On a question of L. V. Keldysh concerning the structure of Borel sets

This article deals with the solution of certain old problems in descriptive set theory and topology connected with the structural properties of the Borel sets in the space of irrational numbers. At the center of this circle of problems is the question of L. V. Keldysh on the universality of an element strictly of class $\alpha$. The author's solution is based on the principle of determinancy, which, as Martin proved, is satisfied for Borel sets. Closely connected with the question of Keldysh are certain problems considered by Lusin, Aleksandrov and Urysohn, and others. In this paper an answer is given, in particular, to a question posed by them on the number of irreducible Borel sets in each Borel class (a Borel set is said to be irreducible if any nonempty open-closed subset of it is indistinguishable from the ground set by some classification of the Borel sets).
Bibliography: 23 titles.