.

successors

Mirna Dzamonja

Introduction

CofinalitiesOne may feel that the above theorems are easy becausewe deal with countable cofinality, so there is a naturalnotion of well-founded trees. However, one should not betoo quick to say that this is obviously the case, sincesome similarities that one may naively think should holdtrue between κ of countable cofinality and ω, in fact do nothold. For example, the analogue of Konig’s lemma fails.

successors

Mirna Dzamonja

Introduction

successors

Mirna Dzamonja

Introduction

successors

Mirna Dzamonja

Introduction

In fact our descriptive set theory theorem works for anycofinality in place of ω, with natural replacement ofwell-founded by ”with no branches of length ...”.

Mekler and Vaananen (FM 1993) proved that under CHboundedness holds in ω1ω1.

successors

Mirna Dzamonja

Introduction

successors

Mirna Dzamonja

Introduction

Cardinal invariantsLet κ be singular, for simplicity again cf(κ) = ω and let〈κn : n < ω〉 be a sequence of regular cardinalsincreasing to κ with κ0 = 0.

Consider the space κκ offunctions, which we can now partially order by lettingf ≤∗κ f ′ if {α < κ : f (α) > f ′(α)} is bounded in κ. Thecardinal invariants of this space are denoted by d(κ) etc.It turns out that one can connect this space with the Bairespace and show that certain of the cardinal invariants arethe same as their analogues in the Baire space.

For α < κ let k(α) be the unique k such thatα ∈ [κk , κk+1). We use these to code κκ into the Bairespace.

For f ∈ κκ let gf ∈ ωω be given by gf (n) = k(f (κn)). Forg ∈ ωω let f g ∈ κκ be given by letting for all α,

f g(α) = κn+1 iff g(k(α)) = n.

successors

Mirna Dzamonja

Introduction

Cardinal invariantsLet κ be singular, for simplicity again cf(κ) = ω and let〈κn : n < ω〉 be a sequence of regular cardinalsincreasing to κ with κ0 = 0. Consider the space κκ offunctions, which we can now partially order by lettingf ≤∗κ f ′ if {α < κ : f (α) > f ′(α)} is bounded in κ.

Thecardinal invariants of this space are denoted by d(κ) etc.It turns out that one can connect this space with the Bairespace and show that certain of the cardinal invariants arethe same as their analogues in the Baire space.

successors

Mirna Dzamonja

Introduction

Cardinal invariantsLet κ be singular, for simplicity again cf(κ) = ω and let〈κn : n < ω〉 be a sequence of regular cardinalsincreasing to κ with κ0 = 0. Consider the space κκ offunctions, which we can now partially order by lettingf ≤∗κ f ′ if {α < κ : f (α) > f ′(α)} is bounded in κ. Thecardinal invariants of this space are denoted by d(κ) etc.

It turns out that one can connect this space with the Bairespace and show that certain of the cardinal invariants arethe same as their analogues in the Baire space.

successors

Mirna Dzamonja

Introduction

Cardinal invariantsLet κ be singular, for simplicity again cf(κ) = ω and let〈κn : n < ω〉 be a sequence of regular cardinalsincreasing to κ with κ0 = 0. Consider the space κκ offunctions, which we can now partially order by lettingf ≤∗κ f ′ if {α < κ : f (α) > f ′(α)} is bounded in κ. Thecardinal invariants of this space are denoted by d(κ) etc.It turns out that one can connect this space with the Bairespace and show that certain of the cardinal invariants arethe same as their analogues in the Baire space.

successors

Mirna Dzamonja

Introduction

For α < κ let k(α) be the unique k such thatα ∈ [κk , κk+1).

We use these to code κκ into the Bairespace.

successors

Mirna Dzamonja

Introduction

successors

Mirna Dzamonja

Introduction

successors

Mirna Dzamonja

Introduction

Dominating

We observe some basic properties of the aboveoperations.

Lemma(1) If f is non-decreasing then gf is non-decreasing.(2) If g is non-decreasing then f g is non-decreasing.(3) k(f gf (κn)) = k(f (κn)) + 1, for every n.(4) If f is non-decreasing and gf ≤∗ g, then f ≤∗κ f g .

Using this type of reasoning ((4) is used for ≤), we obtain

Theoremd(κ) = d.We note that generalised cardinal invariants for regularcardinals can behave quite wildly, this is well documentedin works by various authors.

successors

Mirna Dzamonja

Introduction

Dominating

Lemma(1) If f is non-decreasing then gf is non-decreasing.

