Some Topics in Pure Set Theory

Abstract

Recall the definition of a Borel set. If ℌ is any collection of subsets of a set X, there is a ionique smallest σ-field of subsets of X containing ℌ : the a-field generated by ℌ . (The notion of a field of sets was defined in Problem I.1. A field, F , of subsets of X is called a σ-field iff whenever A_n ∊ F , n = 1,2,..., then $\bigcup\limits_{n = 1}^\infty {{\text{A}}_{\text{n}} \in F}$ and $\bigcap\limits_{n = 1}^\infty {{\text{A}}_{\text{n}} \in F}$) In ℝ, the σ-field generated by the collection of open sets is the σ-field of all Borel sets: a subset of ℝ is a Borel set iff it is a member of this a-field. We show that the Borel sets lie in a ramified hierarchy, and that there are precisely ${\text{2}}^{{{\aleph}}_{\text{0}} }$ many Borel sets.

This chapter contains some fairly complex proofs, which the casual reader can ignore. Chapter V and VI do not depend upon this chapter. For the reader who is interested in seeing what arguments in set theory look like, let us mention that, unless stated otherwise, the sections in this chapter are all independent of each other .