On the Structure of Independent Families a Dissertation Presented To

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On the Structure of Independent Families

A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University

In partial fulﬁllment of the requirements for the degree Doctor of Philosophy

This dissertation titled On the Structure of Independent Families

by MICHAEL J. PERRON

has been approved for Department of Mathematics and the College of Arts and Sciences by

Todd Eisworth Associate Professor of Mathematics

Robert Frank Dean, College of Arts and Sciences 3

ABSTRACT

PERRON, MICHAEL J., Ph.D., April 2017, Mathematics On the Structure of Independent Families Director of Dissertation: Todd Eisworth

A paper by Shelah that established the consistency of i < u gave an independent family with a host of properties This dissertation investigates how these properties give rise to diﬀerent varieties of independent families. We explore the connection between independent families and ultraﬁlters. We use Martin’s Axiom as a guide for looking at the generic existence of independent families. We show that selective independent families are preserved in both Iterated Sacks Forcing and Side-by-side Sacks Forcing. 4

To Kate, Mom, and Dad. 5

ACKNOWLEDGEMENTS

I would like to thank my parents, William and Mary Perron. They gave so much for me to become the person I am. I would like to thank my girlfriend, Kate Suddarth. This was the most diﬃcult thing I have ever done, and her unending support kept me going through it all. I would like to thank my advisor, Dr. Todd Eisworth. I am extremely grateful for the time and eﬀort in helping me achieve a lifelong dream. 6

TABLE OF CONTENTS

Abstract ...... 3

Dedication ...... 4

Acknowledgements ...... 5

1 Preliminaries ...... 8 1.1 Deﬁnition ...... 8 1.2 Size of Independent Families ...... 9 1.3 Maximal Independent Families ...... 11

2 Selective Independent Families and Forcing ...... 16 2.1 Constructing Independent Families from CH ...... 18 2.2 Ideals and Independent Families ...... 20 2.3 Sacks Forcing and Iterated Sacks Forcing ...... 24 2.4 Side-by-Side Sacks Forcing ...... 29

3 Independence and Q ...... 31 3.1 Topological Interpretation of Independent Families ...... 31 3.2 Topology of Countable Independent Families ...... 32 3.3 Dense Independent Families ...... 34 3.4 Independent Families and ♦-like Principles ...... 37

4 Connections to Ultraﬁlter Properties ...... 41 4.1 Nothing Stronger than Selective Independence ...... 41 4.2 P, Q, and P-point ...... 41 4.3 The Dual Filter ...... 44 7

5 Generic Existence ...... 48 5.1 Independent Families, Martin’s Axiom ...... 48 5.2 Extending to Various Independent Families ...... 52 5.3 Independent Families as a Notion of Forcing ...... 55

6 Miscellaneous ...... 59 6.1 Direct and Inverse Images of Independent Families ...... 59 6.2 n-Independent Families ...... 61 6.3 Ultrapowers ...... 64

Bibliography ...... 67 8

Chapter 1: Preliminaries

1.1: Deﬁnition Our work concerns the structure of objects known as independent families of subsets of ω, deﬁned as follows:

Deﬁnition 1. A family I of subsets of ω is called independent if whenever we have distinct X0, ..., Xn,Y0, ..., Ym ∈ I then X0 ∩ ... ∩ Xn ∩ (ω \ Y0) ∩ ... ∩ (ω \ Ym) is inﬁnite. We call the collection of all such intersections the envelope of I, and we label it ENV(I).

The following notation is due to Shelah [1], and gives us a concise way of discussing the envelope of an independent family.

Definition 2. If A is a subset of ω, define A0 = A and A1 = (ω \A). If I is a set, let FF(I) to be the collection of finite partial functions from I to 2. If h ∈ FF(I) then Ih = T{Ah(A) : A is in the domain of h}.

h h1 h2 Notice that ENV(I) = {I : h ∈ FF(I)} and that I ⊆ I if and only if h1 ⊇ h2. All independent families considered in this paper will be infinite. Observe that if I is an infinite independent family, h ∈ FF(I), and x ∈ Ih, then we can extend h to some k ∈ FF(I) where x 6∈ Ik. To see this, find a set A ∈ I \ dom(h) and then find i ∈ 2 with x 6∈ Ai. Let k to be the extension of h to include A in the domain with k(A) = i. This observation will simplify our results going forward in the following way. If E is a member of the envelope and E ⊆∗ A (i.e. E \ A is finite) or if E ∩ A =∗ ∅ (i.e. E has finite intersection with A), then we can extend E to an F ∈ ENV(I) such that F ⊆ A or F ∩ A = ∅. We summarize this discussion in the following proposition. 9

Proposition 3. Let I be an inﬁnite independent family, A is a subset of ω, and E ∈ ENV(I). If E ⊆∗ A or E ∩ A =∗ ∅ then there is an F ∈ ENV(I) with F ⊆ E such that F ⊆ A or F ∩ A = ∅.

Now we can prove the following elementary lemma which pinpoints when we are able to enlarge an independent family while maintaining independence.

Lemma 4. Let I be an independent family and A ⊆ ω with A 6∈ I. Then the following are equivalent:

i) I ∪ {A} is independent.

ii) Both A and (ω \A) have non-empty intersection with every element of ENV(I).

Proof. Trivial

1.2: Size of Independent Families Independent families can be of any size, up to the size of P(ω). Hausdorﬀ de- scribed an independent family of size c in 1935 [11], and many others have also been constructed. For the beneﬁt of the reader we will present several examples here, and more can be found in [2].

Example 1. (Hausdorﬀ) Let F = {(k, A): k ∈ ω, A ⊆ P(k)}. For each X ⊆ ω let X0 = {(k, A) ∈ F : X ∩ k ∈ A}. Then I = {X0 : X ⊆ ω} is an independent family.

Proof. Let X0, ..., Xn,Y0, ..., Ym be diﬀerent subsets of ω. Since we have a ﬁnite num-

ber of diﬀerent sets we can ﬁnd a k ∈ ω large enough that k ∩ X0, ..., k ∩ Xn, k ∩

Y0, ..., k ∩ Ym are all distinct. Let A = {k ∩ X0, ..., k ∩ Xn}. Then A ⊂ P(k) and

0 0 0 0 (k, A) ∈ X0 ∩ ... ∩ Xn ∩ (F \ Y0 ) ∩ ... ∩ (F \ Ym). 10

Example 2. (Martin Goldstern, who attributes it to Menachem Kojman) Let P be the set of all polynomials with rational coeﬃcients. For each real r deﬁne

Ar = {p ∈ P : p(r) > 0}.

Then {Ar : r is real } is an independent family.

Proof. We can always find a polynomial that is positive for finitely many given reals and non-positive for finitely many other given reals.

Our third example requires the notion of an almost disjoint family.

Deﬁnition 5. A family A ⊆ [ω]ω is almost disjoint if the intersection between any two elements are ﬁnite.

We will omit an example, but there is an almost-disjoint family of size c.

Example 3. (KP Hart) Let F be an almost disjoint family of size c. To each A ∈ F deﬁne A0 to be the collection of all ﬁnite subsets of ω that intersect A. Then {A0 : A ∈ F } is an independent family.

Proof. Let X0, ..., Xn,Y0, ..., Ym be diﬀerent sets in A. For each i ≤ n ﬁnd ki ∈ Xi

where ki 6∈ Yj for every j ≤ m. This is possible since each Xi is inﬁnite yet has

only ﬁnite intersection with each of the ﬁnitely many Yj. Then {ki : i ≤ n} ∈

0 0 <ω 0 <ω 0 X0 ∩ ... ∩ Xn ∩ ([ω] \ Y0 ) ∩ ... ∩ ([ω] \ Ym).

This example by KP Hart suggests a connection between almost disjoint families and independent families which will be explored later. Now that we have some examples of independent families of size c we can show a connection to the notion of ultraﬁlters.

Deﬁnition 6. Call a family of sets F a ﬁlter if whenever: 11

1) B ∈ F and B ⊆ A then A ∈ F

2) whenever A, B ∈ F then A ∩ B ∈ F

We call F an ultraﬁlter if whenever A ⊆ ω then either A ∈ F or (ω \ A) ∈ F .

Posp´ıˇsil’sclassic proof that there are 22ℵ0 ultraﬁlters on ω was greatly simpliﬁed by Hajnal and Juhasz [13] using an independent family of size c. We present a version of this proof from Jech in [11] to demonstrate the utility of independent families.

Theorem 7. There are 22ℵ0 ultraﬁlters on ω.

Proof. Any ultraﬁlter on ω is an element of P(P(ω)) so there are at most 22ℵ0 ultra-

ﬁlters. Now let I be an independent family of ω of size 2ℵ0 . For each f : I → {0, 1}

deﬁne Gf = {X : X is co-ﬁnite } ∪ {X : f(X) = 1} ∪ {ω \ X : f(X) = 0}. Since I is

independent then Gf has the ﬁnite intersection property and so we can extend each

Gf to an ultraﬁlter Uf . If f 6= g then there is some X ∈ I such that f(X) 6= g(X). Without loss of

generality f(X) = 1 and g(X) = 0. So X ∈ Uf but (ω \ X) ∈ Ug hence Uf 6= Ug. So each function creates a distinct ultraﬁlter, and since there are 22ℵ0 functions from I to 2 then there are 22ℵ0 ultraﬁlters on ω.

1.3: Maximal Independent Families An easy application of Zorn’s lemma establishes the existence of an independent family I that is maximal in the sense that it is not property contained in a larger independent family, i.e. that if A ⊆ ω with A 6∈ I then I ∪ {A} is not independent. Moreover, this shows that any independent family is contained in a maximal independent family.

Deﬁnition 8. An independent family I is maximal if whenever A ⊆ ω with A 6∈ I then I ∪ {A} is not independent. 12

We have the easy following characterization of a maximal independent families.

Lemma 9. Let I be an independent family. The following are equivalent:

i) I is maximal.

ii) If A ⊆ ω, there is a B in ENV(I) such that B ⊆ A or B ∩ A = ∅.

iii) If f : ω → 2 then there is a set B in ENV(I) such that f B is constant.

Proof. Trivial

A family R ⊆ [ω]ω is called reaping (or unsplittable) if there is no one set A such that any B ∈ R then |A ∩ B| = ℵ0 and |B \ A| = ℵ0. As evidenced by the previous lemma, an independent family I is maximal if and only if the envelope of I is a reaping family. By a simple diagonalization argument we know that no countable independent family an be maximal.

Lemma 10. If I is a countably inﬁnite independent family then there is a set A ⊆ ω where A 6∈ I and I ∪ {A} is independent.

Proof. Since I is countable we know ENV(I) is also countable, so let hEn : n < ωi enumerate ENV(I). Select two increasing sequences han : n < ωi and hbn : n < ωi such that an, bn ∈ In and that an < bn < an+1. This is possible since each En is inﬁnite, and at every stage we have only picked ﬁnitely many elements. Let A =

{an : n < ω}. Then A 6∈ I, and both A and (ω \ A) have non-empty intersection with every element of ENV(I).

On the other hand, any independent family of size c can be extended to a maximal independent of size c. This raises the question of the minimal cardinality of a maximal independent family. We call this cardinal i, the independence number. 13

Deﬁnition 11. Let i be the cardinal such that i = min{|X| : X is a maximal independent family}.

The independence number is thus between ℵ1 and c, and so it is one of the cardinal invariants of the continuum. When looking at the usual cardinal invariants we don’t have an upperbound for i (besides c of course) but we do have some lower bounds. We noted before the connection between maximal independent families and reaping families, so it makes sense to ask about the minimum cardinality of a reaping family. This gives us the cardinal r, the reaping number.

Deﬁnition 12. r is the cardinal such that r = min{|X| : X is a reaping family}.

An easy observation establishes the following well-known result.

Theorem 13. r ≤ i

Proof. If I is a maximal independent family then Lemma 9 shows that ENV(I) is reaping, and notice that |ENV(I)| = |I|.

As a side note, consider the following concept. A family R ⊆ [ω]ω is called σ- reaping if there is no countable family F such that every B ∈ R there is a set A ∈ F

where |A ∩ B| = ℵ0 and |B \ A| = ℵ0. Let rσ be the minimum cardinality of a

σ-reaping family, and note that r ≤ rσ. Since the envelope of a maximal independent family is reaping it seems natural to ask if it is also σ-reaping, but this cannot happen as any countable subset of the maximal independent family will act as such a family

for the envelope. So the σ-reaping cardinal rσ and i are not obviously related. The cardinal i is also known to be related to other standard cardinal characteristics of the continuum such as the dominating number d. 14

Deﬁnition 14. A family D ⊆ ωω is dominating if for any g ∈ ωω there is an f ∈ D such that g ≤∗ f (i.e. g(n) ≤ f(n) for all but ﬁnitely many n). The dominating number d is the size of the smallest dominating family.