(2) If g is non-decreasing then f g is non-decreasing.(3) k(f gf (κn)) = k(f (κn)) + 1, for every n.(4) If f is non-decreasing and gf ≤∗ g, then f ≤∗κ f g .

successors

Mirna Dzamonja

Introduction

Dominating

Lemma(1) If f is non-decreasing then gf is non-decreasing.(2) If g is non-decreasing then f g is non-decreasing.

(3) k(f gf (κn)) = k(f (κn)) + 1, for every n.(4) If f is non-decreasing and gf ≤∗ g, then f ≤∗κ f g .

successors

Mirna Dzamonja

Introduction

Dominating

Lemma(1) If f is non-decreasing then gf is non-decreasing.(2) If g is non-decreasing then f g is non-decreasing.(3) k(f gf (κn)) = k(f (κn)) + 1, for every n.

(4) If f is non-decreasing and gf ≤∗ g, then f ≤∗κ f g .

successors

Mirna Dzamonja

Introduction

Dominating

successors

Mirna Dzamonja

Introduction

Dominating

successors

Mirna Dzamonja

Introduction

Dominating

successors

Mirna Dzamonja

Introduction

Dominating

Theoremd(κ) = d.

We note that generalised cardinal invariants for regularcardinals can behave quite wildly, this is well documentedin works by various authors.

successors

Mirna Dzamonja

Introduction

Dominating

successors

Mirna Dzamonja

Introduction

Comfort’s question and its consequences

What are the topological properties of the space κκ or 2κ

with various box products?

Comfort and Gotchev(including a paper in DM 2016), building also on work bymany authors, have studied this question in detail and formany more general basis spaces than κ. They obtainedvery satisfactory results for the weight function- but thedensity function turned out much more complicated. Infact this question, posed by Comfort, prompted Gitik andShelah (TOPAP 1998) to develop a new technique inforcing and obtain the following result:

TheoremModulo large cardinals, it is consistent to have a singularcardinal κ with countable cofinality such that the densityof the countably supported box product space is κ+ < 2κ.

successors

Mirna Dzamonja

Introduction

with various box products? Comfort and Gotchev(including a paper in DM 2016), building also on work bymany authors, have studied this question in detail and formany more general basis spaces than κ.

They obtainedvery satisfactory results for the weight function- but thedensity function turned out much more complicated. Infact this question, posed by Comfort, prompted Gitik andShelah (TOPAP 1998) to develop a new technique inforcing and obtain the following result:

successors

Mirna Dzamonja

Introduction

with various box products? Comfort and Gotchev(including a paper in DM 2016), building also on work bymany authors, have studied this question in detail and formany more general basis spaces than κ. They obtainedvery satisfactory results for the weight function- but thedensity function turned out much more complicated.

Infact this question, posed by Comfort, prompted Gitik andShelah (TOPAP 1998) to develop a new technique inforcing and obtain the following result:

successors

Mirna Dzamonja

Introduction

with various box products? Comfort and Gotchev(including a paper in DM 2016), building also on work bymany authors, have studied this question in detail and formany more general basis spaces than κ. They obtainedvery satisfactory results for the weight function- but thedensity function turned out much more complicated. Infact this question, posed by Comfort, prompted Gitik andShelah (TOPAP 1998) to develop a new technique inforcing and obtain the following result:

successors

Mirna Dzamonja

Introduction

with various box products? Comfort and Gotchev(including a paper in DM 2016), building also on work bymany authors, have studied this question in detail and formany more general basis spaces than κ. They obtainedvery satisfactory results for the weight function- but thedensity function turned out much more complicated. Infact this question, posed by Comfort, prompted Gitik andShelah (TOPAP 1998) to develop a new technique inforcing and obtain the following result:

successors

Mirna Dzamonja

Introduction

Consequences of Gitik-Shelah

One of the consequences was to suggest a new method,which we developed in a paper with Shelah (JSL 2003),later taken on in a series of paper with coworkersincluding Cummings, Komjath, Magidor and Morgan.

