Descriptive set theory - Definition and Overview

In mathematics, descriptive set theory is the study of certain classes of "well-behaved" sets of real numbers, e.g. Borel sets, analytic sets, and projective sets. A major aim of descriptive set theory is to describe all of the "naturally occurring" sets of real numbers by using various constructions to build a strict hierarchy beginning with the open sets (generated by the open intervals).

More generally, Polish spaces are studied in descriptive set theory; as it turns out, every Polish space is homeomorphic to a subspace of the Hilbert cube.

Many questions in descriptive set theory ultimately depend upon set-theoretic considerations and the properties of ordinal and cardinal numbers.

References

• A. Kechris, Classical Descriptive Set Theory, GTM 156, Springer-Verlag, 1995.
• Y. Moschovakis, Descriptive Set Theory, North-Holland, 1980.