Type: Article

P-points, MAD families and Cardinal Invariants

Guzmán González, Osvaldo;

Abstract:

The main topics of this thesis are cardinal invariants, P -points and MAD families. Cardinal invariants of the continuum are cardinal numbers that are bigger than ?0 and smaller or equal than c. Of course, they are only interesting when they have some combinatorial or topological definition. An almost disjoint family is a family of infinite subsets of ? such that the intersection of any two of its elements is finite. A MAD family is a maximal almost disjoint family. An ultrafilter U on ? is called a P-point if every countable B?U there is X? U such that X?B is finite for every B?B. This kind of ultrafilters has been extensively studied, however there is still a large number of open questions about them. In the preliminaries we recall the principal properties of filters, ultrafilters, ideals, MAD families and cardinal invariants of the continuum. We present the construction of Shelah, Mildenberger, Raghavan, and Stepr?ns of a completely separable MAD family under s?a. None of the results in this chapter are due to the author. The second chapter is dedicated to a principle of Sierpi?ski. The principle (?) of Sierpi?ski is the following statement: There is a family of functions {?n:?1??1?n??} such that for every I?[?1]?1 there is n?? for which ?n[I]=?1. This principle was recently studied by Arnie Miller. He showed that this principle is equivalent to the following statement: There is a set X={f???<?1}??? such that for every g:??? there is ? such that if ?>? then f??g is infinite (sets with that property are referred to as IE -Luzin sets ). Miller showed that the principle of Sierpi?ski implies that non (M)=?1. He asked if the converse was true, i.e., does non (M)=?1 imply the principle (?) of Sierpi?ski? We answer his question affirmatively. In other words, we show that non (M)=?1 is enough to construct an IE -Luzin set. It is not hard to see that the IE -Luzin set we constructed is meager. This is no coincidence, because with the aid of an inaccessible cardinal, we construct a model where non (M)=?1 and every IE -Luzin set is meager. The third chapter is dedicated to a conjecture of Hrušák. Michael Hrušák conjectured the following: Every Borel cardinal invariant is either at most non (M) or at least cov (M) (it is known that the definability is an important requirement, otherwise a would be a counterexample). Although the veracity of this conjecture is still an open problem, we were able to obtain some partial results: The conjecture is