We start from a supercompact cardinal κ, force 2κ largewhile at the same time obtaining a normal measure D onit generated by a small number of sets (this was alsodone by Gitik and Shelah) but we are able to control whatwill happen to κ in the extension by the Prikry forcing withD. We can also do this for Radin forcing and Prikry withinterleaved collapses. J. Davies (APAL, to appear) hasdone it for Radin with interleaved collapses.

successors

Mirna Dzamonja

Introduction

We start from a supercompact cardinal κ, force 2κ largewhile at the same time obtaining a normal measure D onit generated by a small number of sets (this was alsodone by Gitik and Shelah)

but we are able to control whatwill happen to κ in the extension by the Prikry forcing withD. We can also do this for Radin forcing and Prikry withinterleaved collapses. J. Davies (APAL, to appear) hasdone it for Radin with interleaved collapses.

successors

Mirna Dzamonja

Introduction

We start from a supercompact cardinal κ, force 2κ largewhile at the same time obtaining a normal measure D onit generated by a small number of sets (this was alsodone by Gitik and Shelah) but we are able to control whatwill happen to κ in the extension by the Prikry forcing withD.

We can also do this for Radin forcing and Prikry withinterleaved collapses. J. Davies (APAL, to appear) hasdone it for Radin with interleaved collapses.

successors

Mirna Dzamonja

Introduction

We start from a supercompact cardinal κ, force 2κ largewhile at the same time obtaining a normal measure D onit generated by a small number of sets (this was alsodone by Gitik and Shelah) but we are able to control whatwill happen to κ in the extension by the Prikry forcing withD. We can also do this for Radin forcing and Prikry withinterleaved collapses.

J. Davies (APAL, to appear) hasdone it for Radin with interleaved collapses.

successors

Mirna Dzamonja

Introduction

We start from a supercompact cardinal κ, force 2κ largewhile at the same time obtaining a normal measure D onit generated by a small number of sets (this was alsodone by Gitik and Shelah) but we are able to control whatwill happen to κ in the extension by the Prikry forcing withD. We can also do this for Radin forcing and Prikry withinterleaved collapses. J. Davies (APAL, to appear) hasdone it for Radin with interleaved collapses.

successors

Mirna Dzamonja

Introduction

What is this good for?

This type of extension is convenient for:

1 get consistency results about the successor of κ (asdone in the above works, various results aboutgraphs) and

2 get consistency results about cardinal invariants at alarge cardinal, obtained by Garti and Shelah andBrooke-Taylor, V. Fischer, S. Friedman and Montoya.

For example, what can be said about the generalisedBaire space at λ = κ+?

successors

Mirna Dzamonja

Introduction

What is this good for?

This type of extension is convenient for:

1 get consistency results about the successor of κ (asdone in the above works, various results aboutgraphs) and

2 get consistency results about cardinal invariants at alarge cardinal, obtained by Garti and Shelah andBrooke-Taylor, V. Fischer, S. Friedman and Montoya.

For example, what can be said about the generalisedBaire space at λ = κ+?

successors

Mirna Dzamonja

Introduction

Recent progress

Our work at the successor of a singular has been mademore difficult by the fact that the individual forcing thatneeds to be iterated in our techniques is quitecomplicated and the only iteration theorems known aboutiterating θ+-cc forcing at θ regular uncountable involveshowing a very strong combinatorial form of the chaincondition (just θ+-cc is not enough).

In our ongoing work with Cummings and Neeman wehave developed a new iteration method at such cardinals,provided that θ has some large cardinal properties (as itdoes in our applications, where it is supercompact). Wehope to find many applications. So far, an application wehave is to a forcing by Mekler, which lets us provide auniversal graph at θ+ even if 2θ is large.

successors

Mirna Dzamonja

Introduction

Recent progress

In our ongoing work with Cummings and Neeman wehave developed a new iteration method at such cardinals,provided that θ has some large cardinal properties (as itdoes in our applications, where it is supercompact).

Wehope to find many applications. So far, an application wehave is to a forcing by Mekler, which lets us provide auniversal graph at θ+ even if 2θ is large.

successors

Mirna Dzamonja

Introduction

Recent progress

In our ongoing work with Cummings and Neeman wehave developed a new iteration method at such cardinals,provided that θ has some large cardinal properties (as itdoes in our applications, where it is supercompact). Wehope to find many applications.

So far, an application wehave is to a forcing by Mekler, which lets us provide auniversal graph at θ+ even if 2θ is large.

successors

Mirna Dzamonja

Introduction

Recent progress

In our ongoing work with Cummings and Neeman wehave developed a new iteration method at such cardinals,provided that θ has some large cardinal properties (as itdoes in our applications, where it is supercompact). Wehope to find many applications. So far, an application wehave is to a forcing by Mekler, which lets us provide auniversal graph at θ+ even if 2θ is large.

Descriptive and combinatorial set theory at singular...

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