Descriptive Set Theory and Topology

Abstract

In this article we present the foundations of the descriptive theory of sets and topological spaces. One of the most important directions of descriptive set theory is the study of the interdependence between the internal structure of sets and operations by means of which they are constructed starting from sets of a simpler nature. Analyses of operations over sets are also related to this line of approach. The general theory of operations over sets is at the interface between the abstract and descriptive theories of sets. The study of arbitrary sets of the real line did not lead to any definite results. As is well known, Cantor’s Continuum Hypothesis CH, which asserts that every subset of the real line is either countable or has the cardinality of the continuum c = 2No, can neither be proved nor disproved. Open sets and closed sets of the real line are either countable or have cardinality of the continuum. At the beginning of the twenties it was natural to look for a solution of the Continuum Hypothesis by considering more general classes of sets of the real line. For F σ -sets, an affirmative solution of the problem is trivial. In 1906 Young gave an affirmative solution to the problem for G δσ -sets, and Hausdorff in 1914 gave an affirmative solution for G δσδσ -sets. A complete affirmative solution for all B-sets was found by P.S. Alexandroff and Hausdorff in 1916. The Alexandroff-Hausdorff method was based on the definition of an A-operation and the theorem on representing B-sets as the result of an A-operation over an array of closed sets. This fact convincingly illustrates the value of the theory of operations over sets. It should be noted that many results of descriptive set theory turn out to be particular cases of more general theorems on families of sets generated by certain operations.