Nontrivial Point Sets on the Real Line

Author: Mariam Beriashvili
Keywords: Point Sets, Axiom of Choice, Continuum Hypothesis, Martin's Axiom
Annotation:

In the beginning of the XX-th century, extensive studies were carrying out in the theory of functions of a real variable, which substantially transformed it into a new mathematical discipline. As a result of these investigations, descriptive set theory emerged in which various point sets and real-valued functions are studied, primarily, without the aid of the Axiom of Choice. In this direction, certain classes of effectively determined point sets and functions were introduced: Borel sets, Baire functions, Lebesgue measurable functions, analytic sets, co-analytic sets, projective sets, etc. In this report we are dealing with some classical facts and statements of descriptive set theory, which turned out to be closely connected with mathematical logic and foundations of mathematics. In particular, we consider certain types of interesting and important point sets on the real line, such as Vitali sets, Bernstein sets, Hamel bases, Luzin sets, Sierpinski sets, and so on. At present, these point sets are permanently exploited in real analysis, measure theory and general topology. For constructing Vitali sets, Bernstein sets and Hamel bases some uncountable forms of the Axiom of Choice are needed, while for constructing Luzin sets and Sierpinski sets, additional set-theoretical hypotheses are necessary.