Theorem 15. (Shelah [9]) d ≤ i

Sketch. Let I be an independent family with |I| < d (so |ENV(I)| < d). Let J =

0 ω {Jn : n ∈ ω} be a countable subset of I, and I = I \ J. For any f ∈ ω notice that for each n ∈ ω the set J fn has inﬁnite intersection with every element of ENV(I0),

fn+1 fn ∗ fn and J ⊆ J . Then there is a set Bf such that Bf ⊆ J and Bf has inﬁnite intersection with every element of ENV(I0) (by prop 6.24 in [9]).

ω S Fix two disjoint, countable, dense subsets X and Y of ω. Let A = f∈X Bf , and S note that (ω \ A) ⊇ f∈Y Bf . Then I ∪ {A} is independent.

0 To demonstrate let E ∈ ENV(I). Find E0,E1 ∈ ENV(I ), f ∈ X, g ∈ Y , and

fn gk n, k ∈ ω large enough so that E0 ∩ J ⊆ E and E1 ∩ J ⊆ E. Then ∅= 6 E0 ∩ Bf ⊆

E ∩ A, and ∅= 6 E1 ∩ Bg ⊆ E ∩ (ω \ A).

This result has been improved by Balcar, Hernández-Hernández,and Hruˇsákin [4] to cof(M) ≤ i, where cof(M) (the cofinality of the measure ideal) is the minimal cardinal of a family C of meager subsets of R such that any meager subset of R is contained in a member of C. The three inequalities that we have listed can be strict. Eisworth proved (unpublished) that in the Sacks model i < c. Miller forcing can be used to get the consistency of r < i and adding a random real gives a model where d < i. Previously we had noted a connection between almost disjoint families, independent families, and ultrafilters, and we explore the related cardinal invariants.

Deﬁnition 16. A base for a ﬁlter F is a subfamily of F that contains subsets of all the sets in F . Let u be the cardinal such that u = min{|X| : X is a base for an 15

ultraﬁlter}. We call u the ultraﬁlter number.

Deﬁnition 17. Any almost disjoint family can be extended to a maximal almost disjoint (MAD) family by Zorn’s Lemma. Let a be the cardinal such that a = min{|X| : X is a MAD family}. We call a the almost-disjoint number.

The comparison of i to these cardinal characteristics results in either consistency results or open questions. Miller forcing can be used to get the consistency of u < i (see [9]). On the other hand, Shelah demonstrated the consistency of i < u in his paper [1]. Kunen created a MAD family that was resistant to Cohen forcing, which established the consistency of a < i. It is still an open question whether a ≤ i or if it is consistent that i < a. This open question was our motivation, and work done in this thesis provides techniques for ensuring maximal independent families are preserved in forcing extensions. 16

Chapter 2: Selective Independent Families and Forcing

We start by looking at some properties of independent families that are implicit in Shelah’s proof of the consistency of i < u.

Deﬁnition 18. Let I be an independent family. Call I an everywhere maximal independent family if whenever A ⊆ ω and h ∈ FF(I) there is a k ⊇ h where k ∈ FF (I) and either Ik ⊆ A or Ik ∩ A = ∅.

It is apparent by the deﬁnition that if I is everywhere maximally independent then I is maximally independent. The idea behind the name is best explained by the following deﬁnition. If I is an independent family and h ∈ FF(I) then we say that I is maximally independent in Ih if for every A ⊆ Ih there is a h0 ⊇ h, h0 ∈ FF(I) such that either Ih0 ⊆ A or Ih0 ∩ A = ∅. Notice that I is maximally independent if and only if I is maximally independent in Ih for some h ∈ FF(I) while I is everywhere independent if and only if I is maximally independent in Ih for every h ∈ FF(I). We get a stronger notion of independent families by modifying (iii) of Lemma 8 to consider functions f : ω → ω rather than f : ω → 2.

Deﬁnition 19. Let I be an independent family. Call I a selective independent family if whenever f : ω → ω then there is a set A ∈ ENV(I) such that f A is either one- to-one or constant. Call I an everywhere selective independent family if whenever f : ω → ω and h ∈

k FF(I) there is a k ⊇ h where k ∈ FF(I) and f I is either one-to-one or constant.

If I is an independent family and h ∈ FF(I) then I is selective independent in Ih

h 0 0 h0 if for every f : ω → ω on I then there is a h ⊇ h, h ∈ FF(I) such that f I is 17

one-to-one or constant. As before, notice that I is selective independent if and only if I is selective independent in Ih for some h ∈ FF(I) while I is everywhere selective independent if and only if I is selectively independent in Ih for every h ∈ FF(I).

Alternatively we can say that I is a selective independent family if whenever E is

an equivalence relation on ω, then there is an A ∈ ENV(I) such that E A has either one equivalence class or is equality. We can also say that I is a selective independent

family if whenever {An : n ∈ ω} is a partition of ω then there is a set A ∈ ENV(I)

such that either A ⊆ An for some n ∈ ω or |A ∩ An| ≤ 1 for every n ∈ ω. From these deﬁnitions we can easily see the relations between our types of independent families.

Lemma 20. Let I be an independent family. Then:

• I is everywhere selective ⇒ I is everywhere maximal ⇒ I is maximal.

• I is everywhere selective ⇒ I is selective ⇒ I is maximal.

Proof. An everywhere selective independent family is clearly a selective independent family, and then by Lemma 9, part (iii), we see that if I is selective then I is maximal. Since selective implies maximal, everywhere selective implies everywhere maximal, and clearly any everywhere maximal independent family is maximal.

It is an open question as to whether everywhere maximal, selective, and everywhere selective independent families exist in ZFC. Under various assumptions we will demonstrate the existence of such independent families. As such the related cardinals are not necessarily well-deﬁned but these will be used for notational convenience.

With that in mind we deﬁne the cardinals ie = min{|X| : X is a everywhere maximal independent family}, isel = min{|X| : X is a selective independent family}, and iesel = min{|X| : X is a everywhere selective independent family}. From the previous lemma we can see that i ≤ ie, isel ≤ iesel. 18

By our previous work we know that each of the cardinals have lower bounds r and

d, but as a side note we have an easy proof for d ≤ isel. If I is selectively independent, then {fE : E ∈ ENV(I) and fE is the unique strictly increasing function from ω onto E} is a dominating family. This is will demonstrated explicitly in Chapter 4. We note the following relative to Theorem 13.

ω Deﬁnition 21. The homogeneity cardinal hom1,c is the smallest family H ⊆ [ω] such

that for any f : ω → ω there is a set H ∈ H where f H is one-to-one or constant.

Theorem 22. If I is a selective independent family then hom1,c ≤ |I|.

Proof. Trivial.

As we will see in Chapter 5, if i = c then everywhere maximal independent families exist which means we can easily establish the consistency of strict inequalities

involving ie by using the results of i. Thus in standard Cohen models we have a < ie,

adding ℵ2-many random reals to a model of CH gives us a model for d < ie, and in

the Miller forcing model we get r < ie and u < ie. We have fewer existence results for selective independent families so we have fewer

consistent inequalities. In [1] Shelah actually demonstrated the consistency of iesel < u

instead of just i < u and later in Chapter 2 we will demonstrate that iesel < c in the Sacks Model.

2.1: Constructing Independent Families from CH

Since ℵ0 < i ≤ c, then the Continuum Hypothesis (the assumption that c = ℵ1) obviously gives us the following result without needing any work. However, we give the following easy inductive construction of a maximal independent family to give us a guide for future constructions.

Theorem 23. CH implies i = ℵ1. 19

Proof. Assume CH, and list {Xα ⊆ ω : α < ω1, such that Xα is co-inﬁnite}. Create by induction: S Let I0 = ∅, and if α < ω1 is a limit deﬁne Iα = β<α Iβ. Now suppose α < ω1

and if Iα ∪ {Xα} is independent deﬁne Iα+1 = Iα ∪ {Xα}. Otherwise Iα+1 = Iα.

Then Iω1 is independent since for any h ∈ FF(Iω1 ) there is an α < ω1 with h ∈

h h FF(Iα), meaning Iω1 = Iα is inﬁnite. Iω1 is of cardinality ℵ1 since we know that any stage we have only considered countably many subsets of ω and by previous work we

know any countable independent family can be increased in size. Iω1 is maximal since

if Xα were a counterexample, then Xα ∈ Iα+1 ⊆ Iω1 .

Thus Iω1 is a maximal independent family of size ℵ1.

The crux of the construction comes from two pieces of information: considering all possibilities (which makes it maximal) and knowing at each stage that we will always be able to add a set (which keeps the construction going). What do we change if we wish this construction to build a selective independent family? We want to consider all possible equivalence relations, and for a given equivalence relation we want to know that we can add a set. A lemma from Shelah in [1] allows us to do just that.

Lemma 24. If I is a countable independent family, h0 ∈ FF (I), and E is an equivalence relation on ω then there is a B ⊆ ω and h1 ⊇ h0 such that B 6∈ I, I ∪ {B} is

h1 independent, h1 ∈ FF (I), and E (I ∩ B) is equality or has 1 equivalence class.

We will prove a generalization of this in Chapter 3, but the above suﬃces for now.

Theorem 25. CH implies the existence of an everywhere selective independent family, so iesel = c.

Proof. List {(E, h): E is an equivalence relation on ω and h ∈ FF(ω1)} as {(Eα, hα):

α < ω1}, and we can assume without loss of generality that dom(hα) ⊆ α. We work by induction: 20

Let I0 be a countable independent family, and if α < ω1 is a limit deﬁne Iα = S β<α Iβ. Now let α < ω1, Bα as given in the previous lemma, and Iα+1 = Iα ∪ {Bα}.

Then Iω1 an everywhere selective independent family.

To see this let E be an equivalence relation on ω and h ∈ FF (ω1). Then for some

α < ω1 we have E = Eα and h = hα. By our lemma we found B ∈ ENV(Iα+1) such that E B is equality or has 1 equivalence class.

This proof created an everywhere selective independent family but we actually demonstrated something stronger. We started with an arbitrary countable independent family and we were able to extend this to an everywhere selective independent family. So we did not create just a single everywhere selective independent family but many. This observation will become important in Chapter 5 so we state this formally.

Lemma 26. Assuming CH, if I is a countable independent family then I can be extended to an everywhere selective independent family.

2.2: Ideals and Independent Families We show in this section that each independent family can be associated with an ideal in a natural way, and properties of the independent family are reﬂected by properties of the ideal. This will be important when we consider the question of whether a maximal independent family is preserved by Sacks forcing. Recall the following deﬁnition.

Deﬁnition 27. Let id ⊆ P(ω).

1) Call id an ideal if:

– Whenever B ∈ id and A ⊆ B then A ∈ id

– Whenever A, B ∈ id then A ∪ B ∈ id. 21

2) If id is an ideal, then id+ = {A ⊆ ω : A 6∈ id} is called the co-ideal.

3) If A is a collection of subsets of ω then A ↑ = {X ⊆ ω : ∃A ∈ A such that A ⊆ X} is called the upward closure of A.

We can now deﬁne our aforementioned ideal.

Deﬁnition 28. If I is an independent family then let idI = {A ⊆ ω : for every B in ENV(I) there is a C ⊆ B where C ∈ ENV(I) such that C ∩ A = ∅}.

Proposition 29. If I is an independent family then idI is an ideal.

Proof. Let B ∈ idI , A ⊆ B, and E ∈ ENV(I). Since B ∈ idI then we can ﬁnd an

F ∈ ENV(I) with F ⊆ E where B ∩ F = ∅. But then A ∩ F = ∅, so A ∈ idI .

Now let A, B ∈ idI , and let E ∈ ENV(I). We can then ﬁnd F ∈ ENV(I) with F ⊆ E where B ∩ F = ∅, and we can then ﬁnd in turn G ∈ ENV(I) with G ⊆ F where A ∩ G = ∅. But then G ⊆ E with (A ∪ B) ∩ G = ∅, so A ∪ B ∈ idI .

Note that a set A is in the ideal idI if and only if for every h ∈ FF(I) then there is

0 0 h0 an h ∈ FF(I) such that h ⊇ h and I . What does membership in idI or its coideal

+ idI look like?

