Special Sets of Reals

Abstract

In this chapter we shall work in ZFC, i.e., we assume that the Axiom of Choice holds true. We shall investigate subsets of a Polish space, especially subsets of the real line with some exceptional, or maybe extremal, properties: small, with nice subsets, with good properties of sequences of real continuous functions on it, good covering properties and also those related to harmonic analysis. According to the extremeness of their properties, one would prefer that some of them do exist (since they are nice) or one would prefer that some of them do not exist (since they are at least strange). As we shall see later, ZFC is not strong enough to answer several questions concerning properties of such sets. However, it turned out that there are close relationships among them, which we shall investigate.