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 Michael J. Perron April 2017 c 2017 Michael J. Perron. All Rights Reserved. 2 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 !, defined as follows: Definition 1. A family I of subsets of ! is called independent if whenever we have distinct X0; :::; Xn;Y0; :::; Ym 2 I then X0 \ ::: \ Xn \ (! n Y0) \ ::: \ (! n 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 !, define A0 = A and A1 = (! nA). If I is a set, let FF(I) to be the collection of finite partial functions from I to 2. If h 2 FF(I) then Ih = TfAh(A) : A is in the domain of hg. h h1 h2 Notice that ENV(I) = fI : h 2 FF(I)g 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 2 FF(I), and x 2 Ih, then we can extend h to some k 2 FF(I) where x 62 Ik. To see this, find a set A 2 I n dom(h) and then find i 2 2 with x 62 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 n A is finite) or if E \ A =∗ ; (i.e. E has finite intersection with A), then we can extend E to an F 2 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 !, and E 2 ENV(I). If E ⊆∗ A or E \ A =∗ ; then there is an F 2 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 ⊆ ! with A 62 I. Then the following are equivalent: i) I [ fAg is independent. ii) Both A and (! nA) 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(!). 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 = f(k; A): k 2 !; A ⊆ P(k)g. For each X ⊆ ! let X0 = f(k; A) 2 F : X \ k 2 Ag. Then I = fX0 : X ⊆ !g is an independent family. Proof. Let X0; :::; Xn;Y0; :::; Ym be different subsets of !. Since we have a finite num- ber of different sets we can find a k 2 ! large enough that k \ X0; :::; k \ Xn; k \ Y0; :::; k \ Ym are all distinct. Let A = fk \ X0; :::; k \ Xng. Then A ⊂ P(k) and 0 0 0 0 (k; A) 2 X0 \ ::: \ Xn \ (F n Y0 ) \ ::: \ (F n 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 = fp 2 P : p(r) > 0g. Then fAr : r is real g 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 ⊆ [!]! 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 2 F define A0 to be the collection of all finite subsets of ! that intersect A. Then fA0 : A 2 F g is an independent family. Proof. Let X0; :::; Xn;Y0; :::; Ym be different sets in A. For each i ≤ n find ki 2 Xi where ki 62 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 fki : i ≤ ng 2 0 0 <! 0 <! 0 X0 \ ::: \ Xn \ ([!] n Y0 ) \ ::: \ ([!] n 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 2 F and B ⊆ A then A 2 F 2) whenever A; B 2 F then A \ B 2 F We call F an ultrafilter if whenever A ⊆ ! then either A 2 F or (! n A) 2 F . Posp´ıˇsil'sclassic proof that there are 22@0 ultrafilters on ! 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 22@0 ultrafilters on !. Proof. Any ultrafilter on ! is an element of P(P(!)) so there are at most 22@0 ultra- filters. Now let I be an independent family of ! of size 2@0 . For each f : I ! f0; 1g define Gf = fX : X is co-finite g [ fX : f(X) = 1g [ f! n X : f(X) = 0g. 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 2 I such that f(X) 6= g(X). Without loss of generality f(X) = 1 and g(X) = 0. So X 2 Uf but (! n X) 2 Ug hence Uf 6= Ug. So each function creates a distinct ultrafilter, and since there are 22@0 functions from I to 2 then there are 22@0 ultrafilters on !. 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 ⊆ ! with A 62 I then I [ fAg is not indepen- dent.