Lemma 30. Let I,J be independent families. Then:

i) If I ⊆ J then idI ⊆ idJ

+ ii) ENV (I) ↑ ⊆ idI

+ iii) If hXn : n ∈ ωi is a strictly decreasing sequence of sets in idI then whenever

∗ X ⊆ Xn for every n ∈ ω we have X ∈ idI .

Proof. Statement (i) is true by the deﬁnitions.

+ + For statement (ii) notice that any set E ∈ ENV(I) is in idI so ENV(I) ↑⊆ idI . 22

For statement (iii) assume hXn : n ∈ ωi is a strictly decreasing sequence of sets

+ ∗ in idI and X ⊆ Xn for every n ∈ ω. Find a strictly decreasing sequence of sets + hEn : n ∈ ωi where En ∈ ENV(I) and En is a witness for Xn ∈ idI (i.e. whenever

E ⊆ En and E ∈ ENV(I) then E ∩ Xn is inﬁnite). Notice then that for every n

En \ En+1 ∈ ENV(I), and that (En \ En+1) ∩ X is ﬁnite. We show that no member of the envelope can act as a witness for X ∈ I+, so let

E ∈ ENV(I). If for some n we have E ∩ En = ∅ then E ∩ X is ﬁnite and so cannot

+ witness X ∈ idI . Otherwise for every n E ∩ En ∈ ENV(I), so ﬁnd n large enough

that E ∩ En ⊂ E, meaning the set (E ∩ En) \ En+1 is almost disjoint from X. Thus

+ E cannot be a witness for X ∈ idI and we must conclude that X ∈ idI .

In order for the co-ideal to coincide with the upward closure we need the independent family to be everywhere independent.

Lemma 31. Let I be an independent family. Then I is an everywhere independent

+ family if and only if idI = ENV(I) ↑.

Proof. Assume I is everywhere independent. We noted before that for any indepen-

+ + dent family ENV(I) ↑ ⊆ idI , so now let X ∈ idI . By definition find B ∈ ENV(I) such that for any C ⊆ B we have C ∩X is infinite. Since I is everywhere independent there is a specific C such that C ⊆ X and thus X ∈ ENV(I) ↑.

+ Now assume idI = ENV(I) ↑. Let X ⊆ ω and E ∈ ENV(I). If E has a subset F ∈ + ENV(I) disjoint from X then we are done. If not then E is a witness for E ∩X ∈ idI . By our assumption there is an F in ENV(I) with F ⊆ (E ∩ X). Thus F ⊆ E and F ⊆ X, hence I is everywhere independent.

Before we continue we need to discuss our use of the term selective. The literature contains many different definitions for the word selective. All of these coincide in the case where the ideal of concern is dual to an ultrafilter, but these equivalences fall 23

apart in our more general context. We collect several of these notions here, and give convenient (though generic) names.

Deﬁnition 32. Let id be an ideal, and id+ its co-ideal. Then:

• id is ﬁrst selective (or selective from [16]) if whenever A0 ⊇ A1 ⊇ A2 ⊇ ... with

+ + An ∈ id then there is a B ∈ id with B \ n ⊆ An.

• id is second selective (or semiselective from [16]) if whenever {Dn : n < ω} is a collection of dense open sets in the partial order (id+, ⊆∗) then for any A ∈ id+

+ there is a B ∈ id with B ⊆ A and B \ n ∈ Dn.

• id is third selective (or Ramsey) if whenever A ∈ id+ and f :[A]2 → 2, there

+ 2 is a B ∈ id such that f [B] is constant.

• id is fourth selective (or selective from [6]) if whenever A ∈ id+ and f : A → ω,

+ there is a B ∈ id such that f B is 1-1 or constant.

+ • id is ﬁfth selective if whenever f : ω → ω, there is a B ∈ id such that f B is one-to-one or constant.

Lemma 33. If id is an ideal, then id is ﬁrst selective ⇒ id is second selective ⇒ id is third selective ⇒ id is fourth selective ⇒ id is ﬁfth selective.

Which versions of selectivity can ideals associated with independent families sat- isfy? It is easy to see that whenever I is an everywhere selectively independent family

then idI a fourth-selective ideal. Similarly ﬁfth-selective ideals arise from selective independent families.

Theorem 34. Let be I be an independent family. Then I is everywhere selective if

and only if I is everywhere maximal and idI is fourth-selective. 24

Proof. The forward direction is clear by the deﬁnitions so let I be everywhere maximal

with idI fourth-selective. Let E be an equivalence relation on ω and A ∈ ENV(I). Let

fE be the function that maps each element of A to its equivalence class in E. Since + idI is fourth-selective then there is a B in idI such that fE B is 1-1 or constant.

Then there is a C ∈ ENV(I) such that C ⊆ B. So fE C is 1-1 or constant and thus E C is equality or has one equivalence class.

2.3: Sacks Forcing and Iterated Sacks Forcing The paper of Baumgartner and Laver [6] that deﬁned iterated Sacks forcing demonstrated that selective ideals are preserved by Sacks forcing, using selective in

the same sense of selective independent families. We can only add ℵ2-many Sacks reals iteratively, so from this we then have a pretty clear road map for how to demonstrate

iesel < c in the Sacks model. First, we demonstrate that selective independent families exist under the Continuum Hypothesis. Second, we show that independent families create ideals, and selective independent families will create selective ideals. Finally, we show that preserving the selective ideal will preserve the selective independent family. Let Seq = S{ n2 : n ∈ ω}. A nonempty subset p of Seq is called perfect if:

(i) If s ∈ p, then s n ∈ p for each n (i.e. closed under initial segments). (ii) If s ∈ p, then there are u, t ∈ p where s ⊆ t, s ⊆ u, t 6⊆ u, u 6⊆ t , i.e. every element of p has two incompatible extensions in p. Let PS = {p ∈ Seq : p is perfect} ordered by inclusion: p ≤ q if and only if p ⊆ q. PS is called Perfect set or Sacks forcing. Using the work of Baumgartner and Laver, we deﬁne another partial order on PS which will help with constructions. For any p, q ∈ PS and m, n ∈ ω deﬁne (p, m) > (q, n) if and only if p ≤ q, m > n and every element of n-th level of q has 25

split by the m-th level of p. From this we get the so-called Fusion Principle, where

if we have an inﬁnite sequence (pi+1, ni+1) > (pi, ni) then there is a q ∈ P such that q ≤ pi for every i ∈ ω.

Iterated Sacks forcing is done inductively using PS. Let P1 = PS. For 1 ≤ α ≤ ω2 N define Pα+1 = Pα PS, the canonical partial ordering associated with the extension by forcing first with Pα and then with PS as defined in the extension via Pα. If α is a limit ordinal, then Pα in the inverse limit of hPβ : β < αi if cf(α) = ω, and the direct limit otherwise.

We will use a version of the Fusion Principle for the iteration. Let 1 ≤ α ≤ ω2, p, q ∈ Pα, m, n ∈ ω, and F is a finite subset of the domain of q. Then (p, m) >F (q, n) iff p ≤ q, m > n and for every β ∈ F p β ` (p(β), m) > (q(β), n). Now, suppose S h(pi, ni,Fi): i ∈ ωi is a sequence where pi ∈ Pα, n1 ∈ ω, Fi ⊆ Fi+1, {Fi : i ∈ ω} = S {dom(pi): i ∈ ω}, and for each i,(pi+1, ni+1) >Fi (pi, ni). If we define p such that S T the domain of p is {dom(pi): i ∈ ω} and p(β) = {pi(β): i ∈ ω, β ∈ dom(pi)}

then p ∈ Pα and p ≤ pi for each i. To ﬁnish our goal of showing that everywhere selective independent families are preserved in Sacks Forcing we make use of a pair of lemmas from [6].

+ Lemma 35. If id is a fourth-selective ideal then PS‘the upward closure of id is the complement of a fourth-selective ideal’.

Lemma 36. If id is a fourth-selective ideal then for each α ≤ ℵ2 Pα ‘the upward closure of id+ is the complement of a fourth-selective ideal’.

However, the proofs given in [6] use an incorrect characterization of fourth-selective. Here we will show the correct characterization and then give a sketch as to where this will be applied.

Deﬁnition 37. Let T be a collection of ﬁnite sequences of ω. 26

• Call T a tree if T is closed under initial subsequences.

• If s ∈ T and k ∈ ω let s + k be the concatenation of s and k.

• If s ∈ T let Ts = {k ∈ ω : s + k ∈ T } be the ramiﬁcation of s in T .

• If I is an ideal call T an I-tree if every ramiﬁcation of elements from T are in I+.

• Call T a strong-I tree if no ﬁnite intersection of ramiﬁcations of T are in I.

• When A ∈ I+ and T is a tree on A then call T an A-large tree if every ramiﬁ- cation on T diﬀers from A on a set in I.

ω • A function f ∈ ω is a branch through a tree T if ∀n f n ∈ T , and f is an I-branch if range(f) ∈ I+.

• Call I a weak T -ideal if whenever I is a strong-I tree then T has an I branch.

• Call I a very-weak T -ideal if whenever A ∈ I+ and T is an A-large tree then T has an I branch.

The error in [6] comes from a confusion between a weak T -ideal and very-weak T -ideal. From Grigorieﬀ [18], Proposition 14 of the appendix, we get the following result.

Lemma 38. If I is an ideal then I is fourth selective iﬀ I is a very-weak T -ideal.

This allows us to correct the result from lemmas 4.1 and 4.2 from [6].

Proof of Lemma 35. Let J˙ be the complement of the upward closure of id+. J˙ is an ideal, so we need to show it is fourth-selective. Let f˙ name a function, and assume

˙ + that for some p ∈ P S p ‘f maps A to ω’ for some A ∈ I . Without loss of 27

generality we can assume that for every n ∈ ω {k ∈ ω : f(k) ≤ n} ∈ J˙, meaning we

+ ˙ need to find q ∈ PS with q ≤ p and a set B ∈ id such that q ‘f B is 1-1’. For every q ≤ p and n, k ∈ ω define Z(q, n, k) = {i ∈ A : there is r ∈ PS, m, l ∈ ω ˙ such that (r, m) > (q, n) and r ‘k ≤ f(i) < l’}. Note that A \ Z(q, n, k) ∈ I, as demonstrated in [6]. We create a tree T of finite sequences by induction. Let T have the empty sequence

∅, and deﬁne p∅ = p, n∅ = k∅ = 0. For any sequence s in T , let s ∪ {i} = t ∈ T

if and only if i > max(s) and i ∈ Z(ps, ns, ks); then let pt, nt, kt be a witness for

i ∈ Z(ps, ns, ks). Any ramiﬁcation on T diﬀers from A on a set in I so T is A-large.

Since I is fourth-selective then we can ﬁnd an I-branch g such that g k ∈ T for every k ∈ ω, and let B = range(g).

Thus (pg(k+1), ng(k+1)) > (pgk, ngk), so by fusion we ﬁnd q ∈ PS such that ˙ q ‘f B is 1-1’.

Proof of Lemma 36. We work by induction on α. When α = 1 or α = β + 1 we can apply the previous lemma.

+ ˙ Now let α be a limit ordinal. Let A ∈ I , p ∈ Pα, and p α ‘f maps A to ω’ for some A ∈ I+, and for every n ∈ ω {k ∈ ω : f(k) ≤ n} ∈ J˙. For every q ≤ p, F a ﬁnite subset of the domain of q, and n, k ∈ ω deﬁne

Z(q, n, F, k) = {i ∈ A : there is r ∈ Pα, m, l ∈ ω such that (r, m) >F (q, n) and ˙ r ‘k ≤ f(i) < l’}. Note that A \ Z(q, n, F, k) ∈ I, as demonstrated in [6]. We can create a tree as before and with the same result.

Up to this point we have been informal in our goal of preserving independent families so we present a formal deﬁnition.

Deﬁnition 39. Let P be a poset and I a maximal independent family. Then I is

P-indestructible if P ‘I is a maximal independent family’. 28

It is an easy observation that I is P -indestructible if and only if whenever p ∈ P and τ is such that p ‘τ : ω → 2’ then there is a q ≤ p and B in ENV(I) such that q ‘τ B is constant’. The preservation of other varieties of independence families is deﬁned similarly. Just like in our construction, we ﬁrst show how we preserve maximal independence so that we my modify it to show how we preserve everywhere selective independence.

Lemma 40. If I is an everywhere selectively independent, then for each α ≤ ℵ2 Pα ‘I is maximally independent’.

