ABSTRACT
This introductory chapter fixes the notation and presents some auxiliary facts from set theory, general topology, classical descriptive set theory, measure theory, and real analysis. The symbol ZF denotes the Zermelo–Fraenkel set theory, which is one of the most important formal systems of axioms for the whole of modern mathematics. The basic notions of the Zermelo–Fraenkel system are sets and the membership relation between them. The symbol ZFC denotes the Zermelo–Fraenkel theory with the Axiom of Choice. At the present time, it is widely known that ZFC theory is a basis of modern mathematics, i.e., almost all fields of mathematics can be developed by starting with ZFC. The Axiom of Choice is a very powerful set-theoretical assertion which implies many extraordinary and interesting consequences. The Axiom of Dependent Choices is usually denoted by DC. Actually, the statement DC is a weak form of the Axiom of Choice.
