Selection Principles and Special Sets of Reals

This chapter provides an overview of selection principles and special sets of reals. The field of selection principles in mathematics started with Scheepers' identification and classification of common prototypes for selection hypotheses appearing in classical and modern works. The field of selection principles studies the interrelations among all the properties defined by some selection prototypes as well as similar ones, and properties that do not fall into this category but can be related to the properties that do. The chapter discusses about the Scheepers diagram problem and presents the examples without special set-theoretic hypotheses as well as the examples from CH or MA. The chapter also explains the concepts related to preservation of properties, modern types of covers, splittability, function spaces and local-global principles, and cardinal characteristics of the continuum and additional problems. While all problems mentioned in the chapter are about sets of real numbers, some of them deal with the sets of reals that are not defined by selection principles, and belong to the more classical era of the field.