+ Proof. In V [G], let J be the complement of the upward closure of idI and τ be a function from ω → 2. Since J is selective (by Lemma 36) there is a B in J + such

+ that τ B is constant. But then B contains a set in idI , which in turn contains an element of the ENV(I). So I is maximally independent in V [G].

Lemma 41. If I is an everywhere selectively independent, then for each α ≤ ℵ2 Pα ‘I is an everywhere selectively independent’.

Proof. In V [G] let E be an equivalence relation on ω and A ∈ ENV(I). Let {Ei : i <

ω} be the equivalence classes of E, and deﬁne the function fE mapping each element

of A to the index of its equivalence class. (i.e. fE : A → ω). + Let J be the complement of the upward closure of idI . Since J is selective (by + Lemma 36) there is a B ∈ idI ↑ such that fE B is either 1-1 or constant. But + idI is the upward closure of ENV(I) so there is some C ∈ ENV(I) such that C ⊆ B and thus fE C is either 1-1 or constant, so E C is equality or has one equivalence class.

From these results we know that an everywhere selectively independent families are preserved by both Sacks forcing and iterated Sacks forcing and we get the following: 29

Theorem 42. ‘ i = ℵ < ℵ = c’. Pω2 esel 1 2

2.4: Side-by-Side Sacks Forcing Iterated Sacks forcing is not the only way of adding many Sacks reals to a model. Instead of them iteratively, we can instead add them simultaneously. Adding reals in this manner is in some sense easier and will allow us to make c arbitrarily large. We define side-by-side forcing with PS. For a cardinal κ define P (κ) = {p : p is a function mapping a countable subset of κ into PS }. For p, q ∈ P (κ) define p ≤ q if and only if dom(q) ⊆ dom(p) and if α ∈ dom(q) then p(α) ≤ q(α) in PS. Similar to what we have in the case of iteration we have a fusion lemma for side-by-side Sacks forcing. Let p, q ∈ P (κ), m, n ∈ ω, and F is a finite subset of the domain of q. Then (p, m) >F (q, n) iff p ≤ q, m > n and for every β ∈

F (p(β), m) > (q(β), n). Now, suppose h(pi, ni,Fi): i ∈ ωi is a sequence where S S pi ∈ Pα, n1 ∈ ω, Fi ⊆ Fi+1, {Fi : i ∈ ω} = {dom(pi): i ∈ ω}, and for each i, S (pi+1, ni+1) >Fi (pi, ni). If we deﬁne p such that the domain of p is {dom(pi): i ∈ ω} T and p(β) = {pi(β): i ∈ ω, β ∈ dom(pi)} then p ∈ P (κ) and p ≤ pi for each i.

Lemma 43. If id is a fourth-selective ideal and κ is cardinal then P (κ)‘the upward closure of id+ is the complement of a fourth-selective ideal’.

+ ˙ + Proof. Let A ∈ I , p ∈ P (κ), and p P (κ) ‘f maps A to ω’ for some A ∈ I , and for every n ∈ ω {k ∈ ω : f(k) ≤ n} ∈ J˙. For every q ≤ p, F a ﬁnite subset of the domain of q, and n, k ∈ ω deﬁne

Z(q, n, F, k) = {i ∈ A : there is r ∈ P (κ), m, l ∈ ω such that (r, m) >F (q, n) and ˙ r ‘k ≤ f(i) < l’}. Note that A \ Z(q, n, F, k) ∈ I. We can create a tree as before and with the same result.

Having established this then we can ﬁnd the preservation of maximal and everywhere selective independent families exactly like we had in the iterated model. 30

Lemma 44. If I is an everywhere selectively independent, then for cardinal κ P (κ) ‘I is maximally independent’.

Lemma 45. If I is an everywhere selectively independent, then for cardinal κ P (κ) ‘I is an everywhere selectively independent’.

Theorem 46. Assume CH. For every cardinal κ P (κ) ‘ iesel = ℵ1’. 31

Chapter 3: Independence and Q

We have another method for showing that everywhere selective independent families are preserved in the Sacks Model. Independent families connect ω and Q, and through this connection we will find a ♦-like statement that will both imply the existence of independent families and is preserved by Sacks Forcing. The basis for this work is found in [4] and [5], where Balcar, Hruˇsák,et. al. were investigating analogies between FIN, the ideal of finite subsets of ω, and NWD, the nowhere dense subsets

of Q. Most of the cardinal characteristics translate well, and they uncovered some surprising results. We revisit their work here in order to extend some of their results to selective independent families.

3.1: Topological Interpretation of Independent Families If I is an independent family notice that ENV(I) satisﬁes the deﬁnition of a basis

for a topology. Let TI be the topology generated using ENV(I) as the basic open sets on ω. It is easy to see how our properties translate into topological terms.

Lemma 47. Let I be a independent family, and TI be the corresponding topology.

i) If A ⊆ ω and A 6∈ I, then I ∪{A} is independent if and only if both A and ω \A

are dense in (ω, TI ).

ii) If A ⊆ ω, then A ∈ idI if and only if A is nowhere dense (ω, TI ).

iii) I is countable if and only if the topology TI is ﬁrst countable.

iv) TI is without isolated points.

iv) TI zero-dimensional.

Proof. Statement (i) is true by Lemma 4 and (ii) is true by the deﬁnitions of idI and nowhere dense. Statement (iii) is easily true by how TI is deﬁned. 32

Since we can modify any set in I by a ﬁnite set without losing independence, then we can assume that I separates points (i.e. if i 6= j there is some A ∈ I such that i ∈ A and j 6∈ A).

Members of I and their complements form a clopen subbasis for TI so it is zero- dimensional.

From statement (i) in this previous lemma we can see that for a maximal independent family I it is not possible to write ω as the union of two disjoint sets

that are dense in TI . Any topological space with this property is called irresolvable. The literature is rich with references to irresolvable topological spaces so we give characterizations for the other varieties of independent families.

h An independent family I is everywhere maximal if and only if (I ,TIh ) is irresolvable for every h ∈ FF(I), where TIh is the subspace topology. If every open subset of X is irresolvable we call it strongly irresolvable, so I is everywhere maximal if and only if (ω, TI ) is strongly irresolvable. We can call a topological space (X,T ) selective if whenever X is written as the countable union of sets with empty interiors, then there is some open set that intersects each piece at most once. So I is selective if and only if (ω, TI ) is selective, and

h I is everywhere selective if and only if (I ,TIh ) is selective for every h ∈ FF(I). 3.2: Topology of Countable Independent Families An interesting thing happens when we restrict our attention to a countable independent family. By Sierpi´nski,as detailed in [3], Q is the unique (up to homeomorphism) topological space that is countable, ﬁrst countable, regular, and without isolated points. Recall that any zero-dimensional topological place is completely regular and thus regular, meaning there is a relationship between countable independent families on ω, and Q. 33

This is summarized by the the following lemma, which is a part of Proposition 2.6 in [4].

Lemma 48. I is a countable independent family if and only if (ω, TI ) is homeomorphic to Q.

Proof. Use Lemma 47, statements (iii), (iv) and (v).

Notice that when I is a countable independent family each of the subspaces

h (I ,TIh ) will also be countable, ﬁrst countable, regular, and without isolated points.

h Thus each (I ,TIh ) is homeomorphic to Q. This leads to the following easy result.

Corollary 49. Let I be an independent family. The following are equivalent:

• I is a countable independent family

• (ω, TI ) is homeomorphic to Q.

h • For every h ∈ FF(I), (I ,TIh ) is homeomorphic to Q.

h • For some h ∈ FF(I), (I ,TIh ) is homeomorphic to Q.

We can still use this connection to Q even when I is an uncountable independent family by taking a countable subset J of I and looking at (ω, TJ ). This is then homeomorphic to Q so we see how the sets of I \ J behave.

Lemma 50. Let I be an independent family and J a countable subset of I. Then:

i) If A ∈ I \ J then both A and ω \ A are dense in (ω, TJ ).

ii) If A ⊆ Q is nowhere dense in (ω, TJ ) then A ∈ idI .

iii) If E ∈ ENV(I) then E is not nowhere dense in (ω, TJ ).

iv) If E ∈ ENV(I \ J) then E is dense in (ω, TJ ).

Proof. Trivial. 34

3.3: Dense Independent Families As we’ve observed this connection gives an interesting property to independent

sets from ω when we translate them to Q. If I is a countable independent family,

A ⊆ ω is independent from I, and H is the homeomorphism from (ω, TI ) to Q then H(A) is dense in Q. This allows us to deﬁne a type of independent family unique to Q. Call a collection of dense sets I of Q dense independent if every element of ENV(I) is dense in Q. Because of this connection between ω and Q we have a method for increasing the size of countable independent families, particularly if there is a non-nowhere dense set that we wish to be a part of the envelope. This is help us construct maximal and then everywhere maximal independent families.

Lemma 51. If I is a countable independent family and A is a subset of Q, then there is a set B ⊆ ω and a set E ∈ ENV(I) such that I ∪ {B} is independent and

H(E ∩ B) ⊆ A or H(E ∩ B) ∩ A = ∅ where H is the homeomorphism from (ω, TI ) to

−1 Proof. If A is nowhere dense, then H (A) is in idI , so we can ﬁnd E ∈ ENV(I) such that H(E) ∩ A = ∅. We can let B be any set independent with I.

−1 + Assume A is not nowhere dense in Q, so then H (A) is in idI . Let E ∈ ENV(I) witness, so for every F ∈ ENV(I) with F ⊆ E then F ∩ H−1(A) 6= ∅. So either there is some F ∈ ENV(I) with F ⊂ E and F ⊆ H−1(A), or every F ∈ ENV(I) with F ⊆ E then F ∩ (ω ∩ H−1(A)) 6= ∅. In the ﬁrst case we are done and since I is countable we can let B be any set such that I ∪ {B} is independent. In the second we can ﬁnd an independent set B such that E ∩ B ⊆ H−1(A).

Notice that we actually proved something stronger, where we see when we will have H(E ∩ B) ⊆ A and when H(E ∩ B) ∩ A = ∅. 35

Corollary 52. If I is countable independent family, A is not nowhere dense in Q, and h0 ∈ FF(I) then there is a set B ⊆ ω and h1 ⊇ h0 such that I ∪ {B} is independent,

h1 h0 and H(I ∩ B) ⊆ A where H is the homeomorphism from (I ,T h ) to . I0 Q

A countable independent family has a countable envelope which allows us to repeatedly apply this, resulting in the following.

Corollary 53. If I is a countable independent family and A is a subset of Q then

there is a countable collection of sets {Bn : n < ω} such that J = I ∪ {Bn : n ∈ ω} is

h1 independent and for every h0 ∈ FF(J) there is an h1 ∈ FF(J) such that H(J ) ⊆ A

or H(J h1 ) ∩ A = ∅.

Maximal dense independent families on Q exist by Zorn’s Lemma, and are in

essence equivalent to maximal independent families on ω. Deﬁne iQ = min{|X| : X

is a maximal dense independent family }, and i = iQ as demonstrated in [4] and we give a proof here.

Lemma 54. There is a maximal independent family of size κ if and only if there is a maximal dense independent family of size κ.

Proof. Let I be maximal independent family. Let J ⊆ I be a countable subset, and

let H be the homeomorphism from (ω, TJ ) to Q. Then {H(X): X ∈ I \ J} must be a maximal dense independent family.

Now let I be maximal dense independent family in Q. Let C be a countable independent family in ω, and let H be the homeomorphism from ω to Q. Then J = C ∪ {H−1(X): X ∈ I} is an independent family. Assume J is not maximal independent, and let A ⊆ ω witness. So H(A) ⊆

Q and H(A) ∪ I is dense independent, which contradicts I being maximally dense independent. 36

With r as a lower bound for i we have a similar (but surprisingly not the same) lower bound in Q. Deﬁne the rational reaping number rQ as the cardinality of the

smallest reaping family on P(Q) \ NWD. r, d ≤ rQ ≤ i as demonstrated in [4]. Note

that the sets of rQ are non-nowhere dense, which will be important in our constructions in the next section. Likewise, call a collection of dense sets I dense selective independent if I is dense

h independent and for every f : Q → Q there is an h ∈ FF(I) such that f I is 1-1

or constant. Deﬁne isel,Q = min{|X| : X is a dense selective independent family }.

It follows in similar fashion from Lemma 54 that isel = isel,Q.

We can deﬁne hom1,Q on P(Q) \ NWD similar to hom1,c. It then follows that

rQ ≤ hom1,Q ≤ isel,Q. As far as the author is aware, this cardinal has not been deﬁned

or explored previously so in the interest of completion we show that hom1,Q is well deﬁned.

