Higher Baire Spaces: Cardinal Characteristics, Higher Reals & Bounded Spaces

Abstract

This thesis investigates cardinal characteristics on higher Baire spaces (also known as generalised Baire spaces), with a particular focus on generalising cardinal characteristics of the Cichoń diagram.

Chapter 2 forms the background to the dissertation and contains no new results by the author. We formally define higher Baire spaces and discuss their properties. We furthermore define the main cardinal characteristics using the framework of relational systems. Relational systems help us in giving concise ZFC-results regarding our cardinal characteristics. We define both the cardinal characteristics of the classical and higher Cichoń diagrams and give some (sketches of) proofs of the relations between these cardinal characteristics, and an overview of unknown relations.

In Chapter 3 we introduce bounded higher Baire spaces, as well as bounded versions of the cardinal characteristics of the higher Cichoń diagram. We prove ZFC-results regarding the relation between the bounded and unbounded cardinal characteristics of the higher Cichoń diagram. Finally, we discuss the influence of the choice of bound (and of other parameters) on these cardinals, in particular for which choices the cardinals do not consistently lie strictly between κ and κ+. This leads to several interesting open questions as well.

In Chapter 4 we discuss forcing notions associated with higher Baire spaces. We give properties of such forcing notions and how these properties will influence our cardinal characteristics. We do this by investigating new elements of κ^κ with certain generic combinatorial properties over the ground model. We will also give some simple independence results, most of them old, some of them new.

The last two chapters deal with more complex independence results. In Chapter 5 we show the consistency of a large family (of size κ+) of localisation cardinals with distinct cardinalities. In Chapter 6 we show the existence of a family (of size κ) of antiavoidance cardinals, any finite subset of which yields a forcing extension where all of the cardinals in the finite subset are distinct.