Lemma 55. If f : Q → Q then there is a set B ⊆ Q that is not nowhere dense and

f B is either 1-1 or constant.

Proof. Let {On : n ∈ ω} enumerate a countable base on Q. We have two cases to

consider. Either for every n ∈ ω f On has inﬁnite range, or for some n ∈ ω f On

has ﬁnite range. In the ﬁrst case, inductively choose bn ∈ On where f(bn) 6= f(bi) for

i < n. Then B = {bn : n ∈ ω} must be dense in Q since it intersects every open set.

−1 In the second case, if {k0, ..., kl} is the range of f On then deﬁne Bi = f (ki) ∩ On S for i ≤ l. If each of these sets were nowhere dense then i≤l Bi = On would be nowhere dense, a contradiction.

Lemma 56. hom1,c ≤ hom1,Q

Proof. Let X ⊆ Q such that for every f : Q → Q there is an A ∈ X such that f A is 1-1 or constant. Let H : ω → Q witness Q as countable, and let Y = {H−1(A): 37

A ∈ X}.

Now let f : ω → ω. Deﬁne g : Q → Q via g(a) = H(f(H−1(a)). Then there is a

−1 set A ∈ X such that g A is 1-1 or constant, so f H (A) is 1-1 or constant.

3.4: Independent Families and ♦-like Principles Instead of trying to iterate through all the subsets of ω like we did when assuming

CH, what if we only needed to iterate through ℵ1 many ‘important’ subsets? Of

course it depends on which sense of ‘important’ we mean, but having only ℵ1 many sets would allows us to build using countable independent families like before.

Deﬁnition 57. Let κ be a cardinal and C ⊆ κ.

• Call C is unbounded in κ if sup(C) = κ.

• Call C closed if whenever α ∈ C is a limit ordinal then sup(α ∩ C) = α.

• If C is both closed and unbounded we call C a club set.

• If S ⊆ κ call S a stationary set if S has non-empty intersection with every club set of κ.

Deﬁnition 58. In the style of [5], let A and B be sets and E a relation between A and B.

• Call (A, B, E) an invariant if ∀a ∈ A ∃b ∈ B where aEb, and ∀b ∈ B ∃a ∈ A where aE6 b.

• Call hA, B, Ei = min{|X| : X ⊆ B and ∀a ∈ A ∃b ∈ X where aEb} the evaluation of (A, B, E).

• An invariant (A, B, E) is a Borel invariant if A,B and E are Borel subsets of some Polish space. 38

We can write rQ as hP(Q) \ NWD, P(Q)\ NWD, ‘does not split’ i and hom1,Q as h{f : f is a function from Q to Q}, P(Q)\ NWD, ‘is constant or 1-1 on’ i. Recall from earlier that rQ ≤ i and hom1,Q ≤ isel. If a cardinal can be written as (A, B, E) then we deﬁne ♦(A, B, E) as the following statement:

<ω1 For every Borel function F : 2 → A there is a g : ω1 → B such that for

every f : ω1 → 2 the set {α ∈ ω1 : F (f α)Eg(α)} is stationary.

The function g is called a ♦(A, B, E) sequence for F , and ♦(A, B, E) implies that

hA, B, Ei is ℵ1.

From [5] we see that if our ‘important’ sets are decided by rQ then we can construct an independent family. The following result is Lemma 7.11 from [5]:

Lemma 59. ♦(rQ) implies i = ω1.

If we redo the proof of this lemma using Corollary 53 then we can directly build an everywhere maximal independent family.

Theorem 60. ♦(rQ) implies ie = ω1.

Proof. Abbreviate rQ as hA, B, Ei. Deﬁne F as follows: if X ⊆ ω and I = hIα : β < αi, then F (X,I) = Q if α is ﬁnite, I is not independent, or if there is some k ∈ FF (I) such that Ik ∩ X = ∅ or Ik ⊆ X; otherwise F (X,I) = H(X), where H

is the homeomorphism from (ω, TI ) to Q. Let g be a ♦(rQ) sequence for F and we proceed by induction.

Enumerate FF(ω1) as {hα : α < ω1}, and without loss of generality assume

dom(hα) ⊆ α for every α < ω1.

Let Iω = {In : n < ω} be a countable independent family, and when α is a limit S define Iα = β<α Iβ. If Iα = {Iβ : β < α} is defined then define Iα+1 = Iα ∪ Jα where

Jα is the collection of sets as given by Corollary 53 to decide g(α). 39

Assume Iω1 is not everywhere independent family, and let X ⊆ ω and h0 ∈ FF (ω1) witness. Then {α ∈ ω1 : F (X,Iα) 6= Q, dom(h0) ⊆ α} is a club set, so for some α ∈ ω we know H(X) 6= Q and H(X) E g(α). But then we found h1 ∈ FF(Iα+1) such that

h1 h1 h1 Iα+1 ⊆ g(α). Thus Iω1 ∩ X = ∅ or Iω1 ⊆ X.

Theorem 61. ♦(hom1,Q) implies isel = ω1.

Proof. Abbreviate hom1,Q as hA, B, Ei. Deﬁne F as follows: if f maps ω to ω and I

= hIβ : β < αi, then F (f, I) is the identify function on Q if α is ﬁnite, or if f is not

one to one or constant for any element in ENV(I); otherwise F (f, I) = HI (f), where

HI is the homeomorphism from (ω, TI ) to Q.

Let g be a ♦(hom1,Q) sequence for F . We build a selective independent family

by induction. Let Iω = {In : n < ω} be a countable independent family. Assume

Iα = {Iβ : β < α} is deﬁned, then by Corollary 52 ﬁnd Iα with Cα ∈ ENV(Iα ∪ {Iα})

and Hα(Cα) ⊆ g(α) and deﬁne Iα+1 = Iα ∪ {Iα}.

Now to demonstrate that that Iω1 is selectively independent: Let f map ω to ω

and assume that F (f, Iα) 6= Q. Then Hα(f) E g(α), so f E Cα.

If we combine the ideas from Theorem 60 and 61, we can explicitly build a everywhere selective independent family and thus:

Theorem 62. ♦(hom1,Q) implies iesel = ω1.

Like our construction using CH, we started with a countable independent family and were able to extended it to various independent families.

Lemma 63. Assume I is a countable independent family.

♦(rQ) implies I can be extended to an everywhere maximal independent family of

size ℵ1 and ♦(hom1,Q) implies I can be extended to an everywhere selective indepen-

dent family of size ℵ1. 40

One of the biggest achievements from [5] is giving conditions for when ♦-like statements are preserved in forcing extensions. The full theorem and proof goes beyond what is necessary for this look at selective independent families in Sacks Forcing, but one of the lemmas is directly applicable. The following is Lemma 6.12 in [5].

Lemma 64. Let (A, B, E) be a Borel invariant such that hA, B, Ei ≤ ω1. If T is a tree of height ω1 which is nowhere c.c.c and does not add reals then T forces ♦(A, B, E).

As such this gives an indirect proof wherein instead of creating an independent family and showing that it is preserved by Sacks Forcing we see that a principle is preserved which then creates an independent family in the Sacks Model.

Theorem 65. P forces ♦(hom ), so ‘ i = ℵ < ℵ = c’. ω2 1,Q Pω2 esel 1 2 41

Chapter 4: Connections to Ultraﬁlter Properties

In Chapter 2 we listed several properties that all lay claim to the name selective. All of these are equivalent in the context of ultraﬁlters. In this section we investigate the properties in the context of maximal independent families. In addition we look at other properties usually studied in the context of ultraﬁlters.

4.1: Nothing Stronger than Selective Independence To start, we recall from section 2.2 that selective independent families created ﬁfth- selective ideals and everywhere selective independent families created fourth-selective ideals.

To see how no ideal of the form idI for I and independent family can be third-

selective we need to make use of a connection to Q. We use a coloring of Q found in [12].

Lemma 66. If I is an inﬁnite independent family then there is a function f :[ω]2 → 2

such that every B ⊆ ω homogeneous for f is in idI .

0 Proof. Let I = {In : n < ω} ⊆ I. Let H be the homeomorphism from (ω, TI0 ) to Q. For each pair a, b ∈ ω with a < b deﬁne f({a, b}) = 0 if and only if H(a) < H(b). Let B be any subset of ω that is constant on f. If B is ﬁnite then obviously

B ∈ idI , so assume B is inﬁnite. Clearly then H(B) is order-isomorphic to ω. It must also be nowhere dense since for any a, b ∈ B there can be only ﬁnitely elements of H(B) between H(a) and H(b). Since H(B) is nowhere dense then by a previous

lemma we know B ∈ idI .

4.2: P, Q, and P-point As we showed there cannot be independent families with stronger properties than selective but we may consider many weaker versions. 42

Deﬁnition 67. Let F be a family of sets, and I an independent family. Then:

• F is a q-family if whenever f ∈ ωω is ﬁnite to one then for some A ∈ F we

have f A is one to one.

• F is a p-family if whenever {Xn : n < ω} is a subset of F , then there is some

∗ X ∈ F such that X ⊆ Xn for each n.

ω • F is a p-point family if whenever f ∈ ω, then for some A ∈ F we have f A is bounded or ﬁnite to one.

• I is q-independent, p-independent, or p-point-independent if ENV (I) ↑ is a q, p, or p-point, respectfully.

• I is everywhere q-independent, everywhere p-independent, or everywhere p-point- independent if for every h ∈ FF (I){Ik : k ∈ FF (I), k ⊇ h} ↑ is a q, p, or p-point, respectfully.

Note that an ultraﬁlter is a p-family if and only if it is p-point. This equivalence breaks in a strong way for independent families.

Lemma 68. No independent family I can be p-independent.

Proof. If I is p-independent, let {In : n < ω} be a countable subset of I, and deﬁne

T ∗ Xn = k≤n Ik. Since I is p-independent there is an X ∈ ENV(I) ↑ such that X ⊆ Xn for each n. However, notice that X must be in idI by Lemma 27 which contradicts X ∈ ENV(I) ↑.

Lemma 69. If I is p-point-independent then I is maximal. If I is everywhere p-point- independent, then I is everywhere maximal. 43

Proof. Let be I p-point-independent, A ⊆ ω be co-inﬁnite, and enumerate A as

{an : n < ω}. Deﬁne a function f by setting f(an) = n and f ω \ A is identically 1. By assumption there is a set E ∈ ENV(I) such that f E is either bounded or ﬁnite-to-one. So either E ⊆∗ A or E ⊆∗ (ω \ A), and thus I is maximal independent. Everywhere p-point independent implies everywhere maximal independent in a similar argument.

So to sum up our properties in reference to independent families:

Everywhere Selective −−−→ Everywhere p-point −−−→ Everywhere maximal       y y y Selective −−−→ p-point −−−→ Maximal (4.1) Q-independent families will provide the other piece of selectivity that allows us to connect selectivity and p-independent families, and this also provides a distinction between selectivity with ultraﬁlters and selectivity with independent families. The selective ultraﬁlters are those that are exactly both p and q, while they don’t quite match up in independent families.

Lemma 70. Let I be an independent family. Then

i) If I is selective then I is p-point-independent and q-independent.

ii) If I is p-point-independent and everywhere q-independent then I is selective.

iii) I is everywhere p-point-independent and everywhere q-independent if and only if I is everywhere selective.

Proof. Statement (i) is true by the deﬁnitions. For statement (ii) let f : ω → ω. Since I is p-point-independent there is an

h0 h0 ∈ FF (I) such that f I is ﬁnite-to-one or constant. If constant we are done, 44

h1 and if it is finite-to-one then we can find h1 ⊇ h0 such that f I is one-to-one since I is everywhere q-independent. For statement (iii) the forward direction is true by a similar argument to statement (ii), and the reverse is true by the definitions.

This division of selectivity into p-point and q properties is not a frivolous distinction. In Chapter 1 we noted that selective independent families are easily at least of size d, and this comes from their q-independence:

Lemma 71. If I is q-independent, then the enumerating functions of members of ENV(I) form a dominating family.

Proof. For E ⊆ ω let fE denote the increasing enumerator of E. The collection

{fE : E ∈ ENV(I) } forms a dominating family. To see this, it suﬃces to dominate all strictly increasing functions so let f ∈ ωω be strictly increasing. Deﬁne a function g : ω → ω via g(k) = 0 for k ∈ [0, f(0)] and g(k) = k for k ∈ (f(k), f(k + 1)]. The

function g is then ﬁnite-to-one, so we ﬁnd E ∈ ENV(I) such that g E is 1-1, and

thus f ≤ fE.

In contrast to p-point independent families, q-independent families need not be maximal. Assume CH and let I be a q-independent family. If we add a random real, then I is still a q-independent family since every real in the extension is bounded by a real in the model, and therefore the enumerations of ENV(I) still form a dominating family. But I is not a maximal independent family since the random real is independent from I.

4.3: The Dual Filter

+ The previous section looked a properties of the co-ideal idI . In this section we

consider instead the dual ﬁlter to idI . 45

Deﬁnition 72. Let F il ⊆ ω. Then

1) Call F il an a ﬁlter if:

– Whenever B ∈ F il and B ⊆ A then A ∈ F il.

– Whenever A, B ∈ F il then A ∩ B ∈ F il.

2) If F il is a ﬁlter, then F il+ = {A ⊆ ω : A 6∈ F il} is called the co-ﬁlter.

3) If A is a collection of subsets of ω then {(ω \ X): X ∈ A} is called the dual of A.

Notice that the dual of a ﬁlter is an ideal, and the dual of an ideal is a ﬁlter.

Deﬁnition 73. If I is an independent family then we deﬁne F ilI to be {A ⊆ ω : for every B in ENV(I) there is a C ⊆ B where C ∈ ENV (I) such that C ⊆ A}.

Notice that F ilI is indeed the ﬁlter dual to idI , and also that no member of the envelope can be a member of the ﬁlter.

Lemma 74. If I is an independent family then F ilI is not a p-point ﬁlter.

Proof. Let I be independent, and let X ∈ I. Enumerate (ω \ X) = {an : n > 0}, and let f : ω → ω such that f(X) = 0 and f(an) = n. Any set A such that is f A is one to one or constant means A ⊆ X or A ⊆ (ω \ X), so A 6∈ F ilI .

Lemma 75. If I is an independent family and F ilI is a q-ﬁlter then I is everywhere q-independent.

Proof. Trivial.

There does exist a model without any q-ﬁlters, (as detailed in [16]), so the existence

of an independent family I for which F ilI is a q-ﬁlter is independent of ZFC. It is an open question if an independent family with a p-ﬁlter is demonstrable from ZFC. 46

If we assume CH then it is possible for an independent family to have its filter to be both p and q. This establishes that our properties of everywhere selective, q-filter, and p-filter are all compatible. To demonstrate this we use Lemma 2.2 from [1] in order to explicitly build such an independent family.

Lemma 76. i) Assume I is a countable independent family, and A ⊆ ω such that

h |A∩I | = ℵ0 for every h ∈ FF(I). Then there is a set X ⊆ A such that I ∪{X} is independent.

h ii) Assume I is a countable independent family, and A ⊆ ω such that |A ∩ I | = ℵ0

for every h ∈ FF(I). If E is an equivalence relation on ω with ﬁnite equivalence

∗ h classes then there is a set B ⊆ A such that |B ∩ I | = ℵ0 for every h ∈ FF(I)

and E B is equality.

iii) Assume hIn : n < ωi is an increasing sequence of countable independent families

and hAn : n < ωi is an almost decreasing sequence of subsets of ω such that

h S |An ∩ I | = ℵ0 for every h ∈ FF(In). Let I = n<ω In. Then there is a set

∗ h A ⊆ ω such that A ⊆ An for every n, and |A ∩ I | = ℵ0 for every h ∈ FF(I).

Proof. For (i), enumerate FF(I) as {hn : n ∈ ω}. Select two increasing sequences

hn han : n < ωi and hbn : n < ωi such that an, bn ∈ A ∩ I and that an < bn < an+1.

This is possible since each A ∩ Ihn is inﬁnite, and at every stage we have only picked

ﬁnitely many elements. The set X = {an : n < ω} works as needed.

For (ii), enumerate FF(I) as {hn : n ∈ ω}. Select an increasing sequence han :

hn n < ωi such that an ∈ A ∩ I and that ai E aj if and only if i = j. This is possible

since each A ∩ Ihn is inﬁnite, each equivalence relation is ﬁnite, and at every stage we

have only picked ﬁnitely many elements. The set B = {an : n < ω} works as needed.

For (iii), enumerate FF(I) as {hn : n ∈ ω} and we can assume without loss of T generality that dom(hn) ⊆ n. Deﬁne Bn = k

hn han : n < ωi such that an ∈ Bn ∩ I . The set A = {an : n < ω} works as needed.

Lemma 77. Assume CH. Then there is an independent family I for which F ilI is both a p and q.

Proof. Enumerate the equivalence relations on ω with ﬁnite equivalence classes as

{Eα : α < ω1}. We create an increasing sequence of independent families hIα : α < ω1i and an almost decreasing sequence of sets hAα : α < ω1i such that if α > β then

∗ Aα ⊇ Aβ and |Aα ∩ E| = ℵ0 for every E ∈ ENV(Iα).

Let I0 be a countable independent family and A0 = ω. If α < ω1 is a limit, deﬁne S Iα = β<α Iβ and Aα is as given in (iii) of the previous lemma. Let {Xα : α < ω1} enumerate the co-inﬁnite subsets of ω.

Now let α < ω1. If Aα ∩Xα has inﬁnite intersection with every member of ENV(I)

∗ ∗ then deﬁne Aα+1 ⊆ Aα ∩ Xα given by (ii). Otherwise ﬁnd Aα+1 ⊆ Aα given by

(ii). Then ﬁnd Bα as given in (i) and deﬁne Iα+1 = Iα ∪ {Bα}. Then I = Iω1 is an independent family.

The collection F = {Aα : α < ω1} ↑ is a ﬁlter. Notice that F is a q-ﬁlter by (ii),

∗ and F is a p-ﬁlter since Aα ⊇ Aβ whenever α > β. We show that F = F ilI .

Let X ∈ F ilI , so for some α < ω1 X = Xα. Since X has inﬁnite intersection with

every member of ENV(I), then we found Aα+1 ⊆ Xα. Thus X ∈ F . Now let Y ∈ F

and h ∈ FF(I). Then there is some α < ω1 such that Aα ⊆ Y . Find β < α such that

h ∗ ∗ β 6∈ dom(h). Then I ∩ Bβ ⊆ Iβ ⊆ Iα ⊆ Y .

If we modify this construction where we also pick our sets to get an everywhere selective independent family as in Chapter 2 then we get the independent family Shelah describes in Lemma 2.3 in [1].

Theorem 78. Assume CH. Then there is an everywhere selective independent family

I where F ilI is both a p-ﬁlter and a q-ﬁlter. 48

Chapter 5: Generic Existence

Repeatedly throughout this paper we have been able to take independent families and use a variety of conditions to extend them to a variety of types of independent families. The paper [15] explored this phenomenon for ultraﬁlters and we use this as a guide in the study of independent families.

Deﬁnition 79. We say that (everywhere) selective (or p-point or q or maximal) independent families exist κ-generically if whenever I is an independent family with |I| < κ then I can be extended to an (everywhere) selective (or p-point or q or maximal) independent family.

In the case that κ = c we drop the reference to κ and say the objects exist generically. Using this deﬁnition we can restate some earlier results.

Lemma 80. • Lemma 26: CH implies everywhere selective independent families exist generically.

• Theorem 60: ♦(rQ) implies everywhere maximal independent families exist ℵ1- generically.

• Theorem 62: ♦(hom1,Q) implies everywhere selective maximal independent fam-

ilies exist ℵ1-generically.

5.1: Independent Families, Martin’s Axiom We have established CH is strong enough for selective independent families to exist generically and the reason for it seems to be that our constructions only dealt with a countable independent family at each stage. We explore an assumption weaker than CH that still allows selective independent families to exist generically. 49

Deﬁnition 81. Let P = (P ,≤) be a partial order.

• If p, q ∈ P , then p and q are compatible if there is some r ∈ P with r ≤ p, q.

• If C ⊆ P , call C an anti-chain on P if any two elements are not compatible.

• P has the Countable Chain Condition (ccc) if there are no uncountable anti- chains on P .

Deﬁnition 82. Let m be the largest cardinal such that if P is a ccc partial order and D is a collection of less than κ-many dense sets on P , then there is a ﬁlter F ⊆ P where F ∩ D 6= ∅ for every D ∈ D.

Martin’s Axiom is the assumption that m = c and we abbreviate it as MA. CH implies MA but MA does not imply CH. The following theorem in a known result but only as a consequence of the stronger result that MA implies the lesser cardinal p = c (see [9]). We again give a detailed proof so that it may be modiﬁed as needed.

Theorem 83. MA implies i = c

Proof. If we list {Xα ⊆ ω : α < c, such that Xα is co-inﬁnite} then we can create our independent family by induction just like before. We need only check to make sure that at any stage we can still add to our independent family. Let I be an independent family and |I| = κ < c. Let P = {h ∈ FF(ω)} under the ordering that p ≤ q if and only if p ⊇ q. Then for every n ∈ ω the set Dn =

{h ∈ P : n ∈ dom(h)} is dense in P . Also, for E ∈ ENV(I) the set DE = {h ∈ P : h−1(0) ∩ E 6= ∅ and h−1(1) ∩ E 6= ∅} is dense in P . MA then gives us a generic ﬁlter G on P , and notice S G is a function g : ω → 2

−1 where for every E ∈ ENV(I) g E is not constant. Let A = g (0), and so I ∪ {A} is independent, and thus I is not maximal. 50

When building a selective independent family using CH we used a lemma from Shelah, and we need a similar lemma for MA.

Lemma 84. Assume MA. If I is an independent family, |I| < c, h0 ∈ FF (I), and f : ω → ω then there is a set A and h1 ∈ FF (I) such that A 6∈ I, I ∪ {A} is

h1 independent, h1 ⊇ h0, and f (I ∩ A) is 1-1 or equality.

Proof. We have two cases to consider.

h1 • Case 1: For some h1 ∈ FF(I) with h1 ⊇ h0 f I ﬁnite to 1.

• Case 2: Not case 1.

Case 1: We may assume without loss of generality that every k ⊇ h1 with k ∈

k h k FF(I) f I has the same range as f I 1 . If not true, f I has a smaller yet still

ﬁnite range, and make h1 = k. We need only repeat this process a ﬁnite number of times until our assumption becomes true.

−1 Let P = {k ∈ FF(ω): f k (0) is constant } and use the same ordering as before. As before we have the same dense sets in P , and if G is the ﬁlter on P as given by MA then S G is a function g :→ 2 and let A = g−1(0).

h As before I ∪ {A} is independent so we show that f (I 1 ∩ A) is constant.

h1 Let n, m ∈ (I ∩ A), so there is k1, k2 ∈ G with n ∈ dom(k1), m ∈ dom(k2). Since

−1 G is a ﬁlter on P there is an k3 ∈ P such that k3 ≤ k1, k2 i.e. f k3 (0) is constant

h1 with k3(n) = 0 = k3(m). Thus f(n) = f(m) so f (I ∩ A) is constant. −1 Case 2: Let P = {k ∈ FF(ω): f k (0) is 1-1 } and use the same ordering as before. As before we have the same dense sets in P , and if G is the ﬁlter on P as given by MA then S G is a function g :→ 2 and let A = g−1(0).

h1 As before I ∪ {A} is independent, so let h1 = h0 and we show f (I ∩ A) is 1-1.

h1 Let n, m ∈ I ∩ A, so there is k1, k2 ∈ G with n ∈ dom(k1), m ∈ dom(k2). Since G 51

−1 is a ﬁlter on P there is an k3 ∈ P such that k3 ≤ k1, k2 i.e. f k3 (0) is 1-1, and

h1 k3(n) 6= k3(m). Thus f(n) 6= f(m) so f A ∩ I is 1-1.

Theorem 85. MA implies the existence of an everywhere selective independent family, and thus iesel = c

Proof. List {(E, h): E is an equivalence relation on ω and h ∈ FF(c)} as {(Eα, hα):

α < c}. We may assume without log of generality that hα ⊆ α. Create by induction:

Let I0 be an independent family of size |I| < c. If α < c is a limit, deﬁne Iα = S β<α Iβ. Now let α < c, Aα as given in the previous lemma, and Iα+1 = Iα ∪ {Aα}.

Then Ic is an everywhere selective independent family.

Again, during this proof we use an arbitrary independent family of size less than c so we have created many everywhere selective independent families and they must exist generically.

Corollary 86. MA implies everywhere selective independent families exist generically.

If we examine our proofs involving MA we actually ﬁnd that in each of these constructions we used a countable partial order which trivially makes each partial order ccc. So we are able to make use of a weaker assumption.

Deﬁnition 87. Let mc be the largest cardinal such that if P is a countable partial order and D is a collection of less than κ-many dense sets on P , then there is a ﬁlter F ⊆ P where F ∩ D 6= ∅ for every D ∈ D.

Martin’s Axiom for countable partial orders is the assumption that mc = c and

we abbreviate it as MAc. 52

The cardinal mc is equal to the more famous cardinal cov(B), the covering number of Baire category, which is the cardinality of the smallest family of meager subsets of

R whose union covers R.

Corollary 88. If cov(B) = c then everywhere selective independent families exist generically.

Reversing the implication is true for ultraﬁlters. Is it true for independent families?

Question 1. Does the generic existence of everywhere selective independent families imply that cov(B) = c?

5.2: Extending to Various Independent Families

We are able to reduce CH to MAc in order for everywhere selective independent families to exist generically, and we have weaker assumptions that give us weaker versions of selectivity.

Lemma 89. If i = c then everywhere independent families exist generically.

Proof. Let {(hα,Aα): α < c} enumerate FF(c) × P(ω). Without loss of generality

assume dom(hα) ⊆ α. Let I0 be an independent family with |I| < c. If α a limit S deﬁne Iα = β<α Iβ.

k Now assume Iα is deﬁned, and we create Iα+1. If for some k ⊇ h we have Iα ⊆ Aα

k k k or Iα∩Aα = ∅ we let Iα+1 = Iα. Otherwise for every k ⊇ hα Aα∩Iα and (ω\Aα)∩Iα are

both inﬁnite, and since |Iα| < i then we can ﬁnd a set Bα such that Iα+1 = Iα ∪ {Bα}

hα and Iα ∩ Bα = Aα.

Then Ic is everywhere independent. To see, let h ∈ FF(Ic) and A ⊆ ω. Then

h = hα and A = Aα for some α < c, and then there is some k ⊇ hα such that

k k Iα+1 ⊆ Aα or Iα+1 ∩ Aα = ∅. 53

The assumption that d = c implies the existence of p-point-ultraﬁlters (see [15]) and we see that this works for independent families as well.

Lemma 90. If I is an independent family with |I| < d, and f ∈ ωω. Then there is a

set B ⊆ ω such that I ∪ {B} is independent and for some E ∈ ENV (I ∪ {B}) f E is bounded or ﬁnite-to-one.

S −1 Proof. For each n ∈ ω deﬁne An = {f (k): k ≥ n}. Then this is a decreasing

sequence of sets and A1 = ω. We have two cases to consider.

1) Every An has non-empty intersection with every member of ENV(I).

2) Some An has empty intersection with some member of ENV(I).

In case 1, we can apply prop 6.24 in [9] (just like in our proof of d ≤ i) to ﬁnd a

∗ set B such that B ⊆ An for each n ∈ ω and B has inﬁnite intersection with every

∗ member of ENV(I). Since B ⊆ An for each n ∈ ω then f B is ﬁnite-to-one, and since B has inﬁnite intersection with every member of ENV(I) we can assume without loss of generality I ∪ {B} is independent.

In case 2, ﬁnd n ∈ ω such that An has non-empty intersection with every member

of ENV(I) while An+1 does not. So for some E ∈ ENV(I) we know E ⊆ An \ An+1.

Thus f E is bounded, so we may ﬁnd B since I is not maximal.

Theorem 91. If d = c then p-point-independent families exist generically.

ω Proof. List ω as {fα : α < c} and we build by induction. Let I0 be an independent S family of size |I| < c. If α < c is a limit, deﬁne Iα = β<α Iβ. Now let α < c, Aα as

given in the previous lemma, and Iα+1 = Iα ∪{Aα}. Then Ic is a p-point-independent family. 54

If we combine our ideas from Lemma 89 and Theorem 91 then we can create everywhere p-point independent families.

Theorem 92. If d = c then everywhere p-point-independent families exist generically.

Now in view of our previous work it should be pretty easy to work out what we need for q-independent families. Selective needs p-point and q, and the generic existence of selectivity requires cov(B) = c. For p-point we needed d = c, so q-independence should require cov(B) = d. To prove this we use the following concept. For a function f we say that a set A is f-rare if whenever a, b ∈ A with a < b then f(a) < b.

Lemma 93. Assume cov(B) = d. If I is an independent family, |I| < d, and f ∈ ωω is ﬁnite-to-one, then there is a set A ⊆ ω such that I ∪ {A} is independent and some E ∈ ENV (I ∪ {A}) is f-rare.

Proof. Let P be the collection of all pairs (a, b) of ﬁnite disjoint subsets of ω where a is f-rare. Deﬁne a partial order on P where (a, b) ≤ (c, d) if and only if a ⊇ c and

b ⊇ d. For each E ∈ ENV(I) the set DE = {(a, b) ∈ P : a ∩ E 6= ∅ and b ∩ E 6= ∅} is dense in P . Since there are less than d-many of these dense sets let G be a ﬁlter S on P that meets each one, and deﬁne A = {a :(a, b, F ) ∈ G}. Then I ∪ {A} is independent and A is f-rare.

Theorem 94. If cov(B) = d then q-independent families exist d-generically.

Proof. Let {fα : α < d} be a dominating family and we build by induction. Let I0 be S an independent family of size |I| < d. If α < c is a limit, deﬁne Iα = β<α Iβ. Now

let α < c, Aα as given in the previous lemma, and Iα+1 = Iα ∪ {Aα}. Then Id is a q-independent family. 55

To see let f be a ﬁnite-to-one function. Deﬁne a function f¯ ∈ ωω via f¯(x) = ¯ max{n ∈ ω : f(n) = f(x)}. Then there is some fα that dominates f, and there is ¯ a set Eα ∈ ENV(I) such that Eα is fα-rare. Since fα dominates f then Eα is also ¯ ¯ f-rare. Then f Eα must be 1-1 since if a, b ∈ Eα with a < b then f(a) < b so f(a) 6= f(b).

Again, as before, we can combine the ideas of Lemma 89 with Theorem 94 to create everywhere q-independent families.

Theorem 95. If cov(B) = d then everywhere q-independent families exist d-generically.

5.3: Independent Families as a Notion of Forcing We investigate two natural forcing constructions arising from our work. For the ﬁrst construction, we note that FF(I) (for independent I of size κ) can be viewed as a notion of forcing upon setting h ≤ g if and only if h ⊇ g. P = (FF(I), ≤) is isomorphic to the forcing adding |I| Cohen reals, and has the countable chain condition. Thus forcing with P preserves all cardinals and coﬁnalities. The following are dense in P:

i) Dα = {h ∈ P : α ∈ dom(h)} where α ∈ κ.

ii) Df = {h ∈ P : ∃i < κ where f(i) 6= h(i)} where f is any function in V such that f : κ → 2.

h iii) DA = {h ∈ P : I ∩ A = ∅} where A ∈ idI .

The dense sets in (i) and (ii) are dense sets whenever adding κ Cohen reals, and dense sets in (iii) are true by deﬁnition of the ideal idI . Let G be the generic ﬁlter adjoined by this notion of forcing. Notice that since

h G is a filter then FG = {I : h ∈ G} is a family of sets in ENV(I) with the finite intersection property, meaning that this will be the basis for a filter in the extension. 56

If we strengthen our independent family to one that is everywhere maximal then we have more dense sets in P.

h h iv) DA = {h ∈ P : I ∩ A = ∅ or I ⊆ A} where A ⊆ ω.

Notice that I is everywhere maximal if and only if the sets of (iv) are dense in P.

Definition 96. A filter U in the extension V [G] is a V -ultrafilter, if whenever A ⊆ ω with A ∈ V then either A ∈ U or (ω \ A) ∈ U.

The ﬁlter FG is clearly an V -ultraﬁlter if and only if the sets of (iv) are dense in

P. We can therefore say that FG is an V -ultraﬁlter if and only if I is an everywhere maximal independent family. If we again strengthen our everywhere independent family to an everywhere p-

point-independent family then we have more dense sets in P.

h ω v) Df = {h ∈ P : f I is ﬁnite-to-one or bounded} for f ∈ ω.

Notice that I is everywhere p-point if and only if the sets of (v) are dense in P.

Deﬁnition 97. A V -ultraﬁlter U is V -p-point if whenever f ∈ ωω with f ∈ V then

there exists a set A ∈ U such that f A is ﬁnite-to-one or bounded.

The ﬁlter FG is clearly an V -p-point-ultraﬁlter if and only if the sets of (v) are

dense in P. We can therefore say that FG is an V -p-point-ultraﬁlter if and only if I is an everywhere p-point independent family. If we again strengthen our independent family to an everywhere selective inde-

pendent family, then we have more dense sets in P.

h ω vi) Df = {h ∈ P : f I is one-to-one or constant} for f ∈ ω.

Notice that I is everywhere selective if and only if the sets of (vi) are dense in P. 57

Deﬁnition 98. A V -ultraﬁlter U is V -selective if whenever f ∈ ωω with f ∈ V then there exists a set A ∈ U such that f A is one-to-one or constant.

The filter FG is clearly an V -selective-ultrafilter if and only if the sets of (vi) are dense in P. We can therefore say that FG is an V -p-point-ultrafilter if and only if I is an everywhere p-point independent family. We summarize all of this in the following proposition.

Proposition 99. Let I be an independent family, P = (FF(I), ≤), G be the generic

ﬁlter adjoined by P, and FG the ﬁlter in V [G] created by G. Then:

• I is everywhere maximal if and only if FG is a V -ultraﬁlter.

• I is everywhere p-point if and only if FG is a V -p-point-ultraﬁlter.

• I is everywhere selective if and only if FG is a V -selective-ultraﬁlter.

Our second construction follows the lead of [15]. If κ is an uncountable cardinal with κ let Qκ be the collection of all independent families on ω of size < κ. Say that

I ≤ J whenever I ⊇ J. The generic object adjoined by Qκ will be a family of sets, but Qκ adds no new subsets of ω. In fact, Qκ is κ-closed, so it adds no new sets of κ either.

If κ ≤ i then Qκ then G is an independent family of size κ. G is not only maximal but everywhere maximal since for every A ⊆ ω and h ∈ FF(κ) the set

k k DA,h = {I ∈ Qκ : there is a k ∈ FF(I) such that I ∩ A = ∅ or I ⊆ A} is dense in

Qκ. Notice that these dense sets corresponds directly to our deﬁnition of everywhere maximal independent families existing κ-generically (hence the motivation behind the name). Under various assumptions and sizes of κ we will be able to adjoin various types of independent families. 58

So if κ ≤ d then Qκ adjoins an everywhere p-point independent family. If κ ≤ cov(B) then Qκ adjoins an everywhere p-point independent family. Note that since cov(B) ≤ d then if κ ≤ cov(B) we actually adjoin an everywhere selective independent family since it would be both everywhere p-point and everywhere q.

By extension, if we assume cov(B) = c then Qc adjoins an everywhere selective independent family. If we assume d = c then Qc adjoins an everywhere p-point- independent family. If we assume cov(B) = d then Qd adjoins an everywhere q- independent family. 59

Chapter 6: Miscellaneous

Now based oﬀ the work we’ve done in the previous chapters we have some questions that arise quite naturally.

6.1: Direct and Inverse Images of Independent Families Let I be an independent family and f mapping ω into ω. When is f(I) = {f(X): X ∈ I} an independent family? If f is a one-to-one function then clearly f(I) is still a independent. On the other hand, we see that we need f to be close to a one-to-one function as even a two-to-one onto map may fail to preserve independence. Let X ∈ I and enumerate X = {an : n < ω} and ω \ X = {bn : n < ω}. Then f mapping ω into ω via f(an) = f(bn) = n makes f(X) = f(ω \ X) = ω and so f(I) cannot be independent. Is the inverse image of an independent family still independent? Not necessarily:

Let X ∈ I and enumerate ω \ X = {an : n < ω}. Then f mapping ω into ω via

−1 −1 f(n) = an makes f (X) = ∅ so f (I) cannot be independent. However, so long as we can avoid this case, where the inverse image of a member of the envelope is empty, the inverse image of an independent family can be preserved.

Lemma 100. Let I an inﬁnite independent family and f a map from ω into ω. Then f −1(I) is independent if and only if f −1(Ih) 6= ∅ for every h ∈ FF (I).

Proof. Our conclusion comes easily from the observation that if A, B ⊆ ω then f −1(A ∩ B) = f −1(A) ∩ f −1(B), so for any h ∈ FF(I)(f −1(I))h = f −1(Ih).

An easy example shows that we need to strengthen this result in order to preserve maximal independence. If I is independent and f is a two-to-one onto function from

−1 −1 ω to ω then f (I) is independent. However for each n ∈ ω f (n) = {an, bn} and so

−1 A = {an : n < ω} is independent from f (I). 60

We know then that in order to preserve an independent family we need f to be close to an onto function, and to preserve a maximal independent family f must also be close to a one-to-one function.

Deﬁnition 101. Let I an inﬁnite independent family and f a map from ω into ω.

• Call f I-almost-onto if ω \ f(ω) ∈ idI .

−1 + • Call f I-partially-1-1 if {n : |f (n)| = 1} ∈ idI .

−1 • Call f I-almost-1-1 if {n : |f (n)| > 1} ∈ idI .

Note that f is I-almost-onto if and only if f(ω) ∩ E 6= ∅ for every E ∈ ENV(I).

Lemma 102. Let I be a maximal independent family and f a map from ω that is I-almost-onto. If f −1(I) is a maximal independent family then f is I-partially-1-1.

−1 Proof. Assume f is not I-partially-1-1. So {n : |f ({n})| = 1} ∈ idI and call this set C. Deﬁne A = { min(f −1({n}): n 6∈ C}. Notice that both A and ω \ A have non- empty intersection with f −1({k}) for k ∈ (ω \ C). So if E ∈ ENV(I) then E \ C 6= ∅, meaning both A and (ω \ A) intersect f −1(E). Thus f −1(I) is not maximal.

Lemma 103. Let I be an everywhere independent family and f : ω → ω be I-almost- onto. Then f −1(I) is everywhere independent if and only if f is I-almost-1-1.

−1 + Proof. Assume f is not I-almost-1-1. So {n : |f (n)| > 1} ∈ idI , and call this set C. Find h ∈ FF(I) such that Ih ⊆ C. Let A = {min(f −1(n)) : n ∈ Ih}. Then both A and f −1(Ih) \ A have inﬁnite intersection with f −1(Ik) when k ⊇ h. Thus f −1(I) is not everywhere independent. Now assume f is I-almost 1-1, and we show f −1(I) is everywhere independent.

Let A ⊆ ω and h0 ∈ FF(I). Since f is I-almost 1-1 then we can ﬁnd h1 ∈ FF(I)

−1 h1 where h1 ⊇ h0 and f f (I ) is 1-1. Then f(A) ⊆ ω and we can ﬁnd an h2 ∈ FF(I) 61

h1 h1 −1 h2 where h2 ⊇ h1 such that f(A) ∩ I = ∅ or I ⊆ f(A). But then A ∩ f (I) = ∅ or f −1(I)h2 ⊆ A.

We can also deﬁne an ordering analogous to to the Rudin-Keisler (RK) ordering of ultraﬁlters,

Deﬁnition 104. Let I,J be independent families.

• Deﬁne I ≥ J if there is a function f such that I = f −1(J).

• Deﬁne I ≡ J if I ≥ J and J ≥ I.

Lemma 105. Let I,J be independent families such that I ≥ J.

If I is (everywhere) maximal (or selective or p-point or q) then J is (everywhere) maximal (or selective or p-point or q).

Proof. We demonstrate the first, as all other results have similar proofs. If we have A ⊆ ω then f −1(A) ⊆ ω, so find h ∈ FF(I) such that Ih ∩f −1(A) = ∅ or Ih ⊆ f −1(A). Since I ≥ J then there is a k ∈ FF(J) such that Ih = f −1(J k). In the first case f −1(J k) ∩ f −1(A) = ∅ so J k ∩ A = ∅. In the second case f −1(J k) ⊆ f −1(A), so J k ⊆ A.

6.2: n-Independent Families

Any subset A of ω creates a partition PA of ω into 2 sets, A and ω \ A. Any independent family I then can be viewed as a collection of partitions P = {PX : X ∈

I} of ω into 2 sets, where if P0,P1, ..., Pk ∈ P and any A0 ∈ P0,A1 ∈ P1, ..., Ak ∈ Pk then |A0 ∩ A1 ∩ ... ∩ Ak| = ℵ0. If we were to instead partition ω into more than 2 sets we can still apply the same principle. 62

Deﬁnition 106. Let 1 < n < ω and P be a collection of partitions of ω into n many inﬁnite sets. Then P is an n-independent family if P0,P1, ..., Pk ∈ P and Ai is any

set in Pi then |A0 ∩ A1 ∩ ... ∩ Ak| = ℵ0.

Define FFn(I) as the collection of finite functions from I to n. If P ∈ I, define

k h T h(P ) P as the k-th set in P . If h ∈ FFn(I) deﬁne I = {P : P ∈ dom(h)}. We can deﬁne ENV(I) exactly as before.

Notice that any independent family referenced up to this point have been 2- independent families. Viewed in that context, we can replace any previous men- tion of independent families in this paper with an n-independent family and almost everything would remain true. We only need to establish the existence of n-independent families of size c, and we can do so by induction.

Lemma 107. If I is an inﬁnite n-independent family, then there is an (n+1)- independent family J of the same size.

Proof. For every P ∈ I ﬁx AP ∈ P , and enumerate I = {Pα : α < |I|}. Let Qα be the

same partition as P2α except that AP2α is split into AP2α ∩AP2α+1 and AP2α ∩(ω\AP2α+1 ).

Then J = {Qα : α < |I|} is (n + 1)-independent, and |I| = |J|.

Corollary 108. For every 1 < n < ω there is a an n-independent family of size c.

Since n-independent families share all the qualities of the 2-independent families we’ve explored, the only question of interest is how an n-independent family relates to a k-independent family.

Let in = min{|I| : I is a maximal n-independent family } and notice the i2 = i.

Lemma 109. Let I be an n-independent family, and P is a partition of ω into n-many inﬁnite sets with P 6∈ I. Then the following are equivalent: 63

i) I ∪ {P } is n-independent.

ii) Each element of P has inﬁnite intersection with every element of ENV(I).

Proof. This is just a translation of Lemma 2.

Lemma 110. Let I be an n-independent family. TFAE:

i) I is maximally n-independent.

ii) If P is a partition of ω into n-many inﬁnite sets, then for some A ∈ P and B ∈ ENV (I) we have A ∩ B = ∅.

Proof. This is just a translation of Lemma 5.

Lemma 111. If 1 < n, k < ω, then ink = in.

Proof. Assume I = {Pα : α < |I|} is maximally n-independent. Notice that k-many

k n-partitions P1,P2, ..., Pn creates a single n -partition {A1 ∩ A2 ∩ ... ∩ An : Ai ∈ Pi}.

Write I as the disjoint union of Qα with |Qα| = k, let Rα = ENV(Qα), and J = {Rα : α < |I|}. Notice that ENV(J) is a dense subset of ENV(I). Then J is a maximal nk-independent family.

k To see, let R = {A1,A2, ..., Ank } be an n -partition of ω. Let B = An ∪ ... ∪ Ank

and P = {A1,A2, ..., An−1,B}. Then P is an n-partition of ω so for E ∈ ENV(I) there is come C ∈ P such that E ∩ C = ∅. But since ENV(J) is dense in ENV(I) we can ﬁnd F ∈ ENV(J) with F ⊆ E, and so F ∩ C = ∅.

k For the other direction, assume I = {Pα : α < |I|} is maximally n -independent.

We create κ-many n-partitions from each member of I. For each α < |I| ﬁx hα : Qk Pα → i=1 n , i.e. hα is an enumeration of Pα. Deﬁne πj(i) = {P ∈ Pα : the j-th k S coordinate of h(P ) = i}. For each 0 ≤ i < n, let Qα = { πj(i) : 0 ≤ i < n}.

j Then J = {Qα : α < |I|, 0 ≤ j < k} is an n-independent family and notice ENV(J) is dense in ENV(I) so J is a maximal n-independent family. 64

We can extend our deﬁnition of independence to partitions of ω into ω-many sets, and call these ω-independent families. We can ﬁnd ω-independent families of size c.

Example 4. Let P be the set of all polynomials with rational coeﬃcients. For each

n i real r deﬁne Ar = {p ∈ P : n ≤ |p(r)| < n + 1} and then deﬁne Pr = {Ar : i ∈ ω}.

Then {Pr : r is real } is an ω-independent family.

So then by Zorn’s Lemma we know that there exist maximal ω-independent fam-

ilies, and deﬁne the related cardinal as iω. By a diagonalization argument we know

that iω > ℵ0.

Lemma 112. If I is a countable ω-independent family then there is a partition X of ω into ω-many sets such that X 6∈ I and I ∪ {X} is ω-independent.

Proof. Enumerate ENV(I) as {En : n ∈ ω}. We can ﬁnd an increasing sequence

ha(0, 0), a(0, 1), a(1, 1), a(0, 2), a(1, 2), a(2, 2), a(0, 3), ...i where a(i, j) ∈ Ei. Let Xk =

{a(m−k, m): m ≥ k}, and then for all n and k in ω En ∩Xk 6= ∅. Let X = {Xk : k ∈ ω} and without loss of generality X is a partition of ω since we can add anything not

covered by X to any one of the Xk sets without losing independence. Then I ∪ {X} is ω-independent.

Any partition P = {Pn : n ∈ ω} of ω into a ω-many sets can be viewed as a

ω function fP ∈ ω where fP (k) = n where k ∈ Pn. We can then go further with regards to this deﬁnition of an ω-independent family. Call a collection of functions

F independent if whenever f1, f2, ..., fn ∈ F and k1, k2, ..., kn ∈ ω then |{j ∈ ω :

f1(j) = k1, ..., fn(j) = kn}| = ℵ0. The concept of ω-independent partitions of ω and independent functions are thus equivalent.

6.3: Ultrapowers As stated in the introduction, one of the motivating ideas behind the paper was to explore if it is consistent that i < a. Even though the resulting paper focused 65

on developing a systematic study of independent families we do have something to contribute to this idea. The use of ultrapowers of ccc forcing has been used to create models where a is large. In particular, this approach was used by Shelah [9] to prove d < a. We demonstrate that approach will not work for the consistency of i < a because ultrapowers destroy the maximality of independent families. Let P be a ccc notion of forcing, and let κ be a measurable cardinal with D as its witness (i.e. D is a non-principle κ-complete ultrafilter on κ). Define P κ = {p : p is a function from κ into P }. We can define an equivalence relation on P κ where p ∼ q iff {α : p(α) = q(α)} ∈ D. The ultrapower P κ/D = {[p]: p ∈ pκ} is the collection of these equivalence classes, and since P is ccc then the ultrapower will also be ccc. Similar to the method of Lemma 0.3 in [8], we show that maximal independent families are not preserved.

I˙ Lemma 113. If is a name for an independent family of size at most κ, then P κ/D ‘ I˙ is not maximally independent’.

I˙ ˙α ˙ α Proof. Without loss of generality we may assume P κ/D ‘ = {I : α < κ}’. Let X be the characteristic function for I˙α i.e. X˙ α : ω → 2 where X˙ α(n) = 1 iﬀ n ∈ X˙ α.

α α Since P is ccc, then for a fixed α and a fixed n we can find pn,i ∈ P , kn,i ∈ 2, such α ˙ α α α that pn,i ‘X (n) = kn,i’ and {pn,i : i < ω} is a maximal anti-chain. κ α α ˙ ˙ α In P /D, find [pn,i] = {pn,i : α < κ}/D, kn,i = {kn,i : α < κ}/D, and X = {X : ˙ α < κ}/D. Note that {[pn,i]: i < ω} is a maximal anti-chain. Take I to be the set named by X˙ . We will show that I˙ and I˙c have infinite intersection with each member of the envelope of I˙. ˙ I˙ α α α Fix E ∈ ENV( ). Then for all but finitely many α < κ we can find qn,i, an,i, bn,i α α α α α ˙ ˙α such that {qn,i : i < ω} is a maximal anti-chain, n < an,i, bn,i, and qn,i ‘ an,i ∈ E∩I , 66

α ˙ ˙α c bn,i ∈ E ∩(I ) ’. As before, form [qn,i], an,i, bn,i, and note {[qn,i]: i < ω} is a maximal ˙ ˙ ˙ ˙ c anti-chain with [qn,i] ‘E ∩ I 6⊆ n and E ∩ (I) 6⊆ n’. 67

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